Reactionary fringe meets mutation-biased adaptation. 6. What "limits" adaptation?

This is part of a continuing series of posts by Arlin Stoltzfus on the role of mutation as a dispositional factor in evolution. In this post Arlin discusses the role of adaptation and what determines the pathway that it will take over time. Is it true that populations will always adapt quickly to any change in the environment? (Hint: no it isn't!) Click on the links in the box (below) to see the other posts in the series.

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Reactionary fringe meets mutation-biased adaptation.
6. What "limits" adaptation?
by Arlin Stoltzfus
According to the hatchet piece at TREE, theoretical considerations dictate that biases in variation are unlikely to influence adaptation, because this requires small population sizes and reciprocal sign epistasis.

Yet, we have established that mutation-biased adaptation is real (see The empirical case and Some objections addressed). If theoretical population genetics tells us that mutation-biased adaptation is impossible or unlikely, what is wrong with theoretical population genetics?

Adaptation, before Equilibrium Day

Reactionary fringe meets mutation-biased adaptation
Introduction
1. The empirical case
2. Some objections addressed
3. The causes and consequences of biases in the introduction process
4. What makes this new?
5. Beyond the "Synthesis" debate
    -Thinking about theories
    -Modern Synthesis of 1959
    -How history is distorted
    -Taking neo-Darwinism
      seriously
    -Synthesis apologetics
6. What "limits" adaptation?
7. Going forwardActually, the arguments in TREE's hatchet piece do not represent the best that evolutionary theory has to offer. They emerge from a tradition of treating evolution as an idealized process in which the best form emerges deterministically. This approach-- "the false view of evolution as a process of global optimizing" (see Lewontin in the left sidebar of SandWalk)-- simplifies things, and leads to powerful analytical methods. Levin, et al, (2000) mock this approach hilariously:

To be sure, the ascent and fixation of the earlier-occurring rather than the best-adapted genotypes due to this bottleneck-mutation-rate mechanism is a non-equilibrium result. On Equilibrium Day deterministic processes will prevail and the best genotypes will inherit the earth

Clearly, Equilibrium Day has not arrived. For instance, in the set of 35 pairs of populations studied by Storz, et al. (2019), 15 of the highland populations do not show detectable changes in hemoglobin oxygen affinity.  Are we to imagine that, in 20 of the highland populations, hemoglobin adaptation has gone to equilibrium, reaching adaptive perfection, and in the other 15 cases, there has not been time for the first change?

In reality, for various reasons, we expect natural adaptation to be a process that does not go to equilibrium, but whose outcome reflects the initial state and subsequent kinetics, including kinetic biases imposed by mutation.

Theoreticians may refer to such influences as "limits to adaptation," but we must be perfectly clear that this concept of limits is explanatory, not causal. There are no biological processes preventing the process of adaptation from being adaptation: it is what it is. Beneficial mutations feed adaptation, even when their occurrence is rare.

To make our consideration of "limits" concrete, I will refer to the conditions of altitude adaptation in Storz, et al. (2019). Why would we find that a hemoglobin has undergone some change that is merely convenient, but not the best possible change?

First, let us consider the effect of population size, via the quantity uN. If every mutation has a rate u such that uN >> 1, better alleles will win even if they start out at lower frequencies, thus the efficacy of biases in mutation will be reduced. If uN << 1, new mutations will be appearing at long intervals, giving mutationally favored alleles an advantage.  

This theory applies to idealized panmictic populations.  When the demographics include spatial structure, mutational effects are amplified (Ralph and Coop, 2015).

What is uN for the case of hemoglobin adaptation? Consider a hemoglobin gene with 450 nucleotide sites, and 3 x 450 = 1350 mutant neighbors, thus about 1000 non-synonymous mutations. Given a mutation rate of 10^-10 to each neighbor, the total rate is 10^-7. Suppose that 100 of the 1000 mutations improve hemoglobin affinity and thus are beneficial.

Given these numbers, and given a highland population of N = 10⁶ birds (an over-estimate, thus a conservative estimate), uN = 0.0001 for each beneficial mutation, and uN = 0.01 for the set of 100, i.e., a new affinity-enhancing mutation is introduced every 100 generations. Thus, uN is small.

