# Dissipation-Driven Selection under Finite Diffusion: Hints from Equilibrium and Separation of Time Scales

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Phase-Transition for Selection in Two-State Systems

## 3. Simplest Case for Selection: A Three-State System

## 4. Time-Scale Separation and Equilibrium Hints

#### 4.1. Fast-Dissipation Chemical Sub-Networks in Two-Box Models

#### 4.2. Fast-Dissipation Ensemble Distribution in Two-Box Models

#### 4.3. Numerical Results and Energy Landscapes

#### 4.4. Fast-Dissipation Chemical Sub-Networks for Continuous Systems

## 5. Diffusion-Controlled Switch of Selection

## 6. Equilibrium Hints for Entropy Production

## 7. Discussion and Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. The Maximum Possible Ratio of Two-State System

## References

- Gardiner, C. Stochastic Methods; Springer: Berlin, Germany, 2009; Volume 4. [Google Scholar]
- Rao, R.; Esposito, M. Nonequilibrium thermodynamics of chemical reaction networks: Wisdom from stochastic thermodynamics. Phys. Rev. X
**2016**, 6, 041064. [Google Scholar] [CrossRef] [Green Version] - Pascal, R.; Pross, A.; Sutherland, J.D. Towards an evolutionary theory of the origin of life based on kinetics and thermodynamics. Open Biol.
**2013**, 3, 130156. [Google Scholar] [CrossRef] [Green Version] - Assenza, S.; Sassi, A.S.; Kellner, R.; Schuler, B.; De Los Rios, P.; Barducci, A. Efficient conversion of chemical energy into mechanical work by Hsp70 chaperones. elife
**2019**, 8, e48491. [Google Scholar] [CrossRef] - Goloubinoff, P.; Sassi, A.S.; Fauvet, B.; Barducci, A.; De Los Rios, P. Chaperones convert the energy from ATP into the nonequilibrium stabilization of native proteins. Nat. Chem. Biol.
**2018**, 14, 388–395. [Google Scholar] [CrossRef] [Green Version] - Zwicker, D.; Seyboldt, R.; Weber, C.A.; Hyman, A.A.; Jülicher, F. Growth and division of active droplets provides a model for protocells. Nat. Phys.
**2017**, 13, 408–413. [Google Scholar] [CrossRef] - Horowitz, J.M.; England, J.L. Spontaneous fine-tuning to environment in many-species chemical reaction networks. Proc. Natl. Acad. Sci. USA
**2017**, 114, 7565–7570. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Maes, C.; Netočnỳ, K.; de Galway, W.O. Low temperature behavior of nonequilibrium multilevel systems. J. Phys. Math. Theor.
**2013**, 47, 035002. [Google Scholar] [CrossRef] [Green Version] - Basu, U.; Maes, C. Nonequilibrium response and frenesy. J. Phys. Conf. Ser.
**2015**, 638, 012001. [Google Scholar] [CrossRef] [Green Version] - Maes, C. Non-Dissipative Effects in Nonequilibrium Systems; Springer: Berlin/Heidelberg, Germany, 2017. [Google Scholar]
- Busiello, D.M.; Liang, S.; Piazza, F.; De Los Rios, P. Dissipation-driven selection of states in non-equilibrium chemical networks. Commun. Chem.
**2021**, 4, 1–7. [Google Scholar] [CrossRef] - Dass, A.V.; Georgelin, T.; Westall, F.; Foucher, F.; De Los Rios, P.; Busiello, D.M.; Liang, S.; Piazza, F. Equilibrium and nonequilibrium furanose selection in the ribose isomerisation network. Nat. Commun.
**2021**, 12, 2749. [Google Scholar] [CrossRef] - Astumian, R.D. Kinetic asymmetry allows macromolecular catalysts to drive an information ratchet. Nat. Commun.
**2019**, 10, 1–14. [Google Scholar] [CrossRef] [Green Version] - Schnakenberg, J. Network theory of microscopic and macroscopic behavior of master equation systems. Rev. Mod. Phys.
