arXiv subjects

Lucie Laurence

Publications and source records attributed to Lucie Laurence.

5 recordsLinked to original sources

arXiv2026

Thermodynamic Limits of Stochastic Chemical Reaction Networks with Phosphorylation

In this paper we investigate the stability properties of a fundamental mechanism of biological cells called phosphorylation. The system is a chemical reaction network (CRN) for which a chemical species, {\em the substrate}, can be sequentially transformed into two phosphorylated forms, by the activity of two types of enzymes, one type for phosphorylation, the other for dephosphorylation. We investigate a stochastic representation of this model, under the mass action kinetics. The total mass of the substrate is fixed at $N$, while the total mass of enzymes scales proportionally to $N$. The asymptotic behavior, when $N$ is large, of the concentrations of all chemical species is studied. We investigate the possible {\em stable} subsets of chemical species for the kinetics of the law of mass action. A stable subset is such that, with a convenient initial state, the number of copies of the species of this subset remains $O(1)$ on any finite time interval as $N$ gets large. The role of the twelve reaction rate constants, {\em the catalytic constants} of the CRN, is investigated from this point of view. An averaging principle of the corresponding Markov process is established for several regimes of the CRN. It is shown in particular that there exists a regime with three equilibrium points, with two of them stable. The proofs of the results rely on stochastic calculus with Poisson processes, convenient couplings of subsets of coordinates of the Markov process, technical results on $M/M/\infty$ queues, and a stability analysis of a dynamical system in $\mathbb{R}_+^4$.

math.PR↗

arXiv2025

Scaling Methods for Stochastic Chemical Reaction Networks

The asymptotic properties of some Markov processes associated to stochastic chemical reaction networks (CRNs) driven by the kinetics of the law of mass action are analyzed. The scaling regime introduced in the paper assumes that the norm of the initial state is converging to infinity. The reaction rate constants are kept fixed. The purpose of the paper is of showing, with simple examples, a scaling analysis in this context. The main difference with the scalings of the literature is that it does not change the graph structure of the CRN or its reaction rates. Several CRNs are investigated to illustrate the insight that can be gained on the qualitative properties of these networks. A detailed scaling analysis of a CRN with several interesting asymptotic properties, with a bi-modal behavior in particular, is worked out in the last section. Additionally, with several examples, we also show that a stability criterion due to Filonov for positive recurrence of Markov processes may simplify significantly the stability analysis of these networks.

math.PR↗

arXiv2025

Noise-induced stabilization in a chemical reaction network without boundary effects

We present a chemical reaction network that is unstable under deterministic mass action kinetics, exhibiting finite-time blow-up of trajectories in the interior of the state space, but whose stochastic counterpart is positive recurrent. This provides an example of noise-induced stabilization of the model's dynamics arising due to noise perturbing transversally the divergent trajectories of the system that is independently of boundary effects. The proof is based on a careful decomposition of the state space and the construction of suitable Lyapunov functions in each region.

math.PR↗

arXiv2024

Analysis of Stochastic Chemical Reaction Networks with a Hierarchy of Timescales

We investigate a class of stochastic chemical reaction networks with $n{\ge}1$ chemical species $S_1$, \ldots, $S_n$, and whose complexes are only of the form $k_iS_i$, $i{=}1$,\ldots, $n$, where $(k_i)$ are integers. The time evolution of these CRNs is driven by the kinetics of the law of mass action. A scaling analysis is done when the rates of external arrivals of chemical species are proportional to a large scaling parameter $N$. A natural hierarchy of fast processes, a subset of the coordinates of $(X_i(t))$, is determined by the values of the mapping $i{\mapsto}k_i$. We show that the scaled vector of coordinates $i$ such that $k_i{=}1$ and the scaled occupation measure of the other coordinates are converging in distribution to a deterministic limit as $N$ gets large. The proof of this result is obtained by establishing a functional equation for the limiting points of the occupation measure, by an induction on the hierarchy of timescales and with relative entropy functions.

math.PR↗

arXiv2024

Stochastic Chemical Reaction Networks with Discontinuous Limits and AIMD processes

In this paper we study a class of stochastic chemical reaction networks (CRNs) for which chemical species are created by a sequence of chain reactions. We prove that under some convenient conditions on the initial state, some of these networks exhibit a discrete-induced transitions (DIT) property: isolated, random, events have a direct impact on the macroscopic state of the process. If this phenomenon has already been noticed in several CRNs, in auto-catalytic networks in the literature of physics in particular, there are up to now few rigorous studies in this domain. A scaling analysis of several cases of such CRNs with several classes of initial states is achieved. The DIT property is investigated for the case of a CRN with four nodes. We show that on the normal timescale and for a subset of (large) initial states and for convenient Skorohod topologies, the scaled process converges in distribution to a Markov process with jumps, an Additive Increase/Multiplicative Decrease (AIMD) process. This asymptotically discontinuous limiting behavior is a consequence of a DIT property due to random, local, blowups of jumps occurring during small time intervals. With an explicit representation of invariant measures of AIMD processes and time-change arguments, we show that, with a speed-up of the timescale, the scaled process is converging in distribution to a continuous deterministic function. The DIT analyzed in this paper is connected to a simple chain reaction between three chemical species and is therefore likely to be a quite generic phenomenon for a large class of CRNs.

math.PR↗