The Questionable Authority

Yesterday’s post on evolutionary speed limits and Haldane’s Dilemma has sparked some interesting discussion, and some of the comments have already started to move beyond the very simple scenario that I outlined. Next week, I’ll post a couple of more complex examples, and look at the effect of things like a lower frequency of mutants in the starting population, what can happen with two mutations being selected at the same time, and whether mutations need to be fixed to be evolutionarily meaningful. I’ll also go over a couple of basic concepts that might help in understanding those scenarios.

Today, I’m just going to respond to part of one of the comments that was left on the last post. This is mostly because it’s an interesting question that deserves a thorough response, partly because the question involves some basic concepts that should be explained before I dive into more detail, and partly because it’s Friday and I really don’t want to spend the time plugging numbers in to work up another example.

Caligula, fairly early on in the comments, raises a point that involves a concept that is very basic to evolutionary biology: fitness:

Now let’s apply your scenario. Assume the initial fitness 1.0, and assume that a benefical mutation occurs. But what is the physical interpretation of a mutation which increases one’s absolute fitness beyond 1.0? Now, I do think that such an interpretation is possible, and it does not need to be ridiculous. In terms of viability, absolute fitnesses 1.1 and 1.0 behave the same, of course. There is no 110% chance of survival, after all. However, fitness 1.1 means that the offspring of a sexual parent can afford to lose a beneficial allele in segregation without necessarily suffering from reduced viability. So, under gene selection, fitness above 1.0 might still make a difference.

A distinction needs to be made here between relative fitness and absolute fitness. Absolute fitness measures (in some form – there are a couple of different measures that are possible) the reproductive output of a given form. Relative fitness measures how well different forms of the same trait do when compared with each other. The two measures look at very different things, and cannot be compared.

I’m going to write a post this weekend on fitness, as part of the series of basic concepts posts here at Scienceblogs. (John Wilkins has already written one, but he’s looking at the concept from a slightly different perspective, so I’m going to do another. ) For the moment, the important thing to note is that the comment above seems to be confusing relative and absolute fitness a bit.

But even if fitness beyond 1.0 might be somewhat useful, I believe you understand my point. I think your scenario is a mirror image of Haldane’s, rather than something qualitatively new. You likely are not suggesting that fitness can climb beyond 1.0. In order to ensure this, you must implicitly assume that the population is less-than-optimally adapted. So, when you say the a beneficial mutation with coefficient 0.1 occurs, you are probably implicitly assuming that the mean fitness of the population at that moment is at most 0.9. Compare to Haldane’s scenario above to see that the math is now identical, even if the biological “background story” might be slightly different.

I see no problem at all with assuming that the population is not “optimally” adapted. Actually, I see more of a problem with assuming that any population is ever optimally adapted. All of the members of a population might, at a particular point in time, have the best available genetic makeup for a particular trait, but that does not necessarily mean that they have the best possible genotype.

This is actually where the difference between relative and absolute fitness becomes very important, because it shows a situation where the relative fitness of a geneotype can change without the need for any corresponding change in the absolute fitness. This is exactly what happened in the second example I showed, where the size of the population increased rapidly because of the increased number of survivors among those carrying the mutation. The absolute fitness of the old population did not change at all. The offspring of the “normal” individuals were just as likely to survive to reproduce ten generations in as they were before the first “mutant” showed up. The relative fitness of the old population, on the other hand, changed a great deal. Initially, it was the most fit genotype present (becuase it was the only genotype present). When the new form appeared, however, it was more fit than the “normal” form, and therefore had a higher relative fitness.

Comments

  1. #1 caligula
    January 26, 2007

    Mike,

    Thank you for your response.

    I believe I made it clear in my comment, as a whole, that I recognize what absolute fitness is, compared to relative fitness. I emphasize early on that my counter-example assumes an absolute fitness 1.0, and that I use such a thought experiment to make a point.

