Field of Science

Evolution, sex, and mixability
Last Friday Christos Papadimitriou gave a seminar at UC Santa Barbara in the Computer Science department. The title of his talk was Computational Insights and the Theory of Evolution [announcement].

Covertly computational ideas have influenced the Theory of Evolution from the very start. This talk is about recent work on Evolution that was inspired and informed by computation. Considerations about the performance of genetic algorithms led to a novel theory of the role of sex in Evolution based on the concept of mixability, while the equations describing the evolution of a species can be reinterpreted as a repeated game between genes. The complexity of local optimality informs the mystery of variation, and a theorem on Boolean functions helps us understand better the emergence of complex adaptations.

Papadimitriou is a very accomplished computer scientist, a member of the National Academy of Sciences and has written several textbooks and has received many awards.

The work he presented at the seminar was on evolutionary theory. Of the 60 minutes he spoke, he spent the first 30 talking about the science of evolution before Darwin, about the work of Wallace and Darwin, Mendel's work, and then briefly about Fisher, Wright, and Haldane.

Following that he then spoke about Mixability. He met Adi Livnat at Berkeley in 2006, and together they ended up publishing a paper on Livnat's theory of mixability in PNAS in 2008: A mixability theory for the role of sex in evolution [1]. In short, mixability is a theory that states that in sexually reproducing organisms, fitness alone is not what is optimized, but fitness plus population entropy is. What this means is that in addition to fitness, variation (as measured by entropy) is also optimized given the rate of recombination. Recombination introduces genetic variance by mixing the genetic material of two parents (rather than making a clone), and since genetic variance is the fuel of evolution, this is the benefit of sex, so the story goes.

Following this 2008 paper, three more papers have appeared on the same subject by the same two authors (and various others, including Marc Feldman, by the way). Now, the first PNAS paper from 2008 was edited by Dan Hartl. This means that it had a independent editor who sent the paper out for review. (As we shall hear soon, it was sent to six reviewers.) The next three papers appeared in PNAS (2010), Journal of Theoretical Biology (2011), and PNAS (2014) [2-4]. Both of these PNAS papers were contributed by Papadimitriou. Because he is a member of the NAS, he is allowed to contribute four papers per year to PNAS, which means that he acts as the editor, and is thus the person choosing which reviewers to send the paper to for review [source]:
Papers listed as “Contributed” by NAS members (at least one of the authors is an NAS member) account for only 5% of submissions and less than a quarter of published papers. NAS members are only allowed up to four Contributed papers per year. Member-contributed papers must have at least two independent reviews and are evaluated by the Editorial Board.
So only two reviewers chosen by one of the authors, and the editor/author makes the final decision anyway. Hmm. Just saying*...

Anyhow, Papadimitriou's presentation ends, and I eagerly raise my hand to ask my question. Check out this recording of the exhange.
Papadimitriou: I will pray for answers, but please give me your questions.

Me: So, you said you met Adi Livnat in 2006...

Papadimitriou: Yes.

Me: In 2005 there's a paper also in PNAS by Guy Sella and Aaron Hirsch [5]. Are you familiar with this paper?

Papadimitriou: Yes, I am familiar.

Me: They introduced the concept of free fitness analogous to the free energy.

Papadimitriou: Yes.

Me: And that seems to me to be curiously identical to mixability.

Papadimitriou: Not really, no. I mean...

Me: The point is that they update both fitness and entropy at the same time, yes?

Papadimitriou: We have two conclusions that seem to... First of all they came up with [?], supposedly do that. We came to this conclusion 1) through simulations. So, we came up with the concept of mixability, in other words that the alleles that are favored are not the ones that have the best combinations, but the ones that have the best average performance. By analyzing fitness landscapes. Please, when we published our paper we got six reviews. Two of them, they could have been from that group [i.e., Sella and Hirsch]. One of them said that this is well known, and the other one said it is wrong. The point is that once you have to articulate the theory. Let's take Kimura's neutral theory. Of course you'll find hunters in the literature. The point is that theories are cheap. Coming up with a model that says "well we know physics, so let's apply the free energy principle to evolution". The point is that it's the force of argument informed by the theory that eventually becomes the cradle of science.

Me: But you don't even cite them, which is...

Papadimitriou: Yeah, we didn't know at the time. So the second part is where we explicitly say it's a derivation. It's a derivation from the equations of Fisher and Wright. In other words, its a proof. Not a postulation. You understand what I say?   

Me: I Understand what you're saying, but it also rests on the fact that - something that biologists all agree on, which is that variation in a population is a good thing. And it's important for evolution. Without it evolution basically does not work, and you're saying that you are getting that through sex, but you're also ignoring mutations, which you said in the beginning as well. Everybody knows that this is something that happens as well, and especially in the bacterial populations that you were talking about where you said in the beginning that they are all sexual with conjugation and so on. But largely they are not. They are getting a lot of their variation from mutations, and this is known from experimental evolution as well.

