arXiv subjects

Hong Qian

Publications and source records attributed to Hong Qian.

At least 91 records · Page 5Linked to original sources

arXiv2017

Emergent Lévy behavior in single-cell stochastic gene expression

Single-cell gene expression is inherently stochastic; its emergent behavior can be defined in terms of the chemical master equation describing the evolution of the mRNA and protein copy numbers as the latter tends to infinity. We establish two types of "macroscopic limits": the Kurtz limit is consistent with the classical chemical kinetics, while the Lévy limit provides a theoretical foundation for an empirical equation proposed in [Phys. Rev. Lett. 97:168302, 2006]. Furthermore, we clarify the biochemical implications and ranges of applicability for various macroscopic limits and calculate a comprehensive analytic expression for the protein concentration distribution in autoregulatory gene networks. The relationship between our work and modern population genetics is discussed.

q-bio.MN↗

arXiv2016

A framework towards understanding mesoscopic phenomena: Emergent unpredictability, symmetry breaking and dynamics across scales

By integrating 4 lines of thoughts: symmetry breaking originally advanced by Anderson, bifurcation from nonlinear dynamics, Landau's theory of phase transition, and the mechanism of emergent rare events studied by Kramers, we introduce a possible framework for understanding mesoscopic dynamics that links (i) fast lower level microscopic motions, (ii) movements within each basin at the mid-level, and (iii) higher-level rare transitions between neighboring basins, which have rates that decrease exponentially with the size of the system. In this mesoscopic framework, multiple attractors arise as emergent properties of the nonlinear systems. The interplay between the stochasticity and nonlinearity leads to successive jump-like transitions among different basins. We argue each transition is a dynamic symmetry breaking, with the potential of exhibiting Thom-Zeeman catastrophe as well as phase transition with the breakdown of ergodicity (e.g., cell differentiation). The slow-time dynamics of the nonlinear mesoscopic system is not deterministic, rather it is a discrete stochastic jump process. The existence of these discrete states and the Markov transitions among them are both emergent phenomena. This emergent stochastic jump dynamics then serves as the stochastic element for the nonlinear dynamics of a higher level aggregates on an even larger spatial and slower time scales (e.g., evolution). This description captures the hierarchical structure outlined by Anderson and illustrates two distinct types of limit of a mesoscopic dynamics: A long-time ensemble thermodynamics in terms of time $t$ tending infinity followed by the size of the system $N$ tending infinity, and a short-time trajectory steady state with $N$ tending infinity followed by $t$ tending infinity. With these limits, symmetry breaking and cusp catastrophe are two perspectives of the same mesoscopic system on different time scales.

cond-mat.stat-mech↗

arXiv2016

A Discrete Stochastic Formulation for Reversible Bimolecular Reactions via Diffusion Encounter

The classical models for irreversible diffusion-influenced reactions can be derived by introducing absorbing boundary conditions to over-damped continuous Brownian motion (BM) theory. As there is a clear corresponding stochastic process, the mathematical description takes both Kolmogorov forward equation for the evolution of the probability distribution function and the stochastic sample trajectories. This dual description is a fundamental characteristic of stochastic processes and allows simple particle based simulations to accurately match the expected statistical behavior. However, in the traditional theory using the back-reaction boundary condition to model reversible reactions with geminate recombinations, several subtleties arise: it is unclear what the underlying stochastic process is, which causes complications in producing accurate simulations; and it is non-trivial how to perform an appropriate discretization for numerical computations. In this work, we derive a discrete stochastic model that recovers the classical models and their boundary conditions in the continuous limit. In the case of reversible reactions, we recover the back-reaction boundary condition, unifying the back-reaction approach with those of current simulation packages. Furthermore, all the complications encountered in the continuous models become trivial in the discrete model. Our formulation brings to attention the question: With computations in mind, can we develop a discrete reaction kinetics model that is more fundamental than its continuous counterpart?

