It worked for me quite a while back; the Good MD is immune
to logic, and highly resistant to appropriate snippage.
--
Bob C.
"Evidence confirming an observation is
evidence that the observation is wrong."
- McNameless
[snip]
Wherein I reduce the Dear Dr.Dr.'s garbage to its crucial elements.
>
> The probability function I derived to compute the probability of two
> mutations occurring is applicable to detrimental, neutral or
> beneficial mutations.
It is appropriate only when and if the Dear Dr. Dr. could actually understand the conditions where it is correct. And the division by 4 part of his "derivation" of the binomial probability distribution (for that *is* what he derived although he apparently is unaware of that fact) is wrong under any conditions.
> What distinguishes whether the mutation is
> detrimental, neutral or beneficial is how the subpopulation with the
> particular mutation responds over generations.
Agreed. One uses the terms "beneficial" or "detrimental" to describe statistically significant changes in the fraction of the population with a particular genetic state from generation to generation relative to its alternative genetic state under specific environmental conditions. Such changes only occur when the two genetic states produce a *phenotypic difference* that matters wrt relative reproductive success. One uses the term "neutral" to describe the state when either the different genetic states produce no phenotypic difference that the environment can use to discriminate between the genetic states on the metric of relative reproductive success or when the phenotypic difference produced is irrelevant on the metric of relative reproductive success. Empirically, this is identified by the fact that the generation to generation changes in fraction of the population having a particular state varies by no more than the expected amount of variance due to chance alone. The percentage amount of expected
chance variance is a function of population size. Typically the 95% confidence level is used to distinguish chance differences generation to generation from statistically significant differences generation to generation.
> If the mutation is
> beneficial, the subpopulation will increase in number,
More important than number is the change in fractional distribution. Take antibiotic resistance. Upon selection (the addition of the antibiotic), the population size drastically decreases because of the deaths of the sensitive cells. What matters is that the fraction of the population with the genetic state of antibiotic resistance increases from 10^-8 to 1.0. Whether or not one continues with the selective condition, subsequent growth (and growth is all that is required to increase numbers at this point) will not change the fraction of the population that has the 'genetic resistant state' significantly until there is mutation to a different genetic state that has a significant reproductive advantage over the resistant state under the environmental conditions at these post-selection times.
> if the mutation
> is neutral, the subpopulation size will remain relatively constant
> over generations
Again, it is the fraction of the population that is important, not the number. Under neutral conditions (conditions where the two genetic states have no reproductive advantage relative to each other, either because they don't produce a phenotypic difference or because the phenotypic difference does not affect reproductive success), the generation to generation differences in the frequency of the two genetic states will be due to chance alone. *Because chance has no memory* (a point you forget), this means that the frequencies of the two genetic states will undergo neutral drift (a drunkard's walk) that will only end by fixation of one trait or the other. The percentage effect of such chance variation is higher in small populations. Random walks will happen because one of the assumptions of the Hardy-Weinberg rule is that the population is of infinite size and real populations are not of infinite size.
So, assuming a large enough population, the generation to generation effect of chance variation in the fraction of the population with a particular allele will be close to whatever the original fraction was. If that fraction is 10^-8, one would expect the next generation to have a fraction close to that. New mutation will not have a significant effect on this fraction because new mutations will have only a [1-(1/2N)] chance of not going to extinction by chance. If, by chance, one new neutral mutation has increased in frequency to a significant level, new mutations of the same particular type will be an insignificant fraction of the frequency of that mutation type in the population.
> and if the mutation is detrimental, the subpopulation
> size will decrease over time.
No. In this case, the frequency of the mutant genotype will essentially equal the rate of new mutation to that genotype. Any particular new mutant genotype that is deleterious will be driven to extinction, but new mutations of that genotype will occur every generation. Eventually, you will have an equilibrium level where the loss of m deleterious mutant alleles each generation equals the gain of m deleterious mutant allleles each generation. Thus the *fraction* of the population having a deleterious allele will remain constant. [This is slightly different in the case of diploid organisms, where a 'recessive' allele can be selectively neutral in the heterozygous state and only harmful in the homozygous state. But the rule that an equilibrium between gain and loss of the allele each generation still holds.]
> The mathematical significance of this
> relates to the probability of the next beneficial mutation occurring
> at the proper locus (position on the genome).
The probability of any subsequent or second mutation occurring is a function of the rate of mutation to that state and the number of individuals in a population that have the needed previous genetic state. The number of individuals in a population that have the requisite genetic state is a function of both selection for that state and the number of generations of growth under selective conditions for that state.
Thus, the precise order of selective events matters. If the genetic state of antibiotic resistance is selectively neutral or detrimental under conditions where there is no antibiotic, then the steady-state fraction of the population having that state is essentially equal to the mutation rate assuming that there has not been sufficient time for a drunkard's walk to, by chance, having increased the frequency. And in the examples we have been using, where the initial population was grown from a double-sensitive organism, there hasn't been sufficient time for a significant amount of drift. Thus the *number* of individuals with resistance to the antibiotic is equal to the mutation rate to that genetic state times the population size examined for that genetic state. At generation one of selection for the genetic state of antibiotic resistance, the *number* of individuals with the resistant genetic state has not changed. But the *fraction* of the population with the resistant state has changed dramatically und
er these conditions, from 10^-8 to 1.0. After 30 generations of population doubling, almost regardless of whether or not one continues to use the selective conditions of antibiotic present, the *fraction* of the population resistant to the antibiotic remains essentially unchanged (1.0, with only back mutation providing the sensitive genetic state), but the *number* has increased.
I certainly agree that the probability of finding a double-mutant is importantly a function of the number of individuals with the first mutation selected for. The odds of finding a double-mutant for a second mutation in the cells selected for resistance to the first antibiotic is low if you do that selection in the first few generations after selection. But it becomes increasingly possible in larger cells precisely because the size of the population with the first genetic resistant state is now large. It is large precisely because of the earlier selection changing the fraction of cells with the first resistant state and subsequent selective growth of that subfraction.
>
[snip]
>
> Evolutionists for decades have used the Poisson distribution function
> in an attempt to describe the mutation and selection phenomenon. I
> believe this is not the correct probability distribution to use
> because the random mutation is not a Poisson random variable.
The Poisson is only used as an estimate of the binomial probability distribution when certain conditions apply. They apply in the cases I used it in. So your real problem is that you disagree with the use of the binomial probability distribution. You do so even though, with the exception of the division of 4, the equation you "derived" is nothing but the binomial probability distribution and is based on its assumptions.
> In
> addition, the Poisson distribution does not properly relate population
> size, number of generations and mutation rate for computing the number
> of trials for a particular mutation. I have derived what I believe is
> the correct probability function for computing the probability of two
> mutations A and B to occur. I�ll repeat the derivation here for you.
