# Gear Shifting in Biological Energy Transduction

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Variation of the Force Ratio May Induce Catabolic Gear Shifting

#### 2.2. Gear Shifting Simulations

#### 2.3. Simulation of ATP Synthesis Flux through a Dual Pathway

#### 2.4. Discontinuous Optimal Gear Shifting

#### 2.5. Reproducibility and Accessibility of the Data

## 3. Results

#### 3.1. Mosaic Non-Equilibrium Thermodynamics and How the Variation of the Force Ratio Induces Gear Shifting of Catabolism

_{1}> 0 molecules of ATP. The remaining driving force $\left(\Delta {G}_{c}-{n}_{1}\xb7\Delta {G}_{p}\right)$ equals the catabolic Gibbs energy minus the stoichiometry (n

_{1}) multiplied by the Gibbs energy of ATP synthesis. As is customary in linear non-equilibrium thermodynamics [8,28,29], the flux is then assumed to be proportional to that remaining driving force. We use L

_{1}for the proportionality constant (‘catalytic capacity’) and write for the dependence of its flux (${J}_{c1}>0$) on the two relevant Gibbs energies:

_{1}, and the catalytic capacity of catabolism (${L}_{1}>0)$ is written as fraction $\left(1-\phi \right)$ of a total catalytic capacity $L\stackrel{\mathrm{def}}{=}{L}_{1}+{L}_{2}>0$ of catabolism. The explicit mention of the stoichiometry ${n}_{1}$ reflects that we here use the more mechanistic ‘mosaic non-equilibrium thermodynamics’ of Westerhoff and Van Dam [27].

#### 3.2. Phenomenological Non-Equilibrium Thermodynamics: Gear Shifting Affects the Phenomenological Stoichiometry as Well as the Degree of Coupling

#### 3.3. Variations in the Phenomenological Stoichiometry Could Improve ATP Synthesis and Growth

#### 3.4. Gear Shifting and Varied Relative Pathway Capacities Could Improve ATP Synthesis and Growth

## 4. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(

**a**) Dependence of ATP synthesis flux ($-{J}_{p}$; normalized by maximal catabolic flux,$L\xb7\Delta {G}_{c}$), variable flux–ratio stoichiometry $n$ = $(-{J}_{p}/{J}_{c}$) and thermodynamic efficiency $\left(\eta \right)$ on the counteracting force ratio $X=\Delta {G}_{p}/\Delta {G}_{c}$. Equation (9) was used for the simulation of $-{J}_{p}/\left(L\xb7\Delta {G}_{c}\right)$ as function of force ratio $X$; Equations (8) and (9) were used for the calculation of $-{J}_{p}/{J}_{c}$; the equation $\eta =-({J}_{p}/{J}_{c})\xb7X$ was used for simulation of $\eta $ as function of force ratio $X$. (

**b**) Flux–ratio stoichiometry versus ATP synthesis. Increase of the variable stoichiometry $n$= ($-{J}_{p}/{J}_{c}$) with increasing ATP synthesis flux. At a force ratio of 0.75, the output flux of ATP synthesis reverted to ATP degradation, while catabolism continued. This corresponds to a car still using gasoline to try to move forward and upward but being forced back down due to gravity. Consequently, the stoichiometry and the thermodynamic efficiency become negative. At the force ratio of 0.83, catabolism also inverted. At this and higher force ratios, the model simulates reverse operation where ATP hydrolysis would drive reversal of catabolism, which is not often realistic. (

**c**) Fractional flux through two pathways when varying the force ratio $X$. Equations (3) and (9) were used for the calculation of ${J}_{p1}/{J}_{p}$; Equations (5) and (9) were used for the calculation of ${J}_{p2}/{J}_{p}$. Results of computations for two parallel catabolic pathways with $\phi $ = 0.2, ${n}_{1}$ = 1, ${n}_{2}$ = 2, $L$ = 1, $\Delta {G}_{c}$ = 1, ${L}_{p}^{\ell}=0$.

