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There are two simple caricatures of evolutionary dynamics: the phenotypic caricature focuses on continuous and predictable selection on variability of quantitative traits, whereas the genotypic caricature focuses on discrete, stochastic mutations. Although the apparent contradictions between these pictures were reconciled long ago, our quantitative understanding of the interplay between them is still surprisingly primitive. Indeed, even the simplest models of the dynamics of large asexual populations in which many alleles and many new mutations contribute to the evolving fitness have resisted solution. The PNAS paper by Hallatschek (1) is a substantial advance in the development of the mathematical methods needed to analyze these and more complex models. What is the source of the basic difficulty? The Fundamental Theorem of Natural Selection describes, quantitatively, how a diverse population subject to selective pressure evolves. It states that the rate of change of the average of a quantitative trait is proportional to the hereditable part of the variance of that trait times the strength of selection for it. But what determines the variance of a trait that is under selection? Initially, the frequencies of the various alleles that contribute to it: if there are many of these, the sum of their positive and negative contributions will give rise to a roughly normal bell-shaped distribution of the fitness. However, as selection proceeds, some of the alleles will rise to fixation and others die out. Thus, under steady selection, after a modest time, only the beneficial alleles will remain and there will be no variance at all: the increase in fitness then comes to a grinding halt. But of course there can be new beneficial mutations. If only one such mutation arose at a time, it would sweep through the population before another occurred: this is the simple population genetics caricature of discrete mutation-selection events. …

Leading the dog of selection by its mutational nose