Single molecule dynamics and statistical fluctuations of gene regulatory networks: A repressilator

We develop a probabilistic method for analyzing global features of a cellular network under intrinsic statistical fluctuations, which is important when there are finite numbers of molecules. By making a self-consistent mean field approximation of splitting the variables in order to reduce the large number of degrees of freedom, which is reasonable for a not very strongly interacting network, we discovered that the underlying energy landscape of the mitogen-activated protein kinases (MAPKs) signal transduction network (with experimentally measured or inferred parameters such as chemical reaction rate coefficients in the network) is funneled toward a global minimum characterized by the nonequilibrium steady-state fixed point of the system at the end of the signal transduction process. For this system, we also show that the energy landscape is robust against intrinsic fluctuations and random perturbation to the inherent chemical reaction rates. The ratio of the slope versus the roughness of the energy landscape becomes a quantitative measure of robustness and stability of the network. Furthermore, we quantify the dissipation cost of this nonequilibrium system through entropy production, caused by the nonequilibrium flux in the system. We found that a lower dissipation cost corresponds to a more robust network. This least dissipation property might provide a design principle for robust and functional networks. Finally, we find the possibility of bistable and oscillatory-like solutions, which are important for cell fate decisions, upon perturbations. The method described here can be used in a variety of biological networks.funnel ͉ stability ͉ potential landscape ͉ function

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“…The potential energy function U(x) (14,(18)(19)(20)(21)(22)(23)33) can be related to steady-state probability:…”

Section: Resultsmentioning

confidence: 99%

“…From the engineering perspective, efforts have been made to understand the network from control perspectives with robust yet fragile natures (10). There have been a number of studies attempting to determine why networks are robust in their biological function among perturbations (11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24).…”

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confidence: 99%

Intrinsic noise, dissipation cost, and robustness of cellular networks: The underlying energy landscape of MAPK signal transduction

Lapidus

Han

Wang

2008

Proc. Natl. Acad. Sci. U.S.A.

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“…This leads to the corresponding projections to the protein concentration variable at ξ = − 1 and ξ = 1 giving the different results at the two different locations. Therefore, two peaks for the distribution at the protein concentration space are expected when ω is small in the nonadiabatic regime, which is the possibility ignored in the conventional view of gene switches (9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21). Fig.…”

Section: Theory For Nonequilibrium Eddy Currentmentioning

confidence: 99%

Eddy current and coupled landscapes for nonadiabatic and nonequilibrium complex system dynamics

Zhang

Sasai

Wang

2013

Proc. Natl. Acad. Sci. U.S.A.

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Physical and biological systems are often involved with coupled processes of different time scales. In the system with electronic and atomic motions, for example, the interplay between the atomic motion along the same energy landscape and the electronic hopping between different landscapes is critical: the system behavior largely depends on whether the intralandscape motion is slower (adiabatic) or faster (nonadiabatic) than the interlandscape hopping. For general nonequilibrium dynamics where Hamiltonian or energy function is unknown a priori, the challenge is how to extend the concepts of the intra-and interlandscape dynamics. In this paper we establish a theoretical framework for describing global nonequilibrium and nonadiabatic complex system dynamics by transforming the coupled landscapes into a single landscape but with additional dimensions. On this single landscape, dynamics is driven by gradient of the potential landscape, which is closely related to the steadystate probability distribution of the enlarged dimensions, and the probability flux, which has a curl nature. Through an example of a self-regulating gene circuit, we show that the curl flux has dramatic effects on gene regulatory dynamics. The curl flux and landscape framework developed here are easy to visualize and can be used to guide further investigation of physical and biological nonequilibrium systems.nonequilibrium landscape | adiabaticity | nonadiabaticity H eterogeneity among coupled processes is a hallmark of complex behaviors of physical and biological systems. A photoexcited molecule, for example, may relax into different low-energy states depending on the difference among time scales of electronic and atomic motions. Molecular motors are fueled by ATP hydrolysis and move into the different structural states, where the heterogeneous distribution of time scales of reactions and structural changes should characterize the motor performance. Dynamical complexity owing to the interplay of heterogeneous processes with multiple time scales can give rise to rich phenomena.In a system having multiple time scales, some dynamical quantities may take discrete values whereas others are continuous. An example of such heterogeneity is found in the electron-transfer reaction, in which the electronic state is discrete and atomic motions are continuous (1). Then, the system can be represented by multiple electronic energy surfaces and the change in atomic positions is motion along individual surfaces. Complex behaviors of the system are explained by the combined process of intrasurface motion along the same and intersurface hopping between different electronic energy surface(s). If the intrasurface motion is slower (faster) than the intersurface hopping, the process is called adiabatic (nonadiabatic). We here extend this notion, the significance of adiabatic and nonadiabatic effects, beyond the Hamiltonian systems to general nonequilibrium problems. Indeed, important complex systems such as reaction networks, gene switches, and molecular motors are co...

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“…The variational method has been already applied to stochastic gene regulatory networks, and it has given qualitatively good results [2,16,17]. However, the original formulation of the variational method has been based on the Poisson ansatz, in which we assume that the probability distribution of the system is described by the Poisson distribution.…”

Section: Introductionmentioning

confidence: 99%

“…However, the original formulation of the variational method has been based on the Poisson ansatz, in which we assume that the probability distribution of the system is described by the Poisson distribution. Hence, it is necessary to use a Hartree approximation, and then only restricted fluctuation effects can be included [16,17]. In order to improve the variational method, "a superposition ansatz" has been proposed in ref.…”

Section: Introductionmentioning

confidence: 99%

Approximation scheme for master equations: Variational approach to multivariate case

Ohkubo

2008

The Journal of Chemical Physics

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We study an approximation scheme based on a second quantization method for a chemical master equation. Small systems, such as cells, could not be studied by the traditional rate equation approach because fluctuation effects are very large in such small systems. Although a Fokker-Planck equation obtained by the system size expansion includes the fluctuation effects, it needs large computational costs for complicated chemical reaction systems. In addition, discrete characteristics of the original master equation are neglected in the system size expansion scheme. It has been shown that the usage of the second quantization description and a variational method achieves tremendous reduction in the dimensionality of the master equation approximately, without loss of the discrete characteristics. We here propose a new scheme for the choice of variational functions, which is applicable to multivariate cases. It is revealed that the new scheme gives better numerical results than old ones and the computational cost increases only slightly.

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Single molecule dynamics and statistical fluctuations of gene regulatory networks: A repressilator

Cited by 30 publications

References 27 publications

Intrinsic noise, dissipation cost, and robustness of cellular networks: The underlying energy landscape of MAPK signal transduction

Intrinsic noise, dissipation cost, and robustness of cellular networks: The underlying energy landscape of MAPK signal transduction

Eddy current and coupled landscapes for nonadiabatic and nonequilibrium complex system dynamics

Approximation scheme for master equations: Variational approach to multivariate case

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