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Author Topic:   Rebuttal To Creationists - "Since We Can't Directly Observe Evolution..."
Kleinman
Member (Idle past 335 days)
Posts: 2142
From: United States
Joined: 10-06-2016


Message 233 of 2926 (898542)
09-25-2022 4:37 PM
Reply to: Message 232 by ringo
09-25-2022 3:18 PM


Re: Apples and oranges
Kleinman:
Mathematically, microevolutionary events don't add up. They are random events so the joint probability of microevolutionary events occurring is computed using the multiplication rule.
ringo:
Raindrops are random micro-water. They can't add up to macro-water (rivers, lakes, oceans)?

ringo, you are making the same logical inconsistency that Taq makes in his attempt to model human evolutionary fitness improvement. Adaptive mutations are particular mutations, not any mutation. The correct analogy for your raindrop concept would be, what is the probability of two particular raindrops ending up in the same body of water? You might try and argue that raindrops are indistinguishable but mutations are distinguishable. And you must model DNA evolution and the accumulation of adaptive mutations in a lineage using the multiplication rule.

This message is a reply to:
 Message 232 by ringo, posted 09-25-2022 3:18 PM ringo has replied

Replies to this message:
 Message 238 by ringo, posted 09-26-2022 11:42 AM Kleinman has not replied
 Message 239 by Taq, posted 09-26-2022 12:20 PM Kleinman has replied

  
Kleinman
Member (Idle past 335 days)
Posts: 2142
From: United States
Joined: 10-06-2016


Message 234 of 2926 (898549)
09-26-2022 8:11 AM
Reply to: Message 175 by Percy
09-22-2022 10:10 AM


Re: Video not available
Further response to Percy's Message 175
Kleinman:
There are a couple of ways to model adaptation. You can use the "at least one rule from probability theory as shown here in this paper:
The basic science and mathematics of random mutation and natural selection
Percy:
Instead of providing a link to a paper, please describe the model adaptation process yourself and use the link as a supporting reference.
Kleinman:
Or you can use a Markov chain random walk calculation to compute the probability of an adaptive mutation occurring as show here:
The Kishony Mega-Plate Experiment, a Markov Process
Percy:
Again, instead of providing a link to a paper, please describe the model adaptation process yourself and use the link as a supporting reference.



OK. I'll use the The basic science and mathematics of random mutation and natural selection reference since that is the formulation I used to compute the probabilities in the Lenski paper, Fixation and Adaptation in the Lenski E. coli Long Term Evolution Experiment
The Haldane frequency equation gives us a way to compute the subpopulation sizes of the more and less fit variants. We know that the less fit variants will ultimately be driven to extinction so only members of the subpopulation of the more fit variant are candidates for a beneficial mutation "A" that must occur at some site in its genome. We start with the probability that a mutation will occur at that site in a single replication. (Note that this math applies to every site in the genome, not just the site(s) that is/are candidates for adaptive mutations. That is why this is an exhaustive search for every possible mutation). That probability is the mutation rate, call it "mu". But we also need to consider that the mutation that occurs may not be a beneficial mutation. It is possible that the wrong base substitution occurs. In other words, the mutation itself is a random trial with multiple possible outcomes. We can write the set of possible outcomes as follows:
P(Ad) + P(Cy) + P(Gu) + P(Th) + P(iAd) + P(iCy) + P(iGu)+ P(iTh) + P(del) + ... = 1
where the first four terms are possible base substitutions, the next four terms represent insertions of bases, the ninth term is the probability of a deletion and the ellipsis represents any other form of mutation you can imagine.
Let P(BeneficialA) represent the probability of the mutation that gives improved reproductive fitness. Note that P(BeneficialA) is some number between 0 and 1 and for most cases will have a value of about 1/3 to 1/4. Then, the probability of mutation A occurring at the particular site in a single replication is written:
P(A) = P(BeneficialA)*mu (1)
One could think of P(BeneficialA)*mu as the "beneficial mutation rate", a probability value slightly lower than the mutation rate "mu". It should be clear that the probability of the A mutation occurring in a single replication is very low.
The next step is to compute the probability of that mutation A occurring at least once in "n" replications of the more fit variant. This is done using the "at least one rule". It is a very simple rule to apply and understand. I do a step-by-step derivation in this paper The basic science and mathematics of random mutation and natural selection
for this case. If there are any questions on how to derive or apply this rule I'll try and answer them. When this rule is applied to equation (1) you get the following probability equation:
P(A) = 1 − (1 − P(BeneficialA)*mu)^n (2)
Equation (2) is evaluated using the population size generated by the Haldane frequency/fixation equation. I plotted the results for several different fitness parameters in the Lenski fixation/adaptation paper if you are interested. Solving Haldane's frequency/fixation and equation (2) gives the correct mathematical description of the Lenski experiment. It demonstrates mathematically why competition slows the adaptation process. Competition slows the accumulations of replications the more fit variant can do by limiting the energy available to that variant. Note that the more fit variant accumulated replications (the random trial for the next adaptive mutation) most rapidly after the variant fixes in the population. If Lenski used a larger volume in his experiment (increases the carrying capacity), it might not be necessary for that variant to fix in order to achieve the necessary number of replications to give a reasonable probability of the next adaptive mutation to occur on some member of that subset. If the carrying capacity is much larger such as in the Kishony experiment, you can have multiple different lineages taking different evolutionary trajectories to adaptation in the same environment. The math for each of these lineages is the same.
If there aren't any questions or comments on this math or how to apply this math to the Lenski experiment, I'll go onto more of Percy's comments, in particular, how to do the mathematics of adaptation using a Markov process random walk model.

This message is a reply to:
 Message 175 by Percy, posted 09-22-2022 10:10 AM Percy has seen this message but not replied

  
Kleinman
Member (Idle past 335 days)
Posts: 2142
From: United States
Joined: 10-06-2016


Message 240 of 2926 (898566)
09-26-2022 12:51 PM
Reply to: Message 235 by Taq
09-26-2022 10:56 AM


Re: Apples and oranges
Kleinman:
Is the population of 100,000 exact clones of each other or are there different variants in the population with different sets of mutations?
Taq:
They would have variation just like your average sampling of mammal species.

Are some of the variants more fit than other variants and are they engaged in biological evolutionary competition?
Kleinman:
Are all 100,000 individuals on the exact same evolutionary trajectory and do all their descendants over generations remain on that same evolutionary trajectory where each individual gets the same set of mutations as every other individual or do the different individuals get different sets of mutations and the population is genetically diverging?
Taq:
Again, mutations spread through a sexually reproducing population.

