1
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Ertel B, Seifert U. Estimator of entropy production for partially accessible Markov networks based on the observation of blurred transitions. Phys Rev E 2024; 109:054109. [PMID: 38907510 DOI: 10.1103/physreve.109.054109] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/13/2023] [Accepted: 03/25/2024] [Indexed: 06/24/2024]
Abstract
A central task in stochastic thermodynamics is the estimation of entropy production for partially accessible Markov networks. We establish an effective transition-based description for such networks with transitions that are not distinguishable and therefore blurred for an external observer. We demonstrate that, in contrast to a description based on fully resolved transitions, this effective description is typically non-Markovian at any point in time. Starting from an information-theoretic bound, we derive an operationally accessible entropy estimator for this observation scenario. We illustrate the operational relevance and the quality of this entropy estimator with a numerical analysis of various representative examples.
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Affiliation(s)
- Benjamin Ertel
- II. Institut für Theoretische Physik, Universität Stuttgart, 70550 Stuttgart, Germany
| | - Udo Seifert
- II. Institut für Theoretische Physik, Universität Stuttgart, 70550 Stuttgart, Germany
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2
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Remlein B, Seifert U. Nonequilibrium fluctuations of chemical reaction networks at criticality: The Schlögl model as paradigmatic case. J Chem Phys 2024; 160:134103. [PMID: 38557838 DOI: 10.1063/5.0203659] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/15/2024] [Accepted: 03/15/2024] [Indexed: 04/04/2024] Open
Abstract
Chemical reaction networks can undergo nonequilibrium phase transitions upon variation in external control parameters, such as the chemical potential of a species. We investigate the flux in the associated chemostats that is proportional to the entropy production and its critical fluctuations within the Schlögl model. Numerical simulations show that the corresponding diffusion coefficient diverges at the critical point as a function of system size. In the vicinity of the critical point, the diffusion coefficient follows a scaling form. We develop an analytical approach based on the chemical Langevin equation and van Kampen's system size expansion that yields the corresponding exponents in the monostable regime. In the bistable regime, we rely on a two-state approximation in order to analytically describe the critical behavior.
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Affiliation(s)
- Benedikt Remlein
- II. Institut für Theoretische Physik, Universität Stuttgart, 70550 Stuttgart, Germany
| | - Udo Seifert
- II. Institut für Theoretische Physik, Universität Stuttgart, 70550 Stuttgart, Germany
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3
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Guislain L, Bertin E. Discontinuous phase transition from ferromagnetic to oscillating states in a nonequilibrium mean-field spin model. Phys Rev E 2024; 109:034131. [PMID: 38632801 DOI: 10.1103/physreve.109.034131] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/20/2023] [Accepted: 02/22/2024] [Indexed: 04/19/2024]
Abstract
We study a nonequilibrium ferromagnetic mean-field spin model exhibiting a phase with spontaneous temporal oscillations of the magnetization, on top of the usual paramagnetic and ferromagnetic phases. This behavior is obtained by introducing dynamic field variables coupled to the spins through nonreciprocal couplings. We determine a nonequilibrium generalization of the Landau free energy in terms of the large deviation function of the magnetization and of an appropriately defined smoothed stochastic time derivative of the magnetization. While the transition between paramagnetic and oscillating phase is continuous, the transition between ferromagnetic and oscillating phases is found to be discontinuous, with a coexistence of both phases, one being stable and the other one metastable. Depending on parameter values, the ferromagnetic points may either be inside or outside the limit cycle, leading to different transition scenarios. The stability of these steady states is determined from the large deviation function. We also show that in the coexistence region, the entropy production has a pronounced maximum as a function of system size.
