Do you really understand the mathematics of evolution?

I think you already saw it:"The Poisson distribution is ok."If this is a throw away argument, I really don't want to go through the effort of digging through all the formulae and doing the math. Is there a point you want to make with all of this? You still need valid assumptions to get ballpark estimates.

A Poisson distribution is a binomial distribution:"A familiar nongenetics example of a binominal probability distribution is a coin flip: heads and tails are two discrete outcomes, and the probability of each is 0.5 on any single flip. One type of binomial distribution is the Poisson distribution, which expresses the probability of a given number of occurrences of an event that occurs during a fixed time period."
https://www.genetics.org/content/genetics/202/2/371.full.pdf So what is the ultimate point you are trying to make? Explain to us why it is worth our time to do the calculations? If we do the calculations, will you admit that people do understand the calculations and stop posting in the thread?

Going with a per base mutation rate of 1E-9 and a genome size of 4.6E6 bases, that would be one mutation per 217 replications. If we are looking for a specific substitution mutation that would be 1 mutation out of 4.6E6 bases and 3 possible mutations per base for a total of 1.38E7 mutations. multiply the number of replications per mutation by the number of possible mutations and you get 3E9 replications. So you would need about 3 billion replications to have a reasonable chance of getting any specific substitution mutation in E. coli with that specific mutation rate and genome size.E. coli in saturated liquid culture is about 1E6 to 1E7 per ml. If we added 1 bacterium to 100 ml to 1 liter of culture and let it reach stationary phase we should get our mutant.Now what?

I think we can all agree that needing two mutations at once is going to make adaptation nearly impossible if you start with zero variation. Is there a point beyond that?What would this look like if you start out with a certain amount of variation? I knew as much when I started, but sometimes it is easier to follow a train of thought instead of employing algebra to eliminate redundancies.In the same vein, the neutral fixation rate is equal to the mutation rate no matter the size of the population.Edited by Taq, : No reason given.

The same calculations would apply, starting with the first bacterium to gain resistance to a single drug. You would need another 3 billion descendants from that first resistant bacterium to get the mutation for resistance to the 2nd drug. That doesn't tell us what the genetic variation is. A population of 3 billion that recently descended from a common ancestor is going to have less genetic variation than a sample of 3 billion individuals from a population that has been dividing for a long time period. It is similar to the genetic variation in 10 close family members vs. the genetic variation of 10 individuals randomly sampled from across the globe. I did the math. Your only complaint seems to be that I didn't do the math the way you wanted me to. Just in case you were curious.

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For a diploid population of size N and neutral mutation rate mu , the initial frequency of a novel mutation is simply 1/(2N), and the number of new mutations per generation is 2N*mu * (1/2N)=mu . Since the fixation rate is the rate of novel neutral mutation multiplied by their probability of fixation, the overall fixation rate is mu. 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)

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I would assume it would be the multiplicative, so 3E9^2. Genetic variation is an observation. The Kishoni experiment does start with a single bacterium, but I am asking a different question. What if we start with a wild population that already has genetic variation? I did the math. I started it AND FINISHED IT. What I didn't do is construct all of the equations and cancel out variables using algebra. It is applicable to molecular evolution. If you ask for the math of evolution part of that math is the rate of fixation of neutral mutations. You will also get neutral mutations moving towards fixation in the Kishony experiment.

How is it incorrect? How are they conditional and not independent?Let's see your math. We aren't going back in time. I am talking about right now. What if we gathered E. coli from around the world and put them in a single population? What then?

I was under the impression that you were combining the antibiotics into one region which would require adaptations to both drugs in order to adapt to the single new region. Is this not the case? This would mean that even if we get a mutation for resistance against one of the drugs, that mutation won't be selected for. How is the genetic variation within a population not a factor in evolution?

We could talk about the population of E. coli in a single person's gut, if we wanted to. Would you agree that the genetic variation of the E. coli population in your gut is probably greater than that used in the Kishony experiment? Doesn't the starting genetic variation of a population affect how that population evolves, and how the math of population genetics applies to it?

We also have to be careful not to fall victim to the Sharpshooter fallacy. It is entirely possible for a new mutation to be neutral in one genetic background and beneficial in another genetic background. This beneficial phenotype would be dependent on two mutations.So how many neutral mutations can become beneficial mutations when combined with new mutations? I don't think we can really know this number for any genome. If there are millions of possible beneficial interactions, then it isn't surprising that a beneficial phenotype emerges that requires two mutations, one of which is neutral all by itself. It would be incorrect to draw a bulls eye around this phenotype and then claim that it is highly improbable that such a trait emerged.

There is a universe that exists outside of the Kishony experiment. Antibiotic resistance isn't the only beneficial adaptation that exists in the universe. I would also suspect that there are examples of antibiotic resistance where there are multiple mutated bases that can give rise to the same phenotype.I would also agree that a mutation doesn't have to reach fixation in order to interact with new mutations.

Not all adaptations are the same. I think it is relatively rare for there to be only a single substitution mutation within the entire genome that will confer increased fitness in a given environment. I understand that just fine. Evolution is very Malthusian in that there will be winners and losers, and this is true of neutral mutations as well.

That needs context. If we are talking about positive or negative selection, then competition does change the rate of fixation for those mutations under selection. You tell us.

The number of possible beneficial mutations would be a good start. If there are more possible beneficial mutations then you need fewer divisions in order to see an increase in fitness. Why don't you tell us? Say what?Why do we see sequence conservation within exons when we compare genomes between species, but a lack of sequence conservation in introns? You first.

It would take half as many divisions. It would be nice to see some reciprocation. All of biology boils down to thermodynamics, as do all physical processes. In a simplistic model, energy flows from the Sun to photsynthesizers to herbivores to carnivores. There is about 10% energy transfer at each trophic level. The total energy is limited in our solar system, and energy can't increase in our isolated solar system. Imperfect replicators competing for limited resources is what drives evolution. When you compare functional genes between species you will see fewer differences between exons than you will introns. I understand selective pressures just fine. "Slower" is a relative term. What are you comparing to? Are you comparing it to a population that increases exponential towards infinity? The number of replications is the same per culture because they are observed to reach the same density, and are started from the same number of bacteria. You are assuming that the mutations have to come in a specific order. If there are multiple beneficial mutations then you can have a mix of those beneficial mutations in the population simultaneously.