52 research outputs found
Single molecule simulations in complex geometries with embedded dynamic one-dimensional structures
Stochastic models of reaction-diffusion systems are important for the study
of biochemical reaction networks where species are present in low copy numbers
or if reactions are highly diffusion limited. In living cells many such systems
include reactions and transport on one-dimensional structures, such as DNA and
microtubules. The cytoskeleton is a dynamic structure where individual fibers
move, grow and shrink. In this paper we present a simulation algorithm that
combines single molecule simulations in three-dimensional space with single
molecule simulations on one-dimensional structures of arbitrary shape.
Molecules diffuse and react with each other in space, they associate to and
dissociate from one-dimensional structures as well as diffuse and react with
each other on the one-dimensional structure. A general curve embedded in space
can be approximated by a piecewise linear curve to arbitrary accuracy. The
resulting algorithm is hence very flexible. Molecules bound to a curve can move
by pure diffusion or via active transport, and the curve can move in space as
well as grow and shrink. The flexibility and accuracy of the algorithm is
demonstrated in four numerical examples
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
Convergence of methods for coupling of microscopic and mesoscopic reaction-diffusion simulations
In this paper, three multiscale methods for coupling of mesoscopic
(compartment-based) and microscopic (molecular-based) stochastic
reaction-diffusion simulations are investigated. Two of the three methods that
will be discussed in detail have been previously reported in the literature;
the two-regime method (TRM) and the compartment-placement method (CPM). The
third method that is introduced and analysed in this paper is the ghost cell
method (GCM). Presented is a comparison of sources of error. The convergent
properties of this error are studied as the time step (for updating
the molecular-based part of the model) approaches zero. It is found that the
error behaviour depends on another fundamental computational parameter , the
compartment size in the mesoscopic part of the model. Two important limiting
cases, which appear in applications, are considered: (i) \Delta t approaches 0
and h is fixed; and (ii) \Delta t approaches 0 and h approaches 0 such that
\Delta t/h^2 is fixed. The error for previously developed approaches (the TRM
and CPM) converges to zero only in the limiting case (ii), but not in case (i).
It is shown that the error of the GCM converges in the limiting case (i). Thus
the GCM is superior to previous coupling techniques if the mesoscopic
description is much coarser than the microscopic part of the model
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
On the Reaction Diffusion Master Equation in the Microscopic Limit
Stochastic modeling of reaction-diffusion kinetics has emerged as a powerful
theoretical tool in the study of biochemical reaction networks. Two frequently
employed models are the particle-tracking Smoluchowski framework and the
on-lattice Reaction-Diffusion Master Equation (RDME) framework. As the mesh
size goes from coarse to fine, the RDME initially becomes more accurate.
However, recent developments have shown that it will become increasingly
inaccurate compared to the Smoluchowski model as the lattice spacing becomes
very fine. In this paper we give a new, general and simple argument for why the
RDME breaks down. Our analysis reveals a hard limit on the voxel size for which
no local RDME can agree with the Smoluchowski model
Site fidelity and range size of wintering Barnacle Geese Branta leucopsis
Barnacle Geese restrict their movements to relatively few key sites and exhibit considerable variation in ranging behaviour. To examine individual and seasonal variation in site fidelity, habitat use, range size and foraging
strategies of Barnacle Geese Branta leucopsis, the movements of 18 male Barnacle Geese tagged in two discrete areas were tracked for 3–6 months from late autumn until departure on the spring migration.
Tagged geese concentrated their feeding in a relatively small proportion of apparently suitable habitat. Geese moved increasingly further afield in midwinter, and there was a clear predeparture shift to the largest area of relatively undisturbed, and possibly more nitrogen-rich, saltmarsh on the Solway. Birds from one of the two capture sites tended to be more sedentary and have smaller home ranges
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