It is well known (to the sort of people to whom such things are well known) that the Scottish engineer Fleeming Jenkin was the first to formalize a toy model of Darwin’s evolutionary theory—with results that were most unfavorable to Darwin: The model predicts that random improvements, even when they confer survival advantages, still tend to disappear over the course of a few generations. This was in 1867.
It seems to be far less well known that Jenkin’s model also predicts that all life on earth dies out after a few generations, which would seem to cast doubt on its assumptions. Jenkin was apparently unaware of this, and so, presumably, was Darwin, who gave considerable credence to the Jenkin model in the final edition of The Origin of Species. This was in 1872.
It seems to be even less well known that the inadequacy of Jenkin’s model was identified in a little-noticed letter to the editor of Nature by the mathematician Arthur Sladen Davis. In that letter, Davis corrected Jenkin’s error and supplied an alternative model that he believed was favorable to Darwin. This was in 1871, but apparently Darwin never heard about it.
And it seems to be known only to me (and now, to the readers of this blog!) that Davis’s model is also flawed, in the opposite direction from Jenkin’s, in that it predicts that any species population must grow without bound following the appearance of a beneficial mutation. And as a result of this, the Davis model undercuts Darwin more than it supports him.
I’ve adjusted Davis’s model much as Davis adjusted Jenkin’s, and gotten a result that could be considered favorable to Darwin. In fact, it’s more or less the result that Davis thought he’d gotten, but hadn’t.
Here comes the more technical part. If this is not your cup of tea, stop reading now. Do come back tomorrow though. I’m not always like this.
Both Jenkin and Davis start with a “blending” theory of inheritance that it is thoroughly at odds with the modern understanding of heredity: They assume that the traits of the offspring are some kind of average of the traits of their parents. These models, then, are therefore thoroughly irrelevant to modern evolutionary theory, with its discrete units of inheritance (i.e. genes). But they remain relevant to the question: What ought Darwin’s contemporaries have thought of Darwin’s theory?
Jenkin begins with the assumption that each individual has a “fitness level”. When two individuals mate, their offspring inherit the average of their fitness levels. An individual with fitness level, say, 1.2, produces 1.2 times as many surviving offspring as an individual with fitness level 1. Let’s make this concrete by saying that your fitness level is defined to be the length of your beak, and the number of your surviving offspring is proportional to your beak length.
Initially, everyone’s beak has length 1. Then along comes a mutant (or, in 19th century terminology, a “sport”), with beak length 2. The sport necessarily mates with an individual of beak length 1, producing offspring with beak length 1.5; they in turn mate (almost surely) with individuals of beak length 1 and produce offspring of beak length 1.25. It’s not hard to show that no matter how long this process runs, the sport can never have more than about 2.38 descendants, whose beak lengths approach 1 over time. The mutation has essentially no lasting effect.
Unfortunately, Jenkin made the phenomenal blunder of assuming that to maintain a fixed population size, each individual must have just one surviving offspring. But because each offspring has two parents, this a recipe for rapid extinction.
Davis corrected the error by assuming that each pair of non-mutant parents produce two offspring, which is indeed a recipe for stable population size as long as there are no mutants. From this assumption he finds (though he doesn’t put it quite this way) that over time the population-wide excess beak length approaches 2.38. In other words, if several generations down the line you’ve got 1000 creatures, their total beak length will be 1002.38. This looks like a small but lasting effect, which presumably can be magnified by further mutations. On this basis, Davis seems to think he’s resurrected Darwin.
The problem, though, is that in Davis’s model, all those sports cause the population to grow without bound, while the excess beak length stays constant. Starting with a population of 100, in generation 10, we have 2439 sports, each with a beak length of 1.0098, for a total excess beak length of about 2.38. In generation 15, we have 78,124 sports, each with a beak length of 1.00003, again for a total excess beak length of about 2.38. Those sports are going to be pretty hard to distinguish from a creature that never mutated in the first place. So Davis’s model predicts that mutations can lead to population growth, but not that they can alter a species.
By way of repairing this, I’ve modified Davis’s model by culling the population every generation to keep it constant (with sports culled in proportion to their numbers). The result: Once again the total excess beak length goes to 2.38, but this time it’s shared among individuals of a fixed population size. This seems to be the pro-Darwin result that Davis thought he’d gotten but really hadn’t.
