The Relationship between Symmetry and Specific Properties of Supramolecular Systems

Abstract

Released agent is a supramolecular system formed around nanobubbles in highly diluted solutions of substances subjected to technological processing. Released agent retains the structure and symmetry of the supramolecular system of the dissolved substance. It has two specific properties: (1) when combined with the original substance, it modifies its effect on an organism; (2) and it could cause an atypical pathological reaction in some individuals. These properties could be due to the transmission of symmetry from released agent to the receptor in the molecular recognition reaction by deforming the receptor structure. A theoretical analysis of these properties was carried out using methods of statistical thermodynamics of complex systems, chemical thermodynamics of supramolecular systems, and analytical mechanics. The released agent’s modifying effect is a phenomenon of the receptor pre-organization in the molecular recognition reaction. The selective ability and free energy of complex formation of released agent (from highly diluted antibodies to the beta-subunit of the insulin receptor) with the insulin receptor was evaluated. The atypical pathological reaction of some individuals to released agents resembles a resonance phenomenon that occurs when the symmetries (structures) of released agent and the receptor coincide. The obtained results show a possibility to investigate released agents’ properties not only empirically, but also using the methods of theoretical physics.

1. Introduction

Currently, the use of drugs based on highly diluted (HD) substances, including the so-called “released-active” drugs, is expanding in various countries. Released-active drugs have proven to be safe and effective against a wide range of diseases [1,2,3,4]. Nevertheless, disputes about the fundamental possibility of biological effects of high dilutions still persist. The concept of an aqueous solution as a simple liquid with a homogeneous and isotropic structure, where no trace of the original (soluble) substance remains due to repeated dilutions, hinders the development of physics of HD solutions. In fact, HD aqueous solutions are supramolecular systems that are able to transmit their structure and symmetry when interacting with supramolecular structures (receptors) of the organism. This transmission of the structure and symmetry is the mechanism of activation and training of the immune system. Specific physical-chemical, biochemical, and biophysical properties of supramolecular systems of HD aqueous solutions have been determined (Epstein [5]). The combination of such specific properties serves as experimental proof of the supramolecular systems existence in HD aqueous solutions.

This research was aimed to establish a connection between the symmetry of supramolecular systems (released agents) and the specific properties of these systems. The concept of released agent was used by the author following the work by Epstein O.I. (Epstein [5]), where he postulated that HD causes biological systems to respond at the molecular level. Specifically, receptors in an organism participate in a reaction of molecular recognition of HD aqueous solutions. It is important to note that, based on calculations, HD do not contain molecules of the original substance subjected to the dilution procedure, i.e., the qualitative properties of HD are not due to the original substance. As in (Epstein [5]), the author uses the terms “released-active dilutions” or “released-active drugs”, and postulates that their properties are due to the presence of released agent.

Released agent has the properties of integrity and symmetry. In the process of production of released-active drugs, aqueous solutions are subjected to a procedure based on a special technology. This processing promotes self-assembly (association) of water molecules into supramolecular structures (Epstein [5]. A fairly popular concept has been developed about the “memory” of the solvent, i.e., the presence of a dynamic material matrix in dilutions that contains information about the eliminated original substance, which is based, according to a computer model, on the restructuring of the hydrogen bonds of the solvent (Drozdov [6]).

It is difficult to explain the specific properties of released agents based on the classical theory of solutions, since water is not a “simple” liquid, especially after being subjected to technological processing, which takes place during high dilutions. Such aqueous solutions are nonequilibrium multiphase systems containing nanoscale bubbles and impurities [4,5,6,7,8,9,10]. The presence of released agents in HD aqueous solutions affects (as evidenced by experimental data) the magnitude of the zeta potential, conductivity, and sound velocity [5,11,12,13].

In addition to quantitative changes in physical and physical-chemical properties, the presence of released agents in aqueous solutions is manifested in two specific properties. One of them is the ability to modify the original substance. Modification results in significant enhancement of the efficacy of interaction between the original substance and the supramolecular matrix (receptor) of the organism [14,15,16]. Another specific property appears in the abnormal specific pathological reactions of some individuals to the effects of released agents (Epstein [5]).

O.I. Epstein [5] proposed the hypothesis of spatial homeostasis, where he pointed at the connection between these specific properties of released agents with the symmetry of the supramolecular matrices of released agents themselves and the supramolecular matrices of the organism. He suggested that the specificity of the interaction of the supramolecular systems of released agents and the organism is due to the proximity of the spatial symmetries of these systems. As a result of this interaction, the supramolecular systems of an organism can slightly change its symmetry. Thus, there is a translation of symmetry from the released agent to the organism.

The purpose of this article is to describe the symmetry of the supramolecular system using quantitative parameters and to establish the relationship between these parameters and the specific properties of released agents: modification of the original substance and the abnormal pathological reaction of some individuals to low doses of released agents. To achieve this purpose, methods of statistical thermodynamics, chemical thermodynamics of supramolecular systems, and analytical mechanics were used.

First, Liouville’s theorem was proven for a complex system that consists of several interacting subsystems. Based on this theorem, an invariant was obtained that characterizes the degree of deformation of the phase volume of a complex system and serves as a measure of the change in the structure and symmetry of a complex (supramolecular) system. Then the change in this invariant was calculated for lactose hydrates obtained using an HD of aqueous solutions. A connection was established between a change in the structure and symmetry of the supramolecular system of the receptor and the phenomenon of receptor preorganization in the molecular recognition reaction (the modifying effect of the released agent on the receptor). Finally, a model that relates the change in the value of the invariant to the abnormal response of the supramolecular receptor system to the action of released agent is presented.

Thus, the author identified the parameter that characterizes the symmetry of the supramolecular system and is directly related to the interaction of released agents with receptors in an organism, since certain molecular symmetry of ligands is a condition for their specific interaction with the corresponding receptors (Flisova-Villasnor [17]).

