Deciphering the metabolic functions of organisms requires understanding the dynamic responses of living cells upon genetic and environmental perturbations, which in turn can be inferred from enzymatic activity. In this work, we investigate the optimal modes of operation for enzymes in terms of the evolutionary pressure driving them toward increased catalytic efficiency. We develop a framework using a mixed-integer formulation to assess the distribution of thermodynamic forces and enzyme states, providing detailed insights into the enzymatic mode of operation. We use this framework to explore Michaelis-Menten and random-ordered multi-substrate mechanisms. We show that optimal enzyme utilization is achieved by unique or alternative operating modes dependent on reactant concentrations. We find that in a bimolecular enzyme reaction, the random mechanism is optimal over any other ordered mechanism under physiological conditions. Our framework can investigate the optimal catalytic properties of complex enzyme mechanisms. It can further guide the directed evolution of enzymes and fill in the knowledge gaps in enzyme kinetics.

Understanding the dynamic responses of living cells upon genetic and environmental perturbations is crucial to decipher the metabolic functions of organisms. The rates of enzymatic reactions and their evolution are key to this understanding, as metabolic fluxes are limited by enzymatic activity. In this work, we investigate the optimal modes of operations for enzymes with regard that the evolutionary pressure drives enzyme kinetics toward increased catalytic efficiency. We use an efficient mixed-integer formulation to decipher the principles of optimal catalytic properties at various operating points. Our framework allows assessing the distribution of the thermodynamic forces and enzyme states, providing detailed insight into the mode of operation. Our results confirm earlier theoretical studies on the optimal kinetic design using a reversible Michaelis-Menten mechanism. The results further explored the optimal modes of operation for random-ordered multi-substrate mechanisms. We show that optimal enzyme utilization is achieved by unique or alternative modes of operations depending on the reactant's concentrations. Our novel formulation allows investigating the optimal catalytic properties of all enzyme mechanisms with known elementary reactions. We propose that our novel framework provides the means to guide and evaluate directed evolution studies and estimate the limits of the direct evolution of enzymes.