The choice between deterministic and stochastic modeling can lead to dramatically different theoretical conclusions regarding the steady state behavior of a gene regulatory network (GRN). This is particularly interesting when low-molecular counts and slow TF-gene binding /unbinding lead to the emergence of new phenotypes in the stochastic model that are not reflected in the corresponding deterministic model. This work uncovers a mechanism that underlies this emergence of multiple modes under slow slow promoter kinetics, and studies it theoretically. Mathematical tools from singular perturbation theory are employed in order to analytically characterize stationary distributions of Chemical Master Equations for GRN's, in the limit of slow switching, as a mixture of Poisson distributions This approach, which may be interpreted as a finite-dimensional reduction of a countable Markov chain, offers a rigorous framework to explain phenomena such as non-genetic population heterogeneity and transcriptional bursting . As illustrations, the theory is used in order to tease out the role of cooperative binding in stochastic models in comparison to deterministic models, and applications are given to various model systems, including isolated or populations of toggle switches and a trans-differentiation network.
Phenotypical variability in the absence of genetic variation often reflects complex energetic landscapes associated with underlying gene regulatory networks (GRNs). In this view, different phenotypes are associated with alternative states of complex nonlinear systems: stable attractors in deterministic models or modes of stationary distributions in stochastic descriptions. We provide theoretical and practical characterizations of these landscapes, specifically focusing on stochastic slow promoter kinetics, a time scale relevant when transcription factor binding and unbinding are affected by epigenetic processes like DNA methylation and chromatin remodeling. In this case, largely unexplored except for numerical simulations, adiabatic approximations of promoter kinetics are not appropriate. In contrast to the existing literature, we provide rigorous analytic characterizations of multiple modes . A general formal approach gives insight into the influence of parameters and the prediction of how changes in GRN wiring, for example through mutations or artificial interventions, impact the possible number, location, and likelihood of alternative states. We adapt tools from the mathematical field of singular perturbation theory to represent stationary distributions of Chemical Master Equations for GRNs as mixtures of Poisson distributions and obtain explicit formulas for the locations and probabilities of metastable states as a function of the parameters describing the system . 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, a synthetic oscillator and a trans-differentiation network.
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The choice between deterministic and stochastic modeling can lead to dramatically different theoretical conclusions regarding the steady state behavior of a gene regulatory network (GRN). This is particularly interesting when low-molecular counts and slow TF-gene binding /unbinding lead to the emergence of new phenotypes in the stochastic model that are not reflected in the corresponding deterministic model. This work uncovers a mechanism that underlies this emergence
Phenotypical variability in the absence of genetic variation often reflects complex energetic landscapes associated with underlying gene regulatory networks (GRNs). In this view, different phenotypes are associated with alternative states of complex nonlinear systems: stable attractors in deterministic models or modes of stationary distributions in stochastic descriptions. We provide theoretical and practical characterizations of these landscapes, specifically focusing on stochastic slow promoter kinetics, a time scale relevant when transcription factor binding and unbinding are affected by epigenetic processes like DNA methylation and chromatin remodeling. In this case, largely unexplored except for numerical simulations, adiabatic approximations of promoter kinetics are not appropriate. In contrast to the existing literature, we provide rigorous analytic characterizations
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under slow slow promoter kinetics, and studies it theoretically. Mathematical tools from
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GRN's, in the limit of slow switching, as a mixture
GRNs as mixtures
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This approach, which may be interpreted as a finite-dimensional reduction of a countable Markov chain, offers a rigorous framework to explain phenomena such as non-genetic population heterogeneity and transcriptional bursting
and obtain explicit formulas for the locations and probabilities of metastable states as a function of the parameters describing the system
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including isolated or populations of toggle switches
such as toggle switches in isolation or in communicating populations, a synthetic oscillator