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Affiliation(s)
- Laura Guislain
- Université Grenoble Alpes, CNRS, LIPhy, 38000 Grenoble, France
| | - Eric Bertin
- Université Grenoble Alpes, CNRS, LIPhy, 38000 Grenoble, France
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4
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Ohga N, Ito S, Kolchinsky A. Thermodynamic Bound on the Asymmetry of Cross-Correlations. PHYSICAL REVIEW LETTERS 2023; 131:077101. [PMID: 37656850 DOI: 10.1103/physrevlett.131.077101] [Citation(s) in RCA: 3] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/11/2023] [Accepted: 06/08/2023] [Indexed: 09/03/2023]
Abstract
The principle of microscopic reversibility says that, in equilibrium, two-time cross-correlations are symmetric under the exchange of observables. Thus, the asymmetry of cross-correlations is a fundamental, measurable, and often-used statistical signature of deviation from equilibrium. Here we find a simple and universal inequality that bounds the magnitude of asymmetry by the cycle affinity, i.e., the strength of thermodynamic driving. Our result applies to a large class of systems and all state observables, and it suggests a fundamental thermodynamic cost for various nonequilibrium functions quantified by the asymmetry. It also provides a powerful tool to infer affinity from measured cross-correlations, in a different and complementary way to the thermodynamic uncertainty relations. As an application, we prove a thermodynamic bound on the coherence of noisy oscillations, which was previously conjectured by Barato and Seifert [Phys. Rev. E 95, 062409 (2017)PRESCM2470-004510.1103/PhysRevE.95.062409]. We also derive a thermodynamic bound on directed information flow in a biochemical signal transduction model.
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Affiliation(s)
- Naruo Ohga
- Department of Physics, Graduate School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
| | - Sosuke Ito
- Department of Physics, Graduate School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
- Universal Biology Institute, Graduate School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
| | - Artemy Kolchinsky
- Universal Biology Institute, Graduate School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
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5
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Aguilera M, Igarashi M, Shimazaki H. Nonequilibrium thermodynamics of the asymmetric Sherrington-Kirkpatrick model. Nat Commun 2023; 14:3685. [PMID: 37353499 DOI: 10.1038/s41467-023-39107-y] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/29/2022] [Accepted: 05/26/2023] [Indexed: 06/25/2023] Open
Abstract
Most natural systems operate far from equilibrium, displaying time-asymmetric, irreversible dynamics characterized by a positive entropy production while exchanging energy and matter with the environment. Although stochastic thermodynamics underpins the irreversible dynamics of small systems, the nonequilibrium thermodynamics of larger, more complex systems remains unexplored. Here, we investigate the asymmetric Sherrington-Kirkpatrick model with synchronous and asynchronous updates as a prototypical example of large-scale nonequilibrium processes. Using a path integral method, we calculate a generating functional over trajectories, obtaining exact solutions of the order parameters, path entropy, and steady-state entropy production of infinitely large networks. Entropy production peaks at critical order-disorder phase transitions, but is significantly larger for quasi-deterministic disordered dynamics. Consequently, entropy production can increase under distinct scenarios, requiring multiple thermodynamic quantities to describe the system accurately. These results contribute to developing an exact analytical theory of the nonequilibrium thermodynamics of large-scale physical and biological systems and their phase transitions.
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Affiliation(s)
- Miguel Aguilera
- BCAM - Basque Center for Applied Mathematics, Bilbao, Spain.
- IKERBASQUE, Basque Foundation for Science, Bilbao, Spain.
- School of Engineering and Informatics, University of Sussex, Falmer, Brighton, United Kingdom.
| | - Masanao Igarashi
- Graduate School of Engineering, Hokkaido University, Sapporo, Japan
| | - Hideaki Shimazaki
- Graduate School of Informatics, Kyoto University, Kyoto, Japan
- Center for Human Nature, Artificial Intelligence, and Neuroscience (CHAIN), Hokkaido University, Sapporo, Japan
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6
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Guislain L, Bertin E. Nonequilibrium Phase Transition to Temporal Oscillations in Mean-Field Spin Models. PHYSICAL REVIEW LETTERS 2023; 130:207102. [PMID: 37267541 DOI: 10.1103/physrevlett.130.207102] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/14/2022] [Revised: 03/14/2023] [Accepted: 04/24/2023] [Indexed: 06/04/2023]
Abstract
We propose a mean-field theory to describe the nonequilibrium phase transition to a spontaneously oscillating state in spin models. A nonequilibrium generalization of the Landau free energy is obtained from the joint distribution of the magnetization and its smoothed stochastic time derivative. The order parameter of the transition is a Hamiltonian, whose nonzero value signals the onset of oscillations. The Hamiltonian and the nonequilibrium Landau free energy are determined explicitly from the stochastic spin dynamics. The oscillating phase is also characterized by a nontrivial overlap distribution reminiscent of a continuous replica symmetry breaking, in spite of the absence of disorder. An illustration is given on an explicit kinetic mean-field spin model.
