Summations Only

This post is a response to posts 140, 141

quote:

---

The point of doing the probability calculations is to determine the population size necessary based on a given mutation rate to determine the probability of a beneficial mutation occurring. Fixation of a given variant is neither necessary nor sufficient for this process to work. It is even possible for the relative frequency of a variant in a population to decrease yet the variant is still able to evolve to the selection pressure.

quote:

---

The hardest part is knowing how many beneficial mutations there are to begin with. Looking backwards, we can only see the beneficial mutations that did occur. Jumping to the much larger conclusion that these beneficial mutations are the only ones possible is what leads to bad conclusions. Like the old saying goes, there is more than one way to skin a cat, and that usually applies to be beneficial mutations.

---

---

You can have more than one beneficial mutation for a given selection pressure, the Weinreich experiment demonstrates that. However, each particular beneficial mutation gives rise to a lineage that is on a different evolutionary trajectory than those variants with a different beneficial mutation. What each of the different variants share in common is that they must amplify (increase in number) before there is a reasonable probability of another beneficial mutation occurring on a member of that lineage.

quote:

---

So do you want to try to compute the probability that a single beneficial mutation will occur on some member of a lineage?

quote:

---

The problem is determining how many beneficial mutations are possible in a given lineage in a given environment.

​

For example, let's look at flight in terrestrial animals. Many different lineages have evolved the ability to fly, and each independent lineage found different ways to achieve that ability. Bats, birds, and dragonflies all have different adaptations. There are many, many possible ways of achieving the benefit of flying. There is no way that we can currently know how many beneficial mutations are possible in a lineage to attain flight. They are probably nearly infinite in number, given background DNA sequences and possible solutions.

​

So how in the world do we compute the odds of beneficial mutations occurring when we can't even know which mutations would be beneficial?

---

---

What determines if a mutation is beneficial or not is whether the variant can amplify (increase in number). In a very limited sense, amplification does not have to occur to improve the probability of a beneficial mutation occurring, a small number lineage which doesn't grow in size over the generations can have enough replications (the random trial) over many generations to improve the probability of another beneficial mutation occurring on a member of that lineage.

quote:

---

Easy peasy, just like the original text, only quoted.

---

Thank you

quote:

---

Kleinman writes:

​

Yes. But the probability problem you must solve is the probability of a beneficial mutation occurring in a given number of replications. The analogous dice rolling problem would be for example the probability of rolling at least a single 1 in a given number of rolls.

The probability of a beneficial mutation occurring is nearly guaranteed since no lineage is perfectly adapted to their environment. There are multiple adaptations that could occur, and multiple mutations that can achieve each adaptation.

​

The problem you keep having is that you are committing the Sharpshooter fallacy. What you have is a person firing a bullet into a forest 1 km away. When the bullet strikes a tree, you paint a tiny little bullseye around it, and tell everyone just how improbable it is that the sharpshooter could hit that tiny target.

---

My mathematical model predicts the behavior of every real, measurable and repeatable example of rmns. If you think I'm cherry picking the data, post a real, measurable and repeatable example of rmns that doesn't obey my mathematics.

This post is a response to posts 146, 147, 150

quote:

---

Kleinman writes:

​

Theodoric, my argument is that randommutationandnaturalselectioncan'tdoit. And the reason rmns can't do it is the multiplication rule of probabilities.

Let's use the lottery as an analogy to show how you are improperly using probabilities.

​

Let's say that the odds of winning our example lottery is 1 in 1 million. In 10 drawings there are 10 winners. What is the probability that those specific people are the winners?

​

If, as you claim, we multiply the probabilities that those specific people would win, then it is 1 million to the 10th power, or 1x10^60. That's a 1 with 60 zeros after it. The odds of winning real lotteries is even less than 1 in 1 million. If we took the odds of the last 10 winners of the Powerball lottery being the winners, we would have an even larger number on our hands. The fact that those specific people won the lottery is nearly impossible, yet it happened.

​

In reality, the odds of those people winning the lottery is 1 in 1, BECAUSE IT HAPPENED. That's the part you keep ignoring.