>
> Probability of two beneficial mutations occurring (not simultaneously)
> at two loci as a function of population size and number of
> generations.
>
> The following are the definition of the variables used.
> n -- is the total population size
> nA -- is the fraction of the total population size with mutation A
> nGA � is the number of generations for beneficial mutation A to occur
> nGB � is the number of generations for beneficial mutation B to occur
> mA -- the probability that in one organism in one generation, a
> mutation A will affect a specific locus in the genome
Ordinarily this would be considered the "mutation rate." But the Dr. Dr. does not use mA as the mutation rate. He uses mA/4 as the mutation rate. That is the only difference between the Dr. Dr.'s derived probability function and the binomial derived probability function.
Interestingly enough, the Dr. Dr. doesn't ever tells us how he is able to identify that a *mutation A* has (or will) "affect[ed] a specific locus in the genome". That is, he doesn't tell us what change occurred at this site so that he can identify it as a mutation. Moreover, he doesn't tell us how he identifies the "specific locus in the genome". Later it appears that what means by "specific locus" is "specific nucleotide site". And that *after* he identifies somehow that there "will be" (his wording) a mutation in this nucleotide site, he then tosses a 4-sided dice to determine what that mutation is. Which means that even the original nt at that site can be considered a "mutational change".
This, of course, makes no sense at all. The only way it makes sense is if the Dr. Dr. is under some delusion that a nt site remains empty of any nt until there is some event (a mutation event) that, apparently, is determined by ESP and then the nt for that empty site is randomly chosen from equimolar pools of the 4 possible nts.
> mB -- the probability that in one organism in one generation, a
> mutation B will affect a specific locus in the genome
> P(A) is the probability that beneficial mutation A will occur at a
> particular locus
> P(Ac) is the probability that beneficial mutation A will NOT occur at
> a particular locus
> P(B) is the probability that beneficial mutation B will occur at a
> particular locus
> P(Bc) is the probability that beneficial mutation B will NOT occur at
> a particular locus
>
> First, compute the probability the beneficial mutation A will NOT
> occur at a particular locus
>
> divide mA by four
>
> mA/4 -- the probability that in one organism in one generation, a
> mutation A will turn a specific locus into a specific nucleotide other
> than the one it already is -- for instance, turn G, C, or T into A.
Yet, by dividing by 4, he is assuming that all 4 nts (*including* the original one at that site) are all equally likely to fill this apparently empty nt site, but *only* when you already know that a "mutation" has occurred at that site. After all mA is "the probability that in one organism in one generation, a mutation A will affect a specific locus in the genome". And apparently this event can be determined without any knowledge of the phenotypic difference produced or even what genotypic event occurred, not even that it occurs at a specific nt.
My guess is that he really does believe the creationist nonsense that genes are made by assembling nts by pure chance one nt at a time. That, after all, is the assumption behind the creationist nonsense about the probability of assembling a 747 in a tornado. That is what underlies the creationist idea of the chance of evolution of a gene being one in 4^n where n is the number of nucleotides. That is, he does not believe in reproduction, where genes are assembled by an imperfect copy mechanism. Otherwise, I cannot think of a single reason for dividing the mutation rate by 4.
If one is talking about mutation that produces a *selectively relevant* change, one can talk about change in phenotype due to change in genotype, since what nature works on during selection -- beneficial or detrimental -- is phenotypic difference. In the case of selectively neutral mutation, one must still know the initial starting point and end point, be that a phenotypic difference that has no selective effect or a change in a gene that can only be identified by knowing that a *change* in sequence has occurred.
> subtract that result from 1
>
> 1-(mA/4) -- the probability that in one organism in one generation,
> the specific mutation in question will NOT occur
> raise that result to the power of n
>
> (1-(mA/4))^n -- the probability that in the entire population in one
> generation, the specific mutation will NOT occur in ANY individual.
That is (again, except for the divided by 4 bit) essentially the binomial probability under the assumption that n = number of trials/generation and the number of generations actually tested = 1. n is not necessarily the *entire* population of organisms; it is only the number of individuals tested for the "event", be that 'event' mutation to A-resistance or both A-and B-resistance.
Again, remove the divided by 4 bit, and this is the assumption of binomial probability.
> raise that result to the power of nGA
nGA, as I recall, is = number of generations in which n (or a mean of n) individuals/generation are tested.
Or, mathematically equivalently, multiply n, the number tested/generation by nGA, the number of generations in which n individuals are tested. The result of that multiplication is more easily identifiable as "the total number of trials performed" that is in the binomial probability distribution. Using an exponent is not wrong, but it is obfusticating the terms to hide the fact that you have a binomial probability distribution (except for the division of the probability of the "event" -- aka, the mutation event -- by 4).
> ((1-(mA/4))^n)^nGA = (1-(mA/4))^(n*nGA) -- the probability that in the
> entire population in nGA generations, the specific mutation will NOT
> occur in ANY individual
Again, remove the divided by 4 and use the mathematically identical multiplication form instead of the exponential, and the equation is not different from the expectations of one or more mutants in the binomial probability distribution, which, under conditions we are using, can be estimated to more than 6 significant figures by use of the Poisson.
>
> Then by the complementation rule of probabilities P(A) = 1 � P(Ac)
> where P(A) is the probability that the specific mutation will occur at
> a particular locus in nGA generations in the population size n and
> P(Ac) is the probability that a specific mutation will NOT occur at a
> particular locus in nGA generations in the population size n. Gives:
> P(A) = 1 - (1-(mA/4))^(n*nGA)
> is the probability that a specific mutation will occur at a particular
> locus in nGA generations in a population size n.
>
> Now, compute the probability the beneficial mutation B will NOT occur
> at a particular locus
>
> divide mB by four
>
> mB/4 -- the probability that in one organism in one generation, a
> mutation B will turn a specific locus into a specific nucleotide other
> than the one it already is -- for instance, turn G, C, or T into A.
>
> subtract that result from 1
>
> 1-(mB/4) -- the probability that in one organism in one generation,
> the specific mutation B in question will NOT occur
> raise that result to the power of nA
>
> (1-(mB/4))^nA -- the probability that in the subset of the population
> with mutation A in one generation, the specific mutation B will NOT
> occur in ANY individual.
> raise that result to the power of nGB
> ((1-(mB/4))^nA)^nGB = ((1-(mB/4))^(nA*nGB) -- the probability that in
> the entire population in nGB generations, the specific mutation will
> NOT occur in ANY individuals of the nA subgroup.
