arXiv subjects

Hao Ge

Publications and source records attributed to Hao Ge.

68 records · Page 4Linked to original sources

arXiv2011

Thermodynamics and Geometry of Reversible and Irreversible Markov Processes

Master equation with microscopic reversibility ($q_{ij}\neq 0$ iff $q_{ji}\neq 0$) has a {\em thermodynamic superstructure} in terms of two state functions $S$, entropy, and $F$, free energy: It is discovered recently that entropy production rate $e_p=-dF/dt+Q_{hk}$ with both $-dF/dt=f_d, Q_{hk} \ge 0$. The free energy dissipation $f_d\ge 0$ reflects irreversibility in spontaneous self-organization; house-keeping heat $Q_{hk}\ge 0$ reveals broken time-symmetry in open system driven away from equilibrium. In a Riemannian geometric space, the master equation is a geodesic flow when $Q_{hk}=0$; here we show that the $e_p$ decomposition is orthogonal: $e_p$, $f_d$, $Q_{hk}$ forms a pythagorean triples. Gradient flow means {\em maximum dissipation principle} outside Onsager's regime. The presence of $Q_{hk}$ makses gradient flow no longer generally true. Thermodynamics of stochastic physics requires a new geometric perspective.

cond-mat.stat-mech↗

arXiv2011

Markov processes follow from the principle of Maximum Caliber

Markov models are widely used to describe processes of stochastic dynamics. Here, we show that Markov models are a natural consequence of the dynamical principle of Maximum Caliber. First, we show that when there are different possible dynamical trajectories in a time-homogeneous process, then the only type of process that maximizes the path entropy, for any given singlet statistics, is a sequence of identical, independently distributed (i.i.d.) random variables, which is the simplest Markov process. If the data is in the form of sequentially pairwise statistics, then maximizing the caliber dictates that the process is Markovian with a uniform initial distribution. Furthermore, if an initial non-uniform dynamical distribution is known, or multiple trajectories are conditioned on an initial state, then the Markov process is still the only one that maximizes the caliber. Second, given a model, MaxCal can be used to compute the parameters of that model. We show that this procedure is equivalent to the maximum-likelihood method of inference in the theory of statistics.

cond-mat.stat-mech↗

arXiv2011

Maximum Entropy Principle, Equal Probability a Priori and Gibbs Paradox

We prove that information-theoretic maximum entropy (MaxEnt) approach to canonical ensemble is mathematically equivalent to the classic approach of Boltzmann, Gibbs and Darwin-Fowler. The two approaches, however, "interpret" a same mathematical theorem differently; most notably observing mean-energy in the former and energy conservation in the latter. However, applying the same MaxEnt method to grand canonical ensemble fails; while carefully following the classic approach based on Boltzmann's microcanonical {\em equal probability a priori} produces the correct statistics: One does not need to invoke quantum mechanics; and there is no Gibbs paradox. MaxEnt and related minimum relative entropy principle are based on the mathematical theorem concerning large deviations of rare fluctuations. As a scientific method, it requires classic mechanics, or some other assumptions, to provide meaningful {\em prior distributions} for the expected-value based statistical inference. A naive assumption of uniform prior is not valid in statistical mechanics.

cond-mat.stat-mech↗

arXiv2010

van't Hoff-Arrhenius Analysis of Mesoscopic and Macroscopic Dynamics of Simple Biochemical Systems: Stochastic vs. Nonlinear Bistabilities

Multistability of mesoscopic, driven biochemical reaction systems has implications to a wide range of cellular processes. Using several simple models, we show that one class of bistable chemical systems has a deterministic counterpart in the nonlinear dynamics based on the Law of Mass Action, while another class, widely known as noise-induced stochastic bistability, does not. Observing the system's volume ($V$) playing a similar role as the inverse temperature ($β$) in classical rate theory, an van't Hoff-Arrhenius like analysis is introduced. In one-dimensional systems, a transition rate between two states, represented in terms of a barrier in the landscape for the dynamics $Φ(x,V)$, $k\propto\exp\{-VΔΦ^‡(V)\}$, can be understood from a decomposition $ΔΦ^‡(V) \approxΔϕ_0^‡ Δϕ_1^‡/V$. Nonlinear bistability means $Δϕ_0^‡>0$ while stochastic bistability has $Δϕ_0^‡<0$ but $Δϕ_1^‡>0$. Stochastic bistabilities can be viewed as remants (or "ghosts) of nonlinear bifurcations or extinction phenomenon, and $Δϕ_0^‡$ and $Δϕ_1^‡$ as "enthalpic" and "entropic" barriers to a transition.

physics.chem-ph↗

arXiv2009

The Physical Origins of Entropy Production, Free Energy Dissipation and their Mathematical Representations

