Complexity of evolutionary equilibria in static fitness landscapes

A fitness landscape is a genetic space – with two genotypes adjacent if they differ in a single locus – and a fitness function. Evolutionary dynamics produce a flow on this landscape from lower fitness to higher; reaching equilibrium only if a local fitness peak is found. I use computational complexity to question the common assumption that evolution on static fitness landscapes can quickly reach a local fitness peak. I do this by showing that the popular NK model of rugged fitness landscapes is PLS-complete for K ≥ 2; the reduction from Weighted 2SAT is a bijection on adaptive walks, so there are NK fitness landscapes where every adaptive path from some vertices is of exponential length. Alternatively – under the standard complexity theoretic assumption that there are problems in PLS not solvable in polynomial time – this means that there are no evolutionary dynamics (known, or to be discovered, and not necessarily following adaptive paths) that can converge to a local fitness peak on all NK landscapes with K = 2. Applying results from the analysis of simplex algorithms, I show that there exist singlepeaked landscapes with no reciprocal sign epistasis where the expected length of an adaptive path following strong selection weak mutation dynamics is e ) even though an adaptive path to the optimum of length less than n is available from every vertex. The technical results are written to be accessible to mathematical biologists without a computer science background, and the biological literature is summarized for the convenience of non-biologists with the aim to open a constructive dialogue between the two disciplines. ∗[email protected] 1 ar X iv :1 30 8. 50 94 v1 [ qbi o. PE ] 2 3 A ug 2 01 3 Nothing in biology makes sense except in the light of evolution. Theodosius Dobzhansky