# Entropy Density Acceleration and Minimum Dissipation Principle: Correlation with Heat and Matter Transfer in Glucose Catabolism

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## Abstract

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## 1. Introduction

_{i}/dt where S

_{i}is the entropy of the system with T the temperature. We get dΨ/dt ≤ 0 that tends to be minimum (zero) in a steady-state (and as a special case of steady state at global thermodynamic equilibrium) [7]. In this study, we prove Prigogine’s minimum energy dissipation principle via the calculation of a novel physical quantity, the time derivative of the rate of entropy density production. This quantity is defined as the entropy density acceleration generated by heat and matter transfer inside cells and with the intercellular environment in irreversible reactions such as the ones characterizing glucose catabolism. The introduction of the entropy density acceleration stems from the mechanical concept of acceleration that plays a crucial role to characterize the dynamics of rigid bodies in kinematics and mechanics. Basing on the analogy between the behavior of mechanical systems and thermodynamic systems, we redefine this concept in out-of-equilibrium thermodynamics. We remind that this concept derived from mechanics and applied to thermodynamics is not an end in itself and the model developed does not represent a mere theorization but tries to describe reality, as it is consistent with the experimental data of several works [11,12,13,14,15].

## 2. Methods

#### 2.1. Entropy Density Acceleration for Glucose Catabolism

_{i}(x,t) + r

_{e}(x,t). Here, r

_{i}= ∂s

_{i}/∂t ≥ 0 is the RIEDP with, in a compact form, s

_{i}= S

_{i}/V (S

_{i}is the internal entropy and V is the volume of the thermodynamic system) the internal entropy density, and r

_{e}= ∂s

_{e}/∂t is the REEDP with, in a compact form, s

_{e}= S

_{e}/V (S

_{e}is the external entropy) the external entropy density. This latter quantity is often called in the literature the entropy flow or entropy flow rate or entropy production rate of the system [18,19,22]. However, for the sake of simplicity, we have adopted the same nomenclature of the internal contribution.

_{i}(x,t) = r

_{i Q}(x,t) + r

_{i D}(x,t) + r

_{i r}(x,t), with r

_{i Q}(x,t) the contribution due to heat flow and transfer inside the cell, r

_{i D}(x,t) the one associated to molecules diffusion and internal transport and r

_{i r}(x,t) the one due to irreversible chemical reactions occurring inside the cell. In the special case studied, glucose catabolism occurs in two compartments, cytoplasm and mitochondria characterized by different internal structure morphology. In particular, some reactions occur in the cytoplasm (compartment 1), giving rise to internal entropy production dS

_{1}and other reactions in the mitochondria (compartment 2), giving rise to internal entropy production dS

_{2}. According to the local formulation of the second principle of thermodynamics, due to the extensive nature of entropy, the total entropy production is dS

_{tot}= dS

_{1}+ dS

_{2}, that is dS

_{tot}is the sum of the contributions of its subparts [18]. We have applied this scheme to each internal heat and matter contribution. It was then reasonable to divide each contribution by the average volume of the cell that is different for normal and cancer cells calculating first the rate of internal entropy density production ∂s

_{i}/∂t for heat and mass contributions and then the internal entropy density accelerations. Hence, even though it does not appear explicitly, the compartmentalization of the cell has been taken into account in the calculations.

_{i}(x,t) = ∂r

_{i}(x,t)/∂t for the internal entropy density acceleration (IEDA) with a

_{i}(x,t) = a

_{i Q}(x,t) + a

_{i D}(x,t) + a

_{i r}(x,t). Instead, r

_{e}(x,t) = r

_{e Q}(x,t) + r

_{e}

_{exch}(x,t) with r

_{e Q}(x,t) (r

_{e}

_{exch}(x,t)) is the contribution due to the heat transfer (matter exchange) from the cell to the intercellular environment. As a result, the external entropy density acceleration (EEDA), a