---

Taq, your analogy is not correct, with rmns, only when a member of a particular lineage gets the beneficial mutation (wins the lottery) does it improve fitness. And if there is more than a single selection pressure targeting a single gene that particular member may have to win two or more lotteries at the same time to improve fitness. Once in a while when the population is huge, that happens, like with HIV and Malaria. That's why these replicators need more than two targeted selection pressures to suppress the evolution of drug resistance

quote:

---

Kleinman writes:

​

That's the point. When selection pressures target more than a single gene simultaneously, the beneficial mutations must appear simultaneously in order to improve fitness.

Can you point to any two vertebrate species where two simultaneous mutations had to happen in one of those lineages since the time that they shared a common ancestor?

​

If not, it seems that your line of argument is completely irrelevant, at least where the evolution of vertebrates is being discussed.

---

Here's an example: Rodenticide - Wikipedia

quote:

---

Kleinman writes:

​

And it takes huge populations and/or large numbers of generations in order for the probabilities to become realistic in these situations.

You only need to shuffle a deck once to get a highly improbable event to occur.

---

Are you suggesting that if you really shuffle a reptiles' genome you get a bird?

quote:

---

Kleinman writes:

​

And don't get me wrong, evolution does occur, but the mechanisms of evolution, in particular, rmns, can not make the genetic transformations necessary for the theory of evolution to be true.

Theories are not "true" or "false". They are good or bad, complete or incomplete, useful or not useful, etc. The Theory of Evolution is a good explanation of how evolution happens. It is also the only explanation we have, which makes it by default the best. It is fairly complete. It is very useful.
I'm sorry, but you can't use mathematics to trump that. If the mathematics disagrees with reality, it's the mathematics that you've got wrong.

---

The theory of evolution doesn't explain anything. It doesn't explain how rmns works, it doesn't explain how recombination works. It's a theory which takes the concept of common descent and says every living thing we see today came from some replicator from the primordial soup. This is a belief system made up by someone who doesn't understand the consequences of the multiplication rule of probabilities. Somebody better explain correctly how rmns works if you want to understand how to prevent drug resistant microbes and failed cancer treatments.

quote:

---

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.

---

Ok, any mutation that is detrimental causes the loss or reduced fitness of that member.

quote:

---

* 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.

---

Ok, but you must also assume the mice have adequate food, adequate water, no disease, no thermal stress...

quote:

---

* 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.)

---

I don't agree with your terminology or concept. Increasing food sources reduces selection pressures on populations and therefore increases the diversity of populations. Variants that would not otherwise have sufficient fitness to reproduce in the reduced food source environment would die out. While increasing the food sources will allow them to reproduce. This doesn't mean that rmns can't work in a scenario like this. A variant which can survive and reproduce on 2 of the 3 food sources could improve fitness by getting a mutation which would allow it to use the 3rd food source as well. This would be particularly important when the first two food sources disappear.

quote:

---

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.

---

There is no such thing as pressures of opportunity. If a member of a population is being preyed upon, running faster, being able to jump and being able to fly can give improve fitness to reproduce. There are empirical examples of this where reptiles evolve longer legs so as to run faster and escape the predator. But this is no more an example of rmns as the evolution of Great Danes and Chihuahuas. If the reptile must evolve the alleles to escape the predator by rmns, they are dead meat. What the predator does is kill all the slower members of the population and the remaining members by recombination change the expression of existing alleles to give longer legged variants.

quote:

---

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.

---

The phenotypes of populations can be altered markedly by recombination. Consider the variants seen in the canine family in just a few thousand years of selective breeding. However, the creation of new alleles by rmns is an extremely slow process, even under ideal circumstances with the correct selection pressures. And if you have multiple directional selection pressures acting simultaneously, the process only slows further.

quote:

---

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.

---

Take a closer look at the Lenski starvtion selection pressure experiment, he maintains his populations at e7-e8. Yet is still takes over a thousand generations per beneficial mutation. Do you think that rmns will work more quickly if he subjects his populations to thermal stress as well as starvation stress concurrently?

quote:

---

Kleinman writes:

​

If you want to make realistic interpretations of the fossil record, you need to take into account the mechanisms of genetic transformation.

We already have. The divergence between the genomes of species matches up with the age of morphological divergence seen in the fossil record when we include population sizes, mutation rates, and the rest.

