Patent Application
20220369958
The present group of inventions refers to the medicine and medical technology and more particularly to methods and devices for monitoring blood glucose level by measuring the calorimetric method of the thermal effect and metabolic rate of the local area of living tissue. Also, the present invention relates to methods of producing a biopolymer molecule with predetermined biological properties and a predetermined spatial structure. The use of the group of inventions will allow non-invasive monitoring of blood glucose levels in diabetic patients, including early diagnosis of diabetes mellitus.
One of the prevalent and dangerous diseases of the times is diabetes mellitus, the most common form of which, accounting for over 90-95% of all diabetics, is type 2 diabetes mellitus (DM2), which is one of the most pressing problems of modern medicine. The social importance of this chronic disease is determined by several factors, the most important of which is the high prevalence of the disease throughout the world, while the methods of its treatment are still under development. According to the forecast of the International Diabetes Federation (IDF), the number of patients with diabetes by 2035 will increase up to 592 million compared with 422 million in 2018.
U.S. Federal health statistics center shows that 12.3% of Americans aged 20 and more are living with diabetes. And about 3% are not even aware of this, thereby aggravating their condition. The other 37% live with pre-diabetes (a condition characterized by elevated blood glucose levels). By comparison, 10 years ago this figure was only 27% instead of 37%. Researchers from the US Centers for Disease Control say, if people with pre-diabetes do not change their lifestyle for the better, in 30% of cases they will develop type 2 diabetes within 5 years. According to the prognosis of International Diabetes Federation (IDF), spending on diabetes treatment will rise up to 5936 billion by 2035, from $586 billion in 2012.
Diabetes often leads to the development of complications such as blindness, renal disorders, nervous diseases and cardiovascular diseases. Diabetes is a leading disease resulting in blindness at the age of 20 to 74 years. Approximately from 12,000 to 24.000 persons annually loss vision because of diabetes. Diabetes is a leading cause of renal diseases in about 40% of new cases. About 40 to 60% of patients with diabetes are predisposed to different forms of nervous diseases, which can result in amputation of limbs. Patients with diabetes are approximately 2 to 4 times more predisposed to cardiac diseases, in particular myocardial infarction.
Diabetes is a disease associated with insufficient production or inefficient use of insulin by cells of the body. In spite of the fact that causes of the disease are not completely understood, some factors such as genetic, environmental, viral have been identified.
There are two main forms of diabetes: type 1 and type 2.
Type 1 diabetes (known as insulin-dependent diabetes) is an autoimmune disease wherein insulin production completely terminates; and it most often develops in childhood and youth. Patients with type 1 diabetes need daily insulin injections.
Type 2 diabetes is a metabolic disease caused by that the body cannot produce a sufficient amount of insulin or utilization thereof is inefficient. Patients with type 2 diabetes make up about 90 to 95% of a total amount of diabetics. Morbidity of type 2 diabetes in the USA approaches an epidemiologic threshold, mainly due to increase in the number of elderly Americans and a significant prevalence of a sedentary lifestyle leading to obesity.
Insulin promotes glucose penetration into a cell with subsequent cleavage thereof to obtain energy for all metabolic processes. In diabetics, glucose cannot penetrate into a cell, it accumulates in blood and cells experience energetic hunger.
Patients with type 1 diabetes inject insulin to themselves using a special syringe and a cartridge. Continuous subcutaneous injection of insulin through an implanted pump is also possible. Insulin is typically prepared from swine pancreas or is synthesized chemically.
Generally accepted medical methods of diabetes treatment prescribe patients taking insulin to carry out independent monitoring of blood glucose level. Knowing blood glucose level, patients can adjust insulin dose in subsequent injection. Adjustment is necessary, since due to different reasons, blood glucose level fluctuates during a day and from day to day. In spite of the importance of such monitoring, several conducted studies showed that a portion of patients who carry out such monitoring at least once a day, diminishes with age. This fall occurs mainly because the currently used method of monitoring is associated with invasive drawing a blood sample from a finger. Many patients consider drawing a blood sample from a finger to be a more painful procedure that insulin injection.
Research and development (R&D) aimed at addressing the problem of diabetes mellitus are actively pursued in the following technological areas:
1) Creation of a non-invasive (bloodless) method for continuous monitoring of the patients' blood glucose level;
2) Creation of invasive methods of blood glucose level measurement based on new-generation biosensors;
3) Development of wearable electronic devices for preventive medicine for early diagnosis of diabetes mellitus based on new-generation biosensors and biotechnologies;
4) Creation and implementation into clinical practice of novel effective and safe anti-diabetic drugs.
Thus, creation of a non-invasive (bloodless) method for monitoring blood glucose level in diabetic patients and a medical device on the basis thereof capable of replacing industrial invasive glucometers is one of the topical unsolved problems of modern medicine and public healthcare. The task of developing new methods and devices for preventive medicine for early diagnosis of diabetes mellitus based on noninvasive continuous recording of glucose curves is closely related to the development of wearable electronic devices for noninvasive continuous blood glucose level monitoring based on new-generation biosensors and biotechnology.
A commercial glucose meter for measuring blood glucose levels in an invasive manner, is a biosensor that uses a glucose oxidase enzyme for blood glucose cleavage. During the cleavage process, the enzyme first oxidizes glucose and uses two electrons to reduce FAD (a component of the enzyme) to FAD-H2, which, in turn, is oxidized in several steps by the electrode. The resulting current is proportional to the concentration of glucose. In this case, the electrode is the transducer and the enzyme is the bioselective element.
Methods and devices for noninvasive measuring blood glucose level are known: [1-10].
Method of calorimetry is known to be widely used in biology for the study of thermal processes at the molecular and cellular levels [1-10]. The microcalorimetry method is also successfully used to study thermal processes (heat release and absorption) in separate organs, in particular, in active muscles and nerve fibers.
Increased activity over the past decade in studies aimed at creating the microcalorimetry method for studying in physiological conditions in vivo, the thermal processes associated with the metabolism of a local area of human living tissue, is largely due to the work on the development of a noninvasive method for monitoring blood glucose level (noninvasive glucometer) based on monitoring of heat production of a local area of human tissue.
The direct calorimetry and indirect calorimetry are the known methods of physiological calorimetry. The direct calorimetry method
contemplates immediate determination of a total amount of irradiated heat using a calorimetric chamber for live objects.
The indirect calorimetry method allows for determining an amount of irradiated heat in an indirect way based on accounting respiratory gas exchange dynamics using respiratory chambers and different systems. Two possible modifications of the indirect calorimetry method are distinguished: a method of complete gas analysis (accounting absorbed O2 and evolved CO2) and a method of incomplete gas analysis (accounting absorbed O2).
A known method of measuring the basal metabolic rate of the human body using a wholebody calorimeter (direct calorimetry) is described in U.S. Pat. No. 4,386,604 A, Publ. Jul. 6, 1983. By change in an air temperature and a total water amount evaporating from the whole-body surface, a total whole-body heat irradiation is determined and the basal metabolic rate is calculated. The main drawback of this method consists in that for embodiment thereof, cumbersome, stationary and expensive whole-body calorimetric chambers are required. Furthermore, the direct calorimetry method is characterized by a low accuracy.
The method and device for microcalorimetric measurement of the rate of local metabolism of tissue, water content of intercellular tissue, concentration of biochemical components of blood and pressure in the cardiovascular system, described in [11], is known, wherein the measurement of heat effect and a rate of metabolism of a local tissue site determine the blood glucose level (RU 2396897). The metabolic rate is determined by measuring the total amount of water evaporating during an insensible perspiration from the surface of a local area of the skin and measuring the temperature and humidity of the ambient air. The method of measurement makes it possible to realize a high accuracy in laboratory conditions with controlled parameters of a microclimate (temperature and humidity of the room) wherein the measurement is carried out, with constant values of climatic parameters of the environment (temperature and humidity of the environment, atmospheric pressure). The main disadvantage of this method, limiting its practical application, is that the results of measurements depend on the physic-and-climatic factors of the environment; in each case, a new calibration of the measuring channel is required when the climatic factors change, with constant values of the microclimate parameters of the room where measurements are taken.
Spectral methods for non-invasive measurement of blood glucose concentration are known [1]. The main disadvantage of known non-invasive devices for measuring blood glucose concentration based on diffuse reflectance spectroscopy and absorption spectroscopy is a high measurement error due to the low ratio of the registered signal to noise, which limits the accuracy of spectral measurements at characteristic frequencies of blood glucose molecules in blood capillaries located 0.5-1 mm deep from the epidermal surface.
The well-known method of optical-calorimetric spectroscopy is based on light absorption with excitation of energy levels of the substance molecules of a biological object, followed by emission-free relaxation of levels and heating of the object. In contrast to the method of isothermal calorimetric spectroscopy described above, in the method of optical-calorimetric spectroscopy, the temperature of the substance is a variable; the information parameter in this method is the change in the temperature of the testing sample. The degree of heating is determined by the magnitude of the substance absorption capacity, the light intensity and the efficiency of competing processes (fluorescence, photochemical and photoelectric effects). The main advantage of optical-calorimetric methods is the possibility of recording absorption spectra of highly scattering media, which is very important in biology and medicine. Since the measured parameter is a change in temperature, non-selective radiation receivers are used as detectors, i.e. no restrictions on the receiver wavelengths are presented.
However, none of the existing non-invasive continuous blood glucose monitoring techniques allows accurate real-time monitoring of the content and temporal dynamics of the biochemical components of living tissue, which are the main factor used for early diagnosis and therapy of diabetes mellitus.
A technical task and technical result of the present invention consist in the development of a fundamentally new method of living tissue spectroscopy under in vivo conditions—absorption isothermal calorimetric spectroscopy, which allows to determine the content of biochemical components of living tissue in real time, and a device that allows this method to be carried out.
The developed method and device for non-invasive continuous monitoring of blood glucose level are used for early diagnosis of diabetes mellitus and monitoring of blood glucose level in the process of hypoglycemic therapy and are based on the use of new-generation biosensors.
An additional technical task and a technical result of the present invention is the creation of a new class of biosensors and biological catalysts to be used in a variety of fields of human activity, in addition to early diagnosis of diabetes: in the fields of new-generation drug development, biomedicine and protein and genetic engineering, food industry, safety and environmental quality monitoring.
One of the promising application areas for the group of inventions is the creation and introduction into clinical practice of new effective and safe medicinal products, in particular, glucokinase is one of the promising targets for the creation of new antidiabetic drugs. An additional technical task and technical result of the present invention is the development of a method for obtaining a biopolymer molecule with specified biological properties and a specified spatial structure. The proposed method for obtaining a biopolymer molecule allows to determine the primary structure of a biopolymer molecule, which, with a given biochemical composition of a solvent, under certain conditions (temperature, pressure, acidity) can collapse into a biopolymer molecule with a heterophase structure, having the functions of a biosensor and/or biological catalyst, with bioselectivity to a given solvent molecule S, catalyzing a biochemical reaction involving the given molecule S. The polymer chain of the biopolymer molecule with specified properties, the structure of which is determined by the proposed method, is synthesized by protein and/or genetic engineering methods at the next stage of the technological chain. On the basis of biopolymer molecules with a heterophase structure obtained by the proposed method, a new generation of biological catalysts and effective and safe drugs for the treatment of, in particular, diabetes mellitus and other chronic socially significant diseases can be created. In addition, the proposed method includes the stage of determining the three-dimensional spatial structure and biological properties of the biopolymer molecule. The proposed method for determination of the three-dimensional spatial structure and biological properties of the biopolymer molecule based on the known primary structure of the polymer chain has important practical application for the development of new drugs and new methods of treatment. The problem of predicting the spatial structure of the biopolymer molecule and its relation with biological function is one of the biggest unsolved problems of modern science, solution to which has made it possible to create principally new biotechnologies that can be applied in protein and gene engineering.
The proposed method and device for its implementation, focused on non-invasive continuous monitoring of blood glucose level, allows to determine the blood glucose level by measuring with the help of fundamentally new method of in vivo absorption isothermal calorimetric spectroscopy, which allows determining the content of biochemical components of a living tissue in real time. The magnitude of electromagnetic radiation power at the characteristic frequency of the biochemical component under study, absorbed in a local volume of substance of a living tissue, is measured not by the change in tissue temperature, but by the change in osmotic pressure value (or amount of water in the intercellular space), characterizing the volume of intercellular substance, which is in heterophase condensed state, at the phase transition temperature (corresponding to the thermodynamic equilibrium
of the crystalline and liquid phases of the intercellular substance). In this case, the amount of absorbed power of electromagnetic radiation falling on the surface of the studied local area of living tissue epidermis is determined by the amount of intercellular substance that has changed its phase state (melting of the globular phase of hyaluronic acid), by measuring the intercellular substance osmotic pressure in real time.
The proposed method of early diagnosis and therapy of diabetes mellitus can be implemented using a fundamentally new method of in vivo absorption isothermal calorimetric spectroscopy of a living tissue, which allows to determine the content of biochemical components of a living tissue in real time.
The said technical result is achieved by implementing a method of isothermal calorimetric spectroscopy of biochemical components of a patient's living tissue, comprising the following steps:
In particular embodiments of the invention, the intensity of the electromagnetic radiation may be constant or variable, varying at a constant rate and/or modulated frequency and/or amplitude.
In particular embodiments of the invention, the physiological parameter characterizing the thermodynamic phase state of the intercellular substance is the osmotic pressure of the intercellular substance and/or the amount of water in the intercellular space and/or the elastic pressure of the living tissue under the applicator.
In particular embodiments of the invention, the biochemical components of intercellular and/or intracellular substance are selected from the group consisting of water, hyaluronic acid, glucose, triglycerides and other biochemical components of intercellular substance, cells and blood.
In particular embodiments of the invention, the wavelength of the electromagnetic radiation applied to the tissue region is selected from the range of electromagnetic radiation, which is determined from the characteristic absorption frequencies of the living tissue biochemical components in the optical and/or near infrared and/or mid and far infrared and/or terahertz and/or microwave range.
In particular embodiments of the invention, the concentration of the biochemical component of the intercellular substance and/or intercellular fluid and/or blood is additionally measured based on diffuse reflectance spectroscopy.
In particular embodiments of the invention, the concentration of the biochemical component of the intercellular substance and/or intercellular fluid and/or blood is additionally measured based on Raman spectroscopy.
In particular embodiments of the invention, the biochemical component is blood glucose, the concentration of which is determined from the concentration of glucose bound to the polymer chain monomers of the intercellular substance.
In particular embodiments of the invention, the osmotic pressure of the intercellular substance or the amount of water in the intercellular space is determined based on spectral measurements at wavelengths corresponding to characteristic absorption frequencies of biochemical components of the intercellular and/or intracellular substance.
In particular embodiments of the invention, the spectral measurements are based on a two-frequency spectroscopy method, wherein the wavelength of the electromagnetic radiation is selected by the characteristic frequency of the hyaluronic acid intercellular tetramer in the terahertz region, and the wavelength of the intercellular osmotic pressure spectrometer is selected by the characteristic frequencies of water absorption in the stratum corneum in the infrared region.
In particular embodiments of the invention, the osmotic pressure of the intercellular substance or the amount of water in the intercellular space is determined from the amount of water in the stratum corneum based on measurements of physical characteristics of the stratum corneum, which are selected from the group consisting of electrophysical characteristics, spectral and optical-acoustic characteristics, and thermal physical characteristics.
In particular embodiments of the invention, the osmotic pressure of the intercellular substance or the amount of water in the intercellular space is determined from the amount of water in the stratum corneum based on measurements of spectral characteristics of the stratum corneum at wavelengths corresponding to characteristic frequencies of water in the stratum corneum, using a spectral method that is selected from the group consisting of: IR spectroscopy, Raman spectroscopy, optical-acoustic spectroscopy, double-beam spread spectroscopy.
In particular embodiments of the invention, the osmotic pressure of the intercellular
substance or the amount of water in the intercellular space is determined from the amount of water in the stratum corneum by measuring the electrical characteristics of the stratum corneum, which are selected from the group consisting of the transverse electrical conductivity of the stratum corneum at DC and/or AC currents, as well as the dielectric permittivity.
In particular embodiments of the invention, the blood glucose concentration is determined based on the temporal dynamics of a physiological parameter, the physiological parameter being the osmotic pressure of the intercellular substance of the tissue region below the applicator, with the intercellular substance below the applicator being a natural biosensor with a heterophase structure with selectivity to the glucose molecule and sensitivity to external heat flows, and the individual calibration of the patient's local tissue area is determined by the content of hyaluronic acid in the intercellular substance and the amount of water in the tissue volume under the applicator, measured by the spectral method.
In particular embodiments of the invention, the method is applicable for early diagnosis of diabetes mellitus based on the temporal dynamics of the biochemical component concentration.
In particular embodiments of the invention, the method is applicable for early diagnosis of diabetes mellitus based on the temporal dynamics of intercellular osmotic pressure.
The specified technical result is also achieved through the development and creation of a device for isothermal calorimetric spectroscopy of biochemical components of living tissue, containing a beat—and waterproof applicator, made with the possibility of application to the skin of a patient with dosed pressure, a temperature sensor, one or several sensors of physiological parameters, which characterize the thermodynamic phase state of the intercellular substance under the applicator, one or several electromagnetic radiation sources, a device for creating a calibration effect, with the sensors of physiological parameters located under the applicator, with the signals from the above sensors coming in series to the inputs of an amplifier module and/or a synchronous detector and/or an analog-digital converter, installed on the upper surface of the applicator, to the information processing unit and information display unit.
In particular embodiments of the invention, the electromagnetic radiation source is made to emit with constant and/or modulated intensity and/or frequency modulation.
In particular embodiments of the invention, sensors of physiological parameters characterizing the thermodynamic state of the intercellular substance under the applicator are made as sensors of osmotic pressure of the intercellular substance or sensors of amount of water in the intercellular tissue space in the local volume under the applicator and/or sensors of elastic pressure of the living tissue under the applicator.
In particular embodiments of the invention, a device for creating a calibration effect is a source of thermal power, made in the form of a resistor and/or a Peltier element and/or a source of electric current and/or voltage; a device for creating dosed pressure on the surface of the applicator.
In particular embodiments of the invention, the radiation range of electromagnetic radiation sources is selected from a group including optical and/or near infrared band; middle infrared band; far infrared and/or terahertz band; microwave radiation band.
In particular embodiments of the invention, an osmotic pressure sensor of the intercellular substance or a sensor of the amount of water in the intercellular space of tissue in a local volume under the applicator is an electrometric sensor based on measuring the electrophysical characteristics of the stratum corneum and tissue under the applicator, the measurement principle of which is selected from a group including measurement of the transverse electrical conductivity of the stratum corneum at a constant and/or alternating current; measurement of the dielectric constant of the stratum corneum; measurement of electrical conductivity and/or dielectric permittivity of the tissue under the applicator.
In particular embodiments of the invention, an osmotic pressure sensor of the intercellular substance or a sensor of the amount of water in the intercellular space of tissue in a local volume under the applicator is an electrometric sensor based on measuring the electrophysical characteristics of the stratum corneum and tissue under the applicator, the measurement principle of which is selected from a group including measurement of the transverse electrical conductivity of the stratum corneum at a constant and/or alternating current; measurement of dielectric permittivity of the stratum corneum; measurement of electrical conductivity and/or dielectric permittivity of the tissue under the applicator.
In particular embodiments of the invention, the wavelengths of electromagnetic radiation from the source of the spectral sensor are determined by the characteristic absorption frequencies of the biochemical components of the intercellular substance and/or intracellular substance, which are selected from a group including water, hyaluronic acid, glucose and other biochemical components of the intercellular substance and intercellular fluid.
In particular embodiments of the invention, the osmotic pressure sensor of the intercellular substance is a spectral sensor based on spectral measurements of the intercellular substance characteristics in the terahertz range at a wavelength corresponding to the cross-coupling energy between the monomers of the hyaluronic acid polymer chain.
In particular embodiments of the invention, the spectral sensor is based on
spectral measurements of the amount of water in the stratum corneum by the characteristic frequencies of water in the stratum corneum based on a spectral method that is selected from a group including isothermal calorimetric spectroscopy, absorption spectroscopy, diffusion reflectance spectroscopy, Raman spectroscopy, optical-acoustic spectroscopy.
In particular embodiments, the device according to the invention additionally contains a spectral device including a source and receiver of electromagnetic radiation for spectral measurements based on diffusion reflectance spectroscopy.
In particular embodiments of the invention, the sensor of physiological parameters characterizing the thermodynamic phase state of the intercellular substance under the applicator is made using a piezoelectric measurement method.
In particular embodiments of the invention, the device is applicable for early diagnosis of diabetes mellitus based on the time dynamics of the biochemical parameter concentration and/or the value of the osmotic pressure of the intercellular substance.
The specified technical result is also achieved by implementing a method for obtaining a biopolymer molecule with specified biological properties and a specified spatial structure consisting of one or several subunits capable of spontaneous folding into a spatial configuration with a heterophase structure having specificity to one or several specified substrate S molecules, with a certain solvent composition, under certain conditions for temperature and pressure, which consists in the fact, that the number of subunits is determined based on the three-dimensional spatial structure of a macromolecule, while the primary structure of each subunit of a biopolymer molecule is determined a: follows:
In particular embodiments of the invention, the polymer chain monomer A has a positive charge, and the solvent ion M has a negative charge.
In particular embodiments of the invention, the solvent ion M is a monovalent positively charged metal ion.
In particular embodiments of the invention, the primary structure of a biopolymer macromolecule consisting of one subunit is a polysaccharide chain, the repeating monomer of which contains at least one type of disaccharide pair formed by one monosaccharide A with a negatively charged group and one neutral monosaccharide B with no charged group.
In particular embodiments of the invention, the primary structure of a biopolymer macromolecule consisting of a single subunit is a hyaluronic acid polymer chain, the substrate molecule is a glucose molecule, the monovalent metal ion M is a sodium ion, while the chemical composition of the solvent is close to the chemical composition of blood plasma and the intercellular medium of a living system, under physiological conditions in vivo.
In particular embodiments of the invention, a divalent ion and/or a substance is added to the solvent, the molecule of which has the ability to form a divalent ionic bond with negatively charged monomers of the hyaluronic acid polymer chain.
In particular embodiments of the invention, a biopolymer molecule consisting of one subunit is a polypeptide chain of amino acids, with repeating monomers containing at least one amino acid with a negatively charged group Ra, and one amino acid with a neutral uncharged group Rb. The monovalent metal ion M is the potassium ion, while the chemical composition of the solvent is close to the chemical composition of the intracellular environment of the living system under physiological conditions in vivo.
In particular embodiments of the invention, additional divalent metal ions or divalent compounds are added to the solvent, which are selected from a group including magnesium ion Mg+2, manganese Mn+2, ions of other metals and compounds.
In particular embodiments of the invention, the substrate is a D-glucose molecule and/or a molecule of another monosaccharide.
In particular embodiments of the invention, a biopolymer molecule with a heterophase structure is a biosensor with selectivity to a given substrate molecule S, active in an aqueous medium, converting a signal of substrate concentration S into a signal proportional to the volume of the macromolecule and/or intramolecular osmotic pressure, which is measured by an electronic device.
In particular embodiments of the invention, the biopolymer molecule is a hyaluronic acid polysaccharide, which has selectivity to the glucose molecule.
In particular embodiments of the invention, a biopolymer molecule with specified biological properties, consisting of two subunits, capable of spontaneous folding in a spatial configuration with a heterophase structure, having specificity to a given substrate molecule S, with a certain solvent composition, under certain conditions of temperature and pressure, and the method is that for each of the two subunits, the primary structure is determined as follows:
In particular embodiments of the invention, the subunits are polypeptide chains of amino acids.
In particular embodiments of the invention, for each subunit, the location of repeating monomers in the chain is determined, characterizing the interaction between subunits, forming a unique amino acid configuration of the active center of a protein molecule, which is formed during spatial folding in the region of space between the domains of subunits.
In particular embodiments of the invention, a biopolymer molecule with a heterophase structure is a biological catalyst with specified characteristics and selectivity to a given substrate molecule S, active in an aqueous medium, catalyzing biochemical reactions involving a substrate molecule S.
In particular embodiments of the invention, a biopolymer molecule is an enzyme with specified characteristics based on a biopolymer molecule with a heterophase structure.
In particular embodiments of the invention, the primary structure of a biopolymer molecule with specified biological properties is determined by the characteristics of the hexokinase enzyme, the equilibrium constant and the rate of the biochemical reaction catalyzed by the enzyme.
In particular embodiments of the invention, the primary structure of a biopolymer molecule with specified biological properties is determined by the characteristics of the glucokinase enzyme, the equilibrium constant and the rate of the biochemical reaction catalyzed by the enzyme.
In particular embodiments of the invention, with a given solvent composition and a given primary structure of a biopolymer molecule consisting of 2 or more subunits, the quaternary spatial structure of a macromolecule is determined as follows:
In particular embodiments of the invention, the biopolymer molecule, the primary structure of which consists of 2 or more subunits, is an oligomeric protein.
The present invention also includes a biopolymer molecule with a heterophase structure, which is a biosensor with selectivity to a given substrate molecule and obtained by the method of the invention, showing activity in an aqueous medium.
The present invention also includes a biopolymer molecule with a heterophase structure, having selectivity to a given biological molecule of the substrate S, which is a biological catalyst for catalyzing a biochemical reaction involving a substrate S molecule, obtained by the method of the invention. A biopolymer molecule with a heterophase structure, having selectivity to a given substrate S molecule, which is an enzyme for catalyzing a biochemical reaction involving a substrate S molecule, obtained by the method of the invention.
In particular embodiments of the invention, the biopolymer molecule is characterized by specificity to the glucose molecule, as well as the by the following:
In particular embodiments of the invention, the biopolymer molecule is characterized by specificity to the glucose molecule, and differs in that the biopolymer molecule is a polypeptide chain, the repeating monomer of which contains an amino acid with a negatively charged side group, which is selected from a group consisting of glutamic acid, aspartic acid, glycine, cysteine, tyrosine, while an amino acid with an uncharged side group is selected from a group containine serine, asparagine, threonine, glutamine.
In particular embodiments of the invention, the amino acid with a negative side group charge is the glutamic acid; the amino acid with an uncharged side group is asparagine; the substrate of the biochemical reaction is D-glucose.
In particular embodiments of the invention, the amino acid with a negative side group charge is the aspartic acid; the amino acid with an uncharged side group is the amino acid that is selected from the group of asparagine, threonine, cysteine, serine; the substrate of the biochemical reaction is D-glucose.
In particular embodiments of the invention, the polypeptide chain additionally contains amino acids with positively charged side groups, which are selected from the group of lysine, arginine, histidine.
In particular embodiments of the invention, the solvent additionally contains a divalent metal ion, which is selected from a group including magnesium ion Mg+2, manganese ion Mn+2, ions of other metals and compounds or another compound.
In particular embodiments of the invention, the polypeptide chain of the first subunit contains an amino acid with a negatively charged side residue group, and the polypeptide chain of the second subunit contains an amino acid with a positively charged side residue group.
In particular embodiments of the invention, the amino acids of the active center of the protein are serine, the side residue of which has a negatively charged group, and histidine, the side residue of which has a positively charged group.
In particular embodiments of the invention, a biopolymer molecule is a molecule for creating a biopolymer substance or a pharmaceutical composition based on hyaluronic acid for the treatment of wounds and tissue inflammation.
In particular embodiments of the invention, a biopolymer molecule is a molecule for creating a drug or pharmaceutical composition for the treatment of diabetes mellitus.
The accompanying drawings, which are incorporated in and constitute a part of this specification, illustrate embodiments of the inventive group and, together with a general disclosure of the group of the invention and the following detailed disclosure of the embodiments, serve to explain the principles of the present invention.
Physical basis of heat exchange of living tissue with the environment.
Heat exchange is a spontaneous and irreversible process of heat transfer caused by temperature gradient. The following forms of heat exchange are distinguished: heat conductivity, convection, radiant heat exchange, heat exchange in phase transitions.
Heat transfer is heat exchange between the body surface and a medium (liquid, gas) contacting therewith.
Evaporative cooling is heat exchange between tissue and the environment caused by evaporation of water delivered to the epidermis from deep tissue layers. Heat flow density is determined by the product of evaporation heat (steam generation heat) by flow density of water evaporating from the surface.
Radiant heat exchange (radiation heat exchange, radiant transfer) is energy transfer from one body to another caused by the processes of emission, propagation, scattering and absorption of electromagnetic radiation. Each of these processes adhere to definite regularities.
Thus, under the conditions of equilibrium heat radiation emission and absorption adhere to the Plank's law of radiation, to the Stephan-Boltzmann law, to the Kirgoff law of radiation.
Essential difference of radiant heat exchange from the other forms of heat exchange (convection, heat conductivity) consists in that it can occur in the absence of a material medium separating surfaces of heat exchange, as electromagnetic radiation also propagates under vacuum.
The Plank's law of radiation establishes relation between radiation intensity, spectral distribution and temperature of the black body. In elevation of temperature, radiation energy rises. Radiation energy depends on wavelength. A total energy irradiated by the black body and measurable by a contact-less infrared thermometer is a total energy irradiated at all wavelengths. It is proportional to the Plank's equation integral by wavelengths and it is described in physics by the Stephan-Boltzmann's law.
Radiant heat exchange between tissue surface and the environment is determined by the ratio:
ΔWR=β×(Tskin4−Tair4)=W0×[4×(Tskin−Tair)/Tskin],
ΔT<<Tskin, Tskin is skin surface temperature. Tair is ambient air temperature.
W
0
=β×T
skin
4.
ΔWR is heat radiation from tissue surface to the environment.
Heat conductivity is one of the forms of heat transfer from more heated body parts to less heated parts.
Heat conductivity results in leveling temperature. In heat conductivity, energy transfer is effected as a result of direct energy transfer from particles having a greater energy to particles with a lower energy. If relative change in T at an average free run distance length of particles is small, then the main heat conductivity law (the Fourier's law) is fulfilled: heat flow density is proportional to temperature gradient ΔT: WΔT=−λ×ΔT,
Where λ—is heat conductivity coefficient of heat conductivity independent on ΔT. The λ—coefficient depends on aggregate state of a substance, molecular structure, temperature, pressure, composition thereof etc.
Convection is heat transfer in liquids and gases by substance flows. Convection results in leveling substance temperature. In stationary heat delivery to the substance, stationary convection flows occur therein. Intensity of convection depends on difference of temperatures between layers, heat conductivity and viscosity of medium.
Evaporative cooling is heat exchange between tissue and the environment caused by evaporation of water delivered to the epidermis surface from deep tissue layers through intercellular water transfer (by intercellular space). Heat flow density is determined by the product of evaporation heat (steam generation heat) by flow density of water evaporating from the surface.
In a temperature (thermal) neutral zone under normal conditions, water transport by sweating is known to be practically absent and the main contribution into evaporative cooling process is determined by water transport to the body surface. In physiology and medicine, this process is known as insensible (non-perceivable) perspiration.
Intercellular insensible perspiration is observed under so called “thermal comfortable (neutral) conditions”:
Ambient temperature: 18-23° C.,
Atmospheric pressure: 730-770 mm Hg (millimeters of mercury).
Intensity of evaporative cooling process under thermal comfortable (neutral) conditions is known to make up 400 to 700 mL/day or 10−8-10−7 g/second×cm2. This corresponds to values of heat flows of 1 to 10 mW/cm2.
It is known that the physiological process of intercellular insensible perspiration has a high sensitivity towards the external heat fluxes. Threshold of the process sensitivity to heat fluxes is 0.1 mW/cm2, which is equivalent to sensitivity of the process to ambient temperature equal to 0,005 degrees.
An important consequence of high sensitivity of the physiologic process to an insensible perspiration towards external heat fluxes practical importance is dependence of the physical characteristics of the human skin epidermis on external physical and climatic factors, in particular, on the temperature and humidity of the environment and atmospheric pressure. In particular, the results for measurements of the physiological parameter of human skin epidermis—speed of imperceptible perspiration, with constant values of microclimate of the room, which is used for measurements, are dependent on the physical and climatic factors of the environment (temperature and humidity of the environment, atmospheric pressure). Low-frequency fluctuations in the temperature of the outside air within even one degree, with constant parameters of the microclimate of the room, which is used for measurements, lead to significant low-frequency fluctuations of the measured physiological parameter and, as a result, increase in measurement error.
Physiological and biochemical basis of heat production of human living tissue
The process of oxidation of glucose which is one of the main energy suppliers in the body, occurs in accordance with the equation that may be presented in the following equation:
Glucose+Oxygen⇒CO2+H2O.
The change of the standard free energy in this reaction under physiological conditions is equal:
ΔG=−686 kcal/mol.
For comparison, a male weighing seventy kilograms who goes upstairs for an hour, expends about 1,000 kcal. From this, it is clear that 686 kcal, mentioned above are a vast amount of energy. Work done by man is of course much less than energy expended during this work as in irreversible process, not all change in free energy is converted into work. Real efficacy of this conversion (as will be described below) is not higher than 40%. Moreover, food is not “burned” immediately in oxygen releasing energy in the form of heat and this release occurs in steps and includes a number of rather complex chemical conversions each of which gives a small “portion” of energy.
Glucose is oxidized in the body forming carbon dioxide and water: this is one of the most universal processes underlying respiration and digestion processes.
In breaking each glucose molecule accompanied by lowering free energy, energy is released that is sufficient to form 93 ATP molecules by binding of phosphate groups to ADP molecules. Not all 93 molecules appear to be actually formed. At the same time, all the process includes a large number of enzymatic reactions. Nutrients (carbohydrates, fatty acids and amino acids) enter into a series of reactions forming the Krebs cycle (or the cycle of tricarboxylic acids) during which carbonic backbone of molecules is broken down with formation of CO2 but ATP is not formed here. On the following reaction steps transfer of electrons using special enzymes (respiratory chain) occurs. At these steps, ATP is synthesized and the last step on the way of a long process of electron transfer consists in binding thereof to molecular oxygen. Generally, electron transfer process along the respiratory chain resulting in accumulation of energy in ATP molecules is called oxidative phosphorylation. As a result of this process, 38 molecules of ATP as calculated per every consumed glucose molecule are formed. Efficacy of such transformation equals to 38/93=40%.
A value of heat production or heat power of the body can be quantitatively assessed originating from the following simple considerations.
