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Tuesday, July 02, 2019

Reactionary fringe meets mutation-biased adaptation. 3. The causes and consequences of biases in the introduction process

This is the fourth in a series of guest posts by Arlin Stoltzfus on the role of mutation as a dispositional factor in evolution.

Reactionary fringe meets mutation-biased adaptation. 3. The causes and consequences of biases in the introduction process
by Arlin Stoltzfus

As discussed previously, mutation-biased adaptation occurs in the laboratory and in nature. In the cases that have been examined, modest several-fold mutational biases have modest several-fold effects on the changes involved in adaptation.

Reactionary fringe meets mutation-biased adaptation
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

    -Synthesis apologetics
6. What "limits" adaptation?
7. Going forward
How can this happen? Classical thinking says that mutation is a weak pressure easily overcome by selection. This "opposing pressures" argument was invoked by Fisher (1930), Haldane (1933) and Wright (1931), as well as Huxley, Ford, Stebbins, Simpson and others. On this basis, it is assumed that the effects of mutation bias will be seen only in neutral evolution, where the opposing pressure of selection is absent, or with unusually high mutation rates.

Yet, Yampolsky and Stoltzfus (2001) showed that biases in variation can influence evolution in a completely different way that does not require absolute constraints, neutral evolution, or high mutation rates.

Climbing Mount Probable

Imagine, as an analogy for evolution, a climber operating on a rugged landscape (figure). A human climber would plan a path to the peak, but we must imagine a blind climber operating with strictly local rules. Suppose the robotic climber moves by a two-step proposal-acceptance mechanism. In the "proposal" or "introduction" step, the climber reaches out with one of its limbs to grasp a nearby hand-hold or foot-hold. Each time this happens, there is a chance that the climber will use this point of leverage to move.

If we bias the acceptance step, such that higher points of leverage are more likely to be chosen, this will cause the climber to ascend, to go upward.

What if we also impose a bias on the proposal step? Suppose that the robot has longer or more active limbs on the left side, and therefore grasps more points of leverage on the left than on the right. If so, the joint probability of proposal and acceptance is greater on the left, so that the trajectory of the climber will be biased, not just upwards, but to the left. If the landscape is rough, the climber will tend to get stuck on a local peak upwards and leftwards of the starting point. If the landscape is a smooth hemisphere, the climber will spiral upwards and leftwards, ascending to the single summit.

A population-genetic model

To demonstrate a principle, one chooses the simplest case. In the simplest case, the climber has only 2 choices: up and to the left, up and to the right. The non-trivial case is when one course is more strongly advantageous, and the other is more strongly favored by mutation. That is, we are imagining an evolving system at the point where it faces 2 beneficial options (figure: black arrows), with one-- here the leftward mutation-- having a higher mutation rate u2 > u1, but a lower fitness benefit s2 < s1.

Let the leftward bias in mutation be B = u2 / u1, with the upward bias in selection coefficients K = s1 / s2.  How often will evolution climb to the left, instead of the right?  How strong is the influence of B vs K, and how do they depend on the absolute size of mutation rates and selection coefficients?

The left figure shows the results of simulating this model, with the bias in outcomes (left vs. right) as a function of the bias in mutation, and considering population size N from 100 to a million, B from 1 to 1000, and fixed values of s1 = 0.02 and s2 = 0.01, so that K = 2. For full details, see Yampolsky and Stoltzfus (2001).

All of the lines are going up, which means that the bias in outcomes increases with the bias in mutation.  When a line goes above y = 1, the mutationally favored outcome prevails.  For N = 100 or N = 1000, the bias in outcomes is approximately B / K (dashed grey line), the outcome expected from origin-fixation dynamics.

Traditional "forces" thinking says that population genetics is explained by the action of forces or pressures. More pressure means more effect, right? So, is this a matter of the pressure of mutation overcoming weak selection? No, not at all. Selection is strong in most of these population: Ns ranges from about 1 to about 10,000.

In fact, the effect depends on the smallness of the mutation rate. The figure at right shows bias in outcomes as a function of absolute mutation rate for N = 1000 and K = 2 with B values of (from top to bottom) 64, 16, 4 and 1.  As mutation rate increases to the right, the effect of mutation bias diminishes because, to the extent that both alleles have been introduced repeatedly (due to the higher rate) and have escaped stochastic loss, selection tends to ensure that the fitter allele prevails, even if it starts out at a lower frequency.

