Bad takes #5. It’s just contingency

Unfamiliar ideas are often mis-identified and mis-characterized. It takes time for a new idea to be sufficiently familiar that it can be debated meaningfully. We look forward to those more meaningful debates. Until then, fending off bad takes is the order of the day! See the Bad Takes Index.

A common “stages of truth” meme holds that successful disruptive ideas are first (1) dismissed as absurd, then (2) resisted— the idea is declared unlikely and the evidence is strenuously disputed—, and finally (3) regarded as trivial and attributed to long tradition. Haldane’s version is that “The process of acceptance will pass through the usual four stages: (i) this is worthless nonsense; (ii) this is an interesting, but perverse, point of view; (iii) this is true, but quite unimportant; (iv) I always said so.” The QuoteInvestigator piece on the stages-of-truth meme has this version:

For it is ever so with any great truth. It must first be opposed, then ridiculed, after a while accepted, and then comes the time to prove that it is not new, and that the credit of it belongs to some one else

In their ruthless parody of Bad Synthesis Apologetics, Svensson and Berger (2019) model all the stages of truth in the same paper: (1) they dismiss strawman versions of the theory as absurd (see Bad Takes #3 and Bad Takes #4), (2) they present a clumsy and botched version of the actual theory but dispute the evidence and insist that the phenomenon is unlikely for reasons of population genetics, and (3) finally, implicitly admitting that the phenomenon is real and that the theory we proposed is correct, they describe it as trivial and familiar:

These studies therefore only exemplify how historical contingency and mutational history interact with selection during adaptation to novel environments [31, 38, 52], entirely in line with standard evolutionary theory and the uncontroversial insight that different genomic regions contribute differentially to adaptation driven by selection, with mutations merely providing the genetic input [53].

In this way, the reader is guided through the stages of truth from patent absurdity to yesterday’s news.

However, our focus here is only on the end-point of this progression, in which Svensson and Berger (2019) give the impression that the new work on mutation-biased adaptation represents ordinary textbook knowledge, so that these new results induce no changes in evolutionary reasoning, raise no new questions, and suggest no new priorities for research. The specific implication of the passage above is that these findings are merely a matter of “contingency” and present nothing original or new relative to the contents of references 31, 38 and 52.

[figure legend: A recent exploration of “contingency” by Wong (2019), revealing the lack of a precise meaning other than something vaguely to do with chanciness.]

Yet contingency is not a causal theory: it is an explanatory concept indicating that a system is non-equilibrium, so that the state of the system cannot be predicted without knowing the initial conditions and detailed dynamics. The notion of contingency, by itself, does not provide a theory of the dynamics. If we try to answer the odd question, “what does contingency predict about how the mutation spectrum shapes the spectrum of adaptive substitutions?” then we will get nowhere without a theory for the dynamics, and this theory will not be about contingency (an explanatory concept, not a cause of anything), but about the dynamical issue of how the details of mutation rates influence the spectrum of adaptive substitutions.

References 31 and 38 are from the field of quantitative genetics, and simply do not provide any such dynamical theory, e.g., here is the abstract to reference 31:

The introduction and rapid spread of Drosophila subobscura in the New World two decades ago provide an opportunity to determine the predictability and rate of evolution of a geographic cline. In ancestral Old World populations, wing length increases clinally with latitude. In North American populations, no wing length cline was detected one decade after the introduction. After two decades, however, a cline has evolved and largely converged on the ancestral cline. The rate of morphological evolution on a continental scale is very fast, relative even to rates measured within local populations. Nevertheless, different wing sections dominate the New versus Old World clines. Thus, the evolution of geographic variation in wing length has been predictable, but the means by which the cline is achieved is contingent.

Reference 52 is Good, et al (2017), a deep sequencing study of samples from Lenski’s LTEE (long-term evolution experiment). This is mainly an empirical analysis of allele trajectories and clonal interference and so on. There are no explicit claims for an effect of mutation bias on the spectrum of adaptive substitutions (mutation bias is mentioned only in relation to mutators, but they generate a lot of hitch-hikers so this does not establish an effect of mutation bias on adaptation). Indeed, the presentation of results indicates in various places (e.g., the comments on parallelism) that the authors are not paying attention to the issue of how mutation bias influences probabilities of beneficial changes.

What is going on here? The richness of the satire by Svensson and Berger (2019) may be overwhelming to the naive reader, who surely will become dizzy and lose track of all the ways that the authors avoid facing the relevant scientific issues. And that is precisely the point: they are illustrating how to avoid addressing the novelty of (1) a formal pop-gen theory that focuses on the introduction process, and which makes novel predictions about evolution based on tendencies of variation (addressing aspects of parallelism, trends, GP maps, findability, etc), in a way that directly contradicts the classic Haldane-Fisher “mutation pressure” argument, and (2) empirical results confirming a distinctive prediction of this theory, namely effects of mutation biases on adaptation (not requiring neutrality or high mutation rates), contradicting a long neo-Darwinian tradition of dismissing internal biases in evolution.

