1
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Teza G, Yaacoby R, Raz O. Eigenvalue Crossing as a Phase Transition in Relaxation Dynamics. PHYSICAL REVIEW LETTERS 2023; 130:207103. [PMID: 37267560 DOI: 10.1103/physrevlett.130.207103] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/14/2022] [Accepted: 04/07/2023] [Indexed: 06/04/2023]
Abstract
When a system's parameter is abruptly changed, a relaxation toward the new equilibrium of the system follows. We show that a crossing between the second and third eigenvalues of the relaxation operator results in a singularity in the dynamics analogous to a first-order equilibrium phase transition. While dynamical phase transitions are intrinsically hard to detect in nature, here we show how this kind of transition can be observed in an experimentally feasible four-state colloidal system. Finally, analytical proof of survival in the thermodynamic limit of a many body (1D Ising) model is provided.
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Affiliation(s)
- Gianluca Teza
- Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 7610001, Israel
| | - Ran Yaacoby
- Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 7610001, Israel
| | - Oren Raz
- Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 7610001, Israel
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2
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Owen JA, Horowitz JM. Size limits the sensitivity of kinetic schemes. Nat Commun 2023; 14:1280. [PMID: 36890153 PMCID: PMC9995461 DOI: 10.1038/s41467-023-36705-8] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/18/2022] [Accepted: 02/10/2023] [Indexed: 03/10/2023] Open
Abstract
Living things benefit from exquisite molecular sensitivity in many of their key processes, including DNA replication, transcription and translation, chemical sensing, and morphogenesis. At thermodynamic equilibrium, the basic biophysical mechanism for sensitivity is cooperative binding, for which it can be shown that the Hill coefficient, a sensitivity measure, cannot exceed the number of binding sites. Generalizing this fact, we find that for any kinetic scheme, at or away from thermodynamic equilibrium, a very simple structural quantity, the size of the support of a perturbation, always limits the effective Hill coefficient. We show how this bound sheds light on and unifies diverse sensitivity mechanisms, including kinetic proofreading and a nonequilibrium Monod-Wyman-Changeux (MWC) model proposed for the E. coli flagellar motor switch, representing in each case a simple, precise bridge between experimental observations and the models we write down. In pursuit of mechanisms that saturate the support bound, we find a nonequilibrium binding mechanism, nested hysteresis, with sensitivity exponential in the number of binding sites, with implications for our understanding of models of gene regulation and the function of biomolecular condensates.
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Affiliation(s)
- Jeremy A Owen
- Department of Physics, Massachusetts Institute of Technology, Cambridge, MA, 02139, USA. .,Department of Chemistry, Princeton University, Princeton, NJ, 08540, USA.
| | - Jordan M Horowitz
- Department of Biophysics, University of Michigan, Ann Arbor, MI, 48109, USA. .,Center for the Study of Complex Systems, University of Michigan, Ann Arbor, MI, 48104, USA. .,Department of Physics, University of Michigan, Ann Arbor, MI, 48109, USA.
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3
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Horii H, Lefevere R, Itami M, Nemoto T. Anomalous fluctuations of renewal-reward processes with heavy-tailed distributions. Phys Rev E 2022; 106:034130. [PMID: 36266861 DOI: 10.1103/physreve.106.034130] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/18/2022] [Accepted: 09/07/2022] [Indexed: 06/16/2023]
Abstract
For renewal-reward processes with a power-law decaying waiting time distribution, anomalously large probabilities are assigned to atypical values of the asymptotic processes. Previous works have revealed that this anomalous scaling causes a singularity in the corresponding large deviation function. In order to further understand this problem, we study in this article the scaling of variance in several renewal-reward processes: counting processes with two different power-law decaying waiting time distributions and a Knudsen gas (a heat conduction model). Through analytical and numerical analyses of these models, we find that the variances show an anomalous scaling when the exponent of the power law is -3. For a counting process with the power-law exponent smaller than -3, this anomalous scaling does not take place: this indicates that if we only consider the standard deviation from the expectation, any anomalous behavior will not be detected. In this case, we argue that anomalous scaling appears in higher order cumulants. Finally, many-body particles interacting through soft-core interactions with the boundary conditions employed in the Knudsen gas are studied using numerical simulations. We observe that the variance scaling becomes normal even though the power-law exponent in the boundary conditions is -3.