You aren't answering my question about whether all 100,000 individuals are on the same evolutionary trajectory. But, let's get some more detail on your statement that mutations spread through a sexually reproducing population. Is it possible that a beneficial mutation is lost in a sexually reproducing population? What happens if the beneficial allele is heterozygous rather than homozygous?
Kleinman:
There will already be mutant variants at the target site. On the other hand, with meiosis, you have parents each passing half the genome. If I understand your argument correctly, you are claiming that one parent passes beneficial alleles from their set of chromosomes and the other parent passes their beneficial alleles from their set of chromosomes. If I understand your argument correctly, then your population must be diverse and not clones.
Taq:
It means that mutations from different lineages are combined. You keep asking how these beneficial mutations are put in the same lineage. This is how.

What is the probability of beneficial alleles in a diverse population recombining in the same descendant?
Kleinman:
How many beneficial alleles are in your population? What is the frequency of the different beneficial alleles in your population? Which members have the beneficial alleles? Are these beneficial alleles homozygous or heterozygous in each of the members? And how do you compute the probability that a descendant will get these beneficial alleles from any two parents in your population of 100,000?
Taq:
This is going to differ based on a myriad of conditions. There is no single answer for any of those questions. Even in the Lederberg experiment there was a 1,000 fold difference in the beneficial mutation rate for two different phenotypes. The very fact that you pretend there is a beneficial mutation rate only highlights your misunderstandings of how evolution works.

Tell us how you compute the probability that a descendant will get beneficial alleles by recombination for the conditions of your model.
Kleinman:
Let's say you have "m" possible beneficial mutations. What is the probability of at least one of those "m" possible beneficial mutations occurring in "n" replications? It doesn't change the number of replications much from only a single beneficial mutation.
Taq:
Really? Even with a very small population of 100,000 individuals it only too 90,000 years to get all possible SNP's with 360 million births. If there are 20 million possible beneficial mutations this would be 20 million out of 18 billion possible SNP's. This gives us a 1 in 900 chance of getting a beneficial mutation. This means we only need 18 births to get a beneficial mutation. 18 is a lot different than 360 million.

How do 20 million possible beneficial mutations end up in the lineages of all humans? Recombination? Do all 20 million possible beneficial mutations give an equal improvement in fitness to your different lineages, or is there biological evolutionary competition that causes the loss of some of the less fit variants?
Kleinman:
What????
Taq:
Its from the Lenski paper you keep citing. It's the first sentence in the abstract.
quote:
When large asexual populations adapt, competition between simultaneously segregating mutations slows the rate of adaptation and restricts the set of mutations that eventually fix.
Distribution of fixed beneficial mutations and the rate of adaptation in asexual populations
Taq:
Are you telling me you don't understand the paper you keep citing?



Try reading beyond the abstract:
quote:
In larger populations the situation becomes more complicated, as many distinct mutations segregate simultaneously, but only one lineage can fix in the absence of recombination. Many of the mutations are therefore wasted, and a complicated process of interference arises between the mutations competing for fixation. This process is also found to a degree in sexual populations (the Hill–Robertson effect) if recombination cannot act quickly enough to place all of the competing mutations on the same genetic background (16).
Kleinman:
I already pointed out previously that Haldane's frequency equation was a conservation of energy process based on the principle that it takes energy to replicate. However, Flake and Grant demonstrate this mathematically in the following paper:
An Analysis of the Cost-of-Selection Concept
Taq:
You will notice that their model applies to species with haploid genomes, not diploid genomes like that seen in primates.
quote:
It is shown for a continuous haploid model that the common standard assumptions usedin calculating the cost of gene substitution, namely, large constant population size and small constant selective value, are unnecessary.


OK, Haldane's frequency equation for a sex-linked diploid is:
pn^2AA + 2pnqnAa + qn^2aa = 1
Which of the variants fix in the population, AA, Aa, or aa?
Kleinman:
This is done because it is the number of replications of the more fit variant that determines the probability of the next adaptive mutation occurring in this subset of the population.
Taq:
Which doesn't apply in sexually reproducing species.

Taq, you are still confused on this point. Try thinking of it this way. It's the number of replications of a particular allele that determines the probability of an adaptive mutation occurring at some site in that allele. If that allele is homozygous in that diploid then you will get two replications of that allele with every creature replication. If the allele is heterozygous at that site, you will have only one replication of each particular allele.

This message is a reply to:
 Message 235 by Taq, posted 09-26-2022 10:56 AM Taq has replied

Replies to this message:
 Message 242 by Taq, posted 09-26-2022 1:28 PM Kleinman has replied

  
Kleinman
Member (Idle past 335 days)
Posts: 2142
From: United States
Joined: 10-06-2016


Message 241 of 2926 (898569)
09-26-2022 1:02 PM
Reply to: Message 239 by Taq
09-26-2022 12:20 PM


Re: Apples and oranges
Kleinman:
The correct analogy for your raindrop concept would be, what is the probability of two particular raindrops ending up in the same body of water?
Taq:
A raindrop falls and runs downhill to a small stream. Small streams flow into a larger river. Larger river dumps into ocean. Same body of water.

Sexual reproduction is analogous. Lineages flow into each other and form the larger body of variation that is the population.

If you understood how sexual reproduction works you would be able to figure these things out.​

I get it now! Sexual reproduction works by gravity! I'm impressed by your understanding of the laws of physics.
Kleinman:
And you must model DNA evolution and the accumulation of adaptive mutations in a lineage using the multiplication rule.
Taq:
Not for sexually reproducing populations.

Tell us how you think adaptive alleles are evolved in sexually reproducing populations.

This message is a reply to:
 Message 239 by Taq, posted 09-26-2022 12:20 PM Taq has replied

Replies to this message:
 Message 243 by Taq, posted 09-26-2022 1:33 PM Kleinman has not replied

  
Kleinman
Member (Idle past 335 days)
Posts: 2142
From: United States
Joined: 10-06-2016


Message 244 of 2926 (898579)
09-26-2022 2:35 PM
Reply to: Message 242 by Taq
09-26-2022 1:28 PM


Re: Apples and oranges
Kleinman:
Are some of the variants more fit than other variants and are they engaged in biological evolutionary competition?
Taq:
Beneficial mutations on separate genes would not be in competition with each other because they could merge into the same lineage.