    My point is that your scenario, unless you specifically define it to do otherwise, still concerns adaptation of a population to its environment. Yet you say that we can forget environmental changes. Now, assuming that the environment does not change, unavoidably adaptation will at some point reach its goal: absolute fitness 1.0, or complete adaptation to the enviroment. Otherwise the fitness can grow no more. In terms of relative fitness: after the population reaches the “optimal genotype”, then what? The environment needs to change in order to provide further challenges for adaptation. And if the environment does change, well, then the scenario becomes mathematically identical with Haldane’s scenario. One might argue that the former scenario is somehow more “gentle”, but I don’t think it is true. If it is more gentle, it merely assumes a gentler “genetic load” which means a slower substitution rate.

    However. I do not dispute that there could be selection which does not involve adaptation to external challenges. I call it “intraspecific competition” (borrowed from Nunney(2003)), but perhaps I should call it “pure intraspecific competition” (because intraspecific competition might manifest itself in some form even when a population is adapting to its environment). In the “pure” scenario, beneficial alleles exclusively help individuals to gain benefits at the expense of others in the same population. Such alleles might, for example, (a) help an individual to become a clan leader in a harem species, (b) win a much-contested territory, etc. (Entirely fair examples are not easy to come up with, because no trait probably fits exclusively in one class of challenges.) In such a case, one can truly say that the environmental challenges are essentially not involved. I have simulated this scenario and verified that, as I’ve said at PT, the intensity of selection can really jump through the ceiling. I also think that it might be quite relevant in the evolution of Homo sapiens, because it is possible that during “good times” (DaveScot’s scientific term), selection in hominid societies has mainly concerned things like the following.
    Individual selection (competition within a clan):
    – winning status
    – winning allies to gain more status
    – eventually, winning the leadership and a right to reproduce
    – gaining the status of the leading female

    Group selection (competition between clans):
    – gaining territory

    But this scenario is hardly “orthodox”, and I’m currently consulting an authority concerning its biological feasibility. If it is acceptable, I’m sure I will be given references to works that handle it more rigorously than my amateurish treatment.

  2. #2 caligula
    January 26, 2007

    “I also think that it might be quite relevant in the evolution of Homo sapiens, because it is possible that during “good times” (DaveScot’s scientific term), selection in hominid societies has mainly concerned things like the following.”

    A fumble. It has little to do with “good times” or “bad times”. Pure intraspecific competition can take place during both, and at a remarkable intensity. Pure intraspecific competition simply does not affect the absolute mean fitness of the population.

  3. #3 BC
    January 26, 2007

    So, the definition of “absolute fitness 1.0″ means that 100% of the children born will survive to adulthood and have the maximum number of children possible (and that number is the same for all individuals of fitness “1.0”) – or, at least, death is meted out completely randomly to the population? That doesn’t really sound like something that happens in the real-world. Further, if this did happen, it means that no “forward” selection pressure could exist. (Although deleterious mutations that dropped an individual’s absolute fitness below 1.0 could be culled from the population.) The population would enter into a “stasis” in that case. I don’t really see the particular problems with suggesting that populations could enter a stasis. But, I think the bigger problem is claiming that an individual could have an absolute fitness of 1.0. It might be easy to write that number down in your calculations, but I think the world is a complex enough place that absolute fitnesses of 1.0 generally don’t exist.

  4. #4 caligula
    January 27, 2007

    BC:
    Note that the optimal genotype in each iteration of Mike’s scenario keeps increasing its absolute fitness compared to the optimal genotype of the previous iteration. The optimal genotype will hit absolute fitness 1.0, and it will do so quite fast if we assume a decent rate of substitutions with decent coefficients. Then what? (And if we tuned down coefficients, we would eventually be discussing nearly neutral theory where sampling effect rules over selection. And we still might not get as many adaptations as we want (thousands upon thousands)!) I maintain that the environment has to change if we are to keep adapting to it.

