Block-tridiagonal state-space realization of Chemical Master Equations: A tool to compute explicit solutions

Borri, Alessandro; Carravetta, Francesco; Mavelli, Gabriella; Palumbo, Pasquale

doi:10.1016/j.cam.2015.10.008

Cited by 14 publications

(17 citation statements)

References 42 publications

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“…Following the approach illustrated e.g. by Borri et al (2016), we truncate the master equation to a finite lattice {0, 1, ..., s max } × {0, 1, ..., X max }, and calculate the (unique) normalised steady-state solution; this amounts to finding a nullvector of a sparse square matrix of large order (s max +1)(X max +1). The upper bound for the active protein is set to s max = 20, while the upper bound X max for the inactive protein is set to X max = 4 x + /ε , where x + is the uppermost steady state of the ODE (8).…”

Section: Master Equationmentioning

confidence: 99%

“…Explicit solutions to the master equation, especially at steady state, can be found for models with few components Zhou and Liu, 2015) and/or with special structural properties (Kumar et al, 2015;Anderson and Cotter, 2016). Generally, however, explicit solutions are unavailable or intractable and one resorts to stochastic simulation or seeks a numerical solution to a finite truncation of the master equation (Munsky and Khammash, 2006;Borri et al, 2016;Gupta et al, 2017). An alternative approach, which often provides useful qualitative insights into the model behaviour, is based on reduction techniques such as quasi-steadystate (Srivastava et al, 2011;Kim et al, 2014) and adiabatic reductions (Bruna et al, 2014;Popovic et al, 2016), piecewise-deterministic framework (Lin and Doering, 2016;Lin and Buchler, 2018), linear-noise approximation (Schnoerr et al, 2017;Modi et al, 2018), or moment closure (Singh and Hespanha, 2007;Andreychenko et al, 2017;Gast et al, 2019).…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Mixture distributions in a stochastic gene expression model with delayed feedback

Bokes

Borri

Palumbo

et al. 2019

Preprint

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Noise in gene expression can be substantively affected by the presence of production delay. Here we consider a mathematical model with bursty production of protein, a one-step production delay (the passage of which activates the protein), and feedback in the frequency of bursts. We specifically focus on examining the steady-state behaviour of the model in the slow-activation (i.e. large-delay) regime. Using a quasi-steady-state (QSS) approximation, we derive an autonomous ordinary differential equation for the inactive protein that applies in the slow-activation regime. If the differential equation is monostable, the steady-state distribution of the inactive (active) protein is approximated by a single Gaussian (Poisson) mode located at the globally stable steady state of the differential equation. If the differential equation is bistable (due to cooperative positive feedback), the steady-state distribution of the inactive (active) protein is approximated by a mixture of Gaussian (Poisson) modes located at the stable steady states; the weights of the modes are determined from a WKB approximation to the stationary distribution. The asymptotic results are compared to numerical solutions of the chemical master equation.

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Section: Master Equationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Mixture distributions in a stochastic gene expression model with delayed feedback

Bokes

Borri

Palumbo

et al. 2019

Preprint

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“…The drawback of such a powerful modeling tool is the so called curse of dimensionality which, in many cases, although CMEs are linear equations, prevents from explicitly computing the solutions. Therefore, even when looking for the stationary distribution, one usually sets the problem in order to implement efficient algorithms [3], or resort to approximate solutions (like e.g. the Finite State Projection algorithm [20]), or to numerical Monte Carlo approaches, like the Gillespie Stochastic Simulation Algorithm (SSA) [16].…”

Section: Subtractor Model Settingmentioning

confidence: 99%

Noise Propagation in Chemical Reaction Networks: Analysis of a Molecular Subtractor Module

Cosentino

Manes

Palombo

et al. 2019

2019 18th European Control Conference (ECC)

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The realization of embedded molecular control systems is a challenging aim in Synthetic Biology, where a major goal is to design synthetic biological circuits performing specific tasks. In this field, the novel emergent approach is to assemble the circuit in a modular fashion, possibly restraining reciprocal interactions from interconnected modules (zero-retroactivity). Within this framework, recent results have been proposed, dealing with the realization of an embedded subtractor module, with the idea of exploiting it in a more general chemical reaction network that resembles a classical control scheme. So far, this research has been carried out according to the deterministic approach. More sophisticated analysis requires the use of stochastic models, which play a paramount role in investigating noise propagation in chemical reaction networks, especially when the species copy number is low and the intrinsic stochasticity of the phenomena under investigation cannot be neglected. This note deals with a first analysis of the subtractor module, according to the stochastic approach. To this end, Chemical Master Equations are exploited to model one of the possible molecular circuits implementing the subtractor, and moment equations are written in order to evaluate how noise propagates with respect to different values of the inputs and different model parameter settings.

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“…Explicit solutions to the master equation, especially at steady state, can be found for models with few components (Bokes et al 2012;Zhou and Liu 2015) and/or with special structural properties (Kumar et al 2015;Anderson and Cotter 2016). Generally, however, explicit solutions are unavailable or intractable and one resorts to stochastic simulation or seeks a numerical solution to a finite truncation of the master equation (Munsky and Khammash 2006;Borri et al 2016;Gupta et al 2017). An alternative approach, which often provides useful qualitative insights into the model behaviour, is based on reduction techniques such as quasi-steady-state (Srivastava et al 2011;Kim et al 2014;Plesa et al 2019) and adiabatic reductions (Bruna et al 2014;Popovic et al 2016), piecewise-deterministic framework (Lin and Doering 2016;Lin and Buchler 2018), linear-noise approximation (Schnoerr et al 2017;Modi et al 2018), or moment closure (Singh and Hespanha 2007;Andreychenko et al 2017;Gast et al 2019).…”

Section: Introductionmentioning

confidence: 99%

“…Generally, however, explicit solutions are unavailable or intractable and one resorts to stochastic simulation or seeks a numerical solution to a finite truncation of the master equation (Munsky and Khammash 2006 ; Borri et al. 2016 ; Gupta et al. 2017 ).…”

Section: Introductionmentioning

confidence: 99%

Mixture distributions in a stochastic gene expression model with delayed feedback: a WKB approximation approach

et al. 2020

View full text Add to dashboard Cite

Noise in gene expression can be substantively affected by the presence of production delay. Here we consider a mathematical model with bursty production of protein, a one-step production delay (the passage of which activates the protein), and feedback in the frequency of bursts. We specifically focus on examining the steady-state behaviour of the model in the slow-activation (i.e. large-delay) regime. Using a formal asymptotic approach, we derive an autonomous ordinary differential equation for the inactive protein that applies in the slow-activation regime. If the differential equation is monostable, the steady-state distribution of the inactive (active) protein is approximated by a single Gaussian (Poisson) mode located at the globally stable fixed point of the differential equation. If the differential equation is bistable (due to cooperative positive feedback), the steady-state distribution of the inactive (active) protein is approximated by a mixture of Gaussian (Poisson) modes located at the stable fixed points; the weights of the modes are determined from a WKB approximation to the stationary distribution. The asymptotic results are compared to numerical solutions of the chemical master equation.

show abstract

Block-tridiagonal state-space realization of Chemical Master Equations: A tool to compute explicit solutions

Cited by 14 publications

References 42 publications

Mixture distributions in a stochastic gene expression model with delayed feedback

Mixture distributions in a stochastic gene expression model with delayed feedback

Noise Propagation in Chemical Reaction Networks: Analysis of a Molecular Subtractor Module

Mixture distributions in a stochastic gene expression model with delayed feedback: a WKB approximation approach

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