5,448 research outputs found
Error Analysis of Diffusion Approximation Methods for Multiscale Systems in Reaction Kinetics
Several different methods exist for efficient approximation of paths in
multiscale stochastic chemical systems. Another approach is to use bursts of
stochastic simulation to estimate the parameters of a stochastic differential
equation approximation of the paths. In this paper, multiscale methods for
approximating paths are used to formulate different strategies for estimating
the dynamics by diffusion processes. We then analyse how efficient and accurate
these methods are in a range of different scenarios, and compare their
respective advantages and disadvantages to other methods proposed to analyse
multiscale chemical networks.Comment: 17 pages, 8 figure
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
Jump-Diffusion Approximation of Stochastic Reaction Dynamics: Error bounds and Algorithms
Biochemical reactions can happen on different time scales and also the
abundance of species in these reactions can be very different from each other.
Classical approaches, such as deterministic or stochastic approach, fail to
account for or to exploit this multi-scale nature, respectively. In this paper,
we propose a jump-diffusion approximation for multi-scale Markov jump processes
that couples the two modeling approaches. An error bound of the proposed
approximation is derived and used to partition the reactions into fast and slow
sets, where the fast set is simulated by a stochastic differential equation and
the slow set is modeled by a discrete chain. The error bound leads to a very
efficient dynamic partitioning algorithm which has been implemented for several
multi-scale reaction systems. The gain in computational efficiency is
illustrated by a realistically sized model of a signal transduction cascade
coupled to a gene expression dynamics.Comment: 32 pages, 7 figure
Constrained Approximation of Effective Generators for Multiscale Stochastic Reaction Networks and Application to Conditioned Path Sampling
Efficient analysis and simulation of multiscale stochastic systems of
chemical kinetics is an ongoing area for research, and is the source of many
theoretical and computational challenges. In this paper, we present a
significant improvement to the constrained approach, which is a method for
computing effective dynamics of slowly changing quantities in these systems,
but which does not rely on the quasi-steady-state assumption (QSSA). The QSSA
can cause errors in the estimation of effective dynamics for systems where the
difference in timescales between the "fast" and "slow" variables is not so
pronounced.
This new application of the constrained approach allows us to compute the
effective generator of the slow variables, without the need for expensive
stochastic simulations. This is achieved by finding the null space of the
generator of the constrained system. For complex systems where this is not
possible, or where the constrained subsystem is itself multiscale, the
constrained approach can then be applied iteratively. This results in breaking
the problem down into finding the solutions to many small eigenvalue problems,
which can be efficiently solved using standard methods.
Since this methodology does not rely on the quasi steady-state assumption,
the effective dynamics that are approximated are highly accurate, and in the
case of systems with only monomolecular reactions, are exact. We will
demonstrate this with some numerics, and also use the effective generators to
sample paths of the slow variables which are conditioned on their endpoints, a
task which would be computationally intractable for the generator of the full
system.Comment: 31 pages, 7 figure
Transition manifolds of complex metastable systems: Theory and data-driven computation of effective dynamics
We consider complex dynamical systems showing metastable behavior but no
local separation of fast and slow time scales. The article raises the question
of whether such systems exhibit a low-dimensional manifold supporting its
effective dynamics. For answering this question, we aim at finding nonlinear
coordinates, called reaction coordinates, such that the projection of the
dynamics onto these coordinates preserves the dominant time scales of the
dynamics. We show that, based on a specific reducibility property, the
existence of good low-dimensional reaction coordinates preserving the dominant
time scales is guaranteed. Based on this theoretical framework, we develop and
test a novel numerical approach for computing good reaction coordinates. The
proposed algorithmic approach is fully local and thus not prone to the curse of
dimension with respect to the state space of the dynamics. Hence, it is a
promising method for data-based model reduction of complex dynamical systems
such as molecular dynamics
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