By contrast, as mutation rate decreases going from right to left, the effect of the bias increases toward B / K. That is, under limiting conditions, an invariant relationship emerges. This relationship is the expectation of an origin-fixation process for modestly beneficial alleles, in which the rate of evolution is proportional to both Nu (rate of origin) and to 2s (probability of fixation), so that the bias in rates is proportional to B = u2 / u1, and inversely proportional to K = s1 / s2, because the relative origin-fixation rates are (Eqn 2 of Yampolsky and Stoltzfus).

R2 / R1 = 4Nu2s2 / (4Nu1s1) = (u2 / u1)(s2 / s1)

The sushi conveyor and the buffet

At the opposite extreme of the origin-fixation regime is the "gene pool" regime, in which all variants are guaranteed to be present at low frequencies in the initial population. This is a crucial distinction, because the gene-pool regime is the source of the expectations of classical population genetics.

The difference can be explained intuitively by comparing two styles of self-service restaurant-- the buffet and the sushi conveyor. At the buffet, we begin with an abundance of choices ready at hand, and fill our plate with the desired amount of each; at the sushi conveyor, we iteratively make a yes-or-no choice of the chef's latest creation as it passes by our table. In either case, we exercise choice (we select), and we may end up with a satisfying meal.

Biases in the food choices have different effects in the two regimes. Suppose that the buffet has 5 apple pies and 1 cherry pie. This quantitative bias makes no difference to a rational customer who prefers cherry pie: this customer will always choose cherry pie. The only relevant kind of bias is the complete absence of cherry pie, i.e., an absolute constraint. The effect of a bias is different for the sushi conveyor. Suppose that there is a 5 to 1 ratio of salmon to tuna. Even a customer who would prefer tuna in a side-by-side comparison may eat salmon more often, because a side-by-side comparison simply is not part of the process.

We have seen the sushi conveyor regime. Is there a buffet regime? The answer is yes, as shown in the figure on the right. The upper line shows the results for N = 1000 shown earlier. The lower lines show what happens when the alternative alleles are present in the initial gene pool with frequencies of 0.005 (open circles) or 0.01 (open squares): selection nearly always ensures that the fitter variant prevails, regardless of mutation bias.

The power of selection is greatly magnified in the buffet regime, where the 2-fold difference between s1 = 0.02 and s2 = 0.01 means an almost deterministic preference for the favored allele. In the sushi-conveyor regime, the same difference corresponds merely to a 2-fold difference in the chances of fixation.

Three implications

If biases in the introduction process are important in evolution, this has a number of implications, e.g., it suggests that evolution takes place in a regime that is more mutation-limited than what theoreticians typically imagine. But, before we get to these implications, which depend on empirical importance, let us consider some abstract implications.

1. The potential for directional trends

The model above addresses what theoreticians would call a 1-step adaptive walk. Is it possible for a sustained bias in the introduction of new variation to cause a sustained trend over many steps?

For instance, organisms with high GC-content at synonymous sites and inter-genic regions also have proteins enriched for amino acids encoded by high-GC codons (Gly, Ala, Arg, Pro) and depleted for those encoded by low-GC codons (Phe, Tyr, Met, Ile, Asn, Lys; see Singer and Hickey, 2000). Such differences usually are attributed to mutational biases operating in the context of neutral evolution. However, we can now understand the potential for mutation biases to operate in non-neutral evolution.

Stoltzfus (2006) considered the effect of GC:AT bias on the amino acid composition of a protein, using an abstract NK fitness model to represent interactivity of amino acids in a protein. The results tend to confirm what we expect from the analogy of Climbing Mount Probable above. On a perfectly smooth landscape, the trajectory of evolution is deflected initially by mutation bias, but the ultimate destination-- the optimal sequence-- is unaffected.  On rough landscapes, a mutation-biased composition evolves along a trajectory to a local peak.  The rougher the landscape, the shorter the trajectory and the greater the per-step effect of mutation bias.

2. The causal efficacy of developmental biases

A genotype:phenotype or GP map may induce biases in the accessibility of alternative phenotypes by mutation, as represented in the figure at left. This idea is frequently invoked in the evo-devo literature (e.g., Alberch and Gale, 1985; Emlen, 2000). Here, a 4-fold bias in mutational paths is induced: one mutation to A at the a locus leads to alternative phenotype P1, and 4 different mutations at loci b1, b2, b3 and b4 lead to phenotype P2.

We can define natural models of this effect using the genetic code, a GP map relating codon genotypes to amino acid phenotypes. In the figure below, we see two amino acid phenotypes accessible from the current genotype, GAT, which encodes Asp. The alternative Glu phenotype can be reached by 2 different transversion mutations, whereas Val can be reached by only 1.