2. An Invariant Characterizing the Symmetry of a Complex System (Supramolecular System)

2.1. Liouville’s Theorem for a Complex System

The proposed theory (presented further in the paper) takes into account the specific properties of supramolecular structures as a special case of complex systems. In the terminology of statistical thermodynamics, these specific properties are expressed as non-additivity of thermodynamic parameters, that is, the presence of binding energy between parts of such structures (integrity) along with the presence of many metastable states (configuration structural states with low barriers of transitions between them).

Supramolecular structures (systems) can be described as complex systems composed of several interacting subsystems. Supramolecular systems are chemical systems linked together through intermolecular (non-covalent) interactions. Thus, released agents formed in HD aqueous solutions are also complex systems because they have a supramolecular structure. Supramolecular systems are integral. Therefore, it is not correct to describe supramolecular systems without taking into account the interaction energy of the subsystems that compose this system. Moreover, the interaction energy can assume not a single, but several possible values during the formation of various conformations of released agent. Thus, these structures can exist not only in the equilibrium (stable) state, but also in various metastable states. Taking into account the interaction energy of subsystems and the possibility of the existence of metastable states requires applying the principles of complex system thermodynamics.

The classical Gibbs thermodynamics (Gibbs [18]) is based on two postulates. The first is the additivity postulate suggesting that the energy of a thermodynamic system is the sum of the internal energies of its subsystems (the interaction energy between the subsystems is neglected). The second is the equilibrium postulate, according to which a thermodynamic system under definite and constant external conditions exists in a state of thermodynamic equilibrium. In equilibrium, the macroscopic parameters of the system remain unchanged for an arbitrarily long time, and the system cannot spontaneously leave this state. Both postulates are incorrect in relation to supramolecular systems.

Statistical thermodynamics of complex systems is a variation of the classical Gibbs statistical thermodynamics (Gibbs [18]), that is, a variation that differs by taking into account the stochastic dependence between the subsystems that make up a complex system. This dependence is a consequence of the presence of the interaction energy between subsystems, the value of which is comparable to the energy of subsystems.

The equation for the energy of a complex system E_c (for example, a system that consists of two subsystems: “1” and “2”; certainly, a larger number of subsystems is possible), considering the interaction energy of these subsystems, can be expressed as:

E_c = E₁ + E₂ + E_(1,2)

(1)

where E₁ is the energy of a part of the first subsystem not participating in the interaction, E₂ is the energy of a part of the second subsystem not participating in the interaction, and E_(1,2) is the energy of interaction between subsystems.

Notably, the interaction energy depends on the state of both the first and second subsystems. The Hamiltonian equations for a complex system were adjusted in this work. The generalized coordinates and generalized momenta of the subsystems of a complex system depend on the interaction energy between the subsystems, which leads to a stochastic dependence of the states of these subsystems. For the first subsystem, the equations for coordinates and momenta can be expressed as:

q_1i = ∂E₁/∂p_1i + ∂E_(1,2)/∂p_1i

p_1i = − (∂E₁/∂q_1i +∂E_(1,2)/∂q_1i)

Equations for the coordinates and momenta of the second subsystem are similar:

q_2j = ∂E₂/∂p_2j + ∂E_(1,2)/∂p_2j

p_2j = − (∂E₂/∂q_2j + ∂E_(1,2)/∂q_2j)

where q is the coordinate, p is the momentum, i is the index of the coordinates and momenta of the first subsystem’s particles, and j is the index of the coordinates and momenta of the second subsystem’s particles.

The Gibbs phase space of a complex system is the space of coordinates and momenta, the dimension of which is 6N, where N is the number of all molecules that make up the complex system (assuming that each molecule has 3 degrees of freedom). The Gibbs phase space has 3N coordinates of molecules and 3N coordinates corresponding to the projections of the momenta of the molecules. The change in time of the state of a complex system is described in the Gibbs phase space as the motion of one point (phase) in 6N dimensional space. J. Willard Gibbs introduced the concept of a phase ensemble, that is, a set of systems obeying the same equations of motion, but with different initial conditions. The phase ensemble fills the phase space.

The density of the number of states (phases) of a complex system in the elementary volume of this phase space D_ci,j, expressed in terms of the density of the number of states of individual subsystems, is described by the equation:

Dc_i,j = D_1i • D_2j • i_(1,2)i,j

(2)

where i is the index of coordinates and momenta of particles belonging to the first subsystem, j is the index of coordinates and momenta of particles belonging to the second subsystem, D_1i is the density of the number of states (phases) of the first subsystem in the elementary volume of its phase space, D_2j is the density of the number of states (phases) of the second subsystem in the elementary volume of its phase space, and i_(1,2)i,j is the parameter characterizing the change of the density of the number of states (phases) of a complex system in the elementary volume of its phase space as a result of the interaction of the first and second subsystems. In relation to released agents, a change in the density of the number of states occurs as a result of the interaction of the subsystems of the supramolecular system.

The meaning of the magnitude i_(1,2)i,j can be explained by considering not the densities of the number of states (phases) in the elementary volume of the Gibbs phase space, but the possible presence of phases in the elementary volume of the Gibbs phase space. Equation (2) will be as follows:

Pc_i,j = P_1i • P_2j • i_(1,2)i,j

(3)

where i is the index of coordinates and momenta of the first subsystem’s particles, j is the index of coordinates and momenta of the second subsystem’s particles, P_1i is the probability of the appearance of the first subsystem phase in the elementary volume of its phase space in the absence of interaction between the subsystems, P_2j is the probability of the appearance of the second subsystem phase in the elementary volume of its phase space in the absence of interaction between subsystems, and i_(1,2)i,j is a parameter characterizing the difference between conditional and unconditional probabilities (according to the probability multiplication theorem: P_2j/1i = P_2j i_(1,2)i,j).

The meaning of Liouville’s theorem in classical statistical thermodynamics was formulated by J.W. Gibbs: “The fact that the quantity, which we have called density-in-phase, is constant in time for any given system, implies therefore that its value is independent of the coordinates which are used in its evaluation” (Gibbs [18]).Therefore, if the phases enclosed in a certain phase volume change in time according to the dynamic laws of motion of a certain system where forces depend only on coordinates or on coordinates and time, then the phase volume that they fill in remains constant. This formulation of the principle of conservation of the Gibbs phase volume is valid for stationary (both equilibrium and metastable) phase states.