> Then by the complementation rule of probabilities P(B) = 1 � P(Bc)
> where P(B) is the probability that the specific mutation will occur at
> a particular locus in nGB generations in the subpopulation size nA and
> P(Bc) is the probability that a specific mutation will NOT occur at a
> particular locus in nGB generations in the subpopulation size nA.
> Gives:
> P(B) = 1 � ((1-(mB/4))^(nA*nGB)
> is the probability that a specific mutation B will occur at a
> particular locus after mutation A has occurred as a function of
> subpopulation size nA and the number of generations nGB after mutation
> A has occurred.
>
> And finally, the probability that mutation B will fall on a member of
> the subpopulation with mutation A by the multiplication rule of
> probabilities is:
>
> P(A)*P(B) = {1 - (1-(mA/4))^(n*nGA)} * {1 � ((1-(mB/4))^(nA*nGB)}
>
> This is the correct probability function for two point mutations. A
> then mutation B occurring not simultaneously as a function of
> population and subpopulation size and the number of generations for
> each event for given mutation rates. This is a significantly different
> probability function than you would see for the Poisson distribution
> function.
Again, except for the division by 4, I recognize this as a binomial probability distribution. And if the event is tested for when P(A) approximately equals 10^-8 (that is, when the frequency of the mutant in a population is 10^-8) rather than 1 (after both selection and growth for A resistance) and P(B) approximately equals 10^-8 the result will be quite different for the same n.
>
[snip]
Quite a roundabout way of conceding defeat, Richard!
Chris
You know that and I know that and just about everybody else here and
everywhere knows that. Somehow Kleinman, MD, PhD, fails to get it.
>>> It's really quite simple. Given various simple assumptions, such as
>>> independent assortment, panmixis, a constant population, and frequencies
>>> p and q for the two alleles, the expected frequency of AB individuals is
>>> just pq. As p and q increase, pq increases. We have already specified
>>> that p and q are increasing. If AB phenotypes are favored over A, B, and
>>> "wild type" phenotypes, p and q will increase faster than they would in
>>> the absence of that advantage.
>
> John is possibly falsely thinking that you were talking about
> eucaryotic recombination involving organisms with a meiotic cycle and
> also are thinking of A and B as alternate alleles of the same gene.
> I think you are probably thinking of A and B as different *genes*
> rather than alleles of the same gene. But it is hard to tell what
> you mean since you have *repeatedly* refused to clarify what you
> mean. Probably because you don't understand the criticisms, in this
> case don't know the difference between "allele" and "gene locus" and
> "nt site".
No, John is possibly falsely thinking that he was talking about
eukaryotic recombination with A and B as alleles at two unlinked loci.
> hersheyh Aug 24, 12:59 pm
> >> On Jul 22, 9:18 pm, hersheyh <her...@yahoo.com> wrote:
> >> > The figure here
> >> > http://www.google.com/url?sa=D&q=http://en.wikipedia.org/wiki/File:Random_genetic_drift_chart.png
> >> > which can be found in this article
> >> > http://www.google.com/url?sa=D&q=http://en.wikipedia.org/wiki/Genetic_drift
> >> > shows the relationship between fixation of one of two alleles at a
> >> > single gene locus that started out at 50% in a population and the size
> >> > of the population. For populations of size 20, it is very clear that
> >> > completely random variation from generation to generation can lead to
> >> > fixation of one or the other allele rather quickly. For larger
> >> > populations, the neutral drift away from the starting point is slower,
> >> > but is still significant after a mere 50 generations. What this graph
> >> > does not show is that the probability of mutation per generation also
> >> > increases with population size. We have gone thru that math several
> >> > times, and each time you have ignored it because you don't understand
> >> > it. Why would it be any different this time?
> >> I ve read this page previously. Your gross over-extrapolation of this
> >> mathematics demonstrates your evolutionist bias.
So tell us why the math is a "gross over-extrapolation" due to "evolutionist bias".
> >So, exactly what is wrong with the math, mathematically? Is it the statement that the probability of
> > a neutral mutation that has just occurred at a nt site becoming fixed in a population = 1/(2Ne),
> > where Ne is the effective population size? Is it the statement that the probability of a mutation at
> > that nt site occurring being 2Ne*u, where u is the mutation rate for that site (again, assuming
> > selective neutrality or near neutrality)? Are you claiming that there are not 3 X 10^9 nt sites in
> >the haploid human genome? Are you claiming that the vast majority of those sites are selectively
> > crucial (a statement that is contrary to evidence since the mean mutation rate for point mutation is
> > around 10^-8), or do you agree that most mutation is selectively neutral?
> There is nothing wrong with the mathematics. What is wrong is your
> extrapolation of this mathematics to multiple neutral mutations
> simultaneously being fixed in the population.
So, are you saying, then, that there is nothing wrong with the calculation that the probability of fixation per nt site is u, the mutation rate? That your problem then comes from the multiplication of the probability of fixation per nt site by the total number of nt sites per haploid genome to get the rate of fixation per generation per haploid genome? Exactly how is that mathematically wrong or an extrapolation? The terms do come out to give the rate of fixation per generation per haploid genome, don't they? It is no different from saying that if the probability of a 6 per roll is 1/6, in 600 independent rolls I would expect to see 100 6's. Or saying, if I line up 100 coins and the odds of heads is 1/2 per coin flip, that I would expect, over all 100 coins flipped to see 50 heads. Are those irrational mathematical extrapolations?
> You have this enormous
> mathematical blind spot in your thinking. You somehow throw out the
> multiplication rule of probabilities for computing the joint
> probability of multiple independent events for every stochastic
> process you see fit. This is not mathematically based science you are
> practicing. This is evolutionist mathematical irrationality.
Exactly where, in the above, did I "throw out the multiplication rule of probabilities" or use it incorrectly? Are you claiming that, if the probability of heads per coin flip is 1/2, that if I flip a hundred coins I should multiply the probability of heads in each coin flip together to get (1/2)^100? Is that what you think the *correct* use of the multiplication rule implies? It sure seems like you are claiming that the *correct* use of the multiplication rule in the above equations would involve u^3000000000, that is, multiplying the probability of fixation per nt by itself 3 billion times. Again, that would be like claiming that the probability of getting heads in flips in 100 coins is (1/2)^100.
> What I am saying is that whether the genetic differences are selective
> or neutral makes no real difference in the mathematics of evolution.
> Let all the genetic differences between humans and chimpanzees be
> selective which gives the most rapid fixation of mutations. You are
> still no where close to being able to do the mathematical accounting
> for these differences in 500,000 generations.
Show your math here: Ooops. All we get is your WAG.
> You can be as derisive
> as possible but that will not give you any scientific or mathematical
> evidence to support your mathematically irrational belief system and
> in the meantime you have bungled the basic science and mathematics of
> the mutation and selection phenomenon and harmed millions of people in
> the process.
