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abstract:
Many processes in systems biology can be modelled as reaction systems
in which $d\in\mathbb{N}$ different species interact via
$r\in\mathbb{N}$ reaction channels. In most applications the time
evolution of such a system can be accurately described in terms of
classical \emph{deterministic reaction kinetics}: The reaction system
is translated into a system of $d$ ordinary differential equations
(the reaction-rate equations), and the solution $y(t)\in\mathbb{R}^d$
indicates how the concentration or amount of each of the $d$ species
changes in time. The traditional model is simple and computationally
cheap, but fails in situations where the influence of stochastic noise
cannot be ignored, and where the amount of certain species is so small
that it must be described in terms of integer particle numbers instead
of real-valued, continuous concentrations. This is the case in gene
regulatory networks, viral kinetics with few infectious individuals,
and many other biological systems.
\emph{Stochastic reaction kinetics} provides a more accurate
description because it respects the discreteness and randomness of the
system. The time evolution is modeled by a random variable
$X(t)\in\mathbb{N}_0^d$ which evolves according to a Markov jump
process. If $X(t)=z$ for some state $z=(z_1, \ldots,
z_d)\in\mathbb{N}_0^d$ then exactly $z_i$ particles of the $i$-th
species exist at time $t$. The object of interest is the probability
$ p(t,z) = \mathbb{P}(X(t)=z) $ that at time $t$ the system is in
state $z\in\mathbb{N}^d$. The corresponding distribution $p$ is the
solution of the chemical master equation, but solving this equation is
a highly nontrivial problem because the solution has to be computed in
each state of a high-dimensional state space.
The idea of \emph{hybrid deterministic-stochastic models} is to
interpolate between the accurate but computationally costly stochastic
reaction kinetics and the simple but rather coarse deterministic
description. In hybrid models, some part of the system (e.g. some of
the species) is treated stochastically while the other part is
represented in the deterministic setting. A particularly appealing
hybrid model has recently been proposed by Andreas Hellander and Per
L\"otstedt. In this talk, we discuss the properties of the
Hellander-L\"otstedt model and sketch an extension which allows to
overcome certain limitations. |
