Do you really understand the mathematics of evolution?

That is incorrect. Do you understand the difference between joint probabilities when the individual probabilities are independent and when they are conditional? Because what you are assuming here is that the mutations in this circumstance are independent when they are actually conditional (dependent). If you want to understand that with words rather than with algebra, what that means is what is the probability that mutation A2 occurs on some member after mutation A1 has already occurred. Your calculation assumes that the mutations have to occur simultaneously. That's the point of correctly understanding the mathematics of evolution. If you go back far enough in time, that variant doesn't exist anywhere in the universe. You have just started doing the mathematics of evolution and that's just for one step of DNA evolution to a single selection pressure. And for a mutation rate of 1e-9, it takes 3e9 replications of a variant. And your math grossly overestimates the number of replications necessary for DNA evolution to 2 simultaneous selection pressures. The number of replications necessary for those evolutionary conditions is about 4-5 orders of magnitude smaller. Fixation is part of molecular evolution but any fixation of neutral mutations in the Kishony experiment will be due to those neutral mutations hitchhiking on the variant with the beneficial mutation. But there is no fixation occurring in the Kishony experiment. The less fit variants (drug-sensitive variants) are still happily growing in the lower drug concentration regions while DNA evolution is proceeding. You need to understand DNA evolution in a non-competitive environment (the Kishony experiment) before you attempt to do the mathematics of DNA evolution in a competitive environment (the Lenski experiment). You should start by learning how to do the mathematics of conditional probabilities because that is the correct mathematics to use when considering 2 or more simultaneous selection pressures.If you want a simple example that demonstrates the difference between joint independent probabilities and joint conditional dependent probabilities, study the example of random card drawing, with and without card replacement. Here's a simple explanation of the difference between independent and dependent probabilities.
https://www.youtube.com/watch?v=aBvWUr9hTLE
When considering the application of these principles to DNA evolution and the Kishony experiment, the single drug experiment gives rise to independent joint probabilities while the multiple drug experiment gives rise to dependent joint probabilities.So, do you want to try to redo the math where you estimate 3E9^2 replications for the 2-drug Kishony experiment?

In the single selection pressure case, the evolutionary steps are independent. Each step is a new binomial probability problem independent of the previous step and a new sample space occurs for each step. You can see this in the Kishony experiment, mutations A1 and A2 occur in separate drug-concentration regions and these regions correspond to the mathematical sample spaces. This paper shows how you do the math in for this DNA evolutionary process:
Just a moment...For the multiple selection pressure situation, mutations A1 and A2 must occur in the same region if two drugs are used. So A1 and A2 are occurring in the same mathematical sample space as well. What makes this a conditional probability situation is that A2 must occur on the reduced sample space of those variants that already have the A1 mutation (or vice versa if mutation A1 occurs on some variant that already has mutation A2). Mutations A1 and A2 occurring on some member that has neither mutation do not give the drug-resistant variant but they will contribute to their respective subsets of either A1 or A2 which is why the joint probability of the A1-A2 variant is much better than 3E9^2 (but still requires about 1e13 replications for that variant to appear). Here's the paper which explains how to do the math for multiple simultaneous selection pressures:
Just a moment... You can do that but then you are not talking about evolution. You are talking about migration. What I'm trying to get across to you is how DNA evolution by common descent works. When DNA replicates with error, you get the diversification of the population. And when adaptation requires the accumulation of specific sets of mutations, that can be achieved by lineages in the population only if a sufficient number of replications at each evolutionary step occur. That process works most efficiently when only a single selection pressure is acting on the population at a time but for a mutation rate of 1e-9, it takes about 3e9 replications for each evolutionary step.The Kishony experiment gives the most ideal environment to demonstrate this because his lineages only face minimal competition and don't require fixation at each evolutionary step. On the other hand, the Lenski experiment is carried out in a highly competitive environment (with bottlenecks every 6-7 generations). This slows the accumulation of the 3e9 replications of the most fit variants to get their next beneficial mutation. Kishony's lineages accumulate their 5 adaptive mutations in a matter of days, Lenski's lineages have taken more than 30 years to accumulate their 100 or so beneficial mutations.

