Do you really understand the mathematics of evolution?

Then, explain to us how epistasis affects the mathematics of either the Kishony or Lenski experiments. And show your math if you think you understand the mathematics of evolution. And the more complex the fitness landscape, the more complex any evolutionary trajectory. And if you want to see a bottleneck in a population, have a big change in the environment, that is if the population is not driven to extinction. Big environmental changes are far more difficult for populations to adapt to. Even the Kishony experiment demonstrates this. If the step change in antibiotic concentration is too large, the experiment does not work. That's because 2 or more mutations are needed to give the necessary improvement in fitness. DNA evolution works most efficiently when only a single mutation gives improved fitness in the given environment. The reason for this is the multiplication rule which requires exponentially more replications for that evolutionary process to have a reasonable probability of occurring. I didn't say the beneficial mutations have to occur at the same time. But the multiplication rule still applies no matter when the beneficial mutations occur. What improves the probability of this happening is the ability of the particular variant's ability to replicate.Let's use your slot-machine analogy. Let's say there are two possible jackpots, a super jackpot, and a big jackpot. And let the probability of winning the super jackpot is one in ten million and the probability of winning the big jackpot is one in a million with each pull of the arm. Do you understand how to compute the probability of winning both of those jackpots as a function of the number of pulls on the arm? It's essentially the same math as the Kishony experiment if he were to use two drugs.And you are still having difficulty understanding that every evolutionary pathway must obey this math. Kishony's bacteria may have multiple different sets of mutations that can give resistance to the drug used but it still is going to take a billion replications for each evolutionary step on each of these trajectories. These are the mathematical facts of life of DNA evolution. You don't need to do that. Simply take the bacteria that has already adapted to ciprofloxacin and use those as the starter population for the trimethoprim experiment. All you are demonstrating is that DNA evolution works most efficiently a single selection pressure at a time. That's how MRSA was created except in the clinical environment. When one drug failed, go on to the next drug until that drug fails. If you are going to gamble with microbes, you should understand the rules of the game. The limited carrying capacity environment in the Lenski forces his populations to compete for the limited glucose. If Lenski were to run his experiment in 100ml of solution instead of 10ml, the more fit variant will achieve enough replications to have a reasonable probability of another beneficial mutation occurring before the less fit variants are driven to extinction, fixation is not necessary for a beneficial mutation to occur in this case. What you should try to understand is that the human body has a huge carrying capacity for microbes. The population size of these microbes easily achieves the numbers necessary for drug-resistant variants to occur. In other words, the Kishony experiment would not work in a standard-sized petri dish, you need very large populations for DNA evolution to work. Not so. Lenski has run experiments using thermal stress for his selection condition.
https://courses.pbsci.ucsc.edu/...i_&_Bennett_AmNat_1993.pdf

The reason this is the case is that the carrying capacity in the drug-free region is not large enough to accommodate both the wild-type and the drug-resistant variants multiplication. I tried to explain this in an earlier post but I'm just not getting through to you.Try and understand it this way. Imagine you have a vast, unlimited size petri dish with infinite carrying capacity and no antibiotics. You start with a single wild-type bacterium that is sensitive to all antibiotics. This bacterium doubles, the resultant bacteria double, and on and on with members accumulating mutations at the frequency of the mutation rate. After 30 doublings, you will have a billion bacteria and for a mutation rate of e-9, you will have on average, some member with a mutation at every site in the genome. That means there will be about 1 member with a beneficial mutation for ciprofloxacin, another member with a beneficial mutation for trimethoprim but the rest of the population will have mutations at other sites or no mutations at all. Then this population continues to double for 30 more generations. You will then have a billion members with the ciprofloxacin mutation (plus any other mutations accumulated including one for the trimethoprim), a billion members with the trimethoprim mutation (plus any other mutations including one for the ciprofloxacin), and for the remaining population of a billion minus the ciprofloxacin and trimethoprim variants, you will have a quintillion members (minus a few members along the way that get either the ciprofloxacin or trimethoprim mutations). The bands in the Kishony experiment give a region where the drug-resistant variants can grow without having to compete for the depleted resources in the drug-free region. Here's a Venn Diagram of the system: A1 and A2 are the two different variants. That answers that, you don't understand the mathematics of evolution so you snip out the part highlighted in red. What you refuse to accept is that the steps in every evolutionary trajectory are joint by the multiplication rule. And if it takes multiple mutations to improve fitness in an evolutionary step, you have multiple instances of the multiplication rule acting simultaneously. That's why the Kishony experiment won't work on his large petri when two drugs are used or when the step increase in drug-concentration is so large that it requires two or more mutations for that step. You can't explain mathematical behavior of the Kishony or Lenski experiments so now you start storytelling. Why does it take a billion replications for every evolutionary step in the Kishony experiment? Please show your math. I havn't ignored that at all. It doesn't matter which evolutionary trajectory that a lineage takes. It doesn't change the math. That math will work if the mutation rate for your first case is 5e-8 and 5e-7 for you second case. For the Kishony and Lenski experiments, the beneficial mutation closer to about e-9. But the mutation rate is not the major driving factory in DNA evolution, it is the number of selection pressures the population is adapting to because each selection pressure introduces another instance of the multiplication rule in the evolutionary trajectory. This is why despite its very high mutation rate, hiv cannot evolve efficiently to 3 drug therapy.
Just a moment... Is that so? You are obviously not aware that single-drug therapy has been the standard of care for infectious diseases for years. That's why MRSA is so common. What is your logic and reasoning to think that beneficial mutations for one selection pressure would be the same for a totally different selection pressure?