    In the “pure intraspecific competition” scenario, the problem does not exist. Say we have a new mutant whose incresed ability in fighting over territory gives it a better chance to win one, thus increasing its absolute fitness. But because the ecological niche is finite, the success of the new genotype reduces the absolute fitness of other; they are less likely to win a territory, let alone the best territory. Their absolute fitness decreases. In terms of relative fitness, we have a genetic load between an optimal genotype and the genotype of the majority. But note that when the mutant genotype increases its frequency, absolute fitnesses behave in a peculiar way. The absolute fitness of the optimal genotype will start decreasing towards the original mean fitness (absolute fitness before the mutation). When everyone learns the good trick, no one has an edge over others anymore. Thus, the absolute fitness does not increase in the long run. (To be honest, with extreme rates of substitution this scenario does produce artefacts that I call “uber individuals”, however. Uber individuals temporarily climb above 1.0 absolute fitness, because they have luckily inherited a whole bunch of beneficial alleles which still have a low frequency. This makes me wonder if intraspecific competition can be as intense in nature as it is in my simulations — or whether epistatic fitness interaction is needed in modeling a lot of concurrency. And this is why I have had to think about a physical interpretation for absolute fitness 1.1!)

  5. #5 caligula
    January 27, 2007

    BC:
    Oh, and about your question concerning absolute fitness 1.0. In my original counter-argument, I mentioned “random deaths”. These are lethal environmental factors that are so rare that at least most populations don’t have means to adapt to them. (Most species can’t adapt to forest fires, but some plants have.) This effectively removes the concept of absolute 100% survival. With such an addition, I don’t think absolute fitness 1.0 (or even occasionally above that) is necessarily an absurd concept. But this is just me and I leave more qualified people to make the final judgement.

  6. #6 BC
    January 27, 2007

    Wow. I wrote a computer program to run through Haldane’s Dilemma. I’m getting all the numbers out of it that I’m supposed to (so I must be doing something right). But, on close inspection, I really understand why it is that Haldane’s Dilemma isn’t the hurdle that IDists say it is. As long as you run the simulation exactly how Haldane says, “Haldane’s Dilemma” is a problem, but his idea is overly narrow in how he defines natural selection mechanism. Making a small (and very reasonable) adjustment to his idea, and you can get extremely fast propagation of alleles through a population. This so-called “speed limit” is actually a farce.

  7. #7 Joe Felsenstein
    January 27, 2007

    The discussion here seems to have two alternatives: “soft selection” in which selection affects competitive ability (but not growth rate at low population density) and selection in which absolute fitness is fixed (reaching a maximum at 1.0 if it is viability). This is overly restrictive.
    If you have a population with density dependent population size regulation, a mutant that increases relative fitness can be selected both at low and at high population density, without the equilibrium population size being affected much, and without this being soft selection. Thus we can discuss advantageous mutations and whether they create a cost (either in the sense of a need for reproductive excess to prevent extinction, or a necessary reproductive excess to accompany change of gene frequencies) without this being a case of soft selection. Or have I missed this point being made already in this discussion?

  8. #8 Nigel Depledge
    January 27, 2007

    Caligula, you point out that, in an unchanging environment, a population will eventually become optimally adapted. This is not necessarily the case. Recall that selection works only upon variation that exists as a result of mutation and recombination. It may be that the precise set of mutations that could give rise to optimal adaptations never occurs, as this is basically a stochastic process.

    Aside from this, there may be several strategies that could each be equally successful, but require different adaptations for the same habitat.

    Additionally, consider the facts that (a) evolution of a species never occurs in isolation – the predators and prey (or food plants) are also evolving, and (b) individuals of a species compete most strongly amongst one another, so optimal adaptation (absolute fitness 1.0) is far less important than relative fitness among individuals of the same population.

  9. #9 caligula
    January 27, 2007

    “Recall that selection works only upon variation that exists as a result of mutation and recombination. It may be that the precise set of mutations that could give rise to optimal adaptations never occurs, as this is basically a stochastic process.”

    Certainly. But then there would be no substitutions due to selection either, would there? And as I’ve said: gradually tuning down selection coefficients, so as to never reach absolute fitness 1.0, doesn’t help either. This would 1. be unrealistic, 2. decrease the substitution rate, and 3. eventually become nearly neutral substitution, i.e. effectively mere drift.