Your impression is correct and the same mathematics for DNA evolution to 2 drugs applies for the case if a single drug is used but the step increase between regions requires 2 mutations for adaptation. And you are correct, in either case, if a member of the population has only one of the two mutations necessary for growth in the next higher concentration region when two correct mutations are required, that member with only a single mutation would be selected out. If you want to talk about the genetic variation of the entire world-wide population of e coli, you have to do that in the context of the different environments that the different populations are evolving. I don't know how large that world-wide population is but for the sake of discussion, assume it is 1e20. And let's assume that these bacteria are growing in a vast idealized environment such as Kishony's drug-free region. Then, using the mathematics I've presented, you can calculate the diversity of that population if it was a vast single colony as a function of the mutation rate. But that vast idealized environment doesn't exist. In the real world, there are many selection pressures such as thermal stress, starvation, toxins, dehydration, competition from other replicators, predation, etc. These selection pressures reduce the diversity of populations, reducing or eliminating those variants that don't have sufficient reproductive fitness.So, bringing a drug-resistant variant of e coli will change the behavior of the Kishony experiment for that particular drug but it doesn't change the fundamental physics and mathematics of DNA evolution.

You are correct, the carrying capacity of our gut is greater than the Kishony experiment. What that means is that there is a high probability of drug-resistant variants already in that environment. The way to correctly address this was described by Edward Tatum in his 1958 Nobel Laureate lecture:
Edward Tatum – Nobel Lecture - NobelPrize.org For many years, the standard of care taught in medical schools has been the use of single-drug therapy for the treatment of infectious diseases. In most cases, this works ok if the patient being treated has a good functioning immune system which removes any of the resistant variants that the antibiotic does not. But, if the patient has a poorly function immune system, single-drug therapy, especially at low doses is the formula for selecting for drug-resistant variants and treatment failure.

What you are missing is that with a mutation rate of 1e-9 that with 3e9 replications that you have shot at every target possible and hit each target on average once somewhere in the population. And what that means, for example, in the Kishony experiment that you are going to have some member of that population with a beneficial mutation for the Ciprofloxacin environment and a different member of the population with a beneficial mutation for trimethoprim. What determines whether either of these mutations are beneficial is the environment in which that variant is trying to grow. Sure you can know. Again, using the Kishony experiment as the example, we know that in that population of 3e9 that there about 13.8 million variants. The vast majority of those variants cannot grow in the regions where there are drugs. So the vast majority of mutations will be neutral or detrimental. To determine how many beneficial mutations there are would require genetic sequencing. Here is a paper where describes this:
https://www.brown.edu/...Publications/Weinreich_etal2006.pdf
Note that Weinreich makes an error in the fundamental mathematics of DNA evolution (which was not recognized by the peer-reviewers) by stating that each step on the evolutionary trajectory requires fixation The Kishony experiment shows that claim is incorrect.

Do you think that DNA evolution works differently outside of the Kishony experiment? Outside of the Kishony experiment, there a multiple simultaneous selection pressures. Do you think that DNA evolution works more efficiently under those circumstances? Even in the simple Kishony experiment, it takes exponentially more replications for adaptation to occur if 2 drugs are used and likewise if the increase in drug concentration is too large between bands, the population is facing the same mathematical constraints. So what, elsewhere in the universe changes these mathematical facts of life? It is quite likely that in that 13.8 million variants in the Kishony experiment, there is a variant that would have improved fitness for the Lenski experiment. And sure, you can have different variants that achieve resistance to a given selection pressure. These different variants will have the same phenotype but each of these variants will have taken their own particular evolutionary trajectory to achieve their particular genotype. That is what the Weinreich paper is all about. And each step for each of these different evolutionary trajectories will require about 3e9 replications. You do understand that if the different variants in a population are forced to compete in a limited carrying capacity environment will slow DNA evolution?

Does competition and fixation accelerate DNA evolution? If it does, why do the Lenski team say this:
Just a moment... And what is wrong with what they are saying here?