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Affiliation(s)
- Laura Guislain
- Universite Grenoble Alpes, CNRS, LIPhy, 38000 Grenoble, France
| | - Eric Bertin
- Universite Grenoble Alpes, CNRS, LIPhy, 38000 Grenoble, France
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7
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Chun HM, Horowitz JM. Trade-offs between number fluctuations and response in nonequilibrium chemical reaction networks. J Chem Phys 2023; 158:2888610. [PMID: 37144710 DOI: 10.1063/5.0148662] [Citation(s) in RCA: 2] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/02/2023] [Accepted: 04/17/2023] [Indexed: 05/06/2023] Open
Abstract
We study the response of chemical reaction networks driven far from equilibrium to logarithmic perturbations of reaction rates. The response of the mean number of a chemical species is observed to be quantitively limited by number fluctuations and the maximum thermodynamic driving force. We prove these trade-offs for linear chemical reaction networks and a class of nonlinear chemical reaction networks with a single chemical species. Numerical results for several model systems support the conclusion that these trade-offs continue to hold for a broad class of chemical reaction networks, though their precise form appears to sensitively depend on the deficiency of the network.
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Affiliation(s)
- Hyun-Myung Chun
- School of Physics, Korea Institute for Advanced Study, Seoul 02455, South Korea
| | - Jordan M Horowitz
- Department of Biophysics, University of Michigan, Ann Arbor, Michigan 48109, USA
- Center for the Study of Complex Systems, University of Michigan, Ann Arbor, Michigan 48104, USA
- Department of Physics, University of Michigan, Ann Arbor, Michigan 48109, USA
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8
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Oberreiter L, Seifert U, Barato AC. Universal minimal cost of coherent biochemical oscillations. Phys Rev E 2022; 106:014106. [PMID: 35974563 DOI: 10.1103/physreve.106.014106] [Citation(s) in RCA: 5] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/15/2021] [Accepted: 06/13/2022] [Indexed: 06/15/2023]
Abstract
Biochemical clocks are essential for virtually all living systems. A biochemical clock that is isolated from an external periodic signal and subjected to fluctuations can oscillate coherently only for a finite number of oscillations. Furthermore, such an autonomous clock can oscillate only if it consumes free energy. What is the minimum amount of free-energy consumption required for a certain number of coherent oscillations? We conjecture a universal bound that answers this question. A system that oscillates coherently for N oscillations has a minimal free-energy cost per oscillation of 4π^{2}Nk_{B}T. Our bound is valid for general finite Markov processes, is conjectured based on extensive numerical evidence, is illustrated with numerical simulations of a known model for a biochemical oscillator, and applies to existing experimental data.
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Affiliation(s)
- Lukas Oberreiter
- II. Institut für Theoretische Physik, Universität Stuttgart, 70550 Stuttgart, Germany
| | - Udo Seifert
- II. Institut für Theoretische Physik, Universität Stuttgart, 70550 Stuttgart, Germany
| | - Andre C Barato
- Department of Physics, University of Houston, Houston, Texas 77204, USA
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9
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Remlein B, Weissmann V, Seifert U. Coherence of oscillations in the weak-noise limit. Phys Rev E 2022; 105:064101. [PMID: 35854576 DOI: 10.1103/physreve.105.064101] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/14/2022] [Accepted: 05/13/2022] [Indexed: 06/15/2023]
Abstract
In a noisy environment, oscillations lose their coherence, which can be characterized by a quality factor. We determine this quality factor for oscillations arising from a driven Fokker-Planck dynamics along a periodic one-dimensional potential analytically in the weak-noise limit. With this expression, we can prove for this continuum model the analog of an upper bound that has been conjectured for the coherence of oscillations in discrete Markov network models. We show that our approach can also be adapted to motion along a noisy two-dimensional limit cycle. Specifically, we apply our scheme to the noisy Stuart-Landau oscillator and the thermodynamically consistent Brusselator as a simple model for a chemical clock. Our approach thus complements the fairly sophisticated extant general framework based on techniques from Hamilton-Jacobi theory with which we compare our results numerically.
