Length of adaptive walk on uncorrelated and correlated fitness landscapes

We consider the adaptation dynamics of an asexual population that walks uphill on a rugged fitness landscape which is endowed with a large number of local fitness peaks. We work in a parameter regime where only those mutants that are a single mutation away are accessible, as a result of which the population eventually gets trapped at a local fitness maximum and the adaptive walk terminates. We study how the number of adaptive steps taken by the population before reaching a local fitness peak depends on the initial fitness of the population, the extreme value distribution of the beneficial mutations, and correlations among the fitnesses. Assuming that the relative fitness difference between successive steps is small, we analytically calculate the average walk length for both uncorrelated and correlated fitnesses in all extreme value domains for a given initial fitness. We present numerical results for the model where the fitness differences can be large and find that the walk length behavior differs from that in the former model in the Fréchet domain of extreme value theory. We also discuss the relevance of our results to microbial experiments.

Figure 1

Schematic representation of adaptive walks in a four-dimensional sequence space, starting from the same initial sequence. Arrows represent the shift of the population from a sequence to a more fit sequence one mutation away (refer to text for details).

Figure 2

Variation of average walk length with initial fitness in the linear model on uncorrelated fitness landscapes for various $\kappa$. Simulation points are for $L=1000$ and lines are obtained from (31) for all $\kappa <1$, except the line for $\kappa =3/2$, which is a guide for the eye. Inset: Comparison of the average walk length in the full model $(*)$ and the linear model (+) for $(1/\kappa )ln(1+\kappa f_0)=2$. The solid line shows the walk length expressions, (8) (bottom) and (9) (top), for the greedy adaptive walk and the random adaptive walk, respectively.

Figure 3

Plot showing the variation of average walk length with initial fitness for the linear model on correlated fitness landscapes for various $B$ when $\kappa =-1$. Theoretical predictions, (75) and (77) (lines), are compared to simulation data (symbols). Inset: Plot showing the rate function for $\kappa =-1$ obtained using (67) and (70) (symbols) and the analytical formulas, (72) and (73).

Figure 4

Plot showing the variation of average walk length with initial fitness for the linear model on correlated fitness landscapes for various $B$ when $\kappa =2/3$ and $L_B=1000$. The theoretical prediction, (80) (lines), is compared to the simulation data (symbols).

Figure 5

Plot showing the simulation data for the walk length distribution and the index of dispersion (inset) of the walk length for various $\kappa$ when $L=1000$ using the full model on uncorrelated fitness landscapes. Inset: $(1/\kappa )ln(1+\kappa f_0)=2$.