  10. #10 caligula
    January 27, 2007

    “Additionally, consider the facts that (a) evolution of a species never occurs in isolation – the predators and prey (or food plants) are also evolving, and (b) individuals of a species compete most strongly amongst one another, so optimal adaptation (absolute fitness 1.0) is far less important than relative fitness among individuals of the same population.”

    Agreed. But (a) invokes environmental changes, which turns it equivalent with Haldane’s scenario, and (b) is not what we typically understand with adaptation. (b) may be a major factor in evolution, and as you can see from my earlier posts, I’m very interested in and simulating (b). But I would not yet declare that (b) is the most influental factor. I would only declare that if it is, it is very powerful indeed. Adaptation to environment just seems to have its severe limits that are missing from (b), even though nobody, Haldane himself included, assumes that Haldane’s “magic number” applies universally and unavoidably. (Except ReMine, of course.)

  11. #11 JS
    January 28, 2007

    I am not a biologist, so I don’t know the textbook answers to the points caligula raises. I do, however, see two problems with his scenario: It assumes that the habitat is uniform in both time and space. This is obviously not the case.

    Taking the simple issue first, it seems reasonable to assume that any species will fill all available space fairly quickly. ‘All available space’ includes regions that are borderline w.r.t. habitability (think humans in the arctic). For such a habitat, there is no unique fitness function; what works in part of the habitat may not work in other parts. Thus, in different areas of the habitat, different genotypes will have the best relative fitness.

    Further, Haldane’s dilemma involves, IIRC, a situation where the entire habitat changes at the same rate. This is rarely the case. Most of the time, different parts of the habitat changes at different rates. While this might indeed cause extinction in some regions, it will almost certainly leave a viable population in some neigbouring region.

    Under the assumption of rapid migration stated earlier, members of this viable population would spread into the changed environment, thus both preserving sufficient overall population to produce a reasonable mutation frequency and applying selection pressures in excess of what an isolated population could endure.

    The more interesting objection, however, revolves around the assumption that the environment is unchanged in time. From my (admittedly simple) perspective, it would appear that an environment can be both changing and constant at the same time, if there are processes in play that operate on radically different timescales.

    To illustrate this point, suppose you have a population of widgits that lives, reproduces and dies over the course of one minute. To such a species, the environment changes radically over the course of one day.

    A mutation that allows our widgits to better survive being out in the sun would be a massive advantage around noon, and might quickly spread to all the parts of the population that live outdoor.

    When it turns out, several hundred generations later, that the same mutation causes our widgits to glow in the dark, the mutants will be hunted down by night-active birds, and the ‘normies’ will crawl out from their in-door havens and frolic in the near-empty habitat.

    But has the environment changed? The glowing widgits would certainly tell you that yes it has, and they’re none too pleased about it! But on a human timescale (decades), the environment hasn’t changed at all.

    Which genotype of widgits have the higher absolute fitness? The question is meaningless: It depends on the timescale involved. If you look at ‘only’ a hundred generations, and you look during noon, the mutants are clearly the most fit. But if you look at a timescale of days, the normies are clearly the most fit, as long as they do not get pushed out completely during the noon hours.

    Combining these temporal and spacial variations, it becomes clear why you can have rapid evolution and still not reach a fitness of 1.0

    Admittedly, I am not a biologist, and I may have made some fairly outragous errors in my analysis, but I think it seems reasonable on the surface.

    – JS

  12. #12 caligula
    January 28, 2007

    JS:
    Yes, you raise and interesting point which I have also mentioned at UD I think. Assuming non-uniform environmental pressures might indeed create something interesting, especially if we assume that individuals behave in such a way that they favor “subenvironments” where their genotype is the most beneficial. Assume an ecological niche which is divided into N “subenvironments”, each imposing a unique selection pressure missing from the other subenvironments. Also assume N subpopulations, each with a beneficial allele (at separate loci) to counter the unique pressure in one of the subenvironments (but behaving like a nearly neutral allele in the others). The population would still more or less mate randomly, like a whole species; let us assume, for example that males look for a mate in whichever subenvironment, and after each mating return to their favored subenvironment. Lucky offspring of parents favoring different subenviroments might inherit a genome favoring both. Eventually, we could have a population which has adapted to all of these subenvironments (although such a scenario might also result in further migration and specialization, thus weakening random mating and eventually leadin to speciations). The peculiar beneficial alleles we a are studying here, although nearly neutral in most subenvironments, should increase their frequency throughout the population, because subenvironments where they are useful behave as “factories” to produce them much faster than drift. I suppose this scenario might somewhat decrease the costs involved, because inviduals somewhat avoid environmental pressures which would harm their fitness, and yet their offspring seem to get beneficial alleles against these pressures at a pace faster than drift.