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Affiliation(s)
- Hiroshi Horii
- Université Paris Cité, Laboratoire de Probabilités, Statistiques et Modélisation, UMR 8001, F-75205 Paris, France
| | - Raphaël Lefevere
- Université Paris Cité, Laboratoire de Probabilités, Statistiques et Modélisation, UMR 8001, F-75205 Paris, France
| | - Masato Itami
- Center for Science Adventure and Collaborative Research Advancement, Kyoto University, Kyoto 606-8502, Japan
| | - Takahiro Nemoto
- Graduate School of Informatics, Kyoto University, Yoshida Hon-machi, Sakyo-ku, Kyoto 606-8501, Japan
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4
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Lim H, Jung Y. Reaction-path statistical mechanics of enzymatic kinetics. J Chem Phys 2022; 156:134108. [PMID: 35395879 DOI: 10.1063/5.0075831] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We introduce a reaction-path statistical mechanics formalism based on the principle of large deviations to quantify the kinetics of single-molecule enzymatic reaction processes under the Michaelis-Menten mechanism, which exemplifies an out-of-equilibrium process in the living system. Our theoretical approach begins with the principle of equal a priori probabilities and defines the reaction path entropy to construct a new nonequilibrium ensemble as a collection of possible chemical reaction paths. As a result, we evaluate a variety of path-based partition functions and free energies by using the formalism of statistical mechanics. They allow us to calculate the timescales of a given enzymatic reaction, even in the absence of an explicit boundary condition that is necessary for the equilibrium ensemble. We also consider the large deviation theory under a closed-boundary condition of the fixed observation time to quantify the enzyme-substrate unbinding rates. The result demonstrates the presence of a phase-separation-like, bimodal behavior in unbinding events at a finite timescale, and the behavior vanishes as its rate function converges to a single phase in the long-time limit.
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Affiliation(s)
- Hyuntae Lim
- Department of Chemistry, Seoul National University, Seoul 08826, South Korea
| | - YounJoon Jung
- Department of Chemistry, Seoul National University, Seoul 08826, South Korea
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5
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Casert C, Vieijra T, Whitelam S, Tamblyn I. Dynamical Large Deviations of Two-Dimensional Kinetically Constrained Models Using a Neural-Network State Ansatz. PHYSICAL REVIEW LETTERS 2021; 127:120602. [PMID: 34597112 DOI: 10.1103/physrevlett.127.120602] [Citation(s) in RCA: 9] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/18/2020] [Revised: 07/13/2021] [Accepted: 07/28/2021] [Indexed: 06/13/2023]
Abstract
We use a neural-network ansatz originally designed for the variational optimization of quantum systems to study dynamical large deviations in classical ones. We use recurrent neural networks to describe the large deviations of the dynamical activity of model glasses, kinetically constrained models in two dimensions. We present the first finite size-scaling analysis of the large-deviation functions of the two-dimensional Fredrickson-Andersen model, and explore the spatial structure of the high-activity sector of the South-or-East model. These results provide a new route to the study of dynamical large-deviation functions, and highlight the broad applicability of the neural-network state ansatz across domains in physics.
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Affiliation(s)
- Corneel Casert
- Department of Physics and Astronomy, Ghent University, 9000 Ghent, Belgium
| | - Tom Vieijra
- Department of Physics and Astronomy, Ghent University, 9000 Ghent, Belgium
| | - Stephen Whitelam
- Molecular Foundry, Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, California 94720, USA
| | - Isaac Tamblyn
- Department of Physics, University of Ottawa, K1N 6N5, Ontario, Canada
- Vector Institute for Artificial Intelligence, Toronto, M5G 1M1, Ontario, Canada
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6
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Sharpe DJ, Wales DJ. Graph transformation and shortest paths algorithms for finite Markov chains. Phys Rev E 2021; 103:063306. [PMID: 34271741 DOI: 10.1103/physreve.103.063306] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/25/2021] [Accepted: 04/29/2021] [Indexed: 12/20/2022]
Abstract
The graph transformation (GT) algorithm robustly computes the mean first-passage time to an absorbing state in a finite Markov chain. Here we present a concise overview of the iterative and block formulations of the GT procedure and generalize the GT formalism to the case of any path property that is a sum of contributions from individual transitions. In particular, we examine the path action, which directly relates to the path probability, and analyze the first-passage path ensemble for a model Markov chain that is metastable and therefore numerically challenging. We compare the mean first-passage path action, obtained using GT, with the full path action probability distribution simulated efficiently using kinetic path sampling, and with values for the highest-probability paths determined by the recursive enumeration algorithm (REA). In Markov chains representing realistic dynamical processes, the probability distributions of first-passage path properties are typically fat-tailed and therefore difficult to converge by sampling, which motivates the use of exact and numerically stable approaches to compute the expectation. We find that the kinetic relevance of the set of highest-probability paths depends strongly on the metastability of the Markov chain, and so the properties of the dominant first-passage paths may be unrepresentative of the global dynamics. Use of a global measure for edge costs in the REA, based on net productive fluxes, allows the total reactive flux to be decomposed into a finite set of contributions from simple flux paths. By considering transition flux paths, a detailed quantitative analysis of the relative importance of competing dynamical processes is possible even in the metastable regime.
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Affiliation(s)
- Daniel J Sharpe
- Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, United Kingdom
| | - David J Wales
- Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, United Kingdom
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7
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Whitelam S, Jacobson D. Varied phenomenology of models displaying dynamical large-deviation singularities. Phys Rev E 2021; 103:032152. [PMID: 33862814 DOI: 10.1103/physreve.103.032152] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/07/2021] [Accepted: 03/16/2021] [Indexed: 11/07/2022]
Abstract
Singularities of dynamical large-deviation functions are often interpreted as the signal of a dynamical phase transition and the coexistence of distinct dynamical phases, by analogy with the correspondence between singularities of free energies and equilibrium phase behavior. Here we study models of driven random walkers on a lattice. These models display large-deviation singularities in the limit of large lattice size, but the extent to which each model's phenomenology resembles a phase transition depends on the details of the driving. We also compare the behavior of ergodic and nonergodic models that present large-deviation singularities. We argue that dynamical large-deviation singularities indicate the divergence of a model timescale, but not necessarily one associated with cooperative behavior or the existence of distinct phases.
