64,901 research outputs found
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
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 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
The pseudo-compartment method for coupling PDE and compartment-based models of diffusion
Spatial reaction-diffusion models have been employed to describe many
emergent phenomena in biological systems. The modelling technique most commonly
adopted in the literature implements systems of partial differential equations
(PDEs), which assumes there are sufficient densities of particles that a
continuum approximation is valid. However, due to recent advances in
computational power, the simulation, and therefore postulation, of
computationally intensive individual-based models has become a popular way to
investigate the effects of noise in reaction-diffusion systems in which regions
of low copy numbers exist.
The stochastic models with which we shall be concerned in this manuscript are
referred to as `compartment-based'. These models are characterised by a
discretisation of the computational domain into a grid/lattice of
`compartments'. Within each compartment particles are assumed to be well-mixed
and are permitted to react with other particles within their compartment or to
transfer between neighbouring compartments.
We develop two hybrid algorithms in which a PDE is coupled to a
compartment-based model. Rather than attempting to balance average fluxes, our
algorithms answer a more fundamental question: `how are individual particles
transported between the vastly different model descriptions?' First, we present
an algorithm derived by carefully re-defining the continuous PDE concentration
as a probability distribution. Whilst this first algorithm shows strong
convergence to analytic solutions of test problems, it can be cumbersome to
simulate. Our second algorithm is a simplified and more efficient
implementation of the first, it is derived in the continuum limit over the PDE
region alone. We test our hybrid methods for functionality and accuracy in a
variety of different scenarios by comparing the averaged simulations to
analytic solutions of PDEs for mean concentrations.Comment: MAIN - 24 pages, 10 figures, 1 supplementary file - 3 pages, 2
figure
Multiscale reaction-diffusion algorithms: PDE-assisted Brownian dynamics
Two algorithms that combine Brownian dynamics (BD) simulations with
mean-field partial differential equations (PDEs) are presented. This
PDE-assisted Brownian dynamics (PBD) methodology provides exact particle
tracking data in parts of the domain, whilst making use of a mean-field
reaction-diffusion PDE description elsewhere. The first PBD algorithm couples
BD simulations with PDEs by randomly creating new particles close to the
interface which partitions the domain and by reincorporating particles into the
continuum PDE-description when they cross the interface. The second PBD
algorithm introduces an overlap region, where both descriptions exist in
parallel. It is shown that to accurately compute variances using the PBD
simulation requires the overlap region. Advantages of both PBD approaches are
discussed and illustrative numerical examples are presented.Comment: submitted to SIAM Journal on Applied Mathematic
Algorithm for Mesoscopic Advection-Diffusion
In this paper, an algorithm is presented to calculate the transition rates
between adjacent mesoscopic subvolumes in the presence of flow and diffusion.
These rates can be integrated in stochastic simulations of reaction-diffusion
systems that follow a mesoscopic approach, i.e., that partition the environment
into homogeneous subvolumes and apply the spatial stochastic simulation
algorithm (spatial SSA). The rates are derived by integrating Fick's second law
over a single subvolume in one dimension (1D), and are also shown to apply in
three dimensions (3D). The proposed algorithm corrects the derived rates to
ensure that they are physically meaningful and it is implemented in the AcCoRD
simulator (Actor-based Communication via Reaction-Diffusion). Simulations using
the proposed method are compared with a naive mesoscopic approach, microscopic
simulations that track every molecule, and analytical results that are exact in
1D and an approximation in 3D. By choosing subvolumes that are sufficiently
small, such that the Peclet number associated with a subvolume is sufficiently
less than 2, the accuracy of the proposed method is comparable with the
microscopic method, thus enabling the simulation of
advection-reaction-diffusion systems with the spatial SSA.Comment: 12 pages, 9 figures. Submitted to IEEE Transactions on NanoBioscienc
Switching and diffusion models for gene regulation networks
We analyze a hierarchy of three regimes for modeling gene regulation. The most complete model is a continuous time, discrete state space, Markov jump process. An intermediate 'switch plus diffusion' model takes the form of a stochastic differential equation driven by an independent continuous time Markov switch. In the third 'switch plus ODE' model the switch remains but the diffusion is removed. The latter two models allow for multi-scale simulation where, for the sake of computational efficiency, system components are treated differently according to their abundance. The 'switch plus ODE' regime was proposed by Paszek (Modeling stochasticity in gene regulation: characterization in the terms of the underlying distribution function, Bulletin of Mathematical Biology, 2007), who analyzed the steady state behavior, showing that the mean was preserved but the variance only approximated that of the full model. Here, we show that the tools of stochastic calculus can be used to analyze first and second moments for all time. A technical issue to be addressed is that the state space for the discrete-valued switch is infinite. We show that the new 'switch plus diffusion' regime preserves the biologically relevant measures of mean and variance, whereas the 'switch plus ODE' model uniformly underestimates the variance in the protein level. We also show that, for biologically relevant parameters, the transient behaviour can differ significantly from the steady state, justifying our time-dependent analysis. Extra computational results are also given for a protein dimerization model that is beyond the scope of the current analysis
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
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