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Stochastic models for regulatory networks of the genetic toggle switch.

ABSTRACT: Bistability arises within a wide range of biological systems from the lambda phage switch in bacteria to cellular signal transduction pathways in mammalian cells. Changes in regulatory mechanisms may result in genetic switching in a bistable system. Recently, more and more experimental evidence in the form of bimodal population distributions indicates that noise plays a very important role in the switching of bistable systems. Although deterministic models have been used for studying the existence of bistability properties under various system conditions, these models cannot realize cell-to-cell fluctuations in genetic switching. However, there is a lag in the development of stochastic models for studying the impact of noise in bistable systems because of the lack of detailed knowledge of biochemical reactions, kinetic rates, and molecular numbers. In this work, we develop a previously undescribed general technique for developing quantitative stochastic models for large-scale genetic regulatory networks by introducing Poisson random variables into deterministic models described by ordinary differential equations. Two stochastic models have been proposed for the genetic toggle switch interfaced with either the SOS signaling pathway or a quorum-sensing signaling pathway, and we have successfully realized experimental results showing bimodal population distributions. Because the introduced stochastic models are based on widely used ordinary differential equation models, the success of this work suggests that this approach is a very promising one for studying noise in large-scale genetic regulatory networks.

SUBMITTER: Tian T

PROVIDER: S-EPMC1482501 | biostudies-literature | 2006 May

REPOSITORIES: biostudies-literature

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Stochastic models for regulatory networks of the genetic toggle switch.

Tian Tianhai T Burrage Kevin K

Proceedings of the National Academy of Sciences of the United States of America 20060519 22

Bistability arises within a wide range of biological systems from the lambda phage switch in bacteria to cellular signal transduction pathways in mammalian cells. Changes in regulatory mechanisms may result in genetic switching in a bistable system. Recently, more and more experimental evidence in the form of bimodal population distributions indicates that noise plays a very important role in the switching of bistable systems. Although deterministic models have been used for studying the existen ...[more]

PMID: 16714385

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Project description:A toggle switch consists of two genes that mutually repress each other. This regulatory motif is active during cell differentiation and is thought to act as a memory device, being able to choose and maintain cell fate decisions. Commonly, this switch has been modeled in a deterministic framework where transcription and translation are lumped together. In this description, bistability occurs for transcription factor cooperativity, whereas autoactivation leads to a tristable system with an additional undecided state. In this contribution, we study the stability and dynamics of a two-stage gene expression switch within a probabilistic framework inspired by the properties of the Pu/Gata toggle switch in myeloid progenitor cells. We focus on low mRNA numbers, high protein abundance, and monomeric transcription-factor binding. Contrary to the expectation from a deterministic description, this switch shows complex multiattractor dynamics without autoactivation and cooperativity. Most importantly, the four attractors of the system, which only emerge in a probabilistic two-stage description, can be identified with committed and primed states in cell differentiation. To begin, we study the dynamics of the system and infer the mechanisms that move the system between attractors using both the quasipotential and the probability flux of the system. Next, we show that the residence times of the system in one of the committed attractors are geometrically distributed. We derive an analytical expression for the parameter of the geometric distribution, therefore completely describing the statistics of the switching process and elucidate the influence of the system parameters on the residence time. Moreover, we find that the mean residence time increases linearly with the mean protein level. This scaling also holds for a one-stage scenario and for autoactivation. Finally, we study the implications of this distribution for the stability of a switch and discuss the influence of the stability on a specific cell differentiation mechanism. Our model explains lineage priming and proposes the need of either high protein numbers or long-term modifications such as chromatin remodeling to achieve stable cell fate decisions. Notably, we present a system with high protein abundance that nevertheless requires a probabilistic description to exhibit multistability, complex switching dynamics, and lineage priming.

Project description:BACKGROUND: Various computational models have been of interest due to their use in the modelling of gene regulatory networks (GRNs). As a logical model, probabilistic Boolean networks (PBNs) consider molecular and genetic noise, so the study of PBNs provides significant insights into the understanding of the dynamics of GRNs. This will ultimately lead to advances in developing therapeutic methods that intervene in the process of disease development and progression. The applications of PBNs, however, are hindered by the complexities involved in the computation of the state transition matrix and the steady-state distribution of a PBN. For a PBN with n genes and N Boolean networks, the complexity to compute the state transition matrix is O(nN22n) or O(nN2n) for a sparse matrix. RESULTS: This paper presents a novel implementation of PBNs based on the notions of stochastic logic and stochastic computation. This stochastic implementation of a PBN is referred to as a stochastic Boolean network (SBN). An SBN provides an accurate and efficient simulation of a PBN without and with random gene perturbation. The state transition matrix is computed in an SBN with a complexity of O(nL2n), where L is a factor related to the stochastic sequence length. Since the minimum sequence length required for obtaining an evaluation accuracy approximately increases in a polynomial order with the number of genes, n, and the number of Boolean networks, N, usually increases exponentially with n, L is typically smaller than N, especially in a network with a large number of genes. Hence, the computational efficiency of an SBN is primarily limited by the number of genes, but not directly by the total possible number of Boolean networks. Furthermore, a time-frame expanded SBN enables an efficient analysis of the steady-state distribution of a PBN. These findings are supported by the simulation results of a simplified p53 network, several randomly generated networks and a network inferred from a T cell immune response dataset. An SBN can also implement the function of an asynchronous PBN and is potentially useful in a hybrid approach in combination with a continuous or single-molecule level stochastic model. CONCLUSIONS: Stochastic Boolean networks (SBNs) are proposed as an efficient approach to modelling gene regulatory networks (GRNs). The SBN approach is able to recover biologically-proven regulatory behaviours, such as the oscillatory dynamics of the p53-Mdm2 network and the dynamic attractors in a T cell immune response network. The proposed approach can further predict the network dynamics when the genes are under perturbation, thus providing biologically meaningful insights for a better understanding of the dynamics of GRNs. The algorithms and methods described in this paper have been implemented in Matlab packages, which are attached as Additional files.

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Stochastic models for regulatory networks of the genetic toggle switch.

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Stochastic models for regulatory networks of the genetic toggle switch.

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