1
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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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2
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Meibohm J, Esposito M. Finite-Time Dynamical Phase Transition in Nonequilibrium Relaxation. PHYSICAL REVIEW LETTERS 2022; 128:110603. [PMID: 35362998 DOI: 10.1103/physrevlett.128.110603] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/15/2021] [Revised: 01/20/2022] [Accepted: 02/25/2022] [Indexed: 06/14/2023]
Abstract
We uncover a finite-time dynamical phase transition in the thermal relaxation of a mean-field magnetic model. The phase transition manifests itself as a cusp singularity in the probability distribution of the magnetization that forms at a critical time. The transition is due to a sudden switch in the dynamics, characterized by a dynamical order parameter. We derive a dynamical Landau theory for the transition that applies to a range of systems with scalar, parity-invariant order parameters. Close to criticalilty, our theory reveals an exact mapping between the dynamical and equilibrium phase transitions of the magnetic model, and implies critical exponents of mean-field type. We argue that interactions between nearby saddle points, neglected at the mean-field level, may lead to critical, spatiotemporal fluctuations of the order parameter, and thus give rise to novel, dynamical critical phenomena.
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Affiliation(s)
- Jan Meibohm
- Complex Systems and Statistical Mechanics, Department of Physics and Materials Science, University of Luxembourg, L-1511 Luxembourg, Luxembourg
| | - Massimiliano Esposito
- Complex Systems and Statistical Mechanics, Department of Physics and Materials Science, University of Luxembourg, L-1511 Luxembourg, Luxembourg
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3
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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: 15] [Impact Index Per Article: 5.0] [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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4
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Stoof R, Goñi-Moreno Á. Modelling co-translational dimerization for programmable nonlinearity in synthetic biology. J R Soc Interface 2020; 17:20200561. [PMID: 33143595 DOI: 10.1098/rsif.2020.0561] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/02/2023] Open
Abstract
Nonlinearity plays a fundamental role in the performance of both natural and synthetic biological networks. Key functional motifs in living microbial systems, such as the emergence of bistability or oscillations, rely on nonlinear molecular dynamics. Despite its core importance, the rational design of nonlinearity remains an unmet challenge. This is largely due to a lack of mathematical modelling that accounts for the mechanistic basis of nonlinearity. We introduce a model for gene regulatory circuits that explicitly simulates protein dimerization-a well-known source of nonlinear dynamics. Specifically, our approach focuses on modelling co-translational dimerization: the formation of protein dimers during-and not after-translation. This is in contrast to the prevailing assumption that dimer generation is only viable between freely diffusing monomers (i.e. post-translational dimerization). We provide a method for fine-tuning nonlinearity on demand by balancing the impact of co- versus post-translational dimerization. Furthermore, we suggest design rules, such as protein length or physical separation between genes, that may be used to adjust dimerization dynamics in vivo. The design, build and test of genetic circuits with on-demand nonlinear dynamics will greatly improve the programmability of synthetic biological systems.
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Affiliation(s)
- Ruud Stoof
- School of Computing, Newcastle University, Urban Sciences Building, Science Square, Newcastle upon Tyne NE4 5TG, UK
| | - Ángel Goñi-Moreno
- School of Computing, Newcastle University, Urban Sciences Building, Science Square, Newcastle upon Tyne NE4 5TG, UK.,Centro de Biotecnología y Genómica de Plantas (CBGP, UPM-INIA), Universidad Politénica de Madrid (UPM), Instituto Nacional de Investigación y Tecnología Agraria y Alimentaria (INIA), Campus de Montegancedo-UPM, 28223 Pozuelo de Alarcón, Madrid, Spain
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5
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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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6
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Coggan JS, Keller D, Markram H, Schürmann F, Magistretti PJ. Excitation states of metabolic networks predict dose-response fingerprinting and ligand pulse phase signalling. J Theor Biol 2020; 487:110123. [PMID: 31866398 DOI: 10.1016/j.jtbi.2019.110123] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/06/2019] [Revised: 11/08/2019] [Accepted: 12/16/2019] [Indexed: 12/30/2022]
Abstract
With a computational model of energy metabolism in an astrocyte, we show how a system of enzymes in a cascade can act as a functional unit of interdependent reactions, rather than merely a series of independent reactions. These systems may exist in multiple states, depending on the level of stimulation, and the effects of substrates at any point will depend on those states. Response trajectories of metabolites downstream from cAMP-stimulated glycogenolysis exhibit a host of non-linear dynamical response characteristics including hysteresis and response envelopes. Dose-dependent phase transitions predict a novel intracellular signalling mechanism and suggest a theoretical framework that could be relevant to single cell information processing, drug discovery or synthetic biology. Ligands may produce unique dose-response fingerprints depending on the state of the system, allowing selective output tuning. We conclude with the observation that state- and dose-dependent phase transitions, what we dub "ligand pulses" (LPs), may carry information and resemble action potentials (APs) generated from excitatory postsynaptic potentials. In our model, the relevant information from a cAMP-dependent glycolytic cascade in astrocytes could reflect the level of neuromodulatory input that signals an energy demand threshold. We propose that both APs and LPs represent specialized cases of molecular phase signalling with a common evolutionary root.
