How fast can evolution work?

Jan 25 2007 Published by under Uncategorized

There's been a bit of talk about "Evolutionary Speed Limits" over at the Intelligent Design weblog Uncommon Descent. Most of the discussion involves "Haldane's Dilemma." This concept is rooted in an article written by the noted evolutionary geneticist J. B. S Haldane in 1957. There's a lot of math involved, and you can see it over at the Wikipedia page I linked above. The bottom line, for those not interested in the math, is this: according to Haldane's calculations, a species cannot reasonably fix beneficial mutations (a particular mutation becomes "fixed" when it is present in all of the population) at a rate any faster than 1 mutation per 300 generations.

A number of anti-evolutionists have taken this as evidence against evolution. If, they argue, genetic changes can only be fixed at a rate of 1 per 300 generations, how can evolution possibly explain the differences between species like humans and chimps, where not nearly enough generations have passed to account for the number of differences that we observe. There are a number of problems with using Haldane's calculations in this way, and in this post I'm going to look at one of those - the one that I think is the most important. For clarity, I should probably make sure that I am very explicit about what, exactly, the problem is before I start, so here it is:

Using Haldane's 1 substitution per 300 generations as a speed limit for all evolution is wrong because Haldane's calculations and concerns only apply under certain very specific circumstances.

Having just said that, I will now proceed to totally ignore Haldane, his dilemma, and his mathematics for most of the rest of the post. Instead of trying to look at equations, abstract concepts, and other confusing things, I'm going to use a set of very simple examples to show how a new mutation can spread through a population until it reaches a point when all other versions of that gene are gone.

These examples are necessarily going to be very much simplified. The population sizes will be very small and the effects of the mutations will be very large because I want the changes to be obvious very quickly. This is not because I want to trick you into thinking that evolution really moves really, really fast, or make things seem easier than they really are. The simplification has one purpose and one purpose only: it makes the math easy and the results clear.

Let's start with the basic scenario that is going to be used in these examples, and look at the hypothetical population as it exists before any mutant forms arise. The starting population will consist of ten individuals, each of whom is about to have ten children. The whole starting population will die right after reproducing (this is called "discrete generations," and it makes the math easier). The children will be mature in two years - if they live that long. Most won't. Only 20% will make it until the end of the first year, and only 50% of those who do will make it through the second.

Here's how this population changes from generation to generation:

Table 1: The basic population.
generation 1 2 3 4 5
starting size 10 10 10 10 10
# of offspring 100 100 100 100 100
# at age 1 20 20 20 20 20
# at age 2 10 10 10 10 10

It's easy to see in this example that the population size is absolutely constant from generation to generation. Now, let's look at what happens if we add a favorable mutation that works early on in the life history of this population. This mutation will arise in one individual in the new starting population, and will result in offspring having twice as much of a chance to make it through that first year - 40% will survive instead of 20%.

Table 2: The first mutation.
generation 1 2 3 4 5
starting size standard 9 9 9 9 9
mutant 1 2 4 8 16
# of offspring standard 90 90 90 90 90
mutant 10 20 40 80 160
# at age 1 standard 18 18 18 18 18
mutant 4 8 16 32 64
# at age 2 standard 9 9 9 9 9
mutant 2 4 8 16 32

In this example, the population as a whole is growing - at the start, there were 10 individuals (9 "normals" and 1 mutant), and at the end there were 41 (32 "mutants" and 9 "normals"). This means that the frequency of the mutant gene was becoming more common over time. At the start of the scenario, 10% of the population were mutants, and after 5 generations that number had climbed to 78%. At the same time, though, the numbers of "normal" individuals weren't changing much - there were still nine every generation. As long as no mechanism is in place to eliminate them from the population, the mutant gene will never become fully "fixed." The "normal" form will become very, very rare, but it will still be there. (By my calculations, after about 15 generations, less than 0.02% of the population will have the old "normal" form.)

So how do mutants become fixed, then? There are quite a few different ways, all of which work, but I'm just going to look at one of the possibilities. In that last scenario, the population was growing. We know that doesn't normally happen - most populations are relatively stable. So let's add one more assumption: no matter how many individuals make it to reproductive age, only 10 will reproduce, and those 10 will be randomly selected from those who get that far.