In this discussion, we are talking about two evolutionary processes. The first evolutionary process is what Darwin calls the "struggle for existence" or what we usually call competition. And the second evolutionary process is what Darwin calls "adaptation" or what we are calling DNA evolution. The Lenski and Kishony experiments are both DNA evolution processes with the important distinction that the Lenski experiment is being carried out in a highly competitive environment.The only clear gross anatomical changes that is occurring in these experiments is that Lenski's bacteria are getting larger. What the Lenski and Kishony experiments show is that every evolutionary transitional step takes a billion replications and that just for a single selection pressure environment. And the reason for this is the multiplication rule of probabilities.A common misconception is the notion that a series of microevolutionary changes can add up to a macroevolutionary change. This is mathematically incorrect. Microevolutionary changes don't add, they are linked by the multiplication rule of probabilities because each of these mutations are random events. Your basic idea is correct. If I understand your argument correctly, you are saying there should be far more transitional fossils in the fossil record for the theory of evolution to be correct. To get a sense of the numbers, there have been about 50 T rex skeletons discovered and it is highly unlikely that there were billions of T rexes (pinnacle predator). So if it takes a billion replications (under the best of circumstances) for each evolutionary step, where are all these vast numbers of transitional fossils of reptiles transforming into birds and fish transforming into mammals when there are 50 fossils of a single pinnacle predator which most likely existed in very small numbers compared to those replicators lower in the food pyramid? From this link:
How many T-Rex were there in the dino era? - Quora
"The food pyramid of most ecosystems is accurately described relative species abundance as a power of ten, decreasing from, e.g., 10,000 primary producers (plants), to 1000 herbivores to 100 secondary carnivores, to 10 top predators, and finally 1 pinnacle predator."

If you understood the mathematics of evolution, you would understand how much larger the region needs to be for the experiment to work. How much simpler of an evolutionary experiment can you make than either the Kishony or Leski experiments? If you don't know the exact values for the mutation rates and population sizes, write out the mathematics with those numbers as variables. If you can't do it, I'll show you how to do the math. I've never said that the all the steps have to taken all at once. Why don't you try to understand what it takes to make a single evolutionary step? The math isn't that difficult. If I was going to ask you about the mathematical behavior of airplanes and flight, I would start with Newton's 2nd law and ask how you would derive Bernoulli's equation. But we are not talking about flight, we are talking about DNA evolution and competition. So, derive for us the mathematics which describes a lineage accumulating a specific set of mutations allowing adaptation to a selection pressure. When you do that, you will understand the mathematics of the Kishony experiment (and the Lenski experiment as well). You are wrong Taq. The mathematics for any step on an evolutionary trajectory is the same regardless of the selection pressure. The same equation that describes a step of DNA evolution for the Kishony experiment applies to the Lenski experiment. A study was done at one of the hospitals in my area. What they did is swabbed and cultured everyone's (patients, employees, visitors) nose on entry. 50% were carriers of MRSA. This coincides with the numbers in my medical practice. About 50% of the soft tissue infections I treat are community-acquired MRSA. The question you should ask is, why are hospital-acquired MRSA infection resistant to more antibiotics than community-acquired MRSA? mike the wiz has already admitted that he doesn't understand the mathematics of evolution. So, I'll help him with the answer. Are you familiar with Markov chains? I don't think you are so here is a link that explains the concept.
Models of DNA evolution - Wikipedia
This link gives a mathematical definition for a transition, they call it a transition matrix. Read about and try to learn something about the mathematics of DNA evolution.

