44 research outputs found
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
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First passage time density for the Ehrenfest model
We derive an explicit expression for the probability density of the first passage time to state 0 for the Ehrenfest diffusion model in continuous time
Multi-resolution dimer models in heat baths with short-range and long-range interactions
This work investigates multi-resolution methodologies for simulating dimer
models. The solvent particles which make up the heat bath interact with the
monomers of the dimer either through direct collisions (short-range) or through
harmonic springs (long-range). Two types of multi-resolution methodologies are
considered in detail: (a) describing parts of the solvent far away from the
dimer by a coarser approach; (b) describing each monomer of the dimer by using
a model with different level of resolution. These methodologies are then
utilised to investigate the effect of a shared heat bath versus two uncoupled
heat baths, one for each monomer. Furthermore the validity of the
multi-resolution methods is discussed by comparison to dynamics of macroscopic
Langevin equations.Comment: Submitted to Interface Focu
Wound healing angiogenesis: The clinical implications of a simple mathematical model
Nonhealing wounds are a major burden for health care systems worldwide. In addition, a patient who suffers from this type of wound usually has a reduced quality of life. While the wound healing process is undoubtedly complex, in this paper we develop a deterministic mathematical model, formulated as a system of partial differential equations, that focusses on an important aspect of successful healing: oxygen supply to the wound bed by a combination of diffusion from the surrounding unwounded tissue and delivery from newly formed blood vessels. While the model equations can be solved numerically, the emphasis here is on the use of asymptotic methods to establish conditions under which new blood vessel growth can be initiated and wound-bed angiogenesis can progress. These conditions are given in terms of key model parameters including the rate of oxygen supply and its rate of consumption in the wound. We use our model to discuss the clinical use of treatments such as hyperbaric oxygen therapy, wound bed debridement, and revascularisation therapy that have the potential to initiate healing in chronic, stalled wounds
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
Turing pattern or system heterogeneity? A numerical continuation approach to assessing the role of Turing instabilities in heterogeneous reaction-diffusion systems
Turing patterns in reaction-diffusion (RD) systems have classically been
studied only in RD systems which do not explicitly depend on independent
variables such as space. In practise, many systems for which Turing patterning
is important are not homogeneous with ideal boundary conditions. In
heterogeneous systems with stable steady states, the steady states are also
necessarily heterogeneous which is problematic for applying the classical
analysis. Whilst there has been some work done to extend Turing analysis to
some heterogeneous systems, for many systems it is still difficult to determine
if a stable patterned state is driven purely by system heterogeneity or if a
Turing instability is playing a role. In this work, we try to define a
framework which uses numerical continuation to map heterogeneous RD systems
onto a sensible nearby homogeneous system. This framework may be used for
discussing the role of Turing instabilities in establishing patterns in
heterogeneous RD systems. We study the Schnakenberg and Gierer-Meinhardt models
with spatially heterogeneous production as test problems. It is shown that for
sufficiently large system heterogeneity (large amplitude spatial variations in
morphogen production) it is possible that Turing-patterned and base states
become coincident and therefore impossible to distinguish. Other exotic
behaviour is also shown to be possible. We also study a novel scenario in which
morphogen is produced locally at levels that could support Turing patterning
but on intervals/patches which are on the scale of classical critical domain
lengths. Without classical domain boundaries, Turing patterns are allowed to
bleed through; an effect noted by other authors. In this case, this phenomena
effectively changes the critical domain length. Indeed, we even note that this
phenomena may also effectively couple local patches together and drive
instability in this way.Comment: 10 figure