---

So you have the genetic sequences for dinosaurs?

quote:

---

Are you arguing that conservative selection pressures are what transformed dinosaurs into birds.
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."

---

I had a link to a paper which is now dead given to me by Edward Max. In that link, they studied the genomes of reptiles and the genomes of birds and looked at which genes would have to be transformed to transform scales to feathers. They identified at least 8 genes which would have to be transformed. They didn't say how many mutations in each gene.

quote:

---

I have empirical examples of rmns for microbes, plants, insects, rodents, cancers, some of these examples are for clonal replicators, some for sexually reproducing replicators.
How nice for you. Do you also have the data relevant to dinosaur-bird evolution I mentioned in my post?

---

Last I checked, nobody has sequenced the dinosaur genome except in Jurrasic Park.

quote:

---

Even if birds could evolve as quickly as HIV does, the theory of evolution does not have a chance. Are you going to argue that combination therapy doesn't work for the treatment of HIV?
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.

---

The equations I derived are general equations applicable to any example of rmns. They are applicable to all real, measurable and repeatable examples of rmns. If you use combination selection pressures on birds, they will not be able to evolve by rmns any differently than any other replicator.

quote:

---

Of course, you don't know what the selection pressures are which would transform a reptile into a bird because they don't exist.
Well, don't you think that being able to fly is kinda useful to birds?

---

Certainly, but if you are going to take a life form that can not fly and try to give that life form the alleles necessary to fly by rmns, those life forms are already the honored guests at dinner.

quote:

---

And targeted selection pressures are those pressures which target a single genetic loci.
Locus. I'm still not sure what that would have to do with, for example, the evolution of birds from dinosaurs.

---

Thanks, you are correct. My point is, most real, measurable and repeatable examples of rmns are cases where there are targeted selection pressures. For example, antimicrobial resistance, herbicide resistance and so on. Have you ever seen a case where a microbe evolves resistance to iodine? It doesn't happen because iodine reacts with too many biological molecules, too many genetic loci targeted. Starvation and thermal stress target too many genetic loci simultaneously for replicators to evolve efficiently to these kinds of selection pressures. The Lenski experiments are examples of this. And when these pressures are combined, the ability to evolve to these pressures becomes multiplicatively more difficult to evolve to by rmns.

quote:

---

Don't hide your empirical examples, but make sure they are real, measurable and repeatable examples of rmns.
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.

---

Fair enough, it's the directional selection pressures which I am talking about which drive rmns.

quote:

---

So three drug combination therapy does not work for the treatment of HIV? You'd better notify the WHO, CDC, FDA, and NBA
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.

---

Again, fair enough, but I think you now recognize that single drug treatment is useless, two drug therapy works better, three drug therapy handles the vast majority of cases, four drug therapy... My paper on the evolution of drug resistance to multiple simultaneous selection pressures addresses this. There's a pattern which emerges as you add selection pressures. You are forcing lineages to do several orders of magnitude more replications for each additional selection pressure for each beneficial mutation required for adaptation. This is easy for the lineages to accomplish the amplification required when the selection pressures are applied sequentially. However, when done simultaneously, the amplification process is suppressed by the various selection pressures.

quote:

---

I'm going to stop you right there because you are making the same error in physics which Haldane and Kimura make in their models of substitution and fixation. Fixation is neither necessary nor sufficient for rmns to work. Do you understand why?
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".

---

Fixation is a common notion taught in evolutionary biology but it is based on an incorrect understanding of the physics. Fixation is based on the notion that natural selection is a conservative phenomenon. Haldane in his substitution model is based on the assumption that the increase in one variant must be accompanied by a decrease in another variant (hence substitution). Kimura in his diffusion fixation model uses the same basic concept. In order for a variant to be fixed, the other variants must disappear. But that is not what happens with rmns. First, the probabilities of a beneficial mutation occurring are not dependent on the relative frequency of the variant but the actual number of members replicating who would benefit from the particular mutation. Second, rmns can occur with multiple different variants, each taking their own particular evolutionary trajectory to improved fitness and it doesn't matter what the other lineages are doing as long as they are not competing for the resources of the environment. None of the variants need to be fixed in order for this process to happen. Here's a video demonstrating this:
Scientists create video of bacteria evolving drug resistance.
I've sent an email to the people doing this experiment to try the experiment with 2 and then 3 drugs instead of the single drug experiment. Fixation is not the key variable for evolution by rmns, it is amplification to improve the probability of the next beneficial mutation.

quote:

---

Do you think the TOE states birds descended from modern reptiles?