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Affiliation(s)
- Stephen Whitelam
- Molecular Foundry, Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, California 94720, USA
| | - Daniel Jacobson
- Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, California 91125, USA
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8
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Hurtado-Gutiérrez R, Carollo F, Pérez-Espigares C, Hurtado PI. Building Continuous Time Crystals from Rare Events. PHYSICAL REVIEW LETTERS 2020; 125:160601. [PMID: 33124846 DOI: 10.1103/physrevlett.125.160601] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/13/2019] [Accepted: 09/17/2020] [Indexed: 06/11/2023]
Abstract
Symmetry-breaking dynamical phase transitions (DPTs) abound in the fluctuations of nonequilibrium systems. Here, we show that the spectral features of a particular class of DPTs exhibit the fingerprints of the recently discovered time-crystal phase of matter. Using Doob's transform as a tool, we provide a mechanism to build classical time-crystal generators from the rare event statistics of some driven diffusive systems. An analysis of the Doob's smart field in terms of the order parameter of the transition then leads to the time-crystal lattice gas (TCLG), a model of driven fluid subject to an external packing field, which presents a clear-cut steady-state phase transition to a time-crystalline phase characterized by a matter density wave, which breaks continuous time-translation symmetry and displays rigidity and long-range spatiotemporal order, as required for a time crystal. A hydrodynamic analysis of the TCLG transition uncovers striking similarities, but also key differences, with the Kuramoto synchronization transition. Possible experimental realizations of the TCLG in colloidal fluids are also discussed.
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Affiliation(s)
- R Hurtado-Gutiérrez
- Departamento de Electromagnetismo y Física de la Materia, Universidad de Granada, Granada 18071, Spain
- Institute Carlos I for Theoretical and Computational Physics, Universidad de Granada, Granada 18071, Spain
| | - F Carollo
- Institut für Theoretische Physik, Universität Tübingen, Auf der Morgenstelle 14, 72076 Tübingen, Germany
| | - C Pérez-Espigares
- Departamento de Electromagnetismo y Física de la Materia, Universidad de Granada, Granada 18071, Spain
- Institute Carlos I for Theoretical and Computational Physics, Universidad de Granada, Granada 18071, Spain
| | - P I Hurtado
- Departamento de Electromagnetismo y Física de la Materia, Universidad de Granada, Granada 18071, Spain
- Institute Carlos I for Theoretical and Computational Physics, Universidad de Granada, Granada 18071, Spain
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9
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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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10
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Jacobson D, Whitelam S. Direct evaluation of dynamical large-deviation rate functions using a variational ansatz. Phys Rev E 2019; 100:052139. [PMID: 31869879 DOI: 10.1103/physreve.100.052139] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/01/2019] [Indexed: 06/10/2023]
Abstract
We describe a simple form of importance sampling designed to bound and compute large-deviation rate functions for time-extensive dynamical observables in continuous-time Markov chains. We start with a model, defined by a set of rates, and a time-extensive dynamical observable. We construct a reference model, a variational ansatz for the behavior of the original model conditioned on atypical values of the observable. Direct simulation of the reference model provides an upper bound on the large-deviation rate function associated with the original model, an estimate of the tightness of the bound, and, if the ansatz is chosen well, the exact rate function. The exact rare behavior of the original model does not need to be known in advance. We use this method to calculate rate functions for currents and counting observables in a set of network- and lattice models taken from the literature. Straightforward ansätze yield bounds that are tighter than bounds obtained from Level 2.5 of large deviations via approximations that involve uniform scalings of rates. We show how to correct these bounds in order to recover the rate functions exactly. Our approach is complementary to more specialized methods and offers a physically transparent framework for approximating and calculating the likelihood of dynamical large deviations.
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Affiliation(s)
- Daniel Jacobson
- Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, California 91125, USA
| | - Stephen Whitelam
- Molecular Foundry, Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, California 94720, USA
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11
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Pérez-Espigares C, Hurtado PI. Sampling rare events across dynamical phase transitions. CHAOS (WOODBURY, N.Y.) 2019; 29:083106. [PMID: 31472495 DOI: 10.1063/1.5091669] [Citation(s) in RCA: 16] [Impact Index Per Article: 3.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/04/2019] [Accepted: 07/16/2019] [Indexed: 06/10/2023]
Abstract
Interacting particle systems with many degrees of freedom may undergo phase transitions to sustain atypical fluctuations of dynamical observables such as the current or the activity. In some cases, this leads to symmetry-broken space-time trajectories which enhance the probability of such events due to the emergence of ordered structures. Despite their conceptual and practical importance, these dynamical phase transitions (DPTs) at the trajectory level are difficult to characterize due to the low probability of their occurrence. However, during the last decade, advanced computational techniques have been developed to measure rare events in simulations of many-particle systems that allow the direct observation and characterization of these DPTs. Here we review the application of a particular rare-event simulation technique, based on cloning Monte Carlo methods, to characterize DPTs in paradigmatic stochastic lattice gases. In particular, we describe in detail some tricks and tips of the trade, paying special attention to the measurement of order parameters capturing the physics of the different DPTs, as well as to the finite-size effects (both in the system size and in the number of clones) that affect the measurements. Overall, we provide a consistent picture of the phenomenology associated with DPTs and their measurement.
