259 research outputs found
Reaction rates for a generalized reaction-diffusion master equation
It has been established that there is an inherent limit to the accuracy of
the reaction-diffusion master equation. Specifically, there exists a
fundamental lower bound on the mesh size, below which the accuracy deteriorates
as the mesh is refined further. In this paper we extend the standard
reaction-diffusion master equation to allow molecules occupying neighboring
voxels to react, in contrast to the traditional approach in which molecules
react only when occupying the same voxel. We derive reaction rates, in two
dimensions as well as three dimensions, to obtain an optimal match to the more
fine-grained Smoluchowski model, and show in two numerical examples that the
extended algorithm is accurate for a wide range of mesh sizes, allowing us to
simulate systems intractable with the standard reaction-diffusion master
equation. In addition, we show that for mesh sizes above the fundamental lower
limit of the standard algorithm, the generalized algorithm reduces to the
standard algorithm. We derive a lower limit for the generalized algorithm,
which, in both two dimensions and three dimensions, is on the order of the
reaction radius of a reacting pair of molecules
Reaction rates for mesoscopic reaction-diffusion kinetics
The mesoscopic reaction-diffusion master equation (RDME) is a popular
modeling framework, frequently applied to stochastic reaction-diffusion
kinetics in systems biology. The RDME is derived from assumptions about the
underlying physical properties of the system, and it may produce unphysical
results for models where those assumptions fail. In that case, other more
comprehensive models are better suited, such as hard-sphere Brownian dynamics
(BD). Although the RDME is a model in its own right, and not inferred from any
specific microscale model, it proves useful to attempt to approximate a
microscale model by a specific choice of mesoscopic reaction rates. In this
paper we derive mesoscopic reaction rates by matching certain statistics of the
RDME solution to statistics of the solution of a widely used microscopic BD
model: the Smoluchowski model with a mixed boundary condition at the reaction
radius of two molecules. We also establish fundamental limits for the range of
mesh resolutions for which this approach yields accurate results, and show both
theoretically and in numerical examples that as we approach the lower
fundamental limit, the mesoscopic dynamics approach the microscopic dynamics
Simulation of stochastic reaction-diffusion processes on unstructured meshes
Stochastic chemical systems with diffusion are modeled with a
reaction-diffusion master equation. On a macroscopic level, the governing
equation is a reaction-diffusion equation for the averages of the chemical
species. On a mesoscopic level, the master equation for a well stirred chemical
system is combined with Brownian motion in space to obtain the
reaction-diffusion master equation. The space is covered by an unstructured
mesh and the diffusion coefficients on the mesoscale are obtained from a finite
element discretization of the Laplace operator on the macroscale. The resulting
method is a flexible hybrid algorithm in that the diffusion can be handled
either on the meso- or on the macroscale level. The accuracy and the efficiency
of the method are illustrated in three numerical examples inspired by molecular
biology
Local error estimates for adaptive simulation of the Reaction-Diffusion Master Equation via operator splitting
The efficiency of exact simulation methods for the reaction-diffusion master
equation (RDME) is severely limited by the large number of diffusion events if
the mesh is fine or if diffusion constants are large. Furthermore, inherent
properties of exact kinetic-Monte Carlo simulation methods limit the efficiency
of parallel implementations. Several approximate and hybrid methods have
appeared that enable more efficient simulation of the RDME. A common feature to
most of them is that they rely on splitting the system into its reaction and
diffusion parts and updating them sequentially over a discrete timestep. This
use of operator splitting enables more efficient simulation but it comes at the
price of a temporal discretization error that depends on the size of the
timestep. So far, existing methods have not attempted to estimate or control
this error in a systematic manner. This makes the solvers hard to use for
practitioners since they must guess an appropriate timestep. It also makes the
solvers potentially less efficient than if the timesteps are adapted to control
the error. Here, we derive estimates of the local error and propose a strategy
to adaptively select the timestep when the RDME is simulated via a first order
operator splitting. While the strategy is general and applicable to a wide
range of approximate and hybrid methods, we exemplify it here by extending a
previously published approximate method, the Diffusive Finite-State Projection
(DFSP) method, to incorporate temporal adaptivity
MOLNs: A cloud platform for interactive, reproducible and scalable spatial stochastic computational experiments in systems biology using PyURDME
Computational experiments using spatial stochastic simulations have led to
important new biological insights, but they require specialized tools, a
complex software stack, as well as large and scalable compute and data analysis
resources due to the large computational cost associated with Monte Carlo
computational workflows. The complexity of setting up and managing a
large-scale distributed computation environment to support productive and
reproducible modeling can be prohibitive for practitioners in systems biology.
This results in a barrier to the adoption of spatial stochastic simulation
tools, effectively limiting the type of biological questions addressed by
quantitative modeling. In this paper, we present PyURDME, a new, user-friendly
spatial modeling and simulation package, and MOLNs, a cloud computing appliance
for distributed simulation of stochastic reaction-diffusion models. MOLNs is
based on IPython and provides an interactive programming platform for
development of sharable and reproducible distributed parallel computational
experiments
- …