One way to avoid these key issues is to engage in whataboutery, i.e., responding to an issue by demanding attention to a second issue. What about other research? What about selection? Whataboutery provides the writer an opportunity to engage the reader on some related topic, e.g., for purposes of name-dropping. Rather than taking the opportunity to educate readers on the details of a new and exciting — but poorly known — body of work on mutation bias and molecular adaptation, i.e., rather than covering the studies that are the topic of their commentary, Svensson and Berger instead lavish their attention on older and much better known work on related topics by eminent scientists, e.g., the LTEE from Lenski and colleagues, lizard stuff from Jonathan Losos, the famous stickleback Pitx1 example, or David Houle’s work on fly wings.

More generally, Svensson and Berger (2019) are mocking the way that Synthesis apologists do not contemplate the practice of science in terms of falsifiable theories, precise reasoning, or the prospect of striking future discoveries, but are mainly concerned with crafting a narrative of traditions made of people, vague ideas, and flexible themes. They trivialize new work by assigning it to familiar and vague categories that make it seem ordinary (e.g., it’s just chance, it’s just more population genetics), rather than mapping it to the specific issues that motivate it, make it significant, and raise unanswered questions for the future.

Model of Bell’s first telephone from 1875

To understand how this game works, consider a completely unrelated example, namely the invention of a telephone 150 years ago (image). The novelty-hating curmudgeon may object as follows: You say there is something new here? How arrogant to make such a claim! There is nothing new here at all! This is merely an engineered device, and inventors have been crafting devices for centuries! Have you no respect for past work? I could show you 15 devices from just the past few years that are more impressive than this one, with more parts. There is no fundamentally new technology here, merely pieces of wood and metal and wire! I could build something like this in an afternoon for $25. There are no new electrical or mechanical principles at work, merely electrical currents and vibrations controlled by magnets. It looks like other devices I have seen. I could break it easily with a hammer. I doubt that it can fly like an airplane.

The problem is not that these objections are false statements. They could all be true. The problem is that they fail to address the crucial issue: the telephone prototype instantiates a generalizable technology to support remote voice communication through wires, thus over long distances.

Erik and David have done an excellent job of illustrating how to play the irrelevant-objections-to-novelty game. When they argue that new work on mutation-biased adaptation is just another example of contingency, this represents the strategy of describing new work in a trivially general way, like saying that the first telephone is just a device. When they claim that the theory we proposed is already part of the Modern Synthesis, on the grounds that it can be broken down into familiar parts, this is like objecting that the telephone is made of familiar parts and therefore cannot be new.

Of course, the significance of a new device— or a new theory— is not in the list of parts, but in what the assembled whole accomplishes.

What is the actual significance of recent work on mutation-biased adaptation? The essence of neo-Darwinism is a dichotomy of variation and selection, in which variation merely provides raw materials (substance, not form), and selection is the source of order, shape, and direction. Theories of internal biases directly contradict neo-Darwinism. The argument of Haldane and Fisher that such theories are incompatible with population genetics (see Bad takes #2) was eagerly adopted by the architects of modern neo-Darwinism, yet (1) this classic conclusion is unwarranted theoretically and (2) its implications are refuted empirically. These two provocative claims are established by recent work on mutation-biased adaptation; they are not part of textbook knowledge; they are not established in well known studies cited by Svensson and Berger to illustrate scientific name-dropping.


Good BH, McDonald MJ, Barrick JE, Lenski RE, Desai MM. 2017. The dynamics of molecular evolution over 60,000 generations. Nature 551:45-50.

Bad takes #3. Mutation bias as an independent cause of adaptation.

Unfamiliar ideas are often mis-identified and mis-characterized. It takes time for a new idea to be sufficiently familiar that it can be debated meaningfully. We look forward to those more meaningful debates. Until then, fending off bad takes is the order of the day! See the Bad Takes Index.

Svensson and Berger (2019) begin their opinion piece with a multifarious attack on the theory that mutation bias is “adaptive in its own right” or “an independent force” or something that “can explain the origin of adaptations independently of, or in addition to, natural selection.” They enthusiastically invoke versions of this theory in 5 different places, just on the first page (yellow highlights)!

They associate this theory with the line of argument on mutation bias and adaptation developed in work from my group (e.g., Yampolsky and Stoltzfus, 2001; Stoltzfus, 2006a, 2006b, 2012, 2019), and extended by studies such as:

None of these sources promote or assume a theory of mutation bias as an independent cause of adaptation, or a theory of mutation being adaptive in its own right. For instance, in the original twin-peaks model of Yampolsky and Stoltzfus (2001), the peak favored by the higher selection coefficient is accessible only via a lower mutation rate, and vice versa. Stoltzfus and Norris (2015) worked very hard to establish that— contrary to lore— transitions and transversions that change amino acids hardly differ in their fitness effects.

