Summations Only

So far, you have said --- at what is perhaps unnecessary length --- that the probability of a particular mutation at a given locus is the probability of there being any mutation at that locus multiplied by the probability that if there is a mutation at that locus it will be that particular mutation.I don't think there's a single person here who needed to have that explained to them.So my question would be ... when do we get to the dinosaurs?Edited by Dr Adequate, : No reason given.

In an attempt to communicate your's, mine, and other's confusion over the argument being made by Kleinman in this thread, I will attempt to configure the argument in terms of the lottery. It is at this point that we all just scratch our heads and wonder what in the world he is arguing against.Edited by Taq, : No reason given.Edited by Taq, : No reason given.

Hi Kleinman,As Dr Adequate said, what you say is self-evidently true, plus you already provided that equation back in Message 186. Since we're past 200 messages now I think it isn't unfair to call upon you to move ahead more expeditiously. Continuing your focus on the the bacterial example is fine. Dr Adequate is eager to move ahead to your dinosaur-to-bird claim, but one thing at a time is probably a good idea.To help move things along allow me to anticipate a couple questions people might have:

• What does Weinreich et. al. have to do with your simple equation?
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• What leads you to draw different conclusions from Weinreich et. al. than do the authors?

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Hi Kleinman,
As Dr Adequate said, what you say is self-evidently true, plus you already provided that equation back in Message 186. Since we're past 200 messages now I think it isn't unfair to call upon you to move ahead more expeditiously. Continuing your focus on the the bacterial example is fine. Dr Adequate is eager to move ahead to your dinosaur-to-bird claim, but one thing at a time is probably a good idea.

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Hi, Percy, you didn't put in your post that I couldn't reply to you directly so forgive me for taking the liberty. What I'm deriving here are the general equations which govern rmns. I happen to use this bacterial example because we have empirical data. But the governing mathematics is applicable to any arbitrary example of rmns.

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What does Weinreich et. al. have to do with your simple equation

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Weinreich happened to do a good job measuring the empirical data. I could have used an example of the evolution of HIV to drug therapy or the evolution of Malaria (which is haploid/diploid) which I did use for deriving the mathematics or rmns for multiple simultaneous selection pressures.

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What leads you to draw different conclusions from Weinreich et. al. than do the authors?

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I'm not sure what different conclusions you mean? What I've done (and doing here) is describing the physics and mathematics of rmns. I'm just using Weinreich's paper and data as an example.So let's pick up the calculation where we have the probability of beneficial mutation A occurring in a single replication at the particular site:
P(A) = P(BeneficialA)𝜇
In order to compute the probability that mutation A will occur in a population size (let's call the population size n), we first must use the complementary rule of probabilities and compute the probability that mutation A will not occur which gives the equation:
P(Ac) = 1 — P(A) = 1 - P(BeneficialA)𝜇
where P(Ac) is the probability that mutation A will not occur at the particular site. Then to compute the probability that mutation A will not occur in n replications in a single generation, we use the multiplication rule of probabilities and obtain:
P(Ac) = (1 — P(A))^n = (1 - P(BeneficialA)𝜇)^n
Then to compute the probability that mutation A will not occur in G generations (call it nGA), we again use the multiplication rule and obtain:
P(Ac) = ((1 − P(BeneficialA)𝜇)^n)^nGA = (1 − P(BeneficialA)𝜇)^n*nGA
and then to compute the probability that mutation A will occur in a population size n in nGA generations, we again use the complementary rule of probabilities and obtain the following equation:
P(A) = 1 − (1 − P(BeneficialA)𝜇)^(n*∗nGA)So now the mathematical question is, what is the probability that mutation B will occur on some member with mutation A. The mathematics is self evident.
Again, we'll stop at this point for questions, comments, complaints...

The problem is that the bacterial examples you use show that RMNS can produce the observed evolution of adaptations in those bacteria. You have yet to show a real example in a real species where RMNS can not produce the observed differences between two species or a species and its ancestors. You claim that RMNS can not produce the differences we do see, but you have yet to show us a single example where this is true. You need to describe why the physics and mathematics of RMNS disqualifies RMNS as the mechanism behind the divergence of species with reference to specific genetic differences.

