arXiv subjects

Hong Qian

Publications and source records attributed to Hong Qian.

At least 109 records · Page 6Linked to original sources

arXiv2014

Stochastic Dynamics of Electrical Membrane with Voltage-Dependent Ion Channel Fluctuations

Brownian ratchet like stochastic theory for the electrochemical membrane system of Hodgkin-Huxley (HH) is developed. The system is characterized by a continuous variable $Q_m(t)$, representing mobile membrane charge density, and a discrete variable $K_t$ representing ion channel conformational dynamics. A Nernst-Planck-Nyquist-Johnson type equilibrium is obtained when multiple conducting ions have a common reversal potential. Detailed balance yields a previously unknown relation between the channel switching rates and membrane capacitance, bypassing Eyring-type explicit treatment of gating charge kinetics. From a molecular structural standpoint, membrane charge $Q_m$ is a more natural dynamic variable than potential $V_m$; our formalism treats $Q_m$-dependent conformational transition rates $λ_{ij}$ as intrinsic parameters. Therefore in principle, $λ_{ij}$ vs. $V_m$ is experimental protocol dependent,e.g., different from voltage or charge clamping measurements. For constant membrane capacitance per unit area $C_m$ and neglecting membrane potential induced by gating charges, $V_m=Q_m/C_m$, and HH's formalism is recovered. The presence of two types of ions, with different channels and reversal potentials, gives rise to a nonequilibrium steady state with positive entropy production $e_p$. For rapidly fluctuating channels, an expression for $e_p$ is obtained.

physics.bio-ph↗

arXiv2014

Small Open Chemical Systems Theory and Its Implications to Darwinian Evolutionary Dynamics, Complex Self-Organization and Beyond

The study of biological cells in terms of mesoscopic, nonequilibrium, nonlinear, stochastic dynamics of open chemical systems provides a paradigm for other complex, self-organizing systems with ultra-fast stochastic fluctuations, short-time deterministic nonlinear dynamics, and long-time evolutionary behavior with exponentially distributed rare events, discrete jumps among punctuated equilibria, and catastrophe.

physics.chem-ph↗

arXiv2013

The Zeroth Law of Thermodynamics and Volume-Preserving Conservative Dynamics with Equilibrium Stochastic Damping

We propose a mathematical formulation of the zeroth law of thermodynamics and develop a stochastic dynamical theory, with a consistent irreversible thermodynamics, for systems possessing sustained conservative stationary current in phase space while in equilibrium with a heat bath. The theory generalizes underdamped mechanical equilibrium: $dx=gdt+\{-D\nablaϕdt+\sqrt{2D}dB(t)\}$, with $\nabla\cdot g=0$ and $\{\cdots\}$ respectively representing phase-volume preserving dynamics and stochastic damping. The zeroth law implies stationary distribution $u^{ss}(x)=e^{-ϕ(x)}$. We find an orthogonality $\nablaϕ\cdot g=0$ as a hallmark of the system. Stochastic thermodynamics based on time reversal $\big(t,ϕ,g\big)\rightarrow\big(-t,ϕ,-g\big)$ is formulated: entropy production $e_p^{\#}(t)=-dF(t)/dt$; generalized "heat" $h_d^{\#}(t)=-dU(t)/dt$, $U(t)=\int_{\mathbb{R}^n} ϕ(x)u(x,t)dx$ being "internal energy", and "free energy" $F(t)=U(t)+\int_{\mathbb{R}^n} u(x,t)\ln u(x,t)dx$ never increases. Entropy follows $\frac{dS}{dt}=e_p^{\#}-h_d^{\#}$. Our formulation is shown to be consistent with an earlier theory of P. Ao. Its contradistinctions to other theories, potential-flux decomposition, stochastic Hamiltonian system with even and odd variables, Klein-Kramers equation, Freidlin-Wentzell's theory, and GENERIC, are discussed.