_{e}(x,t) = ∂r

_{e}(x,t)/∂t includes two contributions, a

_{e}(x,t) = a

_{e Q}(x,t) + a

_{e}

_{exch}(x,t) where a

_{e Q}(x,t) = ∂r

_{e Q}(x,t)/∂t (a

_{e}

_{exch}(x,t) = ∂r

_{e}

_{exch}(x,t)/∂dt) is the acceleration contribution related to heat (matter) transfer between the cell and the intercellular environment. Therefore, owing to the previous definitions a(x,t) = a

_{i}(x,t) + a

_{e}(x,t) is the total acceleration. We note, according to this framework, that the entropy density acceleration has a space and time dependence and this latter dependence is still one-dimensional as for r(x,t) = r

_{i}(x,t) + r

_{e}(x,t).

#### 2.2. Internal Entropy Density Acceleration for Glucose Catabolism

_{iQ}(x,t) due to heat transfer inside the cell is:

_{k}is the partial molar energy and

**F**

_{u}(x,t) =

**∇**(1/T(x,t)) is the heat thermodynamic force driving ${J}_{u}$ with T(x,t) the temperature distribution. Instead, the IEDA due to mass diffusion and matter exchange inside the cell assumes the form:

**F**

_{k}(x,t) =

**∇**(μ

_{k}(x,t)/T(x,t)) with μ

_{k}the chemical potential of the kth chemical species with k = 1, 2, …, N. The entropy density acceleration generated by the chemical irreversible reactions reads:

_{jk}the stoichiometric coefficients, and v

_{j}= 1/V

_{cell}dξ

_{j}/dt is the velocity of the jth reaction with dξ

_{j}the variation of the jth degree of advancement and V

_{cell}= L

^{3}is the volume of the cubic cell (L is the side of the average cube). We now calculate the IEDA a

_{i Q}(x,t) = ∂r

_{i Q}(x,t)/∂t associated to the heat flow during glucose catabolism recalling the expression of r

_{i Q}(x,t) [43]:

_{i Q}(x,t) includes two terms depending on the trigonometric series: the first term is inversely proportional to the characteristic decay time, while the second one is proportional to the thermal conductivity.

_{i D}(x,t) = ∂r

_{i D}(x,t)/∂t either for respiration or fermentation process recalling the corresponding rate of entropy density [43]:

_{0}is the maximum cell temperature, N

_{α}is the number of chemical species in either the respiration or fermentation process, D

_{k}is the diffusion constant of the kth chemical species, μ

_{k}= u

_{k}

_{∙}

_{e}

^{−(|x − L/2|/L + t/}

^{τ)}is the kth species chemical potential, and N

_{m k}is the number of moles of the kth chemical species. The IEDA associated to matter diffusion inside the cell either for respiration or fermentation process reads:

_{i Dα}(x,t) consists of three contributions each of them weighted by the trigonometric series.

_{ferm}and w

_{resp}expressing the frequency of occurrence of respiration and fermentation process in a normal and in a cancer cell (see Section 3, Results), we write the total acceleration contribution a

_{i D}due to the two metabolic pathways as:

_{i D}(x,t) = w

_{resp}a

_{i D}

_{resp}(x,t) + w

_{ferm}a

_{i D}

_{ferm}(x,t)

_{i r}(x,t) = ∂r

_{i r}(x,t)/∂t caused by irreversible reactions occurring inside the cell during glucose catabolism via the corresponding rate expressed in the form [43]:

_{kin}

_{α}the kinetic constant, ν

_{k}stoichiometric coefficients and N

_{m}

_{Glucose}the number of glucose moles (note the typo error L/2 – x in the argument of the sine in Equations (4), (10) and (11) of [43]). We get a

_{i r}(x,t) either for respiration or for lactic acid fermentation:

_{i r}

_{α}(x,t) consists of three contributions. We express the entropy density acceleration due to irreversible reactions during glucose catabolism in the form:

_{i r}= w

_{resp}a

_{i r}

_{resp}+ w

_{ferm}a

_{i r}

_{ferm},

#### 2.3. External Entropy Density Acceleration for Glucose Catabolism

_{e}(x,t) = ∂r

_{e}(x,t)/∂t includes two contributions, namely a

_{e}(x,t) = a

_{e Q}(x,t) + a

_{e}

_{exch}(x,t). Here, a

_{e Q}(x,t) = ∂r

_{e Q}(x,t)/∂t (a

_{e}

_{exch}(x,t)= ∂r

_{e}

_{exch}(x,t)/∂t) is the acceleration contribution related to heat (matter) transfer between the cell and the intercellular environment. The general expression for a

_{e Q}(x,t) reads:

_{ic}is the intercellular temperature and dQ/dt is the time derivative of the heat. Instead, the general expression for a

_{e}

_{exch}takes the form:

_{pr}is the number of products of the reaction and N

_{m k}is the number of moles of the kth product of reaction and.

_{e Q}contribution to EEDA, we recall the corresponding REEDP for respiration and fermentation process [43]:

_{B}= 1.3805 × 10

^{−23}J/K is the Boltzmann constant, N

_{m}

_{pr resp}(N

_{m}

_{pr ferm}) is the number of moles of the products in respiration (fermentation) process. Note the contribution proportional to 1/t on the second member that breaks the time reversal symmetry [42] as occurs for the other rate contributions. In principle, at small distances from the border of the cell (for small x) the term proportional to 1/t is greater than the term proportional to 1/t

^{2}resulting in a negative r

_{e Q}for the typical intercellular size. However, in our model, we have set ourselves under the hypothesis of large x and small t (1000 μs << 1 s) neglecting the term proportional to 1/t even close to the cell border yielding:

_{e Q}for any x outside the cell and any t. This means that the entropy outside the cell increases because of the heat released by the irreversible reactions occurring inside the cell but this increase leads to a decrease of the rate of entropy of the cell because heat is removed from the cell. Hence, the EEDA a

_{e Q}(x,t) takes the simple form:

_{0}= 10 μm is a characteristic length having the size of a normal cell, dτ

_{β}(β = 1,2 with 1 referred to respiration and 2 to fermentation) is a characteristic time such that 1/dτ

_{1}(1/dτ

_{1}) is 10

^{−5}s

^{−1}(10

^{−4}s

^{−1}) of the order of k

_{kin}, N

_{pr resp}(N

_{pr ferm}) is the number of products of the respiration (fermentation) process. Also this matter contribution has a positive contribution if referred to the intercellular environment but it would be taken as negative if referred to the cell because matter exchange leads to a removal of the products of irreversible reactions from the cell.

_{e}

_{exch}

_{α}(x,t) = ∂r

_{e}

_{exch}(x,t)/∂t resulting from irreversible exchange of matter with the intercellular environment turns out to be:

_{e}

_{exch}

_{α}(x,t) consists of three contributions. We calculate the entropy acceleration related to exchange of matter with the intercellular environment as:

_{e}

_{exch}= w

_{ferm}a

_{e}

_{exch ferm}+ w

_{resp}a

_{e}

_{exch resp},

## 3. Results

_{6}H

_{12}O

_{6}+ 6O

_{2}→ 6 CO

_{2}+ 6 H

_{2}O and leading to the formation of carbon dioxide (CO

_{2}) and water (H

_{2}O). Instead, fermentation process or aerobic glycolysis consists only of glycolytic step, C

_{6}H

_{12}O

_{6}→ 2 C

_{3}H

_{5}O

_{3−}+ 2 H

^{+}and leads to the formation of lactic acid ions (C

_{3}H

_{5}O

_{3−}) and hydrogen ions (H

^{+}).

