Solution of PDEs for First-Order Photobleaching Kinetics using Krylov Subspace Spectral Methods

Abstract

We solve the first order reaction-diffusion equations which describes binding-diffusion kinetics using photobleaching scanning profile of confocal laser scanning Microscopes approximated by a Gaussian laser profile. We show how to solve the first order photobleaching kinetics partial differential equations with prebleach steady state initial conditions using a time-domain method known as a Krylov Subspace Spectral method (KSS method). KSS are explicit methods for solving time-dependent variable-coefficient partial differential equations (PDEs). KSS methods are advantageous compared to other methods because of its high resolution and its superior scalability. We will apply Gaussian Quadrature rules in the spectral domain developed by Golub and Meurant to solve PDEs. We present a simple rough analytical solution, as well as a computational solution that is first-order accurate. We then use this solution to examine short and long time behaviors.