Gokhale CS, Iwasa Y, Nowak MA, Traulsen A.
The pace of evolution across fitness valleys.
Journal of Theoretical Biology
The problem of evolution as I see it is that of a mechanism by which the species may continually find its way from lower to higher peaks [. . .]. In order that this may occur, there must be some trial and error mechanism on a grand scale [. . .]. To evolve, the species must not be under strict natural selection. Is there such a trial and error mechanism? (Wright, 1932).
Wright had asked a very interesting question in theoretical population dynamics. How a population which is stuck at one of the many possible local fitness maxima evolve to the global fitness maximum. Fisher, who thought in a much more geometrical way, envisioned that any local fitness maxima would be a point on the slope of another adaptive hill in a higher dimension. Thus given enough time, a population would finally make it to the global maximum. Wright was more in favour of random drift. He believed that by drift populations could reach the foots of other hills and then selection could take over so as to drive the populations uphill towards the peak. In his Shifting Balance Theory, Wright postulates that a number of sub-populations could explore the fitness landscapes and as the number of sub-populations increases, the chance that one of them finds the global optimum also increases (Wright, 1932). Once at the global optimum, that sub-population will outcompete all the other types of that species and the species as a whole will have reached the global adaptive peak (Ridley, 1996).
We address the question using the mutational landscape. Mutations during individual reproduction are either ultimately lost (when the mutants go extinct) or fixed in a population (when the mutants take over). The probability that a mutation reaches fixation increases with the relative fitness of the mutant. As the fitness landscape is made up of mutants which are one mutational step away from each other, we can ask the question how long it takes until a number of mutations reach fixation. Of course, this depends on the fitness of the mutants. But in addition, the order of mutations is crucial:
If each mutation needs another mutation as a prerequisite to occur, evolution occurs on a single path or ridge in fitness landscape.
If the order of mutations is arbitrary, then there are many paths possible along which the mutations are accumulating and evolution typically proceeds faster.
We address the pace of evolution in the two scenarios (see Fig. 2.1) and show how the size of the population affects the way a population evolves. The developed theory allows us to ask when evolution occurs faster on a narrow ridge or through a broad valley with disadvantageous intermediate mutations.
The framework developed here can serve as a reference case for evolution in real fitness landscapes, as it can be easily extended to incorporate the complexity and variation seen in experimental studies, Fig. 2.2.
This way of approaching the problem lets us explore the different regimes of mutation rate.
For small mutation rates the population evolves by a process where the mutations occur one after the other, which has been termed periodic selection (Atwood et al., 1951) and theoretically described as the strong- selection weak-mutation regime (Gillespie, 1983, 2004 (2nd edition) (see Fig. 2.3 (a)).
For intermediate mutation rates the population does not move from mutational step to the next as a whole. Instead many mutants are present in the population at the same time. This phenomenon of competition amongst multiple mutants termed as clonal interference or stochastic tunneling has been a subject of in-depth theoretical and experimental studies. (Gerrish and Lenski, 1998; Elena et al., 1998; Elena and Lenski, 2003; Iwasa et al., 2004; Park and Krug, 2007). This process is of importance in studies related to cancer initiation (Iwasa et al., 2004; Michor et al., 2004; Beerenwinkel et al., 2007a,b; Bozic et al., 2010) (see Fig. 2.3 (b)).
For high mutation rates the system we can use ordinary differential equations to capture the behavior of the system (see Fig. 2.3 (c)).
This approach has been used to investigate how different strains of pathogens can evolve from one another (Alexander and Day, 2010) or how long does it take for a population to acquire complex adaptive traits (Lynch, 2010a,b).
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