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 |
|---|
| Forlag | Department of Mathematical Sciences, Faculty of Science, University of Copenhagen |
|---|---|
| Antal sider | 145 |
| Status | Udgivet - 2015 |