cond-mat.stat-mech↗

arXiv2016

Mathematical Formalism of Nonequilibrium Thermodynamics for Nonlinear Chemical Reaction Systems with General Rate Law

This paper studies a mathematical formalism of nonequilibrium thermodynamics for chemical reaction models with $N$ species, $M$ reactions, and general rate law. We establish a mathematical basis for J. W. Gibbs' macroscopic chemical thermodynamics under G. N. Lewis' kinetic law of entire equilibrium (detailed balance in nonlinear chemistry kinetics). In doing so, the equilibrium thermodynamics is then naturally generalized to nonequilibrium settings without detailed balance. The kinetic models are represented by a Markovian jumping process. A generalized macroscopic chemical free energy function and its associated balance equation with nonnegative source and sink are the major discoveries. The proof is based on the large deviation principle of this type of Markov processes. A general fluctuation dissipation theorem for stochastic reaction kinetics is also proved. The mathematical theory illustrates how a novel macroscopic dynamic law can emerges from the mesoscopic kinetics in a multi-scale system.

math-ph↗

arXiv2016

Nonlinear Stochastic Dynamics of Complex Systems, II: Potential of Entropic Force in Markov Systems with Nonequilibrium Steady State, Generalized Gibbs Function and Criticality

In this paper we revisit the notion of the "minus logarithm of stationary probability" as a generalized potential in nonequilibrium systems and attempt to illustrate its central role in an axiomatic approach to stochastic nonequilibrium thermodynamics of complex systems. It is demonstrated that this quantity arises naturally through both monotonicity results of Markov processes and as the rate function when a stochastic process approaches a deterministic limit. We then undertake a more detailed mathematical analysis of the consequences of this quantity, culminating in a necessary and sufficient condition for the criticality of stochastic systems. This condition is then discussed in the context of recent results about criticality in biological systems

cond-mat.stat-mech↗

arXiv2016

Nonlinear Stochastic Dynamics of Complex Systems, III: Noneqilibrium Thermodynamics of Self-Replication Kinetics

We briefly review the recently developed, Markov process based isothermal chemical thermodynamics for nonlinear, driven mesoscopic kinetic systems. Both the instantaneous Shannon entropy {\boldmath $S[p_α(t)]$} and relative entropy {\boldmath $F[p_α(t)]$}, defined based on probability distribution {\boldmath $\{p_α(t)\}$}, play prominent roles. The theory is general; and as a special case when a chemical reaction system is situated in an equilibrium environment, it agrees perfectly with Gibbsian chemical thermodynamics: {\boldmath $k_BS$} and {\boldmath $k_BTF$} become thermodynamic entropy and free energy, respectively. We apply this theory to a fully reversible autocatalytic reaction kinetics, represented by a Delbrück-Gillespie process, in a chemostatic nonequilibrium environment. The open, driven chemical system serves as an archetype for biochemical self-replication. The significance of {\em thermodynamically consistent} kinetic coarse-graining is emphasized. In a kinetic system where death of a biological organism is treated as the reversal of its birth, the meaning of mathematically emergent "dissipation", which is not related to the heat measured in terms of {\boldmath $k_BT$}, remains to be further investigated.

cond-mat.stat-mech↗

arXiv2016

Nonlinear Stochastic Dynamics of Complex Systems, I: A Chemical Reaction Kinetic Perspective with Mesoscopic Nonequilibrium Thermodynamics