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Affiliation(s)
- Carlos Pérez-Espigares
- Departamento de Electromagnetismo y Física de la Materia, and Institute Carlos I for Theoretical and Computational Physics, Universidad de Granada, Granada 18071, Spain
| | - Pablo I Hurtado
- Departamento de Electromagnetismo y Física de la Materia, and Institute Carlos I for Theoretical and Computational Physics, Universidad de Granada, Granada 18071, Spain
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12
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Abstract
We study the stochastic dynamics of infinitely many globally interacting units made of q states distributed uniformly along a ring that is externally driven. While repulsive interactions always lead to uniform occupations, attractive interactions give rise to much richer phenomena: We analytically characterize a Hopf bifurcation which separates a high-temperature regime of uniform occupations from a low-temperature one where all units coalesce into a single state. For odd q, below the critical temperature starts a synchronization regime which ends via a second phase transition at lower temperatures, while for even q this intermediate phase disappears. We find that interactions have no effects except below critical temperature for attractive interactions. A thermodynamic analysis reveals that the dissipated work is reduced in this regime, whose temperature range is shown to decrease as q increases. The q dependence of the power-efficiency trade-off is also analyzed.
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Affiliation(s)
- Tim Herpich
- Complex Systems and Statistical Mechanics, Physics and Materials Science Research Unit, University of Luxembourg, L-1511 Luxembourg, Luxembourg
| | - Massimiliano Esposito
- Complex Systems and Statistical Mechanics, Physics and Materials Science Research Unit, University of Luxembourg, L-1511 Luxembourg, Luxembourg
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13
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Garrido PL, Hurtado PI, Tizón-Escamilla N. Infinite family of universal profiles for heat current statistics in Fourier's law. Phys Rev E 2019; 99:022134. [PMID: 30934242 DOI: 10.1103/physreve.99.022134] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/25/2018] [Indexed: 06/09/2023]
Abstract
Using tools from large deviation theory, we study fluctuations of the heat current in a model of d-dimensional incompressible fluid driven out of equilibrium by a temperature gradient. We find that the most probable temperature fields sustaining atypical values of the global current can be naturally classified in an infinite set of curves, allowing us to exhaustively analyze their topological properties and to define universal profiles onto which all optimal fields collapse. We also compute the statistics of empirical heat current, where we find remarkable logarithmic tails for large current fluctuations orthogonal to the thermal gradient. Finally, we determine explicitly a number of cumulants of the current distribution, finding interesting relations between them.
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Affiliation(s)
- P L Garrido
- Departamento de Electromagnetismo y Física de la Materia, and Instituto Carlos I de Física Teórica y Computacional, Universidad de Granada, E-18071 Granada. Spain
| | - P I Hurtado
- Departamento de Electromagnetismo y Física de la Materia, and Instituto Carlos I de Física Teórica y Computacional, Universidad de Granada, E-18071 Granada. Spain
| | - N Tizón-Escamilla
- Departamento de Electromagnetismo y Física de la Materia, and Instituto Carlos I de Física Teórica y Computacional, Universidad de Granada, E-18071 Granada. Spain
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14
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Nemoto T, Fodor É, Cates ME, Jack RL, Tailleur J. Optimizing active work: Dynamical phase transitions, collective motion, and jamming. Phys Rev E 2019; 99:022605. [PMID: 30934223 DOI: 10.1103/physreve.99.022605] [Citation(s) in RCA: 38] [Impact Index Per Article: 7.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/08/2018] [Indexed: 06/09/2023]
Abstract
Active work measures how far the local self-forcing of active particles translates into real motion. Using population Monte Carlo methods, we investigate large deviations in the active work for repulsive active Brownian disks. Minimizing the active work generically results in dynamical arrest; in contrast, despite the lack of aligning interactions, trajectories of high active work correspond to a collectively moving, aligned state. We use heuristic and analytic arguments to explain the origin of dynamical phase transitions separating the arrested, typical, and aligned regimes.