**1976**, 48, 571. [Google Scholar] [CrossRef] - Hänggi, P.; Talkner, P.; Borkovec, M. Reaction-rate theory: Fifty years after Kramers. Rev. Mod. Phys.
**1990**, 62, 251. [Google Scholar] [CrossRef] - Kustova, E.; Giordano, D. Cross-coupling effects in chemically non-equilibrium viscous compressible flows. Chem. Phys.
**2011**, 379, 83–91. [Google Scholar] [CrossRef] - Kolesnichenko, E.G.; Gorbachev, Y.E. Gas-dynamic equations for spatially inhomogeneous gas mixtures with internal degrees of freedom. I. General theory. Appl. Math. Model.
**2010**, 34, 3778–3790. [Google Scholar] [CrossRef] - Busiello, D.M.; Gupta, D.; Maritan, A. Coarse-grained entropy production with multiple reservoirs: Unraveling the role of time scales and detailed balance in biology-inspired systems. Phys. Rev. Res.
**2020**, 2, 043257. [Google Scholar] [CrossRef] - Klich, I.; Raz, O.; Hirschberg, O.; Vucelja, M. Mpemba index and anomalous relaxation. Phys. Rev. X
**2019**, 9, 021060. [Google Scholar] [CrossRef] [Green Version] - Astumian, R.D. Adiabatic operation of a molecular machine. Proc. Natl. Acad. Sci. USA
**2007**, 104, 19715–19718. [Google Scholar] [CrossRef] [Green Version] - Mandal, D.; Jarzynski, C. A proof by graphical construction of the no-pumping theorem of stochastic pumps. J. Stat. Mech. Theory Exp.
**2011**, 2011, P10006. [Google Scholar] [CrossRef] [Green Version] - Busiello, D.M.; Hidalgo, J.; Maritan, A. Entropy production for coarse-grained dynamics. New J. Phys.
**2019**, 21, 073004. [Google Scholar] [CrossRef] - Liang, S.; Busiello, D.M.; Rios, P.D.L. The intrinsic non-equilibrium nature of thermophoresis. arXiv
**2021**, arXiv:2102.03197. [Google Scholar] - Piazza, R.; Parola, A. Thermophoresis in colloidal suspensions. J. Phys. Condens. Matter
**2008**, 20, 153102. [Google Scholar] [CrossRef] - Rahman, M.; Saghir, M. Thermodiffusion or Soret effect: Historical review. Int. J. Heat Mass Transf.
**2014**, 73, 693–705. [Google Scholar] [CrossRef] - Bo, S.; Celani, A. Multiple-scale stochastic processes: Decimation, averaging and beyond. Phys. Rep.
**2017**, 670, 1–59. [Google Scholar] [CrossRef] [Green Version] - Raz, O.; Subaşı, Y.; Jarzynski, C. Mimicking nonequilibrium steady states with time-periodic driving. Phys. Rev. X
**2016**, 6, 021022. [Google Scholar] [CrossRef] [Green Version] - Busiello, D.M.; Jarzynski, C.; Raz, O. Similarities and differences between non-equilibrium steady states and time-periodic driving in diffusive systems. New J. Phys.
**2018**, 20, 093015. [Google Scholar] [CrossRef] - Bai, S.; Ge, H.; Qian, H. Structure for energy cycle: A unique status of the second law of thermodynamics for living systems. Sci. China Life Sci.
**2018**, 61, 1266–1273. [Google Scholar] [CrossRef] [PubMed] - Lervik, A.; Kjelstrup, S.; Qian, H. Michaelis–Menten kinetics under non-isothermal conditions. Phys. Chem. Chem. Phys.
**2015**, 17, 1317–1324. [Google Scholar] [CrossRef] [PubMed] - Jin, X.; Ge, H. Nonequilibrium steady state of biochemical cycle kinetics under non-isothermal conditions. New J. Phys.
**2018**, 20, 043030. [Google Scholar] [CrossRef] - England, J.L. Dissipative adaptation in driven self-assembly. Nat. Nanotechnol.
**2015**, 10, 919–923. [Google Scholar] [CrossRef] - Schrödinger, E. What is Life? The Physical Aspect of the Living Cell and Mind; Cambridge University Press: Cambridge, UK, 1944. [Google Scholar]
- Prigogine, I.; Nicolis, G. Biological order, structure and instabilities. Q. Rev. Biophys.
**1971**, 4, 107–148. [Google Scholar] [CrossRef] [PubMed] [Green Version]

