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Global potential, topology, and pattern selection in a noisy stabilized Kuramoto-Sivashinsky equation. Proc Natl Acad Sci U S A 2020; 117:23227-23234. [PMID: 32917812 DOI: 10.1073/pnas.2012364117] [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: 11/18/2022] Open
Abstract
We formulate a general method to extend the decomposition of stochastic dynamics developed by Ao et al. [J. Phys. Math. Gen. 37, L25-L30 (2004)] to nonlinear partial differential equations which are nonvariational in nature and construct the global potential or Lyapunov functional for a noisy stabilized Kuramoto-Sivashinsky equation. For values of the control parameter where singly periodic stationary solutions exist, we find a topological network of a web of saddle points of stationary states interconnected by unstable eigenmodes flowing between them. With this topology, a global landscape of the steady states is found. We show how to predict the noise-selected pattern which agrees with those from stochastic simulations. Our formalism and the topology might offer an approach to explore similar systems, such as the Navier Stokes equation.
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Terebus A, Liu C, Liang J. Discrete and continuous models of probability flux of switching dynamics: Uncovering stochastic oscillations in a toggle-switch system. J Chem Phys 2019; 151:185104. [PMID: 31731858 DOI: 10.1063/1.5124823] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/15/2022] Open
Abstract
The probability flux and velocity in stochastic reaction networks can help in characterizing dynamic changes in probability landscapes of these networks. Here, we study the behavior of three different models of probability flux, namely, the discrete flux model, the Fokker-Planck model, and a new continuum model of the Liouville flux. We compare these fluxes that are formulated based on, respectively, the chemical master equation, the stochastic differential equation, and the ordinary differential equation. We examine similarities and differences among these models at the nonequilibrium steady state for the toggle switch network under different binding and unbinding conditions. Our results show that at a strong stochastic condition of weak promoter binding, continuum models of Fokker-Planck and Liouville fluxes deviate significantly from the discrete flux model. Furthermore, we report the discovery of stochastic oscillation in the toggle-switch system occurring at weak binding conditions, a phenomenon captured only by the discrete flux model.
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Affiliation(s)
- Anna Terebus
- Richard and Loan Hill Department of Bioengineering, University of Illinois at Chicago, Chicago, Illinois 60607, USA
| | - Chun Liu
- Department of Applied Mathematics, Illinois Institute of Technology, Chicago, Illinois 60616, USA
| | - Jie Liang
- Richard and Loan Hill Department of Bioengineering, University of Illinois at Chicago, Chicago, Illinois 60607, USA
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Terebus A, Liu C, Liang J. Discrete flux and velocity fields of probability and their global maps in reaction systems. J Chem Phys 2018; 149:185101. [PMID: 30441929 DOI: 10.1063/1.5050808] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/14/2023] Open
Abstract
Stochasticity plays important roles in reaction systems. Vector fields of probability flux and velocity characterize time-varying and steady-state properties of these systems, including high probability paths, barriers, checkpoints among different stable regions, as well as mechanisms of dynamic switching among them. However, conventional fluxes on continuous space are ill-defined and are problematic when at the boundaries of the state space or when copy numbers are small. By re-defining the derivative and divergence operators based on the discrete nature of reactions, we introduce new formulations of discrete fluxes. Our flux model fully accounts for the discreetness of both the state space and the jump processes of reactions. The reactional discrete flux satisfies the continuity equation and describes the behavior of the system evolving along directions of reactions. The species discrete flux directly describes the dynamic behavior in the state space of the reactants such as the transfer of probability mass. With the relationship between these two fluxes specified, we show how to construct time-evolving and steady-state global flow-maps of probability flux and velocity in the directions of every species at every microstate and how they are related to the outflow and inflow of probability fluxes when tracing out reaction trajectories. We also describe how to impose proper conditions enabling exact quantification of flux and velocity in the boundary regions, without the difficulty of enforcing artificial reflecting conditions. We illustrate the computation of probability flux and velocity using three model systems, namely, the birth-death process, the bistable Schlögl model, and the oscillating Schnakenberg model.