math-ph↗

arXiv2013

Nonlinear Biochemical Signal Processing via Noise Propagation

Single-cell studies often show significant phenotypic variability due to the stochastic nature of intra-cellular biochemical reactions. When the numbers of molecules, e.g., transcription factors and regulatory enzymes, are in low abundance, fluctuations in biochemical activities become significant and such "noise" can propagate through regulatory cascades in terms of biochemical reaction networks. Here we develop an intuitive, yet fully quantitative method for analyzing how noise affects cellular phenotypes based on identifying a system's nonlinearities and noise propagations. We observe that such noise can simultaneously enhance sensitivities in one behavioral region while reducing sensitivities in another. Employing this novel phenomenon we designed three biochemical signal processing modules: (a) A gene regulatory network that acts as a concentration detector with both enhanced amplitude and sensitivity. (b) A non-cooperative positive feedback system, with a graded dose-response in the deterministic case, that serves as a bistable switch due to noise-induced bimodality. (c) A noise-induced linear amplifier for gene regulation that requires no feedback. The methods developed in the present work allow one to understand and engineer nonlinear biochemical signal processors based on fluctuation-induced phenotypes.

q-bio.QM↗

arXiv2013

Dissipation, Generalized Free Energy, and a Self-consistent Nonequilibrium Thermodynamics of Chemically Driven Open Subsystems

Nonequilibrium thermodynamics of a system situated in a sustained environment with influx and efflux is usually treated as a subsystem in a larger, closed "universe". It remains a question what the minimally required description for the surrounding of such an open driven system is, so that its nonequilibrium thermodynamics can be established solely based on the internal stochastic kinetics. We provide a solution to this problem using insights from studies of molecular motors in a chemical nonequilibrium steady state (NESS) with sustained external drive through a regenerating system, or in a quasi-steady state (QSS) with an excess amount of ATP, ADP, and Pi. We introduce the key notion of {\em minimal work} that is needed, $W_{min}$, for the external regenerating system to sustain a NESS ({\em e.g.}, maintaining constant concentrations of ATP, ADP and Pi for a molecular motor). Using a Markov (master-equation) description of a motor protein, we illustrate that the NESS and QSS have identical kinetics as well as the Second Law in terms of a same positive entropy production rate. The difference between the heat dissipation of a NESS and its corresponding QSS is exactly the $W_{min}$. This provides a justification for introducing an {\em ideal external regenerating system} and yields a {\em free energy balance equation} between the net free energy input $F_{in}$ and total dissipation $F_{dis}$ in an NESS: $F_{in}$ consists of chemical input minus mechanical output; $F_{dis}$ consists of dissipative heat; and the amount of useful energy becoming heat is the NESS entropy production. Furthermore, we show that for non-stationary systems, the $F_{dis}$ and $F_{in}$ correspond to the entropy production rate and housekeeping heat in stochastic thermodynamics, and identify a relative entropy $H$ as a generalized free energy.

cond-mat.stat-mech↗

arXiv2013

A Decomposition of Irreversible Diffusion Processes Without Detailed Balance

As a generalization of deterministic, nonlinear conservative dynamical systems, a notion of {\em canonical conservative dynamics} with respect to a positive, differentiable stationary density $ρ(x)$ is introduced: $\dot{x}=j(x)$ in which $\nabla\cdot\big(ρ(x)j(x)\big)=0$. Such systems have a conserved "generalized free energy function" $F[u]$ $=$ $\int u(x,t)\ln\big(u(x,t)/ρ(x)\big)dx$ in phase space with a density flow $u(x,t)$ satisfying $\partial u_t =-\nabla\cdot(ju)$. Any general stochastic diffusion process without detailed balance, in terms of its Fokker-Planck equation, can be decomposed into a reversible diffusion process with detailed balance and a canonical conservative dynamics. This decomposition can be rigorously established in a function space with inner product defined as $\langleϕ,ψ\rangle=\intρ^{-1}(x)ϕ(x)ψ(x)dx$. Furthermore, a law for balancing $F[u]$ can be obtained: The non-positive $dF[u(x,t)]/dt$ $=$ $E_{in}(t)-e_p(t)$ where the "source" $E_{in}(t)\ge 0$ and the "sink" $e_p(t)\ge 0$ are known as house-keeping heat and entropy production, respectively. A reversible diffusion has $E_{in}(t)=0$. For a linear (Ornstein-Uhlenbeck) diffusion process, our decomposition is equivalent to the previous approaches developed by R. Graham and P. Ao, as well as the theory of large deviations. In terms of two different formulations of time reversal for a same stochastic process, the meanings of {\em dissipative} and {\em conservative} stationary dynamics are discussed.

math-ph↗

arXiv2013

A Hamiltonian-Entropy Production Connection in the Skew-symmetric Part of a Stochastic Dynamics

