Do you really understand the mathematics of evolution?

The way to understand how a population which does recombination on replication affects the DNA evolution of allele(s) involved in adaptation when compared to a clonal haploid population, is still to consider the replications of the particular allele(s). It is easily seen that clonal replication of a haploid is simply the replication of that member. However, with recombination, the particular allele(s) could be laterally transferred (or not transferred) on replication. So, as with the haploid case, the replications of the variant with the particular allele(s) are counted, so are the replications counted of the variant with the particular allele(s) in the recombination case. An experiment analogous to the Lenski experiment has been started which is demonstrating this.
Eukaryotic Adaptation to Years-Long Starvation Resembles that of Bacteria - PubMed
They are in the early stages of this experiment and have not yet measured the effects of recombination but the mathematical argument comes down to this. You have a population of many different types of recombing variants. Let's say you have some members of the population which have a beneficial allele (call it "A") at one genetic locus and other members of the population which have a beneficial allele (call it "B") at a different genetic locus. The remainder of the population has neither beneficial allele A nor beneficial allele B, call those variants "C". For a population size N, what is the probability of an A variant randomly recombining with a B variant to give an offspring with both A and B alleles?

It doesn't change the math much. You showed that if there were two possible beneficial mutations, it is still going to take 1.5e9 replications for at least one of those beneficial mutations to occur. What if there are 10 possible beneficial mutations? Now, if the population has to evolve to 2 or more simultaneous selection pressures simultaneously, that changes the math a lot. In fact, the number of replications for each DNA evolutionary step increases by multiple orders of magnitude. That's why Kishony's experiment will not work with two drugs or with one drug if the increase in concentration of that drug is too large requiring two or more mutations for the required increase in fitness for that region. Kishony will need a vastly larger petri dish than his mega-plate. That's not correct. Let me illustrate with the extreme case. Kishony's population in the drug-free region when they achieve their 3e9 replications will have on average, every possible point mutation. Some of those point mutations will be detrimental, even to the point that it causes the death of that variant. Wouldn't you say that that particular variant has lower reproductive fitness than any of the variants that are still able to replicate? But it is! DNA evolution is a random walk process. In particular, it is mathematically a Markov Chain process. And Markov chains are an entropy process. I gave you a link which explains this:
Entropy rate - Wikipedia
Read the paragraph titled "Entropy rates for Markov chains". And Shannon information is related to entropy:
Entropy (information theory) - Wikipedia
That's how DNA evolution is related to information. Fitness (biology) - Wikipedia
If you sum that term over time (generations), that gives you the total number of replications. You are the one who brought up the subject of recombination while we were still discussing DNA evolution. I know how to do the mathematics of random recombination, I've already published the mathematics. I've told you how to set up the problem, read [MSG=101]. I'll even give you another hint. Let the total population size be "n" and the number of members with beneficial allele A is nA, the number of members with beneficial allele B is nB and the remainder of the population which has neither beneficial allele A nor beneficial allele B is nC. Figure out this math and you will understand why recombination has very little effect on DNA evolution.

Do the math and/or present the empirical evidence! If it takes 2 (and they don't need to be simultaneous) mutations to improve fitness, the number of replications goes into the trillions for there to be a reasonable probability of that happening. Why does the Kishony experiment not work if the difference in drug concentration is too large requiring more than a single beneficial mutation? And that's when the colony has achieved 3e9 replications. Or really? There is no such thing as a detrimental (fatal) mutation unless there is competition? What happens if there is a mutation that causes the failure of some vital metabolic pathway? That variant can still replicate if there is no competition. We are talking in the real world. The first law of thermodynamics pertains to energy. DNA evolution is a second law of thermodynamics process. Obviously you are ignoring the links I've given you to Markov chain DNA evolution models and Markov chain entropy. You can say that DNA evolution is not a thermodynamic process but you would be wrong. Read those links I gave you and find out why you are wrong. I have done the math and it has been peer-reviewed and published and is also in the National Library of Medicine. I'm trying to get you to figure this out. I'll even give you another hint, the math is exactly the same as a random card drawing problem. If you say that you don't know how to do the math, I'll show you how to do it.

I'll say it again and as many times as necessary for you to get it. DNA evolution is a Markov Chain process. From the way you are posting, it appears you have no idea what a Markov Chain process is. Here's a simple introductory video to learn how you do a Markov Chain calculation. This is part 1, it wouldn't hurt for you to watch all three parts.
Entropy (energy dispersal) - Wikipedia
Why don't you tell us all what that difference is. And make sure you show your math. How many times do you want me to post the links to the papers?
So, here's a link to the paper for the mathematics of DNA evolution to a single selection pressure:
Just a moment...
Here's the link to the paper for the mathematics of DNA evolution to multiple simultaneous selection pressures:
Just a moment...
Here's a link to the paper on the mathematics of random recombination:
Just a moment...
Sorry, that paper is behind a paywall.
Here's the paper which explains the mathematics of the Lenski experiment:
Just a moment...Have fun learning how DNA evolution works.