Thus, whereas some other authors have promoted the idea of adaptive mutation (e.g., Cairns, Caporale, Rosenberg), “natural genetic engineering” (Shapiro) or merely some statistical correlations between mutational patterns and fitness effects (Monroe, et al. 2022), the above sources do none of those things. Thus, Svensson and Berger (2019) are illustrating the concept of a strawman argument, criticizing an alternative view by relying on a false representation of that view.

The strawman argument of Svensson and Berger, 2019. The authors use this figure to misrepresent the theory of Yampolsky and Stoltzfus (2001), representing the hypothesis of mutation as an “independent cause of adaptation” by a thick arrow with a question mark. The other parts of this figure are intended to represent conventional thinking.

This form of this strawman argument has a long history. Repeatedly, critics of neo-Darwinism have suggested that observed tendencies or directions of evolutionary change cannot be explained solely by selection, but reflect internal aspects of mutation or development, and advocates of neo-Darwinism have responded by accusing these critics of proposing some form of directed mutation or adaptive mutation, or of engaging in mysticism or teleology. Is this simply a case of intellectual dishonesty? Perhaps, but a more sympathetic view is that this is an error in reasoning induced by an ideology that recognizes only one kind of directionality, so that every argument about directions (e.g., in phenotype space) is treated as an argument about intrinsically adaptive directions.

Because Darwin’s followers have used this strawman argument for over a century, we know how the advocates of internalist theories respond:

I take exception here only to the implication that a definite variation tendency must be considered to be teleological because it is not ‘orderless.’ I venture to assert that variation is sometimes orderly and at other times rather disorderly, and that the one is just as free from teleology as the other. In our aversion to the old teleology, so effectually banished from science by Darwin, we should not forget that the world is full of order . . . If a designer sets limits to variation in order to reach a definite end, the direction of events is teleological; but if organization and the laws of development exclude some lines of variation and favor others, there is certainly nothing supernatural in this” (Whitman, 1919: see p. 385 of Gould, 2002)

In general, when we find that false arguments are maintained and nurtured for generations, this is because they are being used, not to resolve substantive scientific questions, but to maintain conformity within a closed ideological system, i.e., advocates of neo-Darwinism use these arguments to help each other avoid thinking any new thoughts, working together to keep the dream alive. Gatekeepers like Svensson and Berger try to discourage new thinking by treating it as bad science and careerism (in this case, with ironic results).

Of course, we must not confuse theories and people, e.g., a theory supported by bad arguments does not become a bad theory for this reason. The theory and its apologetics are two different things. We can define neo-Darwinism clearly in terms of the dichotomy of the potter (selection) and the clay (variation), and this concept stands by itself. Separate from this, there is a culture or a set of rhetorical practices associated with neo-Darwinian apologetics. It is this neo-Darwinian culture or thought-collective that has a sociological aspect and a self-protective urge that gives rise to the same pathologies as any cult or identity-group.

The false accusations of teleology or directed mutation are part of the conceptual immune system of the neo-Darwinian thought-collective (described in a separate post). This kind of strawman is used in two ways. The first line of defense against alternative views is to claim that they are inherently flawed or absurd: they represent mental errors that can be dismissed immediately, not alternative scientific theories that must be evaluated with research. This is what neo-Darwinians teach about history: the neo-Darwinian view is simply the rational evidence-based approach to evolution, whereas critics behave irrationally and hold views with obvious flaws.

Eventually, however, some of the more intellectually rigorous ones begin to sense that scientific theories are supposed to stand for something substantive and risky, and they begin to consider the plausibility of alternative ideas. In this case, the priests of the religion must offer a more sophisticated argument. In this second line of defense, the alternative is granted as a theoretical possibility, but (1) its importance is minimized, and (2) it is described in different language and grounded in the Darwinian faith tradition itself, by reference to obscure passages in the sacred texts. That is, the doubter is shown that the apostles and disciples of olden times, and even the great Lord Darwin himself, also experienced moments of doubt, posed difficult questions, and considered alternative views. The initiate now has a more sophisticated understanding of the faith, which does not demand inner purity, but merely outward piety. In this case, the faith is shown by creating constructive ambiguity that can be used to shift the focus of discussion from new scientific developments backwards to tradition and to the pantheon of heroes.

Svensson and Berger (2019) illustrate both strategies brilliantly. First they attack claims of mutation-biased adaptation by attacking the straw-man of “independent cause of adaptation,” then they present a more reasonable idea based on population genetics— a butchered version of the theory proposed by Yampolsky and Stoltzfus (2001)—, and present this as conventional wisdom. First they state absurdly that the theory is part of the neutral theory— which in their strangely twisted conception of history was somehow swallowed up by the Modern Synthesis— but they cite no source for this, e.g., the theory of Yampolsky and Stoltzfus (2001) appears nowhere in Kimura’s (1983) seminal book on the neutral theory. They also present a derivation of a key equation in Box 1, which gives a short series of mathematical results and names Haldane, Kimura, and Fisher without naming Yampolsky and Stoltzfus, guiding the reader to conclude falsely that the theory comes from famous dead people.