The infinitesimal transition probability operator for a continuous-time discrete-state Markov process, $\mathcal{Q}$, can be decomposed into a symmetric and a skew-symmetric parts. As recently shown for the case of diffusion processes, while the symmetric part corresponding to a gradient system stands for a reversible Markov process, the skew-symmetric part, $\frac{d}{dt}u(t)=\mcA u$, is mathematically equivalent to a linear Hamiltonian dynamics with Hamiltonian $H=1/2u^T\big(\mcA^T\mcA)^{1/2}u$. It can also be transformed into a Schrödinger-like equation $\frac{d}{dt}u=i\mathcal{H}u$ where the "Hamiltonian" operator $\mathcal{H}=-i\mcA$ is Hermitian. In fact, these two representations of a skew-symmetric dynamics emerge natually through singular-value and eigen-value decompositions, respectively. The stationary probability of the Markov process can be expressed as $\|u^s_i\|^2$. The motion can be viewed as "harmonic" since $\frac{d}{dt}\|u(t)-\vec{c}\|^2=0$ where $\vec{c}=(c,c,...,c)$ with $c$ being a constant. More interestingly, we discover that $\textrm{Tr}(\mcA^T\mcA)=\sum_{j,\ell=1}^n \frac{(q_{j\ell}π_\ell-q_{\ell j}π_j)^2}{π_jπ_{\ell}}$, whose right-hand-side is intimately related to the entropy production rate of the Markov process in a nonequilibrium steady state with stationary distribution $\{π_j\}$. The physical implication of this intriguing connection between conservative Hamiltonian dynamics and dissipative entropy production remains to be further explored.

math-ph↗

arXiv2012

Stochastic Physics, Complex Systems and Biology

In complex systems, the interplay between nonlinear and stochastic dynamics, e.g., J. Monod's necessity and chance, gives rise to an evolutionary process in Darwinian sense, in terms of discrete jumps among attractors, with punctuated equilibrium, spontaneous random "mutations" and "adaptations". On an evlutionary time scale it produces sustainable diversity among individuals in a homogeneous population rather than convergence as usually predicted by a deterministic dynamics. The emergent discrete states in such a system, i.e., attractors, have natural robustness against both internal and external perturbations. Phenotypic states of a biological cell, a mesoscopic nonlinear stochastic open biochemical system, could be understood through such a perspective.

nlin.AO↗

arXiv2012

One-Dimensional Birth-Death Process and Delbrück-Gillespie Theory of Mesoscopic Nonlinear Chemical Reactions

As a mathematical theory for the stochasstic, nonlinear dynamics of individuals within a population, Delbrück-Gillespie process (DGP) $n(t)\in\mathbb{Z}^N$, is a birth-death system with state-dependent rates which contain the system size $V$ as a natural parameter. For large $V$, it is intimately related to an autonomous, nonlinear ordinary differential equation as well as a diffusion process. For nonlinear dynamical systems with multiple attractors, the quasi-stationary and stationary behavior of such a birth-death process can be underestood in terms of a separation of time scales by a $T^*\sim e^{αV}$ $(α>0)$: a relatively fast, intra-basin diffusion for $t\ll T^*$ and a much slower inter-basin Markov jump process for $t\gg T^*$. In the present paper for one-dimensional systems, we study both stationary behavior ($t=\infty$) in terms of invariant distribution $p_n^{ss}(V)$, and finite time dynamics in terms of the mean first passsage time (MFPT) $T_{n_1\rightarrow n_2}(V)$. We obtain an asymptotic expression of MFPT in terms of the "stochastic potential" $Φ(x,V)=-(1/V)\ln p^{ss}_{xV}(V)$. We show in general no continuous diffusion process can provide asymptotically accurate representations for both the MFPT and the $p_n^{ss}(V)$ for a DGP. When $n_1$ and $n_2$ belong to two different basins of attraction, the MFPT yields the $T^*(V)$ in terms of $Φ(x,V)\approx ϕ_0(x)+(1/V)ϕ_1(x)$. For systems with a saddle-node bifurcation and catastrophe, discontinuous "phase transition" emerges, which can be characterized by $Φ(x,V)$ in the limit of $V\rightarrow\infty$. In terms of time scale separation, the relation between deterministic, local nonlinear bifurcations and stochastic global phase transition is discussed. The one-dimensional theory is a pedagogic first step toward a general theory of DGP.