What is the mathematical difference between different adaptations? Does DNA evolution work differently between the Kishony and Lenski experiments? You still haven't mastered the mathematics for DNA evolution to a single selection pressure and only a single beneficial mutation that improves fitness. If you think have mastered that math, tell us how the math changes if there are two or more possible beneficial mutations which give improved fitness to a given selection pressure. Do you understand that fine enough to explain it mathematically? To make that question more specific, how do carrying capacity, selection conditions, and mutation rates affect the DNA evolution mathematical behavior of the Lenski experiment? You are conflating two different physical phenomena. Competition and fixation and DNA evolution are distinctly different phenomena. And you as well as Lenski don't understand that. That is why you cannot put evolution in the correct mathematical context. Let's put my question in the correct context by posting the entire quote: You should understand what is wrong with Lenski's statement. Start with the first five words of the sentence, "When large asexual populations adapt". Are Lenski's populations large? You have already pointed out that our gut has even larger populations. And your calculation for adaptation for the Kishony experiment requires 3e9 replications. So, now try to explain why his populations adapt so slowly. Show your math.

Do the math for the simplest case, assume there are only 2 possible beneficial mutations. That's the kind of vague answer I would expect from nwr and AZPaul3. Do the math that predicts how many fewer divisions in order to see an increase in fitness. Why don't you try first? I'll even give you a hint on how to do the math. Consider the case of 2 possible beneficial mutations. What is the probability of a beneficial mutation occurring at least once at those possible sites? You probably won't understand this but competition and fixation is a first law of thermodynamics process and DNA evolution is a second law of thermodynamics process. And I don't know what you are seeing with introns and exons. You barely understand the basic principles of DNA evolution to single selection pressure. I've already published the math. You can find it here:
Just a moment...
And you should understand why DNA evolution is slowed in a competitive environment. You have already shown that it takes 3e9 replications for a beneficial mutation to occur. So, when you have a populations such as Lenski's populations where many variants are competing for a fixed amount of glucose, that will limit the number of replications for all variants. Then, the most fit variant must drive to extinction the less fit variants in order to have sufficient resources for that most fit variant in that particular lineage to accumulate its 3e9 replications for the next beneficial mutation.Now try to do the math for two or more possible beneficial mutations and learn why this has very minimal effect on the DNA evolution process.

I don't know who taught you probability theory but they failed to teach you the difference between additive and complementary events. The question is, does a beneficial mutation occur or does it not occur, those are complementary outcomes. And then, what happens for the next evolutionary step? Are there multiple possible beneficial mutations for that step as well? I've given you the links where I show how to do the math for single beneficial mutations. I've done the math for multiple beneficial mutations. I just haven't submitted it for peer-review and publication yet. But, if you can't do the math, I'll show you how to do it here. Then you should recognize the thermodynamics for the Lenski experiment. The energy source for the experiment is glucose (and citrate for some of the variants but let's ignore that for the moment). It takes energy (glucose) to replicate. So, any glucose consumed by variants that ultimately go extinct is energy denied to the most fit variant which slows the ability of that variant to replicate. DNA evolution, on the other hand, is an equilibrium problem. Selection is trying to order the genome to give the most efficient replicator. That's a second law problem. Another way of looking at this is that DNA evolution is a Markov chain (random walk). You can read about that here:
Entropy rate - Wikipedia.
Make sure to read further down the page to the section titles "Entropy rates for Markov chains" Do you think that the non-coding regions of genomes are not important? What if those exons do modulation of the introns? You don't understand how to do the mathematics of selection. You have only demonstrated a vague understanding of selection so far. Compare the Kishony experiment with the Lenski experiment. The Kishony experiment variants accumulate 5 beneficial mutations in about 10 days (about 1 beneficial mutation every 2 days). The Lenski variants take between 200 to 1000 generations to accumulate each beneficial mutation. At 6 1/2 doublings/day, that's about 30 to 150 days of growth for every beneficial mutation. But, until fixation occurs, there are variants that ultimately will go extinct consuming resources that the more fit variant could use to replicate. You know, thermodynamics applies to biological systems. No, I'm assuming there is a wide variety of variants in Lenski's populations. The first variant that gets fixed will be the most fit variant. The second variant that gets fixed will be the second most fit variant and so on. In other words, the most beneficial mutation gets fixed first and because it gives the most improvement in fitness and that variant gets fixed most rapidly. The next most beneficial mutation gets fixed next but takes slightly more time for fixation because the improvement in fitness is less than what the first beneficial mutation did. As the experiment has gone on, each evolutionary step is taking longer because the improvement in fitness from the particular beneficial mutation is decreasing at each evolutionary step.Why don't you tell us the total number of replications necessary for an evolutionary step in the Lenski experiment if fixation takes 200 generations?