If G is the number of generations, why don't you just write P(Ac) = ((1 − P(BeneficialA)𝜇)^n)^G = (1 − P(BeneficialA)𝜇)^n*G ? Why call it nGA instead? So far, it's all been exceptionally self-evident, I'll grant you that.Question: when do we get to the dinosaurs?Edited by Dr Adequate, : No reason given.Edited by Dr Adequate, : No reason given.

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Then to compute the probability that mutation A will not occur in G generations (call it nGA), we again use the multiplication rule and obtain:
P(Ac) = ((1 − P(BeneficialA)𝜇)^n)^nGA = (1 − P(BeneficialA)𝜇)^n*nGA
If G is the number of generations, why don't you just write P(Ac) = ((1 − P(BeneficialA)𝜇)^n)^G = (1 − P(BeneficialA)𝜇)^n*G ? Why call it nGA instead?

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Good, I'm glad you are paying attention. I could have simply labeled the number of generations G but you have to keep track of generations for different steps in the evolutionary process. n and nGA are variables but the product of these two numbers is simply the total number of replication trials use to compute the probability that mutation A occurs.However, once mutation A occurs, that member becomes the progenitor for a new lineage which are candidates for mutation B. The population size n is not used for computing the probability of mutation B occurring on some member with mutation A, that population size will be nA, the number of members with mutation A and the generations for this part of the computation will be nGB, the numbers of generations members with mutation A are replicating. This is where amplification becomes critical. If members with mutations A can't either increase in number and or replicate for many generations, you will not have enough trials for there to be a reasonable probability for mutation B to occur.The probability equation for the mutation B is written in the same manner as for mutation A but with different population size and number of generations of replication. I'm sure you can easily do it. And how do compute the joint probability of the two probabilities? I'm sure you know that as well. If you don't want to do it, I'll post the equations.

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The mathematics is self evident. Again, we'll stop at this point for questions, comments, complaints...

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So far, it's all been exceptionally self-evident, I'll grant you that.

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Question: when do we get to the dinosaurs?

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In case you don't recognize it, these probability equations are the general solution for rmns and apply to all replicators. I'll show you how to do the calculation for multiple simultaneous selection pressures after we finish with the single selection pressure model.

There's no reason to believe that at all.

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However, once mutation A occurs, that member becomes the progenitor for a new lineage which are candidates for mutation B

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As I stated earlier it is often not the case that two beneficial mutations have to occur in the same lineage. Therefore any equation that assumes otherwise cannot represent the general case as you claim.

And indeed so far they apply with a small change in terminology to dice and playing cards. In order to apply them to dinosaurs, though, at some point we need to plug in some relevant numbers, such as the population size of the dinosaurs from which birds descended, the mutation rate of dinosaurs, etc.So, without wishing to hustle you I am interested to know when you're going to get on to the good bit.

Hi Kleinman,Weinrich et. al. believe their research indicates that selection constrains evolution to narrower pathways than previously supposed, and that that makes evolution more predictable and repeatable than we might have expected. What is it about their data that leads you to instead conclude that evolution is impossible?I'd like to suggest, strongly, that it isn't necessary to explain simple math and probability in painful detail. You've gone on for almost as long as the Constitution. Time to get to the crux.

You found it. Email addy sent.Enjoy

When will be getting to real replicators and real genomes?

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In case you don't recognize it, these probability equations are the general solution for rmns and apply to all replicators.

There's no reason to believe that at all.

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There is if you understand that rmns is a stochastic process and understand how to do probability calculations.

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However, once mutation A occurs, that member becomes the progenitor for a new lineage which are candidates for mutation B

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As I stated earlier it is often not the case that two beneficial mutations have to occur in the same lineage. Therefore any equation that assumes otherwise cannot represent the general case as you claim.

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The mathematics I'm presenting here is the mathematics of rmns by common descent. How do you think replicators accumulate the mutations necessary to adapt to selection pressures by rmns? Perhaps you think that lateral transfer of genetic material is the way it is done? Try doing the mathematics of random ecombination. If you can't do it, I'll show you.