[Paper Review] Multi-modality in gene regulatory networks with slow gene binding

M. Ali Al-Radhawi, Domitilla Del Vecchio+1 more|arXiv (Cornell University)|May 5, 2017

Gene Regulatory Network Analysis3 citations

TL;DR

This paper develops a singular perturbation framework to analytically characterize stationary distributions in gene regulatory networks with slow transcription factor binding, showing that these distributions emerge as mixtures of Poisson distributions. The method reveals how stochasticity induces multi-modality and non-genetic heterogeneity—phenomena absent in deterministic models—particularly under slow promoter kinetics, offering a rigorous explanation for transcriptional bursting and phenotypic diversity in toggle switches and trans-differentiation networks.

ABSTRACT

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.

Motivation & Objective

• To resolve the discrepancy between deterministic and stochastic models in gene regulatory networks (GRNs), particularly when low molecular counts and slow transcription factor (TF)-gene binding lead to divergent steady-state behaviors.
• To explain the emergence of multi-modal stationary distributions in stochastic GRNs that are not captured by deterministic models, especially under slow promoter kinetics.
• To develop a rigorous analytical method for characterizing stationary distributions of Chemical Master Equations (CMEs) in the limit of slow switching between promoter states.
• To clarify the role of cooperative binding in generating non-genetic population heterogeneity and transcriptional bursting in stochastic versus deterministic frameworks.
• To apply the theoretical framework to concrete biological systems, including isolated and population-level toggle switches and trans-differentiation networks.

Proposed method

• Employing singular perturbation theory to analyze the Chemical Master Equation (CME) for GRNs with slow TF-gene binding kinetics.
• Deriving a finite-dimensional reduction of the countable Markov chain underlying the CME by separating fast and slow timescales in promoter state transitions.
• Approximating the stationary distribution of the full CME as a mixture of Poisson distributions, each corresponding to a slow promoter state.
• Using this decomposition to analytically characterize the multi-modal behavior of the system in the slow switching limit.
• Applying the framework to study cooperative binding effects and to model systems such as toggle switches and trans-differentiation networks.
• Validating the analytical results through theoretical consistency checks and comparisons with known stochastic phenomena like transcriptional bursting.

Experimental results

Research questions

• RQ1 How does slow TF-gene binding kinetics lead to multi-modality in the stationary distribution of a gene regulatory network, even when deterministic models predict a single steady state?
• RQ2 What is the mathematical mechanism by which stochasticity generates non-genetic population heterogeneity in GRNs, and how does it differ from deterministic predictions?
• RQ3 In what way does cooperative binding alter the stationary distribution in stochastic GRNs compared to deterministic models?
• RQ4 How can the stationary distribution of a Chemical Master Equation with slow promoter switching be analytically approximated using a finite-dimensional reduction?
• RQ5 To what extent can this framework explain transcriptional bursting and phenotypic diversity in toggle switches and trans-differentiation networks?

Key findings

• The stationary distribution of a gene regulatory network with slow promoter kinetics is analytically shown to be a mixture of Poisson distributions, each corresponding to a distinct promoter state.
• This mixture structure explains the emergence of multiple modes in the stochastic model—phenomena not present in the corresponding deterministic model.
• The framework rigorously establishes that slow switching kinetics can lead to non-genetic population heterogeneity even in the absence of feedback or cooperative binding.
• The method successfully captures transcriptional bursting as a consequence of slow promoter transitions, providing a theoretical basis for this widely observed phenomenon.
• The theory reveals that cooperative binding enhances multi-modality in stochastic models but has a less pronounced effect in deterministic counterparts, highlighting a key difference in model predictions.
• The approach is validated on model systems such as toggle switches and trans-differentiation networks, where it accurately explains complex multi-stable and bursting behaviors.