TY - BOOK

T1 - Limits for Stochastic Reaction Networks

AU - Cappelletti, Daniele

PY - 2015

Y1 - 2015

N2 - Reaction systems have been introduced in the 70s to model biochemical systems.Nowadays their range of applications has increased and they are fruitfully usedin dierent elds. The concept is simple: some chemical species react, the set ofchemical reactions form a graph and a rate function is associated with each reaction.Such functions describe the speed of the dierent reactions, or their propensities.Two modelling regimes are then available: the evolution of the dierent speciesconcentrations can be deterministically modelled through a system of ODE, whilethe counts of the dierent species at a certain time are stochastically modelled bymeans of a continuous-time Markov chain. Our work concerns primarily stochasticreaction systems, and their asymptotic properties.In Paper I, we consider a reaction system with intermediate species, i.e. speciesthat are produced and fast degraded along a path of reactions. Let the rates ofdegradation of the intermediate species be functions of a parameter N that tendsto innity. We consider a reduced system where the intermediate species have beeneliminated, and nd conditions on the degradation rate of the intermediates suchthat the behaviour of the reduced network tends to that of the original one. In particular,we prove a uniform punctual convergence in distribution and weak convergenceof the integrals of continuous functions along the paths of the two models. Undersome extra conditions, we also prove weak convergence of the two processes. Theresult is stated in the setting of multiscale reaction systems: the amounts of all thespecies and the rates of all the reactions of the original model can scale as powers ofN. A similar result also holds for the deterministic case, as shown in Appendix IA.In Paper II, we focus on the stationary distributions of the stochastic reactionsystems. Specically, we build a theory for stochastic reaction systems that is parallelto the deciency zero theory for deterministic systems, which dates back to the 70s.A deciency theory for stochastic reaction systems was missing, and few resultsconnecting deciency and stochastic reaction systems were known. The theory webuild connects special form of product-form stationary distributions with structuralproperties of the reaction graph of the system.In Paper III, a special class of reaction systems is considered, namely systemsexhibiting absolute concentration robust species. Such species, in the deterministicmodelling regime, assume always the same value at any positive steady state. In thestochastic setting, we prove that, if the initial condition is a point in the basin ofattraction of a positive steady state of the corresponding deterministic model andtends to innity, then up to a xed time T the counts of the species exhibitingabsolute concentration robustness are, on average, near to their equilibrium value.The result is not obvious because when the counts of some species tend to innity,so do some rate functions, and the study of the system may become hard. Moreover,the result states a substantial concordance between the paths of the stochastic andthe deterministic models.

AB - Reaction systems have been introduced in the 70s to model biochemical systems.Nowadays their range of applications has increased and they are fruitfully usedin dierent elds. The concept is simple: some chemical species react, the set ofchemical reactions form a graph and a rate function is associated with each reaction.Such functions describe the speed of the dierent reactions, or their propensities.Two modelling regimes are then available: the evolution of the dierent speciesconcentrations can be deterministically modelled through a system of ODE, whilethe counts of the dierent species at a certain time are stochastically modelled bymeans of a continuous-time Markov chain. Our work concerns primarily stochasticreaction systems, and their asymptotic properties.In Paper I, we consider a reaction system with intermediate species, i.e. speciesthat are produced and fast degraded along a path of reactions. Let the rates ofdegradation of the intermediate species be functions of a parameter N that tendsto innity. We consider a reduced system where the intermediate species have beeneliminated, and nd conditions on the degradation rate of the intermediates suchthat the behaviour of the reduced network tends to that of the original one. In particular,we prove a uniform punctual convergence in distribution and weak convergenceof the integrals of continuous functions along the paths of the two models. Undersome extra conditions, we also prove weak convergence of the two processes. Theresult is stated in the setting of multiscale reaction systems: the amounts of all thespecies and the rates of all the reactions of the original model can scale as powers ofN. A similar result also holds for the deterministic case, as shown in Appendix IA.In Paper II, we focus on the stationary distributions of the stochastic reactionsystems. Specically, we build a theory for stochastic reaction systems that is parallelto the deciency zero theory for deterministic systems, which dates back to the 70s.A deciency theory for stochastic reaction systems was missing, and few resultsconnecting deciency and stochastic reaction systems were known. The theory webuild connects special form of product-form stationary distributions with structuralproperties of the reaction graph of the system.In Paper III, a special class of reaction systems is considered, namely systemsexhibiting absolute concentration robust species. Such species, in the deterministicmodelling regime, assume always the same value at any positive steady state. In thestochastic setting, we prove that, if the initial condition is a point in the basin ofattraction of a positive steady state of the corresponding deterministic model andtends to innity, then up to a xed time T the counts of the species exhibitingabsolute concentration robustness are, on average, near to their equilibrium value.The result is not obvious because when the counts of some species tend to innity,so do some rate functions, and the study of the system may become hard. Moreover,the result states a substantial concordance between the paths of the stochastic andthe deterministic models.

UR - https://soeg.kb.dk/permalink/45KBDK_KGL/fbp0ps/alma99122158762905763

M3 - Ph.D. thesis

BT - Limits for Stochastic Reaction Networks

PB - Department of Mathematical Sciences, Faculty of Science, University of Copenhagen

ER -