^{−4}s as a typical cell decaying time. We have taken as weights for respiration (fermentation) process w

_{resp}= 0.8 (w

_{ferm}= 0.2) in a normal cell and w

_{resp}= 0.1 (w

_{ferm}= 0.9) in a cancer cell. We have also used for both normal and cancer cells the following parameters: thermal conductivity K = 0.600 J/(m s K), thermal diffusivity in water κ

_{H20}= 0.143 × 10

^{−6}m

^{2}/s, diffusion constants at standard conditions: D

_{C6H12O6}= 6.73 × 10

^{−10}m

^{2}s

^{−1}, D

_{O2}= 21.00 × 10

^{−10}m

^{2}s

^{−1}, D

_{CO2}= 19.20 × 10

^{−10}m

^{2}s

^{−1}, D

_{H2O}= 21.00 × 10

^{−10}m

^{2}s

^{−1}, D

_{C3H5O3−}= 9.00 × 10

^{−10}m

^{2}s

^{−1}and D

_{H+}= 45.00 × 10

^{−10}m

^{2}s

^{−1}in aqueous solution. Finally, we have employed the partial molar energies or chemical potentials at t = 0 and x = L/2 and, at standard conditions: μ

_{C6H12O6}= −917.44 kJ/mole, μ

_{O2}= 16.44 kJ/mole, μ

_{CO2}= −385.99 kJ/mole, μ

_{H2O}= −237.18 kJ/mole, μ

_{C3H5O3−}= −516.72 kJ/mole where C

_{3}H

_{5}O

_{3−}is the lactate ion and μ

_{H+}= 0 kJ/mole in aqueous solution.

_{i Q}we have taken as frequency of occurrence of glucose catabolism the value p = 0.85 (0.90) for normal (cancer) cells, while for the calculation of a

_{i r}we have taken as values of the pathway kinetic constants k

_{kin}

^{resp}= 10

^{−5}s

^{−1}and k

_{kin}

^{ferm}= 10

^{−4}s

^{−1}, and N

_{m}

_{Glucose}= 1 as a reference concentration.

#### 3.1. Entropy Density Accelerations for Normal and Cancer Cells: Numerical Calculations

_{e Q}. Indeed, in the first instants of time (0–100 μs), some terms of the entropy density acceleration are positive as depicted in the insets and, due to the appreciable magnitude, this would mask the leading negative trend of the accelerations in the time interval 100–1000 μs. The positive trend of most of the accelerations during the first instants of time is not surprising and is due to the initial increasing behavior of the corresponding rates. Of course, this behavior is only secondary to the leading and most important negative trend characterizing all the entropy density accelerations.

#### 3.2. Internal Entropy Density Acceleration: Numerical Calculations

_{i Q}calculated according to Equation (5) for a normal and a cancer cell, respectively. For both kinds of cells, a

_{i Q}dramatically increases with time close to the cell borders, while in the region close to the cell center exhibits a weak increase and an almost flat profile especially in the cancer cell. For t > 500 μs the spatial and time profile of a

_{i Q}is rather flat passing from the borders to the cell center and approaches zero with increasing time.

_{i D}obtained from Equations (7) and (8) exhibits a strong increase in the central region of the cell for the initial instants of time in both kinds of cells tending to zero for increasing time in the whole cell and exhibiting a flat profile (Figure 2c,d). Note the narrower shape of a

_{i D}in a cancer cell with respect to that in a normal cell, its higher rate of increase at the initial instants of time and a minimum value that is two orders of magnitude less than that of a normal cell. We attribute the general trend to the prevalence of the fermentation process in the cancer cell, while the lesser deep minimum is related to the bigger size of the cancer cell.

_{i r}computed according to Equations (10) and (11). a

_{i r}uniformly increases throughout the whole cell with increasing time but the rate of increase is much higher in a normal cell with respect to a cancer cell. Indeed, for a normal cell a

_{i r}approaches values close to zero for t less than 500 μs, while for a cancer cell this occurs for t more than 500 μs. This slower tendency towards zero in a cancer cell could be due to the prevalence of lactic acid fermentation. Moreover, the minimum value of a

_{i r}in a cancer cell is about three orders of magnitude less than that of a normal cell and this is in part due to the bigger size of the cancer cell.