Table 3: The first mutation, with constant population size
generation 1 2 3 4 5
starting size standard 9 8 7 5 3
mutant 1 2 3 5 7
# of offspring standard 90 80 70 50 30
mutant 10 20 30 50 70
# at age 1 standard 18 16 14 10 6
mutant 4 8 12 20 28
# at age 2 standard 9 8 7 5 3
mutant 2 4 6 10 14
# reproducing standard 8 7 5 3 2
mutant 2 3 5 7 8

I could continue to carry that out, but I think that's enough to show the trend. In this case, the population size stays the same. The gene frequencies change more slowly than they did in the last example, with the unlimited growth, but the trend is the same - the mutant increases in frequency while the standard form decreases. Here, unlike in the previous example, the numbers of "standard" individuals in the population also drop, and the "mutant" will become fixed fairly quickly.

Moving back to Haldane (finally), here are the important things that this example demonstrates:

  • The change in gene frequencies changes (in both versions of the example) without any decrease in the overall population size.
  • The changes in numbers of individuals for each of the two "versions" occur without any increase in the number of births per parent.

The first of those two points is important because Haldane was looking at the maximum practical rate of evolution in cases where the environment had changed, and only the "mutants" were able to survive (and/or reproduce) at the old rate. His 1 substitution per 300 generations was the maximum rate at which the substitution could proceed, under those circumstances, without the population size dropping so far that extinction would become very likely. This very specific situation is sometimes referred to as "hard selection." The situation I outlined does not take place in a changed environment, and does not result in any changes in population size. This is sometimes referred to as "soft selection," and in situations like this the rate of change can be much faster because there is no need to worry about the effects of a shrinking population.

The second point is important because some have claimed that reproduction places speed limits on evolution, and that any substitution requires more reproduction if it is to spread. As the scenario I outlined shows, substitutions can occur without the need for more births, as long as the selection is opertating in a way that makes survival to reproduction more likely.

A second, separate point that has been raised involves the question of how many mutations can be in the process of becoming fixed in a single population. The anti-evolution objections to the speed of evolution assume that only one mutation can be moving toward fixation at a time. This is incorrect, but this post has already run long enough, so I'll save that point for another post in a couple of days.

21 responses so far

  • chezjake says:

    Well done. Quite easy to understand the way you have laid it out.

  • Jaime says:

    I like your analysis; it seems that an important parameter in this example would be the survival improvement factor that the mutation affords. If one lowers this likelihood of survival to a more modest rate (from the current factor of 2 increase) what happens? Is doubling the survival rate (essentially the chances of eventually reproducing) reasonable for a mutation? My own crude calculations (encumbered with assumptions and biases) show that the generational mutation rate is most sensitive to this parameter. In fact if the increased survival rate is only 10% higher (a factor of 1.1) then the mutation NEVER propagates and thus cannot become fixed.
    Great topic and nice work, I'd like to hear your views...

  • jufulu says:

    Risking muddying up the example and missing the point, it occures to me that running multiple discreet mutations at once would accelerate the process of evolution.

  • FhnuZoag says:

    I've had a read of the wikipedia article, and, well, yeah, I don't see how Haldane's Dilemma can be construed as a limit for evolution.
    The critical marker is that Haldane defines 'deaths due to selection' as the difference between the survival rate of the current population with the survival rate of the optimal population - that's where the strong reliance on a situation of sudden adverse change in environment comes in. A strongly advantageous mutation, then, would in Haldane's model create in fact a very *high* selection death value, because there would be a large proportion of the imaginary optimum population could be alive but aren't because the mutation has not propagated. Perhaps a better term would be 'wasted survivability potential', or something.
    The bounds appear because we are holding the top end constant, and looking at how much the bottom end must fall so that the top stays at the same position, something that is irrelevant once we allow the survivability of the optimal population to increase over the original population in its classical situation. Hence, Haldane is actually very useful in explaining mass extinction events, but rather less useful for, say, human evolution, since we can say with a fair degree of certainty that the survivability of modern man is quite a bit enhanced over that of our ancestor species.

  • Creationist Moron says:

    So Haldane's Dilemna isn't really a dilemna? How convenient that it ceases to be so only when those skeptical of Darwinism point the Dilemna out to the public.
    If you Darwinists are going to play Orwellian word games, you shouldn't be so obvious about it.

  • wicker says:

    wasn't there a mathematician in the 19th century who calculated that a train can never go faster than 20 miles per hour, because all air would get sucked out and people would suffocate if it ran faster?
    apparently, ID folks are not aware of the fact that even mathematicians make mistakes.