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Affiliation(s)
- Takahiro Nemoto
- Philippe Meyer Institute for Theoretical Physics, Physics Department, École Normale Supérieure & PSL Research University, 24 rue Lhomond, 75231 Paris Cedex 05, France
| | - Étienne Fodor
- Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom
| | - Michael E Cates
- Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom
| | - Robert L Jack
- Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom
- Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, United Kingdom
| | - Julien Tailleur
- Laboratoire Matière et Systèmes Complexes, UMR 7057 CNRS/P7, Université Paris Diderot, 10 rue Alice Domon et Léonie Duquet, 75205 Paris cedex 13, France
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15
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Katira S, Garrahan JP, Mandadapu KK. Solvation in Space-time: Pretransition Effects in Trajectory Space. PHYSICAL REVIEW LETTERS 2018; 120:260602. [PMID: 30004722 DOI: 10.1103/physrevlett.120.260602] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/02/2017] [Revised: 03/01/2018] [Indexed: 06/08/2023]
Abstract
We demonstrate pretransition effects in space-time in trajectories of systems in which the dynamics displays a first-order phase transition between distinct dynamical phases. These effects are analogous to those observed for thermodynamic first-order phase transitions, most notably the hydrophobic effect in water. Considering the (infinite temperature) East model as an elementary example, we study the properties of "space-time solvation" by examining trajectories where finite space-time regions are conditioned to be inactive in an otherwise active phase. We find that solvating an inactive region of space-time within an active trajectory shows two regimes in the dynamical equivalent of solvation free energy: an "entropic" small solute regime in which uncorrelated fluctuations are sufficient to evacuate activity from the solute, and an "energetic" large solute regime which involves the formation of a solute-induced inactive domain with an associated active-inactive interface bearing a dynamical interfacial tension. We also show that as a result of this dynamical interfacial tension there is a dynamical analog of the hydrophobic collapse that drives the assembly of large hydrophobes in water. We discuss the general relevance of these results to the properties of dynamical fluctuations in systems with slow collective relaxation such as glass formers.
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Affiliation(s)
- Shachi Katira
- Department of Chemical and Biomolecular Engineering, University of California, Berkeley, California 94720, USA
| | - Juan P Garrahan
- School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, United Kingdom
- Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems, University of Nottingham, Nottingham NG7 2RD, United Kingdom
| | - Kranthi K Mandadapu
- Department of Chemical and Biomolecular Engineering, University of California, Berkeley, California 94720, USA
- Chemical Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
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16
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Whitelam S. Large deviations in the presence of cooperativity and slow dynamics. Phys Rev E 2018; 97:062109. [PMID: 30011565 DOI: 10.1103/physreve.97.062109] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/05/2018] [Indexed: 06/08/2023]
Abstract
We study simple models of intermittency, involving switching between two states, within the dynamical large-deviation formalism. Singularities appear in the formalism when switching is cooperative or when its basic time scale diverges. In the first case the unbiased trajectory distribution undergoes a symmetry breaking, leading to a change in shape of the large-deviation rate function for a particular dynamical observable. In the second case the symmetry of the unbiased trajectory distribution remains unbroken. Comparison of these models suggests that singularities of the dynamical large-deviation formalism can signal the dynamical equivalent of an equilibrium phase transition but do not necessarily do so.
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Affiliation(s)
- Stephen Whitelam
- Molecular Foundry, Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, California 94720, USA
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17
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Weber JK, Kang SG, Zhou R. Rare Dissipative Transitions Punctuate the Initiation of Chemical Denaturation in Proteins. Biophys J 2018; 114:812-821. [PMID: 29490243 DOI: 10.1016/j.bpj.2017.12.038] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/09/2017] [Revised: 12/18/2017] [Accepted: 12/27/2017] [Indexed: 10/17/2022] Open
Abstract
Protein unfolding dynamics are bound by their degree of entropy production, a quantity that relates the amount of heat dissipated by a nonequilibrium process to a system's forward and time-reversed trajectories. We here explore the statistics of heat dissipation that emerge in protein molecules subjected to a chemical denaturant. Coupling large molecular dynamics datasets and Markov state models with the theory of entropy production, we demonstrate that dissipative processes can be rigorously characterized over the course of the urea-induced unfolding of the protein chymotrypsin inhibitor 2. By enumerating full entropy production probability distributions as a function of time, we first illustrate that distinct passive and dissipative regimes are present in the denaturation dynamics. Within the dissipative dynamical region, we next find that chymotrypsin inhibitor 2 is strongly driven into unfolded states in which the protein's hydrophobic core has been penetrated by urea molecules and disintegrated. Detailed analyses reveal that urea's interruption of key hydrophobic contacts between core residues causes many of the protein's native structural features to dissolve.
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Affiliation(s)
- Jeffrey K Weber
- IBM Thomas J. Watson Research Center, Yorktown Heights, New York
| | - Seung-Gu Kang
- IBM Thomas J. Watson Research Center, Yorktown Heights, New York
| | - Ruhong Zhou
- IBM Thomas J. Watson Research Center, Yorktown Heights, New York; Department of Chemistry, Columbia University, New York, New York.
