15,366 research outputs found
Hybrid approaches for multiple-species stochastic reaction-diffusion models
Reaction-diffusion models are used to describe systems in fields as diverse
as physics, chemistry, ecology and biology. The fundamental quantities in such
models are individual entities such as atoms and molecules, bacteria, cells or
animals, which move and/or react in a stochastic manner. If the number of
entities is large, accounting for each individual is inefficient, and often
partial differential equation (PDE) models are used in which the stochastic
behaviour of individuals is replaced by a description of the averaged, or mean
behaviour of the system. In some situations the number of individuals is large
in certain regions and small in others. In such cases, a stochastic model may
be inefficient in one region, and a PDE model inaccurate in another. To
overcome this problem, we develop a scheme which couples a stochastic
reaction-diffusion system in one part of the domain with its mean field
analogue, i.e. a discretised PDE model, in the other part of the domain. The
interface in between the two domains occupies exactly one lattice site and is
chosen such that the mean field description is still accurate there. This way
errors due to the flux between the domains are small. Our scheme can account
for multiple dynamic interfaces separating multiple stochastic and
deterministic domains, and the coupling between the domains conserves the total
number of particles. The method preserves stochastic features such as
extinction not observable in the mean field description, and is significantly
faster to simulate on a computer than the pure stochastic model.Comment: 38 pages, 8 figure
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
Hybrid binomial Langevin-multiple mapping conditioning modeling of a reacting mixing layer
A novel, stochastic, hybrid binomial Langevin-multiple mapping conditioning (MMC) model—that utilizes the strengths of each component—has been developed for inhomogeneous flows. The implementation has the advantage of naturally incorporating velocity-scalar interactions through the binomial Langevin model and using this joint probability density function (PDF) to define a reference variable for the MMC part of the model. The approach has the advantage that the difficulties encountered with the binomial Langevin model in modeling scalars with nonelementary bounds are removed. The formulation of the closure leads to locality in scalar space and permits the use of simple approaches (e.g., the modified Curl’s model) for transport in the reference space. The overall closure was evaluated through application to a chemically reacting mixing layer. The results show encouraging comparisons with experimental data for the first two moments of the PDF and plausible results for higher moments at a relatively modest computational cost
Multiscale stochastic reaction-diffusion modelling: application to actin dynamics in filopodia
Two multiscale (hybrid) stochastic reaction-diffusion models of actin dynamics in a filopodium are investigated. Both hybrid algorithms combine compartment-based and molecular-based stochastic reaction-diffusion models. The first hybrid model is based on the models previously\ud
developed in the literature. The second hybrid model is based on the application of recently developed two-regime method (TRM) to a fully molecular-based model which is also developed in this paper. The results of hybrid models are compared with the results of the molecular-based model. It is shown that both approaches give comparable results, although the TRM model better agrees quantitatively with the molecular-based model
The auxiliary region method: A hybrid method for coupling PDE- and Brownian-based dynamics for reaction-diffusion systems
Reaction-diffusion systems are used to represent many biological and physical
phenomena. They model the random motion of particles (diffusion) and
interactions between them (reactions). Such systems can be modelled at multiple
scales with varying degrees of accuracy and computational efficiency. When
representing genuinely multiscale phenomena, fine-scale models can be
prohibitively expensive, whereas coarser models, although cheaper, often lack
sufficient detail to accurately represent the phenomenon at hand. Spatial
hybrid methods couple two or more of these representations in order to improve
efficiency without compromising accuracy.
In this paper, we present a novel spatial hybrid method, which we call the
auxiliary region method (ARM), which couples PDE and Brownian-based
representations of reaction-diffusion systems. Numerical PDE solutions on one
side of an interface are coupled to Brownian-based dynamics on the other side
using compartment-based "auxiliary regions". We demonstrate that the hybrid
method is able to simulate reaction-diffusion dynamics for a number of
different test problems with high accuracy. Further, we undertake error
analysis on the ARM which demonstrates that it is robust to changes in the free
parameters in the model, where previous coupling algorithms are not. In
particular, we envisage that the method will be applicable for a wide range of
spatial multi-scales problems including, filopodial dynamics, intracellular
signalling, embryogenesis and travelling wave phenomena.Comment: 29 pages, 14 figures, 2 table
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
Reactions, Diffusion and Volume Exclusion in a Heterogeneous System of Interacting Particles
Complex biological and physical transport processes are often described
through systems of interacting particles. Excluded-volume effects on these
transport processes are well studied, however the interplay between volume
exclusion and reactions between heterogenous particles is less well known. In
this paper we develop a novel framework for modeling reaction-diffusion
processes which directly incorporates volume exclusion. From an off-lattice
microscopic individual based model we use the Fokker--Planck equation and the
method of matched asymptotic expansions to derive a low-dimensional macroscopic
system of nonlinear partial differential equations describing the evolution of
the particles. A biologically motivated, hybrid model of chemotaxis with volume
exclusion is explored, where reactions occur at rates dependent upon the
chemotactic environment. Further, we show that for reactions due to contact
interactions the appropriate reaction term in the macroscopic model is of lower
order in the asymptotic expansion than the nonlinear diffusion term. However,
we find that the next reaction term in the expansion is needed to ensure good
agreement with simulations of the microscopic model. Our macroscopic model
allows for more direct parameterization to experimental data than the models
available to date.Comment: 13 pages, 4 figure
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