[Here the moderator asked us to take it offline.]

Papadimitriou: So, mutations are paramount. Mutations in eukaryotes are also... relevant mutations are relevant. It's like meteorites: its important to understand what happens in everyday evolution, which means the evolution in populations. I'd be delighted to chat with you after the question session.

I did not stick around to talk with him afterwards. I had other obligations, plus I had basically gotten the answers that I was looking for. My conclusion is, based on this presentation and on reading the papers, that the three components at play in mixability
  1. variation drives evolution
  2. sexual reproduction increases variation through recombination
  3. free fitness is what is (attempted) optimized by populations
were already known prior to 2006. Then, showing this mathematically and by simulations is cool. But claiming that you have invented something new is not. And researchers can say that they just didn't know about the previous work, and that may be true, but as I often say, knowing the literature is half the job. Failure to give credit where it's due is egregious, especially if the following papers also do not cite the right source. None of their papers cite Sella and Hirsch**, even though Papadimitriou is familiar with the paper, and even speculated that one of them could have been a reviewer on the first paper from 2006.

On his Berkeley homepage Papadimitriou writes this about the 2008 paper:
There had been no satisfactory explanation of the advantages of sex in evolution, and yet sex is almost ubiquitous among species despite its huge costs. Here we propose a novel explanation: Using standard models, we establish that, rather astonishingly, evolution of sexual species does not result in maximization of fitness, but in improvement of another important measure which we call mixability: The ability of a genetic variant to function adequately in the presence of a wide variety of genetic partners. 
So the claim to novelty remains, and that seemingly includes a rediscovery of epistasis (the genetic interaction between genes)!

As you can also see from this paragraph, the other part of the story is that mixability is touted as an explanation for the evolutionary origin of sex (i.e., recombination). To that I just want to say that if you postulate a model in which there are no mutations, then yes, recombination saves the day, as it is now the only source of variation. However, this is clearly unrealistic in the extreme, and evades the real question, namely how could sex evolve when clonal growth is so much more efficient and asexual populations (which do exist!) have been doing so well since the dawn of time?

Adit Livnat took everyone by storm (I think) by writing the following final paragraph in 2010:
If sex is tied to the nature of genes, then one may reconsider the question of the origin of sex. Although it is common to imagine evolution as an originally asexual process that became sexual at some point, it is possible that sex had existed in a primitive sense of mixing before the emergence of genes as we know them, and that the interaction of sex and natural selection played a role in the shaping of the genetic architecture [2].
In other words - and I also have this from him verbally at the 2014 Evolution conference - Livnat thinks that sexual reproduction came first, followed by asexual reproduction later in the history of life. Papadimitriou mentioned this in his talk as well, saying that there are basically next to no species that are asexual - even bacteria have sex - and those that lost the ability to have sex are evolutionary dead ends. If you think evolution without sex is impossible, I dare you to explain why Lenski's E. coli populations - which do not exchange  genetic material laterally (i.e., between individual cells) - have been successfully evolving and adapting since 1988 [source].

* I am not the only only one who thinks this track for submissions to PNAS has got to go, but do see this article [6] for another take.

** In researching this topic I also learned that the idea of free fitness originates with Iwasa (1988) [7].

Kyushu University
Department of Biology

[1] Livnat A, Papadimitriou C, Dushoff J, and Feldman MW (2008). A mixability theory for the role of sex in evolution. Proceedings of the National Academy of Sciences of the United States of America, 105 (50), 19803-8 PMID: 19073912

[2] Livnat A, Papadimitriou C, Pippenger N, and Feldman MW (2010). Sex, mixability, and modularity. Proceedings of the National Academy of Sciences of the United States of America, 107 (4), 1452-7 PMID: 20080594

[3] Livnat A, Papadimitriou C, and Feldman MW (2011). An analytical contrast between fitness maximization and selection for mixability. Journal of theoretical biology, 273 (1), 232-4 PMID: 21130776

[4] Chastain E, Livnat A, Papadimitriou C, and Vazirani U (2014). Algorithms, games, and evolution. Proceedings of the National Academy of Sciences of the United States of America, 111 (29), 10620-3 PMID: 24979793

[5] Sella G, and Hirsh AE (2005). The application of statistical physics to evolutionary biology. Proceedings of the National Academy of Sciences of the United States of America, 102 (27), 9541-6 PMID: 15980155

[6] Rand DG, and Pfeiffer T (2009). Systematic differences in impact across publication tracks at PNAS. PloS one, 4 (12) PMID: 19956649

[7] Iwasa, Y. (1988). Free fitness that always increases in evolution Journal of Theoretical Biology, 135 (3), 265-281 DOI: 10.1016/S0022-5193(88)80243-1


  1. I think the idea that early evolution was dominated by mixing wasn't at all new with Livnat in 2010 - it's been bouncing around for ages (see lots from Ford Doolittle).