Let us consider the Liouville equation for non-interacting subsystems, i.e., for a “simple” system. For a system consisting of non-interacting subsystems, the density of the number of phases in the elementary volume of the Gibbs phase space is equal to the product of the densities of the number of non-interacting subsystems’ phases:

D_1i • D_2j = Dc_i,j

(4)

The Liouville equation for the density of the number of phases in the elementary volume of the phase space of a system including two subsystems refers to the equilibrium state of the system. Such an equation describing a state in which the density of the number of phases in the elementary volume of the phase space is constant has the form:

∂Dc_i,j/∂t = 0,

where t—time,

Or (Gibbs [18]):

0 = D_2j∑[(∂D_1i/∂q_1i) • (∂q_1i/∂t) + (∂D_1i/∂p_1i) • (∂p_1i/∂t)] + D_1i∑[(∂D_2j/∂q_2j) • (∂q_2j/∂t) + (∂D_2j/∂p_2j) • (∂p_2j/∂t)]

(5)

Obviously, Equation (5) splits into two equations. For the first subsystem:

∑[(∂D_1i/∂q_1i) • (∂q_1i/∂t) + (∂D_1i/∂p_1i) • (∂p_1i/∂t)] = 0

and for the second subsystem:

∑[(∂D_2j/∂q_2j) • (∂q_2j/∂t) + (∂D_2j/∂p_2j) • (∂p_2j/∂t)] = 0

Since the states of non-interacting subsystems are independent, the phase space of such “simple” system can be considered as separable.

A different situation arises for a complex system consisting of interacting subsystems. There is a relationship (2) between the phase densities of the interacting subsystems in the elementary volume of a complex system:

D_1i • D_2j • i_(1,2) = D_ci,j

The Liouville equation for the density of the number of phases in the elementary volume of the phase space of a complex system including two interacting subsystems has the form (italics highlight the summands of the Gibbs equation that are not present in the system consisting of two non-interacting subsystems) (Gibbs [18]):

∂Dc_i,j/∂t = 0

or:

0 = i_(1,2) • D_2j∑[(∂D_1i/∂q_1i) • (∂q_1i/∂t) + (∂D_1i/∂p_1i) • (∂p_1i/∂t)] +i_(1,2) • D_1i∑[(∂D_2j/∂q_2j) • (∂q_2j/∂t) + (∂D_2j/∂p_2j) • (∂p_2j/∂t)]+ D_1i • D_2j {∑[(∂i_(1,2)/∂q_1i) • (∂q_1i/∂t) + (∂i_(1,2)/∂p_1i) • (∂p_1i/∂t)] + ∑[(∂i_(1,2)/∂q_2j) • (∂q_2j/∂t) + (∂i_(1,2)/∂p_2j) • (∂p_2j/∂t)]}

(6)

It should be noted that the equivalent form of the Liouville Equation (6) will also be used for probabilities P_ci,j, P_1i, and P_2j, since Equations (2) and (3) are equivalent.

Let us analyze Equation (6). The first two summands of this equation correspond up to a multiplier to the Liouville equations for two non-interacting subsystems. If the states of these subsystems are metastable (i.e., these terms of the equation are equal to zero), then the third term of the equation is equal to zero, where the value i_(1,2)i,j appears as a variable. In turn, this means that (∂i_(1,2)/∂t) = 0. That is, the value of i_(1,2)i,j does not change with time (is invariant in time).

Notably, for a complex system, not only equilibrium, but also metastable states arising from the interaction of subsystems are taken into account. The meaning of these additional summands is determined by the value i_(1,2)i,j, which characterizes changes in the density of phases in the elementary volume of the phase space of a complex system as a result of the interaction of subsystems. Since subsystems interact, the coordinates and momenta of one subsystem depend on the coordinates and momenta of the other one. The value i_(1,2)i,j, as shown by Equation (3) and the commentary to it, express the depth of the stochastic dependence between the states of two interacting subsystems. Such interdependence leads to the indivisibility (integrity) of the phase space of a complex system.

Thus, as a result of the interaction of subsystems, the phase space of the complex system is deformed. The depth of this deformation for an elementary volume of phase space is determined by the value i_(1,2)i,j. The value i_(1,2)i,j characterizes not only the depth of deformation, but also the integrity of a complex system.

The following circumstance should be mentioned: any subsystem of a complex system, in turn, can consist of two interacting subsystems. Thus, a complex system can consist not of two, but of three interacting subsystems. Equation (3), according to the probability multiplication theorem, will have the form:

Pc_i,j,k = P_1i • P_2j • P_3k • i_(1,2,3)i,j,k

(7)

where k is the index of coordinates and momenta of the third subsystem’s particles, and P_3k is the probability of the appearance of the third subsystem phase in the elementary volume of its phase space in the absence of interaction between subsystems. In this case, the Liouville equation will break up not into three, but into four terms. The procedure for dividing subsystems can be continued.

Thus, for any number of interacting subsystems, the value i_(1,2,3…) retains the physical meaning of the parameter characterizing the depth of the stochastic connection between the interacting subsystems of a complex system and the parameter characterizing the integrity of a complex system.

2.2. An Invariant Characterizing the Symmetry of a Complex System (Supramolecular Matrix). Proof of Invariance

Symmetry properties are characterized by a parameter that remains invariant to certain transformations. A special type of transformations of the coordinates and momenta of the system is known, where Hamilton’s equations describing the movement of the system retain their form. These are canonical transformations (see paragraph 45 Landau [19]). The movement of the system along its phase trajectory is an obvious special case of canonical transformations. Canonical transformations do not change the volume of the phase space (see paragraph 46 Landau [19]). J.W. Gibbs [18] proved the equivalence of three forms of Liouville’s theorem for phase space:

D = const(t);

G = const(t);

P = const(t)

In stationary states of the system, the density of the number of states (phases) in the elementary volume of the phase space (D) is constant, and the phase volume (G) itself, which is occupied by the phases of the system, is constant in time. In addition, the probability of finding the phase in the elementary volume of the phase space (P) is constant.