However, at the same that they appropriate the theory on behalf of Fisher, et al., they also purport to undermine it by claiming that it depends on unusual conditions including sign epistasis and drift in small populations (these are not actual requirements, but fabrications intended to make the argument seem more respectable).

For more bad takes on this topic

This is part of a series of posts focusing on bad takes on the topic of biases in the introduction of variation, covering both the theory and the evidence. For more bad takes, see the index to bad takes.


Gould SJ. 2002. The Structure of Evolutionary Theory. Cambridge, Massachusetts: Harvard University Press.

Bad takes #1. We have long known.

Unfamiliar ideas are often mis-identified and mis-characterized. It takes time for a new idea to be sufficiently familiar that it can be debated meaningfully. We look forward to those more meaningful debates. Until then, fending off bad takes is the order of the day! See the Bad Takes Index.

An anonymous reviewer responded to the manuscript of Stoltzfus and Yampolsky (2009) with the claim that “we have long known that mutation is important in evolution,” citing the following passage from Haldane (1932) as if to suggest that the message of our paper (emphasizing the dispositional role of mutation) was old news:

A selector of sufficient knowledge and power might perhaps obtain from the genes at present available in the human species a race combining an average intellect equal to that of Shakespeare with the stature of Carnera. But he could not produce a race of angels. For the moral character or for the wings, he would have to await or produce suitable mutations

We included it in the final version of the paper because, actually, this passage demonstrates the opposite of what the reviewer implies. What is Haldane suggesting?

I can’t resist a good story, so let’s begin with this 1930s photo of Italian boxer Primo Carnera, his friend and fellow heavyweight champ Max Baer, and Hollywood actress Myrna Loy. Baer dated Loy in real life. They made a movie together, the three of them (thus the staged publicity photo). Baer, one of the greatest punchers of all time and half-Jewish, became a hero to a generation of Jewish sports fans when he demolished Max Schmeling, the German champion, prompting Hitler to outlaw boxing with Jews. He literally killed one of his opponents, and repeatedly sent Carnera to the floor during their single fight.

Primo Carnera, Myrna Loy, and Max Baer in a publicity photo from the 1930s

But the point of this picture is that, although Baer was a formidable man, Carnera makes him look small. Other fighters were afraid to get in the ring with him. Though enormous — 30 cm taller and 50 kg heavy than the average Italian of his generation —, Carnera was not the aberrant product of a hormonal imbalance. This photo shows a huge man who is stocky but well proportioned, muscular, and surprisingly lean. Again, he was not a misshapen monster, but a man at the far extremes of a healthy human physique, which is precisely Haldane’s point.

Selective breeding to the quantitative extremes of known human ability, Haldane proposes, could produce a race combining the extreme of Carnera’s magnificent stature with Shakespeare’s magnificent verbal ability.

Haldane contrasts this with a different mode of evolution dependent on new mutations, which might produce a race of angels, if one could wait long enough for the mutations to happen. That is, Haldane is contrasting (1) a mode of evolution that could combine the known extremes of human ability with (2) a mode of evolution that could generate imaginary fictitious not-at-all-real creatures. Haldane, Wright and Fisher each argued that a mode of change dependent on new mutations would be too slow to account for the observed facts of evolution. They argued instead that evolution must take place on the basis of abundant standing variation.

That is, in the passage above, Haldane is not endorsing a mode of mutation-dependent evolution, but gently mocking it, in contrast to a mode of evolution that, based on quantitative standing variation, could produce a race of magnificently eloquent champions.

Thus, the reviewer’s comment was a bad take on Haldane, a misinterpretation of Haldane’s meaning.

In addition, the “we have always known” comment represents a more general category of bad take that substitutes, in place of a specific target of criticism, a much broader, fuzzier, or more generic claim. Note what the reviewer does not say: the reviewer might have said

“The authors’ implicit claim of novelty is preposterous. We have long known about the role of biases in the introduction process emphasized in this manuscript. Haldane (1932) and Fisher (1930) explored the theoretical implications of such biases, and Simpson and Mayr repeatedly incorporated a theory of internal variational trends into their interpretations of the fossil record. The authors must cite these sources instead of suggesting that their arguments are original.”

But of course, the reviewer does not say this, because nothing like this ever happened. Think about it.

Certainly, the reviewer is correct that scientists in the mainstream Modern Synthesis tradition have always known that mutation is important in evolution. More precisely, the importance they assigned to mutation was that it is ultimately necessary, because without mutations, evolution would grind to a halt. Haldane, Fisher, Ford, Huxley, Dobzhansky, and others said this explicitly.

However, they did not say that mutation is important as a dispositional factor. Instead, they argued explicitly against this idea, e.g., Haldane (1927) is the original source of the argument that mutation pressure is a weak force (see Bad takes #2).

The theory of biases in the introduction process, by contrast, says that mutation is important in evolution as a dispositional cause, a cause that makes some outcomes more likely than others, and that this importance is achieved (mechanistically) by way of biases in the introduction process.