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Affiliation(s)
- Jay S Coggan
- Blue Brain Project, École Polytechnique Fédérale de Lausanne (EPFL), Geneva CH-1202, Switzerland.
| | - Daniel Keller
- Blue Brain Project, École Polytechnique Fédérale de Lausanne (EPFL), Geneva CH-1202, Switzerland.
| | - Henry Markram
- Blue Brain Project, École Polytechnique Fédérale de Lausanne (EPFL), Geneva CH-1202, Switzerland.
| | - Felix Schürmann
- Blue Brain Project, École Polytechnique Fédérale de Lausanne (EPFL), Geneva CH-1202, Switzerland.
| | - Pierre J Magistretti
- Biological and Environmental Sciences and Engineering Division, King Abdullah University of Science and Technology (KAUST), Thuwal 23955, Saudi Arabia.
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7
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Noa CEF, Harunari PE, de Oliveira MJ, Fiore CE. Entropy production as a tool for characterizing nonequilibrium phase transitions. Phys Rev E 2019; 100:012104. [PMID: 31499824 DOI: 10.1103/physreve.100.012104] [Citation(s) in RCA: 12] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/21/2018] [Indexed: 11/07/2022]
Abstract
Nonequilibrium phase transitions can be typified in a similar way to equilibrium systems, for instance, by the use of the order parameter. However, this characterization hides the irreversible character of the dynamics as well as its influence on the phase transition properties. Entropy production has been revealed to be an important concept for filling this gap since it vanishes identically for equilibrium systems and is positive for the nonequilibrium case. Based on distinct and general arguments, the characterization of phase transitions in terms of the entropy production is presented. Analysis for discontinuous and continuous phase transitions has been undertaken by taking regular and complex topologies within the framework of mean-field theory (MFT) and beyond the MFT. A general description of entropy production portraits for Z_{2} ("up-down") symmetry systems under the MFT is presented. Our main result is that a given phase transition, whether continuous or discontinuous has a specific entropy production hallmark. Our predictions are exemplified by an icon system, perhaps the simplest nonequilibrium model presenting an order-disorder phase transition and spontaneous symmetry breaking: the majority vote model. Our work paves the way to a systematic description and classification of nonequilibrium phase transitions through a key indicator of system irreversibility.
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Affiliation(s)
- C E Fernández Noa
- Instituto de Física da Universidade de São Paulo, 05314-970 São Paulo, Brazil
| | - Pedro E Harunari
- Instituto de Física da Universidade de São Paulo, 05314-970 São Paulo, Brazil
| | - M J de Oliveira
- Instituto de Física da Universidade de São Paulo, 05314-970 São Paulo, Brazil
| | - C E Fiore
- Instituto de Física da Universidade de São Paulo, 05314-970 São Paulo, Brazil
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8
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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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9
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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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10
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Vlysidis M, Kaznessis YN. On Differences between Deterministic and Stochastic Models of Chemical Reactions: Schlögl Solved with ZI-Closure. ENTROPY 2018; 20:e20090678. [PMID: 33265767 PMCID: PMC7513203 DOI: 10.3390/e20090678] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 07/03/2018] [Revised: 08/29/2018] [Accepted: 09/04/2018] [Indexed: 11/16/2022]
Abstract
Deterministic and stochastic models of chemical reaction kinetics can give starkly different results when the deterministic model exhibits more than one stable solution. For example, in the stochastic Schlögl model, the bimodal stationary probability distribution collapses to a unimodal distribution when the system size increases, even for kinetic constant values that result in two distinct stable solutions in the deterministic Schlögl model. Using zero-information (ZI) closure scheme, an algorithm for solving chemical master equations, we compute stationary probability distributions for varying system sizes of the Schlögl model. With ZI-closure, system sizes can be studied that have been previously unattainable by stochastic simulation algorithms. We observe and quantify paradoxical discrepancies between stochastic and deterministic models and explain this behavior by postulating that the entropy of non-equilibrium steady states (NESS) is maximum.