In the previous section, Liouville’s theorem was proven for a complex (supramolecular) system that can be not only in stationary, but also in metastable states. Consequently, the phase volume of a complex G_c system is retained in these states. The phase volumes of the non-interacting subsystems G₁ and G₂ will also be retained. Thus, the value <i_(1,2)i,j˃ will also be retained in the case of canonical transformations:

<i_(1,2)i,j˃ = G_c/(G₁ × G₂) = const (t)

(8)

Here, ˂i_(1,2)i,j˃ is a value that characterizes the depth of deformation of the phase volume of a complex system, i.e., the deformation that results from the interaction of the first and second subsystems.

Therefore, ˂i_(1,2)i,j˃ is a measure of the deformation of the phase volume of a complex system. The phase volume of the system includes the subspace of coordinates of the system and the subspace of momenta of the system. In this equation ˂i_(1,2)i,j˃ describes both the deformation of the subspace of coordinates, which is expressed in the deformation of the spatial configuration of a complex system, and the deformation of the subspace of momenta. The deformation of the subspace of momenta can be expressed by the appearance of resonances, when the vibrations of one part of a complex system resonate (are transmitted and amplified) in another part. The supramolecular system, being a complex system, is characterized by the presence of a spatial structure (order, integrity) and the presence of collective (integral) movements: for example, the ability to produce collective vibrations and resonate.

Liouville’s theorem states that systems that are in a stationary state (equilibrium or metastable) keep the phase volume constant. In this case, the phase volume is an invariant. The interaction of subsystems that make up a complex system leads to the deformation of the phase volume of this complex system. The phase volume of a complex system decreases in comparison with the total phase volume of its non-interacting subsystems. Having proven Liouville’s theorem for a complex system, a parameter that is a measure of the deformation of the phase volume of a complex system was obtained. Then it was proven that this parameter is also an invariant.

The ˂i_(1,2)i,j˃ parameter characterizes the symmetry of a complex, in particular, supramolecular system. A rigorous proof of Noether’s theorem for a complex (supramolecular) system is beyond the scope of this article, which is devoted to physical interpretation, that is, to the identification of physical meaning. Several variations of Noether’s theorem are being actively created at the present time [20,21,22]. It should be noted that the parameter ln˂i_(1,2)i,j˃ is also an invariant of a complex system.

A logical question arises: why is a new invariant required when describing a complex system? Why is there not enough energy, momentum, and angular momentum, which describe the symmetry of systems consisting of non-interacting subsystems? The new invariant was explicitly introduced to describe the bonding and integrity of a complex system. It characterizes exactly the depth of this bonding. Formally, the new invariant results from a new term in Hamilton’s equations that takes into account the energy of the interaction of subsystems.

2.3. The Physical Meaning of an Invariant of a Complex System: The Thermodynamic Coordinate of a Stationary Energy State

In order to find out the physical meaning of the invariant <i_(1,2)i,j˃, it is necessary to apply the formalism of statistical thermodynamics, which links the logarithm of the phase volume with the entropy of the system. As a result, the following expressions for the entropy of a complex system and the entropies of non-interacting subsystems are obtained:

lnG_c = S_c; lnG₁ = S₁; lnG₂ = S₂ ; ln<i(1,2)_i,j˃ = I_s(1:2)

where S_c is the entropy of a complex system; S₁ is the entropy of the first subsystem, S₂ is the entropy of the second subsystem; I_s(1:2) is the value, the meaning of which will be explained below.

Based on Equation (8), the obtained results are:

lnG_c = lnG₁ + lnG₂ + ln(<i(1,2)_i,j˃)

(9)

or:

S_c = S₁ + S₂ + I_s(1:2)

(10)

Thus, the value I_s(1:2) = S_c − (S₁ + S₂) is equal to the difference between the entropy of a complex system and the total entropy of non-interacting subsystems. In general, a complex system includes not two, but more interacting subsystems. Let us denote this value by Is. It should be mentioned that the entropy of a complex system is less than the total entropy of non-interacting subsystems. When deformed by the interaction of subsystems, the phase volume of a complex system is slightly compressed. A decrease in entropy means that a complex system is in a non-equilibrium (metastable) state compared to the state of non-interacting subsystems. The value I_s(1:2) = S_c − (S₁ + S₂) is a parameter that characterizes the level of disequilibrium of the metastable state of a complex system, and a parameter that characterizes the level of deformation of the structure of a complex system that resulted from the transition of a complex system from an equilibrium state to a metastable one, or during the transition from one metastable state to another.

It is impossible to transfer a complex system from an equilibrium state to a nonequilibrium metastable state by heat transfer. Such a transition is possible only as a result of the work applied to a complex system. Therefore, the I_s parameter is not entropy, but it characterizes the process inherent in complex systems only. In the absence of interaction between subsystems, I_s = 0. For description of a specific thermodynamic process, a specific thermodynamic coordinate is required, which characterizes the metastable state of a complex system. Let us propose the term: Is is the thermodynamic coordinate of a stationary energy state. It is “stationary” because it describes not any nonequilibrium, but rather a metastable state of a complex system; “thermodynamic”—because it is obtained by using the tool of statistical thermodynamics; “coordinate”—because it describes a specific state of a complex system.

A quantum mechanical analogue of the I_s value is known. The states of quantum systems are described by the von Neumann entropy. Quantum systems can be in mixed states that are characterized not by a single wave function, but by several mutually orthogonal functions. The mixed states of a quantum system are described by the density matrix (ρ). The von Neumann entropy of such a system (Nielstn [23]) is as follows:

S = −Tr ((ρ)ln (ρ))

Let there be a complex quantum system consisting of two interacting subsystems “A” and “B”, which is described by the density matrix (ρ_ab). Then the joint quantum entropy of the considered complex quantum system is called the magnitude:

S (ρ_ab) = −Tr((ρ_ab) ln (ρ_ab))

For the von Neumann entropy of a complex “entangled” quantum system consisting of two interacting subsystems, the following relation (Nielstn [23]) is applicable:

I_s(a:b) = S (ρ_ab) − (S (ρ_a) + S (ρ_b))

(11)

where S(ρ_a) = −Trln(ρ_a) and S(ρ_b) = −Trln(ρ_b); and (ρ_a), (ρ_b) are density matrices of subsystems “A” and “B”, respectively.

