Ping Ao

Potential in stochastic differential equations: novel construction

There is a whole range of emergent phenomena in non-equilibrium behaviors can be well described by a set of stochastic differential equations. Inspired by an insight gained during our study of robustness and stability in phage lambda genetic switch in modern biology, we found that there exists a classification of generic nonequilibrium processes: In the continuous description in terms of stochastic differential equations, there exists four dynamical elements: the potential function

Calculating biological behaviors of epigenetic states in the phage λ life cycle

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Laws in Darwinian evolutionary theory

, the friction matrix

Emerging of Stochastic Dynamical Equalities and Steady State Thermodynamics from Darwinian Dynamics

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Structure of stochastic dynamics near fixed points

, the anti-symmetric matrix

Global view of bionetwork dynamics: adaptive landscape.

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On the existence of potential landscape in the evolution of complex systems

, and the noise. The generic feature of absence of detailed balance is then precisely represented by

Free Energy, Value, and Attractors

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Transverse force on a quantized vortex in a superfluid.

. For dynamical near a fixed point, whether or not it is stable or not, the stochastic dynamics is linear. A rather complete analysis has been carried out (Kwon, Ao, Thouless, cond-mat/0506280; PNAS, {\bf 102} (2005) 13029), referred to as SDS I. One important and persistent question is the existence of a potential function with nonlinear force and with multiplicative noise, with both nice local dynamical and global steady state properties. Here we demonstrate that a dynamical structure built into stochastic differential equation allows us to construct such a global optimization potential function. First, we provide the construction. One of most important ingredient is the generalized Einstein relation. We then present an approximation scheme: The gradient expansion which turns every order into linear matrix equations. The consistent of such methodology with other known stochastic treatments will be discussed in next paper, SDS III; and the explicitly connection to statistical mechanics and thermodynamics will be discussed in a forthcoming paper, SDS IV.

Existence and construction of dynamical potential in nonequilibrium processes without detailed balance

The biology and behavior of bacteriophage λ regulation have been the focus of classical investigations of molecular control of gene expression. Both qualitative and quantitative aspects of this behavior have been systematically characterized experimentally. Complete understanding of the robustness and stability of the genetic circuitry for the lysis-lysogeny switch remains an unsolved puzzle. It is an excellent test case for our understanding of biological behavior of an integrated network based on its physical, chemical, DNA, protein, and functional properties. We have used a new approach to non-linear dynamics to formulate a new mathematical model, performed a theoretical study on the phage λ life cycle, and solved the crucial part of this puzzle. We find a good quantitative agreement between the theoretical calculation and published experimental observations in the protein number levels, the lysis frequency in the lysogen culture, and the lysogenization frequency for mutants of OR. We also predict the desired robustness for the λ genetic switch. We believe that this is the first successful example in the quantitative calculation of robustness and stability of the phage λ regulatory network, one of the simplest and most well-studied regulatory systems.