    And I don't think that all biologists all agree that variation in a population is a good thing; I certainly have my doubts.

  2. Hi Rosie. Could you elaborate on why you think variation in a population may not be a good thing?

  3. Hi Bjorn. I noticed this a bit late after it was referenced in Carnival of Evolution.

    Parts of this remind me of a mini-debate among computer scientists in the genetic algorithms literature (decades ago) about whether evolution happens by mutation or recombination. The fact that such a debate would happen sounds ridiculous at first glance. However, computer scientists had a habit of initiating genetic algorithms with populations of random bits (whereas no biologist would ever imagine a population simulation of gene X that began with completely random sequences the length of gene X). So you could run a GA and solve problems just from the initial variation, with recombination.

    The other thing that made the "recombination" position superficially plausible to computer scientists is that it aligns well with the rhetoric used by the people that, according to the history we tell ourselves, founded evolutionary theory-- the architects of the Modern Synthesis. If you read Provine, 1971, the key argument that he calls "the effectiveness of selection" is that "selection" (which actually means selection plus recombination) can move a population well beyond the initial range-- for the founders, this was the essence of evolution, and it showed that the mutationist view of evolution by fixation of individual mutations was unnecessary and wrong. For instance, in "The Resistance to Darwinism and the Misconceptions on which it was Based", Mayr (1994) said that "Those authors who thought that mutations alone supplied the variability on which selection can act, often called natural selection a chance theory. They said that evolution had to wait for the lucky accident of a favorable mutation before natural selection could become active. This is now known to be completely wrong. Recombination provides in every generation abundant variation on which the selection of the relatively better adapted members of a population can work." (p. 38) No one inside evolution believes this anymore, but-- due largely to amnesia-- the shift is not addressed.

    Does the mixability theory make any novel predictions? It isn't necessarily a bad thing to introduce a new terminology for old things, because evolutionary biologists have shown an unfortunate tendency to accept, and even to promote, the shifting of terms to cover new findings.

    By the way, in a recent review,, David McCandlish and I discuss the analogy with statistical physics in relation to origin-fixation models. We cite Sella & Hirsch, as well as Iwasa. Interestingly, there was an even earlier treatment by Vogel and Zuckerkandl (1971), although David tells me the math is flawed.

  4. Hi Arlin. That's really interesting that the focus of recombination in GAs comes from starting with a random sequence. As you say, to a biologist that is not a likely situation.

    What are some examples of shifting of terms to cover new findings? Do you think that introducing mixability is beneficial?

  5. I don't know anything about the mixability theory that would make it superior, so I'm asking you. My point is that, given 2 theories that say roughly similar things, we could judge a new theory B superior if it simplifies the world, while the old theory B is a patchwork designed originally to say something different. For instance, the theory of selection plus "constraints" covers everything, because "constraints" covers everything, but this doesn't make it a good theory.

    With regard to shifting terms, the most interesting example is "natural selection", which meant something different to Darwin and his early followers. As historian Jean Gayon argues in _Darwinism's struggle for survival_, "the most important event in the history of Darwinism" was when early geneticists reconceived of selection as a force that shifts the relative frequencies of true-breeding types. True-breeding types were not part of Darwin's evolutionary thinking. Today, for instance, we attach a selection coefficient to an allele, or to the difference it determines phenotypically. Darwin could not have done that. For him, hereditary substances were like fluids, without constancy-- they shifted during growth and development, and then blended together in offspring. There was nothing constant to attach a selection coefficient to.

    In Darwin's thinking, "natural selection" was a compound process that shaped organisms, via hereditary phenotypic fluctuations subject to blending under the struggle for life. This explains why, in the Origin of Species, Darwin often says things that are incompatible with our Mendelian conception of selection, e.g., he says that "natural selection" can act only by infinitesimal differences, and that "natura non facit salta" must be true under his theory. Those statements were true for Darwin's "natural selection", but untrue for our concept.

    Other examples-- "Darwinism", "neo-Darwinism", and the "Modern Synthesis" all have shifted in their meaning. I talk about this "process of amnesia and theory-drift" over the Modern Synthesis in this blog:

    The concept of "random mutation" continues to shift, e.g., Merlin (;view=fulltext) essentially argues that one can accept the arguments of Jablonka and Lamb, and the advocates of directed mutation, and still claim that mutation is "random" because, in all of these cases, mutations are not *inevitably* advantageous.

  6. The paper by Sella and Hirsch is in the weak mutation/strong selection/small population regime, while that of Livnat et al is in the strong recombination/weak selection/large population regime, so the appearance of entropy there applies to entirely different entities.

    1. Hello Lee.

      What difference does it make? The free entropy introduced by Iwasa is a general term that does not depend on population size, etc.


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