  • caligula says:

    Mike,
    Concening your idea that Haldane(1957)'s weakness is that it is a "hostile environment" scenario, I will here give a counter-example. It is admittedly unrealistic as such, but it is a thought experiment to make a point.
    Assume that there were "optimally adapted" organisms (remember: a thought experiment). I.e. there are genotypes with absolute fitness 1.0. (Random deaths notwithstanding, but we ignore them here.) In fact, assume we have a whole population full of such genotypes. Using Haldane's scenario, assume then an environmental change with intensity 0.1. This reduces the fitness of the population to 0.9, save for a tiny subpopulation with an allele to counter this new environmental pressure. (In this case, the allele would originally have to be neutral, because I assume fitness 1.0; in a more realistic scenario, it would have likely been slightly deleterious.)
    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.
    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 think your scenario is a rehash of the old claim that "beneficial alleles have no cost", which is the argument that Walter ReMine has criticized the most. (And his criticism is justified, in my opinion.) Rather, I would concentrate on emphasizing e.g. soft selection and intraspecific competition.

  • caligula says:

    "The situation I outlined does not take place in a changed environment, and does not result in any changes in population size. This is sometimes referred to as "soft selection," and in situations like this the rate of change can be much faster because there is no need to worry about the effects of a shrinking population."
    This is false, in my opinion. I don't think soft selection is related to either changes or non-changes in the environment. Soft selection is related to population density. When a population outgrows the carrying capacity of its ecological niche, reproductive excess starts to go to waste. I.e. excess mortality compensates reproductive excess. This mortality is called "background mortality". Soft selection argues that such mortality, because it is denstity-dependent, is reduced by selection. For example, when predation harvests a population, starvation (due to sheer lack of resources to support and overgrown population) is eased. So, it is said that background mortality acts as a "buffer" for selection, because selection can make room for itself by reducing background mortality.
    Haldane's rather extreme limit 0.1 on the intensity of selection is very likely due to Haldane's view that density-dependent deaths are mostly random; only after they have been applied, selection takes place. This is how Haldane(1957) handles the only example on the effects of density (moths and their larvae), anyway. Thus, selection does not reduce background mortality. Instead, background mortality remains fixed in magnitude and (at least mostly) random in terms of individuals, thus limiting the available intensity of selection (not resulting in extinction).

  • caligula says:

    caligula:
    "Haldane's rather extreme limit 0.1 on the intensity of selection is very likely due to Haldane's view that density-dependent deaths are mostly random"
    I hasten to add that Haldane's limit 0.1 for average intensity of selection is hardly considered universal and unavoidable by Haldane (as opposed to ReMine). Haldane(1957) allows for much more intense selection. I suggest that Haldane's limit is based on the assumptions that populations suffer from density effects most of the time, and that these effects are mostly non-selectable. But I can think of other reasons, too:
    1. Haldane's limit 0.1 is just a hunch! The number 0.1 may additionally be motivated by the fact that math is much more cool when selection is not very intense. Multiplicative fitness can be approximated with additive costs, for starters. More seriously:
    2. Haldane's limit 0.1 is his attempt to estimate the average speed of morphological change during the entire historic evolution of e.g. vertebrates. This includes a remarkable number of lineages with their periods of stasis and rapid evolution. Hence, if punctualism widely applies, Haldane's limit would hardly ever become realized (selection is either much less or much more intense). There is no sound reason to expect that stasis dominated the hominid evolution leading to Homo sapiens.

  • Christophe Thill says:

    I may be wrong on this, but I think that Haldane's model corresponds to only one type of evolution : the one in which a whole species gradually turns into another (ther's a word for it, but it slipped from my mind). The one that creationists love so much to caricature with their "why are there still monkeys?" pseudo-argument.
    But isn't it possible to take another definition of the problem? In this one, we don't care about the new gene taking over the whole population. We just want it to form a stable sub-population, to pass above the theshold where it might become extinct. We may suppose that the mutation prevents interbreeding: for instance, by delaying the mating season a little, or something similar. Then we'll have a branching event.
    Wouldn't this less strict model allow evolution tu run faster?

  • FhnuZoag says:

    "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."
    Well, the trick is to have 1.1 be the relative fitness advantage over the original population. In fact, this does not assume that the original population was not optimal, because due to competition, the original population's fitness would fall once it encounters the new mutated population.
    Why does this avoid Haldane's scenario? From my reckoning, the 300 generation limit arises because of the following:
    First, we set the final mix of mutation to standard to be a 'fixed population'.
    Now, to reach this, we put in conditions that 'death due to selection' can only be so high.
    Finally, we find 300 cycles to be the minimum that would not raise death due to selection so high.
    The thing is, only in Haldane's catastrophe model does death due to selection count as actual deaths, because of the initial drop in fitness across the board. In the beneficial mutation model, it is merely wasted potential. As for the reduction in fitness due to competition, this occurs because of physical constraints which try to keep the population constant, and so there is no extinction threat to the population as a whole.

  • Alex says:

    Also, he assumes that all mutations are of equivalent value.