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Affiliation(s)
- Benedikt Remlein
- II. Institut für Theoretische Physik, Universität Stuttgart, 70550 Stuttgart, Germany
| | - Volker Weissmann
- II. Institut für Theoretische Physik, Universität Stuttgart, 70550 Stuttgart, Germany
| | - Udo Seifert
- II. Institut für Theoretische Physik, Universität Stuttgart, 70550 Stuttgart, Germany
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10
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Behera AK, Junco CD, Vaikuntanathan S. Mechanism for the Generation of Robust Circadian Oscillations through Ultransensitivity and Differential Binding Affinity. J Phys Chem B 2021; 125:11179-11187. [PMID: 34609867 PMCID: PMC8515790 DOI: 10.1021/acs.jpcb.1c05915] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/01/2022]
Abstract
![]()
Biochemical circadian rhythm oscillations
play an important role
in many signaling mechanisms. In this work, we explore some of the
biophysical mechanisms responsible for sustaining robust oscillations
by constructing a minimal but analytically tractable model of the
circadian oscillations in the KaiABC protein system found in the cyanobacteria S. elongatus. In particular, our minimal model explicitly
accounts for two experimentally characterized biophysical features
of the KaiABC protein system, namely, a differential binding affinity
and an ultrasensitive response. Our analytical work shows how these
mechanisms might be crucial for promoting robust oscillations even
in suboptimal nutrient conditions. Our analytical and numerical work
also identifies mechanisms by which biological clocks can stably maintain
a constant time period under a variety of nutrient conditions. Finally,
our work also explores the thermodynamic costs associated with the
generation of robust sustained oscillations and shows that the net
rate of entropy production alone might not be a good figure of merit
to asses the quality of oscillations.
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Affiliation(s)
- Agnish Kumar Behera
- Department of Chemistry, University of Chicago, Chicago, Illinois 60637, United States
| | - Clara Del Junco
- Department of Chemistry, University of Chicago, Chicago, Illinois 60637, United States
| | - Suriyanarayanan Vaikuntanathan
- Department of Chemistry, University of Chicago, Chicago, Illinois 60637, United States.,The James Franck Institute, University of Chicago, Chicago, Illinois 60637, United States
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11
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Seara DS, Machta BB, Murrell MP. Irreversibility in dynamical phases and transitions. Nat Commun 2021; 12:392. [PMID: 33452238 PMCID: PMC7810704 DOI: 10.1038/s41467-020-20281-2] [Citation(s) in RCA: 13] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/29/2020] [Accepted: 11/12/2020] [Indexed: 11/11/2022] Open
Abstract
Living and non-living active matter consumes energy at the microscopic scale to drive emergent, macroscopic behavior including traveling waves and coherent oscillations. Recent work has characterized non-equilibrium systems by their total energy dissipation, but little has been said about how dissipation manifests in distinct spatiotemporal patterns. We introduce a measure of irreversibility we term the entropy production factor to quantify how time reversal symmetry is broken in field theories across scales. We use this scalar, dimensionless function to characterize a dynamical phase transition in simulations of the Brusselator, a prototypical biochemically motivated non-linear oscillator. We measure the total energetic cost of establishing synchronized biochemical oscillations while simultaneously quantifying the distribution of irreversibility across spatiotemporal frequencies.
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Affiliation(s)
- Daniel S Seara
- Department of Physics, Yale University, New Haven, CT, 06511, USA.
- Systems Biology Institute, Yale University, West Haven, CT, 06516, USA.
| | - Benjamin B Machta
- Department of Physics, Yale University, New Haven, CT, 06511, USA.
- Systems Biology Institute, Yale University, West Haven, CT, 06516, USA.
| | - Michael P Murrell
- Department of Physics, Yale University, New Haven, CT, 06511, USA.
- Systems Biology Institute, Yale University, West Haven, CT, 06516, USA.
- Department of Biomedical Engineering, Yale University, New Haven, CT, 06511, USA.
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12
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Oberreiter L, Seifert U, Barato AC. Stochastic Discrete Time Crystals: Entropy Production and Subharmonic Synchronization. PHYSICAL REVIEW LETTERS 2021; 126:020603. [PMID: 33512201 DOI: 10.1103/physrevlett.126.020603] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/17/2020] [Revised: 06/24/2020] [Accepted: 12/22/2020] [Indexed: 06/12/2023]
Abstract
Discrete time crystals are periodically driven systems that display spontaneous symmetry breaking of time translation invariance in the form of indefinite subharmonic oscillations. We introduce a thermodynamically consistent model for a discrete time crystal and analyze it using the framework of stochastic thermodynamics. In particular, we evaluate the rate of energy dissipation of this many-body system of interacting noisy subharmonic oscillators in contact with a heat bath. The mean-field model displays the phenomenon of subharmonic synchronization, which corresponds to collective subharmonic oscillations of the individual units. The 2D model does not display synchronization but it does show a time-crystalline phase, which is characterized by a power-law behavior of the number of coherent subharmonic oscillations with system size. This result demonstrates that the emergence of coherent oscillations is possible even in the absence of synchronization.
