The enzyme cost of given metabolic flux distributions, as a function of logarithmic metabolite levels, is convex

Enzyme costs play a major role in the choice of metabolic routes, both in evolution and bioengineering. Given desired fluxes, necessary enzyme levels can be estimated based on known rate laws and on a principle of minimal enzyme cost. With logarithmic metabolite levels as free variables, enzyme cost functions and constraints in optimality and sampling problems can be handled easily. The set of feasible metabolite profiles forms a polytope in log-concentration space, whose points represent all possible steady states of a kinetic model. We show that enzyme cost is a convex function on this polytope. This makes enzyme cost minimization - finding optimal enzyme profiles and corresponding metabolite profiles that realize a desired flux at a minimal cost - a convex optimization problem.