My calculations, for anyone who wants to see them, are here. Let me add several caveats:
First, I am not an evolutionary biologist, nor anything that could be considered a reasonable approximation thereto. But I think that this might give me a slight advantage here, since the point of the exercise is to view evolution through the 19th century eyes of someone who knows nothing about 20th century evolutionary theory.
Second, neither am I a historian of science. Maybe there was other relevant literature, contemporary with Jenkin and Davis, that I don’t know about. I’ve searched pretty diligently and convinced myself otherwise but I could be badly mistaken.
Third, these calculations are extremely quick-and-dirty. I assumed population sizes large enough so that sports essentially never encounter other sports. This is what Jenkin and Davis assumed, and I’ve maintained the assumption—although in fact it’s quite unwarranted, in Jenkin’s case because the overall populations shrinks so rapidly and in Davis’s because the sport population grows so rapidly. Surely if you corrected for this, some results would change. It’s also possible that the results would change if instead of assuming that the number of your surviving offspring is proportional to your fitness level, we assumed only that the expected number of your surviving offspring is proportional to your fitness level.
Finally, the really interesting question here is: How should all this have affected Darwin’s thought? Was he in any sense remiss in not having done these calculations himself? I am brimming with thoughts on this matter but I think it will probably take me a little while to get them organized and written down. Stay tuned.
In your article “Don’t Vote”, you made a rough estimation of the probability your vote will matter (I think you vastly underestimated, but it was a well-stated start); what are your views on the probability of the simplest self-replicating thing capable of evolution occurring at random, and how many opportunities do you think there have been? In other words, what do you think the order of magnitude of the p-level is for rejecting the hypothesis that life began by chance?
Ben: I think I’d have to know a lot more about chemistry than I do to even begin to think about this question. It’s a good question though.
The core of the argument seems to be that, without discrete genetic transmission, a mutation would be diluted into the general population, even if the mutation has high fitness value, and therefore could not replace the “original”: if black skin has high fitness value, what you end up with is a population with lightly pigmented skin (assuming an initial population with un-pigmented skin and a mutation producing black skin).
It must be pointed out that this is due as much to sexual reproduction as to the “blending” nature of inheritance: with asexual reproduction, the offspring would have the same fitness as the parent, so that an advantageous mutation would eventually replace the original: black skin all around.
There is an error in the derivations for the Jenkin and Davis models: the limit value of f_k (fitness per sport) for k –> infinity is one, not zero. If you correct that error, my intuition is that you’ll get the same limit average sport value for all 3 models. At least, it must be the same for the Jenkin and Davis model, since they only differ in number of offspring, which is normalized away in the computation of average sport value.
Also, contrary to your statement above, you do allow inbreeding of sports in the Landsburg model (though not in the Jenkin and Davis models).
Great catch, Snorri; thanks. I’ve corrected the fitness values (as well as a few other typos) and reposted.
Given “Darwin’s finches”, I’d guess that he’d be interested in the differences between the evolutionary development of separated groups– but that kind of calculaiton requires a random input component. In the modern world we’d forge ahead and do a Monte Carlo simulation, but that’s way beyond 19th century capability.
Unfortunately, Jenkin made the phenomenal blunder of assuming that to maintain a fixed population size, each individual must have just one surviving offspring. But because each offspring has two parents, this a recipe for rapid extinction.
And then he went on to write an episode of Star Trek: Voyager (http://en.wikipedia.org/wiki/Elogium), in which a particular species is revealed to only have one reproductive cycle in their lifetime. I kept expecting some mention that they have litters, but no…
I don’t mind the way Star Trek bends physics, but the way they bend sociology, economics, and evolutionary biology drives me mad. ;-)
(Not that I’m a proponent of the literalism of the Noah’s Ark, but) It is sometimes objected that the Noah’s Ark story of starting all the animal lines back up from what Noah could have had in his Ark is that there would not be sufficient genetic diversity to allow the populations to adapt. Of course, the problem with saying that is that whenever life started in the first place, there would have been absolutely no genetic diversity, and yet life still continued.
It would be interesting (just as a theoretical question and if it hasn’t been done before) to estimate the probability of animal species continuing based upon the numbers that the Bible attributes to the Ark.
Ben, do we have enough of a sample to ask the question? After all, with n=1, we have to figure that a self-replicating bit of goo occurs with probability 1.
