Control analysis of metabolic networks. 1. Homogeneous functions and the summation theorems for control coefficients

1. The summation theorem CJ1 + ... + CJn = 1 for flux control coefficients CJi is shown to be equivalent to the assumption that flux J is a homogeneous function of degree 1 of enzyme concentrations E1,..., En, that is to the assumption J (tE1,..., tEn) = tJ (E1,..., En) for any t not equal to 0. Likewise, the summation theorem CXj1 + ... + CXjn = 0 for concentration control coefficients CXj1 is equivalent to homogeneity of degree 0 of steady-state metabolite concentrations Xj, or Xj (tE1,..., tEn) = Xj (E1,..., En). From this equivalence it is obvious that metabolic control analysis applies only to homogeneous systems. 2. The summation theorem for flux control coefficients is shown to be equivalent to that for concentration control coefficients, provided all reaction rates vi are homogeneous functions of enzyme concentrations Ei. 3. The equivalence between homogeneity of flux J and the summation theorem for flux control coefficients is used to analyse branching of fluxes in metabolic pathways in terms of flux control coefficients.