Now, consider differences in mutation rate.  For instance, given a 4-fold transition rate bias, and remembering that there are twice as many transversions, the rate of introduction for beneficial transversions would be u_vN = 0.0033 and for transitions, u_iN = 0.0067, summing to (u_v + u_i)N = 0.01.

Context effects are an additional source of heterogeneity, e.g., previously we saw a 50-fold range of rates for just 11 nucleotide mutations in MacLean, et al. In our hypothetical example with 100 affinity-improving mutations, some will be introduced at rates an order of magnitude higher or lower than the average of uN = 0.0001. The distribution might look something like this (transversions in black, transitions in red), with a scale covering 3 or 4 orders of magnitude.

A hypothetical distribution of rates of beneficial transversions (black) and transitions (red).

What are the implications? No one knows for sure-- this requires more theoretical work. But it seems to me that, even with population sizes 100 or 1000 times larger than N = 10⁶, we will see mutational effects.

To summarize, we expect uN to be small enough for effects of mutation bias to be important.

We reached this conclusion by considering the specific case of hemoglobin in altitude adaptation, but adaptation may often involve "small" populations, for generic reasons. A population that has invaded a new niche will tend to be small, because the population is new and less well adapted. When pre-adaptive variation enables variant individuals to explore new resources (e.g., higher altitudes, previously toxic foods), the act of exploiting the new resource will tend to segregate the favored individuals from the larger population.

What about negative sign epistasis? Yampolsky and Stoltzfus used reciprocal sign epistasis in their simulation model in order to impose an artificial end-point, so that the population would move left or right and then stop. However, if we have (for instance) an origin-fixation process with beneficial fixations in an infinite genotypic space with no epistasis, the system will just continue moving forever upward in fitness via mutationally favored changes, following Eqn 2:

R₂ / R₁ = (u₂ / u₁)(s₂ / s₁)

If the space is not infinite, but just very large, then this process may go on for a long time.

Thus, reciprocal sign epistasis is not a requirement. One just needs to stop dreaming of Equilibrium Day.

Indeed, the simplest effect we can imagine is just that time is very short, e.g., as suggested earlier, this might explain the lack of change in some upland populations in Storz, et al. (2019). In some other cases we discussed, the scale of adaptation is clearly limited, e.g., experimenters may halt an experiment as soon as the population experiences a jump in fitness. No opportunity is provided for better mutations to arise and take over, much less for various combinations to be tested. In natural or experimental cases of resistance to a toxin or drug, resistant isolates may be identified for study on the basis of having acquired initial resistance, without any requirement for achieving long-term adaptive perfection.

Another condition that will increase the impact of biases due to mutation is diminishing-returns epistasis, in which the effects of two mutations overlap such that the impact of the second is reduced once the first has occurred. For instance, the fitness effect of an affinity-enhancing mutation may be reduced by the presence of another affinity-enhancing mutation. When Equilibrium Day arrives, even small effects will be incorporated into the perfect hemoglobin, but before Equilibrium Day, there may not be enough time.

Diminishing-returns epistasis can limit adaptation in a different way. Consider the potential for altitude-adapting increases in oxygen flow, influenced by the oxygen affinity of hemoglobin, hemoglobin blood concentration, concentrations of effectors, capillary density, lung volume, breathing rate, and so on (e.g., consider the bar-headed goose). To the extent that additive effects predominate, a mutation in one component that increases oxygen supply to tissues will tend to decrease the benefit of other mutations. The general expectation is that adaptation will tend to occur via the most evolvable component, until some other component becomes the most evolvable.

This suggests a second explanation for the observation that some highland populations do not have adaptive hemoglobin changes: perhaps the early steps in adaptation happened in other components, reducing the benefits of hemoglobin adaptation and thus slowing the rate of change in hemoglobin.

Beyond the equilibrium paradigm

The classic focus of theoreticians on equilibria aligns with a classic explanatory paradigm that Sober (1984) literally calls "equilibrium explanation" (Ch. 5 of The Nature of Selection): an observed state is explained as if it were the final end-state of a process, achieved regardless of history. Thus, Reeve and Sherman (1993) argue that, in the adaptationist research program, the target of explanation is "phenotype existence," and the form of an explanation is to show that observed states are more adaptive than neighboring states, "irrespective of the precise historical pathways leading to their predominance."