math.DS↗

arXiv2012

Analytical Mechanics in Stochastic Dynamics: Most Probable Path, Large-Deviation Rate Function and Hamilton-Jacobi Equation

Analytical (rational) mechanics is the mathematical structure of Newtonian deterministic dynamics developed by D'Alembert, Langrange, Hamilton, Jacobi, and many other luminaries of applied mathematics. Diffusion as a stochastic process of an overdamped individual particle immersed in a fluid, initiated by Einstein, Smoluchowski, Langevin and Wiener, has no momentum since its path is nowhere differentiable. In this exposition, we illustrate how analytical mechanics arises in stochastic dynamics from a randomly perturbed ordinary differential equation $dX_t=b(X_t)dt+εdW_t$ where $W_t$ is a Brownian motion. In the limit of vanishingly small $ε$, the solution to the stochastic differential equation other than $\dot{x}=b(x)$ are all rare events. However, conditioned on an occurence of such an event, the most probable trajectory of the stochastic motion is the solution to Lagrangian mechanics with $\mathcal{L}=\|\dot{q}-b(q)\|^2/4$ and Hamiltonian equations with $H(p,q)=\|p\|^2+b(q)\cdot p$. Hamiltonian conservation law implies that the most probable trajectory for a "rare" event has a uniform "excess kinetic energy" along its path. Rare events can also be characterized by the principle of large deviations which expresses the probability density function for $X_t$ as $f(x,t)=e^{-u(x,t)/ε}$, where $u(x,t)$ is called a large-deviation rate function which satisfies the corresponding Hamilton-Jacobi equation. An irreversible diffusion process with $\nabla\times b\neq 0$ corresponds to a Newtonian system with a Lorentz force $\ddot{q}=(\nabla\times b)\times \dot{q}+1/2\nabla\|b\|^2$. The connection between stochastic motion and analytical mechanics can be explored in terms of various techniques of applied mathematics, for example, singular perturbations, viscosity solutions, and integrable systems.

math-ph↗

arXiv2012

The Dynamics of Zeroth-Order Ultrasensitivity: A Critical Phenomenon in Cell Biology

It is well known since the pioneering work of Goldbeter and Koshland [Proc. Natl. Acad. Sci. USA, vol. 78, pp. 6840-6844 (1981)] that cellular phosphorylation- dephosphorylation cycle (PdPC), catalyzed by kinase and phosphatase under saturated condition with zeroth order enzyme kinetics, exhibits ultrasensitivity, sharp transition. We analyse the dynamics aspects of the zeroth order PdPC kinetics and show a critical slowdown akin to the phase transition in condensed matter physics. We demonstrate that an extremely simple, though somewhat mathematically "singular" model is a faithful representation of the ultrasentivity phenomenon. The simplified mathematical model will be valuable, as a component, in developing complex cellular signaling network theory as well as having a pedagogic value.

q-bio.SC↗

arXiv2012

Landscapes of Non-gradient Dynamics Without Detailed Balance: Stable Limit Cycles and Multiple Attractors

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.

math-ph↗

arXiv2012

Stochastic Thermodynamics Across Scales: Emergent Inter-attractoral Discrete Markov Jump Process and Its Underlying Continuous Diffusion

The consistency across scales of a recently developed mathematical thermodynamic structure, between a continuous stochastic nonlinear dynamical system (diffusion process with Langevin or Fokker-Planck equations) and its emergent discrete, inter-attractoral Markov jump process, is investigated. We analyze how the system's thermodynamic state functions, e.g. free energy $F$, entropy $S$, entropy production $e_p$, and free energy dissipation $\dot{F}$, etc., are related when the continuous system is describe with a coarse-grained discrete variable. We show that the thermodynamics derived from the underlying detailed continuous dynamics is exact in the Helmholtz free-energy representation. That is, the system thermodynamic structure is the same as if one only takes a middle-road and starts with the "natural" discrete description, with the corresponding transition rates empirically determined. By "natural", we mean in the thermodynamic limit of large systems in which there is an inherent separation of time scales between inter- and intra-attractoral dynamics. This result generalizes a fundamental idea from chemistry and the theory of Kramers by including thermodynamics: while a mechanical description of a molecule is in terms of continuous bond lengths and angles, chemical reactions are phenomenologically described by the Law of Mass Action with rate constants, and a stochastic thermodynamics.