    However, unless this kind of complex scenario can be presented as a biologically feasible, rigorous mathematical model and/or simulation, it honestly is little more than hand-waving. And I don’t think I could do it; I’m not a biologist either.

  13. #13 caligula
    January 28, 2007

    JS:
    As for your second example, I think one has to concentrate on widgit timescale, not human timescale. In widgit timescale, it seems that the enviroment does change. I believe you agree that when presented as a mathematical model, it would contain the same element as Haldane’s model: adapting to environmental changes.

  14. #14 JS
    January 28, 2007

    On the timescale thing, I think you’re missing the point. From the widgit perspective there are (at least) two relevant timescales: We might call them the ‘generation’ timescale and the ‘species’ timescale, i.e. a time scale of the order of one generation and a timescale on the order of the entire era of the species.

    It is not at all clear to me that we must focus only on the generation timescale if we wish to understand the system. I think it could be reasonably argued that the environment in our example is unchanging – it is, after all, an almost completely periodic variation – and that the non-mutant widgits are the best adapted for their environment.

    Getting a little closer to a realistic example, consider fruit flies instead of our imaginary widgits. Fruit flies have a generation of ~ 1 day. Thus, on a generational timescale, the environment changes radically over the course of a year.

    Yet fruit flies clearly do not become extinct due to this rapid variation. So, if by ‘environment’ we understand only the habitat on a generational timescale, fruit flies seem to contradict outright Haldane’s dilemma.

    The point here is that, ultimately, any successful species will have to be able to endure such periodic variations – hence they can be argued to be part of an ‘unchanging’ environment. On the other hand, if the frequency of the variations it is also reasonable to treat the environment as varying. But focusing solely on one of these aspects seems unjustifiably simplistic.

    Contrary to you, I do think it is possible to make a (semi)meaningful mathematical model of a population in a non-uniform habitat under various models of migration. In fact, I would argue that the basic structure is fairly straightforward:

    First, I would settle upon a number of selective pressures.

    Second, I would distribute them on a two-dimensional surface.

    Third, I would settle upon a number of ways in which our sample population could adapt to these pressures.

    Fourth, I would settle upon certain couplings between different adaptations (camouflage suitable for arctic conditions may not be suitable for tropical and vice versa). These constraints reduce the number of independent adaptations. We may use this fact to eliminate certain adaptations from our equations, since they will be fully described by a combination of other adaptations.

    Fifth, I would settle upon the lethality of each selection pressure and the differential equations governing migration.

    Sixth, we split the habitat and the adaptations into quanta that are small w.r.t. all variations in the input data.

    Seventh, for each of these pieces, we write up the differential equations for rate of change of each genotype, i.e. each combination of adaptations. This would include a reproduction term (which may be affected by different adaptations), a (density- and genotype-dependent) death rate, and a migration term. The first two would change the overall population. The third term would not.

    It is, in principle, possible to solve this numerically in finite time (we end up with a vector equation with many, many terms), but it is, as far as I can see at first glance, an NP problem. So for any realistic size of habitat and variation of species, we would have to hand it over to the compsci people. But it is certainly doable for a small test case.

    I think I get your point about regarding the multiple sub-environments as a combined environment with a combined absolute fitness. But not only does this overlook the fact that parts of the environment may vary (quasi)independently in time, you also implicitly assume that the selective pressures are uncoupled. If some or all of them are coupled, it will, in general, not be possible to achieve 1.0 absolute fitness (recall the polar camo vs. tropical camo example).