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11
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Agazzi A, Dembo A, Eckmann JP. Large deviations theory for Markov jump models of chemical reaction networks. ANN APPL PROBAB 2018. [DOI: 10.1214/17-aap1344] [Citation(s) in RCA: 24] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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12
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Huang S, Li F, Zhou JX, Qian H. Processes on the emergent landscapes of biochemical reaction networks and heterogeneous cell population dynamics: differentiation in living matters. J R Soc Interface 2018; 14:rsif.2017.0097. [PMID: 28490602 DOI: 10.1098/rsif.2017.0097] [Citation(s) in RCA: 16] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/09/2017] [Accepted: 04/18/2017] [Indexed: 12/24/2022] Open
Abstract
The notion of an attractor has been widely employed in thinking about the nonlinear dynamics of organisms and biological phenomena as systems and as processes. The notion of a landscape with valleys and mountains encoding multiple attractors, however, has a rigorous foundation only for closed, thermodynamically non-driven, chemical systems, such as a protein. Recent advances in the theory of nonlinear stochastic dynamical systems and its applications to mesoscopic reaction networks, one reaction at a time, have provided a new basis for a landscape of open, driven biochemical reaction systems under sustained chemostat. The theory is equally applicable not only to intracellular dynamics of biochemical regulatory networks within an individual cell but also to tissue dynamics of heterogeneous interacting cell populations. The landscape for an individual cell, applicable to a population of isogenic non-interacting cells under the same environmental conditions, is defined on the counting space of intracellular chemical compositions x = (x1,x2, … ,xN ) in a cell, where xℓ is the concentration of the ℓth biochemical species. Equivalently, for heterogeneous cell population dynamics xℓ is the number density of cells of the ℓth cell type. One of the insights derived from the landscape perspective is that the life history of an individual organism, which occurs on the hillsides of a landscape, is nearly deterministic and 'programmed', while population-wise an asynchronous non-equilibrium steady state resides mostly in the lowlands of the landscape. We argue that a dynamic 'blue-sky' bifurcation, as a representation of Waddington's landscape, is a more robust mechanism for a cell fate decision and subsequent differentiation than the widely pictured pitch-fork bifurcation. We revisit, in terms of the chemostatic driving forces upon active, living matter, the notions of near-equilibrium thermodynamic branches versus far-from-equilibrium states. The emergent landscape perspective permits a quantitative discussion of a wide range of biological phenomena as nonlinear, stochastic dynamics.
