Control of metabolic pathways by time-scale separation.

The bases underlying the distribution of metabolic control have been elusive, even though many intuitive arguments exist. To analyze this problem, we have applied the structural properties of the control coefficients to systems in which the eigenvalues of the Jacobian of the system are widely separated, that is, systems with time-scale separation. We show that time-scale separation is an effective way to localize metabolic control to only a few enzymes. To achieve time-scale separation, the cell can overproduce most of the enzymes relative to the required activity for the steady state, and control the expression level of only a few enzymes, provided that the overexpressed enzymes do not cause adverse effects. The overexpressed enzymes are responsible for the small response times of the system, and the reactions catalyzed by them are termed 'fast' reactions. The metabolite concentration control coefficients of the 'fast' reactions are always small compared with the 'slow' reactions. Furthermore, the 'fast' reactions do not have effective control on the overall flux. However, the 'fast' reactions may compete with each other at a branch point, leading to significant control coefficients for fluxes to the branches. These results are useful in justifying lumping of 'fast' reactions in mathematical modelling or in the experimental determination of control coefficients. The theoretical results are derived under the assumption that the system possesses a unique, asymptotically stable steady-state and that the reaction steps of the system under analysis are well represented by linear kinetics around the steady state. The application of the results presented in this article are demonstrated with three illustrative examples.