An energetic value of human nutrition is about 2400 kcal/daily. In a first approximation, 2400 kcal=104 J, day (24 hours)=86400 seconds=105 seconds.
Then energy consumed by human body per one second will be 104/105=0.1 kJ×s−1 or 100 J×s−1=100 W;
thus, heat power of a human body is approximately equal to power of an electric bulb having power 100 W.
In muscular contraction, ATP which is energy donor for muscular contraction process, during reaction with myosin, allows for obtaining at most 50 J×g−1 energy. This means that an ideal muscular system (i.e. with efficiency equal to 100%) for lifting a load weighing 1 kg to a 5 in height, would require expenditure of 2×10−3 mole ATP. Actually, muscular efficiency is about 30-40% and the rest portion is released in the form of heat.
Under normal conditions of the body's vital activity, glucose is a main energetic substrate. Normal human blood plasma glucose concentration depending upon nutrition conditions is maintained within the limits of 50 to 120 mg %. Postprandial glucose concentration in the portal vein system during absorption phase can achieve more than 270 mg %. Elevation of blood glucose level always causes increase in insulin secretion.
In resting human body, fasting glucose metabolism rate averages 140 mg/hour per 1 kg body mass, 50% glucose being consumed by the brain, 20% by muscles, 20% by red blood cells and kidneys, and only 10% glucose are left for the rest tissues.
Glucose utilization rate (metabolism rate) in healthy man is a linear function of blood plasma glucose concentration. A mathematical relationship between glucose utilization and blood concentration thereof in normal humans is expressed by the equation:
R
u=0.02554C+0.0785,
And in patients with non-ketotic diabetes:
R
u=0.004448C+2.006,
where Ru is glucose utilization rate, mg/min per 1 kg of body weight, and C—blood plasma glucose concentration, mg %.
The term glucose “utilization” in physiological sense means the rate of glucose transport from blood into a general fund of tissue glucose and exit from it during metabolism. From biochemical point of view, glucose utilization rate is determined by transport through cytoplasmic membrane and by intracellular oxidative phosphorylation of glucose. The terms “turnover rate”, “assimilation” and “consumption” of glucose which are widely spread in the literature are synonyms of the notion glucose “utilization” and they are in any respect equivalent.
Under physiologic conditions, practically in all tissues glucose transport from intercellular medium into a cell is a first limiting reaction in glucose utilization by cells as in the absence of insulin, flow of transportable glucose is always less than glucose phosphorylation rate. Equilibrium between glucose transport and phosphorylation rates is achieved only at high glucose concentrations (400 to 500 mg %). In further increase in glucose concentration, phosphorylation becomes a limiting reaction. In other words, glucose transport rate from intercellular medium through cytoplasmic membrane into intracellular medium is a process limiting glucose utilization rate by a live tissue.
Originating from the above consideration, it appears logical and completely reasoned to draw a conclusion that heat generation as well as glucose utilization rate is a linear function of blood glucose concentration and measurement of local heat generation value allows for determining blood glucose level.
Condensed Matter Physics of the Biopolymer Molecule.
The object of the present study, performed within the work on the group of inventions, is biopolymer molecules capable of spatial self-organization and acquisition of stable molecular structure due to weak intramolecular volume interactions of non-valent nature.
Examples of biopolymer molecules capable of spatial self-organization under physiological conditions are polysaccharides and protein molecules with a polypeptide chain.
The behaviour of proteins is entirely determined by an exceptional, intrinsic spatial structural organization, which is not due to valence interactions and is determined exclusively by weak volumetric interactions. Deprived of it, proteins lose all their biological properties.
Protein chains are capable of spontaneously folding into strictly deterministic structures which geometry and conformational dynamics under physiological (native) conditions are completely determined by the amino acid sequence. The three-dimensional structures of proteins are individualized, very complex and have a strict order, not reducible, however, to periodicity.
Hence, by this the understanding of the extreme complexity of the task of experimental in vitro investigation of the biopolymer molecules properties is explained. Many experimental methods to study the substance structure, effective in the studies of molecules (which structure is determined by covalent bonds), are of limited applicability for the study of biopolymer molecules, the spatial structure of which is determined by weak non-valent interactions.
Modern methods enable determination of the primary structure of biopolymer molecules, consisting of a sequence of monomers linked by strong covalent bonds, while the determination of the spatial structure of such molecules still remains a challenging problem. For example, the number of sequences (primary structure) of protein molecules in the UniPare database (as of August 2016) was over 124 million, while the number of structures in the PDB (Protein Data Bank) database was only just over 121 thousand, representing less than 0.1% of all known sequences, with the gap between the two increasing rapidly and likely to grow further. This strong lag is due to the relative complexity of current methods of structure determination.
Another side of the problem is the lack of understanding of the fundamental nature of weak non-valent intramolecular interactions. By the moment of the present study, humanity with all its computational power and arsenal of experimental data still did not learn to build models that would describe protein folding process and predict the three-dimensional structure of protein on the basis of its primary structure (that is amino acid sequence) that directly indicates the lack of understanding of fundamental nature of weak bulk interactions determining biological properties of biopolymer molecules.
Thus, the spatial structure of biopolymer molecule and its relation to biological function is one of the biggest unsolved problems of modern science, the solution of which is possible under the condition of progress in fundamental studies of physical and chemical nature of weak volume interactions, which determine the three-dimensional structure of biopolymer molecule.
It should be noted that solving the folding problem is directly related to the scientific problem of biological catalysis, a unique phenomenon which physicochemical nature remains unclear.
Hence, the conclusion that solving the problem of folding and biosynthesis is directly connected with the development of new physical and chemical methods for in vivo studies of biopolymer molecules and the possibility of making real progress in the field of condensed matter physics of biopolymer structures.
In the present study, this scientific problem was solved by combining an experimental study that was carried out using a unique experimental method based on noninvasive measurements of the intercellular substance characteristics under in vivo conditions, and a theoretical study of the condensed state of a biopolymer molecule that was performed on the basis of statistical physics methods to generalize the obtained experimental data.
Biosensor Based on Biopolymer Molecule with Heterophase Structure.
A biopolymer molecule of hyaluronic acid polysaccharide turned out to be a unique model object for studies in the field of condensed state physics of biopolymer molecules capable of self-organization of spatial structure. Its choice as an object of study was conditioned by the following considerations:
Hyaluronic acid high molecular weight polysaccharide is the main component of the intercellular substance surrounding a living cell, which plays an important role in the self-organization of physical and chemical processes of cellular metabolism, the most important of which is the process of physical thermoregulation of living tissue, maintaining temperature constancy. As it has been known from experience, all mammalian organisms, which main intercellular component is the biopolymer molecule hyaluronic acid, which structure has remained unchanged during the long biological evolution, have a close body temperature (36-38° C.), which directly indicates the heterophase state of intercellular substance, caused by the thermodynamic properties of the biopolymer molecule of hyaluronic acid.
Experimental study of the thermodynamic properties of hyaluronic acid biopolymer can be performed under in vivo conditions based on non-invasive measurements of the characteristics of human epidermal intercellular substance using sensors placed on the skin epidermis surface.
A biopolymer molecule of hyaluronic acid has the simplest primary structure shown on
In order to study the thermodynamic properties of the intercellular substance, in vivo experimental studies were carried out in which the dependence of the osmotic pressure of the intercellular substance of human living tissue epidermis, characterizing the thermodynamic phase state of the intercellular substance in the local area under the heatproof applicator (applied with dosed pressure to the skin surface and forming an enclosed living tissue system in the local area under the applicator), on the external heat flow, external pressure and blood glucose concentration.
Experimental studies of the dependence of intercellular osmotic pressure on the power of external electromagnetic radiation fluxes falling on the surface of human skin epidermis were carried out in a wide range of frequencies from optical to microwave: optical and near infrared radiation 0.4-0.9 μm; infrared radiation 1-10 μm; microwave radiation 5-10 mm.
It has been experimentally proved that the process of heat exchange between the local area of living tissue epidermis and the environment is reversible and isothermal: reversible changes of incident thermal radiation power in the experiment lead to reversible isothermal changes of osmotic pressure in real time.
As it is known, thermodynamic reversibility and isothermal character of the substance heat exchange with the environment are fundamental properties of the real substance, which appear in the triple point, in which the substance is in the heterophase state.
The example of a system in which thermodynamic equilibrium between three phases of a substance is realized is a mixture of water, ice and steam at a triple point. It is well-known that in a mixture of water and ice at 273.16 K, two phases are in equilibrium, and melting and freezing processes flow reversibly at constant entropy: when the mixture is heated, ice melts, and when cooled, on the contrary, crystallization occurs—the formation of ice from the liquid phase, while the temperature of the water remains constant (isothermal property).
Thus, the results of experiments on measuring the threshold of sensitivity of intercellular substance to heat flows based on noninvasive measurement of osmotic pressure of intercellular substance directly prove the conclusion about the heterophase structure of intercellular substance of living tissue, at physiological body temperature of 36.6° C. Hence, the understanding of the generalization that the same temperature of warm-blooded organisms is a consequence of the heterophase state of the warm-blooded organism's intercellular substance, which is a fundamental property of the intercellular substance of these organisms due to the thermodynamic properties of the biopolymer molecule of hyaluronic acid.
Selectivity of hyaluronic acid glucose molecule. The peculiarity of the biopolymer molecule of hyaluronic acid with heterophase structure is that along with extremely high sensitivity to heat fluxes, the molecule has selectivity to the glucose molecule due to binding by formation of hydrogen bonds between the solvent glucose molecule and polymer chain monomers. In order to investigate the selectivity of the biopolymer molecule of hyaluronic acid to the glucose molecule, experimental studies of the dependence of the osmotic pressure of the epidermis intercellular substance on the blood glucose concentration were carried out.
The property of thermodynamic reversibility of intercellular substance transitions from one state to another has been experimentally proved—reversible changes in blood glucose concentration and, as a consequence, metabolic rate of living tissue leads to reversible changes in osmotic pressure of the intercellular substance: reversible physiological changes in the concentration of glucose in blood lead to reversible changes in osmotic pressure, similar to reversible changes in osmotic pressure in the process of heat exchange of living tissue with the environment—a decrease in glucose concentration leads to an increase in osmotic pressure, and on the contrary, an increase in the concentration of glucose in blood leads to a decrease in osmotic pressure—a response parameter similar to response to the substance cooling.
The property of thermodynamic reversibility of substance transitions from one state to another is a fundamental property of a real substance in a heterophase state at a triple point, therefore, the property of thermodynamic reversibility of the intercellular substance of living tissue, discovered experimentally, is a direct proof of the heterophase state of the intercellular substance under physiological conditions.
The intercellular substance of living tissue, which properties are determined by the thermodynamic properties of a biopolymer molecule of hyaluronic acid with a heterophase structure, is thus a natural glucose biosensor consisting of two elements—a recognition element (molecular recogniser) and a transducer (transducer), which converts changes in glucose concentration into changes in the osmotic pressure of the intercellular substance.
Thus, the results of experimental studies in vivo prove the unique biosensory properties of the intercellular substance of living tissue due to the heterophase structure of the biopolymer molecule of hyaluronic acid—the intercellular substance is a unique natural biosensor with the property of thermodynamic reversibility and bioselectivity to the glucose molecule and extremely high sensitivity to external heat flows.
The theoretical study was carried out in order to clarify the physical and chemical mechanism of selectivity of the biopolymer molecule of hyaluronic acid to glucose and the statistical mechanism of the heterophase condensed state of the combined system of the biopolymer molecule with a solvent near the boundary of the phase transition on the basis of generalization of experimental results. The description of the statistical method and the main results of the study are given below.
The property of thermodynamic reversibility and selectivity is united the polysaccharide molecule of hyaluronic acid with protein molecules, in particular, with enzymes that have a unique selectivity to certain substrate molecules and the property to catalyze biochemical reactions involving these molecules. Therefore, a conclusion is understood that the property of the biological specificity of enzymes, that was confirmed by the results of a large number of biochemical studies, and the selectivity of hyaluronic acid to glucose have a single physical and chemical basis.
Within the framework of theoretical study, a generalization for enzyme molecules was performed using a statistical method, which enabled to obtain solutions for the thermodynamic phases of the substance within the macromolecules. For this purpose, a theoretical study of the physicochemical mechanism of enzymes was carried out using hexokinase as reference enzymes, as they show selectivity to the glucose molecule and catalyze a biochemical reaction involving glucose. The obtained data confirmed heterogeneous statistical mechanism of the enzymatic catalysis.
In this document, the term “substrate S” refers to a solvent molecule that interacts with the monomers of the polymer chain through the formation of a weak non-valent bond. In particular, according to the invention, the solvent is an aqueous solution containing sodium, chlorine, potassium ions and glucose, and the difference between sodium and chlorine concentrations is approximately 40 mmol/L (excess sodium ions), and the glucose content in the heterophase range can vary over a wide range from 3 to 15 mmol/L.
In this document, the term “monomer” refers to the repeating links (structural units) in the biopolymer molecular chain. An example of monomer A is D-glucuronic acid containing a negatively charged group, an example of monomer B is O-Acetylglucosamine, neutrally charged, forming a repeating disaccharide in the hyaluronic acid polymer chain. In case of a polymer chain of a protein molecule, examples of monomers A are negatively charged glutamic and aspartic acids, and positively charged lysine, arginine, histidine; examples of monomers B are uncharged serine, threonine, glutamine, and asparagine.
In this document, the term “macromolecule with a heterophase structure” means . . . the only biopolymer molecule in which, under certain conditions, several thermodynamic phases—crystalline phase (globule), liquid (molten globule), gaseous (coil)—may coexist.
In this document, the term macromolecule with “specified biological properties
and a specified spatial structure” refers to a biopolymer molecule that has a uniquely defined biochemical characteristics (equilibrium constant, biochemical reaction rate) and a uniquely defined three-dimensional spatial structure.
In this document, the term “solvent” means a substance capable of dissolving other solid, liquid, or gaseous substances without causing their chemical changes. Usually, at ambient pressure and room temperature, the solvent is a liquid substance. Examples are water, sea water, glucose solution.
In this document, the term “subunit” means biopolymer
(polypeptide) chain, which together with other components is assembled into a multi- or oligodimensional protein complex. Many natural enzymes and other proteins consist of several protein subunits.
In this document, the term “biological catalyst with specified characteristics” refers to a catalyst with specific rate constant and rate of the biochemical reaction which it catalyzes.
In this document, the term “enzyme with specified characteristics” refers to a protein macromolecule with a uniquely defined three-dimensional spatial structure and specific values of the rate constant and the rate of the biochemical reaction catalyzed by this enzyme.
In this document, the term “biosensor selective to the specified substrate molecule” refers to a biopolymer molecule with selectivity (specificity) to a particular solvent molecule.
Non-Equilibrium Thermodynamics of the Biopolymer Molecule in a Condensed State.
1. Research Objective
The development of a theory of condensed matter describing the thermodynamic behavior near the stability boundary and the phase transition of matter from one ordered state to another is one of the most difficult problems in modern physics [1-4].
The problem of the condensed state of a biopolymer macromolecule is closely related to the problem of macromolecular folding (reversible folding of the biopolymer chain into a strictly defined three-dimensional structure)—one of the major problems in modern science, the solution to which is of great practical importance [5, 6, 7]. The self-organization of protein macromolecules is a vivid example of reversible intramolecular phase transition, which can be realized under certain conditions outside of a living cell as a result of weak interaction of the biopolymer chain with the solvent.
The problem is to describe the biological properties of a macromolecule based on the classical thermodynamic theory on the basis of the known primary structure of the biopolymer chain.
The extreme complexity of the challenge in such formulation is associated with the problem of an accurate statistical description of the interaction between a polymer chain and a solvent (which should give the same results as the thermodynamic theory). It is actually equivalent to the requirement of generalizing the Boltzmann and Gibbs statistical method of ensembles [12, 13, 14] for an accurate statistical description of interactions in a real substance consisting of a large number of positively and negatively charged interacting particles.
In this study, the statistical method of ensembles is generalized for an accurate statistical description of intramolecular weak volumetric interactions of a biopolymer macromolecule forming a system of many positively and negatively charged particles with interaction with a solvent, which occurs in the classical mode as will be shown. Based on the generalization, the statistical behavior of a natural macromolecule of hyaluronic acid near the phase transition temperature is studied [5, 8, 9, 11]. The choice of a macromolecule of hyaluronic acid as a model of a biopolymer macromolecule to solve a common problem is explained by its simplest primary structure consisting of identical repeating monomers (disaccharides) consisting of 2 types of sugar residues (units), in contrast to the more complex primary structure of globular proteins (consisting of 20 types of units), DNA and RNA (consisting of 4 types of units).
High-molecular hyaluronic acid polysaccharide is the main component of the intercellular substance surrounding a living cell, which plays an important role in the self-organization of physicochemical processes of cellular metabolism, the most important of which is the process of physical thermoregulation of living tissue that maintains temperature constancy. It is known that all the diversity of mammals, the main intercellular component of which is hyaluronic acid with a structure that has not changed during the long evolution, have a close body temperature (36-38° C.). It directly indicates the fundamental physical mechanism of thermoregulation based on the unique thermodynamic properties of the intercellular substance.
2. Choice of a Physical Model.
When solving a statistical problem, it is necessary to choose the simplest possible idealized microscopic model in order to obtain the exact solution, because near the phase transition point—a special (in the mathematical sense) point for thermodynamic quantities—any approximate theory may be erroneous.
An understanding of the generalized statistical principle that determines volumetric interactions in a biopolymer macromolecule can be reached on the basis of a statistical description of the interaction between a biopolymer chain of a macromolecule having the simplest chain structure (monomers of which have a charge) and solvent ions, which is discussed below.
2.1. Features of the Chain Structure of Hyaluronic Acid.
A native macromolecule with the simplest chain structure is hyaluronic acid, a high-molecular biopolymer consisting of repeating disaccharides-monomers (mers) (consisting of 2 units). It is soluble in water, in which it forms very viscous solutions. Carboxyl groups of disaccharides are completely ionized and under physiological conditions in a slightly alkaline medium have a negative charge. A macromolecule is a linear chain of monomeric units (disaccharides) interconnected by strong glycosidic bonds (a kind of covalent bond), the number of which in the chain can reach many thousands (N≤25,000). A repeating unit of hyaluronic acid is a disaccharide consisting of residues of D-glucuronic acid and N-acetyl-D-glucosamine connected by a β(1→3) glycosidic bond [5]. Each disaccharide monomer of hyaluronic acid contains three possible sites for modification: hydroxyl, carboxyl, and acetamide groups. The energy of the glycosidic bond is 2 eV (≈50 kcal/mol), which is more than an order of magnitude greater than the characteristic energy of thermal motion at physiological temperature. It is part of the extracellular basic substance of most types of connective tissue of warm-blooded animals, contained in cell membranes. The chemical structure of a hyaluronic acid biopolymer consisting of repeating disaccharide monomers including N-acetylglucosamine and glucuronic acid is shown in
2.2. Statistical Description of Volumetric Interactions of a Polymer Chain with a Solvent.
The polymer chain of natural hyaluronic acid, consisting of a large number of repeating negatively charged disaccharides bound by a strong covalent bond with the energy of Ecov>>kBT (at physiological temperatures), and a solvent with an excessive concentration of Na+ ions form a system of many positively and negatively charged particles, microscopic pair interactions between which determine the features of volumetric interactions of polymer chain with a solvent. Volumetric interactions are those that are not reduced to longitudinal covalent bonds with neighbors along the chain. The problem is to determine the stable configuration of the “polymer chain-solvent” system as the most probable configuration of the statistical system corresponding to the minimum free energy.
As will be shown, the statistical properties of a polymer chain interacting with solvent ions can be described using a statistical method based on the assumption of the short-range nature of the interaction forces between negatively charged chain monomers and positively charged solvent ions. It is assumed that the “polymer chain-solvent” system can be considered as a closed system in thermal and diffusion contact with the reservoir (energy and particles) formed by the polymer chain and solvent.
The statistical behavior of a thermodynamic closed system follows the second law of thermodynamics: if two systems (one of which is a large reservoir) at some point in time are in a configuration different from the most probable one, then the most likely consequence of this will be such a configuration change that at subsequent times the generalized entropy will monotonically increase.
The entropy of a closed “polymer chain-solvent” system can be determined based on the generally accepted thermodynamic definition of this value [12, 18]:
S≡k
B ln g, (1)
where kB is the Boltzmann constant; g is the number of permissible states—random equally probable configurations of a combined “polymer chain-solvent” system.
In the statistical method of Gibbs ensembles, the time averages in the limit of a single system are replaced by the ensemble averages, which are the averages for all systems in the ensemble. Replacing time averages with ensembly averages allows applying a probabilistic model of an accidental event to calculate thermodynamic quantities.
The problem is to describe the interaction of a polymer chain with a solvent based on the ensembles method. Therefore, it is necessary to define an ensemble of systems representing a closed system “polymer chain-solvent” as a set of identically organized systems, each of which is an exact copy of a real system in one of its permissible states.
An ensemble of systems representing a system of a polymer chain interacting with a solvent can be determined based on the chain structure of a macromolecule from the following simple considerations. Negatively charged units of the polymer chain (HA) can be considered as independent nodes that can bind (adsorb) individual sodium Na+ ions of the solvent. It is assumed that the binding energy of the Na+ ion with the node is of the order of the energy of the thermal quantum E0≈kBT. Let the free nodes be denoted by light circles ∘ and the nodes occupied by one Na ion by dark circles ●. A molecular chain consisting of N disaccharide monomers contains N independent nodes. If the electroneutrality condition is fulfilled, then the concentration of sodium ions in the solvent exceeds the concentration of chloride ions by the amount of sodium excess Δ(Na+)═(Na)—(Cl), equal to the concentration of negatively charged polymer chain monomers Δ(Na+)═(N).
It is assumed that the interaction forces between negatively charged monomers and positively charged ions have a short-range nature, which shows itself only in the process of paired interactions of particles that have come closer as a result of a random paired collision. Adsorption (binding) of Na solvent ions by nodes (negatively charged) of the polymer chain during thermal motion occurs as a result of random paired interactions (positively charged chain monomers with negatively charged solvent ions). So a set of microscopic random events occurs, the outcomes of which can be considered as equally possible outcomes of a random experiment—an action that can be repeated many times (a large number of times) under approximately the same conditions, and the results of which cannot be predicted. Therefore, the binding of Na solvent ions by polymer chain nodes in the process of thermal motion can be described on the basis of a probabilistic model of a random experiment [30] as accidental event A “Na adsorption”, which is the combination (sum) of a large number of random microscopic events, the equally possible outcomes of which form the space of elementary outcomes (events):
ΩNa{ωi,{acute over (ω)}j}≡ΩNa{ω1,ω2, . . . ,ωg}+ΩNa{{acute over (ω)}1, . . . ,{acute over (ω)}g},
where ΩNa{ω1, ω2, . . . , ωg} is the set of elementary outcomes of random paired collisions of Na solvent ions with chain nodes (solvent exchange); ΩNa{{acute over (ω)}1, . . . , {acute over (ω)}g} is the set of elementary outcomes of a polymer chain realized as a result of random permutations of ions between nodes (chain exchange), arising from random paired collisions of neutrally charged monomers whose nodes are occupied with Na+ ions; the process of random permutations of ions between the nodes of the chain can be described as a random linear walk of the Na+ ion along the monomers of the polymer chain.
Possible configurations of the distribution of sodium ions across the nodes of the polymer chain (permissible states), which can be realized as a result of a random event ΩNa{ω1, ω2, . . . , ωg} in the process of thermal motion over time, can be described (by analogy with the model system of spins [18]) on the basis of a probabilistic model of a random distribution of N independent nodes located at fixed points along one straight line, each of which may accidentally end up in one of 2 equally possible states.
The total number of equally probable outcomes (configurations) gsolv(N, m) of the set ΩNa{ω1, ω2, . . . , ωg} for N>>1 is described by the Gaussian distribution [18]:
g
solv(N,m)≈[N!/(N/2)!(N/2)!]×exp(−2m2/N)=g(N,0)×exp(−2m2/N);
g
solv(N,0)=[N!/(N/2)!(N/2)!]2N(2/πN)1/2 (2a)
where 2m is the excess of occupied nodes; 2m=(number off occupied nodes)−(number of free nodes).
The excess of occupied nodes determines the amount of energy of interaction of the polymer chain with the solvent. The energy can take many different values, from the minimum εmin=−Nε0 at 2m=N (all nodes are occupied) to max εmax=+Nε0 at 2m=−N (all nodes are free). The interaction between the polymer chain and the solvent arises as a result of a large number of random short-range pair interactions between the negative charges of the chain and the positive charges of the solvent. As a result of each elementary interaction, energy is exchanged between the polymer chain (system) and the solvent (reservoir). When the ion is adsorbed by the monomer of the chain, the energy of the system increases by +ε0 due to a decrease in the energy of the reservoir, when the ion passes from the node to the solvent. On the contrary, the energy of the system decreases due to an increase in the energy of the reservoir by +ε0.
The most probable configuration of the system (polymer chain), realized by the maximum number of ways gsolv(N,0)≈2N(2/πN)1/2, is a configuration with zero excess of occupied nodes m=0, corresponding to zero interaction energy ε=0 of negatively charged chain monomers with solvent ions, in which half of the nodes are occupied ●, the second half of the nodes is free α. N/2 ions are bonded to the nodes of the chain and N/2 free ions are in the solution.
All equally possible configurations with zero interaction energy ε=0 (for n=0) of the set ΩNa{ω1, ω2, . . . , ωg} form a set of possible elementary outcomes ΩNa{{acute over (ω)}1, . . . , {acute over (ω)}g}, which in the equilibrium state are realized with different probabilities. Possible configurations with zero interaction energy, the most probable configuration (left) and the least probable configuration (right), are shown in
The configuration with zero energy (ε=0) shown in
The number of states realizing the most probable configuration of the combined system can be represented as the product of the number of configurations g0rep with positive energy, determining the average energy of volumetric repulsion, by the number of configurations g0attr with negative energy, determining the average energy of volumetric attraction resulting from paired interactions of monomers: gNa(N,0)=g0attr×g0rep.
In the most probable configuration, the number of states with negative energy is equal to the number of states with positive energy: g0attr=g0rep=(N/2)!. The number of states is equal to:
g
Na(N,0)=g0attr×g0rep=(N/2)!2. (2b)
Thus, in the most probable configuration of the polymer chain interacting with the solvent, a uniform distribution of the linear concentration of adsorbed Na ions is realized, corresponding to a uniform linear quantization of the interaction energy of the polymer chain with the solvent and the maximum linear entropy density. Such a configuration corresponds to the minimum free energy F of the interaction of the polymer chain with the solvent:
F=U−TS;U=U
attr
+U
rep≡0;Fmin=−TSmax, (2)
where Uattr=−Nε0/2 is the energy of solvent ions attraction by chain monomers, Urep=+Nε0/2 is the energy of ions repulsion by chain monomers.
In a configuration with minimal interaction energy, the entropy is S=0, so the free energy is not equal to the minimum:
F=−Nε
0
>F
min,at Urep=−Nε0,Uattr=0.
An important characteristic of the most probable configuration of the combined polymer chain-solvent system with the minimum free interaction energy is stability—the property of the system to return to the equilibrium configuration with the minimum free interaction energy Fmin=−TSmax after a random deviation of the system from the equilibrium state as a result of thermal fluctuations. A reasonable characteristic of the stability of an equilibrium linear configuration is the relative deviation of the interaction energy from the average value: δε/Nε0=δ/N=m/N.
The distribution (2a) has a very sharp maximum with a center at m=0, corresponding to the number of states realizing the most probable configuration with zero interaction energy. To numerically estimate the probability of finding a system with a relative deviation δ/N=m/N, the number of N monomers per unit volume equal to the number of Na+ ions under physiological conditions may be taken. The excess concentration of ions in the blood, as is known, is Δ(Na+)=[Na]−[Cl]=40 mmol/l or 0.4×6×1023 monomers/1000 ml=2.4×1020 monomers/ml. Let us put the number of monomers N=1020 and δ=1011, i.e. the relative deviation δ/N=10−9. For such a slight deviation from equilibrium, we have 2δ2/N=200. In this case, the value of g(N, δ) is a fraction equal to e−200≈10−87 of its maximum value. The decrease turns out to be very strong, and the function gchain(N, δ) at N>>1 should be a function of δ with a very, very sharp peak. In reality, this means that the system will never be detected in a state with a deviation δ/N=10−9, no matter how small it may seem.
Here the important conclusion flows out that the most probable stable configuration of the combined system “polymer chain-solvent”, corresponding to the minimum free energy of volumetric interactions, is realized by a uniform linear distribution of adsorbed ions bound to the polymer chain monomers by a weak bond with an energy of ε0≈kBT, which does not change during thermal motion (invariance of the equilibrium configuration of the system to thermal fluctuations), as well as the configuration of the primary structure of the chain in which the monomers are bound by a strong covalent bond Ecov>>kBT.
So the statistical properties of a polymer macromolecule in the most probable stable linear configuration are determined by a relatively small number of configurations corresponding to the minimum free energy of interaction of the polymer chain with the solvent. The average physical characteristics of the macromolecule are very close to the average characteristics of states with only the most probable configuration of the polymer chain. As will be shown, the obtained conclusion is generally valid for a native macromolecule with volumetric interactions.
One of the physical properties of the system is entropy and its additivity property: the entropy of a composite system can be considered equal to the sum of the entropies of the systems included in it, provided that the latter have the most probable configuration. In this case, the entropy of a combined system consisting of a reservoir with the number of states gthermal and a system with the number of states gsystem cannot be calculated by the total number of acceptable states, but only by the number of states implementing the most probable configuration according to the formula:
S=k
B ln(gthermal×gsystem)max.
The configuration with the minimum free energy of a system located in a given volume is a property of a closed (combined) system with respect to the exchange of energy and particles between the system and a large reservoir (thermal bath): a polymer chain is a system in thermal and diffusion contact with a large reservoir (internal thermostat); energy exchange with a quantum of ε0≈kBT between the system and the thermostat occurs in the process of paired interactions without changing the energy of the combined system. As follows from the formula (2b), the average values of free energy and entropy in the configuration with the minimum free energy of interaction of the chain with the solvent are determined only by the number of monomers N in the chain and do not depend on the volume of localization of the macromolecule.
In the case of violation of electroneutrality, volumetric repulsive forces arise due to an excess of negatively charged monomers evenly distributed along the chain. A consequence of the uniform linear concentration of adsorbed ions is the conclusion about the homogeneous distribution of negatively charged monomers in the volume of the macromolecule. The statistical properties of the system (a gas of free negatively charged monomers) are indistinguishable from the statistical properties of an ideal gas, whose entropy depends on the volume, since the probability of realizing a sparse configuration is greater than the probability of a compressed configuration. So the most probable volumetric configuration of the combined system (system+reservoir) is unstable as a result of the effect of volumetric repulsion of negatively charged monomers with a stable linear configuration corresponding to the minimum free energy.
A stable three-dimensional (volumetric) configuration of a gas of negatively charged monomers can be described by analogy with an ideal gas in a vessel under a piston. The equilibrium can be realized only due to external pressure on the piston. Reversible gas expansion at a constant temperature can occur spontaneously due to the work expended by the system on moving the piston, as a transition of the system from a less probable state to a more probable one. The reverse process of isothermal compression cannot proceed spontaneously. The gas cannot spontaneously reduce the volume, but it can increase. Such asymmetry is associated with different probabilities of realization of more sparse and more compressed states. The first of them is more likely, since it can be realized by a larger number of permissible microstates than the second. So, in order to compress, it is necessary to apply a force to the piston, which is counteracted by the elastic (entropic) force of an ideal gas.
Consider the mechanical equilibrium of a polymer chain in a solvent filling the volume of a vessel under a piston, which is made semipermeable-permeable to solvent molecules, but impermeable to a polymer chain to which external pressure is applied. When the solution is separated from the pure solvent by a semipermeable partition, osmotic pressure arises, which is observed in the experiment as the pressure difference between the solution and the pure solvent in equilibrium with it. The osmotic pressure of a weak solution is equal to the pressure of an ideal gas, which at a solution temperature equal to the temperature of an ideal gas occupies the same volume with it and contains a number of moles equal to the number of moles of dissolved substances [15, 16]. The formula for the osmotic pressure of the solution (HA) resulting from the thermal motion of negatively charged polymer chain monomers (without collisions) exactly coincides with the state equation of an ideal gas:
P=RT×N/V, (3)
where R is the universal gas constant, T is the absolute temperature, N is the number of moles of the substance, and V is the volume.
So, the gas pressure of free monomers, intramolecular osmotic pressure, and the pressure of an ideal gas in a vessel under the piston have a single statistical nature due to different probabilities of realization of the rarefied and compressed states and are determined by the equation of state (3). The non-interacting gas of polymer chain monomers, whose statistical behavior exactly obeys the equation of ideal gas (3), can be defined as an ideal gas of free monomers. In practice, none of the real gases follow equation (3) exactly.
So, a stable three-dimensional spatial structure of a macromolecule localized in a limited volume of solvent can be realized only if volumetric attractive forces having a statistical nature arise in the volume of the macromolecule. These forces precisely compensate for the volumetric repulsive forces having a statistical nature and caused by the entropic elasticity of an ideal gas of monomers.
The natural question that has to be answered is about the possible statistical nature of intramolecular attractive forces capable of compensating for the volumetric repulsive forces due to the entropic elasticity of an ideal gas of free polymer chain monomers.
2.3. The Basic Assumption of the Model.
The main assumption of the model is as follows: intramolecular volumetric short-range attractive forces arise as a result of the formation of random hydrogen bonds between monomers and solvent glucose molecules in the process of paired interactions.
Based on the features of the primary structure of hyaluronic acid, the polymer chain of which is formed from repeating disaccharides, it can be assumed that the disaccharides of the chain may have selectivity to the glucose molecule of the solvent. D-glucose molecule (CH2O)6 is the predominant monosaccharide unit of natural polysaccharides, in which the residues of sugar molecules are sequentially connected to each other by glycosidic bonds—a type of covalent bond that connects a sugar molecule with another molecule often with another sugar [5]. So, a weak hydrogen bond may form between the glucose molecule of the solvent and the sugar residue of the hyaluronic acid polymer chain during a pair collision, which is known to be a special case of covalent with delocalization of the electron density along the chain of atoms. This assumption is consistent with the results of biochemical experiments [8-11], which indicate the important role of hydrogen bonds in the formation of the three-dimensional structure of the macromolecule of hyaluronic acid.
