Mutation-bias driving the evolution of mutation rates

In classic game theory, we are often faced with multiple potential equilibria between which to select with no unequivocal way to choose between these alternatives. If you’ve ever heard Artem justify dynamic approaches, such as evolutionary game theory, then you’ve seen this equilibrium selection problem take center stage. Natural selection has an analogous ‘problem’ of many local fitness peaks. Is the selection between them simply an accidental historical process? Or is there a method to the madness that is independent of the the environment that defines the fitness landscape and that can produce long term evolutionary trends?

Two weeks ago, in my first post of this series, I talked about an idea Wallace Arthur (2004) calls “developmental bias”, where the variation of traits in a population can determine which fitness peak the population evolves to. The idea is that if variation is generated more frequently in a particular direction, then fitness peaks in that direction are more easily discovered. Arthur hypothesized that this mechanism can be responsible for long-term evolutionary trends.

A very similar idea was discovered and called “mutation bias” by Yampolsky & Stoltzfus (2001). The difference between mutation bias and developmental bias is that Yampolsky & Stoltzfus (2001) described the idea in the language of discrete genetics rather than trait-based phenotypic evolution. They also did not invoke developmental biology. The basic mechanism, however, was the same: if a population is confronted with multiple fitness peaks nearby, mutation bias will make particular peaks much more likely.

In this post, I will discuss the Yampolsky & Stoltzfus (2001) “mutation bias”, consider applications of it to the evolution of mutation rates by Gerrish et al. (2007), and discuss how mutation is like and unlike other biological traits.

Mutation bias and natural selection

SDE2fig1In the figure at right, adapted from Yampolsky & Stoltzfus (2001), we have a homogeneous population of genotype ab, with fitness 1. Genotypes aB and Ab both have higher fitnesses of 1+s_1 and 1+s_2, respectively. In this case, the mutation rates of a to A (\mu_1) and b to B (\mu_2) become important to whether aB or Ab takes over the population. The two cases of mutation that they consider are given by conditions (1) and (2).

Furthermore, Stoltzfus (2006) showed that on a landscape with sufficiently many local fitness peaks — like the rugged NK model that Artem discussed before — such a mutation bias can, in fact, produce long term evolutionary trends.

The great thing about this idea is that it doesn’t pit mutation bias against natural selection. When that happens, mutation bias tends to lose. That’s the weakness of all the theories of neutral evolution (Kimura, 1984; Nei, 2013) and all its modern descendants (nearly neutral evolution (Ohta, 1992), constructive neutral evolution (Gray et al., 2010), the mutation-bias model of Waxman & Peck (2003) come to mind; there are many others). This idea is really in a different category: mutation bias orients the population toward one fitness hill or another, and natural selection gets us up the hill. Natural selection is the motor but bias is the driver.

But so much is unknown: the actual shape of empirical fitness landscapes, how they shift over time, the stability of G- matrices over time, etc. Can this ever stop being a “good idea” and become a body of mathematics that provides further insight and explanations of important biological phenomena?

Evolution of mutation rates

I think a solution can be found through the literature on the evolution of mutation rates, especially the key paper by Gerrish et al. (2007). Artem will describe the paper in more detail in another post, but basically Gerrish et al. (2007) found a regime in which mutation rates increase indefinitely, through natural selection, until the population melts down. This is shown in the figure below.

Figure from Gerrish et al. (2007)

Figure reproduced from Gerrish et al. (2007). The left panel shows data from numerical simulations of equations used in the paper. The right panel shows data from agent-based simulations. Red lines represent mutation rate, which first slowly increases, then rapidly increases. Blue lines represent fitness, which first steadily increases, then rapidly decreases. Pink and green lines on the right figure are theoretical predictions of fitnesses and mutation rates.

In the above figure, we see a constant increase of mutation rate until a high mutation rate induced cataclysm that rapidly degrades the population’s fitness. The mechanism, expressed in words, is simple. In a clonal population with perfect linkage among fitness genes and genes responsible for mutation rate, the highest fit member of the next generation will come from high mutation rates rather than lower mutation rates, even though on average, higher mutation rates cause lower fitness (through mutation burden and because most mutations are detrimental). Thus, higher mutation rates constantly hitchhike to fixation, until the population has such a high mutation rate that it suddenly goes extinct.

There are several conditions on this, of course, but they are biologically realistic. The most important thing is that mutations should increase mutation rate rather than decrease it. This way, more mutators are produced than non-mutators, which is a sufficient (but not necessary) condition for the next selective sweep to come out of the mutator group rather than the non-mutator group.

As an aside, Artem and I (Xue et al., 2015) were able to rescue this march to infinity by a mechanism called “fidelity drive”. Since strains with low mutation rates have a hard time changing their mutation rates, they are sort of “stuck” in low-mutation land. Strains with high mutation rates, however, have a low, but not impossible, chance of finding large mutations that quickly reduce their mutation rate. The stickiness of low mutation rates manages to stop the evolution to infinity that Gerrish et al. (2007) found, and even produces mutation rate patterns that looks pretty close to empirically observed patterns. However, natural selection had nothing to do with it: again we used a mechanism about how mutation rates are produced, rather than how it is selected.