The inadequacy of this paradigm of perfection has given rise to a modified paradigm of imperfection invoking "chance," "contingency" and "limits" or "constraints" to explain why the end-states are not perfect. Again, these concepts are explanatory, not causal. The attempt to link "constraint" with specific causal mechanisms is futile (see Antonovics and van Tienderen). Constraints do not cause outcomes, but explain non-outcomes; chance is not a cause, but merely indicates a departure from the ideal of determinism; contingency signals a departure from finalism (in which outcomes are pre-determined). Chance, constraints, and contingency are not specific to evolution: they are generic concepts for explaining departures from an ideal, e.g., how the bridge collapsed, how the dog got loose, and so on.

Let us apply this paradigm to Rokyta, et al. (2005), who adapted 20 replicate phage populations, finding that the 4th-most-fit variant (row 4 above) was found the most often (n = 6).

The equilibrium paradigm applies to token cases, thus we must invoke it 20 times. In the single case in which the most-fit alternative (row 1) emerged, adaptive perfection is achieved. In this case, we might invoke the Modern Synthesis of 1959 to hypothesize that selection chose the best variant from an abundant gene pool. In the other 19 cases, there was a constraint, because the best variant was absent, and by chance, some other variant was present; the outcome was contingent on which variant was present initially.

Although this paradigm can be applied to Rokyta, et al. (2005), the results are obviously unsatisfying.

In molecular evolution, we have the Dayhoff paradigm, in which evolution is a process of character-state change with a spectrum of propensities represented by quantities, e.g., a matrix of amino acid replaceability. The quantities may be instantaneous rates or expected differences (sometimes extrapolated out to great distances). The targets of evolutionary explanations are the asymmetries in these quantities. That is, in molecular evolution, we often ask questions or propose hypotheses comparing two types of change X and Y, e.g., transitions vs. transversions.

Within the equilibrium paradigm, the Rokyta results show that chance, constraints and contingency typically prevent perfection. Within the Dayhoff paradigm, the same results reveal short-term evolutionary propensities, and lead to questions about why some types of changes happen more often than others.

Note again that these are explanatory paradigms. Of course we have causal theories in evolution, but we typically invoke them within some story that satisfies our sense that a thing has been explained. To apply a causal model within the Dayhoff paradigm, we might consider the hypothesis that the 4th-most-fit variant happened more often due to a higher rate of mutational introduction, applying an origin-fixation model of kinetics. When we know why the 4th-most-fit variant emerges more than the most fit variant (why X happens more than Y), we have a satisfying explanation within the Dayhoff paradigm.

The same kind of paradigm can be applied to propensities in the evolution of discrete morphological characters, as in Alberch and Gale (1985).

However, the old paradigm was never meant to convey the richness of a world of propensities, leading to statements like this from TREE's hatchet piece:

"These studies therefore only exemplify how historical contingency and mutational history interact with selection during adaptation to novel environments."

Indeed, the old paradigm predates evolutionary biology: creationists also focused on explaining existence in token cases, given their theory that each species emerged by a special process of creation resulting in a state of perfection. The Dayhoff paradigm is more appropriate for science because it focuses on general causes rather than token causes, and it is more appropriate for evolution because it focuses on change instead of end-states.

Moving on

In this post, we have addressed an odd question: what "limits" adaptation relative to an imaginary, idealized world? Why does today look different from Equilibrium Day, when the best genotypes take their rightful places?

In general, today does not look like Equilibrium Day because evolution has slow kinetics. We know this because we can use the distribution of evolved characters to infer the path of history, whereas in systems that have reached equilibrium, history is erased. That is, to the extent that phylogeny inference works on a set of characters, we know that those characters are not at equilibrium. Of course, there may be local quasi-equilibria subject to fast kinetics: these are like small swirling eddies on the surface of a wide slow river.

More specifically, we have considered what might "limit" adaptation in terms of the effects of uN, mutation-rate heterogeneity, spatial structure, time-scale, and the changing focus of adaptation. None of these considerations is definitive: more research is needed to understand the conditions of natural adaptation. The effects of mutation bias may prove useful in probing these conditions.

We also considered an alternative to the classic paradigm in which the target of an evolutionary explanation is the state of an evolved system, considered as an end-product. In the alternative Dayhoff paradigm, the focus is on patterns of change, and there is no need to invoke constraints, chance, and contingency, but only to invoke the causal factors that make one kind of outcome more likely than another.

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