So, the reviewer is making an implicit bait-and-switch argument. The theory of biases in the introduction of variation is a specific population-genetic theory with specific conditions and implications, and the reviewer is responding to this by saying “we have always known that mutation is important,” but this is not the same thing: the traditional importance assigned to mutation is not “dispositional cause that makes some outcomes more likely than others” but “ultimate source of raw materials without which evolution would grind to a halt.”

Finally, this bad take is part of a family of bad takes in which the novelty of a claim X is rejected on the grounds that X sounds vaguely like X’, or that X can be categorized as a member of some larger and fuzzier class of claims (see Bad Takes #5: Contingency). This is often the case with “we have long known” arguments. If the theory was in fact old, the reviewer would not have made a vague “we have long known” argument, but would have cited the original source of the theory, e.g., “Of course the theory of biases in the introduction process is not new, because Haldane proposed exactly the same theory 70 years ago and worked out its implications!” In fact, no such antecedent exists, which is why, to defend tradition, the reviewer must resort to a bait-and-switch argument.


Haldane JBS. 1932. The Causes of Evolution. New York: Longmans, Green and Co.

Stoltzfus A, Yampolsky LY. 2009. Climbing mount probable: mutation as a cause of nonrandomness in evolution. J Hered 100:637-647.

NRC Research Associateship: mutation and evolution

The US National Research Council (NRC) offers competitive Research Associateships for post-doctoral and senior scientists to conduct research in participating federal labs. The awards include a generous stipend as well as benefits (health insurance, travel, relocation), as explained on the program web site.

To apply, you must write a brief research proposal that reflects a plan of your own, or a plan that we develop together, involving some computational approach to molecular evolution. Especially welcome are proposals for empirical or theoretical work on biases in the introduction of variation as a dispositional factor in evolution, building on work such as Yampolsky and Stoltzfus (2001), Stoltzfus and McCandlish (2017) or Stoltzfus and Norris (2016).

The upcoming deadline for proposals is February 1, 2021 (there is another deadline August 1). If you are interested, contact me with a brief introduction, and we’ll go from there.

Arlin Stoltzfus (

Research Biologist, NIST (Data Scientist, Office of Data & Informatics)
Fellow, IBBR; Adj. Assoc. Prof., UMCP;
IBBR, 9600 Gudelsky Drive, Rockville, MD, 20850


Stoltzfus A, McCandlish DM. 2017. Mutational biases influence parallel adaptation. Mol Biol Evol 34:2163-2172

Stoltzfus A, Norris RW. 2016. On the Causes of Evolutionary Transition:Transversion Bias. Mol Biol Evol 33:595-602.

Yampolsky LY, Stoltzfus A. 2001. Bias in the introduction of variation as an orienting factor in evolution. Evol Dev 3:73-83.

PoMo, Oh No! A comment on The Logic of Chance

For a long time I was meaning to write a review of Eugene Koonin’s The Logic of Chance: The Nature and Origin of Biological Evolution.  The book has been out for over 6 years now.  In lieu of an actual review, I’d like to discuss Koonin’s characterization of an emerging view of evolution as a “post-modern” alternative to the “Modern” synthesis.  What could that mean?


A very bad theory about theories

In some venues, this is a familiar trope: a science advocate responds to the criticism that “evolution is just a theory,” explaining that, for scientists, a “theory” is a thoroughly tested, well established explanation.   Yet, for over 200 years, scientists have used “theory” for conjectures that are not well established, even ones whose truth is rejected, e.g., Lamarck’s theory.   What is going on?


Getting “Lamarckian” right

Lamarck’s theory of evolutionary adaptation invokes the inheritance of adaptive responses that emerge by effort.  Over a century ago, this mechanism was rejected by geneticists familiar with results on mutation and inheritance, e.g., Cuenot (1909) rejects Lamarckian modification on the grounds that hereditary variants emerge suddenly, rather than being brought on gradually by use and disuse.

This topic would deserve no further comment, except that the false idea of a rebirth of a Lamarckian mode of evolutionary change has been stirred up repeatedly by contemporary scientists guided by the false dichotomy of Lamarck vs. Darwin.  


Mutation-biased adaptation reaches the mainstream

The most recent issue of PNAS includes a report by Galen, et al linking enhanced mutation at a CpG site to altitude adaptation in Andean house wrens (Troglodytes aedon), based on clear biogeographic and biochemical evidence of adaptation.  I’ve been waiting for this, both in the narrow sense that I’ve been waiting for this particular study to appear in print, and also in the broader sense that I have been waiting for any paper on mutation-biased adaptation to appear in a prominent venue.  Results like these, one hopes, will overturn the “raw materials” doctrine of neo-Darwinism and stimulate the development of a new understanding of the role of mutation in evolution.


Understanding the Mutational Landscape Model

This post started out as a wonky rant about why a particular high-profile study of laboratory adaptation was mis-framed as though it were a validation of the mutational landscape model of Orr and Gillespie (see Orr, 2003), when in fact the specific innovations of that theory were either rejected, or not tested critically.  As I continued ranting, I realized that there was quite a bit to say that is educational, and I contemplated that the reason for the original mis-framing is that this is an unfamiliar area, such that even the experts are confused— which means that there is a value to explaining things.