Here the value I_s, i.e., I_s(a:b), serves as a measure of the “entanglement” (mutual influence) level of the states of interacting quantum subsystems and, at the same time, as a parameter characterizing the integrity and symmetry of a complex “entangled” quantum system.

Thus, in terms of statistical thermodynamics, the value I_s, i.e., the thermodynamic coordinate of the stationary energy state, is a parameter that characterizes the symmetry of a complex system and the level of interconnection of the subsystems, which make up this complex system.

3. The Thermodynamic Coordinate of the Stationary Energy State Is a Measure of the Change in the Structure (Symmetry) of the Supramolecular System of Highly Diluted Aqueous Solutions

The thermodynamic coordinate Is of a stationary energy state is defined as the difference between the total entropy of non-interacting subsystems and the entropy of a complex system. Consequently, different metastable states of a complex system are characterized by different thermodynamic coordinates of the stationary energy state. The level of difference in metastable states, i.e., the level of difference in the structure (symmetry) of a complex system, can be characterized by the difference in the thermodynamic coordinates of a stationary energy state. As for supramolecular systems of HD aqueous solutions (released agents), it means that the presence of released agents themselves in an aqueous solution and the difference between different supramolecular systems of released agents can be determined by the difference in the thermodynamic coordinates of the stationary energy states of these aqueous solutions.

A method of differential scanning calorimetry (DSC) allows entropy to be measured based on a thermogram, an experimental dependence of enthalpy on temperature. However, the heat flux, the effect of which on the solution is necessary for measuring entropy with DSC, destroys supramolecular systems of HD aqueous solutions (released agents). In the manufacturing of drugs based on released agents, a neutral carrier such as lactose is used, which is fluidized with a released agent. It is logical to assume that supramolecular systems of HD aqueous solutions are “reflected” in the structure of lactose hydrates, and the symmetry of these supramolecular systems is transmitted into the symmetry of lactose hydrates. Consequently, the thermodynamic coordinates of the stationary energy state of lactose hydrates of various dilutions correspond, to a certain extent, to the thermodynamic coordinates of supramolecular systems of HD aqueous solutions (released agents).

The paper (Konar [24]) presents the results of experiments performed using DSC of lactose hydrates treated with HD aqueous solutions of Natrum Mur and Sulfur (potencies 30 cH, 200 cH, and 1000 cH). The authors analyzed temperatures and enthalpies calculated according to thermograms of the removal of water of crystallization from lactose (Peak 2). They found that the removal of water of crystallization from lactose in these two preparations required different amounts of heat. The enthalpy change associated with the removal of water of crystallization from lactose depended on the presence of HD aqueous solutions. It was concluded that HD aqueous solutions could modify the number of hydrogen bonds and their strength in the crystal structure of lactose, i.e., a transfer of specific water structures (or released activity) from HD aqueous solutions to solid carrier (lactose) did occur.

Based on thermograms, it is possible to calculate not only the enthalpy of the process (ΔH) and the peak temperature (T_peak), but also the entropy, namely, the entropy difference. The entropy differences of lactose hydrates treated with HD aqueous solutions are thermodynamic coordinates of stationary energy states—the parameters that characterize the structure (symmetry). The use of thermodynamic coordinates of the stationary energy state for processing the obtained thermograms allows for the obtainment of more detailed information. Therefore, the author of the current manuscript used the data published by Konar et al. (Konar [24]) to perform additional calculations. The calculations were carried out using the DSC Tool software [25] Changes in the entropy of lactose hydrates during the removal of water of crystallization from lactose are shown in

Table 1. Changes in entropy of the water of crystallization released from lactose hydrates.

Substance	Control	Natrum Mur 30 cH	Natrum Mur 200 cH	Natrum Mur 1000 cH	Sulfur 30 cH	Sulfur 200 cH	Sulfur 1000 cH
Changes in entropy, J/(kg × K)	342.2 ± 1.2 (ΔSc)	204.1 ± 0.9 (ΔSN/m30)	185.3 ± 0.9 (ΔSN/m200)	235 ± 1.0 (ΔSN/m1000)	207.5 ± 0.9 (ΔSs30)	198.3 ± 0.9 (ΔSs200)	320.1 ± 1.2 (ΔSs1000)

.

The structure (symmetry) of lactose hydrates, which reflects the structure (symmetry) of released agents, is affected by the level of dilution of aqueous solutions and by the type of the original substance. Based on the data presented in Table 1, changes in the thermodynamic coordinate of the stationary energy state have been calculated depending on the level of dilution. The results are presented in

Table 2. Changes in the thermodynamic coordinate of the stationary energy state depending on the number of dilutions of an aqueous solution and type of the original substance.

Number of Dilutions	30	200	1000
Changes in the thermodynamic coordinate of Natrum Mur lactose hydrate J/(kg × K)	138.1 ± 2.1 (ΔIN/m30 = ΔSc − ΔSN/m30)	156.9 ± 2.1 (ΔIN/m200 = ΔSc − ΔSN/m200)	107.2 ± 2.2 (ΔIN/m1000 = ΔSc − ΔSN/m1000)
Changes in the thermodynamic coordinate of Sulfur lactose hydrate J/(kg × K)	134.7 ± 2.1 (ΔIs30 = ΔSc − ΔSs30)	143.9 ± 2.1 (ΔIs200 = ΔSc − ΔSs200)	22.1 ± 2.4 (ΔIs1000 = ΔSc − ΔSs1000)
Changes in the thermodynamic coordinate of lactose hydrate depending on the original substance J/(kg × K)	3.4 ± 4.2 (ΔI30 = ΔIN/m30 − ΔIs30)	13.0 ± 4.2 (ΔI30 = ΔIN/m30 − ΔIs30)	85.1 ± 4.6 (ΔI30 = ΔIN/m30 − ΔIs30)

. In addition, changes in the thermodynamic coordinate of the stationary energy state have been calculated depending on the original substance (Table 2).

