  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.
