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Publications about 'stationary distribution'
Articles in journal or book chapters
and J. K. Kim.
Derivation of stationary distributions of biochemical reaction networks via structure transformation.
Note: Submitted.Keyword(s): stationary distribution,
chemical reaction networks,
biochemical reaction networks,
chemical master equation,
The long-term behaviors of biochemical reaction networks (BRNs) are described by steady states in deterministic models, and stationary distributions in stochastic models. Unlike deterministic steady states, the stationary distributions capturing inherent fluctuations of reactions are extremely difficult to derive analytically. Here, we develop a method for deriving stationary distributions from deterministic steady states by translating given BRNs to have a special network structure. Specifically, we merge and shift nodes and edges so as to make the deterministic steady states complex balanced, i.e., in- and out-flows of each node are equal. Then, using the complex balanced steady states, the stationary distributions of various autophosphorylation and toggle switch systems are derived. This greatly extends a class of BRNs for which stationary distributions can be analytically derived.
M. A. Al-Radhawi,
D. Del Vecchio,
and E. D. Sontag.
Multi-modality in gene regulatory networks with slow gene binding.
PLoS Computational Biology,
chemical master equations.
In biological processes such as embryonic development, hematopoietic cell differentiation, and the arising of tumor heterogeneity and consequent resistance to therapy, mechanisms of gene activation and deactivation may play a role in the emergence of phenotypically heterogeneous yet genetically identical (clonal) cellular populations. Mathematically, the variability in phenotypes in the absence of genetic variation can be modeled through the existence of multiple metastable attractors in nonlinear systems subject with stochastic switching, each one of them associated to an alternative epigenetic state. An important theoretical and practical question is that of estimating the number and location of these states, as well as their relative probabilities of occurrence. This paper focuses on a rigorous analytic characterization of multiple modes under slow promoter kinetics, which is a feature of epigenetic regulation. It characterizes the stationary distributions of Chemical Master Equations for gene regulatory networks as a mixture of Poisson distributions. As illustrations, the theory is used to tease out the role of cooperative binding in stochastic models in comparison to deterministic models, and applications are given to various model systems, such as toggle switches in isolation or in communicating populations and a trans-differentiation network.
and D. Del Vecchio.
Stochastic multistationarity in a model of the hematopoietic stem cell differentiation network.
In Proc. 2018 IEEE Conf. Decision and Control,
chemical master equations.
In the mathematical modeling of cell differentiation, it is common to think of internal states of cells (quanfitied by activation levels of certain genes) as determining different cell types. We study here the "PU.1/GATA-1 circuit" that controls the development of mature blood cells from hematopoietic stem cells (HSCs). We introduce a rigorous chemical reaction network model of the PU.1/GATA-1 circuit, which incorporates current biological knowledge and find that the resulting ODE model of these biomolecular reactions is incapable of exhibiting multistability, contradicting the fact that differentiation networks have, by definition, alternative stable steady states. When considering instead the stochastic version of this chemical network, we analytically construct the stationary distribution, and are able to show that this distribution is indeed capable of admitting a multiplicity of modes. Finally, we study how a judicious choice of system parameters serves to bias the probabilities towards different stationary states. We remark that certain changes in system parameters can be physically implemented by a biological feedback mechanism; tuning this feedback gives extra degrees of freedom that allow one to assign higher likelihood to some cell types over others.
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