An intramolecular hydrogen bond arising between a solvent glucose molecule and a polymer chain disaccharide can be considered a special case of a glycosidic valence bond with electron density delocalization along a chain of atoms. A hydrogen bond can form between polarized sections of the solvent glucose molecule and sections of sugar residues (D-glucuronic acid and N-acetyl-D-glucosamine) of the hyaluronic acid polymer chain and/or polarized sections of the solvent water molecules.
Statistical Model of Biopolymer Molecule.
An idealized model of a natural macromolecule, which reflects the most general properties of natural macromolecules with a chain structure, can be formulated as follows: A native macromolecule of hyaluronic acid, the polymer chain of which consists of a large number N>>1 of repeating monomers (disaccharides) consisting of 2 units, one of which has a negative charge in a slightly alkaline glucose solution with an excess of sodium and potassium ions at constant external pressure. The energy of the longitudinal covalent bond between the monomers is much greater than the characteristic thermal energy at physiological temperatures Ecov>>kBT. Volumetric interactions between monomers of the polymer chain, molecules, and solvent ions have a short-range nature and manifest themselves only in the process of random paired collisions. A glucose molecule can form a random hydrogen bond with a characteristic energy E0 much lower than the energy of the longitudinal covalent bond E0<<Ecov with each unit of each monomer. One of them has a negative charge, and the second has a neutral charge, unlike solvent ions, which can form an ionic bond only with a negatively charged monomer link with the characteristic binding energy ε0≈kBT.
It is assumed that the attraction forces have a short-range nature and manifest themselves only in the process of paired interactions of non-neighboring polymer chain monomers that happen to be neighbors accidentally as a result of a paired collision. The possible long-range influence of the remaining monomers of the system due to the Coulomb (van der Waals) interaction is not taken into account in the model.
As will be shown below, the proposed approach is universal and applicable to describe the properties of an arbitrary biological macromolecule. The generalization of the idealized model of the macromolecule of hyaluronic acid leads to a universal model applicable to any native macromolecule (proteins, DNA, and RNA).
3. The Principle of Interaction of Statistical Type.
Volumetric attractive forces arise as a result of random short-range pair interactions of neutrally and negatively charged links of polymer chain monomers with neutrally charged glucose molecules and solvent ions, which lead to the formation of random hydrogen and weak ionic bonds.
3.1. Statistical Mechanism of Volumetric Interactions.
The main biopolymers are known to function in an aquatic environment: in humans, for example, water is about 60%. The important role of water in living systems is associated with its specific property of forming hydrogen bonds between molecules. The hydrogen bond, according to modern concepts, is defined as a form of association between an electronegative atom and a hydrogen atom H bound covalently with another electronegative atom (O, N, F) [28]. The hydrogen bond can be considered as a pair attractive interaction between atoms of type O, one of which is covalently connected to hydrogen, for example: O—H . . . O. Within the framework of the theory of molecular orbitals, the hydrogen bond is considered as a special case of a covalent bond with delocalization of the electron density along the chain of atoms. The formation of hydrogen bonds between water molecules leads to the emergence of so-called water clusters or complexes. The simplest example of such a cluster is a water dimer: (H2O)2═H2O . . . HOH. The energy of the hydrogen bond is about 0.1 eV (≈2.5 kcal/mol), which is one or two orders of magnitude less than the energy of the covalent bond (1-10 eV), but more than the thermal energy of thermal motion at room temperature 300 K (with the energy of the covalent bond O—H 200 times more thermal energy). Therefore, liquid water can be considered a network of H2O molecules connected by hydrogen bonds (“bound liquid”) [29]. The molecular structure of water at each moment of time is a spatial grid of hydrogen bonds (
Consider the statistical mechanism of volumetric interactions of the hyaluronic acid polymer chain with ions and solvent molecules in a slightly alkaline aqueous solution of the molecule G (hypothetical) with an excess of Na+ ions and positively charged K+ ions (hypothetical).
The concentration of monomers Nmon and the concentrations of ions and solvent molecules G are related by the electroneutrality condition:
N
mon
/V
0mol=(Nmon)=(C)0+(Na)0=(C)0+(K)0, (4)
where V0mol is the average value of the volume of a macromolecule; (C)0 is the concentration of G molecules.
Volumetric attractive forces can occur if the probability of binding of the G molecule by a negatively charged monomer unit is equal to the probability of binding (with energy ε0) of the Na ion, or, in other words, if the equilibrium constant of the adsorption process (with binding energy E0) of the G molecule is equal to the adsorption equilibrium constant of the Na ion (with ionic binding energy ε0). At the same time, the equilibrium constant of the K ion exceeds the equilibrium constant of the Na ion:
(M)(C)/(MC)=(M)(Na)0/(MNa)<(M)(K)0/(KNa). (4a)
As will be shown below, equation (4a) is true at a molecular weight of 180 equal to the molecular weight of the glucose molecule with a binding energy equal to the hydrogen bond energy of 0.1 eV. This indicates the validity of the basic assumption of the model. In this case, the hypothetical K ion is the potassium ion.
Independent nodes of the polymer chain, if equation (4a) is fulfilled, can be formally divided into two groups consisting of NNa nodes (Na) forming an ionic bond with solvent ions, and Ng nodes (G) forming a hydrogen bond with solvent glucose molecules. The concentrations of the nodes of the polymer chain (Na) and (G) are equal to the concentrations of Na ions and glucose molecules of the solvent, respectively:
(NNa)═(Na)0;(Ng)=(C)0;(Nmon)=(NNa)+(Ng)=(C)0+(Na)0, (4b)
where NNa is the number of monomers (Na) and Ng is the number of monomers (G).
When equation (4) is fulfilled, there is 1 glucose molecule per 2 nodes of the monomer (G), while for each negatively charged node (Na), there is 1 Na ion.
Determine the parameter characterizing the distribution of nodes (Na) and (G) of the polymer chain and the critical concentrations of solvent molecules and ions satisfying condition (4b):
γ=NNa/Ng=(Na)0/(C)0=(Na)0/(K)0.
In the process of thermal motion, random configurations of hydrogen bonds between polymer chain monomers and solvent particles can be formed, which can be described as simple reversible chemical reactions:
Hydrogen bonds between the glucose molecule G and 2 water molecules (attraction between the glucose molecule and water molecules):
G+2H2O= . . . OH—H . . . G . . . H—OH . . . (4d)
Hydrogen bonds between the glucose molecule G and the neutrally charged M unit of the chain (attraction between the neutrally charged unit of the monomer and the solvent):
M0+G+H2O=M . . . G . . . H—OH . . . (4f)
Hydrogen bonds between the glucose molecule G and the negatively charged link of the COO− polymer chain (attraction between the negatively charged unit of the monomer and the solvent):
COO−+G+H2O+K+=COO . . . G . . . H—OH . . . K. (4g)
The formation of a hydrogen bond between a glucose molecule and a negatively charged unit of the polymer chain leads to the appearance of an excessive negative charge in the chain of atoms and the formation of an ion pair (between the K+ ion and the negatively polarized OH group of the water molecule), which is formed due to Coulomb forces (weak non-covalent bond): —OH−+K+=—OH−K+.
Hydrogen bonds between the glucose molecule G and 2 non-neighboring monomers of the polymer chain:
COO−+G+M0+K++H2O=COO . . . G . . . M . . . K+H2O. (4h)
The hydrogen bond formation between a glucose molecule and a negatively charged polymer chain unit leads to the formation of an M− K+ ion pair (bound by a weak non-covalent bond) as a result of the appearance of an excessive negative charge in the chain of atoms.
If the exact compensation condition is fulfilled, short-range pair interactions of non-neighboring chain monomers can lead to the formation of chain crosslinking—molecular complexes consisting of non-neighboring monomers and glucose molecules bound by hydrogen bonds. The molecular crosslinking complex can be defined as a tetramer (by the number of 4 valence bonds). The pair interaction with the formation of a tetramer in the volume of the macromolecule can be formally described as an equilibrium chemical reaction consisting of 2 stages. In the first stage, a glucose molecule binds to a monomer to form a dimer; in the second stage, a tetramer is formed as a result of a random paired collision of two dimers.
Arising from the chain structure of the biopolymer macromolecule, the peculiarity of the intramolecular grid of hydrogen bonds is that random hydrogen bonds resulting from random paired interactions of monomers during thermal motion can realize the configuration of a polymer network similar to the polymer network of rubber. It is formed as a result of chemical crosslinking by covalent bonds (during vulcanization) of the natural polymer macromolecule of rubber with impurities of sulfur atoms. The fundamental difference between the polymer network of a natural macromolecule and the polymer network of rubber is that its cross-links are formed by weak (hydrogen) bonds. Cross-links of the polymer mesh randomly arise and are destroyed in the process of thermal motion. This idea is consistent with the known property of hyaluronic acid: a biopolymer that is highly soluble in water, in which it forms very viscous concentrated solutions, capable of turning into a gel-like state. The polymer network fragment and the tetramer configuration with 4 hydrogen bonds are shown schematically in
3.2. The Principle of Minimum Free Energy of Volumetric Interactions
As it was shown above, the equilibrium state of the integrated system of the polymer chain interacting with the solvent is realized as the most probable configuration corresponding to the minimum free energy of interaction of the polymer chain with the solvent (2) with a uniform distribution of the linear concentration of adsorbed Na ions along the circuit of the chain and uniform distribution of free ions in the volume of the solvent.
It is quite reasonable to assume that the principle of the minimum free energy of volumetric interactions (2) is a general principle that determines the statistical properties of a system of many positively and negatively charged particles with a volumetric interaction. In this case, the stable equilibrium spatial configuration of the polymer chain in a glucose solution with an excess of Na and K ions can be realized by the maximum number of states as the most probable configuration corresponding to the minimum free energy of volumetric interactions. The energy of volumetric attraction in this case should be identically equal to the energy of volumetric repulsion (2):
F=U−TS;U=U
attr
+U
rep≡0;Fmin=−TSmax, (2)
where Urep is the repulsive energy caused by the entropic elasticity of an ideal gas of negatively charged monomers; Uattr is the attractive energy caused by paired interactions of monomers; Smax is the entropy, which is determined by the number of states realizing the most probable configuration of the combined system: Smax=kB ln(gthermal×gsystem)max.
It is appropriate to define intramolecular volumetric interactions between polymer chain monomers that satisfy the equation of exact compensation of attraction and repulsion (2) as equilibrium volumetric interactions.
4. Quantization of the Free Energy of Volumetric Interactions.
The problem is to describe intramolecular volumetric interactions as the sum (union) of a very large number of random short-range pair interactions, which can be defined as random events ωi, {acute over (ω)}j, υj, forming a set of elementary outcomes realized in the process of thermal motion:
Ω{ωi,{acute over (ω)}j,υj}=ΩNa{ωi}+ΩG{{acute over (ω)}j}+ΩD{υj},
where ΩNa{ωi} is the set of elementary outcomes of a random event A (paired interaction of Na solvent ions with polymer chain monomers); ΩG{{acute over (ω)}j} is the set of elementary outcomes of a random event B (paired interaction of solvent glucose molecules with polymer chain monomers to form a dimer consisting of a glucose molecule and a monomer); ΩD{υj} is the set of elementary outcomes of a random event D (paired interaction of non-neighboring dimers to form a tetramer configuration); ωi, {acute over (ω)}j, υj are elementary events.
The totality of all equally possible configurations of a macromolecule corresponding to the elementary outcomes of the set Ω{ωi, {acute over (ω)}j, υj} of a random event A+B+D that is the sum of random events A, B, D form an ensemble of systems representing the integrated “polymer chain-solvent” system. In the Gibbs ensembles method, the time averages in the limit of a single system are replaced by ensemble averages, which are the averages for all systems in the ensemble. So, the problem is to determine the ensemble of systems characterizing the volume interactions.
An important consequence of the equation of the exact compensation of the forces of attraction and repulsion (2) is the possibility of forming a stable tetramer configuration—crosslinking of the polymer chain by means of random hydrogen bonds formed during random short-range pair interaction of dimers (event D). The dimer configuration is formed as a result of binding of one glucose molecule of the solvent with the chain monomer. A random event D can be described as a combination of random events D1+D2+D3. The equally possible outcomes of these events form a set of elementary outcomes that can be realized in the process of internal motion of the tetramer resulting from the convergence of monomers in the process of paired collision:
ΩD{Rl,Tl,υj}≡ΩD{Rl}+ΩD{Tl}+ΩD{υj},
where Rl are microscopic forms of a tetramer (G) with a random number of hydrogen bonds l (from 0 to 4) that can be realized as a result of paired interaction of dimers (event D1); Tl are microscopic forms (G) of a tetramer in which hydrogen bonds between dimers cannot be formed (not having energy), realized in the case of paired collisions of dimers (event D2); υj are elementary random events that implement the configuration of the tetramer (Na) that does not coincide with the (R) and (T) configurations of the tetramer (event D3).
Elementary outcomes of the random event D1 are equally possible configurations (G) of the tetramer—microscopic forms of Rl with different numbers of hydrogen bonds l (from 0 to 4) that can be described using the following equilibrium chemical reactions characterized by the same microscopic equilibrium constant kR:
R
0
+
=R
1
R
1
+
=R
2
R
2
+
=R
3
R
3
+
=R
4. (5)
Here, the hydrogen bond (with delocalized electron density) is denoted by the symbol , which usually denotes a localized electron pair of a covalent bond.
The microscopic constant of the equilibrium process kR can be written as the expression:
k
R=(R0)()/(R1)=(R1)(
)/(R2)=(R2)(
)/(R3)=(R3)(
)/(R4). (5a)
Different values of the binding energy of the tetramer E(l) correspond to microscopic forms of Rl with different numbers of hydrogen bonds l, which are multiples of the number of hydrogen bonds: E(l)=lE0, where E0 is the energy of the hydrogen bond; the number of hydrogen bonds l=0, 1, 2, 3, 4. Microscopic forms Rl (l=0, 1, 2, 3, 4) of the tetramer (G) are shown schematically in
In the configuration of the tetramer R1 (l=1) with one hydrogen bond, the configuration of which is described by equation (5a), the effect of attraction between a pair of monomers and one glucose molecule occurs due to a single random hydrogen bond. It can be formed with an equal probability between one of the 2 glucose molecules and one of the 2 monomer sugar residues closest to it. A stable configuration of a pair of monomers and one glucose molecule with a binding energy of E1(l=1)=E0 results from random permutations of a single delocalized hydrogen bond between 1 (one) glucose molecule and 2 (two) monomers.
Generally, in the Rl configuration (with the number of valence bonds l), the binding energy between tetramer molecules resulting from a delocalized valence bond is E(l)=lE0, where l=0, 1, 2, 3, 4.
The possible outcomes of the sum of random events D1+D2—tetramer configurations (G) in conformations (R) and (T) are schematically shown in
The configuration shown in
Each configuration Rl (l=0, 1, 2, 3, 4) with different numbers of hydrogen bonds and different values of binding energies can be realized by different numbers of degenerate states (configurations with the same energy). If equation (2) of exact compensation of attraction and repulsion is fulfilled, the tetramer configuration can be realized in one of the degenerate states in the process of internal motion of the tetramer in the form of random permutations of delocalized electron pairs of hydrogen bonds and random permutations of monomers and glucose molecules.
The number of permissible states (equally possible outcomes) of a tetramer for each microscopic form of Rl with a fixed number of hydrogen bonds (and energy) can be calculated as the sum of random permutations of hydrogen bonds (electron pairs) with a fixed spatial position of monomers and glucose molecules, and of permutations of monomers and glucose molecules with a fixed number of hydrogen bonds.
It can be shown that the total number of random intra-tetramer permutations of hydrogen bonds and molecules gmon(l) can be calculated using the following formula:
g(l)=gmon(l)+g0(l)=6l+4!/(4−l)!l!, (5b)
where the first term gmon(l)=6l determines the number of random permutations of molecules depending on the number of hydrogen bonds l, and the second term g0(l)=4!/(4−l)!l! determines the number of random permutations of hydrogen bonds.
Microscopic forms of Rl with different numbers of hydrogen bonds l are realized by different numbers of states g0(l) (permutations):
g
0(l=0)=1;g0(l=1)=4;g0(l=2)=6;g0(l=3)=4;g0(l=4)=1.
The total number of permutations g(l) implementing microscopic forms of Rl is equal to:
g(l=0)=6×0+g0(l=0)=1;
g(l=1)=6×1+g0(l=1)=6+4=10:
g(l=2)=6×2+g0(l=2)=12+6=18;
g(l=3)=6×3+g0(l=3)=18+4=22;
g(l=4)=6×4+1=24+1=25.
The evolution of the tetramer configuration in the process of internal motion from the less probable configuration R0 (0 bonds) to the most probable configuration R4 (4 bonds) can be described as a sequence of random transitions from one configuration to another R0→R1→R2→R3→R4. The number of equally possible elementary outcomes realizing the evolution (G) of the tetramer from the configuration R0 in the most probable configuration of RA can be calculated using the following expression:
g(4,0)×g(4,1)×g(4,2)×g(4,3)×g(4,4)=1×10×18×22×25≈105.
Thus, when the equation of attraction and repulsion compensation (4) is fulfilled, the stochastic process of tetramer formation (G) proceeds spontaneously with an increase in entropy. The least probable configuration of tetramer (G), the microscopic form R0, which is realized only in 1 state, in the process of internal thermal motion of the tetramer spontaneously evolves into the microscopic form R4, which is realized in the maximum number of 25 states.
Random event D3. The configuration of a tetramer with 4 hydrogen bonds can be realized in the process of thermal motion as a random event D3, which is a combination of event B (glucose adsorption) and event A (Na ion adsorption): D3=B+A. Elementary outcomes of the random event D3 are equally possible configurations of the tetramer (Na) shown schematically in
The tetramers (Na) and (G) differ in the configurations of occupied monomer nodes. In the tetramer (Na), from the two nodes of each monomer (4-5 and 3-6), one node is occupied by a Na ion, and the second node is occupied by a glucose molecule through a single hydrogen bond. In the tetramer (G), on the contrary, both nodes of each monomer are occupied by glucose molecules through hydrogen bonds. Glucose molecules in the tetramer (Na) are associated with 4 similar nodes of monomers 1, 2, 3, 4; therefore, microstates realizing permissible configurations (microscopic forms) with different numbers of hydrogen bonds coincide with the microscopic forms of the tetramer (G) discussed above.
However, in such a tetramer configuration, the number of possible random permutations of monomers during the internal motion of the tetramer (Na) differs from the number of random internal permutations in the tetramer (G). The peculiarity of the internal motion of the tetramer (Na) and the method of counting the number of microstates implementing the configuration of the tetramer (Na) will be discussed in the next section.
4.2. The Ensemble of the Hydrogen Bond Grid.
The problem is to determine the most probable three-dimensional configuration of the grid of hydrogen bonds of a macromolecule in a solvent whose components are connected by equation (4).
The adsorption of Na ions by negatively charged polymer chain monomer nodes (random event A) was discussed in detail above. As shown, at zero concentration of glucose and K ions in the most probable configuration of the polymer chain (
In a solvent with an excess of Na and K ions (assuming that the equilibrium constants characterizing the binding of ions by chain nodes are equal) at zero glucose concentration, the most probable configuration is realized with a uniform distribution of the linear concentration of adsorbed Na and K ions and uniform distribution of free Na and K ions in the solvent volume shown in
Consider the binding features of the glucose molecule (which can be called the G molecule in general). As was shown, in a glucose solution with a composition meeting equation (4), sodium ions and glucose molecules will bind by monomers of the chain. In this case, each of the 2 monomer nodes can bind 1 neutrally charged glucose molecule, and only 1 of these nodes (negatively charged) can bind a positively charged ion. The adsorption of a glucose molecule by a monomer containing 2 nodes is realized as a random event A+B, which is the sum of random events B (paired collision with dimer formation) and D (paired collision with tetramer formation). The possible outcomes are configurations of hydrogen bonds, which are described by equations 4 (f g, h).
Elementary outcomes of random event B (adsorption of glucose molecules by a monomer) are equally possible outcomes: “2 nodes are free”; “1 node is free and 1 node is occupied”; “2 nodes are occupied”. The possible outcome of “2 nodes are occupied” can be realized as a result of the formation of 2 hydrogen bonds between one monomer and a pair of glucose molecules (occupancy of monomer 1). The outcome of “1 node is free and 1 node is occupied” can be realized as a result of the formation of one hydrogen bond between one of the nodes of the monomer and one glucose molecule (occupancy of monomer 1/2). The outcome “2 nodes are free” corresponds to the occupancy of monomer 0. The set of elementary outcomes of event B is denoted by ΩG{0, 1, 2}, where 0, 1, 2 correspond to equally possible elementary outcomes with different numbers of hydrogen bonds 0, 1, 2. It follows from the equal possibility of elementary outcomes that random events of the set elementary outcomes ΩG {0, 1, 2} can be realized with an equal probability of ⅓.
Hence in the most probable linear configuration of the polymer chain corresponding to the uniform linear quantization of the interaction energy of the polymer chain monomers with solvent glucose molecules, an ordered configuration with repeating sections with the same configuration of 3(γ+1) monomers arises, where 3 monomers (G) are with different occupancy values (0, 1, 2) and 3γ monomers (Na) are with alternating free and occupied nodes. An elementary section of a chain of 3(γ+1) monomers consisting of 3 monomers (G) and 3γ monomers (Na) can be defined as an elementary chain. The possible configuration of the elementary chain satisfying the requirement of a uniform linear concentration of populated nodes is shown in
Quantization of the Free Energy of Volumetric Interactions.
Consider the volumetric interaction of a pair of elementary chains, which can be named an elementary volumetric interaction, the result of which is determined by random events B and D—paired interactions of monomers with ions and glucose molecules of the solvent (event B) and paired interactions of monomers with each other (event D).
In the most probable spatial configuration, each of the elementary chains is in the most probable linear configuration corresponding to the minimum free energy of the interaction of the elementary chain with the solvent. The most probable volumetric configuration of elementary chains corresponding to the minimum free energy requirement (2) is realized as a result of the effect of elementary chains pairing, the driving force of which is the system tendency to a volumetric configuration with maximum entropy corresponding to the minimum free energy of volumetric interactions.
The most probable equilibrium configuration of interacting elementary chains is realized as a random event, which is the product of random events B and D, consisting of those and only those outcomes that are common to these events. Elementary random events B (adsorption of glucose molecules by monomers) corresponds to equally possible outcomes with the occupancy of monomers 0, 1, 2, corresponding to the numbers of hydrogen bonds from 0 to 2 (0, 1, 2). As shown, elementary random events D (paired collisions of monomers) corresponds to equally possible outcomes with the number of hydrogen bonds from 0 to 4 (0, 1, 2, 3, 4).
The set of elementary outcomes Ωnet realizing the equilibrium configuration of a pair of interacting elementary chains can be defined as the intersection (common part) of outcomes sets (G) of a single monomer ΩG{0, 1, 2} and sets of outcomes of a single tetramer ΩD{Rl}=ΩD{0, 1, 2, 3, 4}:
ΩG{0,1,2}∩ΩD{0,1,2,3,4}=Ωnet{0,2,4}.
Hence in the equilibrium spatial configuration of the macromolecule, at a critical glucose concentration with an equal probability of ⅓, tetramer configurations R4=Rl (l=4), R2=Rl (l=2), R0=Rl (l=0) with the numbers of hydrogen bonds 4, 2, and 0, respectively, and configurations of hydrogen bonds, which are described by the equations 4d, 4g, 4h, are realized.
The formation of 1 tetramer (G) leads to the crosslinking of 2 elementary chains by means of hydrogen bonds, as a result of which a linked pair of elementary chains with a paired number of non-neighboring monomers that have become closer as a result of the formation of a tetramer is formed. A pair of monomers in a tetramer is appropriately called a monomer pair. The most probable linked configuration of the elementary chains of the polymer chain is schematically shown in
The most probable configuration of interacting elementary chains arises as a result of the chain pairing effect, the driving force of which is the tendency of the unified thermodynamic system consisting of a polymer chain and a solvent to a configuration with maximum entropy corresponding to the minimum free energy of volumetric interactions.
The free energy F of the volume interactions of a linked pair of elementary chains (the free energy of an elementary volume interaction) can be determined using equation (2), in which the maximum entropy S is calculated by the number of states g0pair, realizing the most probable configuration of a linked pair of interacting chains:
F
min
/k
B
T=S
max
/k
B=ln(g0pair)max=ln eσmax=σmax;1/kBT=Δσ/Δε. (6)
where σ is the number of entropy quanta; the interaction energy.
The determination of the number of states through the exponent of the number of entropy quanta is related to the peculiarity of quantization of the interaction energy of a system of many particles with interaction in thermal and diffusion contact with a large reservoir (quantization principle). The change in the number of interaction energy quanta (the change in energy in terms of kBT) is equal to the change in the number of entropy quanta. The most probable configuration of the unified thermodynamic system corresponding to the minimum free energy of volumetric interactions is established with respect to the exchange of energy with quantum kBT between the system (polymer chain) and the reservoir (solvent), which has a discrete nature and at a constant temperature T. The exchange of one quantum of energy is accompanied by the exchange of one quantum of entropy kB (determination of thermodynamic temperature).
A configuration linking a pair of elementary chains by hydrogen bonds can be realized in one of the equally possible configurations R4, R2, R0, realized by the same number of equally possible states. Each of them corresponds to one of 3 equally possible outcomes of random paired interactions: configuration with negative energy (attraction) R4, configuration with positive energy (repulsion) R0, and configuration with zero energy (equilibrium) R2.
As a result of the pairing of elementary chains, each of which can be realized by a maximum number of states g0chain, the free energy of the elementary volume interaction decreases as a result of an increase in the number of states realizing the equilibrium configuration of a connected pair of interacting chains:
σpair=(σchain)2, where σchain—the number of configurations of a connected pair of elementary chains with a fixed configuration of one of the chains.
The number of attraction configurations R4 of a linked pair of elementary chains can be calculated in the following way. The number of attraction configurations (with negative energy) of chain 1 with the frozen configuration of chain 2 can be calculated as the number of random configurations of chain 1 resulting from random permutations (random linear walk) of the monomer (G) forming the configuration of the tetramer R4 (l=4):
σchain1=(γ+1)−3=γ−2.
Here it is taken into account that each of the γ monomers (Na) of the elementary chain can be realized in the configuration of the monomer (G) in one of 3 possible configurations R4, R2, R0.
The number of random configurations of attraction of the chain 2 can be calculated in the same way:
σchain2=(γ+1)−3=γ−2.
The total number of attraction configurations (with negative energy) R4 (l=4) of a connected pair of elementary chains is equal to:
(σpair)attr=σchain1×σchain1=(σchain)2=(γ−2)2.
The total number of repulsion configurations (with positive energy) R0 (l=0) and equilibrium configuration (with zero energy) R2 (l=2) can be calculated in a similar way:
(σpair)rep=(σpair)0=(σchain)2=(γ−2)2.
Thus, in the most probable configuration of a linked pair of elementary chains, the number of entropy quanta with positive energy (repulsion) is equal to the number of entropy quanta with negative energy (attraction) and is equal to the number of entropy quanta with zero energy (equilibrium):
(σpair)attr=(σpair)rep=(σpair)0=(σchain)2=(γ−2)2. (6a)
Hence the number of states realizing the most probable configuration of a coupled pair of interacting elementary chains corresponding to the minimum free energy of volumetric interactions does not depend on the distance between monomers and is determined only by the parameter γ, which can be defined as the interaction constant.
The free energy of the elementary volumetric interaction is thus defined as the sum of two equal contributions corresponding to the free energies of attraction due to the paired interactions of monomers and repulsion due to the entropic elasticity of negatively charged monomers:
F/k
B
T=(σpair)max=−[(σpair)attr+(σpair)rep]=Fattr min+Frep min. (6b)
Thus, the most probable configuration of a macromolecule is realized as a result of the pairing effect of interacting elementary chains, the driving force of which is the system tendency to a configuration with maximum entropy corresponding to the minimum free energy of volumetric interactions. The consequence of the pairing effect of elementary chains is the quantization of the free energy of volumetric interactions, which results in intramolecular ordering and long-range order with the partition of the volume of the macromolecule into equal-sized elementary cells with the same values of maximum entropy corresponding to the minimum free energy of volumetric interactions. The closed system of the macromolecule is divided into identical elementary subsystems with the same volume, composition, and number of particles, which are in thermal and diffusion contact with the reservoir. The total minimum free energy of volumetric interactions is equal to the minimum free energy of the elementary volumetric interaction multiplied by the number of cells in the volume of the macromolecule. The equilibrium state of the macromolecule corresponds to a configuration with a minimum free energy density and a maximum entropy density.
The value of the interaction constant γ (which was defined above as the ratio of the number of ionic bonds to the number of intramolecular hydrogen bonds) does not depend on the spatial configuration of the polymer chain and cannot change during thermal motion with a given constant contour length of the chain and the concentration of solvent ions. Therefore, the interaction constant γ is a topological invariant of the unified thermodynamical system consisting of a polymer chain and a solvent, and an order parameter that determines the quantization of the intramolecular space.
The volume of the elementary cell Vcell where a linked pair of interacting elementary chains with a number 2×3(γ+1) of monomers is localized is spontaneously divided into 3(γ+1) microscopic cells with a volume of V0cell, the number of which is equal to the number of pairs of monomers in a linked pair of elementary chains: Vcell=V0cell×3(γ+1). In this case, for each pair of non-neighboring monomers belonging to different chains of the linked pair, there is one microscopic cell, the volume of which V0cell is determined by the volume of the molecular complex tetramer (G), consisting of 2 monomers and 2 glucose molecules, which is determined by the distance between the monomer and the glucose molecule, depending on the energy of the hydrogen bond E0.
In the most probable equilibrium configuration of the macromolecule, at the critical concentration of solvent glucose determined by equation (4), a homogeneous distribution of monomers (G) and (Na) over microscopic cells in the volume of the elementary cell is realized. Monomers (G) in paired configurations of microscopic forms of tetramer R0, R2, R4, corresponding to equally possible interaction configurations (repulsion, equilibrium, attraction) are randomly distributed over 3 microscopic cells, which can be defined as microscopic cells M(R0), D(R2), T(R4) (by the number of hydrogen bonds in cell 0, 2, 4). Monomers (Na) are randomly distributed in pairs (the node is occupied, the node is free) over the remaining 3γ cells.
The thermal motion of monomers in the volume of a macromolecule can be described as a random three-dimensional walk (through microscopic cells) in a quantized intramolecular space.
The distribution of monomers (G) and (Na) of a linked pair of elementary chains in the quantized space of an elementary cell in one of the equilibrium configurations is schematically shown in
Thus, a closed system of a biopolymer macromolecule in an equilibrium state is divided into elementary cells with the same volume, composition, and a paired number of monomers 2×3(γ+1), forming a set of identically equal closed systems, the properties of which can be described by an ensemble of systems of a single elementary cell. Each system of an ensemble of an elementary cell is an exact copy of a real system in one of its permissible states. It reproduces a real system in one of the permissible states that can be realized in the process of thermal motion over time. It will be shown, that the ensemble of systems of elementary cell allows calculating the thermal averages of thermodynamic quantities characterizing the properties of a macromolecule.
Tetramer configuration (Na). The peculiarity of the tetramer (Na), whose configuration is discussed above, is that it is formed from 6 units that cannot realize the tetramer configuration in the volume of 1 microscopic cell V0cell, which is determined by the average volume of the tetramer (G). The distribution of monomers (Na) of the tetramer is shown in
The total number of configurations (Na) of the tetramer gNa (l) that can be realized as equally possible outcomes of the D3 event can be calculated using the following formula:
So the formation of a non-equilibrium configuration of the tetramer (Na), when equation (4) is fulfilled, cannot occur spontaneously, but can take place due to a change in the free energy (G) of the process occurring spontaneously with an increase in entropy.
5. Statistics of Intramolecular Hydrogen Bonds.
As has been shown, at a critical glucose concentration of C0, the most probable equilibrium configuration of the combined macromolecule system with a maximum entropy density Smax corresponding to the minimum free energy density Fmin of volumetric interactions is realized. At the same time, the number of attraction configurations with negative energy in the volume of the elementary cell is equal to the number of repulsion configurations with positive energy:
(σpair)attr=(σpair)rep=(σpair)0;Fattr=Frep=Fmin/2;
An excessive concentration of glucose molecules of the solvent C>C0 will lead to a spatial configuration (pairs of interacting elementary chains) with an excess of free energy of attraction Fattr>Fmin/2. As a consequence, an excess of the number of tetramers resulting from an increase in the number of configurations (entropy quanta) of attraction with negative energy σattr>σ0attr takes place due to a decrease in the number of repulsion configurations with positive energy σrep<σ0rep with a constant total number of configurations of states σattr+σrep=σ0attr+σ0rep=max corresponding to the minimum free energy of volume interactions and the maximum entropy in the volume of the cell. On the contrary, a negative excess of glucose concentration will lead to a negative excess of the free energy of the interaction Frep<Fmin/2 resulting from an increase in the number of repulsion configurations with positive energy due to a decrease in the number of attraction configurations with negative energy.
The problem is to find the dependence of the thermal average value of the free energy of volumetric interactions ΔF(C)=Fattr−Frep on the concentration of glucose in the solvent.
5.1. Distribution Function of Dimers.
The adsorption of glucose molecules and sodium ions of solvent by polymer chain monomer nodes can be described by analogy with the adsorption of ideal gas atoms by surface nodes. The process at constant temperature is described by the Langmuir adsorption isotherm [18], which is derived for the classical regime from the classical distribution function and relates the proportion of occupied surface nodes to the concentration of ideal gas atoms.
The adsorption isotherm linking the fraction f of the occupied nodes of the polymer chain with the concentration of glucose molecules of solvent C in the classical mode has the following form:
f=C/(C0+C);C0=(1/VQ)×exp(E0/kBT), (8)
where C0 is an equilibrium constant independent of concentration;
VQ is the quantum volume of the glucose molecule [14]: VQ=(h2/2πMkBT)3/2, where h is Planck's constant; M is the molecular weight.