    Further, I am not sure that, given coupled selection pressures, an overall absolute fitness would be very meaningful or interesting. It may well be that a certain configuration has the higher absolute fitness, when viewed over the whole habitat, but still goes extinct, because no matter where it goes it will have lower absolute fitness w.r.t. the local environment than the highly specialised competitors in that particular niche.

    – JS

  15. #15 caligula
    January 29, 2007

    JS:
    On the timescale thing, I think you’re missing the point…It is not at all clear to me that we must focus only on the generation timescale if we wish to understand the system.

    Perhaps we are trying to make a different point here. My point is that I don’t think Mike can escape the risk of extinction, which is limiting the intensity of selection in Haldane’s scenario, by saying: let’s forget about the environment. To get intense selection (and a high substitution rate), a high genetic load is required, which in turn creates the risk of extinction. Period. There — I made my original point without ever mentioning “environment”. (But I still think that in order to maintain a high genetic load for a long time, we need changes in the environment.) I don’t think my point is weakened by your suggestion that we might understand hominid evolution better if we analysed it both under human timescale and under “mountain timescale”. You are doubtless right, but I don’t think it addresses my point.

    you also implicitly assume that the selective pressures are uncoupled

    Yes. I do it by assuming that the beneficial alleles are at separate loci, and that they are nearly neutral in all but one sub-environment. Snow camo and jungle camo, for example, are not compatible with (at least) the latter assumption.

    It may well be that a certain configuration has the higher absolute fitness, when viewed over the whole habitat, but still goes extinct, because no matter where it goes it will have lower absolute fitness w.r.t. the local environment than the highly specialised competitors in that particular niche.

    Agreed. But we aren’t discussing the survival of any particular genotype encountered during the substitution process. We are discussing the realization and fixation of the optimal genotype, which includes segregation. The optimal genotype at least should beat all other genotypes. In case some benefical alleles are coupled, and incompatible with each other, they will have to fight it out, which will slow down substitution. Either one allele wins the competition, or the differences lead to further migration and eventual speciation. In the latter case, I think we would have more than one optimal genotype, or multiple “attractors”, which attract different sub-populations both towards a specific genotype and a specific sub-environment. I don’t see how such “coupling” would speed up substitution. as opposed to a scenario where all beneficial alleles are uncoupled. Nor does it remove the requirement for changes in the environment. The “attractors” still have absolute fitnesses in their own sub-environments, even if it is not meaningful to compare these fitnesses with each other. New attractors are needed in each subenviroment, with a higher abolute fitness than the previous one, in order to drive substitution for a long time.

    As for our ability to model this type of scenario, I remain unconvinced. Neither your description nor mine is (a) a very rigorous mathematical model or (b) necessarily biologically feasible.

  16. #16 caligula
    January 29, 2007

    caligula:
    To get intense selection (and a high substitution rate), a high genetic load is required, which in turn creates the risk of extinction. Period. There — I made my original point without ever mentioning “environment”.

    A fumble. I can hardly omit mentioning the environment, because it is a key factor here. After all, in pure intraspecific competition we can (in principle) have an enormous genetic load without any change in mean fitness, i.e. without risking extinction. I will have to rephrase the above:

    To get intense selection by means of adapting to environment, thus leading to a high substitution rate, a high genetic load in terms of external selection pressures is required. This creates a risk of extinction. Period. There — I made my original point without ever mentioning “changes in the environment”.

    Not sure if it clarifies anything anymore in this latter form, though!

  17. #17 JS
    January 29, 2007

    W.r.t. timescales, you’re right, we were talking about different things. Selection pressures operating on different timescales, if anything, would increase the likelyhood of extinction.

    But I still don’t think you get the point w.r.t. the inhomogeneous environment. The point is that if different parts of the environment change at different rates, local extinction does not equate to species-wide extinction. In other words, it does not directly circumvent the problem of local extinction, but migration makes it much less significant.

    It simply does not matter that a species has only .1 % chance to survive in a given sub-habitat, if there is a continual stream of migrants that is being pushed into the habitat. The habitat may kill off a thousand populations, but if the thousand-and-first takes root. Migration, as the Romans found out to their regret, cannot be beaten by attrition.

