Speed of Evolution

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Gokhale CS, Iwasa Y, Nowak MA, Traulsen A.

The pace of evolution across fitness valleys.

Journal of Theoretical Biology

Aug;259(3):613–20, 2009

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).

Single path and Hypercube. Hypothetical fitness paths where only a fixed sequence of mutation can lead to the state of higher fitness (single path) or a multitude of paths lead to the ultimate state (hypercube). The intermediate states are all assumed to have the same fitness s as compared to the fitness of the initial state considered to be 1 and the final state as r_d..

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:

  1. If each mutation needs another mutation as a prerequisite to occur, evolution occurs on a single path or ridge in fitness landscape.

  2. 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.

Examples of experimentally constructed fitness landscapes. The images collated from: (a) [Weinreich et al. (2006)](#weinreich:2006): Mutational paths in the β lactamase gene conferring resistance to bacteria from β lactam antibiotics where were found to be viable.(b) [Lozovsky et al. (2009)](#lozovsky:2009): The major inferred pathways for the evolution of pyrimethamine resistance. (c) [Lee et al. (1997)](#lee:1997): All paths between the 5srRNA sequences of *Vibrio proteolyticus* and *V. nereis* at the three positions where they differ..

This way of approaching the problem lets us explore the different regimes of mutation rate.

Regimes of mutation rates. (a) If mutation rates are very low then the system is monomorphic most of the time, occasionally a mutation occurs and it can either go extinct or reach fixation. This scenario can be captured analytically. (b) For intermediate mutation rates multiple mutants can coexist and the situation is difficult to characterise analytically. (c) For high mutation rates the dynamics can be approximated by differential equations again yielding tractable results..

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).

Link to Paper

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M. Ridley. Evolution. Wiley-Blackwell, second edition, August 1996.

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E. R. Lozovsky, T. Chookajorn, K. M. Brown, M. Imwong, P. J. Shaw, S. Kamchonwongpaisan, D. E. Neafsey, D. M. Weinreich, and D. L. Hartl. Stepwise acquisition of pyrimethamine resistance in the malaria parasite. Proc. Natl. Acad. Sci., 106(29):12025–12030, 2009.

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N. Beerenwinkel, T. Antal, D. Dingli, A. Traulsen, K. W. Kinzler, V. E Velculescu, B. Vo- gelstein, and M. A. Nowak. Genetic progression and the waiting time to cancer. PLoS Comput. Biol., 3:e225, 2007.

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I. Bozic, T. Antal, H. Ohtsuki, H. Carter, D. Kim, S. Chen, R. Karchin, K. W. Kinzler, B. Vogelstein, and M. A. Nowak. Accumulation of driver and passenger mutations during tumor progression. Proc. Natl. Acad. Sci. U.S.A., Early Edition, September 2010.

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M. Lynch. Scaling expectations for the time to establishment of complex adaptations. Proc. Natl. Acad. Sci. USA, Early edition, 2010.

Speed of Evolution was last modified on Chaitanya S. Gokhale