So, in your model of human evolution, the fixation of beneficial alleles does not occur?
Kleinman:
You aren't answering my question about whether all 100,000 individuals are on the same evolutionary trajectory.
Taq:
Yes I am. Because genes move through the population this puts the population on the same trajectory. The only way this could not be the case is if there is an interruption in gene flow between subpopulations.

In your model, does every human lineage have equal reproductive fitness? Do any human lineages go extinct?
Kleinman:
What is the probability of beneficial alleles in a diverse population recombining in the same descendant?
Taq:
Your father is homozygous for a beneficial mutation in gene A. Your mother is homozygous for a beneficial mutation in gene B. You have a 25% chance of getting both beneficial mutations. Do you understand this or not?

I don't think you are doing your math correctly on that one. But WRT your model, do all the fathers in your model have beneficial allele A and all the mothers have beneficial alleles B?
Kleinman:
How do 20 million possible beneficial mutations end up in the lineages of all humans?
Taq:
Natural selection drives beneficial genes to fixation, and sexual recombination drives the merger of beneficial alleles into the same genome.

How do you compute the probability that all these beneficial mutations end up in a single lineage in your model?
Kleinman:
Try reading beyond the abstract:
Taq:
That is for asexual populations. Primates reproduce sexually.

Try reading this again:
quote:
This process is also found to a degree in sexual populations (the Hill–Robertson effect) if recombination cannot act quickly enough to place all of the competing mutations on the same genetic background (16).
How do you compute the rapid fixation of 20 million beneficial mutations in your model?
Kleinman:
OK, Haldane's frequency equation for a sex-linked diploid is:
pn^2AA + 2pnqnAa + qn^2aa = 1
Which of the variants fix in the population, AA, Aa, or aa?
Taq:
That's for a single allele. What about different genes?

You have yet to show how you do a fixation calculation for your model for a single genetic loci let alone 20 million genetic loci. Are you ever going to get beyond your simple-minded neutral evolutionary model where you assume that 20 million mutations are beneficial?
Kleinman:
It's the number of replications of a particular allele that determines the probability of an adaptive mutation occurring at some site in that allele.
Taq:
In the Lederberg experiment the mutation rate was the same for all bacteria. Spectinomycin resistance occurred once in every 10 billion divisions and phage resistance occurred once in every 10 million divisions. How do you explain this?

Where did the resistance allele come from that the phage transmits? What if the phage transmitted some other allele besides a resistance allele?
Kleinman:
Tell us how you think adaptive alleles are evolved in sexually reproducing populations.
Taq:
Let's use antibiotic resistance as our model.

We have two populations of the same bacterial species. We put one population on a plate that has antibiotic A and another population on a plate with antibiotic B. We get resistant colonies on both plates. We then mix the two populations together in media that has no antibiotic. How many of those bacterial descendants of this mixed population are going to have both mutations for resistance to both drugs? None, if the bacteria are reproducing asexually.

Do the same for a diploid sexually reproducing population. What are the results? Could you find bacterial descendants of the mixed population that has both resistance markers if they are found on different genes? YES!!

Just as a side note to this comment, Kishony's experiment doesn't work when using two drugs simultaneously. Do you understand why?
And to your comment here, you have assumed that the two subsets of your sexually reproducing population each have fixed their respective resistance alleles so they exist at frequencies of 1 in their respective populations. If we assume both subsets have equal population sizes when you combine the frequencies of the resistance alleles for each drug will be 0.5. What happens if the resistance alleles are not at high frequency in the population? What is the probability of a recombination event for the beneficial alleles if their frequencies are f1, f2, and where the remaining members of the population have frequency f3, i.e., resistance to neither drug? And BTW, combination therapy works for the treatment of malaria which can do sexual reproduction. And the malaria population size for someone with hyperparasithemia can reach 1 trillion.

This message is a reply to:
 Message 242 by Taq, posted 09-26-2022 1:28 PM Taq has replied

Replies to this message:
 Message 245 by Taq, posted 09-26-2022 3:40 PM Kleinman has replied

  
Kleinman
Member (Idle past 335 days)
Posts: 2142
From: United States
Joined: 10-06-2016


Message 246 of 2926 (898590)
09-26-2022 5:23 PM
Reply to: Message 245 by Taq
09-26-2022 3:40 PM


Re: Apples and oranges
Kleinman:
So, in your model of human evolution, the fixation of beneficial alleles does not occur?
Taq:
Fixation of beneficial mutations does occur in my model. Why you think otherwise is beyond me.

The math is way beyond you. 20 million beneficial mutations * 300 generations/fixation = 6 billion generations
Kleinman:
In your model, does every human lineage have equal reproductive fitness? Do any human lineages go extinct?
Taq:
There would be variation of fitness across individuals just like in any mammalian population.

How much variation? And how many different lineages in your population?
Kleinman:
Do any human lineages go extinct?
Taq:
The only lineages that actually exist are the y-chromosome and mitochondrial lineages because those are haploid, and those can go extinct.

So you are claiming that no human lineages have ever gone extinct? What happens to the less fit human lineages when the most fit are fixed in the population?
Kleinman:
I don't think you are doing your math correctly on that one. But WRT your model, do all the fathers in your model have beneficial allele A and all the mothers have beneficial alleles B?
Taq:
The math was wrong. It should be 100% if the father and mother are homozygous.

If a father is homozygous for a beneficial mutation in gene A and a mother is homozygous for a beneficial mutation in gene B, 100% of offspring will have one copy of each beneficial mutation. Do you agree or not?

That's still not quite right. What if both the mother and father are homozygous for gene A or homozygous for gene B?
Kleinman:
How do you compute the probability that all these beneficial mutations end up in a single lineage in your model?
Taq:
If they are all being driven to fixation because of selection then that would be population wide for all of the beneficial mutations. If you are asking for the specific equations, those are found in population genetics:

Population Genetics

Percy likes it when you post from your link. Post the equation you think applies.
Kleinman:
How do you compute the rapid fixation of 20 million beneficial mutations in your model?
Taq:
Fixation wouldn't need to be rapid.

Fixation isn't rapid and Haldane's mathematical estimate of 300 generations/fixation has been verified experimentally. And if you are concerned that Lenski's experiment uses asexual replicators, don't worry, Haldane's math includes that for diploid sexual replicators. 20 million beneficial mutations * 300 generations/fixation. How many generations are in your model?
Kleinman:
You have yet to show how you do a fixation calculation for your model for a single genetic loci let alone 20 million genetic loci.
quote:
Natural Selection results in change of allele frequency (q) [read as "delta q"]
in consequence of differences in the relative fitness (W)
of the phenotypes to which the alleles contribute.