---

Doesn't matter, if the alleles don't exist in the lineage, they have to come from someplace. Modern reptiles don't have the alleles to produce feathers, where did "ancient" reptiles get the alleles?

quote:

---

My mathematical model predicts the behavior of every real, measurable and repeatable example of rmns. If you think I'm cherry picking the data, post a real, measurable and repeatable example of rmns that doesn't obey my mathematics.

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.

---

No Doc, what I am saying is that rmns works the same for all replicators. rmns works in a cycle of beneficial mutation followed by amplification of that mutation to improve the probability of another beneficial mutation occurring on that lineage.

quote:

---

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.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.

---

I have done the math and I understand that it's behind a paywall but it's there.
Here's the first step in doing the math, it is determining the possible outcomes for a mutation.
P(−∞ < X < +∞) = P(Ad) + P(Cy) + P(Gu) + P(Th) + P(iAd) + P(iCy) + P(iGu)
+ P(iTh) + P(del)+ = 1
Where the i term denotes insertion, del denotes deletion, ... denotes any other possible mutation you can think of.

quote:

---

This part of the abstract sums it up.

This examines the use of mathematical models to shed light on how biological, pattern-forming gene networks operate and how thoughtless, haphazard, non-design produces networks whose robustness seems inspired, begging the question what else unintelligent non-design might be capable of.

​

In other words "thoughtless, haphazard, non-design" creates complex stuff.

​

Sorry to have to be the bearer of bad news...

---

Why don't you bear us some empirical examples of your bad news?

The following is a response to post 182, 185

quote:

---

Ok, any mutation that is detrimental causes the loss or reduced fitness of that member.

Right, why do you mention it?

---

Just trying to make sure I understand your scenario.

quote:

---

Ok, but you must also assume the mice have adequate food, adequate water, no disease, no thermal stress...

​

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.

---

I guess so, ok so go on.

quote:

---

I don't agree with your terminology or concept. Increasing food sources reduces selection pressures on populations ...

​

Not if (as I explicitly said) "each [...] requires a different adaptation to exploit effectively". Then they impose selection pressures.

---

So some variant can us one food source, other variants use the second food source, and a third variant can us the remaining source?

quote:

---

There is no such thing as pressures of opportunity.

​

Wrong. See my example.

---

You can reduce the selection pressures on a population which will allow for increasing diversity, so if you want to call that an opportunity, I guess so.

quote:

---

The phenotypes of populations can be altered markedly by recombination. Consider the variants seen in the canine family in just a few thousand years of selective breeding. However, the creation of new alleles by rmns is an extremely slow process, even under ideal circumstances with the correct selection pressures. And if you have multiple directional selection pressures acting simultaneously, the process only slows further.

​

Wrong (in general). See my math.

---

I've seen your math and you need to learn something about the concept of fixation, it has no bearing on rmns.

quote:

---

Take a closer look at the Lenski starvtion selection pressure experiment, he maintains his populations at e7-e8. Yet is still takes over a thousand generations per beneficial mutation. Do you think that rmns will work more quickly if he subjects his populations to thermal stress as well as starvation stress concurrently?

​

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.

---

Is that like killing me softly with his song? You need to suggest to Lenski to run his experiment with both thermal stress and starvation stress so he doesn't have to wait a thousand generations per beneficial mutation.

quote:

---

Last I checked, nobody has sequenced the dinosaur genome except in Jurrasic Park.

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.

---

The calculations for rmns are actually quite simple.

quote:

---

The equations I derived are general equations applicable to any example of rmns.

​

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.

---

My calculations also say that if amplification doesn't occur, the probability of another beneficial mutation occurring on that lineage remains low.

quote:

---

Certainly, but if you are going to take a life form that can not fly and try to give that life form the alleles necessary to fly by rmns, those life forms are already the honored guests at dinner.