We distinguish a mechanical representation of the world in terms of point masses with positions and momenta and the chemical representation of the world in terms of populations of different individuals, each with intrinsic stochasticity, but population wise with statistical rate laws in their syntheses, degradations, spatial diffusion, individual state transitions, and interactions. Such a formal kinetic system in a small volume $V$, like a single cell, can be rigorously treated in terms of a Markov process describing its nonlinear kinetics as well as nonequilibrium thermodynamics at a mesoscopic scale. We introduce notions such as open, driven chemical systems, entropy production, free energy dissipation, etc. Then in the macroscopic limit, we illustrate how two new "laws", in terms of a generalized free energy of the mesoscopic stochastic dynamics, emerge. Detailed balance and complex balance are two special classes of "simple" nonlinear kinetics. Phase transition is intrinsically related to multi-stability and saddle-node bifurcation phenomenon, in the limits of time $t\rightarrow\infty$ and system's size $V\rightarrow\infty$. Using this approach, we re-articulate the notion of inanimate equilibrium branch of a system and nonequilibrium state of a living matter, as originally proposed by Nicolis and Prigogine, and seek a logic consistency between this viewpoint and that of P. W. Anderson and J. J. Hopfield's in which macroscopic law emerges through symmetry breaking.

cond-mat.stat-mech↗

arXiv2016

Nonequilibrium Thermodynamic Formalism of Nonlinear Chemical Reaction Systems with Waage-Guldberg's Law of Mass Action

Macroscopic entropy production $σ^{(tot)}$ in the general nonlinear isothermal chemical reaction system with mass action kinetics is decomposed into a free energy dissipation and a house-keeping heat: $σ^{(tot)}=σ^{(fd)}+σ^{(hk)}$; $σ^{(fd)}=-\rd A/\rd t$, where $A$ is a generalized free energy function. This yields a novel nonequilibrium free energy balance equation $\rd A/\rd t=-σ^{(tot)}+σ^{(hk)}$, which is on a par with celebrated entropy balance equation $\rd S/\rd t=σ^{(tot)}+η^{(ex)}$ where $η^{(ex)}$ is the rate of entropy exchange with the environment.For kinetic systems with complex balance, $σ^{(fd)}$ and $σ^{(hk)}$ are the macroscopic limits of stochastic free energy dissipation and house-keeping heat, which are both nonnegative, in the Delbrück-Gillespie description of the stochastic chemical kinetics.Therefore, we show that a full kinetic and thermodynamic theory of chemical reaction systems that transcends mesoscopic and macroscopic levels emerges.

physics.chem-ph↗

arXiv2016

Entropy Production in Mesoscopic Stochastic Thermodynamics: Nonequilibrium Kinetic Cycles Driven by Chemical Potentials, Temperatures, and Mechanical Forces

Nonequilibrium thermodynamics (NET) investigates processes in systems out of global equilibrium. On a mesoscopic level, it provides a statistical dynamic description of various complex phenomena such as chemical reactions, ion transport, diffusion, thermochemical, thermomechanical and mechanochemical fluxes. In the present review, we introduce a mesoscopic stochastic formulation of NET by analyzing entropy production in several simple examples. The fundamental role of nonequilibrium steady-state cycle kinetics is emphasized. The statistical mechanics of Onsager's reciprocal relations in this context is elucidated. Chemomechanical, thermomechanical, and enzyme-catalyzed thermochemical energy transduction processes are discussed. It is argued that mesoscopic stochastic NET provides a rigorous mathematical basis of fundamental concepts needed for understanding complex processes in chemistry, physics and biology, and which is also relevant for nanoscale technological advances.

cond-mat.stat-mech↗

arXiv2016

Mesoscopic Kinetic Basis of Macroscopic Chemical Thermodynamics: A Mathematical Theory