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Affiliation(s)
- Lukas Oberreiter
- II. Institut für Theoretische Physik, Universität Stuttgart, 70550 Stuttgart, Germany
| | - Udo Seifert
- II. Institut für Theoretische Physik, Universität Stuttgart, 70550 Stuttgart, Germany
| | - Andre C Barato
- Department of Physics, University of Houston, Houston, Texas 77204, USA
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13
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Rana S, Barato AC. Precision and dissipation of a stochastic Turing pattern. Phys Rev E 2020; 102:032135. [PMID: 33075863 DOI: 10.1103/physreve.102.032135] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/25/2020] [Accepted: 08/28/2020] [Indexed: 11/07/2022]
Abstract
Spontaneous pattern formation is a fundamental scientific problem that has received much attention since the seminal theoretical work of Turing on reaction-diffusion systems. In molecular biophysics, this phenomenon often takes place under the influence of large fluctuations. It is then natural to inquire about the precision of such pattern. In particular, spontaneous pattern formation is a nonequilibrium phenomenon, and the relation between the precision of a pattern and the thermodynamic cost associated with it remains largely unexplored. Here, we analyze this relation with a paradigmatic stochastic reaction-diffusion model, i.e., the Brusselator in one spatial dimension. We find that the precision of the pattern is maximized for an intermediate thermodynamic cost, i.e., increasing the thermodynamic cost beyond this value makes the pattern less precise. Even though fluctuations get less pronounced with an increase in thermodynamic cost, we argue that larger fluctuations can also have a positive effect on the precision of the pattern.
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Affiliation(s)
- Shubhashis Rana
- Department of Physics, University of Houston, Houston, Texas 77204, USA
| | - Andre C Barato
- Department of Physics, University of Houston, Houston, Texas 77204, USA
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14
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Del Junco C, Vaikuntanathan S. Robust oscillations in multi-cyclic Markov state models of biochemical clocks. J Chem Phys 2020; 152:055101. [PMID: 32035451 DOI: 10.1063/1.5143259] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/26/2022] Open
Abstract
Organisms often use cyclic changes in the concentrations of chemical species to precisely time biological functions. Underlying these biochemical clocks are chemical reactions and transport processes, which are inherently stochastic. Understanding the physical basis for robust biochemical oscillations in the presence of fluctuations has thus emerged as an important problem. In a previous paper [C. del Junco and S. Vaikuntanathan, Phys. Rev. E 101, 012410 (2020)], we explored this question using the non-equilibrium statistical mechanics of single-ring Markov state models of biochemical networks that support oscillations. Our finding was that they can exploit non-equilibrium driving to robustly maintain the period and coherence of oscillations in the presence of randomness in the rates. Here, we extend our work to Markov state models consisting of a large cycle decorated with multiple small cycles. These additional cycles are intended to represent alternate pathways that the oscillator may take as it fluctuates about its average path. Combining a mapping to single-cycle networks based on first passage time distributions with our previously developed theory, we are able to make analytical predictions for the period and coherence of oscillations in these networks. One implication of our predictions is that a high energy budget can make different network topologies and arrangements of rates degenerate as far as the period and coherence of oscillations are concerned. Excellent agreement between analytical and numerical results confirms that this is the case. Our results suggest that biochemical oscillators can be more robust to fluctuations in the path of the oscillator when they have a high energy budget.
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Affiliation(s)
- Clara Del Junco
- Department of Chemistry and The James Franck Institute, University of Chicago, Chicago, Illinois 60637, USA
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15
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Nguyen B, Seifert U. Exponential volume dependence of entropy-current fluctuations at first-order phase transitions in chemical reaction networks. Phys Rev E 2020; 102:022101. [PMID: 32942421 DOI: 10.1103/physreve.102.022101] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/10/2020] [Accepted: 07/07/2020] [Indexed: 12/23/2022]
Abstract
In chemical reaction networks, bistability can only occur far from equilibrium. It is associated with a first-order phase transition where the control parameter is the thermodynamic force. At the bistable point, the entropy production is known to be discontinuous with respect to the thermodynamic force. We show that the fluctuations of the entropy production have an exponential volume-dependence when the system is bistable. At the phase transition, the exponential prefactor is the height of the effective potential barrier between the two fixed-points. Our results obtained for Schlögl's model can be extended to any chemical network.
