Introduction
Darwinian evolution is the motion of populations in the space of all possible heritable types graded by their reproductive value, the fitness landscape \cite{Stadler,Orr_2009,Fragata_2019}. In this space, the interaction of selection and mutations allows populations to "continually find their way from lower to higher peaks" \cite{wright1932}, thereby providing a universal mechanism for open-ended evolution \cite{de_Vladar_2017}. Originally a mere metaphor, fitness landscapes have now been measured in a variety of real molecular \cite{Blanco_2019}, viral \cite{Dolan_2018} or microbial \cite{de_Visser_2014} systems. As a result, the goal of predicting evolution no longer appears wholly out of reach \cite{Weinreich_2006,Lobkovsky_2012,de_Visser_2014,L_ssig_2017,de_Visser_2018}. In essence, if we know the topography of the fitness landscape—its peaks, valleys, ridges, etc.—we should be able to compute where a population is likely to move next. Making such predictions from high-resolution fitness assays is the central challenge of quantitative evolutionary theory.
An array of measures quantifying the topography of fitness landscapes has been developed in support of this program. Especially important from the dynamical perspective, the ruggedness of a landscape can be quantified in a variety of ways, some simple but coarse (density of fitness maxima \cite{Kauffman_1987}, correlation functions \cite{Weinberger_1990,Stadler_1996}), others more detailed (amplitude spectra \cite{Hordijk_1998}, geometric shapes \cite{Beerenwinkel_2007}). Unforeseen by Wright, the neutrality of landscapes—the distribution of plateaus rather than peaks—is another key feature of real landscapes \cite{Cowperthwaite_2007}, indeed one that should be expected in any high-dimensional landscape for combinatorial reasons \cite{MAYNARD_SMITH_1970,Gavrilets_1997}. The NK(p) \cite{Kauffman_1989,barnett1998}, Rough Mount Fuji \cite{Aita_2000,Neidhart_2014}, holey \cite{MAYNARD_SMITH_1970,Gavrilets_1997} and other models in turn provide simple landscapes with tunable ruggedness and/or neutrality, which can be used to fit empirical data and simulate evolutionary trajectories. On these foundations a new subfield of mathematical biology has emerged, the quantitative analysis of fitness landscapes \cite{Szendro_2013}.
What these fitness-centric measures fail to capture, however, is the fact that populations with different mutation rates experience the same fitness landscape differently. This is already clear if we consider the rate of fitness valley crossings, which strongly depends on the mutation rate \cite{VANNIMWEGEN_2000,Weissman_2009} and therefore cannot be computed from topographic data alone. But Eigen's quasispecies theory teaches us that varying mutation rates can also have a qualitative effect on evolutionary trajectories, as illustrated dramatically with error catastrophes \cite{Eigen_1971} or error cascades \cite{Tannenbaum_2004} whereby selection fails to limit the accumulation of deleterious mutations and evolution becomes effective neutral. More subtly, mutational robustness has been shown to evolve neutrally \cite{van_Nimwegen_1999} and to sometimes outweigh reproductive rate as a determinant of evolutionary success ("survival of the flattest") \cite{Wilke_2001,Codo_er_2006}. These evolutionary bifurcations are not mere theoretical curiosities: lethal mutagenesis—an effort to push a population beyond its error threshold—is a promising therapeutic strategy against certain viral pathogens \cite{Eigen_2002}.
More fundamentally, these transitions imply that, unless mutations are so rare that populations are genetically homogeneous at all times (the strong-selection weak-mutation (SSWM) limit \cite{Gillespie_1983}), evolution is not reducible to hill-climbing in the fitness landscape. This assumptions is typically violated in large microbial populations, wherein multiple mutants compete for fixation in a process known as clonal interference \cite{Gerrish_1998}. In yet stronger mutation regimes, natural selection should instead be viewed as acting on clouds of genetically related mutants, and evolution as an intermittent succession of stabilization-destabilization of these clouds \cite{Eigen_2007}; in the presence of neutrality, epochal or "punctuated" evolution can also arise through the succession of slow intra-network neutral diffusion and fast inter-network sweeps \cite{Huynen_1996}.
These results raise fundamental questions regarding the dynamical analysis of fitness landscapes: When is flatter better than fitter? Where are the evolutionary attractors in a given landscape with ruggedness and/or neutrality? What quantity do evolving populations optimize? Can we estimate the time scale before another attractor is visited? More simply, can we predict the future trajectory of an evolving population from its current location, the topography of its landscape, and the mutation rate?
In this paper I outline a novel strategy to address these questions in large, asexual populations, for both genotypic (discrete, high-dimensional) and phenotypic (continuous, low-dimensional) fitness landscapes. Inspired by certain analogies with the physics of disordered systems, I show that the selection-mutation process can be understood as a random walk or diffusion in an effective potential—the same kind of dynamics as, say, protein folding kinetics \cite{Bryngelson_1995}. This representation reduces the a priori difficult problem of identifying evolutionary attractors and dominant trajectories in a complex fitness landscape into the much more familiar problem of Markovian metastability \cite{H_nggi_1990}. In contrast with another classical Markovian model of evolution, Gillespie's adaptive walk model \cite{Gillespie_1983}, my approach is not restricted to the SSWM regime and fully accommodates genotypic and/or phenotypic heterogeneity in evolving populations. Moreover, because the effective potential integrates fitness and mutational robustness in a single function on the space of types, it is also more suited to analyze—and predict—the dynamics of a population than the fitness landscape from which it derives.
Results
Selection-mutation dynamics