**Figure 1.**(

**a**) ${R}_{BA}=\overline{{P}_{B}}/\overline{{P}_{A}}$ as a function of the energetic barrier $\epsilon $ for both finite and infinite diffusion cases. Small insets are sketches of the reaction network in different conditions, where the thickness of the arrows reflects the speed of the corresponding reaction. The solutions in these settings are also reported next to the insets. (

**b**) Sketch of a two-state, two-box reaction network in a temperature gradient, where the inner circular arrow represents the direction of non-equilibrium stationary flux J. (

**c**) The theoretical critical point ${\epsilon}_{d}$ scales linearly with $d/{k}_{0}$, showing a clear agreement with numerical estimations, with small deviations only for large values of d.

**Figure 2.**(

**a**) Three-state, two-box reaction network. The same color corresponds to the same value of the energy. The energy difference between A and B (or C) is $\u2206E$; (

**b**) $\u2206E=1,{T}_{c}=0.1,{\epsilon}_{C}=1,{\epsilon}_{B}=2$. We report the logarithm of the selection parameter, ${R}_{CB}=\overline{{P}_{C}}/\overline{{P}_{B}}$, as a function of d for different values of the thermal gradient. ${R}_{CB}$ is maximised at infinite diffusion. (

**c**) In this case, $\u2206E=0.1$, while all other parameters and the color code are the same as in panel (

**b**). Here, we show a peak in the selection strength for finite diffusion.

**Figure 3.**(

**a**) Two-box model for a complex reaction network. Solid lines represent fast-dissipation reactions, while slow-dissipation reactions are indicated by dashed lines. Gray dashed circles indicate the fast-dissipative sub-networks in the cold box. The distribution inside each sub-network follows the Boltzmann equilibrium. Overall, the probability is redistributed by diffusion so that half of the total particles populate each box; (

**b**) Kullback–Leibler divergence between exact solutions and theoretical predictions for the stationary probability distribution of different reaction networks composed of 40 states each. Networks are generated by randomly assigning fast-dissipation ($\epsilon =1$) and slow-dissipation ($\epsilon =5$) reactions between states that uniformly populate a given energy range, ${R}_{E}$. The Kullback–Leibler divergence is shown for ${10}^{3}$ reaction networks with the same energy range (identified by the same color), and for an increasing energy range (here estimated through the variance of the energies). As the roughness of the landscape increases, the proposed framework starts failing; (

**c**) theoretical and exact probabilities in comparison for ${R}_{E}=1,2$, and $2.5$ from left to right (here different colors represent different networks).

**Figure 4.**Total probability of states B, $\overline{{P}_{B}}=P\left({B}_{1}\right)+P\left({B}_{2}\right)$, and C, $\overline{{P}_{C}}=P\left({C}_{1}\right)+P\left({C}_{2}\right)$, as a function of d. The diffusion coefficient can trigger a switch of the selected state at stationarity, which is due to the competition between dissipation-driven and energetic selection. The diffusion coefficients at which each transition occurs can be estimated by comparison with the chemical reaction rate in the cold box. The upper inset sketch the chemical network here investigated. Parameters are ${T}_{c}=0.1$, ${T}_{w}=0.2$, ${\epsilon}_{C}=1$, and ${\epsilon}_{B}=2$.

**Figure 5.**(

**a**) For a simple three-state system, we compare the exact entropy production (solid red line) with the theoretical approximated one obtained using our framework (blue dashed line), and the formula obtained by a small gradient expansion, Equation (25) (yellow dot-dashed line). In the exact ${\dot{S}}_{\mathrm{tot}}$ the probabilities stem from direct solution of the master equation, while in the theoretical ${\dot{S}}_{\mathrm{tot}}$, and in Equation (25), the probabilities are obtained employing the equilibration of fast-dissipative sub-networks. To consistently apply our approach, we choose an intermediate value of the diffusion with respect to the chemical rates (see also Figure 4), showing an excellent agreement among the curves presented. (

**b**) It is evident that the main contribution to the entropy production comes from the reaction $A\leftrightharpoons C$, ${\dot{S}}_{A\leftrightarrow C}$, which supports a much faster dissipation with respect to the slow-dissipation branch $A\leftrightharpoons B$. Indeed, we also see that ${\dot{S}}_{A\leftrightarrow B}\ll {\dot{S}}_{A\leftrightarrow C}$. Parameters are reported in panel (

**a**).

**Figure 6.**The logarithm of the selection strength as a function of the critical barrier ${\epsilon}_{d}$ for a three-state, two-box model (solid line), as sketched in the inset, and for a time-periodic driven three-state system (dashed line). These two paradigms are qualitatively and quantitatively equivalent to determine non-isothermal selection of states.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Liang, S.; De Los Rios, P.; Busiello, D.M.
Dissipation-Driven Selection under Finite Diffusion: Hints from Equilibrium and Separation of Time Scales. *Entropy* **2021**, *23*, 1068.
https://doi.org/10.3390/e23081068

**AMA Style**

Liang S, De Los Rios P, Busiello DM.
Dissipation-Driven Selection under Finite Diffusion: Hints from Equilibrium and Separation of Time Scales. *Entropy*. 2021; 23(8):1068.
https://doi.org/10.3390/e23081068

**Chicago/Turabian Style**

Liang, Shiling, Paolo De Los Rios, and Daniel Maria Busiello.
2021. "Dissipation-Driven Selection under Finite Diffusion: Hints from Equilibrium and Separation of Time Scales" *Entropy* 23, no. 8: 1068.
https://doi.org/10.3390/e23081068