Stationary diffusion gradients associated with photosynthetic carbon flux—a study of compartmental versus diffusion–reaction models

Metabolic processes usually involve diffusion of compounds in addition to their metabolic reactions. Such processes are adequately described by reaction–diffusion models (in the form partial differential equations) which are usually difficult and tedious to solve. Compartmental models (ordinary differential equations) are much easier to analyse but may be inadequate since they do not allow for spatial gradients. However, a compartmental model can be considered as the limit of a reaction–diffusion model for very fast diffusion (all diffusion coefficients Dj→∞). A compartmental model mc is termed “associated” to the reaction–diffusion model mrd if mc is the limit of mrd for all Dj→∞. From the analytical solutions of a reaction–diffusion model and its associated compartmental model the extent of a diffusion gradient of mrd can be estimated by means of parameters from both mc and mrd. This approach is extended to more complicated models that cannot be solved analytically. Gradients can be neglected and, consequently, the compartmental description be used, if the characteristic length s of the diffusion path is small compared with the distance a particle travels in time Te, where Te is the characteristic time for the compartmental model. This ratio of lengths can also be expressed as the ratio of two times, namely the residence time s2/DX and the turnover time XC/v, where XC and v are the steady-state concentration of X and its import rate, respectively, for the associated compartmental model. Characteristic times are given for several simple reaction–diffusion systems in rectangular and spherical geometries. Intracellular gradients of HCO3− and CO2 are calculated for some flux situations relevant to photosynthetic carbon fixation in green microalgae.