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Affiliation(s)
- Sui Huang
- Institute for Systems Biology, 401 Terry Avenue N., Seattle, WA 98109, USA
| | - Fangting Li
- School of Physics, Center for Quantitative Biology, Peking University, Beijing 100871, People's Republic of China
| | - Joseph X Zhou
- Institute for Systems Biology, 401 Terry Avenue N., Seattle, WA 98109, USA
| | - Hong Qian
- Department of Applied Mathematics, University of Washington, Seattle, WA 98195, USA
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13
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A framework towards understanding mesoscopic phenomena: Emergent unpredictability, symmetry breaking and dynamics across scales. Chem Phys Lett 2016. [DOI: 10.1016/j.cplett.2016.10.059] [Citation(s) in RCA: 21] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/26/2022]
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14
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de Franciscis S, Caravagna G, Mauri G, d’Onofrio A. Gene switching rate determines response to extrinsic perturbations in the self-activation transcriptional network motif. Sci Rep 2016; 6:26980. [PMID: 27256916 PMCID: PMC4891709 DOI: 10.1038/srep26980] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/17/2015] [Accepted: 05/11/2016] [Indexed: 01/01/2023] Open
Abstract
Gene switching dynamics is a major source of randomness in genetic networks, also in the case of large concentrations of the transcription factors. In this work, we consider a common network motif - the positive feedback of a transcription factor on its own synthesis - and assess its response to extrinsic noises perturbing gene deactivation in a variety of settings where the network might operate. These settings are representative of distinct cellular types, abundance of transcription factors and ratio between gene switching and protein synthesis rates. By investigating noise-induced transitions among the different network operative states, our results suggest that gene switching rates are key parameters to shape network response to external perturbations, and that such response depends on the particular biological setting, i.e. the characteristic time scales and protein abundance. These results might have implications on our understanding of irreversible transitions for noise-related phenomena such as cellular differentiation. In addition these evidences suggest to adopt the appropriate mathematical model of the network in order to analyze the system consistently to the reference biological setting.
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Affiliation(s)
| | - Giulio Caravagna
- Università degli Studi di Milano-Bicocca, Dipartimento di Informatica, Sistemistica e Comunicazione, Milano, Italy
- School of Informatics, University of Edinburgh, Edinburgh, UK
| | - Giancarlo Mauri
- Università degli Studi di Milano-Bicocca, Dipartimento di Informatica, Sistemistica e Comunicazione, Milano, Italy
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15
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Jia C, Qian M, Kang Y, Jiang D. Modeling stochastic phenotype switching and bet-hedging in bacteria: stochastic nonlinear dynamics and critical state identification. QUANTITATIVE BIOLOGY 2015. [DOI: 10.1007/s40484-014-0035-5] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/24/2022]
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16
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Chen H, He G, Huang F, Shen C. Entropy Production along Dominant Pathway of Nonequilibrium Phase Transition in Mesoscopic Chemical System. CHINESE J CHEM PHYS 2013. [DOI: 10.1063/1674-0068/26/05/549-552] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022]
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17
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Jenkinson G, Goutsias J. Statistically testing the validity of analytical and computational approximations to the chemical master equation. J Chem Phys 2013; 138:204108. [PMID: 23742455 DOI: 10.1063/1.4807390] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
The master equation is used extensively to model chemical reaction systems with stochastic dynamics. However, and despite its phenomenological simplicity, it is not in general possible to compute the solution of this equation. Drawing exact samples from the master equation is possible, but can be computationally demanding, especially when estimating high-order statistical summaries or joint probability distributions. As a consequence, one often relies on analytical approximations to the solution of the master equation or on computational techniques that draw approximative samples from this equation. Unfortunately, it is not in general possible to check whether a particular approximation scheme is valid. The main objective of this paper is to develop an effective methodology to address this problem based on statistical hypothesis testing. By drawing a moderate number of samples from the master equation, the proposed techniques use the well-known Kolmogorov-Smirnov statistic to reject the validity of a given approximation method or accept it with a certain level of confidence. Our approach is general enough to deal with any master equation and can be used to test the validity of any analytical approximation method or any approximative sampling technique of interest. A number of examples, based on the Schlögl model of chemistry and the SIR model of epidemiology, clearly illustrate the effectiveness and potential of the proposed statistical framework.
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Affiliation(s)
- Garrett Jenkinson
- Whitaker Biomedical Engineering Institute, The Johns Hopkins University, Baltimore, Maryland 21218, USA
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18
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Ren J, Sinitsyn NA. Braid group and topological phase transitions in nonequilibrium stochastic dynamics. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 87:050101. [PMID: 23767466 DOI: 10.1103/physreve.87.050101] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/18/2012] [Indexed: 06/02/2023]
Abstract
We show that distinct topological phases of the band structure of a non-Hermitian Hamiltonian can be classified with elements of the braid group. As the proof of principle, we consider the non-Hermitian evolution of the statistics of nonequilibrium stochastic currents. We show that topologically nontrivial phases have detectable properties, including the emergence of decaying oscillations of parity and state probabilities, and discontinuities in the steady state statistics of currents.
