6 research outputs found
Hybrid framework for the simulation of stochastic chemical kinetics
Stochasticity plays a fundamental role in various biochemical processes, such
as cell regulatory networks and enzyme cascades. Isothermal, well-mixed systems
can be modelled as Markov processes, typically simulated using the Gillespie
Stochastic Simulation Algorithm (SSA). While easy to implement and exact, the
computational cost of using the Gillespie SSA to simulate such systems can
become prohibitive as the frequency of reaction events increases. This has
motivated numerous coarse-grained schemes, where the "fast" reactions are
approximated either using Langevin dynamics or deterministically. While such
approaches provide a good approximation when all reactants are abundant, the
approximation breaks down when one or more species exist only in small
concentrations and the fluctuations arising from the discrete nature of the
reactions becomes significant. This is particularly problematic when using such
methods to compute statistics of extinction times for chemical species, as well
as simulating non-equilibrium systems such as cell-cycle models in which a
single species can cycle between abundance and scarcity. In this paper, a
hybrid jump-diffusion model for simulating well- mixed stochastic kinetics is
derived. It acts as a bridge between the Gillespie SSA and the chemical
Langevin equation. For low reactant reactions the underlying behaviour is
purely discrete, while purely diffusive when the concentrations of all species
is large, with the two different behaviours coexisting in the intermediate
region. A bound on the weak error in the classical large volume scaling limit
is obtained, and three different numerical discretizations of the
jump-diffusion model are described. The benefits of such a formalism are
illustrated using computational examples.Comment: 37 pages, 6 figure
An Unstructured Mesh Convergent Reaction-Diffusion Master Equation for Reversible Reactions
The convergent reaction-diffusion master equation (CRDME) was recently
developed to provide a lattice particle-based stochastic reaction-diffusion
model that is a convergent approximation in the lattice spacing to an
underlying spatially-continuous particle dynamics model. The CRDME was designed
to be identical to the popular lattice reaction-diffusion master equation
(RDME) model for systems with only linear reactions, while overcoming the
RDME's loss of bimolecular reaction effects as the lattice spacing is taken to
zero. In our original work we developed the CRDME to handle bimolecular
association reactions on Cartesian grids. In this work we develop several
extensions to the CRDME to facilitate the modeling of cellular processes within
realistic biological domains. Foremost, we extend the CRDME to handle
reversible bimolecular reactions on unstructured grids. Here we develop a
generalized CRDME through discretization of the spatially continuous volume
reactivity model, extending the CRDME to encompass a larger variety of
particle-particle interactions. Finally, we conclude by examining several
numerical examples to demonstrate the convergence and accuracy of the CRDME in
approximating the volume reactivity model.Comment: 35 pages, 9 figures. Accepted, J. Comp. Phys. (2018
Coupling sample paths to the thermodynamic limit in Monte Carlo estimators with applications to gene expression
Many biochemical systems appearing in applications have a multiscale structure so that they converge to piecewise deterministic Markov processes in a thermodynamic limit. The statistics of the piecewise deterministic process can be obtained much more efficiently than those of the exact process. We explore the possibility of coupling sample paths of the exact model to the piecewise deterministic process in order to reduce the variance of their difference. We then apply this coupling to reduce the computational complexity of a Monte Carlo estimator. Motivated by the rigorous results in [1], we show how this method can be applied to realistic biological models with nontrivial scalings
Efficient simulation techniques for biochemical reaction networks
Discrete-state, continuous-time Markov models are becoming commonplace in the
modelling of biochemical processes. The mathematical formulations that such
models lead to are opaque, and, due to their complexity, are often considered
analytically intractable. As such, a variety of Monte Carlo simulation
algorithms have been developed to explore model dynamics empirically. Whilst
well-known methods, such as the Gillespie Algorithm, can be implemented to
investigate a given model, the computational demands of traditional simulation
techniques remain a significant barrier to modern research.
In order to further develop and explore biologically relevant stochastic
models, new and efficient computational methods are required. In this thesis,
high-performance simulation algorithms are developed to estimate summary
statistics that characterise a chosen reaction network. The algorithms make use
of variance reduction techniques, which exploit statistical properties of the
model dynamics, to improve performance.
The multi-level method is an example of a variance reduction technique. The
method estimates summary statistics of well-mixed, spatially homogeneous models
by using estimates from multiple ensembles of sample paths of different
accuracies. In this thesis, the multi-level method is developed in three
directions: firstly, a nuanced implementation framework is described; secondly,
a reformulated method is applied to stiff reaction systems; and, finally,
different approaches to variance reduction are implemented and compared.
The variance reduction methods that underpin the multi-level method are then
re-purposed to understand how the dynamics of a spatially-extended Markov model
are affected by changes in its input parameters. By exploiting the inherent
dynamics of spatially-extended models, an efficient finite difference scheme is
used to estimate parametric sensitivities robustly.Comment: Doctor of Philosophy thesis submitted at the University of Oxford.
This research was supervised by Prof Ruth E. Baker and Dr Christian A. Yate
Approximation and inference methods for stochastic biochemical kinetics - a tutorial review
Stochastic fluctuations of molecule numbers are ubiquitous in biological
systems. Important examples include gene expression and enzymatic processes in
living cells. Such systems are typically modelled as chemical reaction networks
whose dynamics are governed by the Chemical Master Equation. Despite its simple
structure, no analytic solutions to the Chemical Master Equation are known for
most systems. Moreover, stochastic simulations are computationally expensive,
making systematic analysis and statistical inference a challenging task.
Consequently, significant effort has been spent in recent decades on the
development of efficient approximation and inference methods. This article
gives an introduction to basic modelling concepts as well as an overview of
state of the art methods. First, we motivate and introduce deterministic and
stochastic methods for modelling chemical networks, and give an overview of
simulation and exact solution methods. Next, we discuss several approximation
methods, including the chemical Langevin equation, the system size expansion,
moment closure approximations, time-scale separation approximations and hybrid
methods. We discuss their various properties and review recent advances and
remaining challenges for these methods. We present a comparison of several of
these methods by means of a numerical case study and highlight some of their
respective advantages and disadvantages. Finally, we discuss the problem of
inference from experimental data in the Bayesian framework and review recent
methods developed the literature. In summary, this review gives a
self-contained introduction to modelling, approximations and inference methods
for stochastic chemical kinetics.Comment: 73 pages, 12 figures in J. Phys. A: Math. Theor. (2016