Time-dependent product-form Poisson distributions for reaction networks with higher order complexes

Anderson, David F.; Schnoerr, David; Yuan, Chaojie

doi:10.1007/s00285-020-01485-y

Cited by 12 publications

(11 citation statements)

References 29 publications

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“…., xm are all the zeros of the polynomial um(z) of degree m defined in Eq. (16). Using the approximation given in Eq.…”

Section: Approximate Eigenvalues For the Cases Of Fast And Slow Prmentioning

confidence: 99%

“…Using elementary transformations, it is easy to prove that the eigenvalues of Q 1 are given by the roots of the polynomial equation um(z) = 0, where um(z) is the polynomial defined in Eq. (16). Using the approximation given in Eq.…”

Section: Appendix C: Approximate Eigenvalues For the Case Of Fast Promentioning

confidence: 99%

“…Under certain conditions, for an initial joint distribution given by a product of Poissons, the transient joint distribution remains a product of Poissons for all times. 10,16 However, the conditions for this result to hold are very restrictive and not applicable to most systems of biological relevance. To our knowledge, the only exact time-dependent solution for an auto-regulatory feedback loop is the one derived by Ramos et al, 17 where the authors obtained a Heun function for the generating function of a stochastic model neglecting translational bursting, binding fluctuations, and cooperativity.…”

Section: Introductionmentioning

confidence: 99%

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

Chen

Grima

2020

The Journal of Chemical Physics

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While 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 the 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 behavior, 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 noiseinduced phenomenon that occurs when the protein expression is sufficiently bursty, and we use a theory to estimate the observation time window when it is manifested.

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“…., xm are all the zeros of the polynomial um(z) of degree m defined in Eq. (16). Using the approximation given in Eq.…”

Section: Approximate Eigenvalues For the Cases Of Fast And Slow Prmentioning

confidence: 99%

Section: Appendix C: Approximate Eigenvalues For the Case Of Fast Promentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Chen

Grima

2020

The Journal of Chemical Physics

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“…The time-dependent solution for stochastically modelled chemical reaction systems has received comparatively very little attention. Under certain conditions, for an initial joint distribution given by a product of Poissons, the transient joint distribution remains a product of Poissons for all times [9,14]. However, the conditions for this result to hold are very restrictive and not applicable to most systems of biological relevance.…”

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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“…In general, finding the exact values of p ν t (•) is extremely difficult. More precisely, the authors are not aware of any general class of models for which p ν t can be solved for explicitly, with the exception of linear, or first-order, models [25] or models that admit a special Poisson structure [7]. However, there are many numerical methods that give an estimate.…”

mentioning

confidence: 99%

Conditional Monte Carlo for Reaction Networks

F.¹,

Ehlert²

2019

Preprint

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Reaction networks are often used to model interacting species in fields such as biochemistry and ecology. When the counts of the species are sufficiently large, the dynamics of their concentrations are typically modeled via a system of differential equations. However, when the counts of some species are small, the dynamics of the counts are typically modeled stochastically via a discrete state, continuous time Markov chain.A key quantity of interest for such models is the probability mass function of the process at some fixed time. Since paths of such models are relatively straightforward to simulate, we can estimate the probabilities by constructing an empirical distribution. However, the support of the distribution is often diffuse across a high-dimensional state space, where the dimension is equal to the number of species. Therefore generating an accurate empirical distribution can come with a large computational cost.We present a new Monte Carlo estimator that fundamentally improves on the "classical" Monte Carlo estimator described above. It also preserves much of classical Monte Carlo's simplicity. The idea is basically one of conditional Monte Carlo. Our conditional Monte Carlo estimator has two parameters, and their choice critically affects the performance of the algorithm. Hence, a key contribution of the present work is that we demonstrate how to approximate optimal values for these parameters in an efficient manner. Moreover, we provide a central limit theorem for our estimator, which leads to approximate confidence intervals for its error.

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Time-dependent product-form Poisson distributions for reaction networks with higher order complexes

Cited by 12 publications

References 29 publications

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

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

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

Conditional Monte Carlo for Reaction Networks

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