Fixation isn't occurring in the Kishony experiment. Write out the mathematics of DNA evolution for the Kishony experiment. It appears you don't understand the difference between competition and adaptation. It isn't, but if you think it is, just do the mathematics for the first step for the Kishony experiment. I'll make it as easy as possible, just do the mathematics for the first beneficial mutation for the Kishony experiment. It isn't. You just haven't learned how DNA evolution works yet. Try doing the math for the first beneficial mutation in the Kishony experiment. It's clear you need a hint to get started. Start with a single bacterium that doesn't have the correct mutation and starts replicating in the drug-free region. Then write out the probability equation of that particular mutation occurring at least once in N replications for a given mutation rate. None of the employees or visitors were on antibiotics, and yes there is great variability in the immune status of the individuals. That should give you a big clue on why hospital-acquired MRSA is resistant to more antibiotics than community-acquired MRSA. You touched on a key reason, the immune status of inpatients vs outpatients.And I noticed that you snipped out the portion about Markov Chains and the transition matrix. It's probably too much for you at this point. Try to understand the "at least on rule" from probability theory. That's a little easier way to understand DNA evolution.

Sorry, I thought you were familiar with the concept of molecular evolution. Read this paragraph:
Molecular evolution - Wikipedia I've given you enough specificity, you just don't understand molecular evolution well enough to formulate the probability equation that relates the mutation rate and number of replications of a particular variant to determine that probability. Read the above paragraph I linked to and think about it for a while. The mutation rate is a variable, call it "mu". And for simplicity, assume the mutation rate is the same for transitions and transversion. And initially assume there is only a single beneficial mutation. Once you understand the simple case, I'll show you how to do the math for multiple beneficial mutations. More than a single possible beneficial mutation changes the math only slightly. First do the math for the simpler case. Start with one. Once you do that math, then you can try to do the math when there are multiple possible beneficial mutations. Try doing the math for the simple lab experiments first. And in any mathematical model, you want to include only those variables that have a significant effect on the predictability of that model. If we were studying Newton's laws, I wouldn't expect you to predict the motion of a bridge or building in an earthquake. You need to start your study with the motion of a pendulum or a mass and spring. Once you master the simple cases, you can go on to the more complex cases. You've got a point there. So do the mathematics for a single beneficial mutation with a constant mutation rate as a function of the number of replications of a particular variant. Hint: Use the "at least one rule" from probability theory.

Then explain to us the following quote from this link (and show your math):
Molecular evolution - Wikipedia Yes, I can answer this question. If the draw of the tiles is random and the probability of drawing any particular tile is equal (same size, same shape, etc.), then that probability is 1/N, where N is the total number of tiles in the bag. See Taq, that's how you use variables in a probability calculation. OK That's a little different way of looking at the math but that is essentially correct. And if mu=1e-9, 1.38E7/4.6E6*mu)=3.00E+09. The reason why your estimate is 3x larger than my estimate, I'm assuming the mu is the "beneficial" mutation rate and you are assuming the "total" mutation rate where there are 3 possible incorrect mutations at the particular site.Now, what happens to that 1 variant member of the 3e9 population with the beneficial mutation in the Kishony experiment? Not significantly. When we finish with the simpler model, we can discuss how multiple beneficial mutations affect the math. That's a problem for you, not for me. The purpose of mathematical modeling is to do an analysis of a physical system. For example, if you want to do the analysis of a bridge, the geometry of the bridge is a variable, the physical properties are a variable, the loading forces are variables. So if you change the material used for making the cables which have a different tensile strength, you can just change that one variable and see how that affects the stresses in the structure. The math you did above is ok but it is not as flexible as writing the full governing equations of the evolutionary process. For example, what happens if the bacteria used has a different genome length? Do you think that will change the number of replications for the beneficial mutation to occur?