  • caligula says:

    "due to competition, the original population's fitness would fall once it encounters the new mutated population"
    Fair enough. Then we would be including "intraspecific competition". I kind of feel that the idea of intraspecific competition should be explicitly emphasized when it is used, because the typical scenario is a population adapting to its environment, whether or not the environment suddenly changes. So perhaps Mike implicitly thinks like this: when a subpopulation becomes better at avoiding predation, predators will pester others in the population even more; when a subpopulation becomes better at hunting their food, there will be less food for others. Hence, the mean fitness of the population does not need to increase during evolution, because genotypes often compete with each other in such a way that the success of one genotype may actively decrease the success of another, in terms of absolute fitness.

  • I also think the IDists here are ignoring the evolution of evolvability. The evolutionary process is able to tune itself for optimal adoption of novel mutations to some extent using mechanisms such as plasmid/phage gene transfer (in bacteria), sexual reproduction, mutation rate control, etc.

  • JS says:

    I like your analysis; it seems that an important parameter in this example would be the survival improvement factor that the mutation affords. If one lowers this likelihood of survival to a more modest rate (from the current factor of 2 increase) what happens? Is doubling the survival rate (essentially the chances of eventually reproducing) reasonable for a mutation? My own crude calculations (encumbered with assumptions and biases) show that the generational mutation rate is most sensitive to this parameter. In fact if the increased survival rate is only 10% higher (a factor of 1.1) then the mutation NEVER propagates and thus cannot become fixed.

    Just for giggles, I ran that analysis (i.e. same assumptions as Mike, except for a carrying capacity of 1000 and a starting mutation count of 10, i.e. 1 %). In my computation, 90.6 % of the population was mutant after 73 generations, after which the normies declined exponentially, or near enough as makes no matter.
    Of course, for a better analysis, one could write a program that randomised the reproduction (I assumed negligible statistical noise), and try it out for N_mutant(0) = 1
    That would no doubt be interesting, but would also be beyond the scope of this response.
    - JS

  • JS says:

    For further giggles, I fitted a logistical growth function to the population of mutant widgits in our example, and it matches almost perfectly. The growth factor is .099 pr. generation.
    Oh, and by way of comparison, the mutant species growing without competition from the normies (under the assumption that 11 % live to reproduce and that the max number of parents/generation remains 1000) grows to fill the habitat in ca. 50 generations.
    This number is almost certainly too high, since the reproductive rate in a competition-free environment is most certainly higher than in a fully saturated one, but it demonstrates that the fixation of our hypothetical mutation operates on roughly the same timescale as the filling of an empty habitat.
    - JS

  • MrDarwin says:

    I'm always amazed that "Haldane's Dilemma" gets so much traction when it's based on 50 year old assumptions that Haldane himself knew would almost certainly need to be revised. For example, in Haldane's day, nobody knew how many genes there were in any particular organism, nor how many genes (much less which ones) differ between any two species, nor HOW they differ. Thus Haldane's calculations were based entirely upon semi-educated guesses and assumptoins. Those who are promoting Haldane's "dilemma" as any kind of problem for evolution are operating on the very shaky assumption that we haven't learned anything about genetics or speciation in the last 50 years. Has anybody gone back and figured out just how many of Haldane's most basic assumptions were anywhere near right, and how many were wildly wrong?

  • Darby says:

    I'm coming from a bit of ignorance, but let me throw some numbers into the mix.
    Say that the number of mutations that clearly differentiates a descendant species from the ancestral one is 10, but to have any 9 out of the 10 makes a population member just slightly like its ancestor - nowhere near enough to seem a clear "throwback." If the differentiating ancestral alleles are extremely rare, then the new ones don't need to fix - the chances of a modern individual carrying a critical number of the "old" alleles, enough to make its very species suspect, becomes vanishingly small.
    What differentiates a modern human from an ancestor - hip structure, brain size, hairyness, throat anatomy, blood vessel distribution? Can we not find people who have conditions in these areas that reflect the ancestral condition? The idea of "evolutionary throwbacks" was discarded a while back, but that was based on the assumption at play here, that a new species is "genetically pure," and does that even make sense as a starting premise?

  • slpage says:

    I outline a few of the problems with the creationist (i.e., ReMine's) use of Hasldane's dilemma on the Wiki Haldane's dilemma discussion page, which I will sum up here:
    ReMine does not know what the ancestral population was; he does not know, therefore, what traits it lacked that humans possess; he does not know how many mutations are required to produce those traits (if he knew what they were).
    His entire argument is built on quicksand, and his followers are too dim to see it.

  • Venkatramanan says:

    Hi,
    it's a cool article. Can I post your article in my personal home page/blog. Thanks and keep posting