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