The time scale of evolutionary trajectories

A fundamental question in biology is the following: what is the time scale that is needed for evolutionary innovations? There are many results that characterize single steps in terms of the fixation time of new mutants arising in populations of certain size and structure. But here we ask a different question, which is concerned with the much longer time scale of evolutionary trajectories: how long does it take for a population exploring a fitness landscape to find target sequences? Our key variable is the length, $L$, of the sequence that undergoes adaptation. In computer science there is a crucial distinction between problems that require algorithms which take polynomial or exponential time. The latter are considered to be intractable. Here we develop a theoretical approach that allows us to estimate the time of evolution as function of $L$. We show that adaptation on many fitness landscapes takes time that is exponential in $L$, even if there are broad selection gradients and many targets uniformly distributed in sequence space. These negative results lead us to search for specific mechanisms that allow evolution to work on efficient time scales. We study a regeneration process and show that it enables evolution to work in polynomial time