Fitness is a phenotype of individual organisms.
Fitness is determined genetically (at least in part).
Fitness is related to success at survival AND reproduction.
Fitness can be measured & quantified (see below).
i.e., the relative fitness of genotypes can be assigned numerical values.

The consequences of natural selection depend on the dominance of fitness:
e.g., whether the "fit" phenotype is due to a dominant or recessive allele.

Then, allele frequency change is predicted by the General Selection Equation:

delta q = [pq] [(q)(W2 - W1) + (p)(W1 - W0)] / Wbar

where W0, W1, & W2 are the fitness phenotypes
of the AA, AB, & BB genotypes, respectively

Theory of Natural Selection
Taq:
The equations are all over the internet. It's not like it's a secret.


Plug in a selection coefficient and tell us how many generations to fixation. Then you only have 19,999,999 more fixations to go.
Kleinman:
Where did the resistance allele come from that the phage transmits? What if the phage transmitted some other allele besides a resistance allele?
Taq:
Phage resistance doesn't come from the phage genome. It is due to mutations in the gene tonB. I will ask again.

In the Lederberg experiment the mutation rate was the same for all bacteria. Spectinomycin resistance occurred once in every 10 billion divisions and phage resistance occurred once in every 10 million divisions. How do you explain this?

So the resistance allele has to evolve in the bacteria and the phage acquires the gene and transmits it to drug-sensitive bacteria. Are you claiming this is the mechanism that gives humans a reproductive advantage over chimps?
Kleinman:
And to your comment here, you have assumed that the two subsets of your sexually reproducing population each have fixed their respective resistance alleles so they exist at frequencies of 1 in their respective populations.
Taq:
If they didn't have the resistance marker then they wouldn't have grown on the antibiotic plate. 100% of the bacteria on each plate carry the mutation for each antibiotic.

It appears the selection coefficient is extremely high, all the drug-sensitive variants are killed off in a single generation. Is that how your model for human evolution works?
Kleinman:
If we assume both subsets have equal population sizes when you combine the frequencies of the resistance alleles for each drug will be 0.5.
Taq:
Exactly. The two lineages have merged. Half the population has both mutations. If they are now challenged by both antibiotics half of the population will survive, and 100% will have both mutations in the same lineage, and it wouldn't have required iterative mutations.

That's the problem, physicians have been taught for years an incorrect way of using antimicrobial agents and it has resulted in multidrug-resistant microbes. Doctors have been taught to use antimicrobial agents as single drug therapy. When one drug fails, go onto the next, and the next, and the next,... Microbiologists need to do a better job teaching physicians how drug resistance evolves. Do you know any microbiologists that know how drug resistance evolves?
Kleinman:
And BTW, combination therapy works for the treatment of malaria which can do sexual reproduction.
Taq:
What would happen if one village used one drug and a village close by used a different drug? Could you have resistance to each drug develop in each village, and then find both resistance markers in the offspring between those two resistant populations? The answer is yes. The same mutations wouldn't have to repeat themselves in each population.

The probability of that happening depends on the frequency of the different resistance alleles in the population. I'm still waiting for you to figure out that math. I'll even give you a couple of hints. It is the same math as a random card drawing problem except you only have 3 different kinds of cards in the deck. Call one card "A" for the first resistance allele, a second card "B" for the second resistance allele, and the third card "C" for members of the population that have neither the "A" nor "B" resistance alleles. Assume all the members of the population are homozygous at the respective genetic loci to make the math a bit easier and you have nA members in the population, nB members in the population, and nC members so that nA + nB + nC = n, the total population size. What's the probability of drawing an A and B parents to give an AB offspring?

This message is a reply to:
 Message 245 by Taq, posted 09-26-2022 3:40 PM Taq has replied

Replies to this message:
 Message 247 by Taq, posted 09-26-2022 6:05 PM Kleinman has replied

  
Kleinman
Member (Idle past 335 days)
Posts: 2142
From: United States
Joined: 10-06-2016


Message 248 of 2926 (898598)
09-26-2022 7:32 PM
Reply to: Message 247 by Taq
09-26-2022 6:05 PM


Re: Apples and oranges
Kleinman:
The math is way beyond you. 20 million beneficial mutations * 300 generations/fixation = 6 billion generations
Taq:
The neutral fixation rate in diploid organisms is essentially the mutation rate, which is 50 mutations per generation. Beneficial mutations would fix at a higher rate than neutral mutations.

That is hilarious.
Kleinman:
How much variation? And how many different lineages in your population?
Taq:
Every time you ask for lineages in a sexually reproducing population you demonstrate you don't know what you are talking about.

Taq had no parents, he is a product of a population.
Kleinman:
So you are claiming that no human lineages have ever gone extinct? What happens to the less fit human lineages when the most fit are fixed in the population?
Taq:
I am saying that asking for lineages in a sexually reproducing population makes no sense because of the intermingling of alleles and mutations.

All that intermingling of alleles and mutations and still they fix at a rate of greater than 50/generation. The Mexican Salamander has a genome length of 32 billion base pairs. Does that replicator fix 500 mutations/generation?
Kleinman:
What happens to the less fit human lineages when the most fit are fixed in the population?
Taq:
If the most fit are fixed in the population then there is no less fit. They are all equally fit.

That's fitting, the less fit aren't fit for life anymore.
Kleinman:
That's still not quite right.
Taq:
It is. Father is AAbb, mother is aaBB. All offspring will be AaBb. 100% of offspring will have the beneficial mutation for both gene A and gene B.

How did all the fathers end up AAbb and all the mothers aaBB?
Kleinman:
Fixation isn't rapid and Haldane's mathematical estimate of 300 generations/fixation has been verified experimentally.
Taq:
The difference is that in sexually reproducing species you can have more than one mutation moving towards fixation at a time. They are moving towards fixation in parallel.

Do you think that Haldane was wrong when he wrote this:
Haldane:
It is suggested that, in horotelic evolution, the mean time taken for each gene substitution is about 300 generations. This accords with the observed slowness of evolution.
Haldane posts data from sweet peas in his paper. Sweet peas do sexual replication. And do you have any experimental data that shows that fixation can occur more rapidly in sexual replicators than in asexual replicators?
Kleinman:
Plug in a selection coefficient and tell us how many generations to fixation.
Taq:
You don't even understand how sexual reproduction works. Why don't you start there.