**Figure 2.**Gear shifting. (

**a**) Normalized ATP synthesis flux versus force ratio at three magnitudes of the phenomenological stoichiometry $Z$, as well as (green line) their sum total and (purple line) ATP synthesis for ‘variomatic gear shifting’ optimal with respect to maximal ATP synthesis flux. Equation (17) was used for calculation of $-{J}_{p}/\left(L\xb7\Delta {G}_{c}\right)$ as a function of force ratio $X$. In these simulations, $q$ = 0.9 and various values of $Z$ (0.5, 1, 2, and the optimal variomatic $Z={q}^{2}/X$) were used as indicated. $-{J}_{p}/\left(L\xb7\Delta {G}_{c}\right)$

_{total}was calculated as the sum of three. (

**b**) Flux ratio stoichiometry as a function of the force ratio. Equation (19) was used for simulation of $n$. (

**c**) Flux ratio stoichiometry versus ATP synthesis flux.

**Figure 3.**Gear shifting of different pathways at different relative pathway activities $\phi $. (

**a**) Simulated ATP synthesis flux through a dual pathway (pathways 1 and 2 with two different ATP stoichiometries) as function of force ratio at various relative pathway capacities. (

**b**) The fractional flux through pathways with two different ATP stoichiometries when varying the force ratio X under different $\phi \u2019s$ (1, 0.375). (

**c**) The fractional flux through pathways with two different ATP stoichiometries when varying the force ratio X at different $\phi $ (0.25, 0). In these simulations, both $Z$ and $q$ varied as a function of $\phi $, which was kept constant at any of four values, while ${\rm X}$ was varied. Equation (9) was used for the simulation of $-{J}_{p}/\left(L\xb7\Delta {G}_{c}\right)$ as function of force ratio ${\rm X}$. Equations (3) and (9) were used for the calculation of ${J}_{p1}/{J}_{p}$; Equations (5) and (9) were used for the calculation of ${J}_{p2}/{J}_{p}$. In these simulations, ${n}_{1}$ = 1 and ${n}_{2}$ = 3, $L$ = 1, $\Delta {G}_{c}$ = 1, ${L}_{p}^{\ell}=0$, whilst for (

**a**) four values of $\phi $ (1, 0.375, 0.25, and 0) were used. $\phi $ is the catalytic activity of pathway 2 as compared to the catalytic capacity of the two combined.

**Figure 4.**Discontinuous optimal gear shifting. ATP synthesis as function of counteracting force ratio for four different stoichiometries, all together, as well as the optimal gear shifting case with the corresponding gear settings. The equation $-{J}_{p}=\left(1-\phi \right)\xb7n\xb7L\xb7{G}_{c}\xb7\left(1-n\xb7X\right)$ was used for simulation of $-{J}_{p}$ as function of force ratio ${\rm X}$ at four fixed values of $n$. In these simulations, ${n}_{1}$, ${n}_{2},{n}_{3},$ and ${n}_{4}$ were taken equal to 1, 2, 3, and 4, $\phi $ = 0, $L$ = 1, $\Delta {G}_{c}$ = 1. ${J}_{ptotal}$ is the sum of the four ${J}_{p}$’s. The yellow line (consisting of four smaller straight lines at angles with each other) represents the effect of operating always (i.e., at any force ratio ${\rm X}$) only a single one of the four gears, i.e., the one with the highest flux of ATP synthesis. The difference of the optimal gear setting with that of Figure 2a is that the shifting is not continuous but only between integer values of $Z$ as shown by the purple line labelled ‘gear’.

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**MDPI and ACS Style**

Zhang, Y.; Westerhoff, H.V.
Gear Shifting in Biological Energy Transduction. *Entropy* **2023**, *25*, 993.
https://doi.org/10.3390/e25070993

**AMA Style**

Zhang Y, Westerhoff HV.
Gear Shifting in Biological Energy Transduction. *Entropy*. 2023; 25(7):993.
https://doi.org/10.3390/e25070993

**Chicago/Turabian Style**

Zhang, Yanfei, and Hans V. Westerhoff.
2023. "Gear Shifting in Biological Energy Transduction" *Entropy* 25, no. 7: 993.
https://doi.org/10.3390/e25070993