Regulation of the intracellular concentration of substrates is essential for the maintenance of a stable cellular environment. Diffusion and reaction processes supply and consume substrates within cells and determine their steady-state concentrations. To realistically represent these processes by computer simulation they must be modeled in three dimensions. Yet three-dimensional models are inherently computing intensive. This study describes a method, which substantially simplifies the modeling of diffusion into a polyhedral body (a cube), that was used as a model representation of a cell. The method is applied to a case study of oxygen diffusion into nitrogen-fixing, rhizobia-infected cells in legume nodules. The method involved generating a one-dimensional representation of the three-dimensional problem to provide a "surface area profile" of three-dimensional diffusion. The one-dimensional models were significantly easier to program, several orders of magnitude faster to solve and in this study were validated by assessing their results against those of comparable three-dimensional models of diffusion into the same body. The results show the one-dimensional method to be a close approximation of a three-dimensional source-sink problem with systematic differences below 10% for fractional oxygenation of leghemoglobin, cell respiration and nitrogenase activity. Larger differences between models (up to 45%) in the predicted average and innermost O2concentrations had no effects on the physiological conclusions of the study, but were attributed to the poorer resolution of the three- than the one-dimensional model, and to an inherent simplification in the derivation of the one-dimensional surface area profiles. The one-dimensional modeling approach was found to be a simple, yet powerful tool for the study of diffusion and reaction in biological systems.
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