The absolute activity of glucose molecules of a solvent with a concentration of C is determined by the expression for the activity of an ideal monatomic gas λ=CVQ, which is valid in the classical mode (λ≤≤1) for the classical distribution function [18]. It can be shown that the activity of a glucose molecule with a molecular weight of 180 at a concentration of C=40 mmol/l exceeding the physiological norm of glucose concentration by 8 times is equal to λ=CVQ≈10−4<<1.
The equilibrium constant C0 coincides with the microscopic equilibrium constant KR, which was determined above (5a):
K
R=(R0)()/(R1)=(R0)(C)/(R1)=C0=(1/VQ)×exp(E0/kBT). (8a)
Hence the different values of the binding energy E2>E2>0 correspond to different equilibrium constants, the ratio of which is determined by the Boltzmann factor:
C
02(E2)/C01(E1)=exp(E2/kBT)/exp(E1/kBT). (8b)
The expression for the activity of the glucose molecule at λ<<1 has the following form:
λ=CVQ=(C/kR)×exp(E0/kBT), where E0 is the energy of the hydrogen bond, which is negative if energy is required to remove the molecule from the node. Equation (6b) agrees with the condition of the classical mode at concentrations C=kR: λ≈exp(E0/kBT)<<1.
The adsorption of sodium ions by negatively charged polymer chain nodes can also be described using the absorption isotherm (8).
The equilibrium constant KNa of Na ions is determined using equation (8b):
KNa=(1/VQNa)×exp(ε0/kBT), where ε0 is the energy of the ionic bond of sodium ions with a monomer; VQNa is the quantum volume of the Na ion. The equilibrium constants of glucose molecules and ions are related by the ratio:
K
Na
/K
G=(VQG/VQNa)×exp(−E0+ε0/kBT)=(mNa/mG)2/3×exp(ΔE/kBT), (8d)
where mG=180 is the molecular weight of the glucose molecule: mG=11 is the atomic weight of the sodium atom.
Hence the condition under which the Na ions and the glucose molecule have equal values of the binding constant by the negatively charged link of the disaccharide monomer of the polymer chain is determined as:
K
Na
/K
G=1=(mNa/mG)3/2×exp(ΔE/kBT);
ΔE/kBT=(3/2)×ln(mG/mNa)=(3/2)×ln(16.36)≈(3/2)×2.7948=4.19.
So the energy of the ionic bond at E0=5.19kBT corresponding to the characteristic value of a weak hydrogen bond (0.17 eV or 3.9 kcal/mol) is equal to:
ε0=E0−4.19kBT≈kBT,at E0=5.19kBT≈0.17 eV (3.9 kcal/mol).
ΔE/kBT=(3/2)×ln(mG/mNa)=(3/2)×ln(2×8.18/3)=(3/2)×ln(16.36/3)≈(3/2)×1.69=2.54.
ε0=E0−2.54kBT≈kBT,at E0=3.54kBT=0.116 eV (2.7 kcal/mol).
The ratio of the equilibrium constants of ions Na and K can be calculated in the same way:
K
Na
/K
K=(mNa/mK)3/2×exp(ΔE/kBT)=(mNa/mK)3/2=(11/19)3/2≈1/23/2≈1/e.
5.2. Distribution Function of Tetramers.
A slow increase in the glucose concentration C>C0 will lead to reversible compression of the macromolecule (in the process of work performed due to the excess of free energy of attraction of volumetric interactions), which will be accompanied by an increase in the number of tetramers ΔNR=NR−NR0>0 in the elementary cell, as a result of the transformation of an unrelated pair of dimers in the microscopic cell D into tetramer configuration. On the contrary, a decrease in the glucose concentration C<C0 will lead to a reversible expansion (in the process of work performed due to a negative excess of the free energy of repulsion of volumetric interactions). It will lead to a negative excess of the concentration of tetramers ΔVR<0 in the unit cell due to the transformation of the tetramer in cell T into a dimer configuration. The problem is to find the dependence of the thermal average value of the free energy of volumetric interactions ΔF(C)=Fattr−Frep on the glucose concentration of the solvent, which can be calculated as the thermal average number of entropy quanta or the thermal average number of tetramers in the elementary cell volume on the number of glucose molecules.
The average number of tetramers in an elementary cell is always between 0 and 2. With a positive excess of glucose molecules, the average number of tetramers in cell D is between 0 and 1 (occupancy <1, number of bonds <4), with a constant average number of 1 tetramer in cell T; with a negative excess of glucose, the average number of tetramers in cell T is between 0 and −1 (occupancy <1, number of bonds <4), while the average number of dimers in cell D is between 2 and 0.
The problem of the thermal average number of tetramers in an elementary cell is thus reduced to the task of the thermal average occupancy of a single tetramer in cell D, which can be calculated as the average over the ensemble of systems of an elementary cell, which is discussed above.
The probability P(D1) of the tetramer realization in one of several permissible states (configurations) of Rl can be determined using a large statistical sum as the sum of the probabilities P(Rl) of all permissible microstates of Rl:
P(D1)=ΣP((Rl)=,
where =
R+
T is a large statistical sum over all the numbers of tetramers and microstates in the elementary cell, which consists of the sum over the numbers of tetramers and microstates Rl with energies 0, 1 E0, 2E0, 3E0, 4E0, corresponding to the configurations of the tetramer in the conformation (R), and over all the numbers of tetramers and microstates Tl, corresponding to tetramer configurations in the conformation (T); λ≡exp(μ/kBT), where μ is the chemical potential;
C=Σλ1 exp (−lE0/kBT) is the statistical sum for all numbers of tetramers and for all permissible microstates implementing the tetramer configuration in conformation (R) in a microscopic cell D at an arbitrary glucose concentration of solvent C. With this determination, the probability is the ratio of the Gibbs factor to a large statistical sum P(Rl)=[λ1 exp(−lE0/kBT)]/
, corresponds to the absolute probability of occurrence of a tetramer in the microstate Rl with the number of valence bonds; and energy E(l)=lE0.
Consider the problem in the general case. Assume that the tetramer can be realized in each of the permissible conformations (R) and (T) characterized by different equilibrium constants KR and KT. In this case, the required solution will be obtained with KT=∞, as a special case.
The statistical sum R for all the numbers of tetramers and states in the elementary cell can be defined as the product of the thermal average number of tetramers ΔNg in the elementary cell by the statistical sum
RD for all states Rl with energies Et corresponding to different numbers of hydrogen bonds l of a single tetramer in the cell D:
R
/ΔN
g=RD=1+4λexp(−E0/kBT)+6λ2exp(−2E0/kBT)+4λ3exp(−3E0/kBT)+λ4exp(−4E0/kBT). (9)
As shown above, the expression for the activity of glucose molecules in the classical mode (λ<<1) for the classical distribution function [18] has the following form:
λ=CVQ=(C/kR)×exp(E0/kBT), where activity is by default equal to λ≡exp(μ/kBT).
So for a system in the classical mode, the Gibbs factor can be represented as a dimensionless parameter α—glucose concentration C, expressed in units of the microscopic equilibrium constant KR, corresponding to the relative probability of a state with energy E0 and activity λ:
λexp(−E0/kBT)=(C/kR)=α
Hence equation (9) for the statistical sum R can be written in the following simple form:
R
=ΔN
R×[1+4(C/kR)+6(C/kR)2+4(C/kR)+(C/kR)4]=ΔNR×[1+4α+6α2+4α3+α4]=ΔNR×(1+α)4, (9a)
where the glucose concentration C is expressed as a dimensionless parameter α=C/C0=(C)/KR.
It can be shown that the distribution function Θ(α, γ) is the thermal average number of tetramers in a microscopic cell D. Taking into account equation (12d), it has the following form:
Θ(α,γ)=[α(1+α)3−γ]/[(1+α)4+γ]. (14)
The distribution function Θ(α, γ), which depends on the dimensionless variable α and dimensionless order parameter (interaction constant) γ, describes the dependence of the number of quanta of entropy corresponding to the free energy of the volumetric interactions of the polymer chain on the composition of the solvent. At the critical concentration α=α0 and the value of the interaction constant (order parameter) of γ=α0×(1+α0)3, the distribution function is determined as Θ(α, γ)=0.
The distribution functions of tetramers Θ(α, γ=8) and Θ(α, γ=139.69 . . . ) are shown in
The equilibrium constant is L (α)=γ+γ/Θ→∞ at α→α0. Hence Θ is a special (in the mathematical sense) point (α=α0) for thermodynamic quantities corresponding to the point of phase transition of matter. For α>α0, the value of the function is equal to the thermal average of the tetramer occupancy in cell D. For α<α0, the value of the function is equal to the thermal average of the tetramer occupancy in cell T.
5.3. Intramolecular Phase Transition.
As has been shown, in the equilibrium state at a critical glucose concentration, the equilibrium configuration of the distribution of monomers in the elementary cell corresponding to the concentration of 2 monomers/microcell is realized. In the volume of the microcell, which is determined by the volume of the tetramer (G), the configuration (Na) of the tetramer of 6 units (shown in
It can be shown that the conformational transition of the (Na) tetramer, which can be realized at a critical concentration of 3 monomers/1 microcell, is accompanied by a decrease in free energy by the value ΔF=kBT. The conformation (s) can be formed at a concentration of monomers of 3 monomers/1 microcell as a result of the convergence of nodes (1, 2, 3, 4), connected by hydrogen bonds with glucose molecules, and increasing the distance between nodes (5, 6) occupied by Na ions. In the configuration (s) resulting from the convergence of nodes (1, 2, 3, 4), diagonal (13, 24) and longitudinal permutations 14 and 23 can be implemented. In this case, the number of states implementing the (s) conformation of the tetramer coincides with the number of states implementing the (G) configurations of the tetramer, which, as shown above, is 6×4+1=25. Hence the change in the free energy of the transition (Na) of the tetramer from the conformation (0) to the isomeric conformation (s) is equal to:
TΔS=k
B
T×ln(25/9)=kBT×ln 2.777≈kBT. (15)
Thus, the conformational transition of a tetramer (Na) to an isomeric conformation (s) with a decrease in free energy by kBT can be realized at a critical concentration of monomers of 3 monomers/1 microcell.
The distribution of monomers in the volume of the elementary cell at concentrations 2 and 3 monomers/1 microcell is shown schematically in
Thermodynamic states of matter in the volume of a macromolecule corresponding to conformations (λ) and (s) of tetramer (Na) realized at different concentrations of monomers in a microscopic cell 2 and 3 monomers/i microcell, can be defined as condensed states (thermodynamic phases): (λ) is the liquid state (phase) and (s) is the crystalline or solid state (phase).
Hence the folding of the biopolymer chain of a macromolecule into a strictly defined three-dimensional spatial structure (folding of a macromolecule) is realized under certain conditions (critical values of temperature, external pressure, and concentrations), during the conformational transition of a macromolecule from a condensed liquid conformation (molten globule) to a condensed crystalline conformation (globule).
The critical glucose concentration of the solvent, at which the concentration of monomers becomes critical 3 monomers/1 microcell, can be determined using the distribution functions (8) and (14) in the following way. Normalization of the x-axis is chosen from the condition that the glucose concentration (C) is scaled in units of the critical concentration C0 corresponding to the microscopic equilibrium constant kg2=C0 of the process (G). With such normalization, the singular point of the distribution function Θ(α, γ)=0 of the process (G) corresponds to the value of the dimensionless concentration α=αg=C0/kg2=1. The process (λ) is a conjugate process proceeding due to the free energy of the process (G). Therefore, the microscopic equilibrium constant and the distribution function of the process (λ) coincide with the equilibrium constant and the distribution function of the process (G): kλ2=kg2=C0.
As shown above (8a), the microscopic equilibrium constants (kλ2 and ks2) characterizing the conformations (λ) and (s) of the (Na) tetramer and the change in free energy Es−Eλ=kBT during the conformational transition of a macromolecule (from one elementary cell configuration to another) are related by the Boltzmann factor:
k
s2
/k
λ2=exp(Es/kBT)/exp(Eλ/kBT)=exp[(Es−Eλ)/kBT]=exp(kBT/kBT)=e. (16)
The critical glucose concentrations (Cλ and Cs) in phases (λ) and (s) are determined as Cλ=kλ2 and Cs=ks2 respectively. So the ratio of critical glucose concentrations for condensed phases (λ) and (s) of the substance is: Cs/Cλ=ks2/kλ2=e=αs/αλ, where Cλ=C0 is a critical glucose concentration for the condensed phase (λ); Cs is a critical concentration for the condensed phase (s).
Hence, with the chosen normalization, the singular point of the process (s) corresponds to the concentration value α=αs=e. The critical glucose concentration for phase (s) is equal to the equilibrium constant of the process (s) Cs=ks2=eC0=ekλ2. The distribution functions of the processes (λ) and (s) describing the dependence of the average value of the excess of entropy corresponding to the excess of free energy of volumetric interactions are shown in
The functions of the entropy, the free energy of interaction of elementary volume (taken with a minus sign) of the glucose concentration of the solvent for the condensed phases (λ) and (s), are shown in
As shown above, the average relative concentration of dimers is determined by the dependence of the free energy of the elementary volume interaction on the glucose concentration and coincides with the distribution function shown in
As follows from the plots in
The phase transition of a macromolecule is realized as a collapse of the intramolecular grid of hydrogen bonds, which is accompanied by a sharp intramolecular rearrangement. A jump-like increase in the concentration of monomers (density) caused by a jump-like change in the configuration of the grid of hydrogen bonds caused by a jump-like increase in the number of microscopic cells in the volume of an elementary cell with 27 (at γ=8) up to 422.07 (if γ=139.740 . . . ) takes place. A step-like dependence of the concentration of polymer chain monomers on the concentration of solvent glucose resulting from a jump-like change in the free energy of the elementary bulk interaction is shown in
The microscopic equilibrium constant kg2 characterizing the process (G) in the condensed phase (s) can be determined from the requirement that the singular point of the phase transition (α=e) is the singular point of the process (G) for the condensed phase (s) in which (λ) and (s) conformations of the (Na) tetramer have the same probability of realization. To the left of the phase transition point (α<e), (Na) tetramers are realized with a higher probability in the (λ) conformation, and to the right of the transition (α>e), tetramers are realized with a higher probability in the (s) conformation. As shown, at a particular point in the process, the average number of dimers (Ng)0 in the unit cell volume (concentration) is equal to the average number of free solvent glucose molecules (concentration) (C)0 not bound in dimers and tetramers:
(Ng)0=(C)0=kλ2.
As follows from
N
g
/N
g0=1+Θ(α=e,γ=8)×9/8=1.7443 . . .
From the invariance (immutability) of the configuration of the elementary cell at a singular point with a change in the concentration of solvent glucose follows that an increase in the concentration of dimers in the microscopic cell D by 1.7443 . . . times corresponds to the same increase in the concentration of glucose in cell M. Thus, the critical concentration of solvent glucose (Cg)0 at a singular point of the process (G) in condensed phase (s) is equal to:
(Cg)0=(Ng)(α=e)=(1.7443 . . . )×(Ng)0=(1.7443 . . . )×(C)0=(1.7443 . . . )×kλ2=kg2.
5.4. Equilibrium Constants of Conjugate Processes (λ), (g), (s).
As shown, the tetramer configuration is formed in the process of random paired interactions, whose energies for each conformation (λ, g, s) can be calculated using the universal distribution function Θ(α, γ) as thermal averages, if for each of the conjugate processes (λ, g, s) microscopic equilibrium constants characterizing the process can be determined.
It can be shown that the microscopic equilibrium constants of conjugate thermodynamic processes (λ, g, s) are related by the following equations:
a
λ
=k
s2
/k
λ2
=e;b
λ
=k
λ1
/k
s2
=e/1.7443 . . . ;
a
g
=k
s2
/k
g2
=e/1.7443 . . . ;bg=kg1/ks2=e;
a
s
=k
s2
/k
s2=1;bs=ks1/ks2=e2/1.7443. (19)
The change in the free energy of the tetramer transition (G) from conformation R1 with 1 bond to conformation R4 with 4 bonds is equal to: ΔE=E2−E1=[2−ln(kg1/kg2)]×kBT=(2−ln 1.7443)×kBT 3kBT/2. The diagram of the energy levels of tetramers (λ, g, s) in the equilibrium configuration of the polymer mesh of a macromolecule is schematically shown in
6. Distribution Functions of Hyaluronic Add.
The distribution functions of free (unoccupied) dimers Θ1(α) for the processes (λ, s, g) can be found using the glucose adsorption isotherm of polymer chain monomers (8).
The functions Θ1(α) for the processes (λ, g, s) have the following form:
Θg1(a)=1−f=C0/(C0+C)=bg/(bg+α), where bg is the equilibrium constant of the process (g) in terms of the microscopic equilibrium constant kλ2 of the process (λ); α=C/kλ2 is dimensionless glucose concentration in terms of kλ2.
Θλ1(α)=C0/(C0+C)=bλ/(bλ+α), where bλ is the equilibrium constant of the process (λ) in terms of the microscopic equilibrium constant kλ2 of the process (λ); α=C/kλ2 is dimensionless glucose concentration in terms of kλ2.
Θs1(a)=C0/(C0+C)=bs/(bs+α), where bs is the equilibrium constant of the process (s) in terms of the microscopic equilibrium constant kλ2 of the process (λ); α=C/kλ2 is dimensionless glucose concentration in terms of kλ2.
The distribution function Θ2(α, γ) of unoccupied tetramers (λ, s, g)—the ensemble average number of unoccupied tetramers—can be obtained from the universal distribution function Θ(α, γ):
Θ2(α,γ)=1−(2/3)×[Θ(α)+γ/(γ+1)]/[1+γ/(γ+1)]. (20)
The multiplier 2/3 in the expression Θ2(α, γ) takes into account the proportion of configurations with hydrogen bonding—the proportion of glucose molecules bound by polymer chain monomers. As was shown, in the equilibrium configuration of the elementary chain (in the volume of the elementary cell), the total proportion of dimers and tetramers in which glucose molecules are bound with monomers of the chain by means of hydrogen bonds is equal to 2/3. At the same time, the proportion of free monomers and glucose molecules is equal to 1/3.
The distribution functions Θ1(α) and Θ2(α, γ) for conjugate (λ, s, g) processes, determining the thermal average number of unoccupied dimers and tetramers, reduced to a single dimensionless variable α=C/kλ2=C/C0, are given below.
Distribution functions of the process (g):
Θ1g(α)=e/(α+e);
Θ2g(α,γ)=1−{[agα(1+agα)3−γ]/[γ+(1+agα)4]+γ/(γ+1)}×(2/3)×(γ+1)/(2γ+1);
γ=e×(1+e)3;ag=e/1.7443;
Θ3g(α)=Θ1g(α)/Θ2g(α); (21)
Here it is taken into account that the concentration of free monomers (G) with an increase in the concentration of glucose in the solvent changes due to two processes. First, an increase in the proportion of bound (in the dimer configuration) glucose molecules, which is described by the function Θ1g(α). Second, an increase in the concentration of monomers (G) as a result of a decrease in the volume of the cell due to an increase in the proportion of tetramers, which is described by the function Θ2g(α). The thermal average value of the partial volume Vg(α) of monomers (G) in the volume of an elementary cell at a glucose concentration α is determined by the expression:
V
g(α)/V0g=Θ2g(α), (21a)
where V0g is the partial volume of monomers (G) in the volume of the elementary cell at a concentration of α=1.
The distribution functions of the process (g) are shown in
The distribution functions of the processes (λ, s) are determined similarly using equations (19).
The distribution functions of the process (λ) of tetramers (Na) and (G):
Θ1λ(α)=(e/1.7443)/[α+(e/1.7443)];
Θ2λ(α)=1−{[aλα×(1+aλα)3−γ]/[γ+(1+aλα)4]+γ/(γ+1)}×(2/3)×(γ+1)/(2γ+1);
γ=e×(1+e)3;aλ=e. (22)
The distribution functions of the process (s) of tetramer (Na):
Θ1s(α)=(e2/1.7443)/(α+e2/1.7443);
Θ2s(α)=1−{[asα×(1+asα)3−γ]/[γ+(1+asα)4]+γ/(γ+1)}×(2/3)×(γ+1)/(2γ+1);
γ=e×(1+e)3;as=1. (23)
The distribution functions (λ, s, g) of free tetramers are shown in
7. Energy of Equilibrium Volumetric Interactions.
The problem is to find for each process (λ, s, g) the thermal average values of the interaction energies Uattr and Urep, which determine the value of the free energy Fmin of the elementary volume interaction (4), using the distribution functions of processes (λ, s, g) obtained above.
7.1. Repulsive Energy.
As shown, the volumetric repulsive forces in the volume of a macromolecule are determined by the entropic elasticity of an ideal gas of free monomers due to the different probabilities of realization of more rarefied and more compressed states. The energy of volumetric repulsion is equal to the energy of an ideal gas, which is determined in accordance with the equation of state (3) by the concentration of free negatively charged monomers.
In general, if the concentration is α≠0, the energy Ugrep(α) of repulsion of monomers per unit volume (in kBT terms) in accordance with the state equation (3) is equal to:
U
grep(α)=P/kBT=Nmon/2V+Ng×Θ4g(α)/2V=ng/2+(n0gg/2)×Θ4g(α);
Θ4g(α)=[Θ1g(α)−Θ2g(α)]/Θ2g(α)=Θ1g(α)/Θ2g(α)−1=Θ3g(α)−1, (24a)
where Θ4g(α) is the excess concentration of free negatively charged monomers (not bound in the configuration of dimers and tetramers) in phase (G); the difference between the concentration of dimers and the concentration of tetramers (with an equal total number of dimers and tetramers n0g/2); Θ1g(α) and Θ2g(α) are the thermal averages the number of dimers and tetramers per unit volume; here it is taken into account that the partial volume Vg(α) of phase (G) monomers per unit volume in accordance with (21a) is equal to: Vg(α)/V0g=Θ2g(α).
Equation (24a) can be transformed by subtracting from the total energy Ugrep(α) the constant component of the energy ng/2 and then presented as follows:
U
0g rep(α)=[Ugrep(α)−ng/2]/(n0gg/2)=Θ4g(α). (24)
The energy of volume repulsion Uλrep(α) per unit volume of the phase (λ) is determined in the same way by the concentration of free (not bound in dimers and tetramers) monomers in the phase (λ), which is equal to the difference between the concentration of dimers and tetramers in the concentration phase (λ)—the excess of the concentration of free monomers in the phase (λ):
U
0λ rep(α)=[Uλ rep(α)−nλ/2]/(n0gλ/2)=Θ4λ(α)=Θ1λ−Θ2λ(α). (24b)
Here it is taken into account that the concentration of monomers in the phase (λ) changes as a result of a change in the volume of the elementary cell with a change in the concentration of glucose in the solvent.
The energy of volumetric repulsion (s) of monomers Us rep(α) per unit volume of phase (s) is determined similarly by the concentration of free (not bound to dimers and tetramers) monomers in phase (s), which is equal to the difference between the concentration of dimers and the concentration of tetramers in phase (s)—the excess concentration of free monomers in phase (s):
U
0s rep(α)=[Us rep(α)−n2/2]/(n0gs/2)=Θ4s(α)==[Θ1s(α)−Θ2s(α)]×ln(ks2/kg2)=[Θ1s(α)−Θ2s(α)]×ln(e/1.7443). (24c)
As follows from the diagram in
As follows from equations (24), the values of the energy and partial pressures of an ideal gas in phases (λ) and (s) have different values, the difference between which determines the magnitude of the repulsive energy. The thermal average of the energy of the elementary volumetric interaction (repulsion) of free monomers in phases (λ, s) is determined by the free excess of monomers—the excess concentration of free negatively charged monomers (λ) and (s) that are not bound in the configurations of tetramers and dimers in phases (λ, s):
U
λs rep(α)=U0λ rep(α)−U0s rep(α)=Θ4λ(α)−Θ4s(α). (25)
The repulsive energy function of free monomers Uλs rep(α) is shown in
7.2. The Energy of Attraction.
The energy of attraction Uattr between non-neighboring monomers with a concentration of t is equal to the energy of paired interactions, which is determined by the probability of paired collisions of dimers during thermal motion:
U
attr
=V k
B
T×Bn
2, where B—viral factor,V—volume.
In the problem under consideration, the energy of attraction between non-neighboring monomers of the polymer chain is determined exclusively by paired collisions of free (not bound into tetramers and dimers) monomers in equilibrium conformations (R) and (T). As it has been shown, the most probable volume configuration of monomers in an elementary cell is formed as a result of the pairing of elementary chains, during which a hydrogen bond can be realized between 1 (one) glucose molecule of the solvent and 2 (two) monomers of the chain, but cannot be formed between 1 molecule and 3 monomers, therefore the contribution of triple collisions to free energy is equal to zero.
Expressions for the energy of paired interactions per monomer (G)—the energy normalized to the thermal energy of an ideal gas of negatively charged monomers (G), for condensed phases (λ, s, g), have the following simple form:
U
0g attr(α)=[(γ+1)/2]×{[γ+Θ3g(α)]2/(γ+1)2−1};
U
0λ attr(α)=[(γ+1)/2]×{[γ+Θ4λ(α)]2/(γ+1)2−1};
U
0s attr(α)=[(γ+1)/2]×{[γ+Θ4s(α)]2/(γ+1)2−1}. (28a)
where U0g attr(α)=Ug attr(α)/n0gg; U0λ attr(α)=Uλ attr(α)/n0gλ; U0s attr(α)=Us attr(α)/n0gs.
The functions of the energy of paired interactions (attraction) (28) and the energy of an ideal gas (24) for each condensed phase (λ, s, g) are shown in
U
g attr(α)≡Ug rep(α).
Similar identities are fulfilled for all (λ, s, g) of the processes (λ) and (s), as well as for the free energy function of equilibrium interactions:
U
g attr(α)+Ug rep(α)≡0;
U
s attr(α)+Us rep(α)≡0;
U
λ attr(α)+Uλ rep(α)≡0. (29)
The free energy of attraction is the energy of paired interactions of free monomers not connected in tetramer configurations (λ, s) and is defined by the expression:
U
λs attr(α)=Uλ attr(α)−Us attr(α). (30)
Plots of the thermal average energy of volumetric interactions of monomers in the tetramer configuration (Na)—the energy of attraction and repulsion are shown in
The height of the energy barrier separating the isomeric conformations (λ) and (s) of tetramers (Na) can be calculated using the energy level diagram shown in
ΔU=U2−U1=(3−ln1.7443)×kBT≈5kBT/2.
Thus, equation (4) of exact compensation of volumetric interactions is fulfilled at any concentrations of glucose in the solvent. The force of attraction is exactly equal to the force of repulsion at any concentrations of glucose at any point of the energy barrier. Therefore, the most probable spatial configuration corresponding to the minimum free energy of volumetric interactions is a stable state of indifferent (neutral) equilibrium.
The free energy of interaction of negatively charged monomers in the gaseous phase (G) at zero glucose concentration (α=0), at which hydrogen bonds cannot form between monomers (a polymer chain in a solution with an excess of sodium and potassium ions), can be represented as follows:
F
0gas(α=0)/kBT=σ0gas=(γgas)2. (31)
At a non-zero glucose concentration, the energy of interaction between monomers of the gaseous phase increases as a result of the convergence of monomers due to a decrease in the volume of localization of the polymer chain. The value of free energy of interaction in the gas phase with a critical concentration α=e can be calculated using the distribution functions, which are obtained from the universal Θ(α, γ) at values of the order parameter γ=γλ=8, corresponding to the singular point of the process (λ). It is the starting point of the function that is reduced to zero by parallel transport functions on the y-axis at 1+(γ+1)/γ=1+8/9:
F
0gas(α=e)/kBT=2.7443 . . . ×(γgas)2. (31a)
In the condensed phase (s), the free energy of the interaction of monomers in cell M (gaseous phase) and the interaction energy in the repulsion configuration R0 in cell D should be equal:
F
0gas(α=e)=F0s(α=e).
On the other hand, as follows from
F
0λ(α=1)=F0s(α=e);V(α=1)×σ0λ(α=1)=V(α=1)×σ0s(α=e).
Hence a decrease in the volume of the elementary cell by e times, as a result of an increase in the concentration of glucose by e times, leads to an increase in the entropy density by e times:
V(α=1)/V(α=e)=e;
σ0s(α=e)=e×σ0λ(α=1)=e×(γ−2)2.
Hence follows the expression linking the interaction energy of monomers in the gaseous phase with the interaction energy of monomers in the tetramer configuration in the condensed phase (s):
F
0gas(α=e)/F0s(α=e)=2.7443 . . . ×(γgas)2/e×(γ−2)2≡1;
γgas=(γ−2)×(e/2.7443 . . . )1/2=137.740210 . . . ×(2.71828 . . . /2.7443 . . . )1/2=137.740210 . . . ×(0.9905185 . . . )1/2=137.74021 . . . ×0.995247=137.08566 . . . .
The obtained value of the interaction constant closely coincides with the inverse value of the fine structure constant—the fundamental constant of electromagnetic interaction, which is a measure of electromagnetic interaction equal to 1/137.0359990 . . . The close coincidence of the interaction constant γgas obtained on the basis of the statistical method indicates the statistical nature of the fundamental interaction constant (fine structure constant), which should be investigated as part of the generalization of the statistical method for an accurate description of the interaction in the quantum regime.
8. Intramolecular Osmotic Pressure.
As shown, a stable spatial configuration of a macromolecule is realized under the condition of exact compensation of the forces of attraction and repulsion, which can be realized if the osmotic pressure, identically equal to the pressure of an ideal gas (excess) of free negatively charged monomers in the phases (λ) and (s), is exactly equal to the intramolecular pressure resulting from the formation of tetramers (G) in phase (g).
The problem is to calculate the amount of reversible compression work performed due to the free energy of volumetric interactions with a slow increase in glucose concentration, which can be calculated as the product of osmotic pressure by the volume change of the macromolecule.
It can be shown that the expression for osmotic pressure can be represented in the following form:
P
osm
×V
0g=−(2/kλ2)×(2×[γ+Θ3g(α)]/(γ+1)2)×[dΘ3g(α)/dα], (34)
where V0g is a unit volume calculated for 1 monomer of phase (G), which is related to the volume of the macromolecule by the following expression: Posm×[V0mol/3n0gg(γ+1)]=Posm×(V0mol/ng)=Posm×V0g.
The characteristic of the thermal energy of an ideal gas of free monomers in the volume of a microscopic cell Posm×V0g is shown in
Thus, a stable spatial configuration of a macromolecule at constant temperature and external pressure is realized as a result of precise compensation of the forces of attraction and repulsion. The pressure (osmotic) of an ideal gas Posm of free monomers in the volume of a macromolecule is equal in magnitude (and opposite in sign) to the intramolecular pressure Pg resulting from paired interactions of monomers (G) in phase (G): Pg+Posm=0.
The magnitude of the reversible compression work performed due to the free energy of paired interactions in phase (G) can be determined using the expression:
A=Pg(α)×V(α), where V(α) is the partial volume of monomers (G) in the tetramer configuration (cell T) in phase (G) at a concentration α, which is determined by the following expression: V(α)=V0g×(γ+1)×Θ2g(α).
Thus, the expression for the reversible compression work performed due to the free energy of phase (G) monomers over an ideal gas of negatively charged free monomers in phases (λ) and (s) under reversible compression has the following form:
A=P
g(α)×V(α)=−Posm×V=(2/kλ2)×{2×[γ+Θ3g(α)]/(γ+1)2}×[dΘ3g(α)/dα]×(γ+1)×Θ2g(α). (35)
9. Condensed State of the Macromolecule.
Consider the thermal function (enthalpy, heat content) of an ideal gas (Na) of monomers, which can be determined based on the generally accepted definition of this thermodynamic quantity [18]:
H≡U
λs(α)+Pg(α)×V(α), (36)
where Uλs(α) is the energy of elementary volumetric interaction (30); Pg(α)×V(α) is the compression work performed due to the free energy of volumetric interactions; with an increase in glucose concentration (chemical work performed on the system), it is equal to:
A=P
g(α)×V(α)=−Posm×V.
The expression for enthalpy can be represented as follows:
H=U
λs(α)−Posm(α)×V(α). (37)
Equation (37), which relates the energy of volumetric interactions and reversible work performed due to the free energy of volumetric interactions, describes the equilibrium states of a polymer macromolecule, depending on the concentration of solvent glucose. Therefore, it can be defined as the state equation of a macromolecule.
Plots of the free energy functions of the gas of free (Na) monomers Uλs(α) and the work A performed under reversible compression due to the free energy of free (G) monomers are shown in
As was shown, the expression for osmotic pressure includes the parameter kλ2—the microscopic equilibrium constant of the process (λ), the numerical value of which determines the type of function (34). In a thermodynamic process occurring at constant entropy dS≡0 under condition (4), the critical value of the microscopic equilibrium constant kλ2 can be determined from the requirement dH≡0. It is resulting from the law of energy conservation, according to which, for a reversible process occurring at constant entropy dS≡0, the differential of thermal function (36) dH is equal to zero:
dH≡dU
λs
+DA
osm≡0. (38)
Equation (38) expresses the law of energy conservation during reversible transitions of a macromolecule from one equilibrium state to another with slow changes in the concentration of solvent glucose with a constant number of polymer chain monomers. The difference between the change in the internal energy of the gas of free monomers dUλs and the external work DAosm, performed on the system due to the free energy of volumetric interactions is equal to the amount of the heat DQ transferred to the system with an infinitesimal change:
TdS≡dQ≡dH≡0;dUλs+DAosm≡0. (39)
The critical value of the microscopic equilibrium constant kλ2, which satisfies the requirement dH ≡0 in the interval [1, e] of the conformational transition of tetramers (λ⇔s), is kλ2=5.032 . . . The plot of the thermal function at kλ2=5.032 . . . is shown in
As can be seen from the plot, the enthalpy dependence curve on glucose concentration at a constant temperature contains a horizontal section, in the interval of which the average enthalpy value of a polymer macromolecule does not depend on the glucose concentration of the solvent. It has a form similar to the pressure-volume isotherm for a real gas at a temperature at which liquid and gaseous forms of a substance can coexist. In the case of real gas in the vessel under the piston, in the two-phase region of the liquid+gas system, the pressure is constant but the volume may vary. At a given temperature, there is only one pressure value at which the liquid and its vapor are in equilibrium. If the piston is moved down at this pressure, then a certain amount of gas condenses, but the pressure remains constant until all the gas condenses.