>
> >> You try to take this
> >> model and impose the results derived on John Harshman s 40,000,000
> >> differences between human and chimpanzee genomes.
> >Quite successfully.
> If you want to call it a mathematically irrational extrapolation that
> throws out the multiplication rule of probabilities for the joint
> probabilities of multiple independent events for a stochastic process,
> it�s a perfect fit for your mathematically irrational belief system.
>
> >> On average, to
> >> account for these differences requires the fixation of dozens of
> >> neutral mutations generation after generation for hundreds of
> >> thousands of generations.
> >Yes. But fixation is actually a fuzzy boundary when you have a population of 6 billion people because that size almost >guarantees new point mutation at every site. Basically, all that is required for fixation is that the last step from Ne-1 (or >several) individuals having an originally new mutant allele that was first acquired long ago become Ne - 0 by loss of the >few individuals having the original w.t. allele. When you look at the chimp compared to human genome and the time >available since last divergence, the amount of difference seen is that expected if most of the genome is selectively >neutral. That is, the mathematics appears to work in the real world under the assumption that most of the nt's in the >human and chimp genome are selectively neutral (any of the 4 possibilities will have the same functional effect). We >*know* that not all the sequence differences are due to drift (the slowest mechanism for producing a difference). Some >(small) fraction of difference is of selective importa
nc
>
> e.
> Hersheyh, you play fast and loose with population sizes. Do you think
> that five million years ago there was a population of 6 billion
> progenitors?
Not at all. In fact, the best estimate is that during most of its existence, human populations were closer to 10,000 individuals. But I was just commenting on the difficulty of defining "fixation" in a large population. I wasn't using the number 6 billion anywhere in any equation. Can you possibly try to read for comprehension?
> This is why your analysis is a crock of hot steaming bs.
> Why don�t you try doing the analysis of the fixation of two neutral
> mutations in a similar manner as the fixation of a single neutral
> mutation and present the algebra to us? Oh, I forgot, all you know how
> to do is blah, blah, blah and plug in numbers in the wrong probability
> distribution.
No probability distribution. But if I correctly calculated the probability of fixation for a single nt site, the probability for fixation in one of two nt sites would be twice the probability of fixation for a single nt site. I am not interested in the probability that both of the two nts have a fixation. I am interested in how many fixation events there will be if I look at N nts. That is equivalent to asking the probability of getting a six in six flips of the dice. That answer is one, and that is an answer to a different question than asking the probability of getting a six in all six flips of the dice, which is the question you are asking. The mathematical answer to the first is (1/6)(probability of a six per flip)*6(number of flips) = 1 six per six flips. The mathematical answer to the second is (1/6)^6 = 0.0000214 the probability that 6 flips will give a 6 every time.
> >> This drift model only takes into account the
> >> fixation of one of two alleles as you describe above, not the fixation
> >> of dozens of neutral alleles every generation and when in reality, you
> >> have more than two possible alleles at a single locus.
It is just correctly multiplying the probability of an event per trial times the number of trials to get the expected mean number per that many trials. Like multiplying the probability of a 6 per dice throw by the number of dice throws to get the expected mean number of 6's in that larger number of throws.
> >We are talking about point mutational changes in nt's, not alleles or alternate forms of genes. Learn
> > the meaning of genetic terminology, why don't you -- at least before you say more ignorant things?
> > In most genes (say, a coding sequence for a 300 aa protein, thus 900 nt), neutral drift is 1) less
> > likely since the protein must function and there is more constraint on nt sequences, 2) when it
> > occurs, is more likely to be a point mutation that does not change the aa sequence encoded, 3)
> > when an aa is changed by neutral fixation, it will tend to be similar in characteristics (e.g.,
> >hydrophobicity) or in an unnecessary part of the protein, 4) will, given the time of divergence
> > between chimps and humans, produce an average of somewhat less than one aa change per
> > average size protein (in a Poisson distribution, btw), 4) nt changes will be somewhat less frequent
> >than in non-coding regions.
> Your sloppy analysis by blah, blah, blah doesn�t cut it. And the
> Poisson distribution is not the correct probability distribution for
> the mutation and selection phenomenon.
A Poisson distribution of aa changes in protein (per 350 aa lengths) is what an intelligent person with mathematical knowledge would predict if most aa changes in proteins were selectively neutral. However, it is a fact that larger proteins tend to differ more than smaller proteins in which more of the sequence is functionally relevant so the observed data may not be exactly in a Poisson distribution, but I would expect the distribution of aa differences to be close to one.
http://genome.cshlp.org/content/15/12/1746.long
> A random mutation is not a Poisson random variable.
Where did I ever claim that random mutation is a Poisson distribution? I do claim that, to the extent that mutation is a random binomial event (with not mutant being the alternative state), it also meets the criteria for being able to use a Poisson to estimate the values of the distribution. That is, under these conditions, there is little difference between a value calculated by the Poisson and the binomial distributions. Of course, you don't use a binomial distribution in which one determines the mutation rate and uses that. You somehow (it remains unexplained) determine the mutation rate and then divide it by 4. Other than that, you are using a binomial distribution in your equations. But you still have to determine the mutation rate somehow. How do you determine the rate you then divide by 4. ESP?
> Are you too ignorant to look up the
> derivation of that probability distribution to understand why this is
> not the correct way to do the mathematics of this phenomenon?
I most certainly have looked up both the binomial probability distribution and the Poisson approximation and when it can be used. I have pointed the web sites out to you. I even pointed out to you that in some cases the actual probability distribution differs from both the binomial and Poisson because it violates the assumption that every trial has the same probability of generating an event (and that is the reason why we have a Luria-Delbruck probability distribution). But the probability distribution you "derived" from false assumptions about what mutation means and how one determines a mutation rate led you to divide the *real* mutation rate by 4.
> You need
> to rise above this mathematically irrational dogmatism that you�ve
> been indoctrinated with and learn how to do the mathematics of
> mutation and selection properly.
I *am* doing it properly. You have derived a probability distribution based on false assumptions and never tell us how you would determine the mutation rate you divide by 4. If you can't measure the *actual* mutation rate or mutation probability, you can't divide it by 4. But if you have the actual mutation rate, why do you need to divide it by 4?
>
[snip rest to make this readable]
And farmers. Feeding antibiotics labelled as "growth promoters" to
> >>> Really? You honestly believe that is the claim? That the same mutation
> >>> occurs in every person on Earth at the same time? You believe that
> >>> "evolutionists", which is to say, biologists, believe that?