Taq, I've taken multiple biology courses and even 2 years of microbiology. I know how meiosis works. So show us how you use the equations you posted to compute the number of generations to fixation for a single allele.
Kleinman:
So the resistance allele has to evolve in the bacteria and the phage acquires the gene and transmits it to drug-sensitive bacteria.
Taq:
No. I am simply acting as if bacteria were suddenly diploid and sexually reproducing. Phage is not moving any genes around. In one population you get antibiotic resistance through mutation. In another you get phage resistance through mutation (not through any transport of DNA from phage). When you bring the two bacterial diploid sexually reproducing populations you get offspring with both antibiotic and phage resistance.

Why should it surprise you when a scientist puts an agent into a population that transfers a resistance allele? This is simply a breeding program with a phage with a known allele and a population of bacteria. Are you claiming that the human reproductive advantage came about due to breeding?
Kleinman:
Do you know any microbiologists that know how drug resistance evolves?
Taq:
This microbiologist does understand it just fine.

Tell us how the Kishony experiment works. And show your math.
Kleinman:
The probability of that happening depends on the frequency of the different resistance alleles in the population.
Taq:
That's not what happens in the Lenski and Kishony experiments, is it?

You still haven't figured out the difference between DNA evolution, biological evolutionary competition, and recombination. Don't worry, I'll be patient with you till you get it.

This message is a reply to:
 Message 247 by Taq, posted 09-26-2022 6:05 PM Taq has replied

Replies to this message:
 Message 250 by Taq, posted 09-27-2022 10:40 AM Kleinman has replied

  
Kleinman
Member (Idle past 335 days)
Posts: 2142
From: United States
Joined: 10-06-2016


Message 249 of 2926 (898611)
09-27-2022 9:45 AM
Reply to: Message 175 by Percy
09-22-2022 10:10 AM


Re: Video not available
Further response to Percy's Message 175
Percy, I tried to write the matrix equations using Latex. They appear correctly in the Latex editor I used but don't appear correctly here.
Picking up the discussion on computing the probability of a particular mutation occurring using a Markov Process
Kleinman:
Or you can use a Markov chain random walk calculation to compute the probability of an adaptive mutation occurring as show here:
The Kishony Mega-Plate Experiment, a Markov Process
Percy:
Again, instead of providing a link to a paper, please describe the model adaptation process yourself and use the link as a supporting reference.
Kleinman:
Note that either means of computing the probability of an adaptive mutation occurring gives the same result.
Percy:
Please show us your work where the same result is produced.



OK. First, for those readers that don't know what a Markov Chain is, here's a definition from Wikipedia:
Markov chain - Wikipedia
quote:
A Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event.[1][2][3] Informally, this may be thought of as, "What happens next depends only on the state of affairs now."
The first application of a Markov Chain model to DNA evolution was done by Jukes and Cantor and is called the Jukes-Cantor model. The way one does such a calculation is first to draw a state transition diagram of the system of interest. The state transition diagram for a single site in a genome looks as follows:
In this instance, the only possible mutations are base substitutions, no insertions or deletions. The circles represent the possible states, one for each possible base. The P's with the subscripts that label the arrows are the probabilities for the possible state transitions. An AC subscript means an A base has mutated to a C base, a GT subscript means a G base has mutated to a T base, an AA subscript means that no mutation has occurred, and so on for the other transition probabilities.
Then, a transition matrix can be written to describe the evolutionary change in time t is = (pij) where the pij gives the probabilities of change from the state Ei to Ej at time t + Δt where Δt is a replication. If we neglect insertions, deletions, transpositions, and other types of mutations (that is substitutions only), the transition matrix would look as follows:
P (t)=( pAA pAC pAG pAT
pCA pCC pCG pCT
pGA pGC pGG pGT
pTA pTC pTG pTT )

\left[ {\begin{array}
p_{AA} & p_{AC} & p_{AG} & p_{AT}\\
p_{CA} & p_{CC} & p_{CG} & p_{CT}\\
p_{GA} & p_{GC} & p_{GG} & p_{GT}\\
p_{TA} & p_{TC} & p_{TG} & p_{TT}\\
\end{array} } \right
\][/latex]-->
If one assumes that the mutation rates are constant and have the same value for DNA transitions and transversion, we obtain the Jukes-Cantor model.
P (t)=[1−μ μ/3 μ/3 μ/3
μ/3 1−μ μ/3 μ/3
μ/3 μ/3 1−μ μ/3
μ/3 μ/3 μ/3 1−μ]

\left[ {\begin{array}{cccc}
1−μ & μ/3 & μ/3 & μ/3\\
μ/3 & 1−μ & μ/3 & μ/3\\
μ/3 & μ/3 & 1−μ & μ/3\\
μ/3 & μ/3 & μ/3 & 1−μ\\
\end{array} } \right]
[/latex]-->
The Jukes-Cantor model implicitly assumes a population of one. If one wants to compute the frequency distribution of different variants as a population grows "N", it is done as follows:
P (t)=[1−μ/N μ/(3∗N) μ/(3∗N) μ/(3∗N)
μ/(3∗N) 1−μ/N μ/(3∗N) μ/(3∗N)
μ/(3∗N) μ/(3∗N) 1−μ/N μ/(3∗N)
μ/(3∗N) μ/(3∗N) μ/(3∗N) 1−μ/N)]