Steve, as an explanatory model this seems pretty good, but as a model for the whole process it seems over-simple. Consider, for example, the case in which the mutation happens and Pippi Long-Beak is eaten by a grue before breeding.
It makes me think thoughts of cellular automata and Markov systems, but I haven’t had enough coffee yet to consider opening Mathematica….
Kevin, if God is going to destroy all life except for the breeding pairs on the Ark, wouldn’t God have the foresight to ensure that the first generations have special diverse sperm and oocytes?
Sorry, but I am still unconvinced (can you tell that I have reviewed lots of scientific papers?). In the Davis models, both the sports and the “muggles” [spot the quotation] reproduce at twice the rate of the Jenkin model; therefore, at generation k there are 2^k as many individuals in the Davis model as in the Jenkin model, but the proportion of sports and muggles is the same in both models. Since the sports have the same fitness in both models (for a given generation), it follows that the average sport value must also be the same in both models (for a given generation).
I note that you have not given a derivation for the final average sport values, and that might be the source of the error.
Snorri—You’re right. I’ve used the phrase “average sport value” to mean “(total sport value)/(# of sports)” in some places and to mean “(total sport value)/(#of sports plus muggles)” in others; hence the confusion. I’ll fix this tonight.
It’s an interesting thought experiment but my understanding is that the very first premise is incorrect, because offspring do not get the average of two different traits possessed by their parents, they get either 100% of the trait of the mother or 100% of the trait of the father. Now, if your parents have many differing traits with different fitness levels, then your inherited traits will indeed have half from one and half from the other, so your fitness will be about the average of your parents’ fitnesses. However, if an adaptive trait adds 0.1 to a creature’s expected offspring count, then the child will either inherit the trait or not, so they’ll either have 0.1 added to their expected offspring count, or 0.
This is discussed in Richard Dawkins’s essay “Light Will Be Thrown” in his book “A Devil’s Chaplain”. Darwin’s evolutionary models were “pre-Mendelian” which means they assumed blended inheritance. But Darwin’s critics pointed out that blended inheritance was impossible, because it would mean that by now, all humans would look essentially alike, after all the averaging out in every generation! And it wasn’t until Mendel that people realized offspring simply either inherited a trait or they didn’t.
With traits either being inherited or not, I think it would be easy to show that a genetically dominant trait which adds 0.1 to your expected offspring count, will soon sweep the population.
For a recessive trait, I assume that what happens is that in the first individuals who possess it, it has no positive or negative effect on survival (because it’s recessive so it doesn’t manifest itself at all), so it just spreads through random genetic drift. But eventually you reach a tipping point where individuals who have the gene, have a realistic chance of mating with other individuals who have it, and so now for the first time they will produce offspring who actually *have* the trait and enjoying the advantages, and now it starts to spread faster and eventually sweeps through the population as well. (I don’t know anything about this though so I’m just assuming that’s what happens because it’s the only way to me that it seems logically possible. However, there might be another way that I just didn’t think of.)
Snorri:
Here, I think, are the necessary corrections and clarifications:
1) You are right that I’ve used the phrase “average sport value” inconsistently in the chart on page 3 of the link. In Jenkin’s case, the population eventually stabilizes at 2.38 while the sport value per individual continues to be halved each generation, so in the limit, the average sport value should be 0, not 2.38. I will correct this in the link, together with a footnote noting the original error so that your comments will still make sense.
2) Davis’s model, taken as literally as I have taken it, implies that the number of muggles eventually becomes negative (because the increasing population of sports continues to mate with muggles until eventually there are no muggles left to mate with each other). Obviously, one does not want to take this prediction too literally.
Thanks very much for making me get this right. (And do let me know if I’m still missing something.)
Bennett: I suspect you responded before reading the entire post. The point here is to see things through the eyes of a nineteenth century reader who does not know about discrete units of heredity. The ultimate question I want to get at is: Ought people in 1872 have believed that Darwin had things basically right? I plan to come back to this question in later posts.
It’s not exactly true that a trait is either inherited or not since some traits(phenotypes) are expressed as a function of the environment/this is unknowable a priori. And fits well with 19th century and Lamarckian predjudices.Also Model has to account for random wipeouts of genetic bearers not yet reproductive (infant mortality.)also very 19th century.
