Describing the movement of molecules in reduced-dimension models

TL;DR

This paper examines how reduced-dimension models in biological systems can accurately approximate full molecular movement, proposing conditions for validity and a finite difference method to improve model accuracy.

Contribution

It identifies conditions for accurate approximation of molecular movement in reduced-dimension models and introduces a finite difference method to enhance their precision.

Findings

Conditions for valid reduced-dimension approximations

A finite difference method for 2D diffusion in 1D models

Improved accuracy of simplified biological models

Abstract

When addressing spatial biological questions using mathematical models, symmetries within the system are often exploited to simplify the problem by reducing its physical dimension. In a reduced-dimension model molecular movement is restricted to the reduced dimension, changing the nature of molecular movement. This change in molecular movement can lead to quantitatively and even qualitatively different results in the full and reduced systems. Within this manuscript we discuss the condition under which restricted molecular movement in reduced-dimension models accurately approximates molecular movement in the full system. For those systems which do not satisfy the condition, we present a general method for approximating unrestricted molecular movement in reduced-dimension models. We will derive a mathematically robust, finite difference method for solving the 2D diffusion equation within a 1D reduced-dimension model. The methods described here can be used to improve the accuracy of many reduced-dimension models while retaining benefits of system simplification.