The crux of the matter is that the Gillespie-Orr “mutational landscape” model has some innovations, but also draws on other concepts and older work.  We’ll start with these older foundations.

Origin-fixation dynamics in sequence space

First, the mutational landscape model draws on origin-fixation dynamics, proposed in 1969 by Kimura and Maruyama, and by King and Jukes (see McCandlish and Stoltzfus for an exhaustive review, or my blog on The surprising case of origin-fixation models).

In origin-fixation models, evolution is seen as a simple 2-step process of the introduction of a new allele, and its subsequent fixation (image).  2_step_cartoonThe rate of change is then characterized as a product of 2 factors, the rate of mutational origin (introduction) of new alleles of a particular type, and the probability that a new allele of that type will reach fixation.  Under some general assumptions, this product is equal to the rate of an origin-fixation process when it reaches steady state.

Probably the most famous origin-fixation model is K = 4Nus, which uses 2s (Haldane, 1927) for the probability of fixation of a beneficial allele, and 2Nu (diploids) for the rate of mutational origin.  Thus K = 4Nus is the expected rate of changes when we are considering types of beneficial alleles that arise by mutation at rate u, and have a selective advantage s.  But we can adapt origin-fixation dynamics to other cases, including neutral and deleterious changes. If we were applying origin-fixation dynamics to meiotic bursts, or to phage bursts, in which the same mutational event gives rise immediately to multiple copies (prior to selection), we would use a probability of fixation that takes this multiplicity into account.

In passing, note that origin-fixation models appeared in 1969, and we haven’t always viewed evolution this way.  The architects of the Modern Synthesis rejected this view— and if you don’t believe that, read The Shift to Mutationism is Documented in Our Language or The Buffet and the Sushi Conveyor.   They saw evolution as more of an ongoing process of “shifting gene frequencies” in a super-abundant “gene pool” (image).  1_step_cartoon Mutation merely supplies variation to the “gene pool”, which is kept full of variation.  The contribution of mutation is trivial.   The available variation is constantly mixed up by recombination, represented by the egg-beaters in the figure.  When the environment changes, selection results in a new distribution of allele frequencies, and that’s “evolution”— shifting gene frequencies to a new optimum in response to a change in conditions.

This is probably too geeky to mention, but from a theoretical perspective, an origin-fixation model might mean 2 different things.  It might be an aggregate rate of change across many sites, or the rate applied to a sequence of changes at a single locus.  The mathematical derivation, the underlying assumptions, and the legitimate uses are different under these two conditions, as pointed out by McCandlish and Stoltzfus.  The early models of King, et al were aggregate-rate models, while  Gillespie, (1983) was the first to derive a sequential-fixations model.

Second, the mutational landscape model draws on Maynard Smith’s (1970) concept of evolution as discrete stepwise movement in a discrete “sequence space”.  More specifically, it draws on the rarely articulated locality assumption by which we say that a step is limited to mutational “neighbors” of a sequence that differ by one simple mutation, rather than the entire universe of sequences.  The justification for this assumption is that double-mutants, for instance, will arise in proportion to the square of the mutation rate, which is a very small number, so that we can ignore them.  Instead, we can think of the evolutionary process as accessing only a local part of the universe of sequences, which shifts with each step it takes. In order for adaptive evolution to happen, there must be fitter genotypes in the neighborhood.

This is an important concept, and we ought to have a name for it.  I call it the “evolutionary horizon”, because we can’t see beyond the horizon, and the horizon changes as we move.  horizonNote two things about this idea.  The first is that this is a modeling assumption, not a feature of reality.  Mutations that change 2 sites at once actually occur, and presumably they sometimes contribute to evolution.  The second thing to note is that we could choose to define the horizon however we want, e.g., we could include single and double changes, but not triple ones.   In practice, the mutational neighbors of a sequence are always defined as the sequences that differ by just 1 residue.

Putting these 2 pieces together, we can formulate a model of stepwise evolution with predictable dynamics.

mut_landscape_numberline Making this into a simulation of evolution is easy using the kind of number line shown at left. Each segment represents a possible mutation-fixation event from the starting sequence.  For instance, we can change the “A” nucleotide that begins the sequence to “T”, “C” or “G”.  The length of each segment is proportional to the origin-fixation probability for that change (where the probability is computed from the instantaneous rate).  To pick the next step in evolution, we simply pick a random point on the number line. Then, we have to update the horizon— recompute the numberline with the new set of 1-mutant neighbors.

Where do we get the actual values for mutation and fixation?  One way to do it is by drawing from some random distribution.  I did this in a 2006 simulation study. I wasn’t doing anything special.  It seemed very obvious at the time.