Let's put the quote back up which I ask you to explain:
Molecular evolution - Wikipedia Taq, you have actually started to explain why mutation accumulates very slowly across generations in your previous post when your math shows that it takes 3e9 replications for the first beneficial mutation in the Kishony experiment. So, how many replications of that new variant for the next beneficial mutation to occur on that variant? Hint: Just apply your math again. And note, that is the regular pace for a single selection pressure. You got me on that one. You have got to watch your assumptions. 13.8 million are the total number of possible variants (if we consider substitutions only) from the initial founder wild-type bacterium in the Kishony experiment. How many replications necessary for the variant with the first beneficial mutation expected to occur? And how many replications of that variant necessary for the next beneficial mutation to occur? I understand this discussion is difficult for you. It doesn't bother me if I don't get all the information necessary to give an exact answer. If fact, in the world that I work in, that is the usual case. When you asked your tile question, I gave an answer and made an implicit assumption that the tiles had different numbers and you showed my implicit assumption could be wrong. So, here's a specific question. How many replications necessary for the 13.8 million variants to occur in an evolutionary step in the Kishony experiment?

I'm not asking you to write VOLUMES. I'm just asking you something very specific. How many replications for each evolutionary step in the Kishony experiment? And how do you compute that number of replications? It only happens in a matter of days because the generation (doubling) time for bacteria is usually minutes. But if it is happening so quickly, why won't you tell us the number of replications that occur in those matter of days? No, I'm not asking for the mathematics for an entire field, I'm asking for the mathematics for the number of replications for the Kishony experiment. How much more specific do you want me do get? How about just giving us the mathematics for the number of replications for a single evolutionary step in the Kishony experiment? The Poisson distribution is ok. Are you aware that the Poisson distribution is the limiting case of the binomial distribution as the mutation rate approaches 0 and the number of replications goes to infinity?
https://www.youtube.com/watch?v=ceOwlHnVCqo
So use either distribution and tell us how many replications for a single evolutionary step in the Kishony experiment. You are the one having difficulty answering a specific question. Why won't you tell us how man replications it takes for a beneficial mutation to occur for a single step in the Kishony experiment? You can even use the Poisson distribution for your answer even though the binomial distribution is the correct distribution. I can show you how to do that math if you continue to have difficulty. Why? It's a very rare thing in life when you understand something exactly. Sometimes, all you can do is give a ballpark estimate. So far, you refuse to even call Uber to get a ride to the stadium. When are you going to give us the mathematics which estimates the number of replications necessary for a single evolutionary step in the Kishony experiment? You can even use the Poisson distribution to make your estimate. Just use the mean value for the Poisson distribution and you can easily compute the variance (and therefore the standard deviation) of that value to see what the spread of values are. Here is a short video that explains how to do it.
https://www.youtube.com/watch?v=vrQ9JsAPTUc
If you have difficulty applying this math to the Kishony experiment, I'll show you how to do it. So tell us how many replications required for a single evolutionary step in the Kishony experiment using the Poisson distribution.

It seems that nwr is not the only one with a vague understanding of DNA evolution, Taq doesn't do much better. So let's see if we can help. So, if the genome length is 4.6E6 and if mu = 1E-9, it would take about 218 replications of the original founder bacterium of Kishony's drug sensative lineage before we would expect to see the first mutation in one of those 218 members, somewhere it its genome. The remaining 217 members should be exact clones.

The reason why the Poisson distribution is ok but not correct because the probability of success (the mutation rate) is very small and population sizes are very large (the number of trials). Under these circumstances, the Poisson distribution gives similar results as the binomial distribution (the correct distribution function for this probability problem). There isn't a lot of work to go through to answer the question of how many replication necessary for the Kishony experiment to get a variant to adapt to the next higher drug-concentration region. You can even use the Poisson distribution to do your calculation. I've already given you a link that explains how to do the math. If you can't do the math for this simplest of evolutionary experiments, how are you going to do the math for the more complex evolutionary experiments? Quite correct. So, do the math for the simplest case. Assume there is only one possible beneficial mutation at a single site in the genome that will give the necessary improvement in fitness for that new variant to grow in the next higher drug-concentration. Assume you start with a single wild-type bacterium in the drug-free region. What is the number of replications of that wild-type variant needed for that improved fitness variant expected to appear? Please show your math.

Use whatever distribution you think is appropriate. Tell us how many replications are needed to give a reasonable probability of a drug-resistant variant occurring able to grow in the next higher drug-concentration region in the Kishony experiment and show your math. Don't you think that correctly describing the evolution of drug-resistance is important? And this subject is important in understanding how to treat cancer, especially as targeted cancer therapies are developed. And why is it so important to you to stop this thread? What is so terrible about correctly describing the physics and mathematics of evolution? My understanding of evolution has been helped by discussions like this.I think that Darwinian evolution is qualitatively correct. What's wrong with quantitating his theory?

