Numerical Solution Of Reaction-diffusion Equations Research Articles

Novel Approximation of a Stationary Single-Channel Ca2+ Nanodomain

We consider the stationary solution for the Ca2+ concentration near a point Ca2+ source describing a single-channel Ca2+ nanodomain, in the presence of a single mobile buffer with one-to-one Ca2+ binding. Previously, a number of Ca2+ nanodomains approximations have been developed, such as the Excess Buffer approximation, the Rapid Buffering approximation, and the Linear approximation, each valid for appropriate buffering conditions. Apart from providing a simple method of estimating Ca2+ and buffer concentrations without resorting to computationally expensive numerical solution of reaction-diffusion equations, such approximations proved useful in revealing the dependence of nanodomain Ca2+ distribution on crucial parameters such as buffer mobility and its Ca2+ binding properties. Here we present a novel form of analytic approximation, which is based on matching the short-range Taylor series of the nanodomain concentration with the long-range asymptotic series expressed in inverse powers of distance from channel location. Namely, we use a “dual” Padé rational function approximation to simultaneously match terms in the short- and the long-range series, and show that this provides an accurate approximation to the nanodomain Ca2+ and buffer concentrations. We compare this approximation with the previously obtained approximations, and show that it yields a better estimate of the free buffer concentration for a wide range of buffering conditions. The drawback of the presented method is that it has a complex algebraic form above the lowest, bilinear order, and cannot be readily extended to multiple Ca2+ channels. However, it may be possible to extend at least some of the features of this method to the case of cooperative Ca2+ buffers with two Ca2+ binding sites (e.g. calretinin), the case which existing analytic methods do not address. This work was supported in part by the National Science Foundation grant DMS-1517085.

Padé Approximation of a Stationary Single-Channel Ca2+ Nanodomain

We consider the stationary solution for the Ca2+ concentration near a point Ca2+ source describing a single-channel Ca2+ nanodomain, in the presence of a single mobile buffer with one-to-one Ca2+ binding stoichiometry. Previously, a number of Ca2+ nanodomains approximations have been developed, for instance the excess buffer approximation (EBA), the rapid buffering approximation (RBA), and the linear approximation (LIN), each valid for appropriate buffering conditions. Apart from providing a simple method of estimating Ca2+ and buffer concentrations without resorting to computationally expensive numerical solution of reaction-diffusion equations, such approximations proved useful in revealing the dependence of nanodomain Ca2+ distribution on crucial parameters such as buffer mobility and its Ca2+ binding properties. In this study, we present a different form of analytic approximation, which is based on matching the short-range Taylor series of the nanodomain concentration with the long-range asymptotic series expressed in inverse powers of distance from channel location. Namely, we use a “dual” Padé rational function approximation to simultaneously match terms in the short- and the long-range series, and we show that this provides an accurate approximation to the nanodomain Ca2+ and buffer concentrations. We compare this approximation with the previously obtained approximations and show that it yields a better estimate of the free buffer concentration for a wide range of buffering conditions. The drawback of our method is that it has a complex algebraic form for any order higher than the lowest bilinear order, and cannot be readily extended to multiple Ca2+ channels. However, it may be possible to extend the Padé method to estimate Ca2+ nanodomains in the presence of cooperative Ca2+ buffers with two Ca2+ binding sites, the case that existing methods do not address.

Numerical solutions of reaction-diffusion equations: Application to neural and cardiac models

We describe the implementation of the explicit Euler, Crank-Nicolson, and implicit alternating direction methods for solving partial differential equations and apply these methods to obtain numerical solutions of three excitable-media models used to study neurons and cardiomyocyte dynamics. We discuss the implementation, accuracy, speed, and stability of these numerical methods.

An Efficient Wavelet Based Approximation Method to Steady State Reaction–Diffusion Model Arising in Mathematical Chemistry

The mathematical model of Rahamathunissa and Rajendran (J Math Chem 44:849-861, 2008) in an amperometric biosensor response is discussed. In this paper, we have applied the shifted second kind Chebyshev wavelets (CW) to obtain the numerical solutions of reaction-diffusion equations containing a nonlinear term related to Michaelis-Menton kinetics of the enzymatic reaction. The application of the shifted second kind CW operational matrices for solving initial and boundary value problems is presented. The obtained numerical results demonstrate efficient and applicability of the proposed method. The power of the manageable method is confirmed. Moreover the use of shifted second kind CW method is found to be simple, efficient, accurate, small computation cost, and computationally attractive.

An IMEX scheme for reaction-diffusion equations: application for a PEM fuel cell model

Abstract An implicit-explicit (IMEX) method is developed for the numerical solution of reaction-diffusion equations with pure Neumann boundary conditions. The corresponding method of lines scheme with finite differences is analyzed: explicit conditions are given for its convergence in the ‖·‖∞ norm. The results are applied to a model for determining the overpotential in a proton exchange membrane (PEM) fuel cell.

Two continuum models for the spreading of myxobacteria swarms

We analyze the phenomenon of spreading of a Myxococcus xanthus bacterial colony on plates coated with nutrient. The bacteria spread by gliding on the surface. In the first few hours, cell growth is irrelevant to colony spread. In this case, bacteria spread through peninsular protrusions from the edge of the initial colony. We analyze the diffusion through the narrowing reticulum of cells on the surface mathematically and derive formulae for the spreading rates. On the time scale of tens of hours, effective diffusion of the bacteria, combined with cell division and growth, causes a constant linear increase in the colony's radius. Mathematical analysis and numerical solution of reaction-diffusion equations describing the bacterial and nutrient dynamics demonstrate that, in this regime, the spreading rate is proportional to the square root of both the effective diffusion coefficient and the nutrient concentration. The model predictions agree with the data on spreading rate dependence on the type of gliding motility.

A linearly implicit predictor-corrector method for reaction-diffusion equations

A novel linearly implicit predictor-corrector scheme is developed for the numerical solution of reaction-diffusion equations. Iterative processes are avoided by treating the nonlinear reaction terms explicitly, while maintaining superior accuracy and stability properties compared to the well-known θ methods and linearly implicit Runge-Kutta methods. The proposed method allows the opportunity of solving large systems of reaction-diffusion equations by alleviating the necessity of solving the accompanying large linear systems of algebraic equations due to the natural parallelism which surfaces across the system. Numerical results confirm the enhanced stability, accuracy and efficiency of the method when applied to reaction-diffusion equations arising in biochemistry and population ecology.