\left[ {\begin{array}{cccc}
1−μ/N & μ/(3∗N) & μ/(3∗N) & μ/(3∗N)\\
μ/(3∗N) & 1−μ/N & μ/(3∗N) & μ/(3∗N)\\
μ/(3∗N) & μ/(3∗N) & 1−μ/N & μ/(3∗N)\\
μ/(3∗N) & μ/(3∗N) & μ/(3∗N) & 1−μ/N\\
\end{array} } \right]
[/latex]-->
The initial state of the system is written:
E0 = (A0, C0, G0, T0)
and the state of the system a time ti is:
Ei = (Ai, Ci, Gi, Ti)
and the state of the system going from state Ei to state Ei+1 is computed by simple matrix multiplication.
Ei+1 = Ei [P]
For the Jukes-Cantor model one obtains the equations:
Ai+1 = Ai(1-μ) + Ci*μ/3 + Gi*μ/3 + Ti*μ/3
Ci+1 = Ai*μ/3 + Ci(1-μ) + Gi*μ/3 + Ti*μ/3
Gi+1 = Ai*μ/3 + Ci*μ/3 + Gi(1-μ) + Ti*μ/3
Ti+1 = Ai*μ/3 + Ci*μ/3 + Gi*μ/3 + Ti(1-μ)
And for the variable population model:
Ai+1 = Ai(1-μ/N) + Ci*μ/(3*N)+ Gi*μ/(3*N) + Ti*μ/(3*N)
Ci+1 = Ai*μ/(3*N) + Ci(1-μ/N) + Gi* μ/(3*N)+ Ti*μ/(3*N)
Gi+1 = Ai*μ/(3*N) + Ci*μ/(3*N) + Gi(1-μ/N) + Ti*μ/(3*N)
Ti+1 = Ai*μ/(3*N) + Ci*μ/(3*N) + Gi*μ/(3*N) + Ti(1-μ/N)
Note that A,C,G, and T are frequencies of the particular variants with that particular base at that site. If you multiply any of these frequencies by N, you will get the number of members in the population with that given base at that site.
Then, assume that in the initial condition that the base at that site is T but the beneficial mutation is A. The initial condition is written:
E0 = (A0, C0, G0, T0) = (0,0,0,1)
And do lots of matrix multiplications.
For the Jukes-Cantor model and mutation rate 1e-9, one obtains the following frequency curves:
For the variable population transition model and mutation rate 1e-9, one obtains the following frequency curves:
And for comparison, the probability curves for the "at least one" calculation for a beneficial mutation to occur as a function of population size:
Comparing these 3 graphs shows that the Jukes-Cantor model reaches equilibrium at about 3e9 replications. The variable population size model gets an expected number of A variants equal to 1 at about 1.5e8 and the "at least one" calculation for mutation rate 1e-9 gives a rapidly rising probability at about 1.5e8.
Percy:
And don't leave me hanging about what the different populations are competing for. Was my guess of lebensraum right?
Kleinman:
Populations compete for the energy available in the given environment. That's because it takes energy to survive and replicate. Overcrowding may be a selection condition but in and of itself, it is food (energy) that biological populations compete over.
Percy:
One might even say they're competing for resources.


That's right, but ultimately, I think resources are just individual components of the environment that make the energy in the environment available to the replicator. For example, plants need water to convert the energy from the sun to produce sugar. That's why drought and dehydration is a selection pressure.
So tell me Percy, is the physics and math that I've presented get us into the ballpark on how biological evolution works?

This message is a reply to:
 Message 175 by Percy, posted 09-22-2022 10:10 AM Percy has replied

Replies to this message:
 Message 272 by Percy, posted 09-28-2022 10:34 AM Kleinman has not replied

  
Kleinman
Member (Idle past 335 days)
Posts: 2142
From: United States
Joined: 10-06-2016


Message 251 of 2926 (898626)
09-27-2022 11:38 AM
Reply to: Message 250 by Taq
09-27-2022 10:40 AM


Re: Apples and oranges
Taq:
The neutral fixation rate in diploid organisms is essentially the mutation rate, which is 50 mutations per generation. Beneficial mutations would fix at a higher rate than neutral mutations.
Kleinman:
That is hilarious.
Taq:
For a diploid population of size N and neutral mutation rate u , the initial frequency of a novel mutation is simply 1/(2N), and the number of new mutations per generation is 2Nu. Since the fixation rate is the rate of novel neutral mutation multiplied by their probability of fixation, the overall fixation rate is (2Nu)x(1/2N) = u. Thus, the rate of fixation for a mutation not subject to selection is simply the rate of introduction of such mutations.
Fixation - Wikipedia(population_genetics)


It appears that in your microbiology training, none of your instructors taught you what a mutation rate is. If you are able, prepare yourself to be instructed.
Mutation rate - Wikipedia
quote:
In genetics, the mutation rate is the frequency of new mutations in a single gene or organism over time.[2]
And that rate estimated for humans is:
quote:
Using data available from whole genome sequencing, the human genome mutation rate is similarly estimated to be ~1.1×10−8 per site per generation.[15]
The mutation rate is not the ridiculous claim of 50 mutations per generation that you made in your post. You need to put more effort into understanding the equations you use and how you define the variables in these equations. Now put the correct values in your equation and tell us how many generations to fixation for each mutation and then understand why 20,000,000 adaptive mutations are not going to fix in your model.

This message is a reply to:
 Message 250 by Taq, posted 09-27-2022 10:40 AM Taq has replied

Replies to this message:
 Message 252 by Taq, posted 09-27-2022 11:48 AM Kleinman has replied

  
Kleinman
Member (Idle past 335 days)
Posts: 2142
From: United States
Joined: 10-06-2016


Message 253 of 2926 (898635)
09-27-2022 12:46 PM
Reply to: Message 252 by Taq
09-27-2022 11:48 AM


Re: Apples and oranges
Taq:
1.1x10-8 per site per generation. Let's see how that works out. That is a mutation every 110 million bases. There are 6 billion bases in the human diploid genome. (6E9)/(1.1E8) = 54.5 mutations per person per generation. On top of that, we have directly sequenced the genomes of parents and offspring to directly measure the number of mutations per person, and that is where the figure you cite comes from. The directly measured human mutation rate is around 50 mutations per person.

Do you also agree that the neutral fixation rate is approximately the mutation rate, meaning that for a mutation rate of around 50 mutations per person per generation that we will see 50 neutral mutations fixed per generation?
The correct value to use in your equation is 1.1x10-8, not 50. Your value is off by about 9 orders of magnitude. There are many reasons why the number you are trying to use is wrong but one of the biggest is that nowhere in your equation is the genome length a variable. That's why I brought up the Mexican Salamander example because its genome is 10x larger than the human genome. That doesn't make the number of fixations 10x larger, 500 in every generation! For neutral evolution, it will take 1/(1.1x10-8) or about 90 million generations for each neutral fixation. Haldane's fixation rate of 1 fixation for every 300 generations under selection is much more generous. How you could imagine that every member of a population ends up with the same 50 neutral (or any) mutations every generation is a mystery.

This message is a reply to:
 Message 252 by Taq, posted 09-27-2022 11:48 AM Taq has replied

Replies to this message:
 Message 254 by Taq, posted 09-27-2022 1:10 PM Kleinman has replied

  
Kleinman
Member (Idle past 335 days)
Posts: 2142
From: United States
Joined: 10-06-2016


Message 255 of 2926 (898647)
09-27-2022 1:27 PM
Reply to: Message 254 by Taq
09-27-2022 1:10 PM


Re: Apples and oranges
Kleinman:
The correct value to use in your equation is 1.1x10-8, not 50.
Taq:
1.1x10-8 per nucleotide per generation is the same as 54.5 mutations per person per generation for a 6 billion base diploid genome.