​

Well, you know, non-flying dinosaurs also managed to survive for about 160 million years.

---

Apparently they didn't have to escape from their predators by flying away.

quote:

---

Starvation and thermal stress target too many genetic loci simultaneously for replicators to evolve efficiently to these kinds of selection pressures. The Lenski experiments are examples of this.

​

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."

---

Not bad for 20,000 generations and about 20 beneficial mutations. But considering that 30 generations of doubling should have given about e12 members with a given beneficial mutation, you have very slow amplification. Remember the uproar over Haldanes dilemma, 300 generations per evolutionary step? But Haldane's model is physically incorrect.

quote:

---

Again, fair enough, but I think you now recognize that single drug treatment is useless, two drug therapy works better, three drug therapy handles the vast majority of cases, four drug therapy... My paper on the evolution of drug resistance to multiple simultaneous selection pressures addresses this. There's a pattern which emerges as you add selection pressures.

​

I know, I did the math. And the pattern is different depending on how hard the selection pressures are.

---

Do you think that the evolutionary trajectory is dependent on the intensity of selection?

quote:

---

Fixation is based on the notion that natural selection is a conservative phenomenon.

​

No.

---

You are wrong on this one Doc. Do a careful study of Haldane's and Kimura's work. It's based on the concept that an increase in one variant is linked with a decrease in the other variants. In fact Haldane's substitution model is analogus to a conservation of energy problem. Here's a link to a paper which describes this:
Just a moment...
Kimura's work is directly based on a diffusion equation which is also a conservative phenomenon.

quote:

---

Second, rmns can occur with multiple different variants, each taking their own particular evolutionary trajectory to improved fitness and it doesn't matter what the other lineages are doing as long as they are not competing for the resources of the environment.

​

But eventually they certainly will be.

---

They will be what?

quote:

---

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.

---

You are wrong on this Doc. Fixation of an allele is neither necessary nor sufficient for rmns to occur. If you are so sure you are correct, explain why one variant must decrease in order for another variant to increase.

quote:

---

No Doc, what I am saying is that rmns works the same for all replicators. rmns works in a cycle of beneficial mutation followed by amplification of that mutation to improve the probability of another beneficial mutation occurring on that lineage.

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?

---

What I mean by amplification is simply increase in number of members of a particular lineage. Replication is the principle random trial for rmns. There are two ways to increase the number of random trials, you can increase the number of members in a lineage and you can increase number generations that lineage is able to replicate. Here's an easy way to think of this. Let's say you want to know the probability of rolling at least a single 1 with the roll of 1 die 10 times or 10 dice once, or 5 dice rolled twice etc. They all give the same set of possible outcomes and probabilities. So if you have a large numbers of members in a given lineage, it doesn't take very many generations of replications to have a reasonable probability of getting that beneficial mutation. But once that beneficial mutation occurs on one of the members, it is the progenitor of a new subpopulation (lineage) which must now amplify in order to improve the probability of another beneficial mutation occuring to advance the rmns process.

quote:

---

Here's the first step in doing the math ...

​

Could we move forward to the steps that involve birds and dinosaurs?

---

It's the same math for all replicators.