    – JS

  18. #18 JS
    January 29, 2007

    Gah! Must use preview. Me bad.

    The second-to-last sentence should have read “… but if the thousand-and-first takes root, the species as a whole has survived.”

    – JS

  19. #19 caligula
    January 30, 2007

    I’d like to submit a scenario for criticism, using costs listed by Walter ReMine in his papers published in creationist journals. I handle these costs as multipliers to absolute fitness.

    Assume that an average hominid couple produced 10 offspring, i.e. has a reproduction rate 5.0. Thus, the population can afford having absolute fitness 0.2 without risk of extinction.

    Cost of mutation. Let us use the formulas from Nachman(2000), assuming that harmful mutations are at equilibrium with selection. However, instead of Nachman’s value U=3, based on the estimate that humans have 70,000 genes, let U=1. Thus, the average fitness is reduced to e^-U = 1/e.

    Cost of segregation. Can someone suggest a plausible cost for sex? Until then, I assume no cost.

    Cost of random loss vs. cost of substitution. Let us assume that random losses, although certainly real, are negligible compared to cost of substitution. This is what I call “soft selection”. It is the idea that:
    1. most mortality, save for mortality inherent in evolutionary mechanisms themselves (such as mutations and sex), can be countered by beneficial mutations
    2. such beneficial mutations are so frequent that we can assume they actualize in the population quickly.
    “Hard selection” would be the idea that many causes of mortality, perhaps especially density-dependent ones, can’t be evolved against, at least not by all populations. Although there is one cause of mortality that is difficult to evolve against, namely sheer density-dependent starvation, this mortality is weakened by all other types of mortality, such as predation. I assume there is enough such mortality.

    External selection pressures can safely reduce the absolute fitness of the population from 1/e (~0.37) to 0.2. Normalizing the fitness of the optimal genotype to 1.0, the average fitness after applying environmenal pressures would be 0.2e. These environmental pressures can roughly be countered by six beneficial alleles at separate loci, with multiplicative effects and a coefficient 0.1 each: (1.0 – 0.1)^6 = 0.53 ~ 0.2e.

    In other words, this scenario would tolerate six concurrent substitutions with 0.1 coefficient each. Using Haldane’s pre-existing estimate 300 generations/substitution for each of them, we would get one substitution per 50 generations, on average. Applying this result to ReMine’s formula for his “magic number”, this would mean roughly 10,000 adaptations as opposed to ReMine’s 1,667.

    ReMine would strongly oppose the above practice. He is confident that costs from multiple simultaneous substitutions simply add up, and that this always returns the substitution rate back to “normal”. But both assumptions are based on Haldane’s limit 0.1. First, I don’t think simultaneous costs “add up” when the intensity of selection is much higher than Haldane’s limit 0.1, because multiplicative fitness and additive fitness cease to be approximately equal. Second, since we have exceeded Haldane’s estimate for average intensity of selection, we can expect a higher rate of substitution as well.

    Additionally, ReMine seems to be fond of a cost called “cost of unsuccessful substitution”. It is true that drift kills most mutations before they have time to build up enough critical mass so that selection will start to dominate over drift. However, I think this cost can be countered by simply assuming that the rate of beneficial mutations is high enough to compensate for such losses.

    For comparison. I simulated “pure intraspecific” competition with 1,000 loci and a population of 10,000. I generated beneficial mutations with random coefficients 0.01 to 0.10 at a very high rate: one beneficial mutation per 1,000,000 loci, on average. (All mutations were considered to be unique, i.e. the same mutation was never repeated twice.) After an initial delay I started to get about one substitution per 4 generations. To save simulation time, I tend to use a high mutation rate and a fairly small population; I hope this is a decent emulation of a bigger population with a lower mutation rate.

  20. #20 JS
    February 1, 2007

    I’m sceptical of Haldan’s 300 generations – I think a better approach would be to model the population numerically, but otherwise it does not look like a bad model. Whether it says something biologically interesting or not, however, is a question I’ll leave for those better informed than myself.

    – JS

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