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Affiliation(s)
- Anna Terebus
- Department of Bioengineering, University of Illinois at Chicago, Chicago, Illinois 60607, USA
| | - Chun Liu
- Department of Applied Mathematics, Illinois Institute of Technology, Chicago, Illinois 60616, USA
| | - Jie Liang
- Department of Bioengineering, University of Illinois at Chicago, Chicago, Illinois 60607, USA
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Potential landscape of high dimensional nonlinear stochastic dynamics with large noise. Sci Rep 2017; 7:15762. [PMID: 29150680 PMCID: PMC5693902 DOI: 10.1038/s41598-017-15889-2] [Citation(s) in RCA: 25] [Impact Index Per Article: 3.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/05/2017] [Accepted: 10/02/2017] [Indexed: 12/14/2022] Open
Abstract
Quantifying stochastic processes is essential to understand many natural phenomena, particularly in biology, including the cell-fate decision in developmental processes as well as the genesis and progression of cancers. While various attempts have been made to construct potential landscape in high dimensional systems and to estimate transition rates, they are practically limited to the cases where either noise is small or detailed balance condition holds. A general and practical approach to investigate real-world nonequilibrium systems, which are typically high-dimensional and subject to large multiplicative noise and the breakdown of detailed balance, remains elusive. Here, we formulate a computational framework that can directly compute the relative probabilities between locally stable states of such systems based on a least action method, without the necessity of simulating the steady-state distribution. The method can be applied to systems with arbitrary noise intensities through A-type stochastic integration, which preserves the dynamical structure of the deterministic counterpart dynamics. We demonstrate our approach in a numerically accurate manner through solvable examples. We further apply the method to investigate the role of noise on tumor heterogeneity in a 38-dimensional network model for prostate cancer, and provide a new strategy on controlling cell populations by manipulating noise strength.
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Yuan R, Zhu X, Wang G, Li S, Ao P. Cancer as robust intrinsic state shaped by evolution: a key issues review. REPORTS ON PROGRESS IN PHYSICS. PHYSICAL SOCIETY (GREAT BRITAIN) 2017; 80:042701. [PMID: 28212112 DOI: 10.1088/1361-6633/aa538e] [Citation(s) in RCA: 28] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/27/2023]
Abstract
Cancer is a complex disease: its pathology cannot be properly understood in terms of independent players-genes, proteins, molecular pathways, or their simple combinations. This is similar to many-body physics of a condensed phase that many important properties are not determined by a single atom or molecule. The rapidly accumulating large 'omics' data also require a new mechanistic and global underpinning to organize for rationalizing cancer complexity. A unifying and quantitative theory was proposed by some of the present authors that cancer is a robust state formed by the endogenous molecular-cellular network, which is evolutionarily built for the developmental processes and physiological functions. Cancer state is not optimized for the whole organism. The discovery of crucial players in cancer, together with their developmental and physiological roles, in turn, suggests the existence of a hierarchical structure within molecular biology systems. Such a structure enables a decision network to be constructed from experimental knowledge. By examining the nonlinear stochastic dynamics of the network, robust states corresponding to normal physiological and abnormal pathological phenotypes, including cancer, emerge naturally. The nonlinear dynamical model of the network leads to a more encompassing understanding than the prevailing linear-additive thinking in cancer research. So far, this theory has been applied to prostate, hepatocellular, gastric cancers and acute promyelocytic leukemia with initial success. It may offer an example of carrying physics inquiring spirit beyond its traditional domain: while quantitative approaches can address individual cases, however there must be general rules/laws to be discovered in biology and medicine.
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Affiliation(s)
- Ruoshi Yuan
- Key Laboratory of Systems Biomedicine, Ministry of Education, Shanghai Center for Systems Biomedicine, Shanghai Jiao Tong University, Shanghai 200240, People's Republic of China
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Roy-Layinde TO, Laoye JA, Popoola OO, Vincent UE. Analysis of vibrational resonance in bi-harmonically driven plasma. CHAOS (WOODBURY, N.Y.) 2016; 26:093117. [PMID: 27781458 DOI: 10.1063/1.4962403] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/06/2023]
Abstract
The phenomenon of vibrational resonance (VR) is examined and analyzed in a bi-harmonically driven two-fluid plasma model with nonlinear dissipation. An equation for the slow oscillations of the system is analytically derived in terms of the parameters of the fast signal using the method of direct separation of motion. The presence of a high frequency externally applied electric field is found to significantly modify the system's dynamics, and consequently, induce VR. The origin of the VR in the plasma model has been identified, not only from the effective plasma potential but also from the contributions of the effective nonlinear dissipation. Beside several dynamical changes, including multiple symmetry-breaking bifurcations, attractor escapes, and reversed period-doubling bifurcations, numerical simulations also revealed the occurrence of single and double resonances induced by symmetry breaking bifurcations.