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Kaiser M, Jack RL, Zimmer J. Canonical Structure and Orthogonality of Forces and Currents in Irreversible Markov Chains. JOURNAL OF STATISTICAL PHYSICS 2018; 170:1019-1050. [PMID: 31258181 PMCID: PMC6566214 DOI: 10.1007/s10955-018-1986-0] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 08/04/2017] [Accepted: 02/06/2018] [Indexed: 06/09/2023]
Abstract
We discuss a canonical structure that provides a unifying description of dynamical large deviations for irreversible finite state Markov chains (continuous time), Onsager theory, and Macroscopic Fluctuation Theory (MFT). For Markov chains, this theory involves a non-linear relation between probability currents and their conjugate forces. Within this framework, we show how the forces can be split into two components, which are orthogonal to each other, in a generalised sense. This splitting allows a decomposition of the pathwise rate function into three terms, which have physical interpretations in terms of dissipation and convergence to equilibrium. Similar decompositions hold for rate functions at level 2 and level 2.5. These results clarify how bounds on entropy production and fluctuation theorems emerge from the underlying dynamical rules. We discuss how these results for Markov chains are related to similar structures within MFT, which describes hydrodynamic limits of such microscopic models.
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Affiliation(s)
- Marcus Kaiser
- Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY UK
| | - Robert L. Jack
- Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA UK
- Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge, CB2 1EW UK
- Department of Physics, University of Bath, Bath, BA2 7AY UK
| | - Johannes Zimmer
- Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY UK
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19
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Tizón-Escamilla N, Pérez-Espigares C, Garrido PL, Hurtado PI. Order and Symmetry Breaking in the Fluctuations of Driven Systems. PHYSICAL REVIEW LETTERS 2017; 119:090602. [PMID: 28949563 DOI: 10.1103/physrevlett.119.090602] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/23/2016] [Indexed: 06/07/2023]
Abstract
Dynamical phase transitions (DPTs) in the space of trajectories are one of the most intriguing phenomena of nonequilibrium physics, but their nature in realistic high-dimensional systems remains puzzling. Here we observe for the first time a DPT in the current vector statistics of an archetypal two-dimensional (2D) driven diffusive system and characterize its properties using the macroscopic fluctuation theory. The complex interplay among the external field, anisotropy, and vector currents in 2D leads to a rich phase diagram, with different symmetry-broken fluctuation phases separated by lines of first- and second-order DPTs. Remarkably, different types of 1D order in the form of jammed density waves emerge to hinder transport for low-current fluctuations, revealing a connection between rare events and self-organized structures which enhance their probability.
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Affiliation(s)
- N Tizón-Escamilla
- Departamento de Electromagnetismo y Física de la Materia, and Institute Carlos I for Theoretical and Computational Physics, Universidad de Granada, Granada 18071, Spain
| | - C Pérez-Espigares
- University of Modena and Reggio Emilia, via G. Campi 213/b, 41125 Modena, Italy
- School of Physics and Astronomy and Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems, University of Nottingham, Nottingham, NG7 2RD, United Kingdom
| | - P L Garrido
- Departamento de Electromagnetismo y Física de la Materia, and Institute Carlos I for Theoretical and Computational Physics, Universidad de Granada, Granada 18071, Spain
| | - P I Hurtado
- Departamento de Electromagnetismo y Física de la Materia, and Institute Carlos I for Theoretical and Computational Physics, Universidad de Granada, Granada 18071, Spain
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20
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Abstract
The intrinsic stochasticity of gene expression can give rise to large fluctuations and rare events that drive phenotypic variation in a population of genetically identical cells. Characterizing the fluctuations that give rise to such rare events motivates the analysis of large deviations in stochastic models of gene expression. Recent developments in non-equilibrium statistical mechanics have led to a framework for analyzing Markovian processes conditioned on rare events and for representing such processes by conditioning-free driven Markovian processes. We use this framework, in combination with approaches based on queueing theory, to analyze a general class of stochastic models of gene expression. Modeling gene expression as a Batch Markovian Arrival Process (BMAP), we derive exact analytical results quantifying large deviations of time-integrated random variables such as promoter activity fluctuations. We find that the conditioning-free driven process can also be represented by a BMAP that has the same form as the original process, but with renormalized parameters. The results obtained can be used to quantify the likelihood of large deviations, to characterize system fluctuations conditional on rare events and to identify combinations of model parameters that can give rise to dynamical phase transitions in system dynamics.
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Affiliation(s)
- Jordan M Horowitz
- Department of Physics, Physics of Living Systems Group, Massachusetts Institute of Technology, 400 Technology Square, Cambridge, MA 02139, United States of America
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21
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Tizón-Escamilla N, Hurtado PI, Garrido PL. Structure of the optimal path to a fluctuation. Phys Rev E 2017; 95:032119. [PMID: 28415174 DOI: 10.1103/physreve.95.032119] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/08/2016] [Indexed: 11/07/2022]
Abstract
Macroscopic fluctuations have become an essential tool to understand physics far from equilibrium due to the link between their statistics and nonequilibrium ensembles. The optimal path leading to a fluctuation encodes key information on this problem, shedding light on, e.g., the physics behind the enhanced probability of rare events out of equilibrium, the possibility of dynamic phase transitions, and new symmetries. This makes the understanding of the properties of these optimal paths a central issue. Here we derive a fundamental relation which strongly constrains the architecture of these optimal paths for general d-dimensional nonequilibrium diffusive systems, and implies a nontrivial structure for the dominant current vector fields. Interestingly, this general relation (which encompasses and explains previous results) makes manifest the spatiotemporal nonlocality of the current statistics and the associated optimal trajectories.
