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This paper presents the results of a theoretical and numerical study on the analysis of bistable behavior of the most studied gene regulatory network, the lac operon, in terms of the model parameters. The boundedness of the state variables for the considered model are demonstrated, the parameter values providing the existence of the multiple equilibria and thus the bistable behavior are determined, and a local stability analysis of the equilibria is performed. The parameter region yielding the existence of multiple equilibria is determined in an algebraic way based on discriminants. The model given in the state equation form is defined by the ordinary differential equations with the rational right-hand sides constituted within Hill and Michaelis-Menten approaches based on enzyme kinetics. The presented method can also be used in the parametric studies of other gene regulatory and metabolic networks given by state equations with rational right hand sides.

This paper proposes aggregation-based, three-stage algorithms to overcome the numerical problems encountered in computing stationary distributions and mean first passage times for multi-modal birth-death processes of large state space sizes. The considered birth-death processes which are defined by Chemical Master Equations are used in modeling stochastic behavior of gene regulatory networks. Computing stationary probabilities for a multi-modal distribution from Chemical Master Equations is subject to have numerical problems due to the probability values running out of the representation range of the standard programming languages with the increasing size of the state space. The aggregation is shown to provide a solution to this problem by analyzing first reduced size subsystems in isolation and then considering the transitions between these subsystems. The proposed algorithms are applied to study the bimodal behavior of the lac operon of E. coli described with a one-dimensional birth-death model. Thus, the determination of the entire parameter range of bimodality for the stochastic model of lac operon is achieved.

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The diffusion of regulatory proteins within the nucleus plays a crucial role in the dynamics of transcriptional regulation. The standard model assumes a 3D plus 1D diffusion process: regulatory proteins either move freely in solution or slide on DNA. This model however does not considered the 3D structure of chromatin. Here we proposed a multi-scale stochastic model that integrates, for the first time, high-resolution information on chromatin structure as well as DNA-protein interactions. The dynamics of transcription factors was modeled as a slide plus jump diffusion process on a chromatin network based on pair-wise contact maps obtained from highresolution Hi-C experiments. Our model allowed us to uncover the effects of chromatin structure on transcription factor occupancy profiles and target search times. Finally, we showed that binding sites clustered on few topological associated domains leading to a higher local concentration of transcription factors which could reflect an optimal strategy to efficiently use limited transcriptional resources.