The analysis of the data presented in Table 2 shows that there is no monotonous dependence of the coordinates of the stationary energy state of lactose hydrates, which reflect the structure (symmetry) of the released agents of Natrum Mur and Sulfur. It can be hypothesized that this nonmonotonicity is a manifestation of the well-known polymodal dependence of the efficacy of released agents on the number of dilutions [5,26]. It should be noted that the use of the coordinates of the stationary energy state made it possible to establish a quantitative measure of the difference existing in the structures of the released agents of Natrum Mur and Sulfur, namely, the structures of their supramolecular matrices and the increase in this difference as the number of dilutions increases.

The typical structural features of the supramolecular system of released agent are manifested in the parameters of the emission spectrum and the absorption spectrum of solutions containing released agents. Consequently, these features are present in the parameters of the emission spectrum and the absorption spectrum of lactose hydrates obtained using aqueous solutions that contain released agents.

Using terahertz time-domain spectroscopy (THz-TDS), the absorption spectrum of an aqueous solution of lactose saturated with a solution of HD antibodies to IFN-γ was recorded in (Penkov [27]). The observed changes in the parameters of the spectrum of aqueous lactose solutions are associated with the presence of released agents and indicate a change in the supramolecular structure of these solutions. As shown above, such a change can be described as a change in the thermodynamic coordinate.

Based not only on theoretical analysis, but also on experimental data, it can be concluded that the coordinate of the stationary energy state is an integral parameter that characterizes the structure (symmetry) of a released agent. It should be noted that when preparing lactose hydrates using HD aqueous solutions, the symmetry of supramolecular systems of aqueous solutions is transmitted into lactose hydrates. In the absence of such transmission, there would be no difference in the coordinates of the stationary energy state of lactose hydrates obtained using different aqueous solutions. It should also be noted that the thermodynamic coordinates of the stationary energy state of lactose hydrates, which reflect the structure (symmetry) of the released agents of Natrum Mur and Sulfur, have been calculated based on thermographic measurements, which emphasizes the thermodynamic nature of these coordinates.

4. Modifying Effects of Released Agent

It should be noted that the most complete characteristic features of the supramolecular system structure of released agent are manifested when the modifying effect of released agent is exerted on the original substance.

The experimental data obtained by THz-TDS indicate the presence of anomalies in the dielectric constant in aqueous solutions of various substances (Penkov [28]). These anomalies may be due to the presence of supramolecular structures. In that work, the objects of study were aqueous solutions of CaCl₂, KBr, KI, and CsCl, together with high dilutions of the same substances added to them. The authors documented an abnormal increase in the dielectric response of an aqueous solution (e.g., CaCl₂) containing specific supramolecular systems (HD CaCl₂ aqueous solution) compared to a CaCl₂ solution containing no supramolecular structures or containing non-specific supramolecular structures. Thus, a significant contribution of orientational relaxation of supramolecular structures to the total dielectric response was found.

It should be noted that the results presented in (Penkov [28]) were obtained only for inorganic original substances, which form supramolecular structures in HD aqueous solutions. The issue of whether similar results would be valid for organic original substances was not studied in (Penkov [28]).

5. The Phenomenon of Pre-Organization in the Reaction of Molecular Recognition

The modifying effect of a released agent is manifested in the molecular recognition reaction. Two supramolecular matrices interact in this reaction: a ligand, i.e., released agent, and a target, i.e., the receptor. The essence of the molecular recognition reaction is the selective interaction of molecules (for example, a receptor and a ligand in biological systems). This selectivity is due to the complementarity of the structures of the receptor and the ligand.

The molecular recognition reaction allows the presence of released agent to be detected that is specific for the receptor in an aqueous solution. This specificity is due to the fact that such released agents obtained from the original substance (for example, the ligand itself, antibodies to the ligand, or antibodies to the receptor of this ligand) by serial dilution. The presence of released agents in an aqueous solution can be evidenced by registering the so-called “preorganization effect”, i.e., preliminary reconfiguration of the receptor as part of a molecular recognition reaction.

The molecular recognition reaction proceeds as follows:

Lig + Rec ↔ Lig × Rec

A quantitative measure of the effectiveness of the molecular recognition reaction is the binding constant (Atwood [29]):

K_L =[Lig × Rec]/{[Lig][Rec]}

(12)

where [Lig × Rec] is the concentration of the bound complex Lig × Rec, [Lig] is the concentration of unbound ligand molecules, and [Rec] is the concentration of unbound receptor molecules.

After preorganization of the receptor, there is an increase in the efficiency of the molecular recognition reaction of the ligand by the receptor. An increase in efficiency is determined by an increase in the concentration of the bound complex Lig × Rec [Lig × Rec] and by a decrease in the concentration of unbound molecules of the ligand [Lig] and the receptor [Rec].

Thus, the binding constant of the molecular recognition reaction of the ligand by the receptor after its preorganization by released agent (K_L+R) will be greater than the binding constant of the molecular recognition of the ligand with the non-preorganized receptor (K_L):

K_L+R ˃ K_L

The coefficient of increase in the efficiency of ligand binding by the receptor as a result of the preorganization of the latter (S) can be determined as follows:

S = K_L+R/K_L

(13)

The theoretical framework of chemical thermodynamics of supramolecular systems allows for the estimation of the free energy of complexation during the molecular recognition reaction (Atwood [29]):

∆G⁰ = −R • T • lnK

(14)

where: ∆G⁰ is standard-state free energy, R is ideal gas constant, T is temperature, and lnK is natural log of the binding constant.

Since receptor preorganization by released agent is also accompanied by energetic interaction leading to a change in the conformation of the receptor molecule, it is possible to evaluate the free energy of such a preorganization:

∆G⁰_p = −R • T • lnS

(15)

An example of the effect of the receptor preorganization by released agent was described by Gorbunov et al. [14]. A culture of mature human adipocytes that express insulin receptor on their surface was used as an in vitro model. In this work, the authors used HD antibodies to the beta subunit of the insulin receptor, which are one of the active substances of an antidiabetic drug, Subetta. The effectiveness of the molecular recognition reaction (i.e., the value of the binding constant insulin to its receptor) was assessed according to the level of phosphorylation of the beta subunit of the insulin receptor (Gorbunov [14]).