From a mathematical model that describes a complex chemical kinetic system of $N$ species and $M$ elementrary reactions in a rapidly stirred vessel of size $V$ as a Markov process, we show that a macroscopic chemical thermodynamics emerges as $V\rightarrow\infty$. The theory is applicable to linear and nonlinear reactions, closed systems reaching chemical equilibrium, or open, driven systems approaching to nonequilibrium steady states. A generalized mesoscopic free energy gives rise to a macroscopic chemical energy function $φ^{ss}(\vx)$ where $\vx=(x_1,\cdots,x_N)$ are the concentrations of the $N$ chemical species. The macroscopic chemical dynamics $\vx(t)$ satisfies two emergent laws: (1) $(\rd/\rd t)φ^{ss}[\vx(t)]\le 0$, and (2)$(\rd/\rd t)φ^{ss}[\vx(t)]=\text{cmf}(\vx)-σ(\vx)$ where entropy production rate $σ\ge 0$ represents the sink for the chemical energy, and chemical motive force $\text{cmf}\ge 0$ is non-zero if the system is driven under a sustained nonequilibrium chemostat. For systems with detailed balance $\text{cmf}=0$, and if one assumes the law of mass action,$φ^{ss}(\vx)$ is precisely the Gibbs' function $\sum_{i=1}^N x_i\big[μ_i^o+\ln x_i\big]$ for ideal solutions. For a class of kinetic systems called complex balanced, which include many nonlinear systems as well as many simple open, driven chemical systems, the $φ^{ss}(\vx)$, with global minimum at $\vx^*$, has the generic form $\sum_{i=1}^N x_i\big[\ln(x_i/x_i^*)-x_i+x_i^*\big]$,which has been known in chemical kinetic literature.Macroscopic emergent "laws" are independent of the details of the underlying kinetics. This theory provides a concrete example from chemistry showing how a dynamic macroscopic law can emerge from the kinetics at a level below.

physics.chem-ph↗

arXiv2015

A Thermodynamic Theory of Ecology: Helmholtz Theorem for Lotka-Volterra Equation, Extended Conservation Law, and Stochastic Predator-Prey Dynamics

We carry out mathematical analyses, {\em à la} Helmholtz's and Boltzmann's 1884 studies of monocyclic Newtonian dynamics, for the Lotka-Volterra (LV) equation exhibiting predator-prey oscillations. In doing so a novel "thermodynamic theory" of ecology is introduced. An important feature, absent in the classical mechanics, of ecological systems is a natural stochastic population dynamic formulation of which the deterministic equation (e.g., the LV equation studied) is the infinite population limit. Invariant density for the stochastic dynamics plays a central role in the deterministic LV dynamics. We show how the conservation law along a single trajectory extends to incorporate both variations in a model parameter $α$ and in initial conditions: Helmholtz's theorem establishes a broadly valid conservation law in a class of ecological dynamics. We analyze the relationships among mean ecological activeness $θ$, quantities characterizing dynamic ranges of populations $\mathcal{A}$ and $α$, and the ecological force $F_α$. The analyses identify an entire orbit as a stationary ecology, and establish the notion of "equation of ecological states". Studies of the stochastic dynamics with finite populations show the LV equation as the robust, fast cyclic underlying behavior. The mathematical narrative provides a novel way of capturing long-term dynamical behaviors with an emergent {\em conservative ecology}.

math-ph↗

arXiv2015

Stochastic Free Energies, Conditional Probability and Legendre Transform for Ensemble Change

This work extends a recently developed mathematical theory of thermodynamics for Markov processes with, and more importantly, without detailed balance. We show that the Legendre transform in connection to ensemble changes in Gibbs' statistical mechanics can be derived from the stochastic theory. We consider the joint probability $p_{XY}$ of two random variables X and Y and the conditional probability $p_{X|Y=y^*}$, with $y^*= $ according to $p_{XY}$. The stochastic free energies of the XY system (fluctuating Y ensemble) and the $X|Y$ system (fixed Y ensemble) are related by the chain rule for relative entropy. In the thermodynamic limit, defined as $V,Y\to\infty$ (where one assumes Y as an extensive quantity, while V denotes the system size parameter), the marginal probability obeys $p_Y(y)\to \exp(-Vϕ(z))$ with $z=y/V$. A conjugate variable $μ=-ϕ(z)/z$ naturally emerges from this result. The stochastic free energies of the fluctuating and fixed ensembles are then related by $F_{XY}=F_{X|Y=y^*}-μy^*$, with $μ=\partial F_{X|Y=y}/\partial y$. The time evolutions for the two free energies are the same: $d[F_{XY}(t)]/dt=\partial [F_{X|Y=y^*(t)}(t)]/ \partial t$. This mathematical theory is applied to systems with fixed and fluctuating number of identical independent random variables (idea gas), as well as to microcanonical systems with uniform stationary probability.

