Cellular Biology in Terms of Stochastic Nonlinear Biochemical Dynamics: Emergent Properties, Isogenetic Variations and Chemical System Inheritability

Qian, Hong

doi:10.1007/s10955-010-0093-7

Cited by 24 publications

(25 citation statements)

References 79 publications

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“…One of the most popular tools, stochastic differential equation (SDE) has a long history of being used to model continuous processes in physics and chemistry [8,23]. Recently, there is a growing trend to apply it in biology, from complex networks [3,4,19,25] to evolution dynamics [11,29].…”

Section: Introductionmentioning

confidence: 99%

Relation of a New Interpretation of Stochastic Differential Equations to Ito Process

et al. 2012

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Stochastic differential equations (SDE) are widely used in modeling stochastic dynamics in literature. However, SDE alone is not enough to determine a unique process. A specified interpretation for stochastic integration is needed. Different interpretations specify different dynamics. Recently, a new interpretation of SDE is put forward by one of us. This interpretation has a built-in Boltzmann-Gibbs distribution and shows the existence of potential function for general processes, which reveals both local and global dynamics. Despite its powerful property, its relation with classical ones in arbitrary dimension remains obscure. In this paper, we will clarify such connection and derive the concise relation between the new interpretation and Ito process. We point out that the derived relation is experimentally testable. *

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Section: Introductionmentioning

confidence: 99%

Relation of a New Interpretation of Stochastic Differential Equations to Ito Process

et al. 2012

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“…However, for cellular biochemical processes inside individual cells, the number of molecules might not be large enough for an accurate continuous description. In this case, there is a unifying stochastic mathematical framework known as the Delbrück-Gillespie process, whose Kolmogorov forward equation (KFE) has been known as the Chemical Master Equation (CME), and whose stochastic trajectories can be computed with the Gillespie, or Doob-Bortz-Kalos-Lebowitz, algorithm [45,46]. Although spatial homogeneity is still assumed in this theory, its depth and richness lie in Kurtz's theorem, stochastic nonlinear bistability [47], and stochastic oscillations.…”

Section: Reactions As Stochastic Processes and Their Two Descriptionsmentioning

confidence: 99%

A discrete stochastic formulation for reversible bimolecular reactions via diffusion encounter

Razo

Qian

2016

Communications in Mathematical Sciences

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Abstract. 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?

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“…For a small biochemical reaction system which is rapidly stirred, the theory of Delbrück-Gillespie process [16,17], with its probability distribution given by the chemical master equation and its trajectories follow Gillespie algorithm, predicts that its stationary probability distribution, in the limit of large size, has the generic form…”

Section: Rapidly Stirred Biochemical Reaction Systemsmentioning

confidence: 99%

Hill’s small systems nanothermodynamics: a simple macromolecular partition problem with a statistical perspective

Qian¹

2012

J Biol Phys

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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 system size-dependent equilibrium constant for the bulk-membrane partition, we obtain Hill's results on differential and integral chemical potentials μ andμ 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 the system's size; it is closely related to the system's fluctuations and inhomogeneity. These results provide a better understanding of nanothermodynamics and clarify its logical relation with the theory of statistical mechanics.

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Cellular Biology in Terms of Stochastic Nonlinear Biochemical Dynamics: Emergent Properties, Isogenetic Variations and Chemical System Inheritability

Cited by 24 publications

References 79 publications

Relation of a New Interpretation of Stochastic Differential Equations to Ito Process

Relation of a New Interpretation of Stochastic Differential Equations to Ito Process

A discrete stochastic formulation for reversible bimolecular reactions via diffusion encounter

Hill’s small systems nanothermodynamics: a simple macromolecular partition problem with a statistical perspective

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