The results presented in (Gorbunov [14]) demonstrate that due to the preorganization of the insulin receptor caused by Subetta, a 1.8-fold increase in the phosphorylation of the beta subunit of the insulin receptor was observed. It is reasonable to assume that the level of phosphorylation of the beta subunit of the insulin receptor is proportional to the concentration of the insulin—insulin receptor complexes. Then, the concentration of the insulin—insulin receptor complexes ([Lig_i × Rec_subetta+i]) will also increase by 1.8 times compared to the concentration of the insulin—not-preorganized receptor complexes ([Lig_i × Rec_i]):

([Lig_i × Rec_subetta+i]/[Lig_i × Rec_i]) ˃ 1.8

With an increase in the number of bound Lig × Rec complexes, the fraction of the free ligand and the unbound receptor obviously decreases. Thus, the proportion of the receptor unbound to the ligand in the molecular recognition reaction with preorganization will be less than the fraction of the free receptor in the same reaction, but without preorganization. Accordingly, the fraction of unbound ligand in the multiple recognition reaction with preorganization will be less than the fraction of unbound ligand in the same reaction, but without preorganization. It follows that an increase in the efficiency of binding of insulin receptors to its ligand as a result of pre-organization of receptors by HD antibodies can be estimated as follows:

S = (K_subetta+i/K_i) ˃ ([Lig × Rec_subetta+i]/[Lig_i × Rec_i])

where K_subetta+i is the binding constant of the molecular recognition of insulin by a preorganized insulin receptor; K_i is the binding constant of the molecular recognition of insulin by insulin receptors in the absence of preorganization.

Therefore, an increase in the efficiency of binding of insulin receptors to its ligand as a result of preorganization can be estimated as follows:

S = (K_subetta+i/K_i) ˃ 1.8

(16)

Using relation (16) and Equation (15), an estimate of the free energy of the pre-organization of the insulin receptor by Subetta is obtained:

∆G⁰_p = −R • T • lnS < − 1.4 kJ/mol

(17)

The author calculated the value of the selectivity and free energy of the complex formation of the preorganization based on the data presented in (Gorbunov [14]). Thus, an increase in the efficiency of the molecular recognition reaction as a result of preorganization of the receptor for the insulin by HD antibodies to the beta subunit of the insulin receptor was revealed. The energy of interaction of released agents with the receptor during preorganization of the receptor was also evaluated. Thus, a theoretical description, obtained using the method of chemical thermodynamics of supramolecular systems, one of the manifestations of the modifying effect of a released agent, has been presented, and a quantitative assessment of this modifying effect has been made.

The experimental data that the author had at his disposal, describing the effect of modification—the phenomenon of preorganization in the reaction of molecular recognition, did not allow for the calculation of the value of the invariant. However, at a qualitative level, such a connection exists between the phenomenon of preorganization and the invariant. Preorganization is accompanied by a change in the spatial configuration of the receptor’s supramolecular system [28,29], which means a deformation of the receptor’s phase space. The measure of deformation of the phase space is the invariant. Consequently, the phenomenon of preorganization is accompanied by a change in the value of the invariant. The calculation of the selectivity and free energy of preorganization complexation performed by the author proved that the release agent does indeed preorganize the receptor. The preorganization of the receptor is carried out by changing (modifying) its structure [28,29]. Consequently, during the preorganization, the configurational entropy and symmetry of the receptor change, therefore, the thermodynamic coordinate of the stationary energy state of the receptor changes, which means that the symmetry has been transmitted from released agent to the receptor.

6. Physical Interpretation of Atypical (Pathological) Individual Reactions to the Effects of Released Agents

An atypical (pathological) reaction to the effects of released agents is manifested in the fact that the reaction of receptors to small doses of drugs in some cases is similar to the reaction to their subtoxic doses. The physical analogue of such a phenomenon is resonance.

Let us consider not even an interpretation, but rather a physical illustration of this phenomenon in that part, where the effect of the released agent on the receptor leads to an abnormally strong deformation of the supramolecular system of the receptor and, consequently, to the transmission of an abnormally strong signal by the receptor to the body. The result of receiving such a signal is an abnormal pathological reaction.

In the current work the author proposes the model according to which supramolecular released agent and receptor systems is a group of interconnected oscillators. In this case, rotational degrees of freedom are also taken into account as the limiting case of oscillations. One group of oscillators belongs to the receptor, the other—to released agent. A group of interconnected oscillators can be considered (see paragraph 29 Landau [19]) as a set of independent oscillators performing normal oscillations with natural frequencies. Translational degrees of freedom can be neglected. The Hamiltonian of one normal oscillator is as follows:

H = p²/2m + mω (∂q/∂t)²/2

where p is the pulse of the oscillator, q is the coordinate of the oscillator, ω is the natural frequency of the oscillator, and m is the reduced mass of the normal oscillator, which coincides with the mass only if q is the Cartesian coordinate (see paragraph 21 Landau [19]). A set of independent oscillators also models the degrees of freedom of a supramolecular system describing the internal rotations of this supramolecular system.

Equation of the phase trajectory in the Gibbs phase space of the oscillator is as follows:

(q/q₀)² + (p/p₀)² = 1

where p₀ is the amplitude of the oscillator pulse; q₀ is the amplitude of the oscillator coordinate. The relationship between these amplitudes:

p₀ = m • ω • q₀

The area of the phase space of one normal oscillator:

S = π • m • ω • q₀²

Since the normal oscillations of the oscillators are independent, the Gibbs phase volume of the system of normal oscillators will take the form of the product of the areas of the phase space of individual oscillators:

G = πⁿ • Π m_i • ω_i • q_0i²

where n is the number of oscillators in the group (in the supramolecular matrix).