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Affiliation(s)
- Basile Nguyen
- II. Institut für Theoretische Physik, Universität Stuttgart, 70550 Stuttgart, Germany
| | - Udo Seifert
- II. Institut für Theoretische Physik, Universität Stuttgart, 70550 Stuttgart, Germany
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16
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Fritz JH, Nguyen B, Seifert U. Stochastic thermodynamics of chemical reactions coupled to finite reservoirs: A case study for the Brusselator. J Chem Phys 2020; 152:235101. [PMID: 32571070 DOI: 10.1063/5.0006115] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Biomolecular processes are typically modeled using chemical reaction networks coupled to infinitely large chemical reservoirs. A difference in chemical potential between these reservoirs can drive the system into a non-equilibrium steady-state (NESS). In reality, these processes take place in finite systems containing a finite number of molecules. In such systems, a NESS can be reached with the help of an externally driven pump for which we introduce a simple model. The crucial parameters are the pumping rate and the finite size of the chemical reservoir. We apply this model to a simple biochemical oscillator, the Brusselator, and quantify the performance using the number of coherent oscillations. As a surprising result, we find that higher precision can be achieved with finite-size reservoirs even though the corresponding current fluctuations are larger than in the ideal infinite case.
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Affiliation(s)
- Jonas H Fritz
- II. Institut für Theoretische Physik, Universität Stuttgart, 70550 Stuttgart, Germany
| | - Basile Nguyen
- II. Institut für Theoretische Physik, Universität Stuttgart, 70550 Stuttgart, Germany
| | - Udo Seifert
- II. Institut für Theoretische Physik, Universität Stuttgart, 70550 Stuttgart, Germany
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17
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Marsland R, Cui W, Horowitz JM. The thermodynamic uncertainty relation in biochemical oscillations. J R Soc Interface 2020; 16:20190098. [PMID: 31039695 DOI: 10.1098/rsif.2019.0098] [Citation(s) in RCA: 26] [Impact Index Per Article: 6.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/30/2022] Open
Abstract
Living systems regulate many aspects of their behaviour through periodic oscillations of molecular concentrations, which function as 'biochemical clocks.' The chemical reactions that drive these clocks are intrinsically stochastic at the molecular level, so that the duration of a full oscillation cycle is subject to random fluctuations. Their success in carrying out their biological function is thought to depend on the degree to which these fluctuations in the cycle period can be suppressed. Biochemical oscillators also require a constant supply of free energy in order to break detailed balance and maintain their cyclic dynamics. For a given free energy budget, the recently discovered 'thermodynamic uncertainty relation' yields the magnitude of period fluctuations in the most precise conceivable free-running clock. In this paper, we show that computational models of real biochemical clocks severely underperform this optimum, with fluctuations several orders of magnitude larger than the theoretical minimum. We argue that this suboptimal performance is due to the small number of internal states per molecule in these models, combined with the high level of thermodynamic force required to maintain the system in the oscillatory phase. We introduce a new model with a tunable number of internal states per molecule and confirm that it approaches the optimal precision as this number increases.
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Affiliation(s)
- Robert Marsland
- 1 Department of Physics, Boston University , 590 Commonwealth Avenue, Boston, MA 02215 , USA
| | - Wenping Cui
- 1 Department of Physics, Boston University , 590 Commonwealth Avenue, Boston, MA 02215 , USA.,2 Department of Physics, Boston College , 140 Commonwealth Avenue, Chestnut Hill, MA 02467 , USA
| | - Jordan M Horowitz
- 3 Physics of Living Systems Group, Department of Physics, Massachusetts Institute of Technology, 400 Technology Square , Cambridge, MA 02139 , USA.,4 Department of Biophysics, University of Michigan , Ann Arbor, MI 48109 , USA.,5 Center for the Study of Complex Systems, University of Michigan , Ann Arbor, MI 48109 , USA
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18
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Del Junco C, Vaikuntanathan S. High chemical affinity increases the robustness of biochemical oscillations. Phys Rev E 2020; 101:012410. [PMID: 32069602 DOI: 10.1103/physreve.101.012410] [Citation(s) in RCA: 10] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/12/2018] [Indexed: 01/01/2023]
Abstract
Biochemical oscillations are ubiquitous in nature and allow organisms to properly time their biological functions. In this paper, we consider minimal Markov state models of nonequilibrium biochemical networks that support oscillations. We obtain analytical expressions for the coherence and period of oscillations in these networks. These quantities are expected to depend on all details of the transition rates in the Markov state model. However, our analytical calculations reveal that driving the system out of equilibrium makes many of these details-specifically, the location and arrangement of the transition rates-irrelevant to the coherence and period of oscillations. This theoretical prediction is confirmed by excellent agreement with numerical results. As a consequence, the coherence and period of oscillations can be robustly maintained in the presence of fluctuations in the irrelevant variables. While recent work has established that increasing energy consumption improves the coherence of oscillations, our findings suggest that it plays the additional role of making the coherence and the average period of oscillations robust to fluctuations in rates that can result from the noisy environment of the cell.