> >> You have John Harshman’s claim that the human and chimpanzee genomes
> >> differ by 40,000,000. You have less than a million generations to
> >> accumulate for those differences.
> >Why can't you seem to answer the most basic questions? Do you or do you
> >not think that the "evolutionist" argument includes neutral mutations
> >spreading quickly through a population? (as opposed to slowly, but in
> >massively parallel fashion)
> Greg, the reason why you and other evolutionists are wrong about this
> is that you are taking a very low probability process and claim that
> millions of these processes are happening simultaneously.
We are correctly calculating the probability of neutral fixation *per nucleotide* in a given generation as u, the mutation rate. You have agreed that that is the correct probability of neutral fixation *per nucleotide* per generation. That is indeed a low probability for any given nucleotide in the human genome. If the human genome only contained a single nucleotide, then the probability of neutral fixation *per genome* would be the same value. But each nucleotide is an independent trial. Thus the probability of neutral fixation is the probability of neutral fixation *per nucleotide* times the number of nucleotides per genome. If this comes out to be greater than one, then it is appropriate to talk about the mean number of fixations per genome per generation.
This is no different than having a row of 100 coins. The probability that any one of these coins will, when flipped, show heads is 1/2. That is the probability of heads *per coin.* But if you flip 100 coins, the probability of heads *per 100 coins* is (1/2)*100 = 50. [That is the expected *mean* value. The probability of getting exactly 50 heads is smaller because you will have a normal curve of distributions of k.] That is, one expects to see a mean value of 50 heads if all 100 honest coins are flipped. One does not calculate the probability of heads per 100 coins by taking the probability of heads per coin flip and multiplying it by itself 100 times, (1/2)^100 = a very small probability, which is how *you* apparently think it should be done. That would be the probability that *every* coin comes up heads.
> This is
> incorrect mathematically because of the multiplication rule of
> probabilities which governs the joint probabilities of random
> independent events and this is wrong conceptually. This is wrong
> conceptually because of common descent. What you are claiming is that
> neutral mutations are appearing all throughout the population
And are you claiming that neutral mutations do not appear throughout the population? Given that there are 40+ mutations per genome in any given individual.
> and that
> they are spreading through to every member of the population
> simultaneously.
No. Most of them will disappear. For any *given* nt, the probabilty that there will be neutral fixation is u. The probability that a given new mutation (and there are 40+ new mutations per generation per individual) will go to fixation is 1/2Ne. It is merely that the sheer number of trials (nucleotides) is so large that even rare events per nucleotide have a high probability of occurring. Think of it like this. You have a die that has 10^8 faces, one and only one of which is labelled "neutral fixation change". The probability that you will have that face show up in a single flip is 10^-8, a very low probability. But if you flip the die 3X10^9 times (or more accurately, have a row of such die that contains 3X10^9 identical dice), the probability of getting more than one of the die showing the "neutral fixation change" face is actually pretty high.
> It just doesn’t make sense that neutral mutations in
> one family line will show up in a different family line now or ever.
Are you actually claiming that humans don't have sex but are clonal organisms? If humans do have sex, mutations will cross family lines via recombination.
> You have to have 20,000,000 neutral mutations show up in a single
> family line. That is mathematically irrational thinking.
Rather the above sentence is completely irrational gibberish unrelated to any kind of mathematical thinking.
>
> >>And you have every weird
> >> mathematically irrational hypothesis coming from the fertile but
> >> mathematically irrational minds of evolutionists to try to explain
> >> away this accounting problem. In the process of bungling the basic
> >> science and mathematics of mutation and selection with these
> >> mathematically irrational hypotheses, evolutionists have managed to
> >> harm millions of people suffering from diseases subject to the
> >> mutation and selection phenomenon.
> >We tend to ignore this little fugue of yours, but that doesn't mean it
> >isn't complete nonsense. Evolutionary theory predicts antibiotic
> >resistance, and indeed predicts that combinations of deadly agents will
> >be difficult to evolve your way out of. If medical and agricultural
> >people have made choices that ignore this, it is certainly not because
> >evolutionary theory does not describe it accurately.
> Greg, that’s a line of crap you are putting out.
No. It is the truth. You have built your whole edifice solely on the sands of a phrase used as a caption to a paragraph that clearly is not about multitoxin therapies but about the creationist idiotic claims that proteins and genes are somehow assembled from scratch by completely random processes.
As I have pointed out, the problem with that particular creationist idiocy is with the underlying assumption that evolution involves assembly of any particular gene by random assembly from equimolar pools of subunits, not in their use of the multiplication rule. Indeed, *if* evolution involved random assembly of genes or proteins from equimolar pools of subunits, the math would be quite correct. But that assumption is not just wrong; it is intentionally idiotic and an insult to one's intelligence. Just like your apparent use of these idiotic ideas in your division of the *actual* mutation rate by 4, which seems to be related to the fact that you believe that genes are assembled by chance from equimolar pools.
> If evolutionists have
> understood this, why haven’t they stepped into the debate and taught
> medical students the correct basic science and mathematics of the
> mutation and selection phenomenon.
Why would they teach med students that functional genes are assembled at random from equimolar pools of nucleotides? Why would they falsely teach med students that mutation rates are determined by ESP (since one has no way to determine a mutation other than by sequencing) and then are divided by 4 because there are 4 possible nucleotides to get a value that means nothing but the mutation probability divided by 4? Why wouldn't they teach students what the word mutation actually means? Why wouldn't they teach students that selection is environmentally contingent and correctly describe the mathematics of selective neutrality? The answer is that they do. If you actually had something to add to that mathematics, you or someone else would have already published it. Clearly what you have come up with is nothing but a binomial probability distribution (if you ignore your division of the actual event probability by 4). And that is neither novel nor correct. That you, despite being pointed out to the math inv
olved and the underlying assumptions, do not recognize that the Poisson is a quite accurate substitute for the binomial probability distribution under the conditions I used it tells me that your grasp on the underlying meaning of the terms and the relationships is weak.
> It hasn’t been taught correctly in
> the past and it still is not taught correctly now. You have hersheyh
> using the Poisson distribution to try to describe the mutation and
> selection phenomenon yet he doesn’t understand why it is the wrong
> distribution function.
Again, I use the Poisson as a mathematically easier substitute for the binomial probability distribution calculation. I agree that if your equation, which differs from the binomial probability distribution by the division of the event probability by 4, were correct, the Poisson would not come close to that number. But, as I keep pointing out, the division by 4 is stupidity based on ignorance of what event you are measuring -- the mutation probability, not the mutation probability divided by 4.
> Evolutionist doctrine has so permeated the
> thinking of biologists that you have Schneider at the National Cancer
> Institute claiming on his government sponsored web site that the
> multiplication rule does not apply to biological evolution when in
> fact that is the reason combination therapy works. That is the
> mathematical irrationality which is being taught by evolutionists.