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Affiliation(s)
- Jie Ren
- Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA.
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19
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20
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Qian H. Cooperativity in Cellular Biochemical Processes: Noise-Enhanced Sensitivity, Fluctuating Enzyme, Bistability with Nonlinear Feedback, and Other Mechanisms for Sigmoidal Responses. Annu Rev Biophys 2012; 41:179-204. [DOI: 10.1146/annurev-biophys-050511-102240] [Citation(s) in RCA: 71] [Impact Index Per Article: 5.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/09/2022]
Affiliation(s)
- Hong Qian
- Department of Applied Mathematics, University of Washington, Seattle, Washington 98195;
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21
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Ge H, Qian H. Landscapes of non-gradient dynamics without detailed balance: stable limit cycles and multiple attractors. CHAOS (WOODBURY, N.Y.) 2012; 22:023140. [PMID: 22757547 DOI: 10.1063/1.4729137] [Citation(s) in RCA: 21] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/01/2023]
Abstract
Landscape is one of the key notions in literature on biological processes and physics of complex systems with both deterministic and stochastic dynamics. The large deviation theory (LDT) provides a possible mathematical basis for the scientists' intuition. In terms of Freidlin-Wentzell's LDT, we discuss explicitly two issues in singularly perturbed stationary diffusion processes arisen from nonlinear differential equations: (1) For a process whose corresponding ordinary differential equation has a stable limit cycle, the stationary solution exhibits a clear separation of time scales: an exponential terms and an algebraic prefactor. The large deviation rate function attains its minimum zero on the entire stable limit cycle, while the leading term of the prefactor is inversely proportional to the velocity of the non-uniform periodic oscillation on the cycle. (2) For dynamics with multiple stable fixed points and saddles, there is in general a breakdown of detailed balance among the corresponding attractors. Two landscapes, a local and a global, arise in LDT, and a Markov jumping process with cycle flux emerges in the low-noise limit. A local landscape is pertinent to the transition rates between neighboring stable fixed points; and the global landscape defines a nonequilibrium steady state. There would be nondifferentiable points in the latter for a stationary dynamics with cycle flux. LDT serving as the mathematical foundation for emergent landscapes deserves further investigations.
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Affiliation(s)
- Hao Ge
- Beijing International Center for Mathematical Research (BICMR) and Biodynamic Optical Imaging Center (BIOPIC), Peking University, Beijing 100871, People's Republic of China.
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22
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Salgado-García R. Resonant response in nonequilibrium steady states. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 85:051130. [PMID: 23004726 DOI: 10.1103/physreve.85.051130] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/30/2011] [Revised: 04/15/2012] [Indexed: 06/01/2023]
Abstract
The time-dependent probability density function of the order parameter of a system evolving toward a stationary state exhibits an oscillatory behavior if the eigenvalues of the corresponding evolution operator are complex. The frequencies ωn, with which the system reaches its stationary state, correspond to the imaginary part of such eigenvalues. If the system (at the stationary state) is further driven by a small and oscillating perturbation with a given frequency ω, we formally prove that the linear response to the probability density function is enhanced when ω=ωn for n∈N. We prove that the occurrence of this phenomenon is characteristic of systems that are in a nonequilibrium stationary state. In particular, we obtain an explicit formula for the frequency-dependent mobility in terms of the relaxation to the stationary state of the (unperturbed) probability current. We test all these predictions by means of numerical simulations considering an ensemble of noninteracting overdamped particles on a tilted periodic potential.
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Affiliation(s)
- R Salgado-García
- Facultad de Ciencias, Universidad Autónoma del Estado de Morelos, Avenida Universidad 1001, Colonia Chamilpa, 62209, Cuernavaca Morelos, Mexico.