And 54.5 is not the correct value to use in your equation. Just because the number of mutations that occur in a replication is 54.5 doesn't mean that all 54.5 are fixed. The correct value to use in your equation is 1.1x10-8.

This message is a reply to:
 Message 254 by Taq, posted 09-27-2022 1:10 PM Taq has replied

Replies to this message:
 Message 256 by Taq, posted 09-27-2022 1:36 PM Kleinman has replied

  
Kleinman
Member (Idle past 335 days)
Posts: 2142
From: United States
Joined: 10-06-2016


Message 257 of 2926 (898659)
09-27-2022 3:50 PM
Reply to: Message 256 by Taq
09-27-2022 1:36 PM


Re: Apples and oranges
quote:
For a diploid population of size N and neutral mutation rate u , the initial frequency of a novel mutation is simply 1/(2N), and the number of new mutations per generation is 2Nu. Since the fixation rate is the rate of novel neutral mutation multiplied by their probability of fixation, the overall fixation rate is (2Nu) * (1/2N) = u. Thus, the rate of fixation for a mutation not subject to selection is simply the rate of introduction of such mutations.
bolding mine
Fixation - Wikipedia(population_genetics)
Taq:
If the number of new mutations in the population is 2 times the population size then the mutation rate has to be number of mutations per person. Also, the last sentence explicitly states that the rate of fixation is simply the rate of the introduction of such mutations. Across a whole genome that would be 50 mutations.

You should watch this video, the derivation of your calculation begins at about the 6:00 minute mark.
https://www.youtube.com/watch?v=l2Y8oC6G1us&ab_channel=Kr...
You are using a definition of mutation rate based on the entire size of the genome. 2N is the total number of alleles at a given locus and 1/2N is the initial frequency of the first mutation in that allele. The neutral mutation rate being used is just for that genetic locus. If that locus has only a single base, then the neutral mutation rate will be 1.1x10-8. If that genetic locus has 1000 bases, the neutral mutation rate will be about 1.1x10-5 (actually lower if you compute the probability of a mutation occurring at least one site when multiple possible sites are considered). The number of generations to fixation for a single neutral mutation case is about 90,000 generations.

Edited by Kleinman, : Correct math


This message is a reply to:
 Message 256 by Taq, posted 09-27-2022 1:36 PM Taq has replied

Replies to this message:
 Message 258 by Taq, posted 09-27-2022 4:44 PM Kleinman has replied

  
Kleinman
Member (Idle past 335 days)
Posts: 2142
From: United States
Joined: 10-06-2016


Message 259 of 2926 (898669)
09-27-2022 5:30 PM
Reply to: Message 258 by Taq
09-27-2022 4:44 PM


Re: Apples and oranges
Kleinman:
You are using a definition of mutation rate based on the entire size of the genome. 2N is the total number of alleles at a given locus and 1/2N is the initial frequency of the first mutation in that allele. The neutral mutation rate being used is just for that genetic locus. If that locus has only a single base, then the neutral mutation rate will be 1.1x10-8. If that genetic locus has 1000 bases, the neutral mutation rate will be about 1.1x10-5 (actually lower if you compute the probability of a mutation occurring at least one site when multiple possible sites are considered). The number of generations to fixation for a single neutral mutation case is about 90,000 generations.
Taq:
Great, let's find the per nucleotide fixation rate. Since 2N cancels out in the equation for fixation we are left with the per nucleotide mutation rate. If the fixation rate is 1.1E-8 fixed mutations per nucleotide per generation then the number of fixed mutations in a full human genome is (6E9)*(1.1E-8) which is 66 fixed mutations per generation.

You are making another mathematical blunder here. 1/2N is the initial frequency of the mutant allele. Only a tiny fraction of the genome has mutations. What is the initial frequency for your calculation? It certainly isn't 1/2N. This model only makes sense when considering a single genetic locus because the entire length of the genome and the total number of genetic loci in that genome does not affect the calculation. Not all the mutations in an entire genome are fixed. In fact, some mutations are lost over generations. Perhaps you think that the entire genome is fixed? You should watch the entire video. It gives a very sensible explanation of the equation you are trying to use. You cannot use the entire genome length to compute the mutation rate and do this calculation correctly. It must be done based on the mutations/locus.
Mutation rate - Wikipedia
quote:
There are several natural units of time for each of these rates, with rates being characterized either as mutations per base pair per cell division, per gene per generation, or per genome per generation.
I put the boldfacing on the correct definition for mutation rate to be used for this calculation. 90,000 generations/fixation, Haldane's estimate of only 300 generations/fixation but that's with selection. So, when are you going to learn how to do the mathematics of adaptive DNA evolution and give a correct description of the Kishony and Lenski experiments? Don't you think a microbiologist should know how to do that math?

This message is a reply to:
 Message 258 by Taq, posted 09-27-2022 4:44 PM Taq has replied

Replies to this message:
 Message 260 by Taq, posted 09-27-2022 6:30 PM Kleinman has replied

  
Kleinman
Member (Idle past 335 days)
Posts: 2142
From: United States
Joined: 10-06-2016


Message 261 of 2926 (898674)
09-27-2022 7:58 PM
Reply to: Message 260 by Taq
09-27-2022 6:30 PM


Re: Apples and oranges
Kleinman:
You are making another mathematical blunder here. 1/2N is the initial frequency of the mutant allele. Only a tiny fraction of the genome has mutations.
Taq:
1/2N is the fraction of the population that has the mutation when the mutation first occurs. That's what that means.

Why are you doing this to yourself? 1/2N is the frequency of the mutant allele. For a diploid population, there are 2N copies of the allele at the particular genetic locus. One of those 2N copies is the first mutant allele. Only 1 member of that population has that mutant allele. The fraction of the population with a mutant allele initially is 1/N.
Kleinman:
This model only makes sense when considering a single genetic locus because the entire length of the genome and the total number of genetic loci in that genome does not affect the calculation.
Taq:
The equation applies equally to every neutral mutation in the genome.