A complete mathematical theory of nonequilibrium thermodynamics of stochastic systems in terms of master equations is presented. As generalizations of isothermal entropy and free energy, two functions of states play central roles: the Gibbs entropy $S$ and the relative entropy $F$, which are related via the stationary distribution of the stochastic dynamics. $S$ satisfies the fundamental entropy balance equation $dS/dt=e_p-h_d/T$ with entropy production rate $e_p\ge 0$ and heat dissipation rate $h_d$, while $dF/dt=-f_d\le 0$. For closed systems that satisfy detailed balance: $Te_p(t)=f_d(t)$. For open system one has $Te_p(t)=f_d(t)+Q_{hk}(t)$ where the housekeeping heat $Q_{hk}\ge 0$ was first introduced in the phenomenological nonequilibrium steady state thermodynamics. Entropy production $e_p$ consists of free energy dissipation associated with spontaneous relaxation, $f_d$, and active energy pumping that sustains the open system $Q_{hk}$. The amount of excess heat involved in the relaxation $Q_{ex}=h_d-Q_{hk} = f_d-T(dS/dt)$.

cond-mat.stat-mech↗

arXiv2009

Thermodynamic Limit of a Nonequilibrium Steady-State: Maxwell-Type Construction for a Bistable Biochemical System

We show that the thermodynamic limit of a bistable phosphorylation-dephosphorylation cycle has a selection rule for the "more stable" macroscopic steady state. The analysis is akin to the Maxwell construction. Based on the chemical master equation approach, it is shown that, except at a critical point, bistability disappears in the stochastic model when fluctuation is sufficiently low but unneglectable. Onsager's Gaussian fluctuation theory applies to the unique macroscopic steady state. With initial state in the basin of attraction of the "less stable" steady state, the deterministic dynamics obtained by the Law of Mass Action is a metastable phenomenon. Stability and robustness in cell biology are stochastic concepts.

physics.chem-ph↗

arXiv2009

Nonequilibrium phase transition in a mesoscopic biochemical system: From stochastic to nonlinear dynamics and beyond

A rigorous mathematical framework for analyzing the chemical master equation (CME) with bistability, based on the theory of large deviation, is proposed. Using a simple phosphorylation-dephosphorylation cycle with feedback as an example, we show that a nonequilibrium steady-state (NESS) phase transition occurs in the system which has all the characteristics of classic equilibrium phase transition: Maxwell construction, discontinuous fraction of phosphorylation as a function of the kinase activity, and Lee-Yang's zero for the generating function. The cusp in nonlinear bifurcation theory matches the tricritical point of the phase transition. The mathematical analysis suggests three distinct time scales, and related mathematical descriptions, of (i) molecular signaling, (ii) biochemical network dynamics, and (iii) cellular evolution. The (i) and (iii) are stochastic while (ii) is deterministic.

physics.bio-ph↗

arXiv2009

Time-dependent Nonequilibrium Thermodynamics: A Master-equation Approach

Master equation could be applied to model various kinds of biochemical systems. A general theory for its time-dependent nonequilibrium thermodynamics is rigorously derived. We not only introduce a concept of general internal energy, but also propose an extension of the equilibrium state. Moreover, we find out an explicit expression for the extended form of the Second Law, which is phenomenologically stated as "the conversion of work to excess heat is irreversible". The theory is carefully applied to several thermodynamic processes, and its consistence with the classical thermodynamics is clearly shown.

cond-mat.stat-mech↗

arXiv2009

Sensitivity Amplification in the Phosphorylation-Dephosphorylation Cycle: Nonequilibrium steady states, chemical master equation and temporal cooperativity

A new type of cooperativity termed temporal cooperativity [Biophys. Chem. 105 585-593 (2003), Annu. Rev. Phys. Chem. 58 113-142 (2007)], emerges in the signal transduction module of phosphorylation-dephosphorylation cycle (PdPC). It utilizes multiple kinetic cycles in time, in contrast to allosteric cooperativity that utilizes multiple subunits in a protein. In the present paper, we thoroughly investigate both the deterministic (microscopic) and stochastic (mesoscopic) models, and focus on the identification of the source of temporal cooperativity via comparing with allosteric cooperativity. A thermodynamic analysis confirms again the claim that the chemical equilibrium state exists if and only if the phosphorylation potential $\triangle G=0$, in which case the amplification of sensitivity is completely abolished. Then we provide comprehensive theoretical and numerical analysis with the first-order and zero-order assumptions in phosphorylation-dephosphorylation cycle respectively. Furthermore, it is interestingly found that the underlying mathematics of temporal cooperativity and allosteric cooperativity are equivalent, and both of them can be expressed by "dissociation constants", which also characterizes the essential differences between the simple and ultrasensitive PdPC switches. Nevertheless, the degree of allosteric cooperativity is restricted by the total number of sites in a single enzyme molecule which can not be freely regulated, while temporal cooperativity is only restricted by the total number of molecules of the target protein which can be regulated in a wide range and gives rise to the ultrasensitivity phenomenon.

