Mutagenic Evidence for the Optimal Control of Evolutionary Dynamics

Elucidating the fitness measures optimized during the evolution of complex biological systems is a major challenge in evolutionary theory. We present experimental evidence and an analytical framework demonstrating how biochemical networks exploit optimal control strategies in their evolutionary dynamics. Optimal control theory explains a striking pattern of extremization in the redox potentials of electron transport proteins, assuming only that their fitness measure is a control objective functional with bounded controls.

Figure 1

The shift in redox potential from the wild-type value (${\epsilon }_{\mathrm{WT}}=0$) for active site mutants of several different cytochromes and iron-sulfur cluster proteins. Maximum likelihood estimation was employed to quantify the extent to which the proteins have evolved toward a redox potential extremum [12]. The likelihood that the true redox potential distribution is unbiased is listed below each protein.

Figure 2

(a) The evolution of the redox potential ${\epsilon }_i(t)$ for the $i$th enzyme within optimal control theory. The early evolutionary period near $t\approx 0$ is unspecified. During evolution, the potential “bangs” from its lower and upper accessible values, ${\epsilon }_i^l(t)$ and ${\epsilon }_i^u(t)$, respectively, at critical times $t_1,t_2,\cdots$ where effective mutations have occurred. The current evolutionary time is $T$. (b) The evolutionary time dependence of the product ${\lambda }_i{g}_i$ of the Lagrange function ${\lambda }_i$ and the control coupling function $g_i$. The zero crossings of ${\lambda }_i{g}_i$ occurs at $t_1,t_2,\cdots$ where the redox potential ${\epsilon }_i(t)$ undergoes evolutionary jumps in (a).