Very good! I hope AZPaul3 is following this discussion because his estimate was between 1 and infinity. 3 billion is definitely in the ballpark. There are several directions this discussion can take but let's stick with the line we are on and try to give the mathematical explanation of why the Kishony experiment won't work with two drugs instead of one. And to do this, consider the 100ml to 1 liter of culture that you described with the bacterium with the beneficial mutation in that population of 3 billion. How large would the growth solution have to be if you took those 3 billion bacteria (with the one member with the resistance mutation) in order to get a variant with the first 2 beneficial mutations? In other words, how large must the population be for a variant to occur with both mutations necessary to grow in the next higher drug-concentration region with two drugs? Assume that you need only a single mutation for each drug.And by the way, the size of the genome is actually not a factor. When you have a mutation rate of 1e-9 and 3 billion replications of that genome, you will have on average, every substitution possible at every site in the genome no matter how long the genome in some member of that population.Edited by Kleinman, : Correct the title

We are not talking about two simultaneous mutations. We are talking about a second particular mutation occurring on some member of the population where a first particular mutation has already occurred. In the context of the Kishony 2 drug experiment, in your 3 billion population, you already have a variant with a mutation for the first drug. You also have a variant with a mutation for the second drug as well. How large must the total population size be for a mutation for the second drug occur on a member that already has a mutation for the first drug or for a mutation for the first drug occur on a member that already has a mutation for the second drug? This is a classical conditional probability problem. I gave you a Venn diagram earlier in the thread to help you understand the problem: A1 is the beneficial mutation for the first drug and A2 is the beneficial mutation for the second drug. The intersection of the two subsets is what you need to compute. The entire rectangular box represents the entire population size of those members which have neither the A1 or the A2 mutations. We know what the variation is for the Kishony experiment, you are starting with a population size of 3e9. Of that 3e9 population, 13.8 million are variants with each of these members having one of the possible mutations but the vast majority of the starting population is still "wild-type" that is they are exact clones of the original founder bacterium. And we have only 1 member with a beneficial mutation for the first drug and only 1 member with a beneficial mutation for the second drug. Now, the 3e9 bacteria start doubling, how large must that population be for there to be a reasonable probability of getting a variant with both the A1 and A2 mutations. I'll give you a hint here to help you understand this problem. In the single drug experiment, that single drug-resistant mutant moves into the next higher drug concentration region and must achieve 3e9 replication for the next beneficial mutation to occur. I don't agree with that. If you can't do the mathematics, the best you can have is a vague understanding of the phenomenon. We are trying to give nwr more than a vague understanding of evolution. That's not correct but since fixation does not pertain to the Kishony experiment, it is not worth discussing this subject at this time.Try to figure out how large the population size has to be for a variant to occur with 2 beneficial mutations for 2 drugs to make one evolutionary step in the Kishony experiment if 2 drugs are used.

That's correct for the single drug experiment. How does the math work for the 2 drug experiment where some variant has to appear in the drug-free region with a beneficial mutation for each of the drugs before it can grow in the next higher drug-concentration region? How large does the founding colony have to be for there to be a reasonable probability of that variant appearing? The math tells us what the variation is. You start with a single wild-type bacterium without any resistance alleles. When we have 3e9 replication with a mutation rate of 1e-9, that gives us on average 1 member in the population with any of the 13.8 million possible substitution mutations, The remaining members will be wild-type clones of the original founder. Of course, somewhere else in the world they may be other variants but try to understand the Kishony experiment first. You have started to do the math but then you said you can explain your train of thought without algebra. That is vague. And your approach to the math was slightly different than my approach but your approach is correct as well. The problem with your approach becomes obvious when you try to do the math that would predict the behavior of the Kishony experiment when using two drugs instead of one. As you try to do the mathematics for the two-drug experiment,you will see why your approach doesn't work. The problem is that you are working with two linked binomial probability problems. Perhaps you can get your approach to work but I don't think so. Fixation is not applicable to the Kishony experiment. When you finally figure out the math for the two-drug Kishony model, we can discuss fixation in the Lenski experiment (except that is fixation for a haploid and that fixation is selective). This will give you practice in applying evolutionary mathematics to real, measurable, and repeatable experimental examples of evolution.So, try to figure out how large the population size has to be for a variant to occur with 2 beneficial mutations for 2 drugs to make one evolutionary step in the Kishony experiment if 2 drugs are used.