Their application of Markov Chain mathematics is incorrect. The rate of evolution they are calculating is based on a stationary transition matrix which assumes the evolutionary process goes to equilibrium. Try to apply their mathematics to the Kishony or Lenski experiment (or for that matter, any other real empirical example of DNA evolution). The correct Markov mathematics to predict DNA evolution requires a non-stationary transition matrix. And I highly doubt you know how to do that math. I get it. Your understanding of thermodynamics doesn't go any further than the first law. There is also a second law of thermodynamics and Markov chains are an example. I am discussing my papers and I'm showing you how to correctly apply mathematics to DNA evolution. And the point I'm trying to make is that your link to 29+ Evidences for Macroevolution is based on an incorrect application of the mathematics. Do you think that the Kishony and Lenski experiments go to equilibrium? Because that's what the authors in your link are assuming.

Sure! DNA evolution doesn't work any different for humans as it does for any other replicator. Identify a selection pressure, a mutation rate and you can predict the number of generations necessary for that genetic transformation to occur. But note, only about 1e11 people have live in our entire history. That doesn't give many beneficial mutations to work with when it takes 3e9 replications for each of those mutations. The second law explains how energy is dispersed. If you are going to relate this to DNA evolution, it takes energy to replicate, replication is the random trial for a beneficial mutation, only those lineages which replicate sufficiently have a reasonable probability of increasing their information to the environmental selection conditions. Correct the model and explain DNA evolution correctly.

That math works ok if all you want to do is compute mutational divergence. But that has virtually nothing to do with DNA evolution. To put that into the context of the Kishony experiment, in the 3e9 replications, you have about 14 million mutations which represent the divergence from the wild-type. Only a tiny fraction of those mutations give improved fitness. When you are talking about adaptation, you have to get a specific set of mutations into a particular lineage. Try doing the same math for the Lenski experiment. The vast majority of mutations occurring in those lineages do not give improved fitness to those variants that get those mutations. That is a minuscule amount of the energy needed to replicate. The member that replicates needs the energy to grow and mature before it ever expends the much smaller amount of energy necessary to duplicate the DNA molecule. Think of it in terms of the Lenski experiment. A bacterium is about to replicate. When it does, the resulting fission of the single bacterium into 2 bacteria each with about half the mass of the parent cell. Those 2 bacteria need to grow to full size before able to replicate again. That's where most of the energy (glucose) is going to in that experiment.

It's not enough to change each base 300 times over. DNA evolution requires a lineage to accumulate specific mutations. If adaptation requires the lineage to accumulate mutations A and B, it's going to take 3e9 replications for a variant to get mutation A and that variant with mutation A will need to do 3e9 replications to get mutation B regardless of how long the genome is. All you are describing here is genetic diversification. Or perhaps you think every one is getting identical mutations. In my universe, I have to deal with drug-resistant infections. Your problem is that you think that DNA evolution works different for different length genomes. It doesn't. It's the information in the DNA sequence that is related to thermodynamics, specifically the second law. Perhaps it would be easier for you to understand if instead of using the terms information and entropy to describe this by using the terms order and disorder. Genetic adaptation to selection conditions is done by increasing the order of the genome. In the context of the Kishony experiment, the mutations are being selected for to increase the order in the genome to grow in the higher drug-concentration regions. It works the same way for any DNA evolution situation. Random mutations without selection increase the disorder in the genome. If it was possible, over time, the genome would become random sequences of bases as these random mutations without selection are accumulated.

If you think that DNA evolution works differently in bacteria than any other replicator, produce your mathematical and empirical evidence to prove this. Bare assertions may work in your universe, but they don't work well in science. If you understood DNA evolution, you would understand that the mutation rate doesn't dominate the process. It is the multiplication rule of probabilities which dominates the DNA evolution process. This is why the Kishony experiment, as designed, will not work with 2 or more drugs. Just make his mega-plate vastly larger then you can achieve the population sizes need for this DNA evolutionary process to work.As for recombination giving an offspring with the two beneficial mutations occurring, are you ready to do that math? Don't be surprised if you see the multiplication rule showing up in the mathematics of that evolutionary process as well. So what? All you have shown is diversification, not DNA evolution. What's the probability of any one of those individuals getting two beneficial mutations. Maybe you think that someone in Australia with a beneficial mutation will meet someone online from Finland with a different beneficial mutation, get married and have a child with both beneficial mutations. Is that how things work in your universe?All it takes, for a mutation rate of 1e9, for there to be 3e9 replications for you to get on average some member of the population with one of every possible point mutation. You did that math! And it doesn't matter what the length of the genome is. Now, what does that do to the use of targeted therapy for a cancerous tumor that has 3e9 cells? Same physics, same math.

