800 research outputs found
From Brownian Dynamics to Markov Chain: an Ion Channel Example
A discrete rate theory for general multi-ion channels is presented, in which
the continuous dynamics of ion diffusion is reduced to transitions between
Markovian discrete states. In an open channel, the ion permeation process
involves three types of events: an ion entering the channel, an ion escaping
from the channel, or an ion hopping between different energy minima in the
channel. The continuous dynamics leads to a hierarchy of Fokker-Planck
equations, indexed by channel occupancy. From these the mean escape times and
splitting probabilities (denoting from which side an ion has escaped) can be
calculated. By equating these with the corresponding expressions from the
Markov model the Markovian transition rates can be determined. The theory is
illustrated with a two-ion one-well channel. The stationary probability of
states is compared with that from both Brownian dynamics simulation and the
hierarchical Fokker-Planck equations. The conductivity of the channel is also
studied, and the optimal geometry maximizing ion flux is computed.Comment: submitted to SIAM Journal on Applied Mathematic
Reactive Boundary Conditions as Limits of Interaction Potentials for Brownian and Langevin Dynamics
A popular approach to modeling bimolecular reactions between diffusing
molecules is through the use of reactive boundary conditions. One common model
is the Smoluchowski partial absorption condition, which uses a Robin boundary
condition in the separation coordinate between two possible reactants. This
boundary condition can be interpreted as an idealization of a reactive
interaction potential model, in which a potential barrier must be surmounted
before reactions can occur. In this work we show how the reactive boundary
condition arises as the limit of an interaction potential encoding a steep
barrier within a shrinking region in the particle separation, where molecules
react instantly upon reaching the peak of the barrier. The limiting boundary
condition is derived by the method of matched asymptotic expansions, and shown
to depend critically on the relative rate of increase of the barrier height as
the width of the potential is decreased. Limiting boundary conditions for the
same interaction potential in both the overdamped Fokker-Planck equation
(Brownian Dynamics), and the Kramers equation (Langevin Dynamics) are
investigated. It is shown that different scalings are required in the two
models to recover reactive boundary conditions that are consistent in the high
friction limit where the Kramers equation solution converges to the solution of
the Fokker-Planck equation.Comment: 23 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
Adaptive finite element method assisted by stochastic simulation of chemical systems
Stochastic models of chemical systems are often analysed by solving the corresponding\ud
Fokker-Planck equation which is a drift-diffusion partial differential equation for the probability\ud
distribution function. Efficient numerical solution of the Fokker-Planck equation requires adaptive mesh refinements. In this paper, we present a mesh refinement approach which makes use of a stochastic simulation of the underlying chemical system. By observing the stochastic trajectory for a relatively short amount of time, the areas of the state space with non-negligible probability density are identified. By refining the finite element mesh in these areas, and coarsening elsewhere, a suitable mesh is constructed and used for the computation of the probability density
Test Input Generation for Red-Black Trees using Abstraction
We consider the problem of test input generation for code that manipulates complex data structures. Test inputs are sequences of method calls from the data structure interface. We describe test input generation techniques that rely on state matching to avoid generation of redundant tests. Exhaustive techniques use explicit state model checking to explore all the possible test sequences up to predefined input sizes. Lossy techniques rely on abstraction mappings to compute and store abstract versions of the concrete states; they explore under-approximations of all the possible test sequences. We have implemented the techniques on top of the Java PathFinder model checker and we evaluate them using a Java implementation of red-black trees
Inverse mass matrix via the method of localized lagrange multipliers
An efficient method for generating the mass matrix inverse is presented, which can be tailored to improve the accuracy of target frequency ranges and/or wave contents. The present method bypasses the use of biorthogonal construction of a kernel inverse mass matrix that requires special procedures for boundary conditions and free edges or surfaces, and constructs the free-free inverse mass matrix employing the standard FEM procedure. The various boundary conditions are realized by the method of localized Lagrange multipliers. Numerical experiments with the proposed inverse mass matrix method are carried out to validate the effectiveness proposed technique when applied to vibration analysis of bars and beams. A perfect agreement is found between the exact inverse of the mass matrix and its direct inverse computed through biorthogonal basis functions
Dynamics of polydisperse irreversible adsorption: a pharmacological example
Many drug delivery systems suffer from undesirable interactions with the host immune system. It has been experimentally established that covalent attachment (irreversible adsorption) of suitable macromolecules to the surface of the drug carrier can reduce such undesirable interactions. A fundamental understanding of the adsorption process is still lacking. In this paper, the classical random irreversible adsorption model is generalized to capture certain essential processes involved in pharmacological applications, allowing for macromolecules of different sizes, partial overlapping of the tails of macromolecules, and the influence of reactions with the solvent on the adsorption process. Working in one dimension, an integro-differential evolution equation for the adsorption process is derived, and the asymptotic behavior of the surface area covered and the number of molecules attached to the surface are studied. Finally, equation-free dynamic renormalization tools are applied to study the asymptotically self-similar behavior of the adsorption statistics
Stochastic modelling of reaction-diffusion processes: algorithms for bimolecular reactions
Several stochastic simulation algorithms (SSAs) have been recently proposed
for modelling reaction-diffusion processes in cellular and molecular biology.
In this paper, two commonly used SSAs are studied. The first SSA is an
on-lattice model described by the reaction-diffusion master equation. The
second SSA is an off-lattice model based on the simulation of Brownian motion
of individual molecules and their reactive collisions. In both cases, it is
shown that the commonly used implementation of bimolecular reactions (i.e. the
reactions of the form A + B -> C, or A + A -> C) might lead to incorrect
results. Improvements of both SSAs are suggested which overcome the
difficulties highlighted. In particular, a formula is presented for the
smallest possible compartment size (lattice spacing) which can be correctly
implemented in the first model. This implementation uses a new formula for the
rate of bimolecular reactions per compartment (lattice site).Comment: 33 pages, submitted to Physical Biolog
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