Mathematical modelling of carbohydrate and protein metabolism in muscle

We propose a mathematical model based on coupled ordinary differential equations (ODEs) for metabolite concentrations with the aim of investigating how modifications to the rates affects outputs from a regulatory network. Our aim is to model the relationships between energy metabolism and the biosynthesis of non-essential amino acids, such as serine. We consider a network of cytosolic glycolysis, the mitochondrial TCA cycle, and the associated serine synthesis pathway, with the aim of modelling the role of metabolic reprogramming as a mechanism to enhance protein synthesis and growth, particularly in skeletal muscle. Our objective is to explore the consequences of overexpressing two key enzymes, phosphoenolpyruvate carboxykinase 2 (PCK2), and phosphoglycerate dehydrogenase (PHGDH), on the TCA cycle and on serine production. We investigate how the rate of serine synthesis is affected by upregulating both enzymes simultaneously, or each one individually. We find a range of steady-states which depend upon input fluxes into the network. As input fluxes are altered, steady states cease to exist due to a bifurcation to one of two states in which some metabolites grow linearly in time whilst others decay to zero. Asymptotic analysis provides approximations for steady-state solutions near these bifurcation points, and conditions on parameter values which determine where in parameter space the system's behaviour changes. We also perform a parameter sensitivity analysis to determine the effect of perturbations to rate constants and input rates. Our numerical simulations show that the up-regulation of PHGDH, the initial rate limiting enzyme in the serine-synthesis pathway, causes an increase in serine production but that, contrary to our hypothesis, increased expression of PCK2 has no effect. This model aids our understanding of both the effects of drugs and changes in enzyme expression or activities which upregulate one or more reactions in a pathway.