cond-mat.stat-mech↗

arXiv2015

Stochastic phenotype transition of a single cell in an intermediate region of gene-state switching

Multiple phenotypic states often arise in a single cell with different gene-expression states that undergo transcription regulation with positive feedback. Recent experiments have shown that at least in E. coli, the gene state switching can be neither extremely slow nor exceedingly rapid as many previous theoretical treatments assumed. Rather it is in the intermediate region which is difficult to handle mathematically.Under this condition, from a full chemical-master-equation description we derive a model in which the protein copy-number, for a given gene state, follow a deterministic mean-field description while the protein synthesis rates fluctuate due to stochastic gene-state switching. The simplified kinetics yields a nonequilibrium landscape function, which, similar to the energy function for equilibrium fluctuation, provides the leading orders of fluctuations around each phenotypic state, as well as the transition rates between the two phenotypic states. This rate formula is analogous to Kramers theory for chemical reactions. The resulting behaviors are significantly different from the two limiting cases studied previously.

q-bio.MN↗

arXiv2015

Thermodynamics of the General Diffusion Process: Equilibrium Supercurrent and Nonequilibrium Driven Circulation with Dissipation

Unbalanced probability circulation, which yields cyclic motions in phase space, is the defining characteristics of a stationary diffusion process without detailed balance. In over-damped soft matter systems, such behavior is a hallmark of the presence of a sustained external driving force accompanied with dissipations. In an under-damped and strongly correlated system, however, cyclic motions are often the consequences of a conservative dynamics. In the present paper, we give a novel interpretation of a class of diffusion processes with stationary circulation in terms of a Maxwell-Boltzmann equilibrium in which cyclic motions are on the level set of stationary probability density function thus non-dissipative, e.g., a supercurrent. This implies an orthogonality between stationary circulation $J^{ss}(x)$ and the gradient of stationary probability density $f^{ss}(x)>0$. A sufficient and necessary condition for the orthogonality is a decomposition of the drift $b(x)=j(x)+ D(x)\nablaφ(x)$ where $\nabla\cdot j(x)=0$ and $j(x)$ $\cdot\nablaφ(x)=0$. Stationary processes with Maxwell-Boltzmann equilibrium has an underlying conservative dynamics $\dot{x}= j(x)\equiv$ $\big(f^{ss}(x)\big)^{-1}J^{ss}(x)$, and a first integral $φ(x)\equiv-\ln f^{ss}(x)=$ const, akin to a Hamiltonian system. At all time, an instantaneous free energy balance equation exists for a given diffusion system; and an extended energy conservation law among a family of diffusion processes with different parameter $α$ can be established via a Helmholtz theorem. For the general diffusion process without the orthogonality, a nonequilibrium cycle emerges, which consists of external driven $φ$-ascending steps and spontaneous $φ$-descending movements, alternated with iso-$φ$ motions. The theory presented here provides a rich mathematical narrative for complex mesoscopic dynamics.

math-ph↗

arXiv2015

Universal Ideal Behavior and Macroscopic Work Relation of Linear Irreversible Stochastic Thermodynamics

We revisit the Ornstein-Uhlenbeck (OU) process as the fundamental mathematical description of linear irreversible phenomena, with fluctuations, near an equilibrium. By identifying the underlying circulating dynamics in a stationary process as the natural generalization of classical conservative mechanics, a bridge between a family of OU processes with equilibrium fluctuations and thermodynamics is established through the celebrated Helmholtz theorem. The Helmholtz theorem provides an emergent macroscopic "equation of state" of the entire system, which exhibits a universal ideal thermodynamic behavior. Fluctuating macroscopic quantities are studied from the stochastic thermodynamic point of view and a non-equilibrium work relation is obtained in the macroscopic picture, which may facilitate experimental study and application of the equalities due to Jarzynski, Crooks, and Hatano and Sasa.