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Affiliation(s)
- Clara Del Junco
- Department of Chemistry and The James Franck Institute, University of Chicago, Chicago, Illinois 60637, USA
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De Martino D, Barato AC. Oscillations in feedback-driven systems: Thermodynamics and noise. Phys Rev E 2020; 100:062123. [PMID: 31962493 DOI: 10.1103/physreve.100.062123] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/10/2019] [Indexed: 11/07/2022]
Abstract
Oscillations in nonequilibrium noisy systems are important physical phenomena. These oscillations can happen in autonomous biochemical oscillators such as circadian clocks. They can also manifest as subharmonic oscillations in periodically driven systems such as time crystals. Oscillations in autonomous systems and, to a lesser degree, subharmonic oscillations in periodically driven systems have been both thoroughly investigated, including their relation with thermodynamic cost and noise. We perform a systematic study of oscillations in a third class of nonequilibrium systems: feedback-driven systems. In particular, we use the apparatus of stochastic thermodynamics to investigate the role of noise and thermodynamic cost in feedback-driven oscillations. For a simple two-state model that displays oscillations, we analyze the relation between precision and dissipation, revealing that oscillations can remain coherent for an indefinite time in a finite system with thermal fluctuations in a limit of diverging thermodynamic cost. We consider oscillations in a more complex system with several degrees of freedom, an Ising model driven by feedback between the magnetization and the external field. This feedback-driven system can display subharmonic oscillations similar to the ones observed in time crystals. We illustrate the second law for feedback-driven systems that display oscillations. For the Ising model, the oscillating dissipated heat can be negative. However, when we consider the total entropy that also includes an informational term related to measurements, the oscillating total entropy change is always positive. We also study the finite-size scaling of the dissipated heat, providing evidence for the existence of a first-order phase transition for certain parameter regimes.
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Affiliation(s)
- Daniele De Martino
- Jozef Stefan Institute, Jamova Cesta 39, 1000 Ljubjlana, Slovenia.,Instituto Biofisika (UPV/EHU, CSIC), University of the Basque Country, Leioa E-48940, Spain.,Ikerbasque, Basque Foundation for Science, Bilbao 48013, Spain
| | - Andre C Barato
- Department of Physics, University of Houston, Houston, Texas 77204, USA
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Zhang D, Cao Y, Ouyang Q, Tu Y. The energy cost and optimal design for synchronization of coupled molecular oscillators. NATURE PHYSICS 2020; 16:95-100. [PMID: 32670386 PMCID: PMC7363412 DOI: 10.1038/s41567-019-0701-7] [Citation(s) in RCA: 18] [Impact Index Per Article: 4.5] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/27/2019] [Accepted: 09/23/2019] [Indexed: 05/24/2023]
Abstract
A model of coupled molecular biochemical oscillators is proposed to study nonequilibrium thermodynamics of synchronization. We find that synchronization of nonequilibrium oscillators costs addition energy to drive the exchange reaction (chemical interaction) between individual oscillators. By solving the steady state of the many-body system analytically, we show that the system goes through a nonequilibrium phase transition driven by energy dissipation, and the critical energy dissipation depends on both the frequency and strength of the exchange reaction. Moreover, our study reveals the optimal design for achieving maximum synchronization with a fixed energy budget. We apply our general theory to the Kai system in Cyanobacteria circadian clock and predict a relationship between the KaiC ATPase activity and synchronization of the KaiC hexamers. The theoretical framework can be extended to study thermodynamics of collective behaviors in other extended nonequilibrium active systems.