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Affiliation(s)
- N Tizón-Escamilla
- Departamento de Electromagnetismo y Física de la Materia, and Instituto Carlos I de Física Teórica y Computacional, Universidad de Granada, E-18071 Granada, Spain
| | - P I Hurtado
- Departamento de Electromagnetismo y Física de la Materia, and Instituto Carlos I de Física Teórica y Computacional, Universidad de Granada, E-18071 Granada, Spain
| | - P L Garrido
- Departamento de Electromagnetismo y Física de la Materia, and Instituto Carlos I de Física Teórica y Computacional, Universidad de Granada, E-18071 Granada, Spain
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22
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Garrahan JP. Simple bounds on fluctuations and uncertainty relations for first-passage times of counting observables. Phys Rev E 2017; 95:032134. [PMID: 28415371 DOI: 10.1103/physreve.95.032134] [Citation(s) in RCA: 48] [Impact Index Per Article: 6.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/10/2017] [Indexed: 06/07/2023]
Abstract
Recent large deviation results have provided general lower bounds for the fluctuations of time-integrated currents in the steady state of stochastic systems. A corollary are so-called thermodynamic uncertainty relations connecting precision of estimation to average dissipation. Here we consider this problem but for counting observables, i.e., trajectory observables which, in contrast to currents, are non-negative and nondecreasing in time (and possibly symmetric under time reversal). In the steady state, their fluctuations to all orders are bound from below by a Conway-Maxwell-Poisson distribution dependent only on the averages of the observable and of the dynamical activity. We show how to obtain the corresponding bounds for first-passage times (times when a certain value of the counting variable is first reached) and their uncertainty relations. Just like entropy production does for currents, dynamical activity controls the bounds on fluctuations of counting observables.
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Affiliation(s)
- Juan P Garrahan
- School of Physics and Astronomy and Centre for the Mathematics and Theoretical Physics of Quantum Non-equilibrium Systems, University of Nottingham, Nottingham NG7 2RD, United Kingdom
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23
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Bonança MVS, Jarzynski C. Conditional reversibility in nonequilibrium stochastic systems. Phys Rev E 2016; 93:022101. [PMID: 26986282 DOI: 10.1103/physreve.93.022101] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/07/2015] [Indexed: 11/07/2022]
Abstract
For discrete-state stochastic systems obeying Markovian dynamics, we establish the counterpart of the conditional reversibility theorem obtained by Gallavotti for deterministic systems [Ann. de l'Institut Henri Poincaré (A) 70, 429 (1999)]. Our result states that stochastic trajectories conditioned on opposite values of entropy production are related by time reversal, in the long-time limit. In other words, the probability of observing a particular sequence of events, given a long trajectory with a specified entropy production rate σ, is the same as the probability of observing the time-reversed sequence of events, given a trajectory conditioned on the opposite entropy production, -σ, where both trajectories are sampled from the same underlying Markov process. To obtain our result, we use an equivalence between conditioned ("microcanonical") and biased ("canonical") ensembles of nonequilibrium trajectories. We provide an example to illustrate our findings.
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Affiliation(s)
- Marcus V S Bonança
- Instituto de Física 'Gleb Wataghin', Universidade Estadual de Campinas, 13083-859, Campinas, São Paulo, Brazil
| | - Christopher Jarzynski
- Department of Chemistry and Biochemistry and Institute for Physical Science and Technology, University of Maryland, College Park, Maryland 20742, USA
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24
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Das J. Limiting Energy Dissipation Induces Glassy Kinetics in Single-Cell High-Precision Responses. Biophys J 2016; 110:1180-90. [PMID: 26958894 PMCID: PMC4788747 DOI: 10.1016/j.bpj.2016.01.022] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/23/2015] [Revised: 12/17/2015] [Accepted: 01/15/2016] [Indexed: 10/22/2022] Open
Abstract
Single cells often generate precise responses by involving dissipative out-of-thermodynamic-equilibrium processes in signaling networks. The available free energy to fuel these processes could become limited depending on the metabolic state of an individual cell. How does limiting dissipation affect the kinetics of high-precision responses in single cells? I address this question in the context of a kinetic proofreading scheme used in a simple model of early-time T cell signaling. Using exact analytical calculations and numerical simulations, I show that limiting dissipation qualitatively changes the kinetics in single cells marked by emergence of slow kinetics, large cell-to-cell variations of copy numbers, temporally correlated stochastic events (dynamic facilitation), and ergodicity breaking. Thus, constraints in energy dissipation, in addition to negatively affecting ligand discrimination in T cells, can create a fundamental difficulty in determining single-cell kinetics from cell-population results.
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Affiliation(s)
- Jayajit Das
- Battelle Center for Mathematical Medicine, The Research Institute at Nationwide Children's Hospital, Columbus, Ohio; Biophysics Program and Departments of Pediatrics and Physics, The Ohio State University, Columbus, Ohio.