cond-mat.stat-mech↗

arXiv2014

Fitness and entropy production in a cell population dynamics with epigenetic phenotype switching

Motivated by recent understandings in the stochastic natures of gene expression, biochemical signaling, and spontaneous reversible epigenetic switchings, we study a simple deterministic cell population dynamics in which subpopulations grow with different rates and individual cells can bi-directionally switch between a small number of different epigenetic phenotypes. Two theories in the past, the population dynamics and thermodynamics of master equations, separatedly defined two important concepts in mathematical terms: the {\em fitness} in the former and the (non-adiabatic) {\em entropy production} in the latter. Both play important roles in the evolution of the cell population dynamics. The switching sustains the variations among the subpopulation growth thus continuous natural selection. As a form of Price's equation, the fitness increases with ($i$) natural selection through variations and $(ii)$ a positive covariance between the per capita growth and switching, which represents a Lamarchian-like behavior. A negative covariance balances the natural selection in a fitness steady state | "the red queen" scenario. At the same time the growth keeps the proportions of subpopulations away from the "intrinsic" switching equilibrium of individual cells, thus leads to a continous entropy production. A covariance, between the per capita growth rate and the "chemical potential" of subpopulation, counter-acts the entropy production. Analytical results are obtained for the limiting cases of growth dominating switching and vice versa.

q-bio.PE↗

arXiv2014

Fluorescence Correlation Spectroscopy and Nonlinear Stochastic Reaction-Diffusion

The currently existing theory of fluorescence correlation spectroscopy(FCS) is based on the linear fluctuation theory originally developed by Einstein, Onsager, Lax, and others as a phenomenological approach to equilibrium fluctuations in bulk solutions. For mesoscopic reaction-diffusion systems with nonlinear chemical reactions among a small number of molecules, a situation often encountered in single-cell biochemistry, it is expected that FCS time correlation functions of a reaction-diffusion system can deviate from the classic results of Elson and Magde. We first discuss this nonlinear effect for reaction systems without diffusion. For nonlinear stochastic reaction-diffusion systems here are no closed solutions; therefore, stochastic Monte-Carlo simulations are carried out. We show that the deviation is small for a simple bimolecular reaction; the most significant deviations occur when the number of molecules is small and of the same order. Our results show that current linear FCS theory could be adequate for measurements on biological systems that contain many other sources of uncertainties. At the same time it provides a framework for future measurements of nonlinear, fluctuating chemical reactions with high-precision FCS. Extending Delbrück-Gillespie's theory for stochastic nonlinear reactions with rapidly stirring to reaction-diffusion systems provides a mesoscopic model for chemical and biochemical reactions at nanometric and mesoscopic level such as a single biological cell.

physics.bio-ph↗

arXiv2014

The Sampling-and-Learning Framework: A Statistical View of Evolutionary Algorithms

Evolutionary algorithms (EAs), a large class of general purpose optimization algorithms inspired from the natural phenomena, are widely used in various industrial optimizations and often show excellent performance. This paper presents an attempt towards revealing their general power from a statistical view of EAs. By summarizing a large range of EAs into the sampling-and-learning framework, we show that the framework directly admits a general analysis on the probable-absolute-approximate (PAA) query complexity. We particularly focus on the framework with the learning subroutine being restricted as a binary classification, which results in the sampling-and-classification (SAC) algorithms. With the help of the learning theory, we obtain a general upper bound on the PAA query complexity of SAC algorithms. We further compare SAC algorithms with the uniform search in different situations. Under the error-target independence condition, we show that SAC algorithms can achieve polynomial speedup to the uniform search, but not super-polynomial speedup. Under the one-side-error condition, we show that super-polynomial speedup can be achieved. This work only touches the surface of the framework. Its power under other conditions is still open.

cs.NE↗