It applies to any mutant allele but only for a single genetic locus.
Kleinman:
Not all the mutations in an entire genome are fixed. In fact, some mutations are lost over generations. Perhaps you think that the entire genome is fixed?
Taq:
I know all of this. If you think I am saying all mutations are fixed then you don't understand what the equation is saying. In a steady population of 100,000 and a mutation rate of 50 mutations per individual you will have 5 million mutations per generation, 50 of which will reach fixation if all 5 million mutations are neutral.

You don't understand this equation. You have even confused the number of copies of alleles with population size. There are 2N copies of an allele in a diploid population size N. And if you somehow want to extrapolate this model to the entire genome means that the entire genome is being fixed. ringo, if you are reading this post, this is GIGO.
Kleinman:
You cannot use the entire genome length to compute the mutation rate and do this calculation correctly. It must be done based on the mutations/locus.
Taq:
Neutral mutations don't have to be in genes in order to reach fixation. That's just silly. Do you really think all of the sequence outside of genes never mutates?

The rate of fixation can be calculated for the whole genome, and there is no reason why it can't be. You seem to be hung up on the idea that alleles are only single chunks of sequence within genes. That simply isn't true. An allele can be any base that differs between two organisms within a population, and that's inside or outside of coding regions or genes.

If you want to do the math for a mutation not in a coding portion of the genome, then the number of bases in that sequence is one and the mutation rate you need to use is 1.1x10-8. That's about 9,000,000 generations to fixation. Post a link to a paper or biology lecture where they do a neutral mutation fixation calculation the way you want to do it. You are just blowing smoke. Nobody does the calculation the way you want because it is nonsense.
Kleinman:
90,000 generations/fixation, Haldane's estimate of only 300 generations/fixation but that's with selection.
Taq:
The size of the effective population numbers I have seen for populations in the human lineage usually don't go above 10,000 which would require 40,000 generations for a neutral mutation to fix. At 25 years per generation, that would be 1 million years. This means the mutations reaching fixation first entered the genome 1 million years before they fix. This would also mean that the initial population that first split off from the chimp branch would be fixing neutral mutations that first appeared 1 million years before the split. Still, in each generation you will still be fixing a number of neutral mutations that is close to the per genome mutation rate.

Fixations aren't adaptation you should understand this by now from the results of the Lenski experiment. Large numbers of replications are what is required for DNA adaptive evolution. Certainly, small populations can have mutations fix more rapidly than large populations. You get all the mutations your parents have plus a few new ones for your own. Populations do exhaustive searches of all possible mutations in order to get just one member with an adaptive mutation. That's why it takes a billion replications for each adaptive mutation in the Kishony and Lenski experiments for a mutation rate of 1e-9. The reason is that "at least one" calculation applies to every site in the genome. The reason why humans have much larger populations than chimps is that humans can do farming on an industrial scale. It is clear that humans had this capability 10,000 years ago. They understood how to irrigate and use animals for labor. You have a population of about 1 billion population with 2 billion chromosome sets replications and use your mutation rate of 1.1x10-8. That doesn't give you many genome replications to work with for adaptive evolution to operate, even if you want to include recombination.
Taq:
For beneficial mutations, this only puts a 300 generation delay on fixation. If there are 5 beneficial mutations in generation 1 then they reach fixation at generation 301. Beneficial mutations that happen in generation 2 reach fixation in generation 302. Beneficial mutations that happen in generation 3 reach fixation in generation 303. See a pattern? It's not as if all mutations stop until the first mutation reaches fixation. Every generation has mutations which start their march towards extinction or fixation starting at that generation.
Taq, the math gets orders of magnitude worse if it takes 2 or more mutations to give an improvement in fitness. It introduces another instance of the multiplication rule for each selection condition the population must adapt to. That's really bad for your belief in universal common descent but really good for the fields of medicine and agriculture. It gives a successful treatment of HIV and inhibition of the evolution of herbicide-resistant weeds and pesticide-resistant insects. I don't know if you are ready for that math yet but if you or any other readers of this thread are interested, you can find that paper here:
The mathematics of random mutation and natural selection for multiple simultaneous selection pressures and the evolution of antimicrobial drug resistance
Don't worry Percy, I'll post equations and quotes in my next few posts if Taq is finished with his neutral fixation model.

This message is a reply to:
 Message 260 by Taq, posted 09-27-2022 6:30 PM Taq has replied

Replies to this message:
 Message 262 by Taq, posted 09-27-2022 9:11 PM Kleinman has replied

  
Kleinman
Member (Idle past 335 days)
Posts: 2142
From: United States
Joined: 10-06-2016


Message 264 of 2926 (898683)
09-27-2022 10:23 PM
Reply to: Message 262 by Taq
09-27-2022 9:11 PM


Re: Apples and oranges
Kleinman:
1/2N is the frequency of the mutant allele.
Taq:
Yes. That is true for every mutation throughout the genome, not just in genes.

You are having trouble doing undergraduate lower division work so I don't know whether I should give you this paper but why not?
THE AVERAGE NUMBER OF GENERATIONS UNTIL FIXATION OF A MUTANT GENE IN A FINITE POPULATION
Kimura carries out the computation of the fixation of a mutant gene. His calculation doesn't depend on the mutation rate.
quote:
This shows that an originally rare mutant gene in a population of effective size Ne, takes about 4Ne generations until it spreads to the whole population if we disregard the cases in which such a gene is eventually lost from the population
Now the effective population size can be slightly smaller than the actual population size under certain circumstances that I'm sure you know what they are. Remind me again what the population size you use is. Wasn't it 100,000? That gives the generations to fixation of that mutant gene of 4*100,000=400,000 generations. That really helps. Your lower division equation gave an estimate of 900,000 generations for the fixation of a neutral mutation. How many generations since the divergence of humans and chimps from the common ancestor?
Kleinman:
Taq, the math gets orders of magnitude worse if it takes 2 or more mutations to give an improvement in fitness. It introduces another instance of the multiplication rule for each selection condition the population must adapt to.
Taq:
For such a claim you would first need to calculate all of the possible combinations of 2 mutations that would be beneficial. If there are billions and billions of possible beneficial combinations then they wouldn't be hard to find.

You still haven't figured out that different combinations of adaptive mutations give different lineages on different evolutionary trajectories. That math is way over your head. You should stick to trying to figure out an undergraduate lower division equation of neutral fixation.

This message is a reply to:
 Message 262 by Taq, posted 09-27-2022 9:11 PM Taq has replied

Replies to this message:
 Message 273 by Taq, posted 09-28-2022 10:42 AM Kleinman has not replied

  
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