The author is interested in the reaction of the receptor to the effect of released agent. Such an effect can be considered as an external force affecting the oscillators of the receptor, which leads to the appearance of forced oscillations of the normal oscillators of the supramolecular receptor system. Variation of oscillations of a single normal oscillator under the influence of an external force can be described by Equation (see paragraph 22 Landau [19]):

q_i′ = q_0i cos(ω_i t + α_i) + f_i (cosγ_i t + β_i t)/(m_i (ω_i² − γ_i²)

(18)

where q_i′ is the coordinate of the oscillations of a single normal oscillator taking into account the influence of an external force; f_i is the external force acting on the receptor oscillator from a released agent oscillator corresponding to it, γ_i is the frequency of this external force (the oscillation frequency of the corresponding oscillator of released agent), and α_i and β_i are the phases of oscillations of the receptor and released agent. Let us assume that f_i is small, i.e., the released agent exerts a weak effect on the receptor. However, with resonance, even a weak effect is significant.

Resonance condition suggests coincidence or extreme proximity of frequencies and phases of free and forced oscillations, i.e., coincidence or proximity of frequencies and phases of normal oscillations of some normal oscillators of supramolecular systems of the receptor and released agent. Obviously, such a coincidence is individual. The coincidence of natural frequencies is possible if configurations (similarity of symmetries) of the receptor and released agent are similar.

Wherein an increase in the amplitudes of normal oscillations of the oscillators of the supramolecular receptor system means an abnormally large change in the volume of its Gibbs phase space, which is described by equation:

G′ = πⁿ • Π m_i* • ω_i • (q_0i + f_i/(m_i • (ω_i² − γ_i²))²

where G′ is the Gibbs phase volume of the receptor during resonant interaction with released agent.

A significant change in the phase volume corresponds to an abnormally strong reaction of the receptor to the effect of released agent, accompanied by a strong signal to the organism—a signal to the beginning of an abnormal pathological reaction.

As long as the thermodynamic coordinate of the stationary energy state of the receptor (I_s,R) is described by Equation (19), then significant change in the phase volume will cause a significant change of I_s,R.

I_s,R = ln G′ − ln G

(19)

Thus, the pathological reaction of the body to the effects of released agents is caused by the coincidence or extreme proximity of the configurations (symmetries) of the supramolecular systems of the receptor and released agent. Such a reaction is similar to the onset of resonance. The author has obtained formulas that relate the value of the invariant to the change in the parameters of the receptor in case of an abnormal reaction of the receptor to the action of release agent. However, the author does not have experimental data for carrying out quantitative calculations using these formulas.

7. Conclusions

The supramolecular system of a released agent consists of several interacting parts, therefore, a supramolecular system is a complex system. A supramolecular system can change its structure and symmetry when its configuration changes, and the change in configuration is accompanied by a change in the energy of interaction of the system parts. Consequently, in this paper, an invariant has been derived that characterizes the symmetry and integrity of the supramolecular system of a released agent. The physical meaning of this invariant has been revealed: the difference between the entropy of a complex system and the total entropy of non-interacting subsystems that make up this complex system. This invariant has been termed “the thermodynamic coordinate of the stationary energy state”.

The obtained theoretical findings allowed for the interpretation of some of the experimental results. Based on DSC thermograms of lactose hydrate preparations obtained using HD aqueous solutions (released agents) of two substances, Natrum Mur and Sulfur, of various levels of dilution (Konar [24]), the thermodynamic coordinates of the stationary energy state for various types of lactose preparations were calculated.

Changes in the thermodynamic coordinate of the stationary energy state show that the structure and the symmetry of released agents reflected in lactose preparations varies depending on the type of the original substance of HD solution and the level of dilution of this solution.

The use of methods of chemical thermodynamics of supramolecular systems allowed the author of this manuscript to describe the phenomenon of modification, that is, the joint effect of the original substance and its released agent on the receptor, as a process of pre-organization of the receptor by released agent.

A model of an atypical pathological reaction of an organism to the effects of released agents has been developed. Thus, an atypical pathological reaction of the organism to the effect of a released agent can be interpreted as a transmission of symmetry from the released agent to the receptor.

These results indicate the existence of a connection between the symmetry of the supramolecular system of a released agent and the specific properties of the released agent. Methods of statistical thermodynamics, chemical thermodynamics of supramolecular systems, and analytical mechanics have been used to identify this relationship. Thus, thermodynamics can explain the existence of supramolecular structures in high dilutions, and such structures can provide the effects of HD aqueous solutions. The use of fundamental sections of theoretical physics to describe the symmetry and specific properties of released agents makes it possible to conduct not only experimental, but also theoretical research of released agents; the effects of HD aqueous solutions do not contradict the laws of thermodynamics.

The invariant obtained in this work is a parameter that characterizes the symmetry of the supramolecular system, and it can also be applied in other studies: for example, in studies of fluctuating asymmetry. Fluctuating asymmetry serves as an indicator of problems in the development of an organism. A change in the phenotype is a consequence of strong environmental influences affecting genome and transcriptome.

It is of interest to establish a connection between the visually detectable manifestations of fluctuating asymmetry (for example, described in (Vilaseka [30])) and the values of the invariant in the genome: DNA molecules in their aqueous environment. The value of this parameter can be determined using DSC.

It is also of interest to establish a connection between changes in supramolecular chirality and changes in the parameter characterizing the symmetry of the supramolecular system. Supramolecular chirality concerns all aspects of chirality (transfer, generation, amplification, memorization, and modulation) in the case of non-covalent interactions. The paper (Fujiki [31]) notes that many physical and chemical properties of macromolecules and polymers are influenced by solvent molecules. Solvent molecules have great conformational freedom, since they are associated with macromolecules and polymers by non-covalent bonds. This conclusion is in line with the main theses of the current work.

Symmetry plays an important role in chiral recognition (Simonneaux [32]). The quantitative evaluation of the chirality of supramolecular systems remains one of the significant topics in biophysics. In the article (Sidorova [33]) the authors have proposed methods for determining the protein’s chirality through quantitative analysis. It is interesting to compare their results with the results of measuring a parameter characterizing the symmetry of the supramolecular structure of a protein with its aqueous environment. The value of this parameter can be determined using DSC.

The results obtained in the current work can be applied in other studies related to the symmetry of supramolecular systems.