Fig. 4

Diminishing-returns and increasing-costs epistasis in simple models of enzyme kinetics. (Left) The dependence of flux on two enzymes in a serial metabolic pathway, given by \(J = a/(1/{E}_{1 }+ 1/{E}_{2} + b)\) [56]. (Right) Michaelis–Menten kinetics. In the unsaturated regime, the downstream flux depends multiplicatively on \({k}_{on}\) and \({k}_{cat}\). Mutations that increase \(\log {k}_{\text{on}}\) experience diminishing-returns, whereas mutations that reduce \(\log {k}_{\text{cat}}\) incur larger costs at larger values of \(\log {k}_{\text{on}}.\)