Effective potential reveals evolutionary trajectories in complex fitness landscapes

Abstract

Growing efforts to measure fitness landscapes in molecular and microbial systems are premised on a tight relationship between landscape topography and evolutionary trajectories. This relationship, however, is far from being straightforward: depending on their mutation rate, Darwinian populations can climb the closest fitness peak (survival of the fittest), settle in lower regions with higher mutational robustness (survival of the flattest), or fail to adapt altogether (error catastrophes). These bifurcations highlight that evolution does not necessarily drive populations "from lower peak to higher peak", as Wright imagined. The problem therefore remains: how exactly does a complex landscape topography constrain evolution, and can we predict where it will go next?Here I introduce a generalization of quasispecies theory which identifies metastable evolutionary states as minima of an effective potential. From this representation I derive a coarse-grained, Markov state model of evolution, which in turn forms a basis for evolutionary predictions across a wide range of mutation rates. Because the effective potential is related to the ground state of a quantum Hamiltonian, my approach could stimulate fruitful interactions between evolutionary dynamics and quantum many-body theory. SIGNIFICANCE STATEMENTThe course of evolution is determined by the relationship between heritable types and their adaptive values, the fitness landscape. Thanks to the explosive development of sequencing technologies, fitness landscapes have now been measured in a diversity of systems from molecules to micro-organisms. How can we turn these data into evolutionary predictions? I show that preferred evolutionary trajectories are revealed when the effects of selection and mutations are blended in a single effective evolutionary force. With this reformulation, the dynamics of selection and mutation becomes Markovian, bringing a wealth of classical visualization and analysis tools to bear on evolutionary dynamics. Among these is a coarse-graining of evolutionary dynamics along its metastable states which greatly reduces the complexity of the prediction problem.

Neutral quasispecies evolution and the maximal entropy random walk

“…In (33), I showed that continuous-time replicator-mutator (or quasispecies) equations can be understood in terms of a derived Markov process, for which the logarithm of the selection-mutation equilibrium plays the role of an effective potential. With discrete generations, this scheme can be reformulated as follows: Given a discrete space X, consider a sequence of probability distributions p t : X → ℝ evolving under the dynamics p t+1 (x ) = (B p t ) (x)…”

“…3 is not readily interpretable in terms of "evolutionary trajectories." This can be remedied by means of the change of variable q t (x) ∝ S(x)p t (x), where S is the (left) eigenvector of B with largest eigenvalue; by the Perron-Frobenius theorem, this vector is positive and its eigenvalue is the spectral radius  of S. Via this transformation, which only depends on t through a global constant, we obtain a Markovian representation of the original dynamical problem with master equation For this derived process, the function U = − 2 log S plays the role of a potential; its analysis reveals the metastable states and preferred trajectories of the original (nonlinear) process (33,47). The equilibrium distribution for Eq.…”

Localization of neutral evolution: selection for mutational robustness and the maximal entropy random walk

“…To understand the limits of neutral evolution in interfering populations we must study the full trajectory p t (x) and not just its asymptotic equilibrium Q (x). For this purpose we can use the framework recently developed in (Smerlak, 2019), which maps the non-linear dynamics 1 onto a Markov chain on G via the change of variables q t (x) ∝ Q(x)p t (x) (SI ). This new distribution satisfies the master equation…”

“…In (Smerlak, 2019) I showed that continuous-time replicator-mutator (or quasispecies) equations can be understood in terms of a derived Markov process, in which the logarithm of the selectionmutation equilibrium plays the tole of an effective potential. With discrete generations, this scheme can be reformulated as follows.…”

“…For this derived process the function U = −2 log S plays the role of a potential; its analysis reveals the metastable states and preferred trajectories of the original (non-linear) process (Smerlak, 2019). The equilibrium distribution for 4 is q ∞ (x) ∝ S 2 (x), and the original distribution can be reconstructed from q t (x) with p t (x) ∝ S(x) −1 q t (x).…”