Stochastic hybrid models of gene regulatory networks – A PDE approach

Kurasov, Pavel; Lück, Alexander; Mugnolo, Delio; Wolf, Verena

doi:10.1016/j.mbs.2018.09.009

Cited by 30 publications

(25 citation statements)

References 44 publications

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“…The LNA cannot capture either type of bimodality because it is valid for those systems whose deterministic rate equations are monostable [16], provided the average number of molecule numbers is large enough. Hybrid methods such as the cLNA and others [45,46,47] can capture bimodality of type (b) because they model gene states discretely. Methods based on PIDE can capture bimodality of type (a) but not (b) since they do not model discrete gene states explicitly [37].…”

Section: Insights From Modelsmentioning

confidence: 99%

Stochastic Modeling of Autoregulatory Genetic Feedback Loops: A Review and Comparative Study

Holehouse

Cao

Grima

2020

Biophysical Journal

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Auto-regulatory feedback loops are one of the most common network motifs. A wide variety of stochastic models have been constructed to understand how the fluctuations in protein numbers in these loops are influenced by the kinetic parameters of the main biochemical steps. These models differ according to (i) which sub-cellular processes are explicitly modelled; (ii) the modelling methodology employed (discrete, continuous or hybrid); (iii) whether they can be analytically solved for the steady-state distribution of protein numbers. We discuss the assumptions and properties of the main models in the literature, summarize our current understanding of the relationship between them and highlight some of the insights gained through modelling. Discrete Models of auto-regulationFrom a biologist perspective, a minimal model of auto-regulation should describe the main biochemical processes describing the flow of information from gene to mRNA to protein and back to the gene. Hence the model should describe transcription and translation (the two steps at the heart of the central dogma of molecular biology), mRNA and protein degradation, and interactions of proteins with genes. For simplicity we consider the case where there is a single gene copy and all processes are modelled as first-order reactions except the protein-gene interactions which naturally follow second-order kinetics.We refer to this model as the full model since it will be our ground truth, i.e. the finest scale model that we shall consider here. The reactions describing this model are GWhile this model is intuitive, it has not been studied extensively because the mathematical description of its stochastic dynamics, as provided by the chemical master equation, is not easy to solve analytically. In fact even in the absence of the feedback loop, i.e. no protein-gene interactions, its master equation has still not been solved exactly [17]. Hence historically, simplified versions of the full model have received much more attention in the literature. These are the models by Hornos et al. [18] in 2005, Grima et al. [19] in 2012 and Kumar et al. [20] in 2014. Henceforth we shall refer to these as the Hornos, Grima and Kumar models. There exist other discrete models e.g. [21] which in certain limits reduce to the aforementioned three.The three models share a few common properties: (i) They only describe protein fluctuations, i.e. there is no explicit mRNA description. (ii) The models are discrete in the sense that protein numbers change by discrete integer amounts when reactions occur. (iii) The chemical master equation for each model admits an exact solution in steady-state conditions. These exact solutions have been obtained using the method of generating functions but in other studies using similar models, the solution was obtained using the Poisson representation [10,23,24,25]. There are however important differences between the models particularly how they describe protein production and protein-gene interactions that are not often spelled out but can be disc...

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Section: Insights From Modelsmentioning

confidence: 99%

Stochastic Modeling of Autoregulatory Genetic Feedback Loops: A Review and Comparative Study

Holehouse

Cao

Grima

2020

Biophysical Journal

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“…For a recent review of these and other similar models, see [10]. For recently developed methods to approximately solve for the steady-state distribution in models of auto-regulation, see [11][12][13].…”

Section: Introductionmentioning

confidence: 99%

Dynamical phase diagram of an auto-regulating gene in fast switching conditions

Chen

Grima

2020

Preprint

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AbstractWhile the steady-state behavior of stochastic gene expression with auto-regulation has been extensively studied, its time-dependent behavior has received much less attention. Here, under the assumption of fast promoter switching, we derive and solve a reduced chemical master equation for an auto-regulatory gene circuit with translational bursting and cooperative protein-gene interactions. The analytical expression for the time-dependent probability distribution of protein numbers enables a fast exploration of large swaths of parameter space. For a unimodal initial distribution, we identify three distinct types of stochastic dynamics: (i) the protein distribution remains unimodal at all times; (ii) the protein distribution becomes bimodal at intermediate times and then reverts back to being unimodal at long times (transient bimodality) and (iii) the protein distribution switches to being bimodal at long times. For each of these, the deterministic model predicts either monostable or bistable behaviour and hence there exist six dynamical phases in total. We investigate the relationship of the six phases to the transcription rates, the protein binding and unbinding rates, the mean protein burst size, the degree of cooperativity, the relaxation time to the steady state, the protein mean and the type of feedback loop (positive or negative). We show that transient bimodality is a noise-induced phenomenon that occurs when protein expression is sufficiently bursty and we use theory to estimate the observation time window when it is manifest.

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“…The dynamics of these processes is therefore well described by stochastic Markov processes in continuous time with discrete state space [15,22,42]. While few-component or linear-kinetics systems [16] allow for exact analysis, in more complex system one often uses approximative methods [12], such as moment closure [4], linear-noise approximation [3,9], hybrid formulations [25,26,33], and multi-scale techniques [38,39].…”

Section: Introductionmentioning

confidence: 99%

Stationary distributions and metastable behaviour for self-regulating proteins with general lifetime distributions

Çelik

Bokes

Singh

2020

Preprint

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Regulatory molecules such as transcription factors are often present at relatively small copy numbers in living cells. The copy number of a particular molecule fluctuates in time due to the random occurrence of production and degradation reactions. Here we consider a stochastic model for a self-regulating transcription factor whose lifespan (or time till degradation) follows a general distribution modelled as per a multidimensional phase-type process. We show that at steady state the protein copy-number distribution is the same as in a one-dimensional model with exponentially distributed lifetimes. This invariance result holds only if molecules are produced one at a time: we provide explicit counterexamples in the bursty production regime. Additionally, we consider the case of a bistable genetic switch constituted by a positively autoregulating transcription factor. The switch alternately resides in states of up-and downregulation and generates bimodal protein distributions. In the context of our invariance result, we investigate how the choice of lifetime distribution affects the rates of metastable transitions between the two modes of the distribution. The phase-type model, being non-linear and multi-dimensional whilst possessing an explicit stationary distribution, provides a valuable test example for exploring dynamics in complex biological systems.

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Stochastic hybrid models of gene regulatory networks – A PDE approach

Cited by 30 publications

References 44 publications

Stochastic Modeling of Autoregulatory Genetic Feedback Loops: A Review and Comparative Study

Stochastic Modeling of Autoregulatory Genetic Feedback Loops: A Review and Comparative Study

Dynamical phase diagram of an auto-regulating gene in fast switching conditions

Stationary distributions and metastable behaviour for self-regulating proteins with general lifetime distributions

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