Summations Only

It is easy to argue that in an ideal case a population will evolve more efficiently if it is presented with a number of selection pressures concurrently than if they are presented consecutively.Consider a set of potential adaptive mutations to n selection pressures, each of which (for convenience) we shall take to have a chance 1/q in each generation of arising and going on to fixation.If the pressures are presented sequentially, then obviously the expected time (measured in generations) for the first mutation to arise is q, after which the expected time for the second to arise is also q ... and so on. The expected time at which the last mutation arises to cope with the last selection pressure is therefore nq. We may therefore say that the rate of evolution is n/nq, which is, unsurprisingly, just 1/q, the number we started with.Now consider the case where the selection pressures are applied simultaneously. Then when we start off, the probability of any one of the mutations arising and going on to fixation is n/q, and so the time we expect to wait is q/n. Then after that any of n - 1 mutations might arise, so the probability is (n - 1)/q and the time is q/(n - 1).Recognizing a pattern, we may say that the time until the last mutation arises will be given by qΣ(1/k), where the sum is taken from 1 to n. This sum is approximately equal to ∫1/kdk, the integral being taken over the same limits, and of course this integral is just equal to ln n. So we may say that the rate of evolution in this case is (1/q)n/ln n).As we know that 1 < n/ln n) for n > 1, we can conclude that evolution goes at a faster rate when the selection pressures are encountered concurrently than when they are encountered consecutively. Indeed, at a much faster rate as n becomes large: to evolve to meet 100 selection pressures, for example, would be over 20 times faster if these pressures are encountered concurrently than if they are encountered consecutively.BUT! WAIT!You might think on that basis that in that case we should abandon combination therapy for pathogens for what one might call "cyclic therapy": give the patient one medicine, wait 'til the pathogen starts becoming resistant, switch to the next medicine, and so forth.But there are a number of problems with this. I shall not list them all, but two stand out.First, our actual objective is not to mess with the evolution of the pathogen as such, but to protect the health of the patient. Combination therapy allows us to drive the pathogen population lower, because we are poisoning the pathogens in more ways; and as the patient is therefore harboring less pathogens, the patient is (gratifyingly) less sick.Second, lowering the population does in fact retard the evolution of the pathogen. Using Haldane's approximation we may say that the probability of fixation of a given advantageous allele at the point when it arises is given by π ≈ 2s for any suitably large population (where s is the fitness advantage of the mutation). But the probability of such an allele arising in any given generation is proportional to the population size. It follows that larger populations will undergo adaptive evolution faster than smaller ones. Simply by holding down the population size we retard the evolution of the pathogen.We must therefore say that the result obtained above holds when selection just affects which of the population will survive and reproduce (soft selection), but begins to break down when the selective pressures start to affect the size of the population (hard selection), and breaks down more the more this is so. This is intuitively obvious when we consider the case in which the selective pressures are strong enough in combination, but not separately, to bring about the actual extinction of the population, which would halt evolution entirely.It follows that (pace Kleinman), combination therapy is, after all, a special case, in that it involves the combination of selection pressures specifically designed and chosen to have the greatest possible impact on the population size of the pathogen, and so to present the hardest case of hard selection that human ingenuity can devise. Footnotes:(1) Conservative selection pressures may be ignored as so much background noise for the purposes of this exercise.(2) As n tends to infinity the difference between the sum and the integral tends toward γ, the Euler-Mascheroni constant, approximately 0.5772.(3) J.B.S. Haldane, "The mathematical theory of natural and artificial selection", Proceedings of the Cambridge Philosophical Society 1927, 23, pp. 838—844.Edited by Dr Adequate, : No reason given.Edited by Dr Adequate, : No reason given.

