## Abstract

Reaction systems have been introduced in the 70s to model biochemical systems.

Nowadays their range of applications has increased and they are fruitfully used

in dierent elds. The concept is simple: some chemical species react, the set of

chemical 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 species

concentrations can be deterministically modelled through a system of ODE, while

the counts of the dierent species at a certain time are stochastically modelled by

means of a continuous-time Markov chain. Our work concerns primarily stochastic

reaction systems, and their asymptotic properties.

In Paper I, we consider a reaction system with intermediate species, i.e. species

that are produced and fast degraded along a path of reactions. Let the rates of

degradation of the intermediate species be functions of a parameter N that tends

to innity. We consider a reduced system where the intermediate species have been

eliminated, and nd conditions on the degradation rate of the intermediates such

that 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 convergence

of the integrals of continuous functions along the paths of the two models. Under

some extra conditions, we also prove weak convergence of the two processes. The

result is stated in the setting of multiscale reaction systems: the amounts of all the

species and the rates of all the reactions of the original model can scale as powers of

N. 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 reaction

systems. Specically, we build a theory for stochastic reaction systems that is parallel

to the deciency zero theory for deterministic systems, which dates back to the 70s.

A deciency theory for stochastic reaction systems was missing, and few results

connecting deciency and stochastic reaction systems were known. The theory we

build connects special form of product-form stationary distributions with structural

properties of the reaction graph of the system.

In Paper III, a special class of reaction systems is considered, namely systems

exhibiting absolute concentration robust species. Such species, in the deterministic

modelling regime, assume always the same value at any positive steady state. In the

stochastic setting, we prove that, if the initial condition is a point in the basin of

attraction of a positive steady state of the corresponding deterministic model and

tends to innity, then up to a xed time T the counts of the species exhibiting

absolute 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 and

the deterministic models.

Nowadays their range of applications has increased and they are fruitfully used

in dierent elds. The concept is simple: some chemical species react, the set of

chemical 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 species

concentrations can be deterministically modelled through a system of ODE, while

the counts of the dierent species at a certain time are stochastically modelled by

means of a continuous-time Markov chain. Our work concerns primarily stochastic

reaction systems, and their asymptotic properties.

In Paper I, we consider a reaction system with intermediate species, i.e. species

that are produced and fast degraded along a path of reactions. Let the rates of

degradation of the intermediate species be functions of a parameter N that tends

to innity. We consider a reduced system where the intermediate species have been

eliminated, and nd conditions on the degradation rate of the intermediates such

that 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 convergence

of the integrals of continuous functions along the paths of the two models. Under

some extra conditions, we also prove weak convergence of the two processes. The

result is stated in the setting of multiscale reaction systems: the amounts of all the

species and the rates of all the reactions of the original model can scale as powers of

N. 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 reaction

systems. Specically, we build a theory for stochastic reaction systems that is parallel

to the deciency zero theory for deterministic systems, which dates back to the 70s.

A deciency theory for stochastic reaction systems was missing, and few results

connecting deciency and stochastic reaction systems were known. The theory we

build connects special form of product-form stationary distributions with structural

properties of the reaction graph of the system.

In Paper III, a special class of reaction systems is considered, namely systems

exhibiting absolute concentration robust species. Such species, in the deterministic

modelling regime, assume always the same value at any positive steady state. In the

stochastic setting, we prove that, if the initial condition is a point in the basin of

attraction of a positive steady state of the corresponding deterministic model and

tends to innity, then up to a xed time T the counts of the species exhibiting

absolute 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 and

the deterministic models.

Originalsprog | Engelsk |
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Forlag | Department of Mathematical Sciences, Faculty of Science, University of Copenhagen |
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Antal sider | 145 |

Status | Udgivet - 2015 |