The horizontal section of the plot of the thermal function—the enthalpy dependence on the glucose concentration of the solvent with high resolution along the ordinate axis is shown in
The result obtained is consistent with the Gibbs phase rule, which allows determining the number of independent variables v of the thermodynamic state of an equilibrium system containing n components distributed between r phases: v=r×(n−1)+2−n×(r−1)=n−r+2. Here, the number of independent variables that must be fixed in order for the state of the system to be fully determined is v=3 (temperature, glucose concentration, and external pressure). The number of components is n=5 (water, sodium ions, glucose, potassium ions, and polymer chain monomers). Hence the conclusion about the four-phase (gaseous, liquid λ, crystalline s, and crystalline g) thermodynamic equilibrium of a system with the number of phases r=4 is consistent with the Gibbs phase rule:
v=r×(n−1)+2−n×(r−1)=n−r+2=5−4+2=3.
The enthalpy value in the three-phase region (the horizontal section of the thermal function in
The thermal power introduced into the local volume of the intercellular substance leads to isothermal melting of the crystalline phase—a phase transition of the first kind, which is accompanied by the release or absorption of latent heat: Qmelt=T×(Sλ−Ss), where Sλ and Ss are the entropies of the liquid (λ conformation) and crystalline phases (s conformation), designed for one tetramer. The driving force of the phase transition is the system tendency to move into a state with minimal free energy. At a constant temperature, one phase displaces the other because in the two phases of matter, free energy is distributed differently between internal energy and entropy. At the temperature of the phase transition of the first kind, due to latent heat and temperature immutability, the derivatives of free energy, in particular, the specific heat capacity ΔQ/ΔT undergoes a “singularity” (turns to infinity). ΔQ is the input or output heat equal to the latent heat of the phase transition, and ΔT=0, since during the phase transition, the temperature T remains constant.
The physical analog of a system in which a three-phase thermodynamic equilibrium of matter is realized is a mixture of water, ice, and steam at a triple point. As is known, in a mixture of water and ice at 273.16 K two phases are in equilibrium and the processes of melting and freezing proceed reversibly at constant entropy. When the mixture is heated, ice melts, and on the contrary, when cooled, ice forms from the liquid phase, while the water temperature remains constant.
The conclusion about the heterogeneous condensed state of the intercellular substance allows explaining the unique property of isothermy (physical thermoregulation) of living tissue, known from experience. It is the ability of warm-blooded living organisms to maintain temperature constancy with changes in ambient temperature. As is known, mammals whose main intercellular component is hyaluronic acid, the structure of which has not changed in the course of evolution, have a body temperature close to that of a human body. The body temperature of all mammals found in the wild is close to the temperature of the human body.
So, the important conclusion can be drawn that the close values of the temperature of warm-blooded animals are determined not by physiological processes, but by the thermodynamic properties of the intercellular substance, which in the living tissue of mammals is in a condensed state at the temperature of the phase transition of the macromolecule of hyaluronic acid, which can be determined by calculation.
9. Critical Temperature of the Macromolecule of Hyaluronic Acid.
The critical temperature at which intramolecular phase stratification of a substance occurs can be determined from the condition of mechanical equilibrium of the spatial structure of the macromolecule at the point of zero osmotic pressure—at a glucose concentration in which the osmotic pressure is zero.
As can be seen from the plot in
The pressure of the ideal gas of free monomers at the point of zero osmotic pressure (α=0.9474 . . . ) is equal to the external pressure Pext, which consists of two components: atmospheric pressure Patm and excessive external pressure Pexc, which must be applied to compensate the excess pressure of the gas of free monomers arising from the excessive concentration ΔCgexc of free monomers (G) in cell M (microscopic form R0) at critical glucose concentration C0=ks2:
P
ext
=P
atm
+P
osm
+P
exc
=P
atm
+P
exc=1 at +Pexc.
The equilibrium temperature of a substance T can be determined from the state equation of an ideal gas (2) at P=1 at +Pexc:
T=(Patm+Pexc)V/R×N=(Patm+Pexc)/Rn0=(1 at +Pexc)/Rn0, (40)
where R is the universal gas constant, T is the absolute temperature, N is the number of moles of polymer chain monomers, n0 is the concentration of polymer chain monomers, and V is the volume of the macromolecule.
The critical value of the microscopic equilibrium constant kλ2, at which a phase separation of a substance occurs, as shown, is equal to kλ2=5.032.
The critical concentration of polymer chain monomers (critical contour length) (at α=0) can be determined from the equilibrium condition at a singular point of the condensed phase (λ) at γ=n0Na/n0g=n0Na/kλ2=1×(1+1)3=8 at α=1:
n
0
=n
0Na=8kλ2=8×5.032 . . . =40.256.
The dimension of the calculated critical concentration of no can be determined by comparison with the amount of sodium excess [Na]—[Cl] that is easy to calculate from the concentrations of [Na] and [Cl] known from experience in human blood plasma: [Na]—[Cl]=40-42 mmol/l. Hence the critical value of the equilibrium constant kλ2 (the critical concentration of glucose molecules C0) and the critical concentration of monomers no in terms of mmol/l are equal:
n
0
=n
Na=40.256 mmol/l;C0=kλ2=5.032 mmol/l.
The resulting normalization gives the value of glucose concentration, which coincides with the concentration of glucose in the blood under normal conditions.
The value of the excess external pressure Pexc due to the excess of free monomers (G) in the elementary cell at a critical concentration of C0=kλ2=5.032 mmol/l can be determined using the equation of state of an ideal gas (2):
P
exc
≡n
exc
k
B
T. (41)
The excess concentration Δnexc of free monomers at a critical glucose concentration of C0 is determined only by the contribution of negatively charged free monomers (G) in the microscopic cell M. The proportion of free monomers (G) in the cell M, unrelated in the configurations of dimers and tetramers in cells D and T, is equal to ⅓ of the total number (G) of monomers in an elementary cell. So the excess concentration of negatively charged free monomers in the cell M is equal to
n
exc=(⅓)×C0/2=C0/6.
The excess osmotic pressure of tetramers in the configuration (R0) is equal to:
P
gexc
=n
exc
k
B
T=C
0
k
B
T/6=kλ2kBT/6=5.032[mmol/l]×kBT/6=(0.83867 mmol/l)×kBT. (42)
Thus, the critical temperature Tcrit, at which phase separation occurs and a heterogeneous condensed state of the intercellular substance appears, is determined by the critical glucose concentration of the solvent:
T
crit
=P
atm
/R×(n0−kλ2/6)=Patm/R×(8kλ2−kλ2/6)=6Patm/47R kλ2=6Patm/47RC0=6Patm/47C0kBNA. (43)
The numerical value of the critical temperature is
T
crit=1 at/0.0821[ar×l/mol×K]×0.0394173 [mol/l]=(1/0.003236)K=309.5975 K=36.447 . . . C.
The value of the external pressure balancing the excessive osmotic pressure of negatively charged free monomers is equal to
Pexc=(0.83867 mmol/1)×kBT=[0.83867/(2×5.032/3+40.256)]×760 mm Hg=[0.83867/(2×5.032/3+40.256)]×760 mm Hg=(0.83867/43.61)×760 mm Hg=(0.83867/43.61)×760 mm Hg.=0.019×760 mm Hg=14.6 mm Hg.
11. The Triple Region of the Intercellular Substance.
The critical values of the macroscopic parameters characterizing the equilibrium condensed state of the natural macromolecule of hyaluronic acid, calculated on the basis of the state equation (43) of the natural macromolecule coincide with the physiological values of the parameters of human homeostasis:
Thus, the results of numerical calculation based on an idealized model closely coincide with the physiological parameters of living tissue in normal. Here important conclusion is that the interval of the physiological norm of living tissue coincides with the interval of the three-phase region of the macromolecule of hyaluronic acid.
As has been shown isothermy in warm-blooded animals—the relative stability of temperature, at reversible physiological changes in the concentration of glucose in blood (which determines cellular metabolism rate and the intensity of heat production) and the thermal balance between the living tissue and the environment in the field of physical thermoregulation of the body, has a physical nature and is based on the thermodynamic properties of the intercellular substance of a local microvolume of the living t issue. The intercellular substance, in the physiological normal range, is in a heterophase condensed state, in which the crystalline solid phase (globule) is in the thermodynamic equilibrium with the metastable liquid phase (molten globule) and the gaseous phase (coil). As a result of the impact of the ΔWinput external heat flux on the local epidermis surface area, the heat introduced into the local volume of the substance leads to the melting of the solid phase—the phase transition of the substance from the solid state (“globule”) to the liquid state (“molten globule”); the melting process of the globular phase of hyaluronic acid proceeds as a first-order phase transition with a change in the volume of the substance (which has changed the physical state) due to a change in the osmotic pressure of the substance. A change in the osmotic pressure of the intercellular substance (the amount of water in the intercellular space) as a result of melting of the globular phase, leads to a change in the intercellular microcirculation intensity—microhydrodynamics (microfluidics) directed towards the surface of the epidermis, which enables isothermal heat exchange between the tissue and the environment during evaporative cooling with an intensity Wout; the heat balance of a reversible isothermal process, at constant entropy, has the following form:
ΔWinput=ΔWout.
The physical analogue of a system in which thermodynamic equilibrium between three phases of a substance is realized is a mixture of water, ice and steam at a triple point. It is well-known [12], in a mixture of water and ice at 273.16 K, two phases are in equilibrium, and melting and freezing processes flow reversibly at constant entropy: when the mixture is heated, ice melts, and when cooled, on the contrary, crystallization occurs—the formation of ice from the liquid phase, while the temperature of the water remains constant (isothermal property).
The process of melting of a substance at the phase transition temperature occurs under isothermal conditions, without a change in temperature; the amount of heat introduced into the system can be determined by measuring the amount of a substance that has transferred into a liquid state during the melting process (isothermal calorimetry—measuring the amount of heat introduced into the volume of a substance at the phase transition temperature). The principle of isothermal calorimetry is used in practice in a well-known device—isothermal phase transition calorimeter, in which the introduced heat does not change the temperature of the calorimetric system, but causes a change in the physical state of a substance that is part of this system (for example, the melting of ice in the Bunsen ice calorimeter).
In the process of melting a substance, an amount of heat equal to the latent heat of melting ΔH is absorbed by the system without a change in the temperature, but with a change in entropy ΔH/T=ΔS. In this case the amount of heat introduced is calculated according to the mass of the substance that have changed the state of aggregation (for example, the mass of melted ice, which can be measured by the change in the volume of the mixture of ice and water), and the heat of the phase transition. The principle of action of the ice calorimeter is based on the fact that the amount of heat introduced into the calorimeter is measured not according to the heating temperature, as it occurs in the variable-temperature calorimeter, but according to the flow rate of a mixture of water at ice melting temperature. The diagram of an isothermal ice calorimeter is shown in
The principle of operation of the isothermal calorimeter of the living tissue is similar to that of the ice calorimeter and is based on the fact that the amount of heat introduced into a local volume of living tissue or formed in this volume during cellular metabolism is measured not by a change in the temperature of the living tissue, as in the variable-temperature calorimeter, but by the change in the volume of the intercellular substance, determined by the volume of the mixture of the globule (crystalline phase), molten globule (liquid phase), at the melting point of the hyaluronic acid globule. Changes in the volume of the intercellular substance can be determined by measuring the osmotic pressure of the intercellular substance. A diagram explaining the principle of an isothermal microcalorimeter of the living tissue is shown in
Intercellular hyaluronic acid biopolymer of the living tissue is a natural isothermal biosensor with high sensitivity to thermal energy due to the low binding energy of ≈3.5 kBT of the tetramer molecular complex: thermal energy (amount of heat) introduced into the volume of the substance causes isothermal melting of the crystalline phase with the absorption of energy, and, as a result, causes a change in the osmotic pressure of the intercellular substance, which can be measured using a sensor located on the surface of the epidermis. As it was shown above, the conformational transitions of the Na-tetramer from conformation (s) of the crystalline phase to conformation (λ) of the liquid phase may occur only with the absorption of an energy quantum of 1kBT per tetramer. The amount of heat that must be transferred to one unit of mass of a crystalline substance in an equilibrium isobaric-isothermal process in order to transfer it from solid (crystalline) to liquid state (the same amount of heat is released during the crystallization of a substance) is defined as the specific heat of melting (enthalpy of melting).
Experimental studies were carried out to investigate the correlation between intercellular osmotic pressure and the power of external electromagnetic radiation fluxes acting on the surface of the human epidermis in a wide frequency range from optical spectrum to microwave: optical and near infrared radiation 0.4-0.9 μm; infrared radiation 1-10 microns; microwave radiation 5-10 mm.
The intercellular fluid flow rate through the epidermis (mass transfer rate) is determined by the osmotic pressure gradient of the intercellular substance. The slope in
Selectivity of intercellular substance to glucose. A specific feature of the isothermal microcalorimeter of living tissue is that the intercellular substance of the living tissue, along with extremely high sensitivity to heat flow, is selective towards the glucose molecule, the concentration of which determines the concentration of tetramers (crosslinks) of the hyaluronic acid polymer chain and, as a result, determines the ratio between crystalline (globule) and liquid (molten globule) thermodynamic phases. An increase in glucose concentration in the liquid phase of the intercellular substance leads to its crystallization, which is accompanied by a decrease in volume and osmotic pressure; a decrease in glucose concentration leads to the reverse process—dissolution, which is accompanied by an increase in volume and osmotic pressure. Therefore, an increase in the concentration of glucose in a substance leads to a response similar to that in case of cooling.
The intercellular substance of the living tissue is a glucose biosensor, which consists of two elements—a molecular recognizer (tetramer) and a transducer, which converts changes in glucose concentration into changes in the osmotic pressure of the intercellular substance.
The substance of the living tissue is a natural glucose biosensor, which consists of
two elements: a molecular recognizer, the role of which is played by the crystalline phase of the macromolecule, and a transducer, the role of which is played by the liquid phase, which converts changes in glucose concentration into changes in the osmotic pressure of the intercellular substance. These changes are recognized using a sensor located on the epidermis surface.
Caloric equation. The caloric state equation, which relates the variables of the thermodynamic state of living tissue in a local volume under heat- and waterproof applicator in an isothermal calorimeter, is based on the fundamental equation of thermodynamics in the following way.
The fundamental equation of thermodynamics, which describes the reversible quasi-static changes in the state of a closed system under the applicator in the time interval Δt=τ, equal to the time constant of the transient process, can be transformed and represented as an equation describing the reversible changes in the system in the time interval Δt=dt→0, where flow rate can be considered constant:
In the equation (44a), which relates the flows of coupled processes of energy and substance transfer, which are measured in a real experiment, the following definitions are used:
WMET—the power of the heat generation process, which is based on the required value
ΔQMET=∫(WMET)dt with integration within the range from 0 to τ.
WΔT—power of the process of transfer of the heat entering the system from the deep layer to surface layer, due to the temperature difference ΔTskin=Tdeep−Tsurface,
which is determined by the requirement ΔQΔT=∫(WΔT) dt=KΔT×∫(ΔTskin×/Δt) dt with integration within the range between 0 and τ.
KM×(ΔM/Δt)=J−the power of the mass transfer process entering the system from the deep layer, due to the difference in the amount of substance ΔM=Mdeep−Msurface, which is determined on the basis of the requirement μΔN=KM×ΔM=∫J(t)dt with integration in the range from 0 to τ, where μ is the chemical potential, N is the amount of substance, J(t) is the substance flow rate.
Kp×Δp=Wblood—is the power of the heat transfer process, carried out by capillary blood flow due to the difference in hydraulic pressure in the microcirculatory system,
Δp=parter−pvenul,which is determined by the requirement
VΔp=∫Kp×Δp dt=∫Wblood(t)dt, with integration within the range between 0 and τ.
The heat transfer component in the volume of tissue under the applicator, due to the difference in blood pressure, depending on the mass transfer flow index, can have a positive value (heat is introduced into the system), negative value (heat leaves the system) or equal to zero. The constant of this process is expressed in the following form: KP=V−V0>0, where V0 is the volume of the intercellular substance, at zero flow rate J=0. The volume of the substance V>V0, at positive flow J>0; volume of the substance V<V0, at negative flow J<0.
KΔT, KM, Kp—constants, phenomenological coefficients, which are determined by calibration.
The equation of state which correlates the power of coupled processes of energy and substance transfer under the applicator, which are measured in a real experiment, is derived from the equation (44a) by dividing all terms by dt and has the following form:
ΔH/τ=WMET+WΔT+KM×(ΔM/Δt)−Wblood=WMET+KΔT×(ΔTskin×/Δt)+KM×(ΔM/Δt)−Kp×Δp (44b)
It should be noted that the fundamental equation of thermodynamics is an integral form of equation (44b), which correlates the energy flow and mass transfer processes. Integration of equation (44a) in the time range from t0=0, corresponding to the application of the applicator with dosed pressure on the surface of the stratum corneum up to t=τ leads to the original form of the fundamental thermodynamic equation (3), which describes reversible changes in the system in the range Δt=τ, equal to the transient time constant:
Equation (4a) can be expressed as follows for calculating the magnitude of the thermal effect of metabolism:
W
MET
=ΔH/τ−K
ΔT
×ΔT
skin
−K
M
×ΔM+K
p
×Δp. (45)
The latter equation (45) may be presented in the following form:
W
MET=(Hdeep−Hsuf)/τ−KΔT×(Tdeep−Tsuf)/Δt)+KM×(Mdeep−Msuf)/Δt)+Kp×(parter−pvenul) (46)
In the abovedescribed equations:
Hdeep—deep layer enthalpy,
Hsurface—superficial layer enthalpy,
Tdeep—deep layer temperature,
Tsurface—superficial layer temperature,
Mdeep—deep layer water content,
Msurface—superficial layer water content,
Parter—hydrostatic pressure at the capillary entry (in the microcirculatory system),
Pvenule—hydrostatic pressure at the capillary exit (in the microcirculatory system),
Parameters KΔT, KM, KP correspond to the coefficients of proportionality and are equal to constant values determined using calibration method.
The value enthalpy is determined by the amount of water in a unit volume, which at the specified climatic parameters depends on the content of hyaluronic acid in the intercellular space. It may vary in diabetes mellitus and other chronic disease, therefore, it is a patient's individual characteristic, that can be determined using calibration method. On the other hand, water content in a unit volume of tissue at the specified hyaluronic acid content, depends on the environmental conditions.
The last equation (46) describes the relationship between the state variables, provided that the volume of the cell and the intercellular space do not change (ΔVtissue=0) with changes in glucose concentration and/or metabolic rate.
In general, changes in glucose concentration and/or the rate of cellular metabolism leads to changes in cell volume and intercellular space (ΔVtissue=ΔVintercell+ΔVcell). The caloric equation in the general case can be represented in the following form:
H
0/τ−(p×ΔVintercell+KV×ΔVcell)/Δt=KΔT×(ΔTskin/Δt)+KM×(ΔMintercell/Δt). (46a)
The blood sugar level C can be determined using the caloric equation (46a) from the condition KC×C=p×ΔVintercell, by measuring the amount of water in the volume of the tissue ΔVtissue=ΔVcell and the amount of water in the intercellular space ΔMintercell under the applicator of microcalorimeter.
Equation (46a) follows from the condition that the change of internal energy ΔUintercell of the intercellular substance occurs as a result of a change in heat transfer from the depth to the surface:
ΔUintercell/Δt=KΔTΔΔTskin/Δt+KM×ΔM/Δt.
It takes into account, that
ΔUintercell=ΔUC+U where ΔUC is the contribution to the change in the internal energy of the intercellular substance due to a change in glucose concentration; ΔUmet is the contribution to the change in the internal energy of the intercellular substance due to a change in heat generation in the cell.
The expression for ΔUmet, as follows from the first law of thermodynamics, has the form:
ΔUmet=ΔQMET−p×ΔVtissue=KV×ΔVcell,reΔQMET−
.
The expression for ΔUC, as shown above (37), has the form:
ΔUC=H0−p×ΔVintercell, where H0 is the heat content of the intercellular substance in the region of the physiological norm.
The Method of Isothermal Calorimetric Spectroscopy of Living Tissue.
Based on the above described isothermal microcalorimetry method (measurement of the magnitude of the electromagnetic radiation power absorbed by the living tissue using an isothermal phase transition microcalorimeter), a fundamentally new method of in vivo spectroscopy of living tissue can be created—absorption isothermal calorimetric spectroscopy, which enables real-time assessment of the content of biochemical components of living tissue on the basis of spectral measurements at the specific frequencies of the target molecules.
The magnitude of electromagnetic radiation power at the characteristic frequency of the biochemical component under study, absorbed in a local volume of substance of a living tissue, is measured not by the change in tissue temperature, but by the change in osmotic pressure value (or amount of water in the intercellular space), characterizing the volume of intercellular substance at the phase transition temperature (corresponding to the thermodynamic equilibrium between crystalline and liquid phases of the intercellular substance).
Characteristic frequencies of biochemical components of the living tissue substance, which include intercellular substance components (hyaluronic acid, water, glucose etc.), as well as intracellular biochemical molecules and organoids, form an electromagnetic absorption spectrum of the living tissue, which can be determined in vivo using isothermal calorimetric spectroscopy (ICS).
The method of isothermal calorimetric spectroscopy of living tissue is based on the detection of the magnitude of the absorbed power of electromagnetic radiation acting on the surface of the target local area of the epidermis of the living tissue, according to the amount of intercellular substance that has changed the phase state (melting of the globular phase of hyaluronic acid), by real-time measurement of the amount of water or osmotic pressure of the intercellular substance.
The source of electromagnetic irradiation at a wavelength corresponding to the characteristic wavelength of the biochemical component of the living tissue, is integrated into an isothermal microcalorimeter. Following definitions were introduced: 1—applicator (heat- and waterproof), which can be applied on the skin with gradual pressure, and formation of the closed thermodynamic living tissue system; 2—source of the electromagnetic irradiation with adjustable wavelength: 3—sensor of the osmotic pressure of the intercellular substance or intercellular hydrodynamic microflow.
The function of the response of the osmotic pressure of the intercellular substance vs wavelength of the irradiation (at a fixed power) is in direct correlation with the absorption spectrum of the substance; the response (changes) of osmotic pressure to the power of the irradiation is proportional to the absorbed power, and has a large value at the wavelengths at which the energy corresponds to the energies of excitation of vibrations in the studied molecules.
In some variants the invention design enables amplitude-modulation of the irradiation. In a spectral scheme using electromagnetic radiation with amplitude modulation in the tissue unit, under the applicator, oscillations of the elastic pressure of the intercellular substance occur due to the oscillations of the osmotic pressure at the irradiation modulation frequency. In this variant, the amount of absorbed power of electromagnetic radiation can be determined by measuring the elastic pressure of the tissue under the applicator at the irradiation modulation frequency. The closest analogue of the described method is the well-known method of optoacoustic (OA) spectroscopy, based on the effect of gas pressure pulse oscillations in a closed volume upon absorption of radiation modulated at an acoustic frequency. Such effect is due to the transformation of a part of energy of the irradiation-excited molecule into a thermal energy of the medium via radiationless transition. The optical-acoustic effect occurs due to the conversion of a part of the absorbed energy into thermal energy, which leads to the development of the acoustic vibrations in the sample itself or in the gas in contact with it. Registration of acoustic vibrations directly in the substance is carried out by a piezoelectric sensor (liquid and solid samples) or a microphone (gases).
In biochemistry, spectral measurement of a multicomponent system is carried out on the basis of a principle of additivity, according to which the absorption of an individual substance does not depend on other substances that have their own absorption. Thus, at a specified wavelength, the spectral density of a solution containing a mixture of non-interacting components is equal to the sum of the spectral densities of the individual components at the same wavelength. To assess the extent of absorption of the test solution containing any compound, the intensity of the irradiation penetrating this solution is compared to the intensity of the irradiation that has passed through the blank reference solution.
In living tissue spectroscopy, to assess the peak absorbance of the test compound spectral signal (irradiation intensity) at the wavelength corresponding to the characteristic frequency of the test compound is compared to the spectral signal of absorbance of the purified water at this wavelength, intensity of which can be analyzed on the basis of spectral measurement at another wavelength, corresponding to the water absorbance peak (not overlapping peaks of absorbance of the test substance).
The possibility of practical implementation of the isothermal calorimetric spectroscopy of living tissue biochemical components was proved (confirmed) by our experiments in which the correlation between the response of the osmotic pressure of the intercellular substance of the epidermis and the wavelength and power of the electromagnetic irradiation was measured; the correlation between osmotic pressure and electromagnetic irradiation in the optical and near-infrared (0.4-0.9 μm), infrared (1 μm-10 μm) and microwave spectra (5 mm-10 mm) was studied. The correlation between osmotic pressure and the power of infrared irradiation (with a maximum spectral power at a wavelength of 1˜10 μm) is shown in
The Method of Tetrahertz (THz) Calorimetric Spectroscopy.
Terahertz (THz) spectroscopy of living tissue is based on the measurement of the absorbed power of terahertz irradiation acting on the surface of the studied local area of the epidermis of living tissue, by the amount of intercellular substance that has changed the physical state, by measuring the amount of water or osmotic pressure of the intercellular substance in real time.
With regard to biological systems, terahertz irradiation enables assessment of the role of water in biological processes, as well as characteristic oscillatory processes that can characterize the state of a biosystem. An important advantage of terahertz spectroscopy from the medical point of view is that this method is non-contact and non-invasive—irradiation in this spectral range does not cause ionization (like ultraviolet) and significant heating of the subject (like microwave). Having a high sensitivity to the phase state of water and the substance of living tissue, the terahertz range allows to determine the extent of linkage of water molecules in biological systems, the level of solvation of ions in the solution, and the level of hydration of surfaces at the phase boundary. It is known that the most effective spectral measurements in the terahertz range are carried out with samples that represent plane-parallel layers of various thicknesses (from a few millimeters to an order of micrometers or hundreds of angstroms).
B of the tetramer molecular complex, determines the characteristic frequency of the tetramer, which corresponds to the wavelength of electromagnetic radiation λ≈152 μm (65.78 cm−1) in the terahertz spectrum range.
The binding energy of the tetramer is determined by the barrier height between the (1, s) conformations of the Na-tetramer: ΔE=Emax−Emin≈3.5 kBT, where Emin is a minimum energy value to the right from the energy barrier.
The wavelength of electromagnetic radiation corresponding to the energy of the transition of the tetramer to the excited state ΔE=Emax−Emin≈3.5 kBT, depends only on the physiological temperature: λ=c×h/ΔE=c×h/3.5kBT=532 μm/3.5≈152 μm (65.78 cm−1), where c—is the speed of light, h—is Planck's constant.
The resulting wavelength value corresponds to the terahertz (THz) range of the electromagnetic spectrum, which is in the wavelength range between 100 μm (far IR) and 1 mm (microwave).
Exposure to electromagnetic radiation at a wavelength of λ≈152 μm corresponding to an excitation energy of the tetramer of 3.5 kBT, leads to the breakage of the hydrogen bonds of the tetramer and the conformational transition of the Na-tetramer from the conformation (s) of the crystalline phase to the conformation (λ) of the liquid phase, which is accompanied by absorption of a quantum of thermal energy, the value of which can be determined using
Sensitivity of the Living Tissue to the Thermal Flux.
A feature of the natural biosensor of living tissue is that the hyaluronic acid tetramer (molecular recognizer) is highly sensitive not only to electromagnetic quantum hv at the characteristic frequency of the tetramer (λ≈152 μm), corresponding to the dissociation energy of the tetramer ΔE=3.5 kBT, but also has an extremely high sensitivity to thermal energy quanta, which are formed in the substance as a result of the absorption of electromagnetic radiation at frequencies that do not coincide with the characteristic frequency of the tetramer (λ≈152 μm). For example, thermal energy quanta are formed in the intercellular substance as a result of absorption of the electromagnetic radiation at the water absorbance bands.
It is well-known that the spatial distribution of biochemical components in the microvolume of living tissue is non-uniform: hyaluronic acid, which is the main component of the intercellular substance, is not part of the intracellular substance; glucose concentration in the intercellular space is higher than inside the cell where glucose oxidation reaction occur; the total amount of water in the intercellular space of a unit volume of living tissue is much less than the total amount of intracellular water, since the volume of the cell is much larger than the volume of the intercellular space. Therefore, a conclusion is understood that the power of electromagnetic radiation at a wavelength corresponding to the characteristic frequencies of the hyaluronic acid tetramer and other fragments of this polymer macromolecule, as well as the frequencies of the glucose molecule, is absorbed mainly in the volume of the-intercellular substance. On the other hand, electromagnetic radiation at a wavelength corresponding to the characteristic frequencies of water is mainly absorbed in the volume of the cell. Radiation energy absorbed by intracellular water (and/or other intracellular molecules) in an isothermal process is converted into the energy of thermal motion of molecules in the cell and the intercellular space surrounding the cell; interaction of the intercellular substance tetramer with a thermal energy quantum (thermal collision of a substance molecule with a tetramer), which was formed in the cell volume, leads to an isothermal change in the osmotic pressure of the intercellular substance, as a result of tetramer dissociation. The thermal energy arising in the intercellular substance as a result of the absorption of the external electromagnetic radiation energy leads to isothermal melting of the crystalline phase of the intercellular substance.
Therefore, a living cell surrounded by intercellular substance, under physiological conditions, is an isothermal calorimetric power detector that converts electromagnetic energy absorbed inside the cell (or produced in the cell during metabolism) into the osmotic pressure of the intercellular substance.
Electromagnetic radiation at a wavelength corresponding to the characteristic frequency of water is absorbed in the cell: the energy of such radiation is converted into the energy of thermal motion of water molecules and free links of the polymer chain, which leads to the dissociation of the hyaluronic acid tetramer and an isothermal change in the osmotic pressure of the intercellular substance. A living cell surrounded by intercellular substance is an isothermal calorimetric power detector that converts electromagnetic energy absorbed inside the cell (or produced in the cell during metabolism) into the osmotic pressure of the intercellular substance.
Thus, in the absorption spectra of a living tissue substance (intercellular and intracellular), along with the characteristic frequency of the hyaluronic acid tetramer in the terahertz range (□□≈152 μm), other frequencies corresponding to different groups of hyaluronic acid, water, glucose and other biochemical molecules will appear. Therefore, the spectral detection scheme should be developed taking into account the mutual influence on the results of spectral measurements of power absorption by adjacent absorption bands, in particular, terahertz radiation near the peak absorption region of the tetramer will also be absorbed by water molecules, the main part of which in a living tissue is located within the cell. For this reason, the response of the osmotic pressure of the intercellular substance of a local area of the living tissue to terahertz radiation at a wavelength (λ≈152 μm), will consist of two contributions: the spectral response of the tetramer of the intercellular substance and the spectral response to the absorption of this radiation by water molecules. The spectral contribution due to the absorption of water exceeds the spectral contribution due to the absorption of electromagnetic energy by the tetramer, since, in a unit volume of living tissue, the volume of intracellular substance significantly exceeds the volume of intercellular substance. Water spectrum in the terahertz region of the intense absorption peak is close to the characteristic frequency of the tetramer; therefore, the use of the differential method of double-beam spectroscopy allows to isolate the osmotic pressure signal due to the absorption of radiation by the tetramer of the intercellular substance by subtracting the contribution from the total signal due to the absorption of intracellular water. To isolate the spectral signal of the tetramer, it is necessary to determine the intensity of the absorption peak of the tetramer using the described method; To increase the accuracy, one can use a triple-beam (three-wavelength) spectroscopy scheme, in which the wavelengths correspond to the local extrema of the absorption peak (λ≈152 μm) and two side frequencies corresponding to the peak minima. In this case, the contribution to the signal due to the absorption of intracellular water is determined at a wavelength corresponding to the side frequencies of the hyaluronic acid tetramer absorption peak. From the resulting spectral distribution of the signal measured using triple-beam method, a constant signal corresponding to the absorption contribution of water molecules is subtracted. The described scheme of spectral measurement assumes that the spectrum of hyaluronic acid does not contain other absorption peaks adjacent to the tetramer absorption band (λ≈152 μm), due to other fragments of hyaluronic acid.
Tetrahertz (THz) Spectroscopy for Measurement of Blood Glucose Levels.
The method of isothermal calorimetric spectroscopy can be used in practice to determine the content of biochemical components (intercellular and intracellular substances) of living tissue in vivo. In particular, the method allows to determine the concentration of glucose in blood by measuring concentration of hyaluronic acid tetramers of the intercellular substance, which is proportional to the concentration of glucose bound by hyaluronic acid. It has been shown that ⅓ of the total number of glucose molecules in the hyaluronic acid macromolecule is in a free state, and the remaining ⅔ of the glucose molecules are bound to the monomers of the hyaluronic acid polymer chain. ⅓ of these glucose molecules are linked directly in the molecular complex of tetramer. Therefore, the concentration of free (unbound) glucose in the intercellular fluid (and in blood) can be determined by measuring the concentration of glucose bound to the tetramers of the intercellular substance (or the concentration of glucose bound to the polymer chain of hyaluronic acid).
Isothermal THz Calorimetric Spectroscopy of Water in the Living Tissue.
It is well-known that in biochemistry, spectral measurement of a multicomponent system is carried out on the basis of a principle of additivity, according to which the absorption of an individual substance does not depend on other substances that have their own absorption. Thus, at a specified wavelength, the spectral density of a solution containing a mixture of non-interacting components is equal to the sum of the spectral densities of the individual components at the same wavelength. To assess the extent of absorption of the test solution containing any compound, the intensity of the irradiation penetrating this solution is compared to the intensity of the irradiation that has passed through the blank reference solution.
When carrying out living tissue spectroscopy, in order to assess the absorption peak of the test biochemical compound, the spectral signal (radiation intensity) at a wavelength corresponding to the characteristic frequency of the test compound is compared to the spectral signal due to the absorption of purified water at the same wavelength.