OK, it appears you are not familiar with the mathematics of stochastic processes. To do the mathematics of multiple possible beneficial mutations, you need to use the "at least one rule" from probability theory. In our case, it is the probability of at least one beneficial mutation occurring among a set of multiple possible beneficial mutations occurring. Here is a short video that explains how this rule is used:
https://www.youtube.com/watch?v=KFtCj_46TzA
So, in our case, we have multiple possible beneficial mutations that occur at multiple possible sites in the genome. Then the "at least one" formula based on success at least 1 site would be:
P(of at least one success in total number sites) =
1-(1-P(of success at 1 site)^(total number of sites) where
P(of success at 1 site) = 1-(1-e-9)^N
The point here is that if N (the number of replications) is large, even one site is sufficient for that beneficial mutation to occur. With additional possible beneficial mutations will very slightly improve the probability that some beneficial mutation will occur but each of these different beneficial mutations will lead to different evolutionary trajectories for each of these variants and once these variants are on their particular evolutionary trajectories, the number of possible sites for beneficial mutations will not be large and empirical evidence shows that it is most likely just 1. Study the Weinreich results which I gave you a link to in [MSG=81].

What's wrong with your math (just not quite correct) is that you are trying to do the mathematics with the averages (the mean) rather than doing the probability calculation correctly as I showed you how to do in the previous post. If you want to have a better understanding of this math, study the binomial distribution because you are using the mean of that distribution for your math. But that distribution has a variance and standard deviation. So when you claim that for the 2 beneficial mutation case that the number of replications will be reduced by half (ie 1.5e9) will fall within the range of the variance of the binomial distribution. We see what happens to the less fit variants in the Kishony experiment. As long as there are sufficient resources in the drug-free region, they happily replicate. And no, the population won't be swamped out by the less fit variants because the more fit variants (without drug-resistance) are also happily replicating. Remember, of the 3e9 replications that occur for the drug-resistant variant to appear, only about 14e6 will be mutants, the vast majority of that population will be exact clones of the founder wild-type. The first law of thermodynamics is deterministic (conservation of energy). The second law of thermodynamics is stochastic. One means of modeling the mathematics to describe DNA evolution is the Markov Chain process:
Models of DNA evolution - Wikipedia
and Markov Chains have an entropy rate associated with that process.
Entropy rate - Wikipedia
In particular, read the section "Entropy rates for Markov chains"
The problem with the Markov Chain models given in the Wikipedia link above is that they are assuming the transition matrix is stationary and that the evolutionary process goes to equilibrium (that is the distribution of bases goes to equilibrium). What this means is the frequency of A, C, G, and T's go to 0.25. That certainly isn't happening in either the Kishony or Lenski experiments. My next paper will explain how to correct these models so that they predict DNA evolution. Don't be so thin-skinned. My explanation for this is that introns are much more important to the phenotype of a replicator than the exons. For example, two species each can have a beta-keratin gene but the expression of that gene (which is determined by the non-coding regions of the genome) determines the phenotype. Both experiments only require single beneficial mutations for improvement in fitness. You should now have some idea what would happen if it takes 2 mutations to improve fitness. The reason why the Kishony experiment evolves so much more rapidly is the much larger carrying capacity. The more fit variants in the Kishony experiment have a new niche to grow in without competition which allows exponential growth. It doesn't take long to accumulate the 3e9 replications necessary for the next beneficial mutation, about 30 doublings of that variant will do it. You have got it backward. You should be able to compute the strength of the selection pressure based on a fixation rate of 200 generations for the Lenski experiment. You know Lenski starts each day with a population of 5e6. At the end of the day, he has a population of 5e8 of which he bottlenecks that population to 1% of the previous day's population. You should be able to compute the intensity of selection from those numbers.