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23
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Abstract
Bistability is a fundamental phenomenon in nature. In biology, a number of fine properties of bistability have been identified. However, these properties are only consequences of bistability at the physiological level, which do not explain why it had to emerge during evolution. Using optimal homeostasis as the first principle, I find that bistability emerges as an indispensable control mechanism. It is the only solution to a dilemma in glucose homeostasis: high insulin efficiency is required to confer rapidness in plasma glucose clearance, whereas an insulin sparing state is required to guarantee the brain's safety during fasting. The optimality consideration renders a clear correspondence between the molecular and physiological levels. This new perspective can illuminate studies on the twin epidemics of obesity and diabetes and the corresponding intervening strategies. For example, overnutrition and sedentary lifestyle may represent sudden environmental changes that cause the lose of optimality, which may contribute to the marked rise of obesity and diabetes in our generation. Because this bistability result is independent of the parameters of the mathematical model (for which the result is quite general), some other biological systems may also use bistability to control homeostasis.
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Affiliation(s)
- Guanyu Wang
- Department of Physics, George Washington University, Washington, DC 20052, USA.
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Zhou D, Qian H. Fixation, transient landscape, and diffusion dilemma in stochastic evolutionary game dynamics. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 84:031907. [PMID: 22060403 DOI: 10.1103/physreve.84.031907] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/28/2011] [Revised: 04/12/2011] [Indexed: 05/31/2023]
Abstract
Agent-based stochastic models for finite populations have recently received much attention in the game theory of evolutionary dynamics. Both the ultimate fixation and the pre-fixation transient behavior are important to a full understanding of the dynamics. In this paper, we study the transient dynamics of the well-mixed Moran process through constructing a landscape function. It is shown that the landscape playing a central theoretical "device" that integrates several lines of inquiries: the stable behavior of the replicator dynamics, the long-time fixation, and continuous diffusion approximation associated with asymptotically large population. Several issues relating to the transient dynamics are discussed: (i) multiple time scales phenomenon associated with intra- and inter-attractoral dynamics; (ii) discontinuous transition in stochastically stationary process akin to Maxwell construction in equilibrium statistical physics; and (iii) the dilemma diffusion approximation facing as a continuous approximation of the discrete evolutionary dynamics. It is found that rare events with exponentially small probabilities, corresponding to the uphill movements and barrier crossing in the landscape with multiple wells that are made possible by strong nonlinear dynamics, plays an important role in understanding the origin of the complexity in evolutionary, nonlinear biological systems.
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Affiliation(s)
- Da Zhou
- School of Mathematical Sciences, Peking University, Beijing 100871, China.
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25
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Qian H, Bishop LM. The chemical master equation approach to nonequilibrium steady-state of open biochemical systems: linear single-molecule enzyme kinetics and nonlinear biochemical reaction networks. Int J Mol Sci 2010; 11:3472-500. [PMID: 20957107 PMCID: PMC2956107 DOI: 10.3390/ijms11093472] [Citation(s) in RCA: 41] [Impact Index Per Article: 2.9] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/15/2010] [Accepted: 09/14/2010] [Indexed: 11/16/2022] Open
Abstract
We develop the stochastic, chemical master equation as a unifying approach to the dynamics of biochemical reaction systems in a mesoscopic volume under a living environment. A living environment provides a continuous chemical energy input that sustains the reaction system in a nonequilibrium steady state with concentration fluctuations. We discuss the linear, unimolecular single-molecule enzyme kinetics, phosphorylation-dephosphorylation cycle (PdPC) with bistability, and network exhibiting oscillations. Emphasis is paid to the comparison between the stochastic dynamics and the prediction based on the traditional approach based on the Law of Mass Action. We introduce the difference between nonlinear bistability and stochastic bistability, the latter has no deterministic counterpart. For systems with nonlinear bistability, there are three different time scales: (a) individual biochemical reactions, (b) nonlinear network dynamics approaching to attractors, and (c) cellular evolution. For mesoscopic systems with size of a living cell, dynamics in (a) and (c) are stochastic while that with (b) is dominantly deterministic. Both (b) and (c) are emergent properties of a dynamic biochemical network; We suggest that the (c) is most relevant to major cellular biochemical processes such as epi-genetic regulation, apoptosis, and cancer immunoediting. The cellular evolution proceeds with transitions among the attractors of (b) in a "punctuated equilibrium" manner.
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Affiliation(s)
- Hong Qian
- *Authors to whom correspondence should be addressed; E-Mails: (H.Q.); (L.M.B.)
| | - Lisa M. Bishop
- *Authors to whom correspondence should be addressed; E-Mails: (H.Q.); (L.M.B.)
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