On the other hand, development of a method for real-time measurement of water content in living tissue is an urgent task of applied and fundamental importance. The method of isothermal calorimetric spectroscopy allows to determine the water content (in the intercellular space and cell volume) of a local area of living tissue in vivo by measuring the electromagnetic absorption spectrum at wavelengths corresponding to the characteristic frequencies of water, which are well-known when using purified water. The absorption spectrum of water in a wide frequency range (from ultraviolet to microwave) is shown in
The energy of the incoming electromagnetic radiation, the wavelength of which corresponds to the characteristic frequencies of water, is absorbed in the cell volume and converted into the energy of the thermal motion of water molecules (the principle of an isothermal calorimeter), which leads to an isothermal change in the osmotic pressure of the intercellular substance as a result of melting of the crystalline phase of the intercellular substance.
For example, let us review the method of spectral detection in the area around the absorption band of water at a characteristic frequency of 3339 cm−1 (2.99 μm) in the range from 3250 cm−1-3750 cm−1 (ref. to
The distribution of the spectral power of the absorption peak can be determined using a triple-beam (three-wavelength) spectroscopy scheme, in which the wavelengths correspond to the local extrema of the peak (the wavelength corresponding to the peak maximum of 3339 cm−1 (2.99 μm) and two side frequencies of 3250 cm−1 and 2750 cm−1, corresponding to the peak minima. The amount of heat absorbed at a given wavelength in the frequency range between 3250 cm−1 and 2750 cm−1 is determined by the magnitude of the change in the osmotic pressure of the intercellular substance.
In case of spectral measurements in the terahertz region (λ≈152 μm), discussed above, the water absorption coefficient can be taken as a constant, since in the water absorption spectrum close to the characteristic frequency (λ≈152 μm) of the intercellular substance tetramer, according to the
Spectral Method of Determination of Glucose Concentration from the Water Absorption Bands.
The osmotic pressure and the amount of water in the intercellular substance is determined by the concentration of glucose and the magnitude of the external heat flux incident on the surface of the epidermis, determined by climatic parameters. The method of isothermal calorimetric spectroscopy that allows to measure the amount of water in a local microvolume of living tissue, can be practically applied to determine the concentration of glucose in the intercellular substance and blood. As it was shown, the concentration of glucose in blood can be determined based on the experimental correlation between the osmotic pressure and the power of the incident infrared thermal radiation flux, as shown in
The method of microcalorimetric spectroscopy described above allows to determine the correlation between the osmotic pressure and thermal power in a different way, using
a modulated electromagnetic radiation (with amplitude or frequency modulation). In the amplitude modulated measurement scheme, radiation is applied at a wavelength corresponding to the selected characteristic frequency of water, with amplitude (intensity) modulation. An important feature of the spectral measurement method using amplitude-modulated radiation is that the method allows to eliminate the dependence of the osmotic pressure signal on low-frequency fluctuations of environmental parameters. The method of spectral measurement with amplitude modulation allows non-invasive calibration of the measuring device by determining the value of the enthalpy and other calibration parameters using the caloric equation (45).
In the frequency modulation variant, spectral measurements are made at several wavelengths corresponding to the characteristic absorption frequencies of water. Measuring the response of the osmotic pressure of the intercellular substance to electromagnetic radiation (with the same power) at several different wavelengths with at the set absorption coefficients allows to determine the correlation between the osmotic pressure signal of the intercellular substance and the power of electromagnetic radiation absorbed in the local area of the living tissue under the applicator.
The correlation between the osmotic pressure and the absorbed power, determined by the described spectral method using modulated radiation, allows to determine the concentration of glucose in blood. An important feature of the method is that the spectral measurement at several frequencies (at least two), which can be performed in real time, allows to eliminate the contribution to the osmotic pressure signal due to low-frequency fluctuations of climatic parameters.
Factors limiting the accuracy of the described spectral method for measuring glucose concentration include the proximity of water absorption peaks to the absorption peaks of other molecules, such as hyaluronic acid and glucose. As it was noted above, in a living biological tissue, the spectral absorption bands of water, the peak intensity of which is determined by the volume of intracellular water, are adjacent to the glucose absorption bands. The resulting osmotic pressure signal is mainly determined by the absorption of radiation power by intracellular water, the contribution of which to the signal exceeds the contribution due to the absorption of electromagnetic energy power by the biochemical components of the intercellular substance (hyaluronic acid and glucose), as the volume of intracellular substance in a unit volume significantly exceeds the volume of intercellular substance. The diagram of spectral measurements at the characteristic frequencies of water, which takes into account the absorption spectrum of glucose, allows to increase the signal-to-noise ratio in the measuring scheme.
Absorption spectra of glucose in the IR range and transmission in the near IR region in the range of wavelengths from 1,428 μm (7000 cm−1) to 20 μm (500 cm−1), which were studied in detail, are shown in
To assess the correlation between the osmotic pressure of the intercellular substance and the power of the incident electromagnetic radiation in a single-beam measurement scheme with amplitude modulation of radiation, the wavelength can be selected in one of the intervals between the glucose absorption peaks, for example, 7000 cm−1 (1,428 μm), 5000 cm−1 (2.0 μm), 4000 cm−1 (2.5 μm).
In a differential double-beam measurement scheme using frequency modulated radiation, at two (or more) wavelengths, it is possible to select frequencies that do not fall within the range of the glucose uptake peaks, the side frequencies of the glucose uptake peak, for example, 7000 cm−1 (1,428 μm), 5000 cm−1 (2.0 μm), 4000 cm−1 (2.5 μm).
In the above described spectral measurement scheme using modulated radiation, the possible contribution to the resulting signal of the absorption peaks of intercellular hyaluronic acid, which are adjacent to the water absorption peaks, is disregarded, but they make a contribution to the resulting signal that is much smaller than the contribution due to absorption at water frequencies, since the volume of intracellular substance, in a unit volume of living tissue, significantly exceeds the volume of the intercellular substance.
Thus, the concentration of glucose in blood can be determined based on isothermal calorimetric spectroscopy in two different ways:
terahertz spectroscopy method—measurement of the concentration of hyaluronic acid tetramers in the intercellular substance (proportional to the concentration of bound glucose in the intercellular substance) at a wavelength of terahertz radiation (λ≈152 μm), corresponding to the characteristic frequency of the tetramer;
infrared spectroscopy method—measurement of the concentration of hyaluronic acid tetramers based on the correlation between the osmotic pressure of the intercellular substance and the absorbed radiation power, which is determined in real time based on spectral measurements using radiation with amplitude modulation at one wavelength (or several wavelengths) corresponding to the characteristic frequency of water in the infrared or other region of the electromagnetic spectrum.
The signal-to-noise ratio in the abovedescribed spectral measurement scheme can be improved by additional measurement of other biochemical components of the extracellular substance, in particular, the content of hyaluronic acid. Spectroscopy of hyaluronic acid and other components of the intercellular substance is discussed below.
Isothermal Calorimetric Spectroscopy of the Intercellular Substance.
The absorption spectrum of the intercellular substance (which is a multicomponent biopolymer) has a more complex structure of spectral absorption bands compared to the spectrum of pure water, since, along with intense absorption peaks of water (mainly intracellular), it contains absorption peaks corresponding to the characteristic frequencies of the biochemical components of the intercellular substance, the main components of which are water, hyaluronic acid and glucose associated with the monomers of the polymer chain of hyaluronic acid, intercellular (tissue) fluid, similar in composition to blood plasma, which consists of water, amino acids, sugars, fatty acids, coenzymes, hormones, neurotransmitters, salts, as well as waste products of cells.
It should also be noted that the characteristic binding energies of molecules of various pharmaceutical compounds, as well as molecules of compounds and foreign substance in condensed media of the intercellular substance that are physiologically incompatible with the metabolism of living tissue, can also be found in the infrared and terahertz range of the electromagnetic spectrum and appear in the absorption spectra of the intercellular substances.
To assess the absorption peaks of the studied biochemical component of the intercellular substance of living tissue, the spectral signal (radiation intensity) at a wavelength corresponding to the characteristic frequency of the studied compound is compared to the spectral signal due to the absorption of pure water at a given wavelength.
A living cell surrounded by an intercellular substance is a natural isothermal power calorimeter that allows to determine the absorption spectrum of the intercellular and intracellular substance of a living tissue based on measuring the correlation between the osmotic pressure of the intercellular substance and the spectral density of electromagnetic radiation at different wavelengths (1, 2, 3, 4, 5, 6, 7): 1—terahertz radiation at the characteristic frequency (λ≈152 μm) of the hyaluronic acid tetramer; 2—radiation at characteristic frequencies of glucose bound to the monomers of the polymer chain of hyaluronic acid; 3, 4—radiation at characteristic frequencies of hyaluronic acid disaccharide: 5, 6—radiation at the characteristic frequencies of water; 7—electromagnetic radiation at the characteristic frequency of an intracellular molecule.
The absorption of terahertz radiation, the wavelength of which corresponds to the characteristic frequency of the hyaluronic acid tetramer (λ≈152 μm), leads to the dissociation of the molecular complex of the tetramer with the conversion of electromagnetic energy into the energy of thermal motion of molecules, which results in an isothermal change in the osmotic pressure of the intercellular substance. The energy of the incident electromagnetic radiation, the wavelength of which differs from the characteristic frequency of the tetramer, but corresponds to the characteristic frequencies of the biochemical components of the intercellular substance (hyaluronic acid, glucose, water, etc.), is absorbed in the volume of the substance and converted into the kinetic energy of the thermal motion of molecules, which leads to a change in the osmotic pressure of the intercellular substance as a result of the dissociation of the hyaluronic acid tetramer.
Glucose Spectra
Spectra of Hyaluronic Acid and Free Fatty Acids.
Infrared spectra of the substances under study that are components of the constituent parts of biological tissues (glucose, hyaluronic acid, protein—egg albumin, lipid—fat stock), in the frequency range 4000 cm−1-600 cm−1 in this study using Fourier IR spectrometer (Tensor 37 by Bruker) are shown in
In the high-frequency IR region of the spectrum 400 cm−1 (2.5 μm)-3000 cm−1 (3.3 μm), in which an intense water absorption peak was found (
The structure of the spectrum in the lower frequency region of the IR spectrum can be analyzed on the basis of the selected fragment of the spectra of all considered components, shown in
The band of carboxyl carbonyls 1743 cm−1 (5.7 μm) is present in the spectrum of hyaluronic acid (
Spectral Method for Hyaluronic Acid Content Measurement.
Measuring the content of hyaluronic acid in living tissue is an urgent task that has independent practical significance in cosmetology, dermatology, regenerative medicine, etc.; The relevance of the issue is due to the need to monitor hyaluronic acid for early diagnosis of diabetes, as well as for non-invasive monitoring of blood glucose in patients with diabetes mellitus. Clinical studies in a group of patients with diabetes mellitus showed the effect of the changes of the content of hyaluronic acid on the temporal changes in the osmotic pressure, at a constant concentration of glucose in blood, in the time interval of the experiment.
As can be seen from the comparison of the spectra (
substance; the purpose of spectral measurement of biochemical components of intercellular substance is a more complex experimental issue vs. spectral measurement of the amount of water per unit volume of living tissue.
IR Spectroscopy of Hyaluronic Acid.
It is well-known that hyaluronic acid is non-homogeneously distributed in the epidermis, its content in the stratum corneum and upper layers of the epidermis is higher than its content in the dermis; the authors of the study [ . . . ] assume that the transport of hyaluronic acid into the stratum corneum is carried out from keratinocytes located directly under the stratum corneum: keratinocytes are the main cells of the human epidermis (approximately 90% of all epidermal cells), which are present in the basal, spiny, granular, shiny stratum corneum of the epidermis.
To measure the content of hyaluronic acid in the upper layers of the epidermis, the method of diffuse reflectance spectroscopy can be applied, which is based on the registration of electromagnetic radiation diffusely reflected from the upper layers of the epidermis. The method of spectral measurement of the content of hyaluronic acid in the epidermis, based on the registration of electromagnetic radiation diffusely reflected from the epidermis, is shown in
To measure the content of hyaluronic acid in the superficial layer of the epidermis, near electromagnetic radiation in the near infrared (or optical) region at a wavelength corresponding to the characteristic frequency of hyaluronic acid can be used. For example, a frequency in the range of the absorption peak of 7000 cm−1 (1.43 μm) in the spectrum near the absorption peak of glucose 6000 cm−1 (1.67 μm) can be used. The spectrum of hyaluronic acid in the near infrared region is shown in
On the other hand, absorption peaks of hyaluronic acid have been noted to occur adjacent to the intense water absorption peaks.
To isolate the spectral signal corresponding to the peak of hyaluronic acid absorption against the background of an intense water absorption peak, a single-beam (one-wavelength) combined spectral method is used, which is a result of a combination of two methods—isothermal calorimetric spectroscopy, which allows to determine the water content in the epidermis (since the intensity of the absorption peak of intracellular water is much higher than the absorption intensity of intercellular hyaluronic acid), and diffuse reflectance spectroscopy, which allows to determine the total content of water and hyaluronic acid in the superficial layer of the epidermis. The contribution of hyaluronic acid is derived from the resulting spectral signal of diffuse reflectance by subtracting temporal changes of the amount of water in the stratum corneum under the heat- and waterproof applicator applied to the surface with dosed pressure, from the total signal. The measurement is carried out according to the scheme of single-beam spectroscopy at one wavelength corresponding to one of the characteristic frequencies of hyaluronic acid, which is selected in the frequency range of the spectrum in which the absorption peaks of hyaluronic acid are overlap (or are adjacent) the absorption peaks of water (UV, optical, near IR, IR, far R).
In the near IR region, one can choose, for example, a wavelength of 7000 cm−1 (1.43 μm), in the IR region—3500 cm−1 (2.8 μm). In the presented variant of the combined scheme, the use of an additional spectral measuring channel based on the registration of electromagnetic radiation diffusely reflected from the superficial layer of the epidermis, as well as the use of the measuring method of synchronous signal detection, allows to increase the signal-to-noise ratio of the measuring channels. Correlation between the osmotic pressure response and the power of the incoming electromagnetic radiation with amplitude modulation at a wavelength corresponding to the characteristic frequency of water allows to determine the concentration of glucose based on the abovedescribed method.
Therefore, a combination of two spectral methods (isothermal calorimetric spectroscopy and diffuse reflectance spectroscopy) in one spectral device enables better signal separation and detection of the content of the three biochemical components of living tissue (glucose, hyaluronic acid and water) using measurements at one wavelength, corresponding to one of the characteristic frequencies of hyaluronic acid (single beam spectroscopy).
Double-Beam Spectral Method for Glucose Content Measurement.
The above described combined spectral method, based on a double-beam spectroscopy (at two wavelengths) corresponding to the characteristic frequencies of glucose and hyaluronic acid (close to the water absorption peak), enables better signal separation and improves signal-to-noise ratio to determine the concentration of glucose, and also the content of two other components, hyaluronic acid and water, the low-frequency temporal changes of which directly affect the signal-to-noise ratio when measuring glucose concentration. Spectral measurement at each wavelength is carried out in two ways (isothermal calorimetric spectroscopy and diffuse reflectance spectroscopy), the combination of which in one spectral device allows to improve signal separation. As an example, in the near-IR region, a wavelength of 6000 cm−1 (1.67 μm) corresponding to the characteristic frequency of glucose and a wavelength of 7000 cm−1 (1.43 μm) corresponding to the characteristic frequency of hyaluronic acid can be selected.
In another version of the double-beam measurement scheme, the glucose concentration is determined using caloric equation (45), in which the enthalpy ΔH, which depends on the content of hyaluronic acid in the intercellular substance (and has individual values for different patients) and the amount of water in a tissue unit volume, which depends on environmental parameters, is determined by the spectral method based on the hyaluronic acid and water absorption bands.
In another variant, the concentration of glucose is determined using triple-beam spectroscopy at 3 wavelengths corresponding to the selected absorption peaks of glucose, water and hyaluronic acid. Spectral measurements are carried out in the following ranges of the spectrum: in the range of 3550 cm−1-3050 cm−1 (2.8-3.3 μm) at one frequency of 3339 cm−1 (2.99 μm) corresponding to the water absorption peak (or three frequencies corresponding to the absorption peaks of water, hyaluronic acid and glucose); at a frequency of 1637 cm−1 (6.08 μm), corresponding to the second peak of glucose absorption; at a frequency of 1553 cm−1 (6.4 μm), corresponding to the third peak of glucose absorption.
The spectral glucose absorption bands in the optical and near-infrared spectral range are also close to the intense water absorption peaks (
The method for measuring glucose concentration in the red and near-IR range is based on the single-beam combined method described above. To isolate the spectral signal corresponding to the glucose absorption peak against the background of an intense water absorption peak, a single-beam combined spectral method is used, which is obtained as a result of a combination of two methods—isothermal calorimetric spectroscopy, which allows to determine the water content in the epidermis (since the intensity of the absorption peak of intracellular water is many times higher than that of intercellular hyaluronic acid), and diffuse reflectance spectroscopy, which allows to determine the total content of water and glucose in the superficial layer of the epidermis. A combination of two spectral methods (isothermal calorimetric spectroscopy and diffuse reflectance spectroscopy) in one spectral device allows to improve signal separation and determine the content of two biochemical components of living tissue, glucose and water, using measurements at one wavelength corresponding to one of the characteristic frequencies of glucose (single-beam spectroscopy).
The measurement channel based on isothermal calorimetric spectroscopy allows to determine the water content in the volume of the epidermis, since the intensity of the peak of intracellular water absorption is many times higher than that of glucose absorption, the concentration of which in the cell volume is significantly lower than its concentration in the intercellular space, as a result of the process of glucose oxidation in the volume of the cell; the second measurement channel, based on diffuse reflectance spectroscopy, allows to determine the total content of water and glucose in the superficial layer of the epidermis.
In a dual beam spectroscopy, to obtain better separation, two wavelengths are used, for example, a wavelength in the range of 930-1050 nm, corresponding to the width of the water absorption peak, and a wavelength in the range of 1000-1100 nm, corresponding to the absorption peak of glucose; additionally, a wavelength in the range of 900-970 nm, corresponding to the second water absorption peak, can be used. In case of triple-beam spectroscopy, the content of hyaluronic acid is additionally determined at a wavelength that is chosen in the same frequency range, or in another near-IR region, for example, 7000 cm−1 (1.43 μm).
The scheme of spectral measurements in the optical range should be developed taking into account the need for individual calibration for the melanin content in the skin.
Method for Measuring Fatty Acid Triglyceride Content.
Electromagnetic radiation at a wavelength corresponding to the characteristic frequencies of free fatty acids will be intensively absorbed in the lipid layer of the epidermis under the stratum corneum, as well as in the fat layer located under the dermis. The lipid layer of the epidermis, located under the stratum corneum, consists of 10-25 layers of cells oriented parallel to the surface and immersed in the lipid matrix, which makes up about 10% of its volume, and has a unique structure and chemical composition. The skin contains both free fatty acids and fatty acids bound with formation of triglycerides, phospholipids and ceramides. Epidermal lipids mainly consist of: ceramides (50%): cholesterol and its esters (25%) saturated fatty acids (10%).
To determine the content of free fatty acids, a spectral method can be used, which is a combination of two spectral methods, isothermal calorimetric spectroscopy (for probing the fatty layer of the skin located under the dermis) and diffuse reflectance spectroscopy, based on the registration of electromagnetic radiation diffusely reflected from the intercellular substance of the lipid layer of the epidermis with a high content of free fatty acids, located under the stratum corneum; which consists of 10-25 layers of cells, oriented parallel to the surface and immersed in the lipid matrix, which is about 10% of its volume, having a unique structure and chemical composition. Epidermal lipids mainly consist of: ceramides (50%); cholesterol and its esters (25%) saturated fatty acids (10%) (
As can be seen (
The content of fatty acid triglyceride and hyaluronic acid in living tissue can be determined using the method of isothermal calorimetric single-beam spectroscopy at wavelengths corresponding to the characteristic frequencies of fatty acid triglyceride 1743 cm−1 (5.7 μm) and/or 3040 cm−1-2853 cm−1 (3.3-3.5 μm).
In another variant, a spectral method for measuring fatty acid triglyceride content is based on the use of a double-beam spectral scheme at wavelengths corresponding to one of the characteristic frequencies of hyaluronic acid (combined method described above), for example, 3500 cm−1 (2.8 μm)-3000 cm−1 (3.3 μm), and one of the characteristic frequencies of the fatty acid triglyceride, for example, 3040 cm−1-2853 cm−1 (3.3-3.5 μm). In this variant, the spectral measurement in the range of 2.8-3.3 μm is carried out on the basis of a combined method (combination of isothermal calorimetric spectroscopy and diffuse reflectance spectroscopy), and the measurement at the triglyceride wavelength is carried out according to the diffuse reflectance spectroscopy scheme, which allows to determine the content of triglyceride in the superficial lipid layer of the epidermis or according to the isothermal calorimetric spectroscopy scheme, which allows to determine the triglyceride content in the fatty layer under the dermis. The described measurement scheme allows to record a signal proportional to the content of triglyceride, or a total signal proportional to the total content of glucose and lipids, characterizing the rate of glucose and free fatty acids metabolism.
A Method of Triple-Beam Spectroscopy of Glucose, Hyaluronic Acid and Fatty Acid Triglyceride.
Glucose concentration can be determined based on a triple-beam spectroscopy method with the use of modulated emission spectroscopy at a wavelength corresponding to the characteristic frequencies of water by adding a double-beam spectroscopy method at wavelengths corresponding to the characteristic frequencies of hyaluronic acid and fatty acid triglyceride. The advantage of this method is that the intensity of the absorption peaks virtually does not change in the experiments throughout the day (and week), so the measurement is done using double-beam spectroscopy at two wavelengths corresponding to the characteristic frequencies of water and hyaluronic acid. A single-beam scheme of a spectral device at a wavelength corresponding to the characteristic frequency of 1743 cm−1 (5.7 μm) can be used to monitor the content of hyaluronic acid in living tissue.
Method of Spectroscopy of Intracellular Biochemical Macromolecules and Organelles
The method of microcalorimetric spectroscopy described above, in general, allows to determine the concentration of macromolecules of an intracellular substance using electromagnetic radiation with frequencies (wavelengths) that coincide with the characteristic frequencies of tetramers of intracellular macromolecules (proteins, RNA, DNA).
The characteristic frequencies of intracellular proteins fall into the far infrared range of the electromagnetic spectrum. For example, the characteristic binding energies in the macromolecule of the globular enzyme glucokinase and other intracellular enzymes are approximately 7 kcal/mol-20 kcal/mol or 10 kBT-30 kBT, corresponding to a wavelength interval between 20 and 70 μm in the far IR region of the spectrum.
The concentration of the main biochemical components of the intercellular substance, in general, can be determined using the differential method according to the four-beam (four wave) spectroscopy method, which uses wavelengths corresponding to the characteristic frequencies of the 3 main components of the intercellular substance, hyaluronic acid, glucose, water molecules, as well as a wavelength corresponding to the characteristic frequency of an intracellular macromolecule or complex; in the course of measurement, each of the four biochemical compounds can be represented by a set of 1 (or more) characteristic frequencies. Therefore, the number of wavelengths that characterize a group of 2 biochemical components of the intercellular substance, a water molecule and 1 intracellular molecule under study can be more than 4.
Absorption of an incident electromagnetic radiation at wavelengths corresponding to the characteristic frequencies of intracellular biochemical components by intracellular macromolecules (and biochemical complexes) will lead to a change in the heat balance in the study area of the living tissue under the applicator (heating) and, as a consequence, to a change in the osmotic pressure of the intercellular substance due to the breakage of the internal intermolecular bonds of hyaluronic acid tetramers of the intercellular substance, which have a characteristic binding energy of the order kBT.
The proposed method for spectroscopy of the intercellular substance of a local area of living tissue can be implemented using a spectral device, which is schematically shown in
The device contains several sensors for measuring physiological parameters, one of which is designed to measure the osmotic pressure of the intercellular substance, as well as one or several sources of electromagnetic radiation with a constant or modulated intensity.
The device consists of an applicator 1 (heat- and waterproof, having an upper and lower surface), which can be applied on the surface of the epidermis with dosed pressure, one or several sources 3 of electromagnetic radiation, with a constant or modulated intensity, with a wavelength corresponding to characteristic frequency of the studied biochemical component of the living tissue, a device for creating a calibration effect on the controlled tissue area under the applicator, an osmotic pressure sensor of the intercellular substance (2), sensors for measurement of temperature and other physiological parameters (4) located under the applicator, from the inside of the applicator on the surface of the stratum corneum (SC), while the signals from the above sensors are transferred sequentially to the terminals of an amplifier unit 5 and/or a synchronous detector and/or an analog-to-digital converter (hereinafter ADC) installed on the upper surface of the applicator 1; a data processing unit 6; a data display unit 7.
The microprocessor for real-time processing of signals from sensors of physiological parameters of the local tissue region under the applicator and sensors of environmental parameters is an information processing unit 11.
In one of the preferred embodiments, information processing unit 6 is located on the upper surface of the applicator 1, in the other—it is remote and is wired with AD converter.
The information display unit 7 is a smartphone or a personal computer. In this connection, communication of the multi-sensor device with the information display unit 7 can be in various embodiments, both wireless and wired (via USB channel).
At least the osmotic pressure sensor 2, which helps determining the amount of water in the intercellular space of tissue in the local volume under the applicator at a controlled depth on which the local thermodynamic equilibrium is being achieved; the SC temperature sensor 4, which allows determining the temperature of skin surface under applicator, the sensor of the elastic pressure of tissue under the applicator is used as sensors for measuring the dynamics of physiological parameters of the local tissue area under applicator.
The signals from the sensors of physiological parameters (osmotic pressure sensor 2, temperature sensor 3 and elastic pressure sensor 4) located on the inner surface of the applicator 1 are transferred to the terminals of instrumental amplifiers and/or an analog-to-digital converter (unit 10), after which they are transferred to the data processing unit 11 for signal processing. Processed information is transmitted to the display device 12, on the screen of which the current values of the measured parameters and/or metabolic rate of the parameters are displayed in real time.
In case of using electromagnetic radiation with amplitude signal modulation, in order to increase the signal-to-noise ratio, the method of coherent detection of AM signals is used, which can significantly increase the signal-to-noise ratio of the recorded signal and reduce its distortion. In this case, coherent detection of the signal at the output of the osmotic pressure sensor is carried out using an instrumentation amplifier (with a built-in synchronous detection circuit) at the AM frequency of the electromagnetic radiation modulation.
For practical implementation of the multi-sensor device, industrial certified sensors for measuring climatic parameters, as well as certified sensors for measuring the temperature of the stratum corneum and the elastic pressure of the epidermis can be used. For example, Honeywell sensors can be used as sensors for measuring the temperature of the stratum corneum.
Piezoelectric element-based sensors can be used as a sensor for measuring elastic pressure.
The osmotic pressure sensor (2), which registers the dynamics of water transfer in the epidermis by recording the temporal dynamics of the water content in the stratum corneum, can be based on different physical and chemical methods based on different physical principles. In particular, the following methods are applicable to measure the amount of water in the stratum corneum-electrometry methods based on measuring the electrophysical characteristics of SC (electrical conductivity, dielectric constant); spectral methods based on measurement of spectral characteristics (reflection and absorption coefficients); optical-acoustic methods; thermophysical methods based on measurement of thermophysical characteristics (thermal conductivity, heat capacity); electrochemical methods, etc. Also, sensor for measuring the amount of water can be based on the measurement of elastic pressure. Devices for measuring the amount of water in the epidermis are described in the patent [10].
In case of using an electrometric sensor, the amount of water in the epidermis can be determined by measuring electrical characteristics (transverse electrical conductivity and/or permittivity) of the stratum corneum on AC or DC current.
If the spectral sensor is used, the amount of water in epidermis can be determined by measuring the spectral characteristics (reflection coefficient and/or absorption coefficient) of the stratum corneum for electromagnetic radiation with a wavelength near the absorption line due to water content in the SC. In particular, temporal dynamics of the process of swelling of intercellular substance can be recorded according to the temporal dynamics of the coefficient of reflection of the stratum corneum. Increase in the amount of water in the intercellular space leads to a change in the spectral characteristic of SC, caused by a change in the amount of water in SC.
The spectral sensor may be based on a method in which the osmotic pressure of the intercellular substance or the amount of water in the intercellular space is determined by measuring the spectral characteristics of the intercellular substance in the range of the electromagnetic radiation spectrum, which is determined based on the characteristic absorption frequencies of the intercellular substance, and is selected from the group including characteristic frequencies of water in the optical and/or infrared and/or microwave range; group frequencies of hyaluronic acid of the intercellular substance in the infrared and/or terahertz and/or microwave range.
In another variant, the spectral sensor is based on the method for determining the osmotic pressure of the intercellular substance by measuring the amount of water in the stratum corneum based on the physical characteristics of the stratum corneum, which are selected from the group including electrophysical characteristics, spectral and optical-acoustic characteristics, thermophysical characteristics.
In the spectral method, the osmotic pressure of the intercellular substance or the amount of water in the intercellular space is determined based on the amount of water in the stratum corneum by measuring the spectral characteristics of the stratum corneum at the characteristic frequencies of water absorption in the stratum corneum using a spectral method, selected from the group including IR spectroscopy. Raman spectroscopy, optical-acoustic spectroscopy, differential double-beam spectroscopy.
The dosage calibration action is selected from the group consisting of external pressure, local decompression, heating, cooling, electric current or voltage and magnetic field.
For example, the device that generates pulses of electrical voltage or current through SC can be used as a device for creating a calibration effect, such device can determine current value and direction of microcurrents of tissue fluid in the intercellular substance in the deep layer of epidermis, by measuring the kinetic characteristics of mass transfer caused by electro-kinetic phenomena and more particularly by electro-osmosis.
In alternative embodiments, the device for generating calibration action is a thermal power source made in the form of a resistor or Peltier element, or a device for creating a dosed pressure on the surface of the applicator.
Industrial laser diodes (based on quantum-well heterostructures and other types) can be used as sources of electromagnetic radiation in the optical and infrared range; they are widely used and are produced in a large variety of irradiation characteristics, tailor-made for specific application. There are no official standards for the types and sizes of laser diode housings, but sometimes large manufacturers agree to unify packages. In addition, there are services for the development of the emitter housing according to customer requirements.
Industrial photo receivers can be used as receivers of electromagnetic radiation in the optical and infrared range. The leading worldwide companies produce quite a few types of microcircuits in which the radiation receiver (photodetector) and amplifier are combined in one housing. Such systems are widely used in IR remote control systems (such modules are also called IR receivers). The leading worldwide companies produce quite a few types of such microcircuits.
Domestic industry, for example, LED Microsensor NT LLC and IBSG Co., Ltd., produce a wide range of mid-infrared optoelectronic devices—light emitting diodes, LED arrays, photodiodes for the spectral range from 1.6 to 5.0 μm, as well as compatible electronic devices (drivers for LEDs and amplifiers for photodiodes).
In the optical and near-IR range, domestic LEDs can also be used: SIM-012ST (950.970 nm); OIS-150-1020 (1020-1060 nm): KM2520F3C03 (930-950 nm): EDEF-1LS3 (740-760 nm): EDEI-1LS3 (830-850 nm). A photodiode sensitive to optical radiation in the wavelength range from 740 to 1060 nm, for example, a BPW34 type photodiode, is used as an optical radiation receiver.
Miniature sources and receivers of terahertz radiation with radiation characteristics acceptable for applications in practice, have not yet been produced by the world industry and are at the stage of research and development. In particular, scientists from Princeton University (Xue Wu; Kaushik Sengupta, Department of Electrical Engineering, Integrated Micro-systems Research Lab, Princeton University, Princeton, N.J., USA) have achieved a radical reduction in the size of equipment for processing of terahertz electromagnetic radiation. Such devices can become an important element of medical diagnostics, communications and drug development. There are two publications on this topic in the IEEE Journal of SolidState Circuits. In the article, the scientists explained how they managed to create a miniature microcircuit for generating terahertz radiation. It still works in a narrow range, but the researchers realized that this issue can be solved by creating several waves of different lengths at once on a single chip and combining them using precise synchronization. Daniel Mittleman, vice chairman of the Society of Infrared, Millimeter. and Terahertz Waves, said that the work is “highly innovative and potentially has many applications” and also noted that there is still a lot of work to be done before these projects will be implemented in routine devices, but they are promising.
The heat- and waterproof applicator (1) with sensors of intercellular osmotic pressure 2 (or sensors of water amount in stratum corneum) fixed to its lower surface, temperature 3 and elastic pressure of the epidermis on the applicator 4, is fixed on the surface of the stratum corneum, for example, on the surface of the hand, with external dosage pressure and is attached in one embodiment using straps. It follows that the measurement can be made on any part of the surface of the human body, free from the hair, with the exception of the areas of the body where the sweat glands are located, which are activated as a result of emotional impact. Palms of hands are the example of such area.
Provisional dimensions, appearance and design of the multisensor device are shown in
Process of measuring (signals recording) starts immediately at the moment of pressing at least one heat- and waterproof applicator to the SC surface or with 10-50 seconds delay from the moment of pressing.
Temporal dynamics of physiological parameters in the local closed tissue area under the applicator is measured in the “microcalorimeter” mode after applying the applicator to the skin surface, in particular, at least the following parameters are being measured—heat flow through the epidermis area under the applicator or surface temperature of the skin under the applicator, elastic pressure of tissue under the applicator, pressure created by the applicator on the surface of the skin, amount of water in the intercellular space of tissue at a controlled depth at the local thermodynamic equilibrium level. Heat flux is measured through a patch of epidermis under the applicator or a temperature at a controlled depth of Tskin, wherein local thermodynamic equilibrium is established, at a fixed pressure Psensor of applicator.
To calculate blood glucose level, a calibration procedure is carried out and calibration parameters are determined to obtain the constant coefficients necessary for calculation of the level of glucose in blood and the thermal effect of metabolism of the local area of living tissue.
The number of sensors of the multi-sensor device described above can be increased by including additional sensors and sensors in the device that characterize the metabolism of a local area of living tissue and more particularly to blood biochemical parameters (for example, lactate in the blood), blood acidity, frequency of cardiovascular contractions. Additional sensors for through-cut measurements can be installed on the inside of the applicator and integrated with the physiological and climatic parameters sensors described above into a single measuring circuit as part of a multi-sensor device.
Methods for Early Diagnosis of Diabetes Mellitus.