(Table 1 from Rokyta, et al)

Surprisingly, researchers almost never measure actual mutation and selection values for relevant mutants.  One exception is the important study by Rokyta, et al (2005), who repeatedly carried out 1-step adaptation using bacteriophage phiX174.  The selection coefficients for each of the 11 beneficial changes observed in replicate experiments are shown in the rightmost column, with the mutants ranked from highest to lowest selection coefficient.  Notice that the genotype that recurred most often (see the “Number” column) was not the alternative genotype with the highest fitness, but the 4th-most-fit alternative, which happened to be favored by a considerable bias in mutation. Rokyta, et al didn’t actually measure specific rates for each mutation, but simply estimated average rates for different classes of nucleotide mutation based on an evolutionary model.

Then Rokyta, et al. developed a model of origin-fixation dynamics using the estimated mutation rates, the measured selection coefficients, and a term for the probability of fixation customized to account for the way that phages grow. This model fit the data very well, as I’ll show in a figure below (panel C in the final figure).

The mutational landscape model

Given all of that, you might ask, what does the mutational landscape model do?

This is where the specific innovations of Orr and Gillespie come in.  Just putting together origin-fixation dynamics and an evolutionary horizon doesn’t get us very far, because we can’t actually predict anything without filling in something concrete about the parameters, and that is a huge unknown.  What if we don’t have that?  Furthermore, although Rokyta, et al implicitly assumed a horizon in the sense that they ignored mutations too rare to appear in their study, they never tackled the question of how the horizon shifts with each step, because they only took one step.  What if we want to do an extended adaptive walk?  How will we know what is the distribution of fitnesses for the new set of neighbors, and how it relates to the previous distribution?  In the simulation model that I mentioned previously, I used an abstract “NK” model of the fitness of a protein sequence that allowed me to specify the fitness of every possible sequence with a relatively small number of randomly assigned parameter values.

Gillespie and Orr were aiming to do something more clever than that.  Theoreticians want to find ways to show something interesting just from applying basic principles, without having to use case-specific values for empirical parameters.  After all, if we insert the numbers from one particular experimental system, then we are making a model just for that system.


(Figure 1 of Rokyta, et al, explaining EVT)

The first innovation of Orr and Gillespie is to apply extreme value theory (EVT) in a way that offers predictions even if we haven’t measured the s values or assumed a specific model.  If we assume that the current genotype is already highly adapted, this is tantamount to assuming it is in the tail end of a fitness distribution.  EVT applies to the tail ends of distributions, even if we don’t know the specific shape of the distribution, which is very useful.  Specifically, EVT tells us something about the relative sizes of s as we go from one rank to the next among the top-ranked genotypes: the distribution of fitness intervals is exponential.  This leads to very specific predictions about the probability of jumping from rank r to some higher rank r’, including a fascinating invariance property where the expected upward jump in the fitness ranking is the same no matter where we are in the ranking.  Namely, if the rank of the current genotype is j (i.e., j – 1 genotypes are better), we will jump to rank (j + 2)/4.

That’s fascinating, but what are we going to do with that information?  I suspect the idea of a fitness rank previously appeared nowhere in the history of experimental biology, because rank isn’t a measurement one takes anywhere other than the racetrack.  But remember that we would like a theory for an adaptive walk, not just 1-step adaptation.  If we jump from j to k = (j + 2)/4, then from k to m = (k + 2)/4, and so on, we could develop a theory for the trajectory of fitness increases during an adaptive walk, and for the length of an adaptive walk— for how many steps we are likely to take before we can’t climb anymore.

Figure 5.3 from Gillespie’s 1991 book. Each iteration has a number-line showing the higher-fitness genotypes accessible on the evolutionary horizon (fitness increases going to the right). At iteration #1, there are 4 more-fit genotypes. In the final iteration, there are no more-fit genotypes accessible, but there is a non-accessible more-fit genotype that was accessible at iteration #2.

The barrier to solving that theory is solving the evolutionary horizon problem.  Every time we take a step, the horizon shifts— some points disappear from view, and others appear (Figure).  We might be the 15th most-fit genotype, but at any step, only a subset of the 14 better genotypes will be accessible, and this subset changes with each step: this condition is precisely what Gillespie (1984) means by the phrase “the mutational landscape” (see Figure).  In his 1983 paper, he just assumes that all the higher-fitness mutants are accessible throughout the walk.  Gillespie’s 1984 paper entitled “Molecular Evolution over the Mutational Landscape” tackles the changing horizon explicitly.  He doesn’t solve it analytically, but uses simulations.  I won’t explain his approach, which I don’t fully understand.  Analytical solutions appeared in later work by Jain and Seetharaman, 2011 (thanks to Dave McCandlish for pointing this out).

The third and fourth key innovations are to (3) ignore differences in u and (4) treat the chances of evolution as a linear function of s, based on Haldane’s 2s.  In origin-fixation dynamics, the chance of a particular step is based on a simple product: rate of origin multiplied by probability of fixation.  Orr’s model relates the chances of an evolutionary step entirely to the probability of fixation, assuming that u is uniform. Then, using 2s for the probability of fixation means that the chance of picking a mutant with fitness si is simply si / sum(s) where the sum is over all mutants (the factor of 2 cancels out because its the same for every mutant).  Then, by applying EVT to the distribution of s, the model allows predictions based solely on the current rank.