If you think that DNA evolution works differently in bacteria than any other replicator, produce your mathematical and empirical evidence to prove this. You don't understand the mathematics of DNA evolution, you don't understand the mathematics of recombination. If you think that DNA evolution works differently for mitosis, meiosis, or fission, explain it. Bare assertions may work in your universe, but they don't work well in science. Are you ready to show us the mathematics of recombination and how it affects DNA evolution? Please show your math (that is if you know how to do the math). Just say you don't know how to do the math and I'll show you how to do the math of random recombination. I've given you so many hints already, it should be easy for you. Try again. The only correction you need to do to your previous math is that humans are diploid so it only takes 1.5e9 member replications to get the 3e9 genome replications. It is clear that math is not your strong suit so let's make this as easy as possible. Imagine that you have a haploid genome of length one single base. For a mutation rate of 1e-9, how many replications of that base for there to be an average of 1 of each of the possible 3 substitutions? Then do the math for a genome length of 2 bases and tell us how the number of replications change based on genome length. You do understand that for a diploid replicator, a single-member replicating gives two genome replications. And another point you are failing to get, humans have achieved the population sizes necessary for the first mutation to occur in a DNA evolutionary process. It's the second and ensuing beneficial mutations in that lineage that have a low probability of occurring. That variant and its descendants with the first beneficial mutation must replicate 1.5e9 times (diploid) for that second beneficial mutation to occur on some member with the first beneficial mutation. What does that mean empirically? What it means is that there is a reasonable probability that somebody in the population will get a sickle cell trait mutation but a very low probability that someone will get a sickle cell trait mutation will also get a thalassemia mutation. Perhaps you want to try an do the math for someone with a sickle cell trait mutation marrying someone with a thalassemia mutation getting an offspring with both mutations?And when are you going to learn that the mutation rate is not the dominant feature of the DNA evolution process, its the multiplication rule which drives DNA evolution. That's why hiv which has a mutation rate on the order of 1e-5 still cannot evolve efficiently to 3-drug therapy (even though the virus does recombination). If you could do the math, you would understand why this happens. Bare assertions may work in your universe, but they don't work well in science. First, learn the physics and how to do the mathematics before you make your bare assertions. When are you going to show us how the mathematics of recombination works? Just tell us you don't know how to do the math. You will then understand why recombination has no effect on the treatment of hiv.

Yes, mutations A and B can occur in two different individuals and that by some form of lateral transfer they then can be combined in a single descendant. But there are some variants of e coli that can do bacterial conjugation, so no, it doesn't differ greatly. What you should try to learn is how to calculate the probability that that lateral transfer of genetic material can occur. And the probability of that occurring depends on the multiplication rule of probabilities. I posted a link to a paper where they reported that DNA evolution for eukaryotes works the same as for bacteria. Did you bother looking at that link? They looked at the DNA evolution of yeast cells subject to starvation and they saw the same evolutionary behavior that Lenski is seeing with his bacterial starvation evolutionary process. If you want to use a mutation rate of 5.56E-9, then your calculation is correct. But you still need to learn that the mutation rate is not the dominant factor in DNA evolution, its the multiplication rule which dominates DNA evolution. I am not ignoring sexual recombination. You make these vague claims about mutation spreading through the population and then members from both lineages mate and produce offspring with both mutations. Why doesn't this happen with hiv causing combination therapy to fail? It is clear you need more hints to do this math. You have a population size "n", in that population you have a variant with beneficial allele A and the number of members with that allele is nA, you have another variant in the population with beneficial allele B and the number of members with that allele in nB, the remainder of the population has neither allele, call them C and the number of members without either allele is nC. Then nA/n + nB/n + nC/n = 1. Write out the distribution function for this situation for the probability of an nA member recombining with an nB member to give an AB descendant. Remember, an nA member can recombine with another nA member, an nB member, or an nC member, likewise for the nB and nC members. Let's see if we can get you out of this vague understanding of DNA evolution, recombination and genetic transformation.