A method for diagnosing diabetes mellitus based on the changes In the glucose curve. Modern guidelines define Impaired Glucose Tolerance (IGT) as blood glucose concentrations (according to the oral glucose tolerance test results. 2 hours after intake of 75 g of glucose—from 7.8 to 11.0 mmol/L) between normal and diabetic values; the glucose concentration during the tolerance test is measured invasively using a glucometer with several blood samples taken during the oral test. IGT is thought to be a sign of a pre-diabetic state, although not all individuals with IGT develop diabetes. In USA every tenth adult individual has IGT the rate thereof increasing with age achieving every fourth among persons aged from 65 to 74 years.
The abovedescribed method of isothermal calorimetry of living tissue is a promising approach for creating methods for the early diagnosis of diabetes mellitus, in particular, it allows to determine the temporal changes of blood glucose in IGT in the continuous monitoring mode (glucose monitoring curve).
A clinical study in a group of patients with diabetes mellitus showed that the glucose monitoring curves in diabetics, recorded continuously (1 measurement per second) using a prototype isothermal microcalorimeter, demonstrate stepwise (step-like, jumping) changes. Characteristic glucose monitoring curves with a stepwise temporal dynamic, recorded in a clinical study, are show in
It is well-known that blood glucose level in healthy subjects is determined by the level of insulin, which is produced by the pancreatic cells proportionally to the level of glucose in blood, which as a rule shows monotonous changes without spikes. Abnormal changes in the pancreas functions can lead to a non-homogeneous stepwise flow of insulin into blood and, as a result, to spiking of blood glucose.
The study showed that the stepwise changes in blood glucose level can lead to the stepwise changes in sugar concentration in the interstitial fluid in the form of sequential single impulses; a characteristic glucose curve is shown in
The method of blood glucose monitoring based on the method of isothermal calorimetry can be applied in practice for the early diagnosis of diabetes mellitus (types 1 and 2) according to the features of the stepwise changes in the glucose monitoring curve.
A Method for Diagnosing Diabetes Mellitus Based on Living Tissue Enthalpy Monitoring.
The method of isothermal microcalorimetry of living tissue enables early detection of the diabetes mellitus (pre-diabetes) in patients based on the content of hyaluronic acid in the epidermis, through monitoring of the osmotic pressure of the intercellular substance (or the amount of water in the intercellular space).
In a clinical study with the use of a prototype isothermal microcalorimeter of living tissue, we have found that the value of the enthalpy of the intercellular substance in virtually healthy patients, determined experimentally (at a specified power of external heat flow due to climatic parameters of the environment) based on a comparison of non-invasive prototype readings with invasive certified blood glucose meter readings (calibration method) do not change or change very slowly over time, with a characteristic time constant exceeding than one month or more. The result obtained experimentally is in good agreement with the conclusions obtained on the basis of a theoretical study of the thermodynamics of the biopolymer molecule in the condensed state, described above, which predict the stability of the enthalpy value in the range of parameters, corresponding to the physiological normal value,
An increase in the number of links in the polymer chain, equivalent to the introduction of an additional amount of hyaluronic acid disaccharides into the solvent, at a fixed volume of the substance, does not lead to an increase in its concentration in the liquid phase (solution supersaturation)—but leads to an increase in the number of monomers in the crystalline phase during crystallization (process separation from the solid phase solution). An increase in the volume of a substance due to an increase in the volume of a solvent (water containing glucose and sodium ions) will lead to a process of reverse crystallization—dissolution (transition of a substance from a crystalline to a liquid phase). The dissolution process, as well as melting, leads to the destruction of the crystal mesh, which is usually accompanied by cooling of the solution. Dissolution can occur under isothermal conditions, but in this case, heat must be supplied from outside.
A clinical study on a group of patients with diabetes mellitus showed that the value of the enthalpy of the intercellular substance, determined by comparing the non-invasive readings of the device prototype with the invasive readings of a certified glucometer (calibration method), in patients with type 2 diabetes mellitus can have abnormal values—abnormally low or abnormally high; thus, the enthalpy of the intercellular substance of the living tissue of patients with diabetes mellitus is characterized by 2 values of enthalpy corresponding to 2 different states of the intercellular substance with different enthalpy values, which in some patients may differ by more than an order of magnitude.
Study data allowed to conduct in-depth analysis with a conclusion that the abnormally high value of the enthalpy of the intercellular substance observed in patients with diabetes mellitus is associated with an increased content (excess) of hyaluronic acid in the intercellular substance of the epidermis of the diabetic subjects. It is well-known that in type 2 diabetes mellitus, the content of hyaluronic acid (HA) in the intercellular substance increases, as a result of an increase in the rate of its synthesis in hyperglycemia (elevated blood glucose level) leading to the production of an excess amount of hyaluronic acid in the crystalline (globular) phase of the intercellular substance of the epidermis, the upper layer of which is characterized by a low solvent content in the intercellular space.
A change in the amount of solvent in the intercellular space of the tissue under the applicator, in the process of transition to a state of local thermodynamic equilibrium, can lead to a phase transition of a substance from a crystalline (globular) to a liquid (molten globule) state with an increase in the intercellular volume as a result of the dissolution of the crystalline (globular) phase, which is in a metastable state; the process of dissolution due to the supply of heat into the volume of tissue under the applicator through the interstitial fluid flow. The disruption of the thermodynamic phase equilibrium of the intercellular substance can also be due to the abrupt changes in the concentration of glucose and mineral ions in blond.
A clinical study in subjects with type 2 diabetes mellitus showed a correlation between the enthalpy value of the intercellular substance and tissue insulin resistance: in type 2 diabetes with high insulin resistance, abnormally low tissue enthalpy values were observed, corresponding to the predominance of the crystalline phase in the intercellular substance; an increase in the proportion of the crystalline component in the substance leads to an increase in the resistance of the intercellular space, due to an increase in the transfer coefficients of the substance for glucose and insulin transfer from the capillary to the cell and an increase in the proportion of bound glucose. In the physical state (molten globule) with an abnormally high level of osmotic pressure, the proportion of free glucose in the local volume of the intercellular substance can increase due to a decrease in the proportion of bound glucose in the crystalline phase without increasing the concentration of insulin and; as a result, it can lead to an increase in the concentration of glucose in the intercellular fluid.
The value of the enthalpy of the intercellular substance, therefore, is an individual characteristic of the patient's muscle tissue, which has a high diagnostic value. Continuous measurement of the intercellular substance enthalpy by monitoring the content of hyaluronic acid and/or monitoring the osmotic pressure during insulin (or other sugar-lowering) therapy can be used in practice for the early diagnosis of diabetes, as well as for real-time diagnosis of insulin resistance in living tissue of patients with diabetes mellitus (and monitoring the effectiveness of sugar-lowering treatment) based on the monitoring of the content of hyaluronic acid in the epidermis.
It is well-known that in insulin resistance, the sensitivity of muscle cells to insulin decreases. As a result, the ability of glucose to penetrate into muscle cells is impaired. Glucose is accumulated in blood plasma. As a result, the pancreas produces even more insulin, and its level in blood increases above normal threshold. Finally, glucose is deposited in the form of fat, because fat cells are highly susceptible to insulin, and glucose needs to go somewhere from blood. A vicious circle is formed.
Due to an increased rate of synthesis of hyaluronic acid an abnormal content (excess) of hyaluronic acid in the intercellular substance in diabetic patients leads to crystallization of the intercellular substance, which is accompanied by an increased binding of glucose in the crystalline phase and an increase in the transfer coefficients of the intercellular substance due to the need for glucose and insulin transfer from the capillary into the cell; therefore it may be one of the causes of development of type 2 diabetes mellitus and other chronic cardiovascular diseases associated with diabetes. The phase transition of the intercellular substance into an aggregate state with an abnormally high osmotic pressure also leads to an increase in insulin resistance and is accompanied by an elevation of blood pressure due to compression (collapse) of the capillaries as a result of an increase in the osmotic pressure of the substance surrounding the capillary.
Thus, the method of isothermal calorimetry enables early detection and prediction of signs of insulin resistance and diabetes in subjects not diagnosed with type 2 diabetes, based on the nature of the temporal changes of the osmotic pressure of the intercellular substance. A method of isothermal calorimetric spectroscopy described above also allows to detect diabetes mellitus at an early stage on the basis of spectral measurements of the content of hyaluronic acid in the local volume of living tissue.
Jumps in the osmotic pressure of the intercellular substance, which are observed in patients with diabetes mellitus, can lead to jumps in blood pressure. In particular, jumps in the osmotic pressure of the extracellular substance in heart and brain can be a possible cause of heart attacks and strokes. Osmotic pressure jumps in the pancreas can cause a non-homogeneous concentration of insulin in blood, as a result of the stepwise changes in the rate of insulin production in the pancreas. Therefore, a conclusion is understood about the nature of the relationship between diabetes mellitus and cardiovascular disease, which is now clinically proven. The method of isothermal calorimetry described above also allows to diagnose cardiovascular diseases caused by diabetes mellitus at an early stage.
The impact of some external factors, physiological and physical, on the tissue can cause reversible transitions of living tissue from one phase state to another. These factors include, in particular, muscle load and temperature effects. External effects, which cause reversible transitions of the tissue under study, include the effects which lead to reversible changes in a phase state of the intercellular substance. The number of such external physical factors includes the following: an external pressure: a local decompression: an external temperature; electromagnetic radiation causing volume heating of a tissue: a weak direct electric current; a constant magnetic field; a local muscular load to a tissue and others. A method of isothermal microcalorimetry, supplemented by the effect of an external physical factor, allows to determine the value of enthalpy and diagnose diabetes mellitus at an early stage, on the basis of measuring the correlation between osmotic pressure and an external physical factor.
Impaired Glucose Tolerance.
The method of isothermal calorimetry of living tissue described above allows to detect IGT and tissue sensitivity to insulin based on the nature of the changes in the osmotic pressure of the intercellular substance (the amount of water in the intercellular space) with time, without determining the absolute value of enthalpy (without calibration). In a clinical study in patient with type 2 diabetes mellitus, we have found a characteristic stepwise changes in the osmotic pressure of the intercellular substance due to the stepwise changes in glucose concentration in the intercellular substance with tissue insulin resistance: in patients with type 2 diabetes mellitus with high insulin resistance, and abnormally low tissue enthalpy values were observed, corresponding to the predominance of the crystalline phase in the intercellular substance.
A consequence of the stepwise changes in the osmotic pressure of the intercellular substance, which is important for practical medicine, is the step-like nature of blood pressure. Osmotic pressure monitoring method is applicable for early diagnosis of cardiovascular diseases, in particular, hypertension.
The method of isothermal calorimetric spectroscopy described above also allows to detect diabetes mellitus and associated cardiovascular diseases at an early stage on the basis if spectral measurements of the content of hyaluronic acid in a local volume of living tissue.
A Diagnostic Method for Tracking Nutrition by Assessing Calorie Intake.
The epidemiological studies carried out in different countries indicate to a close relation between IGT and obesity. For example, the study carried out in the USA, has found that a mean EBW (excessive body weight) in persons, who consequently developed IGT, was significantly higher than in individuals with a normal EBW. The study carried out in Israel has established that a history of a high EBW was accompanied by a raised frequency of IGT development over a 10-year period.
The method of isothermal calorimetric spectroscopy described above allows to determine the energy value of food intake based on spectral measurements of the glucose and free fatty acid triglycerides content in the local volume of living tissue. Energy value or calorie intake is the amount of energy released in the human body from food during digestion after its full absorption. The energy value of a product is measured in kilocalories (kcal) or kiloJoules (kJ) per 100 g of product. Kilocalories, which are used to measure the energy content of food, is also called the “food calorie.” so the prefix “kilo” is often omitted when referring the energy value in kilocalories. The nutritional value of a product is the content of carbohydrates, fats and proteins per 100 grams of the product.
Skin Condition Diagnostic Method.
The method of isothermal calorimetry and spectroscopy described above allows to determine the physiological state of the skin based on spectral measurements of the content of hyaluronic acid and free fatty acid triglycerides (and other biochemical
components) in a local living tissue unit volume.
Examples of Practical Use. Clinical Study Data.
The appearance of the prototype of the isothermal calorimeter of living tissue, which was used in clinical study to prove the method (proof of concept study) is shown in
Assessment of the effectiveness of the method was performed by two different methods described in the clinical trial protocol. A group of 41 patients with diabetes mellitus (types 1 and 2) was divided into two subgroups of 27 patients (subgroup 1) and 14 patients (subgroup 2), each including patients with type 1 and 2 diabetes mellitus. Random sample of patients.
The purpose of the study, involving 27 patients from the first subgroup, was to confirm the effectiveness of the method, to study the reproducibility of test results and to assess the possibility of individual calibration. Each of the patients underwent 5-6 tests with 35-minute duration of each test (with three invasive measurements), with an average of 3 tests per day. In total 144 tests were performed on 27 patients with type 1 and 2 diabetes mellitus, with a total of 434 invasive measurements.
The indicator of the effectiveness in the first assessment method is the correlation, with 15% bias interval, of the results of non-invasive measurements with the results of invasive measurements, at fixed values of the individual scale coefficient of the given patient (individual calibration).
The study showed that each patient with diabetes mellitus is characterized by two values of the individual scale coefficient K01 and K02, which have a constant individual value for this patient; however, the scale coefficient varies over a wide range of values between patients from K01=0.7 (for patient No 1) to K01=480 (for patient No 25). Test results in subgroup 1 are shown in
Test results in subgroup 1 (27 patients) are shown in
In accordance with the international standard ISO 15197 (section 6.3.3. Minimum system accuracy performance criteria; Data analysis of clinical study according to Clause 6.3.3 of ISO 15197:2013 [p 12, pdf]) following criteria for assessing the accuracy of invasive glucometers have been adopted:
95% of the measurement results must fall within area A, the boundaries of which are determined based on the following conditions:
±15% for values of glycemia equal to or greater than 5.55 mmol/liter;
±0.83 mmol/liter for values less than 5.55 mmol/liter;
The purpose of the study, involving 14 patients from the second subgroup, was to evaluate the effectiveness of the blind measurement method. Each patient underwent 6 tests with 35-minute duration (with three invasive measurements) within two days, of which the results of the first 3 tests (1,2,3) were used for calibration. For each subject, an individual calibration was carried out, during which the values of the individual scale factors of this patient K01 and K02 were determined based on the results of the first 3 tests (each lasting 35 minutes).
The next 3 tests (4, 5, 6) on the next day were performed and processed using blinded method. In total, 83 tests were performed on 14 patients with type 1 and type 2 diabetes mellitus, with a total of 249 invasive measurements. Test results are shown in
Therefore, the results of comparative measurements on patients from subgroup 2 using semi-blind and blind methods, are in accordance with the requirements of the ISO 15197 standard for accuracy. Test results in a group of 41 patients (227 tests, 683 invasive measurements), obtained by pooling the results in subgroups 1 and 2, are shown in
The pooled test results: 42 patients with type 1 and 2 diabetes mellitus: 227 tests with a total of 683 invasive measurements. Red curves indicate zone A, brown curves indicate zone B (±25%).
Only 654 measurements out of 664 or 98.3% of the results fall into the A+B zone.
Therefore, the results of a clinical study in a group of 41 patients with type 1 and 2 diabetes mellitus are in accordance with the requirements of the ISO 15197 standard for accuracy.
The Protein Folding Problem.
Protein folding is the process of spatial packaging of a protein molecule, transition of a protein into a strictly defined spatial form in which it realizes its biological functions; the process of self-organization of a protein molecule—spontaneous folding of a polymer chain under specific conditions into a certain spatial structure, is shown schematically in
The ability of a natural polypeptide chain to spatially self-organize and acquire a specific molecular structure, which is the most striking feature of proteins. The process of self-organization of proteins is still largely unclear and represents one of the biggest problems of modern science. It is known that the functions of proteins can mostly be predicted based on their structure, although not always. Therefore, an urgent scientific problem is the prediction of the structure and, as a consequence, the function of a protein, according to the sequence of amino acids in the polymer chain.
Protein molecules are linear polymers consisting of amino acid residues (monomers): also, they may include modified amino acid residues and components of a non-amino acid nature, for example, glycoproteins characterized by a low carbohydrate content, a carbohydrate residue (oligosaccharide with an irregular structure and containing glucose, mannose, galactose and their amino derivatives, as well as N-acetylneuraminic acid) is attached to the amino acids of the protein chain either via N-glycosidic bond to the amide nitrogen of asparagine, or via O-glycosidic bond to the hydroxyl group of serine, threonine, hydroxylysine residues. When a protein is formed, as a result of the interaction of the α-carboxyl group (—COOH) of one amino acid with the α-amino group (—NH2) of another amino acid, peptide bonds are established. The ends of the protein molecule are called the N- and C-terminus, depending on which of the groups of the terminal amino acid residue is free: —NH2 or —COOH, respectively.
It is well known that the glucose molecule can also form a glycosidic valence bond with amino acids of protein molecules. In glycoproteins characterized by a low carbohydrate content, a carbohydrate residue (an oligosaccharide with irregular structure and containing glucose, mannose, galactose and their amino derivatives, as well as N-acetylneuraminic acid) is linked to the amino acids of the protein chain or to the amide nitrogen of asparagine via N-glycosidic bond, or to the hydroxyl group of serine, threonine, hydroxylysine residues via O-glycosidic bond. Structural formulae of glycoside bonds with amino acids are shown in
As it can be seen, the configuration of the tetramer in the biopolymer molecule of hyaluronic acid is formed with the participation of sodium ions, the concentration of which in the intercellular space is much higher than that of potassium ions.
On the other hand, the ionic composition of cells, as long as they exhibit the properties of a living being, differs from the ionic composition of the environment. The most significant difference is the asymmetric distribution of univalent Na+ and K+ ions: cells actively accumulate K and release Na into the environment. This leads to a difference in concentration of
monovalent cations on the cell membrane.
This explains the conclusion that the monovalent metal ion M, which forms an ionic bond with the charged unit of the polymer chain —Rb-M of the protein molecules located in the plasmatic membrane and cytoplasm of the cell, is the monovalent ion K+: Rb-M→—Rb—K.
The universal model of a biopolymer molecule, on the basis of which the properties of a hyaluronic acid macromolecule are described, which, as noted, is a unique model object for assessment the properties of biopolymer molecules, allows to predict the thermodynamic properties of a random natural macromolecule, in particular, macromolecules of proteins and enzymes based on an accurate statistical description.
In general, the structure of a biopolymer protein molecule with a feature of spatial self-organization and the acquisition of a stable molecular heterophase structure due to weak volumetric interactions of a non-valence nature can be described by the following definition: a linear polypeptide chain of amino acids sequentially connected by an amide (peptide) bond —NH—CH(R)—CO—, the repeating monomer of which consists of two (or more) types of monomer units—ARa and ARb amino acids, with different side residues Ra, Rb, one of which (for example, the ARb unit) contains a charged (negatively or positively) Rb group, and the side residue Ra, of the second unit ARa is uncharged. The water-based solvent includes at least 3 components: one component is a (monomeric) molecule of the substrate (solvent) S, for example, glucose molecule, and two (or more) low molecular weight monovalent ions M1 and M2, for example sodium and potassium ions with different affinities to the monomer, the electric charge of which is inverse to the charge of the side residue ARb, the monomeric solvent (substrate) molecule S can form two hydrogen bonds with the monomeric units of the polymer chain —Ra—S—Rb—; —Ra—S—Rb—; —Ra—S—; —S—Rb; low molecular weight monovalent ions M1 and M2 can form an ionic bond with charged units of the polymer chain —Rb-M1. The energy of the longitudinal covalent bonds E0 of the adjacent links of the polymer chain is much higher than the characteristic thermal energy at physiological temperatures E0/kBT>>1. The energy of transverse hydrogen bonds of the Ra and Rb links with the monomer (substrate) molecule S of the solvent is much less than the energy of the longitudinal valence bonds ES<<E0 of the polymer chain. The low molecular weight ion M1 can be adsorbed by the polymer chain monomer Rb, with the formation of a weak ionic bond —Rb-M1.
In general, the repeating fragment of the polymer chain can consist of the same pairs -A-B— or several types of repeating pairs of monomeric links -A-B—, —C—B—, . . . that can form hydrogen bonds with substrate molecules.
Hydrogen bonds determining the most probable configuration of a bound pair of elementary chains, corresponding to the minimum free energy of volumetric interactions, and as a result, the most probable spatial configuration of a tetramer formed by a pair of non-adjacent polymer chain monomers and a pair of solvent molecules with the participation of ions can be defined as conjugated non-valence bonds or conjugated hydrogen bonds of a biopolymer molecule. An elementary cell corresponding to a bound pair of elementary chains (in its most probable configuration) forms a conjugated system—a closed system of conjugated hydrogen bonds with a delocalized electron density; It is well-known, a common feature of all conjugated systems is the spreading of the electron density of hydrogen bonds over the entire conjugated system.
The structural formulas of amino acids that form polypeptide chains of protein molecules are shown in the table in
It is known that magnesium functions as a cofactor in over 300 known enzymatic reactions involved in a wide range of metabolic processes. Energy production, glucose metabolism, fatty acid oxidation and amino acid activation require magnesium. Magnesium is one of the most important minerals; being the main intracellular element, it activates enzymes that regulate carbohydrate metabolism, stimulates the formation of proteins, regulates the storage and release of energy in ATP, reduces excitation in nerve cells, and relaxes the heart muscle.
It was shown that a tetramer in the volume of a macromolecule, the polymer chain of which contains negatively charged groups, can be formed only with the participation of a monovalent ion (Na, K). A bivalent metal ion (Mg2+, Mn2+) cannot participate in the formation of a tetramer—a crosslink of 2 non-adjacent polymer chain monomers, 2 substrate molecules and 2 monovalent solvent cations, however, it can form a cross-link between two non-adjacent polymer chain monomers without participation of solvent molecules.
Common classification. In accordance with the common classification, 4 levels of the structural organization of proteins can be distinguished: primary, secondary, tertiary and quaternary protein structures. The primary structure (sequence of amino acid residues) of a polypeptide is determined by the structure of its gene or genetic code, while higher-order structures are formed during protein folding. Although the spatial structure of a protein is generally determined by its amino acid sequence, it is quite labile and can depend on external conditions, so it is more correct to speak of the preferred or most energetically favorable protein conformation.
The primary structure is the sequence of amino acid residues in a polypeptide chain, which is usually described using one-letter or three-letter designations for amino acid residues. Important features of the primary structure are conservative motifs—stable combinations of amino acid residues with a specific function that are found in many proteins. Conservative motifs are preserved in the course of species evolution; they often allow to predict the function of an unknown protein.
Secondary structure—local arrangement of a fragment of a polypeptide chain, stabilized by hydrogen bonds. The following are some of the most common types of the secondary structure of protein: alpha helix (α-helix)—dense coils around the long axis of the molecule. One turn is 3.6 amino acid residues, the helix pitch is 0.54 nm (0.15 nm per amino acid residue). The helix is stabilized by hydrogen bonds between H and O peptide groups separated by 4 links; β-sheets (folded layers)—several zigzag polypeptide chains in which hydrogen bonds are formed between amino acids relatively distant from each other (0.34 nm per amino acid residue) in the primary structure or different protein chains (not adjacent, as in the α-helix).
Tertiary structure, the spatial structure of the polypeptide chain, is formed when the polypeptide is folded into a compact three-dimensional system (in the case of enzymes, this is usually a spherical globule). Structurally, it consists of secondary structural elements stabilized by various types of interactions, in which hydrophobic interactions play an important role.
As shown above, volumetric interactions of a simple polymer chain with a solvent can be described as a pairwise volumetric interaction of elementary chains forming a bonded pair, the most probable configuration of which is shown in
Investigations of the principles of protein folding have shown that it is convenient to identify and additional level between the secondary structure and the atomic spatial structure—the fold motif (architecture, structural motif). The folding motif is determined by the mutual arrangement of the elements of secondary structure (α-helixes and β-strands) within the protein domain, a compact globule that can either exist independently or be part of a larger protein along with other domains.
Quaternary structure (or subunit, domain)—the mutual arrangement of several polypeptide chains as part of a single protein complex. Protein molecules that make up a protein with a quaternary structure are formed separately on ribosomes and only after the synthesis is finished, they form a common supramolecular structure. A protein with a quaternary structure can contain both identical and different polypeptide chains. The same types of interactions participate in the stabilization of the quaternary structure as in the stabilization of the tertiary one. Supramolecular protein complexes may consist of dozens of molecules.
Protein domain, an element of the protein's tertiary structure, which is a fairly stable and independent substructure of the protein, the folding of which occurs independently of the other parts; the tertiary structure of a protein molecule is formed by folding a polypeptide into a compact three-dimensional system (in enzymes it is usually a spherical globule).
As a rule, the domain includes several elements of the secondary structure. Domains similar in structure are found not only in related proteins (for example, in the hemoglobins of different animals), but also in completely different proteins. Rather often, separate names are assigned to domains, since their presence directly affects the biological functions of the protein—for example, Ca2+— binding domain of caldomulin, homeodomain, responsible for binding to DNA in various transcription factors, etc.
In oligomeric proteins, each of the subunit-forming polypeptide chains is characterized by its own secondary and tertiary structure. However, these proteins have another conformational level called a quaternary structure; this term describes the position of the polypeptide chains that form individual polypeptide chains relative to each other, i.e. a method for their joint arrangement and packaging with the formation of a native conformation of an oligomeric protein.
Since the domains are sufficiently “autonomous” in forming their structure and performing their function, with the help of genetic engineering it is possible to fuse a domain from another protein to the primary one (thus creating a chimeric protein). If you are lucky, such chimaera will combine the functions of both proteins. For example, by fusing the Cas9 DNA-binding domain with various regulatory domains, it was possible to obtain artificial transcription factors (crisprTF) selectively directed to the desired regions of the genome using tailor-made “RNA guides”. Artificial restriction endonucleases, repressors, and epigenome-modifying enzymes such as DNA methylases and demethylases can also be constructed using Cas9.
The Method of Calculation of a Spatial Structure of the Protein.
The most probable linear configuration of the elementary chain differs from the primary structure configuration of the polymer chain and is formed as a result of the interaction of the polymer chain with the solvent in the equilibrium configuration of the macromolecule corresponding to the minimum free energy of spatial interactions. On the other hand, the volume of a substance in a condensed state, under the specified environmental conditions (pressure, temperature, and other characteristics), is determined only by the forces of volumetric interactions, so the three-dimensional spatial configuration of a biopolymer molecule can be calculated on the basis of the configuration of a bound pair of elementary chains that characterizes the equilibrium configuration of an elementary volumetric interactions.
In case of a biopolymer molecule with a simple polymer chain without subunits (protomers), the folding of a protein molecule—the process of formation of a secondary structure, with spatial packing into a compact globule, occurs spontaneously, as a phase transition of the statistical system of many interacting particles into a spatial configuration with a heterophase structure with minimum free energy of volumetric interactions. The intramolecular heterophasic structure and domain volume of a simple-chain protein molecule with a single subunit can be calculated based on the configuration of a linked pair of elementary interaction chains.
In general, the chain of a biopolymer molecule (for example, an oligomeric protein) may have a more complex primary structure, consisting of several interconnected simple chains (subunits, protomers). Oligomeric proteins—proteins containing 2 or more polypeptide chains (up to several thousand subunits). In oligomeric proteins, each of the subunit-forming polypeptide chains is characterized by its secondary and tertiary spatial structure. Examples of biopolymer molecules consisting of several subunits are shown in
Polypeptide chains in oligomeric proteins can be either identical or different. The number of polypeptide chains in an oligomeric protein can be determined by the number of amino-terminal residues per protein molecule. An oligomeric protein consisting of four polypeptide chains, for example hemoglobin, must have four terminal residues, one for each chain. The insulin molecule has two chains linked by covalent cross-links.
Some proteins (complex proteins), in addition to the polypeptide part, also contain a non-protein—prosthetic group, which can be associated with the protein part both with a covalent or weak non-covalent bond.
Interaction Between Macromolecule Subunits.
Based on the current experience, the three-dimensional spatial structure of complex proteins consisting of several subunits can change in the process of interaction of polymer chains belonging to different subunits with solvent molecules.
Volumetric interaction between adjacent domains arises as a result of volumetric interactions between monomers (belonging to chain sections) from different subunits: a substrate molecule forms a hydrogen bond with two monomers that belong to different subunits and have different electric charge. A change in the three-dimensional spatial structure of the hexokinase protein molecule, which consists of two domains, occurs as a result of a phase transition of a substance in a local area between adjacent domains (corresponding to subunits) into a heterophase condensed state. The statistical mechanism of the transition of a substance in the region between adjacent domains to a heterophase state is described in the section Physics of the Condensed State of a Polymer Molecule above.
Interaction Between Biomolecules.
In general, based on the experience, biopolymer molecules can interact not only with small molecules, but can form complexes with biomolecules (most often with a protein, for example, a cellular receptor, but sometimes, for example, with DNA) and exert some biochemical, physiological or pharmacological effects due to such binding. In biochemistry and pharmacology, a chemical compound (which is often, but not always, a small molecule) that forms a complex with a specific biomolecule is called a ligand. In case of binding a ligand to a protein, the ligand is typically a small signaling molecule that binds to a specific binding site on the target protein (eg., receptor). In case of ligand binding to DNA, the ligand is usually also a small molecule or ion or protein that is bound to a double DNA helix. Binding or association of a ligand to a receptor (the so-called “docking” of a ligand into a specific “niche” on the receptor) is usually reversible and short-term. The reverse process is called the dissociation of the ligand from the bond with the receptor.
In 80-90s, studies of biopolymer complexes resulting from the association of several macromolecules linked by non-valence bonds led to the development of a new interdisciplinary field of knowledge, called “supramolecular chemistry”—this is the chemistry of intermolecular bonds that studies the association of two or more chemical molecules, as well as the structure of such associates. It lies outside the boundaries of classical chemistry, which studies the structure, properties and transformations of individual molecules. If the latter deals mainly with reactions where cleavage and formation of valence bonds occur, then the objects of study of supramolecular chemistry are non-valence interactions—hydrogen bonding, electrostatic interactions, hydrophobic forces, “non-bound” structures.
The simplest example of supramolecular structures is host-guest complexes. The host (receptor) is usually a large organic molecule with a cavity in the center, and the guest is a simpler molecule or ion. For example, cyclic polyesters of various sizes (crown ethers) bind alkali metal ions quite strongly.
Supramolecular structures are characterized by the following properties: 1, the presence of not one, but several binding centers in the host (in crown ethers, this role is played by oxygen atoms with non-divided electron pairs), 2. complementarity: the geometric structures and electronic properties of the host and guest complement each other (in crown ethers, this manifests in the fact that the cavity diameter must correspond to the radius of the ion). Complementarity allows the host to selectively bind guests of a strictly defined structure. In supramolecular chemistry, this phenomenon is called “molecular recognition”. 3. Complexes with a large number of bonds between complementary host and guest have a high structural organization.
An example of a supramolecular biochemical structure are host enzyme molecules. The active center of each enzyme is designed in such a way that it may host only the substance (substrate) that corresponds to it in size and energy; the enzyme will not react with other substrates. Another example of supramolecular biochemical structures are DNA molecules, in which two polynucleotide chains are complementarily linked to each other via many hydrogen bonds. Each chain is both a guest and a host for the another one.
The abovedescribed physics of the heterophase condensed state of biopolymer molecules enables interaction of molecules in supramolecular complexes. The interaction between macromolecules with the formation of supramolecular structures is based on the mechanism similar to that of interaction between subunits in an oligomeric protein, as a result of the formation of hydrogen bonds between the ligand and protein molecules and ionic bonds between the solvent ions and the charged regions of the polymer chain monomers.
A Method to Determine the Spatial Structure of a Complex Macromolecule.
Three-dimensional domain spatial structure of a protein is calculated using the abovedescribed method and consists of the following steps:
Heterophase Biological Catalysis.
A well-known effective method to accelerate the chemical reaction of inorganic molecules, which is widely used in practice, is heterophase catalysis—a change in the chemical reaction rate under the influence of catalysts that form an independent phase and are separated from the reacting substances by an interface. For example, the most common case is when a solid catalyst (contact) increases the rate of reaction between gaseous reactants or a reaction in solution. The catalytic reaction usually proceeds on the surface of a solid catalyst and is due to the activation of reagent molecules upon interaction with the surface. Therefore, for the implementation of heterophase catalysis, adsorption of the components of the reaction mixture from the volumetric phase on the catalyst surface is necessary.
Chemical catalysts that accelerate vital biochemical reactions (about ten thousand) in the cell are enzymes—protein molecules, the polymer (polypeptide) chain of which consists of amino acids.
The main features of enzymatic catalysis, which distinguish it from laboratory and industrial catalysis, are strict selectivity and a high rate of a chemical react ion.
Until recent time, it was believed that proteins are essential components of all enzymes. However, enzymes composed from molecules of ribonucleic acids (ribozymes) were discovered and became the object of intensive research. Interest in this group of enzymes has sharply increased with the development of methods for molecular selection of nucleic acids, which allows, in particular, to begin the targeted construction of ribozymes with various types of catalytic activity.
According to modern concepts (IUPAC), the specificity and catalytic activity of an enzyme molecule is determined by the active center—a special section of the molecule with a unique combination of amino acid residues (and possibly non-protein groups) that ensure direct binding of the enzyme molecule to the substrate molecule and direct participation in catalysis. The process of binding the substrate molecule by the active center of the enzyme is accompanied by a change in the spatial structure of the enzyme molecule. The active site of an enzyme, as a rule, has the form of a narrow depression or gap between adjacent domains. Diagram explaining the effect of the change in the conformation of the hexokinase enzyme molecule during binding of the glucose molecule is shown in
Most often, the active centers of enzymes include functional groups of such amino acids:
OH—groups of serine, threonine, tyrosine;
SH—groups of cysteine;
NH—group of hystidine;
COOH—group of glutamate and aspartate:
NH2—groups of arginine and lysine.
Physical nature of the biological catalysis phenomenon is currently unknown and is one of the unsolved problems of modern science.
The physics of the heterophase condensed state of a biopolymer molecule, the main provisions of which are described above, allows to explain and describe, on the basis of an accurate statistical method, the main properties of biological catalysis—a high rate of a chemical reaction and strict selectivity.
