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

• Published in: Journal of theoretical biology 2003-10, Vol.224 (3), p.385-397
• Main Author: Giersch, Christoph
• Format: Article
• Language: English
• Subjects: Biological Transport; Carbon - metabolism; Carbon concentrating mechanism; Carbon supply; Compartmental model; Diffusion; Diffusion gradient; Diffusion-reaction system; Models, Biological; Photosynthesis; Plants - metabolism
• ISSN: 0022-5193; 1095-8541
• DOI: 10.1016/S0022-5193(03)00182-6

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 D j →∞). A compartmental model m c is termed “associated” to the reaction–diffusion model m rd if m c is the limit of m rd for all D j →∞. From the analytical solutions of a reaction–diffusion model and its associated compartmental model the extent of a diffusion gradient of m rd can be estimated by means of parameters from both m c and m rd . 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 T e , where T e 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 s 2/ D X and the turnover time X C / v, where X C 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 HCO 3 − and CO 2 are calculated for some flux situations relevant to photosynthetic carbon fixation in green microalgae.