We might, then, crudely distinguish between three kinds of selective pressure.* Conservative pressures. Imagine a species of (let us say) mice on an island somewhere, that have been there for a long time and are already well-suited to their niche. They are subject to a thousand conservative selection pressures to stay how they are; this does not particularly interfere with their ability to adapt to one more adaptive pressure.* Adaptive pressures of threat. Now introduce three new predators to the island, each of which by its presence reduces the population, and requiring three different adaptations to evade effectively. If their effect on the population is significant, then it might well be the case that the mice would evolve more effectively if we introduced them one at a time, introducing one when the mice have evolved to cope with the other.* Adaptive pressures of opportunity. But suppose instead that we introduced three new food sources to the island, each of which requires a different adaptation to exploit effectively. As this exerts no downward pressure on the population, the mice would adapt faster if they were introduced concurrently than if they were introduced consecutively, by reason of the math presented in my previous post. (By analogy, it is faster to roll 1, 2, and 3 on a die in any order than to roll them in that order.)So when considering the evolution of birds (for example) it would be important to know what the pressures were on dinosaurs. The conservative pressures can be neglected. If the adaptive pressure was a pressure of threat --- if small dinosaurs were all but driven to extinction by the emergence of many threats which applied to anything that couldn't fly --- then the more of these pressures there were, the more slowly they would evolve. But if the pressures were pressures of opportunity, if there were benefits available to dinosaurs which were better at jumping/gliding/flying, then if these pressures were concurrent, then the more of them there are the faster the dinosaurs would evolve.So the nature and not merely the number of the adaptive pressures on the dinosaurs would be crucial to know if we wanted to know how long it would take. Without knowing the nature of the pressures we can't just say "it would be slower if there was more of them", nor can we say "it would be faster if there was more of them" --- and nor should we be counting conservative pressures at all if they are irrelevant to the benefits of flight. This is all very qualitative, and we may note that there are quantitative caveats. If, for example, the pressures of threat only reduced the population by say 1% each, then it would still lead to faster evolution if they were introduced concurrently. This caveat does not apply to combination therapy, since anything that reduced the population of pathogens by a mere 1% would not be considered as a therapy, and no-one would bother to include it in a drug cocktail.Edited by Dr Adequate, : No reason given.

No, of course not. You can tell I'm not arguing that by the way in which instead of saying "Conservative selection pressures are what transformed dinosaurs into birds" I said "But to even start applying this to dinosaurs we'd need to know how many beneficial mutations get your from dinosaurs to birds and how long it took." How nice for you. Do you also have the data relevant to dinosaur-bird evolution I mentioned in my post? No, I'm not going to argue that combination therapy doesn't work for the treatment of HIV. I'm going to argue that you have done no calculations relevant to the evolution of birds. Well, don't you think that being able to fly is kinda useful to birds? Locus. I'm still not sure what that would have to do with, for example, the evolution of birds from dinosaurs. As I said, the evolution of antibiotic resistance in bacteria. If you are going to count the conservative selection pressures on dinosaurs, and claim that these would impede their evolution into birds, then I am going to count the conservative selection pressures on bacteria and point out that these don't impede them at all in evolving resistance to ampicillin. As the paper I cited shows, it doesn't completely stop evolution in HIV. Obviously it works in terms of conferring benefits on the patient, but it works rather less well in bolstering your claims about the way evolution works. Well, I wouldn't like to be like those dunces Haldane and Kimura.Your meaning is rather obscure, since fixation is one not uncommon product of evolution, so writing "fixation is neither necessary nor sufficient for rmns to work" is rather like writing "Traveling to another country is not necessary for airplanes to work".Of course evolution can produce other results, e.g. when it produces genetic polymorphism.However, if we want to think about the evolution of birds from dinosaurs I would respectfully submit that all the traits which make birds birds and not dinosaurs have in fact been fixed in birds.

Well, the young-Earthers think it's lasted four thousand years old.Unless you think we're dealing with freshly-killed dinosaurs, you must admit that there are conditions which sometimes keep organic material in dinosaur bones from decay. Now, decay doesn't happen on its own, it involves bacteria. If a bone can go four thousand years without bacteria getting into it, it can go a hundred million years without bacteria getting into it. Why not?But we digress.

You're not doing mathematics. You're saying that the evolution of birds from dinosaurs must have been like the evolution of HIV to adapt to combination therapy ... only slower, 'cos that's actually quite fast ... and therefore would have taken so long that it can't have happened.And all this without doing any math, or having any data about birds or dinosaurs. It's an example of a classic creationist trope that I call the Non-Quantitative Quantitative Argument. "This number (which I haven't calculated) is too big/too small to agree with the theory of evolution!" At this juncture I usually point out that they haven't calculated the number, and they get all grumpy.