Mutation rate and other traits

Studying this paper really set me to thinking: what sets mutation rate apart from any other trait? Why can such long term trends come about in mutation rates that are driven by natural selection, but not by the specifics of the environment?

The obvious answer is, of course, that natural selection does not act on mutation rate directly. Natural selection acts only on the phenotype that is produced by mutation. Thus, mutation rates undergo what has been called “second order” selection (Tenaillon et al., 2001), hence, even if there were an optimal mutation rate, natural selection cannot be expected to move mutation rate towards that optimum. This is unlike what Kimura (1967) thought in the 60’s. Sniegowski et al. (2002) provide a good summary of this literature from the previous century..

But is this true? Is it true that traits such as evolvability (Woods et al., 2011) — in the traditional biological sense of the word, not Valiant’s (2009) variant that Artem discussed before — and mutation rate are under a different regime of natural selection than something like body size?

I don’t think so. I think most traits are under “second-order” selection. Moreover, I think the mathematics developed in Gerrish et al. (2007) can capture a large range of evolutionary phenomena, and that long term evolutionary trends can happen for most traits in the same way it can happen for mutation rates (the extinction at the end is optional). I think this type of model captures the same intuition as “developmental-bias” and “mutation-bias” as developed by Arthur and Stoltzfus, but it does so in a more rigorous and insightful way.

I will outline how this might be in the next post.


Arthur, W. (2004). The effect of development on the direction of evolution: toward a twenty‐first century consensus. Evolution & Development, 6(4): 282-288.

Gerrish, P. J., Colato, A., Perelson, A. S., & Sniegowski, P. D. (2007). Complete genetic linkage can subvert natural selection. Proceedings of the National Academy of Sciences, 104(15): 6266-6271.

Gray, M. W., Lukes, J., Archibald, J. M., Keeling, P. J., & Doolittle, W. (2010). Irremediable complexity? Science, 330(6006): 920-921.

Kimura, M. (1967). On the evolutionary adjustment of spontaneous mutation rates. Genetical Research, 9(1): 23-34.

Kimura, M. (1984). The neutral theory of molecular evolution. Cambridge University Press.

Nei, M. (2013). Mutation-driven evolution. OUP Oxford.

Ohta, T. (1992). The nearly neutral theory of molecular evolution. Annual Review of Ecology and Systematics, 23, 263-286.

Sniegowski, P. D., Gerrish, P. J., Johnson, T., & Shaver, A. (2000). The evolution of mutation rates: separating causes from consequences. Bioessays, 22(12): 1057-1066.

Stoltzfus, A. (2006). Mutation-biased adaptation in a protein NK model. Molecular Biology and Evolution, 23(10): 1852-1862.

Tenaillon, O., Taddei, F., Radman, M., & Matic, I. (2001). Second-order selection in bacterial evolution: selection acting on mutation and recombination rates in the course of adaptation. Research in Microbiology, 152(1): 11-16.

Valiant, L.G. (2009). Evolvability. Journal of the ACM, 56(1): 3.

Waxman, D., & Peck, J. R. (2003). The anomalous effects of biased mutation. Genetics, 164(4): 1615-1626.

Woods, R. J., Barrick, J. E., Cooper, T. F., Shrestha, U., Kauth, M. R., & Lenski, R. E. (2011). Second-order selection for evolvability in a large Escherichia coli population. Science, 331(6023): 1433-1436.

Xue, J. Z., Kaznatcheev, A., Costopoulos, A., & Guichard, F. (2015). Fidelity drive: A mechanism for chaperone proteins to maintain stable mutation rates in prokaryotes over evolutionary time. Journal of Theoretical Biology, 364, 162-167.

Yampolsky, L.Y., & Stoltzfus, A. (2001). Bias in the introduction of variation as an orienting factor in evolution. Evolution & Development, 3 (2), 73-83 PMID: 11341676

4 Responses to Mutation-bias driving the evolution of mutation rates

  1. Jake Scott says:

    Interesting post. I have a couple thoughts.

    First, this is closely related to the thesis work of a friend from Max Planck called Benedikt Bauer, who wrote on the topic of mutation bias here:

    and Benedikt may be interested.

    Second, I wonder how you think about different kinds of mutations? There is a growing body of literature suggesting that some mutations are beneficial (driver mutations) and some are deleterious (passengers), the latter of which have been suggested to be slightly deleterious to fitness. The work of a friend, Chris McFarland, has centered on this.

    Here’s chris:

    Anyways, I look forward to your next post, and to reading the paper you and Artem coauthored.

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  3. astoltzfus says:

    Julien, I’m glad to see that you are writing about this topic. I have a small correction. Stoltzfus and Yampolsky 2001 actually *do* address development, the section on “developmental biases in variation.” We simply provide a verbal argument that maps a kind of developmental bias (due to phenocopies) directly onto the genetic model, so that the model is now, under this interpretation, a model of developmental bias. Wallace Arthur, who was the editor of the paper, was influenced somewhat by these arguments, and cited them subsequently.

  4. Pingback: Cataloging a year of blogging: complexity in evolution, general models, and philosophy | Theory, Evolution, and Games Group

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