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Abstract
Life is fundamentally a nonequilibrium phenomenon. At the expense of dissipated energy, living things perform irreversible processes that allow them to propagate and reproduce. Within cells, evolution has designed nanoscale machines to do meaningful work with energy harnessed from a continuous flux of heat and particles. As dictated by the Second Law of Thermodynamics and its fluctuation theorem corollaries, irreversibility in nonequilibrium processes can be quantified in terms of how much entropy such dynamics produce. In this work, we seek to address a fundamental question linking biology and nonequilibrium physics: can the evolved dissipative pathways that facilitate biomolecular function be identified by their extent of entropy production in general relaxation processes? We here synthesize massive molecular dynamics simulations, Markov state models (MSMs), and nonequilibrium statistical mechanical theory to probe dissipation in two key classes of signaling proteins: kinases and G-protein-coupled receptors (GPCRs). Applying machinery from large deviation theory, we use MSMs constructed from protein simulations to generate dynamics conforming to positive levels of entropy production. We note the emergence of an array of peaks in the dynamical response (transient analogs of phase transitions) that draw the proteins between distinct levels of dissipation, and we see that the binding of ATP and agonist molecules modifies the observed dissipative landscapes. Overall, we find that dissipation is tightly coupled to activation in these signaling systems: dominant entropy-producing trajectories become localized near important barriers along known biological activation pathways. We go on to classify an array of equilibrium and nonequilibrium molecular switches that harmonize to promote functional dynamics.
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26
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Weber JK, Pande VS. Entropy-production-driven oscillators in simple nonequilibrium networks. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:032136. [PMID: 25871083 DOI: 10.1103/physreve.91.032136] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/09/2014] [Indexed: 06/04/2023]
Abstract
The development of tractable nonequilibrium simulation methods represents a bottleneck for efforts to describe the functional dynamics that occur within living cells. We here employ a nonequilibrium approach called the λ ensemble to characterize the dissipative dynamics of a simple Markovian network driven by an external potential. In the highly dissipative regime brought about by the λ bias, we observe a dynamical structure characteristic of cellular architectures: The entropy production drives a damped oscillator over state populations in the network. We illustrate the properties of such oscillations in weakly and strongly driven regimes, and we discuss how control structures associated with the "dynamical phase transition" in the system can be related to switches and oscillators in cellular dynamics.
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Affiliation(s)
- Jeffrey K Weber
- Department of Chemistry, Stanford University, Stanford, California 94305, USA
| | - Vijay S Pande
- Department of Chemistry, Stanford University, Stanford, California 94305, USA
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27
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Lang AH, Fisher CK, Mora T, Mehta P. Thermodynamics of statistical inference by cells. PHYSICAL REVIEW LETTERS 2014; 113:148103. [PMID: 25325665 DOI: 10.1103/physrevlett.113.148103] [Citation(s) in RCA: 45] [Impact Index Per Article: 4.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/10/2014] [Indexed: 06/04/2023]
Abstract
The deep connection between thermodynamics, computation, and information is now well established both theoretically and experimentally. Here, we extend these ideas to show that thermodynamics also places fundamental constraints on statistical estimation and learning. To do so, we investigate the constraints placed by (nonequilibrium) thermodynamics on the ability of biochemical signaling networks to estimate the concentration of an external signal. We show that accuracy is limited by energy consumption, suggesting that there are fundamental thermodynamic constraints on statistical inference.
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Affiliation(s)
- Alex H Lang
- Physics Department, Boston University, Boston, Massachusetts 02215, USA
| | - Charles K Fisher
- Physics Department, Boston University, Boston, Massachusetts 02215, USA
| | - Thierry Mora
- Laboratoire de physique statistique, CNRS, UPMC and École normale supérieure, 75005 Paris, France
| | - Pankaj Mehta
- Physics Department, Boston University, Boston, Massachusetts 02215, USA
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28
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Gingrich TR, Vaikuntanathan S, Geissler PL. Heterogeneity-induced large deviations in activity and (in some cases) entropy production. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:042123. [PMID: 25375454 DOI: 10.1103/physreve.90.042123] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/24/2014] [Indexed: 06/04/2023]
Abstract
We solve a simple model that supports a dynamic phase transition and show conditions for the existence of the transition. Using methods of large deviation theory we analytically compute the probability distribution for activity and entropy production rates of the trajectories on a large ring with a single heterogeneous link. The corresponding joint rate function demonstrates two dynamical phases--one localized and the other delocalized, but the marginal rate functions do not always exhibit the underlying transition. Symmetries in dynamic order parameters influence the observation of a transition, such that distributions for certain dynamic order parameters need not reveal an underlying dynamical bistability. Solution of our model system furthermore yields the form of the effective Markov transition matrices that generate dynamics in which the two dynamical phases are at coexistence. We discuss the implications of the transition for the response of bacterial cells to antibiotic treatment, arguing that even simple models of a cell cycle lacking an explicit bistability in configuration space will exhibit a bistability of dynamical phases.
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Affiliation(s)
- Todd R Gingrich
- Department of Chemistry, University of California, Berkeley, California 94720, USA
| | | | - Phillip L Geissler
- Department of Chemistry, University of California, Berkeley, California 94720, USA and Material Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA and Chemical Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
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