# The Use of the Statistical Entropy in Some New Approaches for the Description of Biosystems

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## Abstract

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## 1. Introduction

## 2. Entropy and the Role of Correlations of Parts of an Organism as a Whole

^{13})!. For a multicell organism, each cell can be unique (for example, in the brain). In this case the number of combinations must be equal to 1. An example of a biological system with unique arrangement of each cell (neuron) is the brain. In this case entropy S = ln 1 = 0. A similar estimation was made in ref. [24], but, according to the thermodynamic criterion, it can be proposed that any biological system is almost no more ordered than a piece of rock of the same mass. Thus, the entropy of a set of 10

^{13}different one-cell organisms is almost equal to the entropy of the human body containing 10

^{13}cells. This strange statement is based on the analogy with the thermodynamic estimate of entropy, for example, when cooling a relatively large amount of water. But building a living organism (related to the mentioned value of entropy) is a very difficult task because an organism is a complex nonequilibrium system (see further the kinetic model of metabolism).

^{13})! and in the second case ((10

^{12})!)

^{10}if there are 10 different organs and each contains 10

^{12}cells. It is clear that the number of combinations in the second case is less, and therefore entropy is also less.

^{12})!)

^{10}for the Boltzmann formalism. Thus, the thermodynamic probability is (10

^{13})!/((10

^{12})!)

^{10}. For the simplest example, we consider 100 cells. For a set of 100 cells we have 100!. If we have 10 organs of 10 cells each, the number of variants is (10!)

^{10}. The thermodynamic probability is 100!/(10!)

^{10}, and it is greater than one. Note that we use the so-called thermodynamic probability of the macroscopic state. It is equal to the number of ways (the number of microscopic states) in which a given macroscopic state can be realized. This probability is greater than or equal to one. The thermodynamic probability W is related to the entropy S by S = k

_{B}ln W, where k

_{B}is Boltzmann constant.

^{10}(here a constant k

_{B}is omitted for simplicity), we obtain the well-known expression for the entropy, and one can see that this value for a multicellular biological organism is significantly less than the entropy for a set of different cells.

_{1}, …, N

_{m}. The number of complexes (thermodynamic weight) is equal to N!/(N

_{1}! …N

_{m}!). Here m is the number of levels or in terms of biological approach the number of organs.

## 3. Statistical Entropy in Description of Living Systems at the Molecular Level

## 4. Model of Adsorption and Entropy Evaluation

^{N}= 4 possible microstates. Taking 1 and Kc as the statistical weights of an empty and an occupied site, respectively, one can readily construct the grand partition function going over all the states (see Figure 1) [49]:

_{ij}is the probability of finding the lattice in which two sites are in the ith and jth states. The explicit forms of the probabilities are given in Figure 1.

^{2}= 1 competes with the latter state.

_{max}, absolutely coincides with the fundamental Boltzmann equation, up to Avogadro constant, N

_{A}:

## 5. Kinetic Model of Metabolism and Entropy

_{M}is the Maxwellian (equilibrium distribution). Equation (5) is an equation of the so-called BGK type (see ref. [64]); moreover one can imply that a constant characteristic relaxation time appears in Equation (5). Such an equation is really connected to the Boltzmann transport equation, but it is simpler and can potentially reflect relaxation processes in more complex than gaseous media.

_{0}is the mean boundary velocity for x = 0. For the distribution function we construct the analytical solution using a method of expansion in this parameter. For the first order approximation in a small parameter (of the ratio of internal energy to kinetic energy) we obtain

_{M}

_{0}, are computed through the macroscopic parameters of the nonequilibrium distribution upstream. The main term in the exponent that determines the spatial relaxation is the same order as in the zeroth approximation. Thus, l = u

_{0}τ, where l is the characteristic spatial value of decay of the nonequilibrium state, τ is the average time of interactions (collisions). One can generalize the result if considering τ

_{inn}, i.e., the characteristic mean time of the biochemical reactions, f

_{Minn}= f

_{M}(x, e) is the equilibrium function with the temperature T

_{inn}of the biosystem (the body temperature). Here the temperature appears in the standard expression for the equilibrium distribution (Maxwellian) f

_{Minn}.

_{x}is the x-component of molecular velocity. The entropy flux enters through the left boundary and exits (with a larger value of the entropy flux) through the right boundary of the considered one-dimensional system.

_{x})

_{downstream}> (S

_{x})

_{upstream}.

_{x}(x) calculated from Equation (8). That is consistent with Schrödinger’s ideas that life is powered by negative entropy (negentropy). Indeed, the H-function (H = −S), i.e., negentropy decreases. This flux is the entropy pump mentioned above. Therefore, the difference of output and input entropy fluxes reads

_{eq}(x) and S

_{neq}(x) are the values of statistical entropy at equilibrium and nonequilibrium states, respectively. This difference, ΔS, from Equation (10) can give a possible estimate of the complexity of living objects. This formula actually characterizes the “integral distance” (“metric”) between the “living line” and the “unliving line” in Figure 7, i.e., the complexity of a living organism is a measure of its remoteness from the equilibrium state, which is the simplest basic description of the distribution in terms of velocities or energies.

_{M}depends on the density n(x), mean velocity u(x), and temperature T(x) calculated using the distribution f (x, ξ) from Equation (8). The relaxation process ends at any point with equilibrium, and the entropy in Figure 7 approaches the upper curve. This line will be unchanged even if the mean velocity is zero. Indeed, we can obtain that the entropy for equilibrium with the same density and temperature and with different mean velocities is the same. For the equilibrium entropy from Equation (7) we have

_{neq}to a “non-living line” S

_{eq}.

## 6. Discussion and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Four possible states of the system consisting of two-site lattice and binding ligands, and their corresponding probabilities (indices 0 and 1 stand for an empty and an occupied site, respectively).

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**MDPI and ACS Style**

Aristov, V.V.; Buchelnikov, A.S.; Nechipurenko, Y.D.
The Use of the Statistical Entropy in Some New Approaches for the Description of Biosystems. *Entropy* **2022**, *24*, 172.
https://doi.org/10.3390/e24020172

**AMA Style**

Aristov VV, Buchelnikov AS, Nechipurenko YD.
The Use of the Statistical Entropy in Some New Approaches for the Description of Biosystems. *Entropy*. 2022; 24(2):172.
https://doi.org/10.3390/e24020172

**Chicago/Turabian Style**

Aristov, Vladimir V., Anatoly S. Buchelnikov, and Yury D. Nechipurenko.
2022. "The Use of the Statistical Entropy in Some New Approaches for the Description of Biosystems" *Entropy* 24, no. 2: 172.
https://doi.org/10.3390/e24020172