Seemingly stable chemical kinetics can be stable, marginally stable, or unstable

Agazzi, Andréa; Mattingly, Jonathan C.

doi:10.48550/arxiv.1810.06547

Cited by 3 publications

(4 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In general, the agreement between the stochastic and the deterministic model for large volumes only holds on a finite time horizon only. Negative examples where the two modelling paradigms differ asymptotically are discussed in [19,20]. Our result shows that for large V , under the assumptions of Theorem 1, the stochastic and the deterministic models of (2) are in agreement asymptotically.…”

Section: B Volume Scalingmentioning

confidence: 77%

“…π(a + e i |n + 1) = n + 1 n + d π(a|n) π(a − e i |n − 1) = n + d − 1 n π(a|n)Plugging the ansatz(20) and these recurrence relations into (32), we get that equation (32) holds if and only ifδn+ d i=1 κa i a (i+1) d = δ(n+d−1)+ d i=1 κ(a i −1)(a (i+1) d +1).It simplifies to 0 = δ(d − 1) − κd. Such a condition is identically satisfied under the hypothesis of the theorem which guaranteesκ = d − 1 d δ.…”

mentioning

confidence: 99%

See 1 more Smart Citation

Stationary distributions of systems with discreteness-induced transitions

Bibbona

Kim

Wiuf

2020

J. R. Soc. Interface.

View full text Add to dashboard Cite

We provide a theoretical analysis of some autocatalytic reaction networks exhibiting the phenomenon of discreteness-induced transitions. The family of networks that we address includes the celebrated Togashi and Kaneko model. We prove positive recurrence, finiteness of all moments and geometric ergodicity of the models in the family. For some parameter values, we find the analytic expression for the stationary distribution and discuss the effect of volume scaling on the stationary behaviour of the chain. We find the exact critical value of the volume for which discreteness-induced transitions disappear.

show abstract

Section: B Volume Scalingmentioning

confidence: 77%

mentioning

confidence: 99%

Stationary distributions of systems with discreteness-induced transitions

Bibbona

Kim

Wiuf

2020

J. R. Soc. Interface.

View full text Add to dashboard Cite

show abstract

“…For example, it would be interesting to study stationary distributions of autocatalytic CRNs with switching behavior [72], to identify a class of CRNs maintaining product-form Poisson distributions for all times [73] and to find when CRNs show nonexplosive behavior [74]. Another interesting direction will be to study stability of CRNs [75] and to estimate transition times between different attractors in CRNs [76].…”

Section: Discussionmentioning

confidence: 99%

Incorporating age and delay into models for biophysical systems

et al. 2020

View full text Add to dashboard Cite

In many biological systems, chemical reactions or changes in a physical state are assumed to occur instantaneously. For describing the dynamics of those systems, Markov models that require exponentially distributed inter-event times have been used widely. However, some biophysical processes such as gene transcription and translation are known to have a significant gap between the initiation and the completion of the processes, which renders the usual assumption of exponential distribution untenable. In this paper, we consider relaxing this assumption by incorporating age-dependent random time delays (distributed according to a given probability distribution) into the system dynamics. We do so by constructing a measure-valued Markov process on a more abstract state space, which allows us to keep track of the ‘ages’ of molecules participating in a chemical reaction. We study the large-volume limit of such age-structured systems. We show that, when appropriately scaled, the stochastic system can be approximated by a system of partial differential equations (PDEs) in the large-volume limit, as opposed to ordinary differential equations (ODEs) in the classical theory. We show how the limiting PDE system can be used for the purpose of further model reductions and for devising efficient simulation algorithms. In order to describe the ideas, we use a simple transcription process as a running example. We, however, note that the methods developed in this paper apply to a wide class of biophysical systems.

show abstract

“…For example, it would be interesting to study stationary distributions of autocatalytic CRNs with switching behavior [69], to identify a class of CRNs maintaining product-form Poisson distributions for all times [70], and to find when CRNs show nonexplosive behavior [71]. Another interesting direction will be to study stability of CRNs [72] and to estimate transition times between different attractors in CRNs [73].…”

Section: Discussionmentioning

confidence: 99%

Incorporating age and delay into models for biophysical systems

KhudaBukhsh¹,

Kang²,

Kenah³

et al. 2020

Preprint

View full text Add to dashboard Cite

In many biological systems, chemical reactions or changes in a physical state are assumed to occur instantaneously. For describing the dynamics of those systems, Markov models that require exponentially distributed inter-event times have been used widely. However, some biophysical processes such as gene transcription and translation are known to have a significant gap between the initiation and the completion of the processes, which renders the usual assumption of exponential distribution untenable. In this paper, we consider relaxing this assumption by incorporating age-dependent random time delays into the system dynamics. We do so by constructing a measure-valued Markov process on a more abstract state space, which allows us to keep track of the "ages" of molecules participating in a chemical reaction.We study the large-volume limit of such age-structured systems. We show that, when appropriately scaled, the stochastic system can be approximated by a system of Partial Differential Equations (PDEs) in the large-volume limit, as opposed to Ordinary Differential Equations (ODEs) in the classical theory. We show how the limiting PDE system can be used for the purpose of further model reductions and for devising efficient simulation algorithms. In order to describe the ideas, we use a simple transcription process as a running example. We, however, note that the methods developed in this paper apply to a wide class of biophysical systems.

show abstract

Seemingly stable chemical kinetics can be stable, marginally stable, or unstable

Cited by 3 publications

References 15 publications

Stationary distributions of systems with discreteness-induced transitions

Stationary distributions of systems with discreteness-induced transitions

Incorporating age and delay into models for biophysical systems

Incorporating age and delay into models for biophysical systems

Contact Info

Product

Resources

About