Too bad you don't have such precision when it comes to the mathematics of DNA evolution. And I said "some form of lateral transfer". But if you are going to nitpick on points like this, I'm going to nitpick you on the mathematics, only your mathematical errors are not nits. Lenski specifically chose strains of e coli which don't do conjugation, I don't recall what Kishony said about his strain but I suspect he also chose a variant that doesn't do conjugation to prevent that complication. I doubt it would make a difference in either experiment anyway because before you can laterally transfer a resistance allele with a phage, the allele must evolve by the DNA evolutionary process. In your universe, you probably would introduce a resistance allele from outside the experiment on a phage and say see how quickly DNA evolution works when you take into account worldwide populations. By the way, there is plenty of other empirical data that supports the math that I've presented. It happens these two experiments are among the best-measured examples. I am discussing it. In fact, I'm trying to explain to you why recombination has virtually no effect on DNA evolution. I've even gone so far to set up the mathematical problem. The only thing remaining is to take the variables I've given you and formulate the equation. It really is just an example of a random card drawing problem. Have you ever studied probability theory? You should if you want to understand how stochastic processes work. Nope! So, now you are claiming that hiv doesn't do recombination because it isn't a diploid eukaryote? So tell us, plants are eukaryotes. Why do combination herbicides inhibit the evolution of herbicide-resistant weeds? Some plants are even polyploid. And they do recombination. Why doesn't a variant weed with a resistance allele to one herbicide recombine with another variant with a resistance allele to a second herbicide to give a variant with resistance to both herbicides? Don't be silly, I know what kind of virus hiv is and it does recombination. But I can see you need help writing out the probability distribution function for random recombination. So open wide, I'm going to spoon-feed you:f(x,y) = 2!/(x!y!(2‘x‘y)!) *(nA/n)^x *(nB/n)^y *(nC/n)^(2‘x‘y)(x+y 2)And if you want to compute the probability of an A variant and B variant recombining to give an AB offspring, set x=y=1When you do that, what you will find is that the only time you will have a reasonable probability of that happening (randomly) is if the A and B variants are at high relative frequency in the population. If the C variant represent most of the population, most recombination events will be AC, BC, or CC with the majority being CC recombintion. This is why recombination is not a factor in hiv evolution to 3-drug therapy. You just don't have a high enough frequency of the A and B variants to have a reasonable probability of that recombination event.

I've presented it, you just are having difficulty seeing it. Consider what will happen in the Kishony experiment if he uses a diploid eukaryote instead of a haploid prokaryote. For example, yeast, and instead of antibacterial agents, he uses antifungal agents. All else being the same with two antifungal agents being used. Assume a mutation rate of 1e-9.The wild-type yeast has no beneficial mutations and a colony starts to grow in the drug-free region and after 1.5e9 replications, every possible mutation has occurred on different members of that population. In that set of mutants, there is a member that has a beneficial mutation for one antifungal agent (call it A) and a different member has a beneficial mutation for the other antifungal agent (call it B). Now I've shown how to compute the probability of those two members randomly meeting (Message 123) and recombining to give an offspring AB variant. If you plug x=y=1 into that equation from the earlier message, you get:f(1,1) = P(A,B) = 2*(nA/n)*(nB/n)The population size is n=1.5e9, nA=nB=1 then, nA/n=nB/n=1/1.5e9The only way there is going to be a reasonable probability of the AB recombination event occurring is if nA/n and nB/n are at high frequency.

Swamidass is about as familiar with the mathematics of evolution as the posters on this forum. For example, from the exchange we had on that forum: Swamidass is on the faculty of a medical school yet makes an undergraduate level blunder here. He is confusing additive events with complimentary events and it very easy to show why he is wrong. Let's say for population size N, the probability of a beneficial mutation occurring is 0.6. If you double the population size to 2N the probability of a beneficial mutation occurring goes to 1.2????? If you want to understand the mathematics of DNA evolution, you have to understand the fundamentals of probability theory. Swamidass has demonstrated he doesn't understand those fundamentals and so far, none of the posters on this forum have done much better. And by the way, I asked the same question from PhD evolutionary biologist John Harshman and he makes the same blunder. If you want to understand DNA evolution, you need to understand introductory probability theory.

Using your math, the single resistant mutants would emerge after 3e9 genome (gene) replications, it would then take another 3e9 replications of that variant for the second beneficial mutation to occur on the variant with the first beneficial mutation. If the experiment is performed with a single selection pressure as done with the present Kishony experiment, that 3e9 replications will occur in the next higher drug concentration in about 30 generations of doublings. If it requires both mutations before the variant can grow in the next higher drug-concentration region (higher concentration of drugs or two drugs) then the 3e9 replications of that variant must occur in the original founder colony which already has only slightly less than 3e9 members and it will also be doing 30 doublings. That's why the colony size must be in the trillions before the increase in fitness can be achieved. I hope you now understand why recombination does not make a difference in the DNA evolution process. Unless the members with the individual beneficial mutations are at high frequency, the probability of that recombination event will be minuscule. Even if the yeast are replicating sexually, it has a negligible effect on the DNA evolutionary process.