physics.chem-ph↗

arXiv2009

Boolean Network Approach to Negative Feedback Loops of the p53 Pathways: Synchronized Dynamics and Stochastic Limit Cycles

Deterministic and stochastic Boolean network models are build for the dynamics of negative feedback loops of the p53 pathways. It is shown that the main function of the negative feedback in the p53 pathways is to keep p53 at a low steady state level, and each sequence of protein states in the negative feedback loops, is globally attracted to a closed cycle of the p53 dynamics after being perturbed by outside signal (e.g. DNA damage). Our theoretical and numerical studies show that both the biological stationary state and the biological oscillation after being perturbed are stable for a wide range of noise level. Applying the mathematical circulation theory of Markov chains, we investigate their stochastic synchronized dynamics and by comparing the network dynamics of the stochastic model with its corresponding deterministic network counterpart, a dominant circulation in the stochastic model is the natural generalization of the deterministic limit cycle in the deterministic system. Moreover, the period of the main peak in the power spectrum, which is in common use to characterize the synchronized dynamics, perfectly corresponds to the number of states in the main cycle with dominant circulation. Such a large separation in the magnitude of the circulations, between a dominant, main cycle and the rest, gives rise to the stochastic synchronization phenomenon.

q-bio.MN↗

arXiv2009

Generalized Jarzynski's equality of inhomogeneous multidimensional diffusion processes

Applying the well-known Feynman-Kac formula of inhomogeneous case, an interesting and rigorous mathematical proof of generalized Jarzynski's equality of inhomogeneous multidimensional diffusion processes is presented, followed by an extension of the second law of thermodynamics. Then, we explain its physical meaning and applications, extending Hummer and Szabo's work ({\em Proc. Natl. Acad. Sci. USA} {\bf 98}(7), 3658--3661 (2001)) and Hatano-Sasa equality of steady state thermodynamics ({\em Phys. Rev. Lett.} {\bf 86}, 3463--3466 (2001)) to the general multidimensional case.

cond-mat.stat-mech↗

arXiv2009

Synchronized Dynamics and Nonequilibrium Steady States in a Stochastic Yeast Cell-Cycle Network

Applying the mathematical circulation theory of Markov chains, we investigate the synchronized stochastic dynamics of a discrete network model of yeast cell-cycle regulation where stochasticity has been kept rather than being averaged out. By comparing the network dynamics of the stochastic model with its corresponding deterministic network counterpart, we show that the synchronized dynamics can be soundly characterized by a dominant circulation in the stochastic model, which is the natural generalization of the deterministic limit cycle in the deterministic system. Moreover, the period of the main peak in the power spectrum, which is in common use to characterize the synchronized dynamics, perfectly corresponds to the number of states in the main cycle with dominant circulation. Such a large separation in the magnitude of the circulations, between a dominant, main cycle and the rest, gives rise to the stochastic synchronization phenomenon.

q-bio.MN↗

arXiv2009

Waiting Cycle Times and Generalized Haldane Equality in the Steady-state Cycle Kinetics of Single Enzymes

Enzyme kinetics are cyclic. A more realistic reversible three-step mechanism of the Michaelis-Menten kinetics is investigated in detail, and three kinds of waiting cycle times $T$, $T_{+}$, $T_{-}$ are defined. It is shown that the mean waiting cycle times $ $, $ $, and $ $ are the reciprocal of the steady-state cycle flux $J^{ss}$, the forward steady-state cycle flux $J^{ss}_{+}$ and the backward steady-state cycle flux $J^{ss}_{-}$ respectively. We also show that the distribution of $T_{+}$ conditioned on $T_{+} $) and the one of $T_{-}$ conditioned on $T_{-} $) are both just the same as $ $. In addition, the forward and backward stepping probabilities $p^{+},p^{-}$ are also defined and discussed, especially their relationship with the cycle fluxes and waiting cycle times. Furthermore, we extend the same results to the $n$-step cycle, and finally, experimental and theoretically based evidences are also included.

cond-mat.stat-mech↗

arXiv2009

Nonequilibrium Thermodynamics of Time-dependent Langevin Systems: Energy Balance Relation and the Extended Form of the Second Law

A general nonequilibrium thermodynamic theory is developed for time-dependent Langevin dynamics, starting from the common definition of nonequilibrium Gibbs entropy. It is shown that the notations appearing in the First and the Second Law of thermodynamics could be consistently applied to transient nonequilibrium processes. We find out a general equality representing the energy balance relation for the system, and further an explicit mathematical interpretation for the extended form of the Second Law is proposed, which only comes into existence in the time-dependent case. More applications to several classic thermodynamic processes are finally included.

cond-mat.stat-mech↗