While individual genes obey discrete (Mendelian) inheritance and not blending inheritance, there are many genes making up any particular trait, and for those not under purifying selection (eg, IQ, height) the end result is blending and the offspring end up with an average of the parental traits. (Regression towards the mean.)
ref: a reasonable 19th century mind
I’d conclude that a prudent citizen of the 19th cent would have rejected the simple mathematics of Darwin’s idea, that was at that time-not good enough to displace competing ideas available in that period, for which I had “evidence”, even if imperfect, immediately in front of me. It took a long time to work this out, remember. The switching costs were too high, and easily available counter-examples would make one very uneasy with an imperfect model. Ironically, acceptance of Darwin’s theory prematurely would have had the same pitfalls of religious creationism. A supra-rational mind, 19th century or otherwise, would have remained perfectly agnostic–and that is way more work than easy credulity or defaulting to a set of beliefs.
Steven: Sorry, you’re right, I did actually read the entire original post including the sentence about discrete genetics, but I missed the point of that sentence, that the whole thing becomes moot once you introduce discrete genetics.
However, I think Dawkins’s essay makes a good point that is relevant here: Darwin and his contemporaries should have known that blended inheritance was almost certainly wrong anyway, because if that were the case, then we’d all look the same by now! (Well, maybe we wouldn’t look the same if you allow some Lamarckian development of traits during one’s lifetime and passing on of those traits to children. But we should still all look the same with regard to traits that don’t change much during our lives, like eye color, or with regard to traits that we don’t acquire as the result of actually doing anything, like hair color.)
Bennett: But the point of this post is to ask whether, in a toy model, blended inheritance *would* in fact cause us to all look the same. Jenkin’s model says yes, but it also says we’d all be extinct by now. Davis thought his model said no, but I argue that it really said yes. My modification of Davis’s model holds out hope that the answer is no.
I think that’s conflating two different debates. It seems the two questions addressed by each model are:
A) Can blended inheritance allow a beneficial mutation to have a lasting effect on the species? and
B) Can blended inheritance allow differences to persist among species members, without all species members eventually looking *the same*?
If a beneficial mutation spreads throughout the entire population (even in some weakened form as a result of the averaging of blended inheritance), then the answer to A can be yes while B is still no.
Jenkin’s model:
A) No and B) No
Davis’s model:
A) He thought yes, but your calculations show it’s No
B) He thought yes, but your calculations show it’s also No. (Assuming that the beneficial-mutation population grows without bound, dwarfing the original which stays constant, so that effectively “all members” of the species eventually look the same.)
Landsburg’s model:
A) You’re saying Yes
B) I think it’s not clear from your original post. You said: “Once again the total excess beak length goes to 2.38, but this time it’s shared among individuals of a fixed population size.” But, I had assumed that the excess beak length of 2.38 would eventually be evenly distributed among the whole population. Are you saying that it wouldn’t?
In other words, I think that your models can differ on the answer to question A, but I suspect that all 3 models say that the answer to question B is No — that blended inheritance causes us all to look the same eventually. (And so any such model using blended inheritance should have been self-evidently wrong, because we don’t all look the same.)
Sorry for the late entry, but I believe the proper way to model this is NOT by average beak model, because traits are typically not blended or averaged. Traits related to genetic modification are either passed on intact or not.
Thus the proper model is to separately track the populations of sports and normals. So we begin with 10,000 members, 5000 females and 5000 males, with one of them being a sport. Say that in each generation 10,000 offspring are produced, and the sport has about 10% more success with females than do normals; thus produces an average of 2.2 offspring versus their 1.99996. Of course offspring are discrete, so this manifests itself in an extra sport within five generations. The mutation is at risk of dying out early on, but once there are five or more males with the mutation, numbers take over and the sport population will grow at the expense of the normals.
The result is not an average slight increase in beak length; the result is the mutation spreading to 100% of the population and thus a doubling of beak length. By analogy, our current human intelligence is not due to some ancient ancestor being 50 times smarter than us and passing down 2% of that genius. It is the result of some ancient ancestor being maybe 10% smarter than his peers, and using that slight advantage in understanding and strategy and planning to out-compete the dummies and out-mate them, and his children inherited ALL of his advantage and did the same to their peers, until the mutation spread throughout the population and became an ineffective advantage — At which point the next advantage become more important.