The biases in these cases are not precisely mutation biases, in the sense that we cannot describe them without referring to a GP map. However, the effect of this bias on the outcome of evolution will be the same, because we can simply define the mutation rates and selection coefficients in the Yampolsky-Stoltzfus model relative to specific phenotypes (the green arrow vs. the blue arrow), and then all the implications of the model will follow.

3. An interpretation of structuralism

Contemporary structuralists often relate the chances of evolving a phenotype to its accessibility or density in state-space (e.g., Kauffman, 1992, Fontana, 2002).

The figure at left (after Fontana 2002) shows 3 networks, each representing a set of genotypes that have the same phenotype. Evolving systems move neutrally within a network and, more rarely, acquire a new phenotype. The accessibility of alternative phenotypes depends on how often a mutation in the initial network will generate the alternative phenotype. In the example shown here, starting from P0, the network of phenotype P2 is several times more accessible than P1 (green-gray vs. blue-gray boundaries), even though P1 and P2 have the same number of genotypes.

On this basis, we can see that a key concern of structuralism-- understanding the relative chances for the evolution of various forms by considering their distribution and local accessibility in genetic state-space-- invokes an effect of mutational accessibility that we can understand as a reference to biases in the introduction of variation. 


The simulation results above show a cause-effect relationship that we can contrast with classical "pressures" thinking as follows:

  • It applies when fixations are selective, whereas classical thinking limits the efficacy of mutational tendencies to the case of neutral evolution (or to unusually high mutation rates)
  • It poses a directional, quantitative dependence, whereas classical thinking holds that variation is merely a source of substance (i.e., with an amount, not a direction). Under ideal conditions, a B-fold bias can have a B-fold effect.
  • It depends on the rareness of mutations, whereas classical thinking suggests that variation-biased evolution requires high mutation rates.
  • Under ideal conditions, it establishes a condition of parity with selection, whereas classical thinking treats variation as subservient to selection.
  • Under ideal conditions, it establishes a basis for the composition and decomposition of causes, absent from classical thinking

This cause-effect relationship is not invoked in the canonical works of the Modern Synthesis. SFAIK, it is not yet incorporated in present-day textbooks. What makes this theory so new?


Joe Felsenstein said...

Arlin, your arguments in this series are well-taken. However I do just want to point out a case where the directional effect of mutation is much weaker. Suppose we have directional selection, with rare mutations (so that we can use the origin-fixation model). If the effective population size is N and two mutations are possible, with mutation rates u1 and u2, and the first carries the population upwards one step on the phenotypic scale and the second carries it downwards one step, what is the relative probability of the two directions? If selection coefficients for them are respectively s and -s then we can use Kimura's 1962 formula that the fixation probabilities are f = (1-exp(-2s)/(1-exp(-4Ns). Call these f1 and f2. The probabilities that if we see a step that it is up is then u1 f1 /(u1 f1 + u2 f2).

The bias can be extreme: If mutation rates are equal in both directions, N = 100,000 and s = 0.000001, for example, the ratio of f1 to f2 is 1.49182. Not too extreme. But when s = 0.00001 it is 54.597058 and when s = 0.0001 the ratio is 2.3533819 x 10^(17). Even in the first case (s = 0.000001) the net movement on the scale is strongly biased to the right if the scale is (say) 10 steps long in both directions. With the two smaller values of s the bias is much stronger. One needs mutational bias in the down direction to be 54.59x as strong in the s = 0.00001 case to even the two step probabilities out, and in the s = 0.0001 case you need a truly vast bias.

This of course differs from your case in the simulations where you had two positive values of s while here I have made the downward selection coefficient negative.

Joe Felsenstein said...

As I so often do, left out a parenthesis in Kimura's 1962 formula, It should be f = (1-exp(-2s)/(1-exp(-4Ns)).

Joe Felsenstein said...

... and I said "with the two smaller values of s" when I should have said "with the two larger values of s".

Joe Felsenstein said...

Oops-squared. Correctly parenthesized it is actually:

f = (1-exp(-2s))/(1-exp(-4Ns))

Arlin said...

Thanks for your comments, Joe. If I understand this, what you suggest is something like an origin-fixation version of the opposing-pressures argument of Haldane and Fisher (see part 4 (theory)). Mutation bias and fixation bias are either aligned along a single dimension of fitness, in which case, that is the strongly favored outcome, or mutation bias favors an outcome that is deleterious, in which case it is extremely unlikely to happen.

