The economic basis of periodic enzyme dynamics

Periodic enzyme activities can improve the metabolic performance of cells. As an adaptation to periodic environments or by driving metabolic cycles that can shift fluxes and rearrange metabolic processes in time to increase their efficiency. To study what benefits can ensue from rhythmic gene expression or posttranslational modification of enzymes, I propose a theory of optimal enzyme rhythms in periodic or static environments. The theory is based on kinetic metabolic models with predefined metabolic objectives, scores the effects of harmonic enzyme oscillations, and determines amplitudes and phase shifts that maximise cell fitness. In an expansion around optimal steady states, the optimal enzyme profiles can be computed by solving a quadratic optimality problem. The formulae show how enzymes can increase their efficiency by oscillating in phase with their substrates and how cells can benefit from adapting to external rhythms and from spontaneous, intrinsic enzyme rhythms. Both types of behaviour may occur different parameter regions of the same model. Optimal enzyme profiles are not passively adapted to existing substrate rhythms, but shape them actively to create opportunities for further fitness advantage: in doing so, they reflect the dynamic effects that enzymes can exert in the network. The proposed theory combines the dynamics and economics of metabolic systems and shows how optimal enzyme profiles are shaped by network structure, dynamics, external rhythms, and metabolic objectives. It covers static enzyme adaptation as a special case, reveals the conditions for beneficial metabolic cycles, and predicts optimally combinations of gene expression and posttranslational modification for creating enzyme rhythms.