Rate of entropy model for irreversible processes in living systems

In living systems, it is crucial to study the exchange of entropy that plays a fundamental role in the understanding of irreversible chemical reactions. However, there are not yet works able to describe in a systematic way the rate of entropy production associated to irreversible processes. Hence, here we develop a theoretical model to compute the rate of entropy in the minimum living system. In particular, we apply the model to the most interesting and relevant case of metabolic network, the glucose catabolism in normal and cancer cells. We show, (i) the rate of internal entropy is mainly due to irreversible chemical reactions, and (ii) the rate of external entropy is mostly correlated to the heat flow towards the intercellular environment. The future applications of our model could be of fundamental importance for a more complete understanding of self-renewal and physiopatologic processes and could potentially be a support for cancer detection.


OVERVIEW
We restrict ourselves to the study of the thermodynamics in the single cell representing an open thermodynamic system. Since we are dealing with phenomena taking place locally and under conditions of local equilibrium inside and outside a typical cell, all intensive and extensive thermodynamic variables have a space and time dependence [S1]. On this basis, it is useful to define the rate of entropy density production, In the next two sections, we give the details about the calculation of i r and e r , respectively valid for any irreversible processes occurring in a cell, either normal or cancer. The calculation lies on thermodynamic arguments combined with heat and mass transport equations. The expressions are general and valid for any irreversible chemical process and we then apply them to glucose catabolism.
To calculate the rate of entropy density production, for the sake of convenience and without loss of generality, we represent the cell (either normal or cancer) as a cube of volume 3 cell VL  taking as reference the breast epithelium tissue. Here, L is the average size with L = 10 m for a typical normal cell and L = 20 m for a cancer cell [S2] and we assume that the flows occur mainly along the x direction within a 1D model (see Fig. 1 in the main text). In the numerical calculations shown in the following we assume that all irreversible processes take place at x = L/2 and for values of y and z corresponding to the region of the cytoplasm where glucose catabolism occurs (see the main text for more details).
To describe the glucose catabolism we recall the two reactions described in detail in the main text, namely the respiration process and the lactic acid fermentation process involving glucose (C6H12O6) catabolism. The respiration process is summarized as C6H12O6 + 6O2  6 CO2 + 6 H2O leading to the formation of carbon dioxide (CO2) and water (H2O). The lactic acid fermentation process leads to the formation of two lactic acid ions (C3H5 O3-) and two protons (H + ) and is summarized in the 3 simple form C6H12O6 → 2 C3H5 O3-+ 2 H + [S3, S4] (for a more detailed discussion on this point see the main text).

RATE OF INTERNAL ENTROPY DENSITY PRODUCTION
We define the space and time dependent RIEDP     i i ds ,t r ,t dt  x x (with = ( , , ) x y z x and t the time) giving the amount of local increase of entropy in continuous thermodynamic systems. In our special case, we consider a cell (either normal or cancer) and the RIEDP associated to irreversible processes occurring inside it.
In order to do that, we recall its general expression in terms of heat flow and mass flow (see is the thermodynamic force. In our 1D model, without loss of generality, flows are assumed along x; hence, = ( , 0, 0) . Heat flow occurs symmetrically with respect to x = L/2 along the two directions (x and -x).
To calculate the heat flow, we make the assumption that heat diffusion is mainly due to a conduction transport neglecting, in a first approximation, the convection transport present to a much lesser extent inside a typical cell. The heat transport equation in the 1D case, neglecting the term of heat source, takes the well-known form: Here, the solution to Equation (S3) T (x,t) is the temperature distribution function depending both on spatial and time variable, and, for a given material,  = K/(cs) is the thermal diffusivity in m 2 /s with K the thermal conductivity in J/ (m s K), cs the specific heat in J /(Kg K) and  the density in Kg/ 5 m 3 . Both K and  are assumed uniform throughout the cell. We impose the following initial and boundary conditions on the temperature distribution: This choice is not restrictive because the temperature vanishes only exactly at the cell border due to the boundary conditions corresponding to the cell membrane and has a weak dependence on x inside the cell. However, note that the temperature on the cell membrane is not zero as inferred from the expression of the temperature distribution in the intercellular environment (see paragraph B for details) obtained in the absence of boundary conditions. The solution to equation (S3) taking into account the initial and boundary conditions expressed by equation (S4)    For the 1D case where an exponential time decay depending on a typical decay time  has been included to describe the time evolution of the force. Hence, by inserting the solution to heat equation given in equation (S5) . The minus sign on the second member only indicates that heat flows from the region at higher temperature corresponding to the cell centre to the region at lower temperature close to the cell membrane, namely in the direction along which the temperature decreases (in this case it is symmetrical along +x and -x). Explicitly Here, p denotes the frequency of occurrence of the irreversible reaction. In Fig. S3, we display the bidirectional heat flow per unit time and area given in equation (S7) [S5] (see the main text for more details). The trend of JQ is symmetric with respect to the centre for the two types of cells, is less sharp for a cancer cell especially for t ranging between 0 and 100 s and tends to zero with increasing time.  (1)