This is a distortion of what Schneider actually talks about. The multiplication rule of probability and *how* it is properly used in biology depends on the assumptions you make. The use of the multiplication rule by ignorant creationists who claim that evolution of a gene involves the random assembly of long specific sequences by random drawing from equimolar pools is a stupid assumption that no evolutionary biologist would make, but creationists regularly do. The math using the multiplication rule under these assumptions would be better termed GIGO. It is not mathematically wrong, but it is garbage-in and garbage-out that is based on stupid irrelevant assumptions.
Schneider, specifically is NOT talking about combination therapy, where, sometimes, the assumption of the need for two independent events being present simultaneously in a single individual is correct.
>
[snip]
> You want it in a nutshell? The reason why the theory of evolution is
> mathematically irrational is the multiplication rule of probabilities.
That is not an argument. In the case of the stupid creationist idea that evolution requires the building of a 747 in a tornado, they correctly use the math of the multiplication rule to produce GIGO nonsense because they use it under false assumptions. The mere existence of the rule does not mean it is being correctly used in any particular case. I have pointed out, in the three step process involving lethal antibiotics, as opposed to the one-step process you describe, how different processes can produce the same result with radically different probabilities. You, yourself, have presented example after example of cases where multiple serial mutation in a number of genes can work in relatively short time frames. Examples that *never* would have worked if all the mutations had to be present simultaneously in a single individual before selection. I have also pointed out the way in which *sometimes* the probability distribution for mutation can differ from the binomial probability distribution (which is yo
ur probability distribution, or would be if you recognized that your division of the event probability by 4 makes no sense) and produce the skewed distribution that is called the Luria-Delbruck distribution.
I have never argued that the multiplication of probabilities rule does not hold. In fact, I often have used it. I just point out that the probability of a particular genetic state can change depending on prior historical events (either because of selection for or against that state or because of neutral drift). The probabiilty of a particular genetic state is not *always* the probability of mutation to that state.
> It doesn’t matter whether the process occurs with selection or not.
> The random mutation can not make massive genetic transformations. And
> if you understand that the multiplication rule of probabilities is the
> central governing rule for the evolutionary process, it becomes an
> easy matter to understand how to suppress the mutation and selection
> process.
By arranging conditions that require that two independent low-probability events occur in the same trial. Yes. That will certainly work. But, as I have pointed out, the same two events can occur in the same trial if one of them is first converted into a high probability event and the number of trials is also o.k. In that case, one cannot use the multiplication of the two low-probability events, because one of the events is no longer low-probability.
> Simply force the population to evolve against two selection
> pressures simultaneously and then your problems with multidrug
> resistant microbes, multiherbicide resistant weeds, multipesticide
> resistant insects and less than durable cancer treatments have a
> logical solution.
Empirically, however, even that can be overcome by, say, patients that do not take a full course of antibiotics by stopping too early or that don't use them correctly, thus reducing the toxicity of some of the agents (if, for example, one agent requires taking on a full stomach and the other requires an empty stomach). Or (and this is a serious problem with some drugs) getting fake drugs.
> Of course you will not transform reptiles into birds
> or humans and chimpanzees from a common progenitor by this process.
That would only be a problem if you think that evolution works by a modern lizard laying an egg out of which pops a chicken. Or that a chimpanzee must give birth to a modern human. Otherwise, the transformations need not occur simultaneously. And, in fact, the fossil evidence shows that it didn't occur simultaneously.
>
> >>On the other
> >> hand, if you properly apply the theorems of probability theory to the
> >> mutation and selection phenomenon, you can easily derive the
> >> probability function that gives the probability of two mutations
> >> accumulating in a population and you will find that this mathematics
> >> fits the real behavior of mutating and selecting populations. In
> >> addition, if you properly apply the theorems if probability theory to
> >> the random recombination process, you will also understand why HIV
> >> does not recombine mutations to accelerate the mutation and selection
> >> process. This mathematics is above the skill level of most
> >> evolutionists.
> >The math you present isn't even above *my* skill level. And yet you
> >think it is some sort of revelation. Mathematics must be applied
> >properly to the question at hand. That's where your disagreement with
> >standard biology is.
>
> That’s my point Greg. This mathematics is not that difficult. The
> mathematics of mutation and selection is very similar to the
> mathematics of dice rolling yet we have geneticists like hersheyh
> using the Poisson distribution to describe the phenomenon.
Depends on how many sides the die has and the number of trials. The mathematics of dice rolling is basically that of a binomial probability distribution. If the mean probability of an event is low and the number of trials is high, then the Poisson is essentially the same distribution as the binomial probability distribution. The Poisson can be used instead of the binomial probability distribution when n>20 and p <0.05 or, alternatively, when n>100 and np< 10. And the *fact* is that your equation is nothing but the binomial probability distribution except for your erroneous division of the probability of the event by 4.
> Lenski’s
> team is still using the Poisson distribution because this is what is
> being taught and it is wrong. And the mathematics of recombination is
> also quite straightforward. But you don’t see hersheyh posting the
> derivation of that probability function.
And you apparently don't know the difference between recombination in eucaryotes and in procaryotes.
> What are you evolutionists
> going to do after I post the correct derivation for the probability
> function for random recombination?
Depends. If you screw it up as bad as your derivation for the probability of *mutation and selection*, we will point out where it differs from reality. It is highly unlikely that a mind that cannot tell the difference between "mutation probability" and "mutation probability divided by 4" is going to come up with something useful.
> Are you going to claim that that’s
> what you’ve been doing all along? If that’s the case, post the
> derivation now before I post it. What is being taught in biology
> courses now is a collection of mathematically irrational evolutionist
> crap.
Like the use of the Punnet Square for randomly mating populations of sexually-reproducing eucaryotes and the frequencies of different genotypes that would produce?
Sure. But the probability of different genetic states in a population is not a constant and is not always equal to the probability of mutation.
> That’s what all
> the empirical evidence for the random mutation and natural selection
> phenomenon shows. If you choose otherwise, you are choosing to believe
> in a mathematically irrational belief system.
If you could only show us that it *always* is irrational, that would help.
[snip]
It's not that hard to determine m. You take the bacteria, subject them to a
lethal dose of Antibiotics and determine what percentage survive, then , as
this is only one possible nt mutation you multiply the result by 4.
And I am sure the Good Good Dr. Dr. will take this seriously too.
G
How do you know it's a mutation at a single nucleotide site? How do
you know a mutation to each of the different nucleotide is equally
likely? Why would you multiply by 4, given that one of the
nucleotides wouldn't be a mutation? And even assuming you can
determine what the odds of a site changing randomly, why do you care
about that number? The goal of this math is to determine what the
odds of antibiotic resistance occuring, which in your calculation is
the starting number.