Right, why do you mention it? No, just that before the introduction of the predators they maintained their population at the level that it was in fact at in the face of these pressures. Not if (as I explicitly said) "each [...] requires a different adaptation to exploit effectively". Then they impose selection pressures. Wrong. See my example. Wrong (in general). See my math. Yes if this involves soft selection, no if the selection is too hard, i.e. if the additional stress reduces the population to such a level that this outweighs the effect of concurrent evolution.Edited by Dr Adequate, : No reason given.

I didn't say doing the calculation would would be easy or indeed possible. I merely say that you have not in fact done it. And what do they say? Because if all they say is that simultaneous beneficial mutations are unlikely, then this is true but does not do what you want it to, i.e. cast doubt on bird evolution. Well, you know, non-flying dinosaurs also managed to survive for about 160 million years. Except that the bacteria in that experiment did evolve effectively; they are demonstrably fitter than their ancestors. According to WP: "By 20,000 generations the populations grew approximately 70% faster than the ancestral strain." I know, I did the math. And the pattern is different depending on how hard the selection pressures are. No. But eventually they certainly will be.---Again, I would point out that the traits that make birds birds and not dinosaurs are fixed in birds, so it would be perfectly reasonable to measure the speed of dinosaur-to-bird evolution in terms of the rate at which the genes were fixed.

Amplification? You mean, as in this? And this, you say, improves the probability of another beneficial mutation? How?And why would it stop birds from evolving? Could we move forward to the steps that involve birds and dinosaurs?

I said different, not incompatible. But the opportunities increase the selection pressures, because it is now adaptive to be able to take advantage of the opportunities. See my previous posts. No. Too late. That's not what conservative selection means.And I'm sure both Haldane and Kimura were aware that populations grow sometimes. And, knowing that, they did not find the concept of fixation invalid; and nor do I. They will be competing for the resources of the environment. A population cannot grow indefinitely. Again, this is a strange way of putting it. "Travelling to another country is neither necessary nor sufficient for air travel to occur". Sure. This does not prove that the concept of travelling to another country is meaningless or fictitious, or that one cannot do it by plane. This is not in fact necessarily the case; but it would be the case if the population size is steady, as is often the case in nature.

How does this improve the probability of another beneficial mutation occurring? Do you mean just because the population will be larger as a result of the first beneficial mutation, or what? Well, the actual numbers would be different.C'mon, Kleinman. You say that dinosaur-to-bird evolution can't have taken place because it would take too long for birds to evolve from dinosaurs. In order to prove your point, it is necessary for you to calculate how long it would take birds to evolve from dinosaurs, and to show your working. If you can't do that, then while many other aspects of this thread may have been entertaining, you haven't even made a start on proving your point.Edited by Dr Adequate, : No reason given.

The theory of evolution is mutation, selection, recombination, drift, lateral gene transfer, etc; basically, transmission genetics plus the law of natural selection. It is the theory of evolution because it is supposed to account for the facts of evolution --- from pathogens acquiring resistance to the descent of birds from dinosaurs. So far, it seems to be doing quite a good job.

So far, you have said --- at what is perhaps unnecessary length --- that the probability of a particular mutation at a given locus is the probability of there being any mutation at that locus multiplied by the probability that if there is a mutation at that locus it will be that particular mutation.I don't think there's a single person here who needed to have that explained to them.So my question would be ... when do we get to the dinosaurs?Edited by Dr Adequate, : No reason given.

If G is the number of generations, why don't you just write P(Ac) = ((1 − P(BeneficialA)𝜇)^n)^G = (1 − P(BeneficialA)𝜇)^n*G ? Why call it nGA instead? So far, it's all been exceptionally self-evident, I'll grant you that.Question: when do we get to the dinosaurs?Edited by Dr Adequate, : No reason given.Edited by Dr Adequate, : No reason given.

And indeed so far they apply with a small change in terminology to dice and playing cards. In order to apply them to dinosaurs, though, at some point we need to plug in some relevant numbers, such as the population size of the dinosaurs from which birds descended, the mutation rate of dinosaurs, etc.So, without wishing to hustle you I am interested to know when you're going to get on to the good bit.

The mathematics, possibly. Not the arithmetic. Different populations have different properties (size of population, mutation rate, number of selection pressures operating on them) and this will mean that the arithmetic will involve different numbers.