There has been a call to see the mathematics of rmns. So let's do it. To make it a bit clearer, we will do this mathematics in the context of a real example of rmns measured in an experiment done by Weinreich and published here: icommons.harvard.edu In this experiment, he measured the mutations which would give e coli resistance to a particular antibioticWhat Weinreich el al found was that there were a variety of variants which evolved resistance to the antibiotic. It took 5 mutations to achieve high resistance to the drug but different combinations of mutations could accomplish this. What he found was that the first beneficial mutation determined what the other required mutations would be. So for one particular variant, you can label the set of 5 mutations to give resistance A1, B1, C1, D1 and E1, for another variant, you can label the mutations A2, B2, C2, D2 and E2 and for a third variant, you can label the mutations A3, B3, C3, D3 and E3 and so onWhat each of the variants have in common is that the sequence of mutations for a particular variant must always give improved fitness. So here is how you would do the mathematics for the general case of a variant getting mutations A, B, C, D, and E where each additional beneficial mutation in the evolutionary trajectory gives improved fitness to reproduce against the antibiotic selection pressure.There are two random trials in this problem, the principle random trial is the replication where there are two possible outcomes, either a mutation occurs at the particular site or a mutation does not occur at the particular site. The second random trial for this problem is the mutation itself. When a mutation occurs, it may not be the beneficial mutation, it could be a detrimental or neutral mutation, neither would contribute to improved fitness of that member. So to write out the possible outcomes for a mutation at a particular site, we can use the addition rule of probabilities for mutually exclusive event.P(−∞ < X < +∞) = P(Ad) + P(Cy) + P(Gu) + P(Th) + P(iAd) + P(iCy) + P(iGu)
+ P(iTh) + P(del)+ = 1Where Ad, Cy, Gu and Th are the bases, P(Ad) would indicate the probability that Ad was substituted and so on, P(iAd) would indicate that an Ad was inserted, P(del) would indicate that the base was deleted, and ... indicates any other mutation that could possibly occur. What can be said with mathematical certainty about each of the terms in this equation, the value for each of the terms ranges between 0 and 1. One of these terms also represents the beneficial mutation. So define a term, P(BeneficialA) such that if the substitution of Ad is the beneficial mutation, P(BeneficialA) = P(Ad). If the substitution of Cy is the beneficial mutation, P(BeneficialA) = P(Cy) and so on.Now, define a term, 𝜇, the mutation rate the probability (frequency) that an error in replication will occur at a particular site in a single member in one replication. With these definitions, we can compute the probability that mutation A will occur in a single replication of some member of the population.P(A) = P(BeneficialA)𝜇We'll stop at this point for questions, comments, complaints...Edited by Admin, : Break up the large paragraph into smaller paragraphs and add a little additional spacing.

quote:

---

Hi Kleinman,
As Dr Adequate said, what you say is self-evidently true, plus you already provided that equation back in Message 186. Since we're past 200 messages now I think it isn't unfair to call upon you to move ahead more expeditiously. Continuing your focus on the the bacterial example is fine. Dr Adequate is eager to move ahead to your dinosaur-to-bird claim, but one thing at a time is probably a good idea.

---

Hi, Percy, you didn't put in your post that I couldn't reply to you directly so forgive me for taking the liberty. What I'm deriving here are the general equations which govern rmns. I happen to use this bacterial example because we have empirical data. But the governing mathematics is applicable to any arbitrary example of rmns.

quote:

---

What does Weinreich et. al. have to do with your simple equation

---

Weinreich happened to do a good job measuring the empirical data. I could have used an example of the evolution of HIV to drug therapy or the evolution of Malaria (which is haploid/diploid) which I did use for deriving the mathematics or rmns for multiple simultaneous selection pressures.

quote:

---

What leads you to draw different conclusions from Weinreich et. al. than do the authors?

---

I'm not sure what different conclusions you mean? What I've done (and doing here) is describing the physics and mathematics of rmns. I'm just using Weinreich's paper and data as an example.So let's pick up the calculation where we have the probability of beneficial mutation A occurring in a single replication at the particular site:
P(A) = P(BeneficialA)𝜇
In order to compute the probability that mutation A will occur in a population size (let's call the population size n), we first must use the complementary rule of probabilities and compute the probability that mutation A will not occur which gives the equation:
P(Ac) = 1 — P(A) = 1 - P(BeneficialA)𝜇
where P(Ac) is the probability that mutation A will not occur at the particular site. Then to compute the probability that mutation A will not occur in n replications in a single generation, we use the multiplication rule of probabilities and obtain:
P(Ac) = (1 — P(A))^n = (1 - P(BeneficialA)𝜇)^n
Then to compute the probability that mutation A will not occur in G generations (call it nGA), we again use the multiplication rule and obtain:
P(Ac) = ((1 − P(BeneficialA)𝜇)^n)^nGA = (1 − P(BeneficialA)𝜇)^n*nGA
and then to compute the probability that mutation A will occur in a population size n in nGA generations, we again use the complementary rule of probabilities and obtain the following equation:
P(A) = 1 − (1 − P(BeneficialA)𝜇)^(n*∗nGA)So now the mathematical question is, what is the probability that mutation B will occur on some member with mutation A. The mathematics is self evident.
Again, we'll stop at this point for questions, comments, complaints...