True, but for the probability that I defined that doesn't matter. My
statement is only inaccurate to the extent that G isn't 1/4 of all
nucleotides genome-wide or if some transitions are slightly different
in probability to occur than other transitions (or same for
transversions).
> And the number of mutations depends not only on rates given
> particular starting and ending bases but also upon the frequencies of
> those starting bases in the genome.
I agree with that, per my preceding remark.
> Further, if we start at G and end at
> G, is that a mutation at all?
No, it's not.
> What exactly do you mean by "the mutation rate"?
*exactly" is going to depend on the context of the discussion. For
example, if the discussion is about neutral fixation we may be
interested only in mutations among junk DNA, which distinguish child
from parent, and which are only counted when and if the child reaches
maturity. In another context we may simply look at DNA duplication
during mitosis. I don't think there's much difference between the two
though.
That said, I define "mutation rate" by the following "probability
experiment":
Pick a genomic location at random. Watch a duplication event. If the
copy differs from the original, that's a mutation. Repeat many times
and observe the long-run rate of mutations.
To a reasonable approximation, the mutation product will be G whenever
the pre-mutation type was A (because transitions predominate) and
that's about 1/4. Slightly more accurately we could take also
transversions into account which would probably make the answer even
closer (slightly) to 1/4.
That's not actually a sentence.
> What I can tell you is
> that as Haldane s calculations of more than 50 years ago showed, it
> takes about 300 generations to do the substitution of a more
> beneficial allele than a less beneficial allele.
Really? So if I dump some antibiotics into a tube of bacteria the
resistant strains won't emerge for 300 generations?
> So if an evolutionary
> process requires 10 selected mutations, you can figure that the
> beneficial mutation/amplification of beneficial mutation cycle will
> take about 3000 generations. And that s if you have a subpopulation
> size sufficient to give the necessary trials for the beneficial
> mutation. That s why hersheyh uses population sizes of 10^9 when the
> mutation rate is 10^-8. If the population is small like 10^4 or 10^5,
> you can forget it. The mutation and selection process will die on the
> vine due to the lack of trials for the beneficial mutation. Now for
> your hypothetical claim that you have 100 neutral fixations per
> generation, that s a crocrich of evolutionist mathematical
> irrationality.
So what you're saying is that you can't tell, right? Or if you can
tell, why don't you do the math for us?
When are you going to start doing that? Just curious.
> > What you migh try is reading people's posts for comprehension and
> > responding to what they actually say, which might get the thread
> > wrapped up on under 1,000 (or in this case 2,000) posts.
>
> We are already well into our first hundred of the third round and I am
> only now responding to individual posts again. You are dreaming if you
> think this discussion will end in 3000 posts. You evolutionists are
> just very slow learners.
>
Oh, I think it will end as soon as people get bored with your
nonsense, which may be soon.
You don't read counter arguments.
> > > So you want to know if
> > > a neutral mutation could spread through a population by chance? Your
> > > own evolutionist computations show that there is a very small chance
> > > that this will happen equal to the frequency of that allele. And that
> > > model only applies when you only have two neutral alleles for a single
> > > gene. Now what s the probability of two neutral mutations being fixed
> > > by chance? Shouldn t that joint probability of that event be governed
> > > by the multiplication rule of probabilities?
>
> > No. The question isn't what the odds are of two *specific* mutations
> > being fixed is, the question is what the odds are of *any* two (or
> > more) mutations being fixed.
>
> So present your mathematical model which demonstrates your claim.
It's common sense, which apparently you do not have. Deal out two
bridge hands, and calculate the odds of you getting those exact
hands. How was that possible? If your math doesn't describe the real
world, the trouble is with your math.
>You
> are using the model for the fixation of a single neutral allele from a
> gene that has only two neutral alleles. What makes you think that you
> can extrapolate that model to the fixation of any two (or more)
> mutations being fixed simultaneously? Are you one of those
> evolutionists who don t think that the multiplication rule applies to
> the joint probability of two or more events occurring in a random
> process? You must be if you are making the above claim.
>
> > > >> Now I have shown you mathematically why neutral mutations do not
> > > >> spread through populations rapidly if at all.
> > > >No one claims they spread rapidly.
>
> > > They better if you want to do the accounting to explain the 40,000,000
> > > differences between human and chimpanzee genomes in 500,000
> > > generations.
>
> > Why does it have to be 500,000 generations?
>
> This should be clear to you, I m using evolutionist numbers.
>
No you're not.
The math is similar.
Unless you know what the base was before the mutation occurs, you have no way to know you *have* a
mutation.
> > > What distinguishes whether the mutation is
> > > detrimental, neutral or beneficial is how the subpopulation with the
> > > particular mutation responds over generations.
> >
> > Agreed. One uses the terms "beneficial" or "detrimental" to describe statistically significant changes
> > in the fraction of the population with a particular genetic state from generation to generation
> > relative to its alternative genetic state under specific environmental conditions. Such changes only
> > occur when the two genetic states produce a *phenotypic difference* that matters wrt relative
> > reproductive success. One uses the term "neutral" to describe the state when either the different
> > genetic states produce no phenotypic difference that the environment can use to discriminate
> > between the genetic states on the metric of relative reproductive success or when the phenotypic
> > difference produced is irrelevant on the metric of relative reproductive success. Empirically, this is
> > identified by the fact that the generation to generation changes in fraction of the population having
> > a particular state varies by no more than the expected amount of variance due to chance alone. The
> > percentage amount of expected chance variance is a function of population size. Typically the 95%
> > confidence level is used to distinguish chance differences generation to generation from
> > statistically significant differences generation to generation.
> The point you are missing hersheyh is that the same probability
> function for two mutations accumulating in a population applies
> whether the mutations are beneficial, neutral or detrimental.
And you don't understand that the *probability* of a mutation in a population does depend on whether that variant is beneficial. When a variation is beneficial relative to the alternative in a particular environment, the *probability* or *frequency* of that 'beneficial' variant will increase above the frequency of mutation from the original state to the variant state because the 'beneficial' variant will have a reproductive advantage over the original state.
> When
> mutations are neutral, you don't have the benefit of amplification to
> improve the probability that the next mutation will occur on a member
> with the previous mutation. This is why when John Harshman argues that
> hundreds of neutral mutations are being fixed in the population
> simultaneously every generation; you are requiring that hundreds of
> neutral mutations are accumulating simultaneously. The multiplication
> rule of probabilities makes John's claim mathematically irrational
> nonsense.