cond-mat.stat-mech↗

arXiv2012

Mesoscopic Biochemical Basis of Isogenetic Inheritance and Canalization: Stochasticity, Nonlinearity, and Emergent Landscape

Biochemical reaction systems in mesoscopic volume, under sustained environmental chemical gradient(s), can have multiple stochastic attractors. Two distinct mechanisms are known for their origins: ($a$) Stochastic single-molecule events, such as gene expression, with slow gene on-off dynamics; and ($b$) nonlinear networks with feedbacks. These two mechanisms yield different volume dependence for the sojourn time of an attractor. As in the classic Arrhenius theory for temperature dependent transition rates, a landscape perspective provides a natural framework for the system's behavior. However, due to the nonequilibrium nature of the open chemical systems, the landscape, and the attractors it represents, are all themselves {\em emergent properties} of complex, mesoscopic dynamics. In terms of the landscape, we show a generalization of Kramers' approach is possible to provide a rate theory. The emergence of attractors is a form of self-organization in the mesoscopic system; stochastic attractors in biochemical systems such as gene regulation and cellular signaling are naturally inheritable via cell division. Delbrück-Gillespie's mesoscopic reaction system theory, therefore, provides a biochemical basis for spontaneous isogenetic switching and canalization.

q-bio.SC↗

arXiv2011

Circular Stochastic Fluctuations in SIS Epidemics with Heterogeneous Contacts Among Sub-populations

The conceptual difference between equilibrium and non-equilibrium steady state (NESS) is well established in physics and chemistry. This distinction, however, is not widely appreciated in dynamical descriptions of biological populations in terms of differential equations in which fixed point, steady state, and equilibrium are all synonymous. We study NESS in a stochastic SIS (susceptible-infectious-susceptible) system with heterogeneous individuals in their contact behavior represented in terms of subgroups. In the infinite population limit, the stochastic dynamics yields a system of deterministic evolution equations for population densities; and for very large but finite system a diffusion process is obtained. We report the emergence of a circular dynamics in the diffusion process, with an intrinsic frequency, near the endemic steady state. The endemic steady state is represented by a stable node in the deterministic dynamics; As a NESS phenomenon, the circular motion is caused by the intrinsic heterogeneity within the subgroups, leading to a broken symmetry and time irreversibility.

q-bio.PE↗

arXiv2011

An Information Theoretical Analysis of Kinase Activated Phosphorylation Dephosphorylation Cycle

Signal transduction, the information processing mechanism in biological cells, is carried out by a network of biochemical reactions. The dynamics of driven biochemical reactions can be studied in terms of nonequilibrium statistical physics. Such systems may also be studied in terms of Shannon's information theory. We combine these two perspectives in this study of the basic units (modules) of cellular signaling: the phosphorylation dephosphorylation cycle (PdPC) and the guanosine triphosphatase (GTPase). We show that the channel capacity is zero if and only if the free energy expenditure of biochemical system is zero. In fact, a positive correlation between the channel capacity and free energy expenditure is observed. In terms of the information theory, a linear signaling cascade consisting of multiple steps of PdPC can function as a distributed "multistage code". With increasing number of steps in the cascade, the system trades channel capacity with the code complexity. Our analysis shows that while a static code can be molecular structural based; a biochemical communication channel has to have energy expenditure.

q-bio.SC↗

arXiv2011

Hill's Small Systems Nanothermodynamics: A Simple Macromolecular Partition Problem with a Statistical Perspective

Using a simple example of biological macromolecules which are partitioned between bulk solution and membrane, we investigate T.L. Hill's phenomenological nanothermodynamics for small systems. By introducing a {\em systems size dependent} equilibrium constant for the bulk-membrane partition, we obtain Hill's results on differential and integral chemical potentials $μ$ and $\hatμ$ from computations based on standard Gibbsian equilibrium statistical mechanics. It is shown that their difference can be understood from an equilibrium re-partitioning between bulk and membrane fractions upon a change in system's size; it is closely related to the system's fluctuations and inhomogeneity. These results provide a better understanding of the nanothermodynamics and clarify its logical relation with the theory of statistical mechanics.

physics.bio-ph↗

arXiv2011

Fixation, transient landscape and diffusion's dilemma in stochastic evolutionary game dynamics

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.

q-bio.PE↗