4,738 research outputs found
Parallel in Time Simulation of Multiscale Stochastic Chemical Kinetics
A version of the time-parallel algorithm parareal is analyzed and applied to
stochastic models in chemical kinetics. A fast predictor at the macroscopic
scale (evaluated in serial) is available in the form of the usual reaction rate
equations. A stochastic simulation algorithm is used to obtain an exact
realization of the process at the mesoscopic scale (in parallel).
The underlying stochastic description is a jump process driven by the Poisson
measure. A convergence result in this arguably difficult setting is established
suggesting that a homogenization of the solution is advantageous. We devise a
simple but highly general such technique.
Three numerical experiments on models representative to the field of
computational systems biology illustrate the method. For non-stiff problems, it
is shown that the method is able to quickly converge even when stochastic
effects are present. For stiff problems we are instead able to obtain fast
convergence to a homogenized solution.
Overall, the method builds an attractive bridge between on the one hand,
macroscopic deterministic scales and, on the other hand, mesoscopic stochastic
ones. This construction is clearly possible to apply also to stochastic models
within other fields.Comment: 30 pages, 7 figures, extended abstract to appear in proceedings of
ICNAAM 200
A Constrained Approach to Multiscale Stochastic Simulation of\ud Chemically Reacting Systems
Stochastic simulation of coupled chemical reactions is often computationally intensive, especially if a chemical system contains reactions occurring on different time scales. In this paper we introduce a multiscale methodology suitable to address this problem. It is based on the Conditional Stochastic Simulation Algorithm (CSSA) which samples from the conditional distribution of the suitably defined fast variables, given values for the slow variables. In the Constrained Multiscale Algorithm (CMA) a single realization of the CSSA is then used for each value of the slow variable to approximate the effective drift and diffusion terms, in a similar manner to the constrained mean-force computations in other applications such as molecular dynamics. We then show how using the ensuing Stochastic Differential Equation (SDE) approximation, we can in turn approximate average switching times in stochastic chemical systems
Data-driven modelling of biological multi-scale processes
Biological processes involve a variety of spatial and temporal scales. A
holistic understanding of many biological processes therefore requires
multi-scale models which capture the relevant properties on all these scales.
In this manuscript we review mathematical modelling approaches used to describe
the individual spatial scales and how they are integrated into holistic models.
We discuss the relation between spatial and temporal scales and the implication
of that on multi-scale modelling. Based upon this overview over
state-of-the-art modelling approaches, we formulate key challenges in
mathematical and computational modelling of biological multi-scale and
multi-physics processes. In particular, we considered the availability of
analysis tools for multi-scale models and model-based multi-scale data
integration. We provide a compact review of methods for model-based data
integration and model-based hypothesis testing. Furthermore, novel approaches
and recent trends are discussed, including computation time reduction using
reduced order and surrogate models, which contribute to the solution of
inference problems. We conclude the manuscript by providing a few ideas for the
development of tailored multi-scale inference methods.Comment: This manuscript will appear in the Journal of Coupled Systems and
Multiscale Dynamics (American Scientific Publishers
Reduction of dynamical biochemical reaction networks in computational biology
Biochemical networks are used in computational biology, to model the static
and dynamical details of systems involved in cell signaling, metabolism, and
regulation of gene expression. Parametric and structural uncertainty, as well
as combinatorial explosion are strong obstacles against analyzing the dynamics
of large models of this type. Multi-scaleness is another property of these
networks, that can be used to get past some of these obstacles. Networks with
many well separated time scales, can be reduced to simpler networks, in a way
that depends only on the orders of magnitude and not on the exact values of the
kinetic parameters. The main idea used for such robust simplifications of
networks is the concept of dominance among model elements, allowing
hierarchical organization of these elements according to their effects on the
network dynamics. This concept finds a natural formulation in tropical
geometry. We revisit, in the light of these new ideas, the main approaches to
model reduction of reaction networks, such as quasi-steady state and
quasi-equilibrium approximations, and provide practical recipes for model
reduction of linear and nonlinear networks. We also discuss the application of
model reduction to backward pruning machine learning techniques
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