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Affiliation(s)
- T O Roy-Layinde
- Department of Physics, Olabisi Onabanjo University, Ago-Iwoye, Nigeria
| | - J A Laoye
- Department of Physics, Olabisi Onabanjo University, Ago-Iwoye, Nigeria
| | - O O Popoola
- Department of Physics, University of Ibadan, Ibadan, Nigeria
| | - U E Vincent
- Department of Physical Sciences, Redeemers' University, Ede, Nigeria
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Chen YC, Yuan RS, Ao P, Xu MJ, Zhu XM. Towards stable kinetics of large metabolic networks: Nonequilibrium potential function approach. Phys Rev E 2016; 93:062409. [PMID: 27415300 DOI: 10.1103/physreve.93.062409] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/08/2015] [Indexed: 01/21/2023]
Abstract
While the biochemistry of metabolism in many organisms is well studied, details of the metabolic dynamics are not fully explored yet. Acquiring adequate in vivo kinetic parameters experimentally has always been an obstacle. Unless the parameters of a vast number of enzyme-catalyzed reactions happened to fall into very special ranges, a kinetic model for a large metabolic network would fail to reach a steady state. In this work we show that a stable metabolic network can be systematically established via a biologically motivated regulatory process. The regulation is constructed in terms of a potential landscape description of stochastic and nongradient systems. The constructed process draws enzymatic parameters towards stable metabolism by reducing the change in the Lyapunov function tied to the stochastic fluctuations. Biologically it can be viewed as interplay between the flux balance and the spread of workloads on the network. Our approach allows further constraints such as thermodynamics and optimal efficiency. We choose the central metabolism of Methylobacterium extorquens AM1 as a case study to demonstrate the effectiveness of the approach. Growth efficiency on carbon conversion rate versus cell viability and futile cycles is investigated in depth.
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Affiliation(s)
- Yong-Cong Chen
- Key Laboratory of Systems Biomedicine, Ministry of Education, Shanghai Center for Systems Biomedicine, Shanghai Jiao Tong University, Shanghai, 200240, China.,SmartWin Technology, 67 Tranmere Avenue, Carnegie, VIC 3163, Australia
| | - Ruo-Shi Yuan
- Key Laboratory of Systems Biomedicine, Ministry of Education, Shanghai Center for Systems Biomedicine, Shanghai Jiao Tong University, Shanghai, 200240, China
| | - Ping Ao
- Key Laboratory of Systems Biomedicine, Ministry of Education, Shanghai Center for Systems Biomedicine, Shanghai Jiao Tong University, Shanghai, 200240, China
| | - Min-Juan Xu
- Key Laboratory of Systems Biomedicine, Ministry of Education, Shanghai Center for Systems Biomedicine, Shanghai Jiao Tong University, Shanghai, 200240, China
| | - Xiao-Mei Zhu
- Key Laboratory of Systems Biomedicine, Ministry of Education, Shanghai Center for Systems Biomedicine, Shanghai Jiao Tong University, Shanghai, 200240, China.,GeneMath, 5525 27th Avenue N.E., Seattle, Washington 98105, USA
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Tang Y, Yuan R, Chen J, Ao P. Work relations connecting nonequilibrium steady states without detailed balance. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:042108. [PMID: 25974440 DOI: 10.1103/physreve.91.042108] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/30/2014] [Indexed: 06/04/2023]
Abstract
Bridging equilibrium and nonequilibrium statistical physics attracts sustained interest. Hallmarks of nonequilibrium systems include a breakdown of detailed balance, and an absence of a priori potential function corresponding to the Boltzmann-Gibbs distribution, without which classical equilibrium thermodynamical quantities could not be defined. Here, we construct dynamically the potential function through decomposing the system into a dissipative part and a conservative part, and develop a nonequilibrium theory by defining thermodynamical quantities based on the potential function. Concepts for equilibrium can thus be naturally extended to nonequilibrium steady state. We elucidate this procedure explicitly in a class of time-dependent linear diffusive systems without mathematical ambiguity. We further obtain the exact work distribution for an arbitrary control parameter, and work equalities connecting nonequilibrium steady states. Our results provide a direct generalization on Jarzynski equality and Crooks fluctuation theorem to systems without detailed balance.
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Affiliation(s)
- Ying Tang
- Department of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China
- Key Laboratory of Systems Biomedicine Ministry of Education, Shanghai Center for Systems Biomedicine, Shanghai Jiao Tong University, Shanghai 200240, China
| | - Ruoshi Yuan
- School of Biomedical Engineering, Shanghai Jiao Tong University, Shanghai 200240, China
| | - Jianhong Chen
- Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania 16802, USA
| | - Ping Ao
- Department of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China
- Key Laboratory of Systems Biomedicine Ministry of Education, Shanghai Center for Systems Biomedicine, Shanghai Jiao Tong University, Shanghai 200240, China
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Méndez V, Denisov SI, Campos D, Horsthemke W. Role of the interpretation of stochastic calculus in systems with cross-correlated Gaussian white noises. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:012116. [PMID: 25122260 DOI: 10.1103/physreve.90.012116] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/20/2014] [Indexed: 06/03/2023]
Abstract
We derive the Fokker-Planck equation for multivariable Langevin equations with cross-correlated Gaussian white noises for an arbitrary interpretation of the stochastic differential equation. We formulate the conditions when the solution of the Fokker-Planck equation does not depend on which stochastic calculus is adopted. Further, we derive an equivalent multivariable Ito stochastic differential equation for each possible interpretation of the multivariable Langevin equation. To demonstrate the usefulness and significance of these general results, we consider the motion of Brownian particles. We study in detail the stability conditions for harmonic oscillators with two white noises, one of which is additive, random forcing, and the other, which accounts for fluctuations of either the damping or the spring coefficient, is multiplicative. We analyze the role of cross correlation in terms of the different noise interpretations and confirm the theoretical predictions via numerical simulations. We stress the interest of our results for numerical simulations of stochastic differential equations with an arbitrary interpretation of the stochastic integrals.