In the case of transition-transversion bias in the context of amino acid changes, the mutation bias is essentially orthogonal to selection, in the sense that there is no substantial difference in the distribution of acceptability for amino-acid-changing transitions and amino-acid-changing transversions. This is an empirical result as explained in part 2 (objections).

Joe Felsenstein said...

My example was to show that mutation pressure need not dominate. The case I used had opposing pressures, but not standing genetic variation, as I think you see. Which of these scenarios is most frequent is an empirical question.

Anyone who argues that mutation will determine the direction of evolutionary change need to get more quantitative, as my example shows. You have presented a quantitative argument. Fair enough.
But sometimes at this site I seem to hear Larry making arguments about the effect of neutral mutation, without any quantitation.

Joe Felsenstein said...

Let me amend that. Larry often seems to me to be arguing for the effectiveness of genetic drift in determining the outcome, even when fairly strong selection is acting to favor one outcome. People who do that usually do not see that the theory is against that.

Joe Felsenstein said...

@Arlin, if mutational bias has its effect in ways which are "orthogonal" to selection, where does that leave the statement that evolution is directed not by selection, but by mutation? It seems to me that the reason elephants have long trunks is most likely natural selection. Not that mutation just happened to favor long trunks. Of course the molecular details will be affected by mutational biases, but I think that new mutationism is meaning to unseat natural selection as an explanation of how we come to have adaptations.

Joe Felsenstein said...

typo, should be: "but I think that the new mutationism"

Arlin said...

Joe, "evolution is directed not by selection, but by mutation" is not my statement. Nei says something like this using the non-technical term "driven," but I think we have to interpret this as an explanatory claim. He is not making a causal claim that selection cannot drive directional change in a quantitative trait. He is saying something like "to really understand evolution of organisms, we need to understand the distinctive mutations that open up new opportunities for change, and it is not so important what ephemeral selective conditions led them to be favored at the time they rose to prominence."

With respect to effects on amino acid replacements, transition-transversion bias is essentially orthogonal to selection (an empirical result of Stoltzfus and Norris, 2015). Other kinds of biases may not be orthogonal. For instance, there is clearly a bias toward microdeletions relative to microinsertions, but if we tried to examine the effect of this bias on insertion or deletion of amino acids in proteins, we would not be safe in assuming orthogonality.

Joe Felsenstein said...

If fitness were really "orthogonal" to small deletions, then they would be basically neutral, and if there was a bias toward small deletions, the genome would continually get smaller, and smaller, and smaller. Going once, going twice, gone and then we would end up with a genomeless organism.

Arlin said...

If 98 % of insertions are deleterious and 2 % are beneficial, and the same is true of deletions, then the short-term dynamics will favor deletions if there is a deletion bias in mutation, and this will not depend on neutral allele fixations.

However, I think you are using "neutral" here in a different way.

For mutation biases with effects on composition that accrue in some direction, e.g., GC bias, the effects will slow down in a finite world. For instance, mutations to GC deplete the supply AT sites for further GC mutations. So, the steady-state composition under a 10-fold bias toward GC is not 100 % GC, but 91 % GC and 9 % AT. As the steady-state is reached, the total rate of change slows down.

The dynamics of indel biases are peculiar in that we do not expect any slow-down. This is because deleting 1 site decreases the number of sites for further deletions and also *equally decreases* the number of sites for insertions. So there will be no slow-down going towards a genome-length of 0 bp. The chance of going from 2 to 1 is the same as going from 1 to 0.

So, yes, your argument is abstractly correct, but not very practical because what you are doing is to take one aspect of the evolutionary dynamics and extrapolating that out to infinity. For instance, a bias toward deletions among microindels could dictate that protein families generally shrink in size over 100s of millions of years, but eventually the proteins will become minimalistic and the balance will tip so that deletions are more deleterious than insertions.

Furthermore, there are other effects on size. I was just referring to micro-indels. No one knows what are the biases with regard to larger-scale rearrangements. It could be that the mean length of members of individual gene families shrink over time, but this is offset at the genome level, in the long term, by gene and genome duplication.

Joe Felsenstein said...

I was not actually disagreeing with you about indels. I agree that if they were mostly neutral, and deletions were more frequent than insertions, then genomes would simply disappear, which is a bizarre and amusing (or horrifying) thought. Therefore I was agreeing with you that indels are unlikely to be "orthogonal to" selection.

I take your latest comment as a statement that there will continue to be a bias in favor of deletions, but as the proteins get very small the chance that indels are neutral goes toward zero and insertions would come to be favored. Of course the deletions also may not be neutral even at the start.