Biopolymer molecule with a heterophase structure, which has selectivity to the G molecule of the substrate, can catalyze a biochemical reaction involving this molecule. Acceleration of a biochemical reaction occurs due to the approximation of molecules in the crystalline phase of a biopolymer molecule with a heterophase structure. In the unit cell volume of a biopolymer molecule with a heterophase structure, ⅓ of the total number of substrate molecules is in a free state in the solvent, and the remaining ⅔ of the substrate molecules are bound with the monomers of the enzyme polymer chain. The concentrations of all components of the system in the condensed crystalline (globular) phase increase exponentially by a factor of the order parameter, an increase in which leads to a multiple increase in the number of effective collisions due to the proximity effect.
Heterophasic Biological Catalysis Rate.
Let us describe the physico-chemical mechanism of biological catalysis, using the example of a hexokinase enzyme and its isomer glucokinase, which play an important role in the biochemical regulation of glucose metabolism in a living tissue cell.
The enzymatic reaction of the first stage of glycolysis, catalyzed by both of these enzymes (hexokinase and glucokinase), has the following form:
ATΦ+-D-glucose→AΦ+-D-glucose-6-phosphate,ΔG=-4.0 kcal. (50)
The rate of a chemical reaction, in general, is determined by the law of mass action (Guldberg and Waage)—the rate of the chemical reaction A+B→C with the formation of a new substance C from two substances A and B, is proportional to the product of the concentrations of the reacting substances (under otherwise identical conditions):
W=−k×[A][B], (51)
where k is the rate constant, which does not depend on the concentrations of A and B.
The kinetic scheme of reaction (50), during which the phosphate group P is transferred from an ATP molecule to a glucose molecule G (which cannot proceed spontaneously without a catalyst), can be represented as the following enzymatic reaction:
G+E+P→PGE→PG+E, (52)
where E is a polymer chain monomer that can form a weak ionic bond with the monovalent ion M of the solvent or a hydrogen bond with the substrate molecule G of the solvent.
Enzymatic reaction (51), within the framework of the mechanism under consideration, proceeds in several stages in the active center of the enzyme molecule—a local region of the intramolecular space between the domains corresponding to the subunits of the macromolecule.
At the first stage, schematically shown in
At the second stage (
E+G→EG;
2E+2G=EG+EG→E2G2. (53a)
A tetramer complex can be defined as a substrate—an enzyme complex, which for the reaction under consideration can be denoted by the symbol E2G2, where G is a glucose molecule.
This stage is reversible and is not accompanied by any chemical changes in the substrate.
The transition of the phosphate group P from the ATP molecule to the glucose molecule G, which is in a bound state in the E2G2 tetramer complex in the crystalline (globular) phase (
P+GE→PGE. (53b)
At the third stage, schematically shown in
PGE→PG+E.
Within the framework of the model of a biopolymer molecule with a heterophase structure described above, weak non-covalent interactions, i.e., ionic and hydrogen bonds, are involved in the formation of the GE complex, which is consistent with generally accepted concepts, which assume that the formation of an enzyme-substrate complex (ESC. Michaelis complex) occurs with participation of non-covalent interactions.
The rate W of the enzymatic reaction (50) can be calculated by the formula (51) if the rate constant is known; the rate constant k can be calculated, as will be shown below, as the ratio of two concentration-independent constants using the expression: k=kb/K0, where kb is the rate of passage of the product PG through the energy barrier separating crystalline and liquid phases; K0 is the equilibrium constant of the process of formation of a tetramer from 2 dimers; in this case. 1/K0 is the dissociation constant of the tetramer with formation of 2 dimers. Here, it is taken into account that at the moment of reaching the peak of the energy barrier, on which R2 configuration of the tetramer is realized, the mutual attraction between tetramer dimers is determined only by the contribution of internal entropy, the value of which determines the value of the dissociation constant.
The formula for calculation of the rate constant k can be derived based on the following considerations.
Reagents G (glucose molecule) and E (polymer chain monomer) are in equilibrium with EG dimers and E2G2 tetramers. The dissociation constant KG of the tetramer with the formation of 2 dimers is determined by the following equation:
K
G=1/K0=[E2G2]/[GE][GE].
GP and E reagents are in equilibrium with the activated EGP complex (tetramer-phosphate group):
E+GP→EGP→E+GP. (53d)
The equilibrium constant KGP of the reaction (53b) is determined by the following equation:
K
GP=[EGP]/[GP][E].
The equilibrium constant of the reaction G+P→GP, which cannot proceed spontaneously, is equal to the equilibrium constant of the enzymatic reaction (53 d):
[GP]/[G][P]=[EGP]/[GP][E]=[E2G2]/[GE][GE]=KGP.
Therefore, the equation (50) for the reaction rate can be solved with respect to [GP]:
W=[GP]×[barrier penetration rate]=[GP]×kb=kb×KGP×[G][P] or
W=k
b
×K
GP×[G][P]=(kb/K0)×[G][P]=k×[G][P] (54)
Reaction rate (50) is:
k=k
b
/K
0. (54a)
The equilibrium constant K0 is determined using expression (8):
K
0≡(1/VQ)×exp(Eb/kBT)=(1/VQ)×exp(StetrT/kBT)=(1/VQ)×exp(Stetr/kB),
Where Eb is the binding energy of a glucose molecule with a tetramer, which is determined by the internal entropy of the tetramer Stetr:
VQ—is the quantum volume of a glucose molecule, which is determined using the following expression [14]:
V
Q=(h2/2 πM kBT)3/2=5,32×10−21/(MT)3/2 cm3=3,646×10−24 cm3,
where M is a molecular weight of glucose (substrate); h is Planck's constant.
The rate kb of passage of the GP molecule through the energy barrier separating the isomeric conformations of the tetramer in the liquid and crystalline phases is determined by two factors—the height of the energy barrier Ea the value of which determines the probability of the passage of the molecule through the barrier, and the constant v is a constant having a frequency dimension corresponding to that of an electromagnetic quantum with an energy of kBT:
k
b
=v×w
b
=v×exp(−Eo/kBT),
where v=kBT/h; h—Planck's constant.
A hydrogen bond, as a type of covalent bond with a delocalized electron density, is electromagnetic in nature. The energy of a hydrogen bond is determined by the energy of a pairwise interaction of a glucose molecule with a polymer chain monomer, which is quantized by a discrete portion of energy, proportional to the thermal energy quantum kBT. The oscillation frequency v of the valence bond, in general, is determined by the frequency of the electromagnetic quantum, which can be calculated based on the requirement that in the equilibrium state the quantum of the electromagnetic energy of the valence bond hv is equal to a multiple of the number of thermal energy quanta n×kBT, in relation to which equilibrium is established between the electron pair (oscillator) and thermostat:
nhv=nkBT: v=kBT/h=0.75×1013/c, where n=1, 2, 3, . . . —is the proportion of the exchange quantum of energy of electromagnetic interaction in relation to the quantum of thermal energy kBT.
The expression for the rate constant of a chemical reaction is as follows:
k=k
b
/K
0=[VQ×exp(Stetr/kB)]×(kBT/h)×exp(−Ea/kBT)=VQ×(kBT/h)×exp[−(Ea−StetrT)/kBT] (54a)
Thus, the expression (54) for calculation of the reaction rate (50) has the following form:
W=(kb/K0)×[G][P]={VQ×(kBT/h)×exp[−(Ea−StetrT)/kBT]}×[G][P]. (54d)
The formula (54d) for calculation of the reaction rate between reagents [G] and [P] is universal, applicable for calculating the rate of an arbitrary biochemical reaction, the catalyst of which is a biopolymer molecule with a heterophase structure with selectivity to the substrate molecule G.
It should be noted that formula (54 a) can be reduced to a form that coincides with the Arrhenius formula:
k=V
Q×[exp(Stetr/kB)]×(kBT/h)×exp(−Ea/kBT)=Aexp(−Ea/kBT),
where A=VQ×[exp(Stetr/kB)]×(kBT/h).
Glucokinase Enzyme Calculation.
The thermodynamic functions of an arbitrary enzyme, in particular, hexokinase isomers, can be obtained using the universal distribution functions (14), based on the known values of the Michaelis equilibrium constant for these enzymes. Glucokinase [EC 2.7.1.2] is an isotype IV of the hexokinase enzyme (a monomeric protein of 465 amino acids with a molecular weight of 50 kDa), mainly contained in hepatocytes, as well as in pancreatic cells. It catalyzes the phosphorylation only of D-glucose, by transferring the phosphate group to glucose due to the ATP.
Glucokinase is characterized by an equilibrium constant of Km≈10 mmol/L [31] and causes rapid uptake of glucose from blood in the liver during the absorption period. In this case, glucose enters hepatocytes via GLUT2 type transporters, activity of which does not depend on the level of insulin, but which affinity to glucose is low (Km≈10 mmol/L). Insulin affects the absorption of glucose by the liver only indirectly—it induces the synthesis of glucokinase in hepatocytes. Thus, glucokinase is designed to rapidly convert excess blood glucose after a meal into glycogen.
Saturation curves for the glucokinase enzyme can be obtained from the universal distribution function (14), in which the quantum number for calculating the interaction constant γ is determined from the condition of quantization of the free energy of interaction (31), which in general can be represented in the following form:
F
0gas(α=0)/nkBT=F0gas(α=0)/hv=σ0gas=(γgas)2, (52)
where hv is quantum energy of electromagnetic interaction, proportional to the quantum of thermal energy kBT; hv=n kB, where n=1, 2, 3, . . . .
The equilibrium constant Km≈10 mmol/L satisfies the value of the quantum of interaction energy 3kBT, corresponding to the value of the quantum number n=3. Glucokinase enzyme saturation curves obtained using the universal distribution function of the biopolymer molecule, are shown in
The height of the energy barrier Eb=Emax−Emin, separating the crystalline and liquid phases in a glucokinase molecule can be determined directly from the free energy plot Uλs attr(α) of the glucokinase tetramer versus concentration shown in
The height of the energy barrier Ea in units of ε0=3kBT, can be derived from the energy barrier plot presented in dimensionless units in
Energy values according to the plot
E
max=0.1035 . . .
E
min=−0,188 . . .
E
0=−0,160 . . .
E
max
−E
min=0,1035+0,188=0,2915.
E
0
−E
min=−0,160+0,188=0,028=kBT.
Difference between binding energies in the direct and reverse reactions is:
E
min
−E
0=0,188−0,160=0,028=3kBT.
The height of the energy barrier separating the crystalline and liquid phases:
E
a=3,47ε0=3,47×(3kBT)=10,41×kBT.
E
a=(Emax−Emin)/(E0−Emin)=(Emax−Emin)/3kBT=(0,188+0,1035)/0,028=10,41.
To calculate the rate of the glucokinase-catalyzed enzymatic reaction (50) using formula (54d), with a known glucokinase energy barrier height Ea=3.47×(3kBT), it is necessary to determine the dissociation constant of the tetramer with the formation of 2 dimers, which can be calculated using following expression:
K
0=[EG][EG]/[E2G2]=(1/VQ)×exp(Eb/kBT).
Here, Eb—is the binding energy that must be expended to detach two EG complexes (glucose molecule G bonded to monomer E via one hydrogen bond) forming a tetramer complex [E2G2] in the R2 configuration with 2 hydrogen bonds: EG+EG=E2G2.
Binding energy Eb is determined by the internal entropy of the tetramer in the R2 configuration with 2 hydrogen bonds, which can be calculated using the following expression:
−Eb=StetrT=4kBT ln g(l=2)=4kBT ln18=4×2,89kBT,
where Stetr=4kB ln g(l=2)—is the internal entropy of the tetramer; g(l=2)—is the number of states realizing the microscopic form (configuration) of the R2 tetramer with 2 hydrogen bonds. The factor 4 in the formula corresponds to the number of hydrogen bonds that form in the R2 tetramer configuration as a result of electron density delocalization.
Therefore, binding energy, calculated per one hydrogen bond is:
E
b/4=kBT in Δg=kBT ln18=2,89kBT.
It should be noted that the thermodynamic equilibrium of the system is established with respect to the energy exchange between the system and the reservoir with a quantum of thermal energy kBT, in contrast to the electromagnetic equilibrium of interacting molecules, which is established with respect to the quantum of electromagnetic energy ε0=3.47×(3kBT).
Binding energy Eb is equal:
E
b=4kBT ln Δg=kBT ln 18=11,56kBT.
The numerical expression for the equilibrium constant of the reaction is as follows:
K
0=(1/VQ)×exp(Stetr/kB)=1/3,646×10−24 cm3×e11.56=10NA/6,02×3,646 cm3×e11,56;
K
0=4,385 mmol/L,
where NA=6,02×1023—Avogadro's number.
Glucokinase Enzymatic Reaction Rate Calculation.
The rate of the enzymatic reaction (50) catalyzed by glucokinase at a known energy barrier height of glucokinase Ea=3.47×(3kBT) and dissociation constant 1/K0, can be calculated using formula (54d).
Barrier height Ea calculated per one hydrogen bond in the units of quantum interaction energy ε0=3kBT, as it was shown, is Ea=Emax−Emin=3.47 ε0.
Activation energy on 4 bonds per energy quantum ε0=3kBT is:
E
max
−E
min=4×3,443ε0=4×3,47×3kBT≈41.6kBT.
The rate of penetration through energy barrier is:
v×w
b
=v×exp(−Ea/kBT)=0,75×1013/e41.6=0,75×1013/1018=0,75×10−5.
Thus, the numeric value of the rate constant is:
k=v×w
b/1,735 K0=0,75×10−5/1,735×4,385c×mmol/L.
Glucose concentration in the space unit increases proportionally to an increase in the interaction constant (order parameter):
n
0m
/n
0g
=γ=e×(e+1)3=139,74 . . . =1,4×102.
A product of concentrations of reagents [G] and [P] is:
[G][P]=5 mmol/L×1×2,08×103 mmol/L;
where [P]=1 mmol/L—ATP concentration, [G]=5 mmol/L—glucose concentration. Here it is taken into account that glucose and ATP concentrations at α=1 are equal to 5 and 1 mmol/L, respectively.
Reaction rate is:
W=−k[G][P]=(v×wb/1,735 K0)×[G][P]=0,6×10−3 (mmol/L)/s.
Factor 1.1735 in the reaction rate formula appears as a result of the change in the glucose molecular mass after addition of the phosphate group. In the course of the phosphorylation reaction, D-glucose (G molecule) attaches the phosphate group PO3H2 with a molecular weight of 80 and turns into glucose-6-phosphate (GP molecule) with a molecular weight of 260=180+80.
The equilibrium constants of reactions (53a) and (53b) are related by the expression:
K
G
/K
GP=(MGP/MG)3/2=(260/180)3/21,736 . . .
This formula takes into account that the binding energy Eb of a glucose molecule with a tetramer, which is determined by the internal entropy of the tetramer Stetr, and does not depend on the mass of the molecule.
Therefore, KGP=KG/1,736=1/1,736 K0.
The estimated value is in agreement with the experimental data [13], at Km=10:
30×10−3/0,6×102×1()/c=50×10−5=0,5×10−3(
)/c(Vmax=1×10−3).
Hexokinase Reaction Rate.
Hexokinase (ATP-dependent D-hexose-6-phosphotransferase) (EC 2.7.1.1)—is a cytoplasmatic enzyme from the transferase class, first enzyme in the glycolysis pathway.
Molecular weight 100000-50000, optimal pH: 7.6-8.2. The first isozyme, which is most typical for the brain, consists of one subunit with an established amino acid composition. The issue of the subunit structure of the remaining isoenzymes have not been solved yet. To manifest catalytic activity, it needs Mg2+ ions. The intrinsic substrate for this enzyme is not ATP4, but MgATP2 complex.
Activity in various organs (in μmol/l g of wet tissue weight per 1 min): from 23 (brain) to 3.4 skeletal muscles.
The active center contains serine and histidine residues. SH groups of the polypeptide chain play an important role in the mechanism of the catalytic action. In the body, hexokinase is activated by vitamins, insulin, and also by binding to certain biological membranes, such as mitochondrial; it is inhibited by glucose-6-phosphate and corticosteroids.
Unlike glucokinase, the Michaelis constant of hexokinase is 0.01 mmol/L. Therefore, hexokinase localized in the cells of most tissues of the human body literally “catches” glucose in plasma, whereas glucokinase catalyzes glucose phosphorylation reaction only at its high concentrations. Thus, glucokinase and hexokinase enable redistribution of glucose in the body: during the absorption of nutrients in the intestine, the concentration of glucose in plasma increases, and glucose is transferred to the liver, where it is exposed to the effect of glucokinase: at the end of digestion, against the background of a decrease in the concentration of glucose, it is transferred to the skeletal muscles, where undergoes the effect of hexokinase.
Glucose metabolism begins with an irreversible reaction with hexokinase (or glucokinase), in which the transfer of the ATP phosphate group to glucose is catalyzed with formation of glucose-6-phosphate.
Hexokinase reaction serves not only as a starting, but also as the main limiting reaction among other glycolysis-related reactions. Substrate specificity of hexokinase (EC 2.7.1.1) is relative—in addition to glucose, fructose, mannose, galactose, glucosamine and some type 2-deoxyglucose non-metabolizing sugars undergo phosphorylation. Hexokinase is located in the plasmatic membrane, cytoplasm, and is partially associated with mitochondria; in the brain, the activity of this fraction can reach 50% of the total activity of the enzyme. Experimental data have shown that potassium ion takes part in the excitation of hexokinase, and also it is involved in the excitation of other enzymes—pyruvate kinase, which accelerates the transport of phosphoric acid from pyruvate to adenosine diphosphate, enzymes involved in the Krebs cycle, including those assisting in the synthesis of adenosine triphosphate in the course of oxidative phosphorylation. It is quite obvious that the correlation between hexokinase activity and potassium ions indicates a direct correlation between potassium level and the level of macroergic phosphates. Thus, the importance of potassium for the abovementioned processes has been established experimentally, but the principles of activation of enzymatic systems with potassium have not yet been fully elucidated.
The Michaelis equilibrium constant for the hexokinase reaction is Khex=0.02 mmol/L: the ratio of equilibrium constants is Kgl/Khex=15/0.02=750.
It can be seen that the barrier height per hydrogen bond is Emax−Emin=3,4436 ε0. The value of the quantum of interaction energy in quantum thermal energy units, kBT, can be determined from the plot of the energy barrier presented in
Activation energy value can be determined from the plot:
E
max=0,1035 . . .
E
min=−0,1915 . . .
E
0=−0,163 . . .
E
max
−E
min=0,1035+0,1915=0.295.
E
0
−E
min=−0,161+0,1915=0,0305=kBT.
(Emax−Emin)/(E0−Emin)=0,295/0,0305=9,67≈3,5ε0=3,5×ekBT.
Difference between binding energies in the direct and reverse reactions is:
E
min
−E
0=0,1915−0,161=0,0305=E0=e kBT.
As it can be seen, tetramer dissociation constant can be calculated using following equation:
K
0=(1/VQ)×exp(−Stetr/kB)=(1/VQ)×exp(−ln gtetr),
where gtetr is a number of degenerative internal states having the same energy, realizing the steady-state configuration of a tetramer with 4 valence bonds, provided that the conditions for compensation of attractive and repulsive forces are met.
According to the above described principle of free energy quantization (6), at glucose concentrations α≥1, the equilibrium configuration of the tetramer, with a maximum number of microstates gtetr=25, corresponding to the minimum free energy of interaction, is established with respect to the quantum of interaction energy ε0, a multiple of the quantum thermal energy ε0=n kBT, where n=1, 2, 3, . . . ; ε0≥kBT.
The microstates realizing the equilibrium configuration of the tetramer at α≥1 glucose concentrations have similar energy and are degenerative.
As it was shown above (15), the change in the free energy of the transition (Na) tetramer from conformation (λ) into isomeric conformation (s) is:
TΔS=k
B
T×ln(25/9)=kBT×ln 2,777≈kBT.
At glucose concentrations α<1, corresponding to the values of the Michaelis constant <5 mmol/liter, the spatial configuration with the minimum free energy of volumetric interactions cannot be realized with an electromagnetic interaction energy quantum ε0 proportional to the thermal energy quantum kBT.
It can be shown that the equilibrium configuration of the tetramer, at α<1, can be realized at the value of the interaction energy quantum ε0=e kBT, where e is the natural logarithm base, which is a transcendent and irrational number that cannot be represented as the ratio m/n of an integer in to a natural number n, an infinite non-periodic decimal fraction, which is approximately equal to 2,71828 . . . .
The number e can be represented as the Euler formula:
e=1+1/1!+1/2!+ . . . 1/n!+ . . .
In this case, the value of the energy quantum BO can take quantized values ε0n: corresponding to different integer values of the quantum number n:
ε01=(1+1)×kBT=2kBT;
ε02=(1+1+½)×kBT=5kBT/2;
ε03=(1+1+½+⅙)×kBT=8kBT/3;
ε03=(1+1+½+⅙+ 1/24)×kBT=65kBT/24;
ε0n=(1+1+½+⅙ . . . +1/n!)×kBT. (53)
The energy of the hydrogen bond, in this case, is split into an infinite number of interaction quanta, the energies of which are determined by the quantization condition (53), and, as a result, the level of the energy barrier is split into an infinite number of adjacent energy levels.
In thermal motion, tetramer configurations with different values of the distance between the monomers corresponding to different values of the interaction energy ε0n and the hydrogen bond energy can be realized with equal probability.
It should be noted that different values of intermolecular distances in the tetramer of hexokinase, in contrast to glucokinase, can be formed if the amino acid side chains are longer due to the additional monomer and additional covalent bond, which results in the flexibility of the chain of the side group. Therefore, in case of hexokinase, the amino acid with a negative side group charge is glutamic acid (Glu, E), the side group of which consists of 3 monomer units; an amino acid with an uncharged side group is asparagine (Asn, N), the side group of which consists of 3 monomer units: in case of glucokinase, the amino acid with a negative side group charge is aspartic acid (Asn, D), the side group of which consists of 2 monomer units: amino acid with an uncharged side group is an amino acid selected from the group of threonine, cysteine, serine, the side groups of which consist of 2 monomer units.
The active center of hexokinase contains serine and histidine residues. SH groups of the polypeptide chain play an important role in the mechanism of the catalytic action. Hexokinase is activated by vitamins, insulin, and also as a result of binding to certain biological membranes; it is inhibited by glucose-6-phosphate and corticosteroids.
The second difference between hexokinase and glucokinase is that the substrate specificity of hexokinase (EC 2.7.1.10 is relative—in addition to glucose, fructose, mannose, galactose, glucosamine, some non-metabolizing sugars of the 2-deoxyglucose type are subjected to phosphorylation. It can be assumed that the quantization of the hydrogen bond energy (53) may explain the relative substrate specificity of hexokinase by the fact that the listed sugar molecules form hydrogen bonds with the monomers of the polypeptide chain with close, but different energy values, which correspond to different values of the distance between the tetramer monomers.
The equilibrium configuration R2 with 2 hydrogen bonds, corresponding to the minimum free energy at α<1, in accordance with the scale invariance condition ( ), can be realized if the condition dS/dε≡0 is met; this can be realized in a unique way as a result of splitting into e quanta of each quantum internal entropy of the tetramer, the number of which corresponds to the number of possible configurations of hydrogen bonds in the tetramer. Therefore, S/kB=ln eg=g, where g is the number of configurations equal to an integer.
As it was shown, at glucose concentrations of α≥1, the equilibrium configuration of the tetramer, which is realized by the maximum number of microstates gtetr=25, corresponding to the minimum free energy of interaction, is established with respect to the quantum of interaction energy ε0=kBT. The maximum interaction energy (peak of the energy barrier) is achieved at a glucose concentration corresponding to the specific point G of the process; the tetramer is in the R2 conformation with 2 hydrogen bonds, which can be converted to the R4 or R0 conformation with equal probability.
The number of states g (l=2) in which R2 configuration with 2 hydrogen bonds is realized is 18, of which 6 states realize the electronic configuration and 12 states realize the monomer configuration.
The number of states g (l=2) realizing the R2 configuration with 2 hydrogen bonds, which arise upon splitting the energy of a hydrogen bond, at glucose concentrations a<1, is equal to:
g
hex
=g(l=2)=12e6;ln ghex=6+ln 12.
The binding energy Eb that must be expended to detach two EG complexes (glucose molecule G bonded to monomer E via one hydrogen bond) forming a tetramer complex [E2G2] in the R2 configuration with 2 hydrogen bonds can be calculated using following formula:
E
b=4kBT ln g(l=2)=4kBT ln 12e6=4×(6+ln 12)×kBT=4×8,48×kBT,
where g(l=2) is a number of states, realizing the configuration of R2 tetramer with 2 hydrogen bonds.
Multiplier 4 in the formula corresponds to the number of hydrogen bonds in the tetramer complex. Therefore, binding energy, calculated per one hydrogen bond is:
E
b/4=kBT g(l=2)=4kBT ln 12e6=4×(6+ln 12)×kBT=4×8,48×kBT.
Equilibrium constant is:
g
hex
=g(l=2)=12e6;ln ghex=6+ln 12.
K
0
/K
hex=exp(6+ln 12)/exp(ln 18)=e6+2,48/e2,89=e6−0,41=e5,59=267.7.
K
hex
=K
0/267.7=4.4/267.7=0.016 .
g(l=2)=267.7/4=66,925
g(l=4)=e1×24=65,23.
Activation energy on 4 bonds per energy quantum ε0=ekBT is:
E
a
=E
max
−E
min=4×3,443E0=4×3,443×ekBT≈37.4kBT.
Barrier penetration rate:
v×w
b
=v×exp(−Ea/kBT)=0,75×1013/e37.4=0,75×1013/1,8×1016=0,4×10−3.
Reaction rate is:
W=−k[G][P]=(v×wb/1,735 K0)×[G][P]==[0,4×10/1,735×0,2(s×mmol/L)]×0,2×1×1,4×102 (mmol/L)2=0,32×10−1 (mmol/L)/s.
Calculation results are in agreement with the experimental data [13].
Chemical Bond Types in Tetramers of the Natural Macromolecules.
Within the common classification, chemical bond in the tetramer between substrate molecules and monomers of the polymer chain may be defined as delocalized hydrogen bond.
Depending on the degree of generalization of the electronic density of the chemical bond between several atoms, the bond can be classified into localized and delocalized. Covalent bond is considered localized, if its electron pair is in the field of two nuclei and binds only two atoms. If the electronic density of the chemical bond is distributed between three or more nuclei, such bond is called three-center or multicenter bond, and in general—delocalized. Delocalized bond is a bond in which an electron pair is spread between several (more than 2) atomic nuclei (analog of metallic bond). The pattern of delocalization of electrons in chemical bond may differ depending on the spatial dimension. There are bonds delocalized in one dimension, in plane and in three-dimensional space. From the point of the abovedescribed classification, a well-known metallic bond is a short-acting and long-ranging, non-polar, highly delocalized (in three dimensions) bond.
Depending on the distance of appearance, chemical bonds can be classified into short- and long-range. The strength of the long-range bond is much lower than that of the short-range bonds. Far-range bond is also called weak or intermolecular bond. Short-range bond is manifested at the distance close to the size of an atom. It is formed between atoms within the molecule, crystal in the range between 74-400 μm. Short-range bond-breaking energy lies within the range from 40 to 1,000 kJ/mol. Long-range chemical bond is manifested upon transition of the substance from gaseous into liquid or solid state. It is formed between separate atoms, molecules at distances several times exceeding the length of a standard short-range bond. Short-range bonds are formed as a result of interaction where each electron may be described by an independent wave function-single-electron approximation. Long-range bonds are results of the collective electron motion.
Glycoside Bonds.
It was shown that thermodynamic properties of the biopolymer molecule can be described on the basis of a statistical model, as a result of the interaction of polymer chain monomers with solvent molecules and ions, which leads to the formation of tetramers in the volume of the macromolecule—molecular complexes consisting of 2 non-adjacent polymer chain monomers and 2 glucose solvent molecules, linked by a hydrogen bond, which can be considered a type of weak delocalized covalent bond between the glucose molecule and the monomers of the polymer chain.
It is well known that the glucose molecule can also form a glycosidic valence bond with amino acids of protein molecules. In glycoproteins characterized by a low carbohydrate content, a carbohydrate residue (an oligosaccharide with irregular structure and containing glucose, mannose, galactose and their amino derivatives, as well as N-acetylneuraminic acids is linked to the amino acids of the protein chain or to the amide nitrogen of asparagine via N-glycosidic bond, or to the hydroxyl group of serine, threonine, hydroxylysine residues via O-glycosidic bond. Structural formulae of glycoside bonds are shown in
Pharmaceutical Compositions Based on the Hyaluronic Acid.
Pharmaceutical compositions and products on the basis of hyaluronic acid are widely used in medicine and cosmetology: regeneration of skin cells in aesthetics cosmetology: healing of wounds and burns; protecting ocular structures and ensuring the depth of the anterior chamber during ophthalmic surgery and in the production of contact lenses: accelerated growth of bone cells and rapid bone fusion in fractures; delivery of drugs to the site of inflammation in pathological foci and control over the dosed release of the drug; accelerated growth of bone cells and rapid bone fusion in fractures.
New understanding of thermodynamic properties of the intercellular substance, based on the new method of calculation of polymer macromolecule of hyaluronic acid, allows to develop a fundamentally new pharmaceutical compounds on the basis of hyaluronic acid.
The invention is also related to the pharmaceutical compounds containing biopolymer molecule and one or several pharmaceutically acceptable carriers, adjuvants, solvents and/or fillers, that may be introduced into the patient's body together with the biopolymer molecule, which is the essential part of the invention, and which do not inhibit the effect of the biopolymer molecule, and are non-toxic in the amounts sufficient to deliver the biopolymer molecule in necessary amount.
Hyaluronic acid which is currently used in cosmetics and medicine, is obtained through biochemical synthesis. For this purpose, bacterial cultures, in particular, streptococci, cultivated on the herbal base (wheat broth), are used. This method is based on the ability of some organisms to synthesize hyaluronic acid. The biochemical method allows to obtain a large amount of substance having a necessary molecular weight and acceptable structure.
Stabilized hyaluronic acid-based products, which are used in cosmetics, for a long time persist in skin. They constitute a base for the water-accumulating products and stimulate regenerative processes in dermis. Hyaluronic acid molecule is highly sensitive. It sharply reacts to the chemical modification: thermal or mechanic; therefore, the product should be appropriately preserved in the course of chemical reactions. Stabilized hyaluronic acid is obtained using biochemical synthesis method, followed by fusion, which is called stabilization (formation of cross-links between hyaluronic acid molecules). Hyaluronic acid molecules undergo fusion to prevent their rapid degradation. Such hyaluronic acid demonstrates long-term clinical effects after penetration into skin. After fusion the resulting gel undergo purification, which is a meticulous process and a definitive factor in setting a price of the stabilizing hyaluronic acid-based products.
Depending on the degree of stabilization, gels with various viscosity are obtained for correction of different aesthetic issues: weakly stabilized—to remove fine wrinkles, more strongly stabilized—for correction of nasolabial folds and restoration of the lost volume.
Stabilized hyaluronic acid is used in contour correction and face-lifting, as this type of hyaluronate (hyaluronic acid sodium salt) is a potent volume-sparing compound. Thus, if you need to restore the lost volume, for example, on checks, to push the nasolabial folds externally, adjust facial contour and fill the gaps, use hyaluronic acid.
It must be stabilized and have characteristics, which are expressed in the native condition of the living tissue, in which the intercellular substance is in heterophase. As it was shown, normal physiological characteristics of the living tissue are achieved (at physiological temperature and acidity) at following concentrations of biochemical components in the intercellular substance:
Biopolymer molecules described in this invention, may be used for prevention and/or treatment of human diseases, for example, in the form of the following compounds (“biopolymer molecule” is an active ingredient, which is a biopolymer molecule in heterophase state according to the invention).
Active ingredient concentration in the product, in particular, equivalent concentration of the hyaluronic acid monomers 45 mmol/l, is 15 mg/ml.
Glucose concentration in the product, is equivalent to the concentration of free (non-polymer-bound) glucose of 5 mmol/l, is in the range from 5 to 15 mg/ml. Sodium and potassium ion concentrations in the water solution with pH of 7.0-7.5 is 40 mmol/l and 5 mmol/l, respectively.
Enzyme-Based Drug Formulations.
Type 2 diabetes mellitus is an urgent issue in current healthcare. Despite wide range of oral glucose-lowering drugs, currently, there is a high demand in the development of novel effective and safe drug products and their approval for clinical use. One of the potential targets for the new antidiabetic drugs development is glucokinase (GK). It acts as a glucose “sensor” in the pancreatic beta-cells and controls the rate of glycogen synthesis in liver. Glucokinase activator is a novel class of antidiabetic drugs with a reliable effect, that was demonstrated by the clinical data. It is important to note that glucokinase activators do not cause clinically significant hypoglycemia in experimental diabetic models, and do not affect lipid level. They also do not cause weight gain.
In 2003, a first report of a successful application of low molecular weight GK activator (GKA) to reduce blood glucose level in animal models with type 2 diabetes mellitus. Screening of 120,000 compounds allowed to identify the formulation with the properties of allosteric GK activator. Active R enantiomer of this formulation, R082881675, improved enzymatic activity of GK, after oral administration it reduced blood glucose level in wild type mice and in diabetic models, improved glucose metabolism in liver and glucose-induced insulin secretion in isolated pancreatic islets in rats. The search and characterization of other low-molecular weight GKA have become one of the most important directions in the development of antidiabetic drugs over the next decade. Large pharmaceutical companies have got into the process of creation and testing of such formulations. Among them, “Array Biopharm Inc.”, “Astra Zeneca pic. Merck & Co.”, “OSI Pharmaceuticals Inc./Eli Lilly”, “Pfizer Inc.”, “Roche”, “DaiichiSankyio”, “Takeda”, TransTech Pharma Inc./Forest Laboratories Inc./Novo Nordisk A/S”, “Advinus”. “Hua Medicine”. By 2008, more than 90 low-molecular weight GKA were patented. Most of these formulations are based on the pharmacophore (its diagram is shown in
Benjamin, Inc., Massachusetts (1972).
| Number | Date | Country | Kind |
|---|---|---|---|
| 2019136897 | Nov 2019 | RU | national |
| Number | Date | Country | |
|---|---|---|---|
| Parent | PCT/RU2020/050324 | Nov 2020 | US |
| Child | 17747380 | US |