_{i r}for a normal cell that is negative throughout the whole cell and a

_{i r}for a cancer cell that is negative especially in the central part of the cell. In particular, a

_{i Q}for both a normal and a cancer cell and a

_{i r}for a cancer cell exhibit positive values close to the cell borders, while a

_{i D}exhibits remarkable positive values close to the cell centre with some differences as a function of time between a normal and a cancer cell. The positive trend of these contributions reflects the increase of the corresponding rates at the first instants of time. The positive behavior of a

_{i r}in a cancer cell close to the cell borders could be due to the prevalence of the fermentation process with respect to the respiration process.

#### 3.3. External Entropy Density Acceleration: Numerical Calculations

_{e Q}for a normal and a cancer cell, respectively calculated according to Equations (16) and (17). For this entropy density acceleration, we have performed the numerical calculations taking the interval of time 0–1000 μs because a

_{e Q}, unlike the other contributions, does not exhibit a positive trend during the interval of time 0–100 μs.

_{e Q}becomes flat tending to vanish with increasing t.

_{e}

_{exch}for a normal and a cancer cell calculated according to Equations (19) and (20). The general trend is a uniform increase throughout the intercellular environment during the first instants of time. A sharper increase of a

_{e}

_{exch}characterizes the cancer cell because of the prevalence of the fermentation process. However, on average the absolute value of a

_{e}

_{exch}for a cancer cell is less than for a normal cell. For t larger than 500 μs, in both cases a

_{e}

_{exch}approaches zero with increasing time. In the insets of Figure 2c,d, a

_{e}

_{exch}plotted in the first instants of time (interval 0–100 μs) shows an opposite behavior taking positive values for a normal cell, and negative values for a cancer cell. This is not surprising and may be attributed to the different size of the cells.

_{i D}resulting from lactic acid fermentation and respiration processes, respectively and calculated by means of Equation (7). In the first instants of time, there is a strong rate of increase of a

_{i D}in the central region of the cell for both processes where a

_{i D}is strongly negative. However, there is a broader spatial and time dependence of a

_{i D}for respiration leading to a more extended region of the cell having negative a

_{i D}for small t. In addition, also the minimum of a

_{i D}, symmetric on the left and on the right of the center of the cell, is deeper for respiration. For both processes with increasing time a

_{i D}becomes flat and approaches zero. Figure 3c,d displays the corresponding a

_{i r}of the metabolic pathways calculated via Equation (10). Unlike a

_{i D}, there are not relevant differences in the spatial trends of a

_{i r}in the two processes that are uniformly negative throughout the cell even though the minimum for fermentation is much deeper than that for respiration. This trend is in part due to the kinetic constant k

_{kin}

^{resp}that is one order of magnitude less than k

_{kin}

^{ferm}.

_{e Q}for lactic acid fermentation and respiration, respectively calculated using Equation (16). In both cases a

_{e Q}exhibits a deep negative minimum the more the distance from the cell border that is of the same order of magnitude but more pronounced for respiration process. The rate of increase of a

_{e Q}with increasing time is the same approaching zero uniformly in space still at the initial instants of time. In Figure 4c,d, a

_{e}

_{exch}computed according to Equation (18) is displayed for lactic acid fermentation and respiration. Like for a

_{e Q}the order of magnitude of the negative minimum is the same but, with increasing the distance from the cell border, the trend of a

_{e}

_{exch}remains uniform for both processes. More specifically, the rate of increase of a

_{e}

_{exch}is slightly higher for lactic acid fermentation even though, at t > 500 μs, a

_{e}

_{exch}becomes flat and tends to vanish for both processes.