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Affiliation(s)
- Vicenç Méndez
- Grup de Física Estadística, Departament de Física, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain
| | - S I Denisov
- Department of General and Theoretical Physics, Sumy State University, Rimsky-Korsakov Street 2, UA-40007 Sumy, Ukraine
| | - Daniel Campos
- Grup de Física Estadística, Departament de Física, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain
| | - Werner Horsthemke
- Department of Chemistry, Southern Methodist University, Dallas, Texas 75275-0314, USA
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Tang Y, Yuan R, Ao P. Nonequilibrium work relation beyond the Boltzmann-Gibbs distribution. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 89:062112. [PMID: 25019730 DOI: 10.1103/physreve.89.062112] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/03/2014] [Indexed: 06/03/2023]
Abstract
The presence of multiplicative noise can alter measurements of forces acting on nanoscopic objects. Taking into account of multiplicative noise, we derive a series of nonequilibrium thermodynamical equalities as generalization of the Jarzynski equality, the detailed fluctuation theorem and the Hatano-Sasa relation. Our result demonstrates that the Jarzynski equality and the detailed fluctuation theorem remains valid only for systems with the Boltzmann-Gibbs distribution at the equilibrium state, but the Hatano-Sasa relation is robust with respect to different stochastic interpretations of multiplicative noise.
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Affiliation(s)
- Ying Tang
- Department of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China and Key Laboratory of Systems Biomedicine Ministry of Education, Shanghai Center for Systems Biomedicine, Shanghai Jiao Tong University, Shanghai 200240, China
| | - Ruoshi Yuan
- School of Biomedical Engineering, Shanghai Jiao Tong University, Shanghai 200240, China
| | - Ping Ao
- Department of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China and Key Laboratory of Systems Biomedicine Ministry of Education, Shanghai Center for Systems Biomedicine, Shanghai Jiao Tong University, Shanghai 200240, China
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Abstract
The digital nature of genes combined with the associated low copy numbers of proteins regulating them is a significant source of stochasticity, which affects the phase of biochemical oscillations. We show that unlike ordinary chemical oscillators, the dichotomic molecular noise of gene state switching in gene oscillators affects the stochastic dephasing in a way that may not always be captured by phenomenological limit cycle-based models. Through simulations of a realistic model of the NFκB/IκB network, we also illustrate the dephasing phenomena that are important for reconciling single-cell and population-based experiments on gene oscillators.
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12
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Xu S, Jiao S, Jiang P, Ao P. Two-time-scale population evolution on a singular landscape. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 89:012724. [PMID: 24580274 DOI: 10.1103/physreve.89.012724] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/21/2013] [Indexed: 06/03/2023]
Abstract
Under the effect of strong genetic drift, it is highly probable to observe gene fixation or gene loss in a population, shown by singular peaks on a potential landscape. The genetic drift-induced noise gives rise to two-time-scale diffusion dynamics on the bipeaked landscape. We find that the logarithmically divergent (singular) peaks do not necessarily imply infinite escape times or biological fixations by iterating the Wright-Fisher model and approximating the average escape time. Our analytical results under weak mutation and weak selection extend Kramers's escape time formula to models with B (Beta) function-like equilibrium distributions and overcome constraints in previous methods. The constructed landscape provides a coherent description for the bistable system, supports the quantitative analysis of bipeaked dynamics, and generates mathematical insights for understanding the boundary behaviors of the diffusion model.
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Affiliation(s)
- Song Xu
- Department of Biomathematics, University of California at Los Angeles, Los Angeles, California 90095-1766, USA
| | - Shuyun Jiao
- Shanghai Center for Systems Biomedicine, Key Laboratory of Systems Biomedicine of Ministry of Education, Shanghai Jiao Tong University, Shanghai 200240, China and Department of Mathematics, Xinyang Normal University, Xinyang 464000, Henan, China
| | - Pengyao Jiang
- Department of Ecology and Evolution, University of Chicago, 1101 E. 57th Street, Chicago, Illinois 60637, USA
| | - Ping Ao
- Shanghai Center for Systems Biomedicine, Key Laboratory of Systems Biomedicine of Ministry of Education, Shanghai Jiao Tong University, Shanghai 200240, China and Department of Physics, Shanghai Jiao Tong University, Shanghai 200240, China
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