A linearization method for probability moment equations

Vlysidis, Michail; Kaznessis, Yiannis N.

doi:10.1016/j.compchemeng.2018.01.015

Cited by 4 publications

(6 citation statements)

References 29 publications

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“…Oscillatory behaviors are particularly challenging to capture with stochastic models [ 46 ]. We have reported earlier how methods that can capture multistable behaviors fail to appropriately capture oscillatory ones [ 39 ].…”

Section: Resultsmentioning

confidence: 99%

“…All the Lagrange multipliers can be calculated through Equation ( 15 ) except the zeroth-order one

. As discussed earlier,

depends on the rest of the Lagrange multipliers through the relation [ 39 ]:

…”

Section: Materials and Methods/theorymentioning

confidence: 99%

“…When closure order is specified, the system has the same number of equations and unknowns (the Lagrange multipliers) and a root-finding method such as Newton-Rapshon can be used. The residual of the method is

and the Jacobian matrix is

[ 39 ]. With this approach all the Lagrange multipliers can be calculated other than the zeroth one.…”

Section: Materials and Methods/theorymentioning

confidence: 99%

“…The LMEs are calculated with up to sixth-order moments (

). The closure order is chosen based on the findings of [ 39 , 45 ]. For SSA, 500,000 trajectories were used.…”

Section: Figurementioning

confidence: 99%

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Solving Stochastic Reaction Networks with Maximum Entropy Lagrange Multipliers

Vlysidis

Kaznessis

2018

Entropy

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The time evolution of stochastic reaction networks can be modeled with the chemical master equation of the probability distribution. Alternatively, the numerical problem can be reformulated in terms of probability moment equations. Herein we present a new alternative method for numerically solving the time evolution of stochastic reaction networks. Based on the assumption that the entropy of the reaction network is maximum, Lagrange multipliers are introduced. The proposed method derives equations that model the time derivatives of these Lagrange multipliers. We present detailed steps to transform moment equations to Lagrange multiplier equations. In order to demonstrate the method, we present examples of non-linear stochastic reaction networks of varying degrees of complexity, including multistable and oscillatory systems. We find that the new approach is as accurate and significantly more efficient than Gillespie’s original exact algorithm for systems with small number of interacting species. This work is a step towards solving stochastic reaction networks accurately and efficiently.

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Section: Resultsmentioning

confidence: 99%

“…All the Lagrange multipliers can be calculated through Equation ( 15 ) except the zeroth-order one

. As discussed earlier,

depends on the rest of the Lagrange multipliers through the relation [ 39 ]:

…”

Section: Materials and Methods/theorymentioning

confidence: 99%

and the Jacobian matrix is

[ 39 ]. With this approach all the Lagrange multipliers can be calculated other than the zeroth one.…”

Section: Materials and Methods/theorymentioning

confidence: 99%

“…The LMEs are calculated with up to sixth-order moments (

). The closure order is chosen based on the findings of [ 39 , 45 ]. For SSA, 500,000 trajectories were used.…”

Section: Figurementioning

confidence: 99%

See 2 more Smart Citations

Solving Stochastic Reaction Networks with Maximum Entropy Lagrange Multipliers

Vlysidis

Kaznessis

2018

Entropy

Self Cite

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“…Matrices A, A and the vector of constants µ c of the moment equations are generated as described in [6]. The Jacobian matrix is given by: J i,j = ∂µ i ∂λ i = −µ i,j + µ i • µ j , where [8]. The algorithm of the application can be found in [9].…”

Section: Approachmentioning

confidence: 99%

ZICS: an application for calculating the stationary probability distribution of stochastic reaction networks

Vlysidis¹,

C.²,

Kaznessis³

2018

Preprint

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Stochastic formalisms are necessary to describe the behavior of many biological systems. However, there remains a lack of numerical methods available to calculate the stationary probability distributions of stochastic reaction networks. We have previously development a numerical approach to calculate stationary probability distributions of stochastic networks, named ZI closure scheme. In this work, we present ZICS (Zero-Information Closure Scheme), a free applications based on ZI-closure scheme.

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Noise in Biomolecular Systems: Modeling, Analysis, and Control Implications

Briat

Khammash

2023

Annu. Rev. Control Robot. Auton. Syst.

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While noise is generally associated with uncertainties and often has a negative connotation in engineering, living organisms have evolved to adapt to (and even exploit) such uncertainty to ensure the survival of a species or implement certain functions that would have been difficult or even impossible otherwise. In this article, we review the role and impact of noise in systems and synthetic biology, with a particular emphasis on its role in the genetic control of biological systems, an area we refer to as cybergenetics. The main modeling paradigm is that of stochastic reaction networks, whose applicability goes beyond biology, as these networks can represent any population dynamics system, including ecological, epidemiological, and opinion dynamics networks. We review different ways to mathematically represent these systems, and we notably argue that the concept of ergodicity presents a particularly suitable way to characterize their stability. We then discuss noise-induced properties and show that noise can be both an asset and a nuisance in this setting. Finally, we discuss recent results on (stochastic) cybergenetics and explore their relationships to noise. Along the way, we detail the different technical and biological constraints that need to be respected when designing synthetic biological circuits. Finally, we discuss the concepts, problems, and solutions exposed in the article; raise criticisms and concerns about current ideas and approaches; suggest current (open) problems with potential solutions; and provide some ideas for future research directions. Expected final online publication date for the Annual Review of Control, Robotics, and Autonomous Systems, Volume 14 is May 2023. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.

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A linearization method for probability moment equations

Cited by 4 publications

References 29 publications

Solving Stochastic Reaction Networks with Maximum Entropy Lagrange Multipliers

Solving Stochastic Reaction Networks with Maximum Entropy Lagrange Multipliers

ZICS: an application for calculating the stationary probability distribution of stochastic reaction networks

Noise in Biomolecular Systems: Modeling, Analysis, and Control Implications

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