The dynamics of adapting, unregulated populations and a modified fundamental theorem

A population in a novel environment will accumulate adaptive mutations over time, and the dynamics of this process depend on the underlying fitness landscape: the fitness of and mutational distance between possible genotypes in the population. Despite its fundamental importance for understanding the evolution of a population, inferring this landscape from empirical data has been problematic. We develop a theoretical framework to describe the adaptation of a stochastic, asexual, unregulated, polymorphic population undergoing beneficial, neutral and deleterious mutations on a correlated fitness landscape. We generate quantitative predictions for the change in the mean fitness and within-population variance in fitness over time, and find a simple, analytical relationship between the distribution of fitness effects arising from a single mutation, and the change in mean population fitness over time: a variant of Fisher's 'fundamental theorem' which explicitly depends on the form of the landscape. Our framework can therefore be thought of in three ways: (i) as a set of theoretical predictions for adaptation in an exponentially growing phase, with applications in pathogen populations, tumours or other unregulated populations; (ii) as an analytically tractable problem to potentially guide theoretical analysis of regulated populations; and (iii) as a basis for developing empirical methods to infer general features of a fitness landscape.