PROCESS
From equation (S1), we express the contribution to RIEDP related to diffusion process via the term The thermodynamic force that gives rise to matter (chemical species) flow is where g = G/V is the Gibbs free energy density at constant pressure and G is the Gibbs free energy. According to its general definition given above, the chemical potential depends on the temperature and on the pressure.
For the 1D case,

 
(2) (2) k k ,0,0 is the time and space dependent chemical potential of the kth chemical species with  a typical cell decay time; k u is the chemical potential calculated at x = L/2, where it is assumed that the glucose catabolism takes place, that is equal to the partial molar energy. According to this form, the chemical 10 potential decreases passing from the center to the border of the cell and decreases with increasing time. For every irreversible reaction and for every chemical species   According to the initial condition in Equation (S13), it is reasonably assumed that a pulse of solute at t = 0 is present at a given point x that in our special case corresponds to the center of the cell x = L/2. In the special case, the solutes are the molecules that take part either in the cell respiration or in the lactic acid fermentation process. The boundary condition   ,0 n x t    is realistically satisfied for a value of x close to the cell membrane.  By means of equations (S11) and (S15), we get the general expression of ri D (x,t) valid for any irreversible processes, viz. (S16)  after having neglected the term proportional to the sine series at the numerator that is much smaller than the one proportional to the cosine series. Equation (S16) is equation (2) of the main text.
In particular, equation (S16) applied to glucose catabolism in normal and cancer cells reads with wresp and wferm the probability weights associated to respiration and fermentation processes, respectively with wferm = 1wresp and Nresp and Nferm the corresponding numbers of chemical species and   ,0 iD r x t  .

REACTIONS
We now study the term contributing to the RIEDP due to the irreversible chemical reactions. From Equation (S1) this contribution takes the general form: Here, the subscript "r" stands for reactions, the affinity of the jth reaction reads V . In this framework, the affinity plays the role of the thermodynamic force and the velocity that of the corresponding thermodynamic flow associated to irreversible reactions.
In our analysis, we set M = 1 for every chemical reaction and we assume that flows of molecules are along the x direction so that Combining equations (S19), (S20) and the expression of the velocity of a reaction (equation (S21) and (S22) Here p=0, 1, 2, q = 0, 1, 2 and p + q = 1, 2 for first-and second-order irreversible chemical reactions, respectively and Nm A reag/Vcell (Nm B reag/Vcell) is the molar concentration of reagents A and B, respectively taking the volume of the solution equal to Vcell.
We now apply this formalism to glucose catabolism given by a sequence of reactions of glucose catabolism, either as respiration or as lactic acid fermentation process, classified as first-order. For this special case, we write 6 12 6 6 12 6 6 12 6 cell / 1 v 0


Here, wresp and wferm are the probability weights associated to respiration and fermentation processes, respectively with wferm = 1wresp and Nresp and Nferm are the corresponding numbers of chemical species.

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In Fig. S7, we plot the affinity for a normal cell (panel (a)) and for a cancer cell (panel (b)). In the numerical calculations, we have taken the following parameters: wresp = 0.8 (0.1) and wferm = 0.2 (0.9) for a normal (cancer) cell 5 ,  = 10 -4 s, kkin = 10 -4 /s for normal cells and kkin = 10 -5 /s for cancer cells [S6]. The values used for the chemical potentials at x = L/2 and t = 0 (partial molar energy) of the different chemical species are the ones in Table 1 of the main text.
In both cases, the affinity tends to zero with increasing time as should be expected for every thermodynamic system moving towards equilibrium. Looking at Fig. S7, it turns out that the affinity for a normal cell is one order of magnitude greater than the corresponding one for a cancer cell. This means that the thermodynamic force associated to the glucose catabolism reaction in a cancer cell is weaker with respect to that of the corresponding normal cell. Like for other first-order reactions, we write the velocity of the glucose catabolism in the form

RATE OF EXTERNAL ENTROPY DENSITY PRODUCTION
The human cell (either normal or cancer cell) behaves like an open thermodynamic system. This means that it exchanges energy and matter with the intercellular environment. In this respect, it is useful to define the rate re (x,t) of external entropy density production (REEDP) giving the amount of local entropy density outside a cell in the intercellular environment. Specifically, the REEDP has a contribution related to heat diffusion linked to energy exchange between the cell and the intercellular environment and a contribution due to matter exchange with the intercellular environment in terms where, d1 (d2) is a characteristic time such that 1/ d1 (1/ d2) is about 10 -5 /s (10 -4 /s), namely of the order of the pathway kinetic constant of the glucose catabolism reaction in both processes.

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By analysing the behaviour of the three contributions to REEDP it is   lim , 0 e t r x t   . Hence, we have proved that the equilibrium state of an open thermodynamic system like a cell (either normal or cancer), where irreversible processes take place, implies the minimization of both ri and re with increasing time and its vanishing in the limit of infinite time.