A test of the mutational landscape model?

As noted earlier, this post started as a rant about a study that was mis-framed as though it were some kind of validation of Orr’s model.  In fact, that study is Rokyta, et al., described above.  Indeed, Rokyta, et al.  tested Orr’s predictions, as shown in the left-most panel in the figure below.  The predictions (grey bars) decrease smoothly because they are based, not on the actual measured fitness values shown above, but merely on the ranking.  The starting genotype is ranked #10, and all the predictions of Orr’s model follow from that single fact, which is what makes the model cool!


(Figure 2 of Rokyta, et al. A: fit of data to Orr’s model. B: fit of data to an origin-fixation model using non-uniform mutation rates. C: fit of data to origin-fixation model with non-uniform mutation and probability of fixation adjusted to fit phage biology more precisely. The right model is significantly better than the left model.)

If they did a test, what’s my objection?  Yes, Rokyta, et al. turned the crank on Orr’s model and got predictions out, and did a goodness-of-fit test comparing observations to predictions.  But, to test the mutational landscape model properly, you have to turn the crank at least 2 full turns to get the mojo working.  Remember, what Gillespie means by “evolution over the mutational landscape” is literally the way evolution navigates the change in accessibility of higher-fitness genotypes due to the shifting evolutionary horizon.  That doesn’t come into play in 1-step adaptation.  You have to take at least 2 steps.  Claiming to test the mutational landscape model with data on 1-step adaptation is like claiming to test a new model for long-range weather predictions using data from only 1 day.

The second problem is that Rokyta, et al respond to the relatively poor fit of Orr’s model by successively discarding every unique feature.  The next thing to go was the assumption of uniform mutation.  As I noted earlier, there are strong mutation biases at work.  So, in the middle panel of the figure above, they present a prediction that depends on EVT and assumes Haldane’s 2s, but rejects the uniform mutation rate.  In their best model (right panel) they have discarded all 4 assumptions.  They have measured the fitnesses (Table 1, above), and they aren’t a great fit to an exponential, so they just use these instead of the theory.  Haldane’s 2s only works for small values of s like 0.01 or 0.003, but the actual measured fitnesses go from 0.11 to 0.39!  Rokyta, et al provide a more appropriate probability of fixation developed by theoretician Lindi Wahl that also takes into account the context of phage (burst-based) replication.  To summarize,

Assumption 1 of the MLM.  The exponential distribution of fitness among the top-ranked genotypes is tested, but not tested critically, because the data are not sufficient to distinguish different distributions.

Assumption 2 of the MLM.  Gillespie’s “mutational landscape” strategy— his model for how the changing horizon cuts off previous choices and offers a new set of choices at each step— isn’t tested because Rokyta’s study is of 1-step walks.

Assumption 3 of the MLM.  The assumption that the probability of a step is not dependent on u, on the grounds that u is uniform or inconsequential, is rejected, because u is non-uniform and consequential.

Assumption 4 of the MLM. The assumption that we can rely on Haldane’s 2s is rejected, for 2 different reasons explained earlier.


I’m not objecting so much to what Rokyta, et al wrote, and I’m certainly not objecting to what they did— it’s a fine study, and one that advanced the field.  I’m mainly objecting to the way this study is cited by pretty much everyone else in the field, as though it were a critical test that validates Orr’s approach.  That just isn’t supported by the results.  You can’t really test the mutational landscape model with 1-step walks.  Furthermore, the results of Rokyta, et al led them away from  the unique assumptions of the model.  Their revised model just applies origin-fixation dynamics in a realistic way suited to their experimental system— which has strong mutation biases and special fixation dynamics— and without any of the innovations that Orr and Gillespie reference when they refer to “the mutational landscape model.”

Constructive neutral evolution on Sandwalk

The interesting things at Sandwalk always seem to happen when I’m not looking.  On Sunday, while I was out west taking the offspring to start university at UBC, Larry Moran posted a blog on Constructive Neutral Evolution that has elicited almost 200 comments.  Alas, many of the comments are not particularly useful, as Sandwalk is home to an ongoing pseudoscientific debate on intelligent design.

The one point that I would like to make about CNE is that it was not proposed as some kind of law or tendency (i.e., not like “Biology’s First Law” of McShea and Brandon).  Some other people treat CNE as the manifestation or the realization of some kind of intrinsic tendency to complexification. If this were the case, then examples of reductive evolution (e.g., cases involving viruses and intracellular parasites) would raise a question about the generality of the idea.  Obviously evolutionary change occurs in both reductive and constructive modes.  Bateson and Haldane each speculated that reductive evolution would be common because it is so easy to accomplish.

From my perspective, CNE is not a theory about a general tendency of evolution.  Instead it is a schema for generating specific testable hypotheses of local complexification.

One also can imagine a mode of Reductive Neutral Evolution in which simplification occurs. It is simply a matter of the local position of the system relative to the spectrum of mutational possibilities.