That's right, and I'm showing you how to do the calculation more accurately so that you will have a better understanding of what is happening physically. If the carrying capacity of the environment is large enough, all the variants will still replicate and the population will increase in diversity. When the population reaches the carrying capacity of the environment, the competition between variants will occur. This is why it is so important to discuss the Kishony and Lenski experiments together. You can see the distinct difference between the evolutionary processes when you have DNA evolution without intense competition as in the Kishony experiment and DNA evolution with intense competition as seen in the Lenski experiment. The Kishony experiment is more comparable to how drug-resistance occurs in the clinical medical situation as you so rightly pointed out that the carrying capacity of our gut is quite large, large enough for drug-resistant variants to be preexisting without antibiotics ever being used on that subject. Incorrect. The Kimura model of diffusion is a competition/fixation model, not a DNA evolution model. Kimura wrote a different model for DNA evolution. Not only are you having difficulty with the math, but you are also having difficulty with the physics. Here's Kimura's diffusion model of fixation"
https://www.ncbi.nlm.nih.gov/...icles/PMC1210364/pdf/713.pdf
I already gave you the Wikipedia link on the Markov Chain which gives reference to Kimura's DNA evolution model. I happen to prefer Haldane's model of fixation (substitution). Haldane was right about his 300 generations/fixation estimate that he obtained from his analysis. Lenski's experiment demonstrates his claim. What Haldane was incorrect about was his claim "The principle unit process in evolution is the substitution of one gene for another at the same locus". The Kishony experiment shows that this is wrong. I have a general idea of how this works and it wouldn't surprise me if you know a lot more about this particular aspect of biochemistry. My work focuses more on the fundamental basic science of how genetic transformations occur. I'm going to suggest that the same mathematical principles that I'm putting forth here describing the formation of new alleles applies to the non-coding portions of the genome which control and modulate the coding portions of the genome. Lenski's studies might reveal some empirical evidence since he is measuring his mutations everywhere in the genome. You might get some clues on the function of the non-coding regions of the E coli by the mutations getting fixed at those locations. You are conflating 2 concepts here. When considering fitness, with respect to competition and fixation, it is relative fitness, that is the ability of one variant of a population to replicate in comparison to a different variant. When considering fitness, with respect to DNA evolution, it is the absolute fitness to reproduce, that is the ability of a given variant to replicate sufficiently to have a reasonable probability of the next beneficial mutation occurring. For example, consider the Kishony experiment, a member of the population get a beneficial mutation for the drug environment. That variant may very well have a lower relative fitness compared to the wild-type in the drug-free environment but in the drug region, that variant has the fecundity sufficient to get the next beneficial mutation.

As an average, the simple model you are using is ok, but I'm trying to get you to a higher level of understanding of the math and physics. What competition does is remove the less fit variants from a population. The relative frequency of the less fit variant(s) decreases and the relative frequency of the more fit variant increases. If the carrying capacity of the environment is large enough to allow a population to grow without competition occurring, the absolute number of all variants will increase but the most fit variants will have more offspring than the less fit variants in the same time interval. So, the relative frequencies of the different variants can be changing during this process. But remember, if you sum up the relative frequencies of all variants, it will equal 1. You said, "How heat moves through a system is very distantly related to DNA evolution to the point that false analogies start to emerge.". DNA evolution in no way is a diffusion process, competition can be modeled as a diffusion process and the reason this can be done is that competition and fixation is an example of conservation of energy. Here's a paper where they show this mathematically:
https://www.ncbi.nlm.nih.gov/...33847/pdf/pnas00072-0402.pdf
And the Kimura paper on fixation is using a standard heat transfer model which not only has a diffusion term in it, it also has a convection and energy storage term. Somehow you are stuck on this idea that DNA evolution is a function of relative fitness. It isn't. It is the absolute fitness (the number of replications) which determines the probability of a beneficial mutation to occur. And the accumulation of those replications are slowed if the variant must compete with other variants for a limited amount of resources in the environment. Do you really think you are ready to start doing the mathematics of recombination? This mathematics is a little more complicated than Mendelian genetics. I've written a paper on this subject, but sorry that paper is behind a paywall. If you want to read that paper, here's a link to it:
Random recombination and evolution of drug resistance - PubMed
If you don't have access to the journal, I'll show you how to set up the mathematics and see if you can derive the equations yourself. But recombination has very little effect on DNA evolution.

Why does Taq bring up sexual reproduction (that is recombination) when we are discussing DNA evolution? Is Taq aware of something about DNA evolution? And how do you do the mathematics of DNA evolution with a sexually reproducing population and how does this change the math?