Reaction-Diffusion Processes in Muscle Metabolism Parametric and Sensitivity Analysis

Although diffusion is an important phenomenon for intra-cellular metabolite transfer, it is seldom considered while modeling cellular energetics. Weisz (1973) hypothesized that most biological and biochemical systems would most efficiently operate at the edge of the effectiveness factor curve where (1) they are not diffusion limited, but (2) further increases of reaction rate and capacity would lead to diffusion limitations. Hence it is of importance to study the factors that may affect intra-cellular metabolite diffusion limitations. The present study focuses on analysis of the influence of various parameters on intra-cellular metabolite diffusion limitations. In order to do so, a simplified mitochondrial rate law was developed as a function of ADP, O2, and Pi. A reaction-diffusion model was developed to include ADP, ATP, O2, creatine kinase (CK), phosphor-creatine (PCr), myoglobin (Mb), Mb-O2 complex (MbO2), and Pi as metabolites. The volume averaging technique was performed to account for contributions from all the mitochondria in the cell and the effectiveness factor (η) was computed. Simulations showed that muscle fibers are not limited by diffusion up to certain combinations of ATP turnover rate and fiber size, after which the fibers start to be limited by diffusion for any further increase in ATP turnover rate and fiber size. Also, as the mitochondrial volume fraction increases, the cell can sustain higher ATP turnover rates without diffusion limitation. Comparison of model analysis with experimental data revealed that none of the fibers were strongly limited by diffusion. However, while some fibers were near substantial diffusion limitation, most were well within the domain of reaction-control of aerobic metabolic rate. This may constitute a safety factor in muscle that provides a level of protection from diffusion constraints under conditions such as hypoxia. Myoglobin (Mb) and creatine kinase (CK) have been proposed to facilitate the diffusion of O2 and ADP/ATP, respectively, although the conditions where Mb functions as a facilitator are subject to debate. Nevertheless, it is important to establish the significance and to quantify the effects of Mb and Ck in enhancing metabolite diffusion of metabolites when the cell is not limited by diffusion. The aim of the present work was to determine the conditions (parameter space) where the Mb and CK can affect the intra-cellular metabolite diffusion limitations. To do so the reaction-diffusion model was modified for three cases including a) the effects of Mb alone, 2) CK alone, and 3) without Mb and CK, and η was computed for all cases. The effect of Mb on enhancing intra-cellular diffusion was found to be smaller compared to that of CK at moderate cell size and ATP turnover rates. However, both Mb and CK can have significant effects in enhancing diffusion for larger cells with high ATP demand, and also for cells with low mitochondrial volume fractions (εmito) and low O2 concentrations. Comparison of the case with Mb and CK and the case without Mb and CK with experimental data revealed that Mb and CK do not significantly affect the η except in the regions very close to the transition to diffusion limitations which are just beyond the range of biological observations. Sensitivity analysis was performed to identify the key parameters that control intra-cellular metabolite diffusion limitations. It was observed that when the cell is reaction-controlled, the cell is limited by ATP turnover rate, represented by Vmax,ATPase, while when the cell is limited by diffusion, oxygen supply and transport limits ATP production by the mitochondria. The diffusion constant for O2 was the most sensitive of the diffusion constants, implying that the transport of O2 to mitochondria is the limiting step when the cell is limited by diffusion and hence the cell may be prone to hypoxia when the cell is limited by diffusion. To avoid hypoxia, the cell may have adapted itself in such a way as to avoid diffusion limitations, and this may be the reason most cells are reaction controlled while some operate at the edge of diffusion limitation.