Mutagenic Evidence for the Optimal Control of Evolutionary Dynamics

Evolution is guided by the optimization of fitness measures that balance functionally beneficial properties. In modern theories of evolutionary dynamics, such as the quasispecies model [1] and variants thereof, the fitness measure of a biological system plays a role analogous to that of the free energy of a mechanical system. The dynamics of the system, embodied through mutations, seeks to optimize this measure. Recently, with advances in the understanding of molecular biophysics, increasing attention has been paid to characterizing the fitness measures underlying the evolution of proteins. For example, simulations of protein sequence evolution have confirmed that protein cores evolve almost universally to maximize the free energy gap between the folded and denatured states [2]. However, for functional properties of proteins and protein networks, the appropriate biological fitness measures are not so clear [3]. A current challenge in evolutionary theory is to identify how the fitness measures of complex biological systems depend on the physical properties of their constituent proteins. In the hierarchical evolution of protein networks, biological self-organization [4] influences the dynamics that occur on shorter time scales. Although most theories of evolutionary dynamics have modeled evolution as a dynamical system seeking to optimize a potential or free energy, multi-time-scale evolution of protein networks may be modeled within a broader framework as a control problem. Optimal control (OC) theory is generally concerned with the determination of the time-dependent functional form of the Hamiltonian of a controlled dynamical system that maximizes a desired objective function [5]. An important difference between a dynamical system and a control system is that the latter distinguishes between the free dynamics of the system and the dynamics regulated by controls. In the present case, these controls can take the form of functional protein properties. The evolution of a biological system may be modeled as a control system if the regulatory functional properties of its constituent proteins coevolve with the network’s overall function. Should the evolutionary dynamics of such a system demonstrate features indicative of optimal controls, this would constitute evidence that the system’s evolution has attained a sophisticated level of self-organization amounting to the solution of an OC problem. Here, we show that application of this theory to active site mutations in an enzyme network of central importance for metabolism—the electron transport chain [6]—indicates that the redox potentials of electron transport proteins are controlling the evolutionary dynamics of this network in an optimal fashion, providing insight into the self-organization of this system. The mitochondrial electron transport chain (ETC) removes electrons from the high-energy electron donor NADH and passes them to the electron acceptor O2 through a series of redox reactions involving electron transport proteins. These reactions are coupled to the generation of a proton concentration gradient across the mitochondrial inner membrane, which is ultimately used to produce adenosine triphosphate (ATP). Our prior work [7,8] pursued a strategy of examining ‘‘evolution in reverse’’ with the four-helix bundle ETC hemoprotein cytochrome b562. Starting with the evolved protein, variants with replacements at amino acids near the active site heme were created and examined for redox function. We found two general results. First, within this conserved protein architecture, a range of variation in redox potential " 0 of about 160 mV could be obtained within two rounds of (reverse) evolution, involving only four residues. Statistical analysis based on Chebyshev’s theorem indicates that this range represents, with >75% confidence, the total range accessible through mutations at these positions. Second, the wild-type redox potential was not found to be at the middle of the chemically accessible range of reduction potentials [7,8]. Instead, wt b562 exhibits a redox potential (" 0 � 167 mV) at the extreme of the chemically