## 4. Discussion

_{i Q}and a

_{i D}for both kinds of cells, a

_{e}

_{exch}for a normal cell and a

_{i r}for a cancer cell during the first instants of time (interval 0–100 μs), where the entropy density accelerations are positive because of the peculiar time behavior of the terms contributing to these accelerations for small t.

_{i r}contribution that, for the same cell size taken as reference, is a few orders of magnitude (see Figure 3c) larger for lactic acid fermentation process than for respiration and also to the a

_{e}

_{exch}contribution that exhibits a uniform minimum about three times larger for lactic acid fermentation. Indeed, due to the bigger volume of a cancer cell (on average 8 times the one of the normal cell) and to the spatial dependence along x that in a cancer cell is twice the one of the normal cell, a

_{i r}and a

_{e}

_{exch}shown in Figure 1f and Figure 2d, respectively look only apparently of smaller magnitude in a cancer cell than in a normal cell. This finding reiterates the concept that cancer cells, where lactic acid fermentation prevails, are characterized by a higher entropy gain per unit time (rate of entropy) as found in [43] and, therefore, by a bigger negative entropy acceleration during the initial instants of time.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**IEDA generated by heat, matter and irreversible reactions during glucose catabolism for a time interval of 1000 μs. (

**a**) Calculated a

_{i Q}for a normal cell. Inset: calculated a

_{i Q}for a normal cell in the interval 0–100 μs. (

**b**) As in panel (

**a**) but for a cancer cell. (

**c**) Calculated a

_{i D}for a normal cell. Inset: calculated a

_{i D}for a normal cell in the interval 0–100 μs. (

**d**) As in panel (

**c**) but for a cancer cell. (

**e**) Calculated a

_{i r}for a normal cell. Inset: calculated a

_{i r}for a normal cell in the interval 0–100 μs. (

**f**) As in panel (

**e**) but for a cancer cell.

**Figure 2.**EEDA associated to heat and matter transfer between the cell and the intercellular envinronment. (

**a**) Calculated a

_{e Q}for a normal cell. (

**b**) As in (

**a**), but for a cancer cell. (

**c**) Calculated a

_{e}

_{exch}for a normal cell. Inset: calculated a

_{e}

_{exch}for a normal cell in the interval 0–100 μs. (

**d**) As in (

**c**), but for a cancer cell.

**Figure 3.**IEDA associated to matter transfer inside the cell for lactic acid fermentation and respiration. A representative cell having the size of a normal cell is depicted. (

**a**) Calculated a

_{i D}for fermentation process. (

**b**) As in (

**a**), but for respiration process. (

**c**) Calculated a

_{i r}for fermentation process. (

**d**) As in (

**c**), but for respiration process.

**Figure 4.**EEDA associated to heat and matter transfer from inside the cell to the intercellular environment for lactic acid fermentation and respiration. A representative cell having the size of a normal cell is depicted. (

**a**) Calculated a

_{e Q}for fermentation process. (

**b**) As in (

**a**), but for respiration process. (

**c**) Calculated a

_{e}

_{exch}for fermentation process. (

**d**) As in (

**c**), but for respiration process.

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## Share and Cite

**MDPI and ACS Style**

Zivieri, R.; Pacini, N.
Entropy Density Acceleration and Minimum Dissipation Principle: Correlation with Heat and Matter Transfer in Glucose Catabolism. *Entropy* **2018**, *20*, 929.
https://doi.org/10.3390/e20120929

**AMA Style**

Zivieri R, Pacini N.
Entropy Density Acceleration and Minimum Dissipation Principle: Correlation with Heat and Matter Transfer in Glucose Catabolism. *Entropy*. 2018; 20(12):929.
https://doi.org/10.3390/e20120929

**Chicago/Turabian Style**

Zivieri, Roberto, and Nicola Pacini.
2018. "Entropy Density Acceleration and Minimum Dissipation Principle: Correlation with Heat and Matter Transfer in Glucose Catabolism" *Entropy* 20, no. 12: 929.
https://doi.org/10.3390/e20120929