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Affiliation(s)
- Dongliang Zhang
- The State Key Laboratory for Artificial Microstructures and Mesoscopic Physics, School of Physics, Peking University, Beijing 100871, China
| | - Yuansheng Cao
- Department of Physics, University of California, San Diego, La Jolla, California 92093, USA
| | - Qi Ouyang
- The State Key Laboratory for Artificial Microstructures and Mesoscopic Physics, School of Physics, Peking University, Beijing 100871, China
- Center for Quantitative Biology and Peking-Tsinghua Center for Life Sciences, AAIC, Peking University, Beijing 100871, China
| | - Yuhai Tu
- IBM T. J. Watson Research Center, Yorktown Heights, New York 10598, USA
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Lazarescu A, Cossetto T, Falasco G, Esposito M. Large deviations and dynamical phase transitions in stochastic chemical networks. J Chem Phys 2019. [DOI: 10.1063/1.5111110] [Citation(s) in RCA: 27] [Impact Index Per Article: 5.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/27/2023] Open
Affiliation(s)
| | - Tommaso Cossetto
- Complex Systems and Statistical Mechanics, Physics and Material Science Research Unit, University of Luxembourg, L-1511 Luxembourg, Luxembourg
| | - Gianmaria Falasco
- Complex Systems and Statistical Mechanics, Physics and Material Science Research Unit, University of Luxembourg, L-1511 Luxembourg, Luxembourg
| | - Massimiliano Esposito
- CPHT, CNRS, École Polytechnique, IP Paris, F-91128 Palaiseau, France
- Complex Systems and Statistical Mechanics, Physics and Material Science Research Unit, University of Luxembourg, L-1511 Luxembourg, Luxembourg
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Ehrmann A, Nguyen B, Seifert U. Interlinked GTPase cascades provide a motif for both robust switches and oscillators. J R Soc Interface 2019; 16:20190198. [PMID: 31387482 DOI: 10.1098/rsif.2019.0198] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Key Words] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/09/2023] Open
Abstract
GTPases regulate a wide range of cellular processes, such as intracellular vesicular transport, signal transduction and protein translation. These hydrolase enzymes operate as biochemical switches by toggling between an active guanosine triphosphate (GTP)-bound state and an inactive guanosine diphosphate (GDP)-bound state. We compare two network motifs, a single-species switch and an interlinked cascade that consists of two species coupled through positive and negative feedback loops. We find that interlinked cascades are closer to the ideal all-or-none switch and are more robust against fluctuating signals. While the single-species switch can only achieve bistability, interlinked cascades can be converted into oscillators by tuning the cofactor concentrations, which catalyse the activity of the cascade. These regimes can only be achieved with sufficient chemical driving provided by GTP hydrolysis. In this study, we present a thermodynamically consistent model that can achieve bistability and oscillations with the same feedback motif.
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Affiliation(s)
- Andreas Ehrmann
- II. Institut für Theoretische Physik, Universität Stuttgart, 70550 Stuttgart, Germany
| | - Basile Nguyen
- II. Institut für Theoretische Physik, Universität Stuttgart, 70550 Stuttgart, Germany
| | - Udo Seifert
- II. Institut für Theoretische Physik, Universität Stuttgart, 70550 Stuttgart, Germany
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Oberreiter L, Seifert U, Barato AC. Subharmonic oscillations in stochastic systems under periodic driving. Phys Rev E 2019; 100:012135. [PMID: 31499923 DOI: 10.1103/physreve.100.012135] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/31/2019] [Indexed: 06/10/2023]
Abstract
Subharmonic response is a well-known phenomenon in, e.g., deterministic nonlinear dynamical systems. We investigate the conditions under which such subharmonic oscillations can persist for a long time in open systems with stochastic dynamics due to thermal fluctuations. In contrast to stochastic autonomous systems in a stationary state, for which the number of coherent oscillations is fundamentally bounded by the number of states in the underlying network, we demonstrate that in periodically driven systems, subharmonic oscillations can in principle remain coherent forever, even in networks with a small number of states. We also show that, inter alia, the thermodynamic cost rises only logarithmically with the number of coherent oscillations in a model calculation and that the possible periods of the persistent subharmonic response grow linearly with the number of states. We argue that our results can be relevant for biochemical oscillations and for stochastic models of time crystals.
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Affiliation(s)
- Lukas Oberreiter
- II. Institut für Theoretische Physik, Universität Stuttgart, 70550 Stuttgart, Germany
| | - Udo Seifert
- II. Institut für Theoretische Physik, Universität Stuttgart, 70550 Stuttgart, Germany
| | - Andre C Barato
- Department of Physics, University of Houston, Houston, Texas 77204, USA
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