>
> >
> > > If the mutation is
> > > beneficial, the subpopulation will increase in number,
> > More important than number is the change in fractional distribution. Take antibiotic resistance.
> > Upon selection (the addition of the antibiotic), the population size drastically decreases because of
> > the deaths of the sensitive cells. What matters is that the fraction of the population with the genetic
> > state of antibiotic resistance increases from 10^-8 to 1.0. Whether or not one continues with the
> > selective condition, subsequent growth (and growth is all that is required to increase numbers at
> > this point) will not change the fraction of the population that has the 'genetic resistant state'
> > significantly until there is mutation to a different genetic state that has a significant reproductive
> > advantage over the resistant state under the environmental conditions at these post-selection
> > times.
> That�s not correct hersheyh. It is not the frequency of a beneficial
> allele which determines the number of trials for the beneficial
> mutation;
It is *both* the frequency and the number of trials (individuals tested) that is important. The expected mean number of individuals with a particular state (used to calculate the probability of one or more mutants) is the product of the frequency of that state times the number of individuals tested. But you seem to think that the number of trials is something other than the number of individuals tested for the mutant state. I have no idea what you call a 'trial'. If the probability of a given state per trial (individual) is 10^-8 and the number of trials is 10^9, we get a subpopulation having the state = 10 individuals. OTOH, if the probability of a given state per trial is 0.9999999999 and the population is 10^9, we get a subpopulation having the state of 9.99999999 x 10^8.
> it's the number of members in the subpopulation who are able
> to reproduce which determine the number of trials for the next
> beneficial mutation.
How is that different from what I said? Except, of course, that in some cases the subpopulation in question is at a frequency of 1.00 rather than a frequency of 10^-8. Again, what matters is *both* the frequency and number, or, rather, the product of those two numbers (the expected mean number of individuals with a particular genetic state.
> And as the Weinreich experiment demonstrates, you
> can have multiple variants, each with their own subpopulations which
> have to amplify their own particular beneficial mutations.
And how many times do I have to point out that, because the bacteria in these experiments do not undergo recombination, the only mechanism for generating double-mutants involves serial steps.
> > > The mathematical significance of this
> > > relates to the probability of the next beneficial mutation occurring
> > > at the proper locus (position on the genome).
> >
> > The probability of any subsequent or second mutation occurring is a function of the rate of
> > mutation to that state and the number of individuals in a population that have the needed previous
> > genetic state. The number of individuals in a population that have the requisite genetic state is a
> > function of both selection for that state and the number of generations of growth under selective
> > conditions for that state.
>
> You still don't understand how little the mutation rate contributes to
> the behavior of the mutation and selection phenomenon. HIV has a
> mutation rate 3 or 4 orders of magnitude larger than the rate you like
> to use yet this virus still can not evolve efficiently to selection
> pressures which target two genes simultaneously.
That is, when you impose two strongly reproduction-inhibiting selective conditions simultaneously. This means that you are applying conditions where only viruses with the double-mutants can replicate efficiently. Viruses with single resistance mutations have no significant advantage over viruses with none. In fact, viruses with single resistance mutations may even be at a slight disadvantage over viruses with none. This means that your "event" is the "presence of both mutations" and, I certainly agree, that if your event is the presence of both mutations in the viruses before you change the selective conditions, that that event probability is the joint probability of the two independent mutations. The expected mean number of individuals with both mutations would be the product of the probability of the double-mutant times the number of viruses present at the time of selection.
The reason why double-mutant probability is small is *precisely* because, in the non-selective conditions, neither mutant state is "beneficial". Because neither is "beneficial" in the non-selective conditions, the frequency of the mutant state is going to be (unless there has been a long period of directional drift in the case of selective neutrality) essentially the mutation rate. *If* one or both mutants were "beneficial" in the non-selective environment, both mutations would have increased in frequency relative to the non-mutant state and could be as high as a frequency of 1.0.
> The mutation rate
> only determines the frequency at which trials are done for a
> particular mutation.
This is particularly stupid misunderstanding of probability theory. It makes no sense mathematically.
What is your "mutation rate" and how is it determined empirically?
[I say that "mutation rate" or "mutation frequency" (the latter is actually more important, but is typically not much different if the mutation is neutral or deleterious in the non-selective conditions) is empirically measured by counting the number of "mutant" organisms and dividing that by the total number of organisms one has examined for the "mutant" state. This requires that one both know and be able to distinguish the "mutant" state from the "non-mutant state". In most cases, this is done by looking at mutant and non-mutant phenotypes. In our particular case, it is done by counting the number of cells that survive the presence of antibiotic and the total number of cells present initially. This is done by plating serial dilutions on both selective and non-selective plates.]
In probability theory, "trials" refers to the number of times one tests for the presence of the "event". So, in dice flipping, it is the number of times one flips an honest die (whether one is flipping the same die n times or flipping n dice). In coin flipping, it is the number of times one flips an honest coin (whether one is flipping the same coin n times or flipping n coins). In mutation, the number of trials is the number of individuals one tests for the presence of the "mutant state" (and, necessarily, one must also be able to determine the non-mutant state to distinguish between the two).
You seem to be under the delusion that the *number* of trials is actually a ratio, as the mutation rate in fact is. How do you go from determining the "mutation rate" (and how do you calculate it) to determining the number of trials? Your sentence "The mutation rate only determines the frequency at which trials are done for a particular mutation," makes no sense. The *event* we are interested in is the "number of mutants". The mutation rate (actually frequency) is the "number of mutants observed" divided by the "total number of individuals examined". That is how mutation rates are determined.
Your claim, apparently, is that you can determine mutation rates without any knowledge of *anything* about the "mutant" and "non-mutant" states and then use that rate to determine the number (although you say 'frequency' at which trials are done, which makes no sense at all) of trials. Can you explain how you do this in the case of antibiotic resistance?
> When selection pressures target more than a
> single gene,
Simultaneously. Never forget that. Selection which targets more than a single gene in a serial fashion has a different math.
> you need exponentially more trials for the two beneficial
> mutations and larger mutation rates only increase the number of trials
> additively and improve the probability of the events less than
> additively.
>
> >
> > Thus, the precise order of selective events matters. �
>
> And now you should understand why the canned binomial distribution is
> not the correct mathematical formulation for the mutation and
> selection process because in the derivation of that function, the
> order of events was not important and because of that a combinatorial
> term appears in the equation.
As I have shown, your equation is nothing but the "canned binomial distribution" probability distribution used to calcuate the probability of one or more mutants appearing.
>
> > If the genetic state of antibiotic resistance is selectively neutral or detrimental under conditions where there is no antibiotic, then the steady-state fraction of the population having that state is essentially equal to the mutation rate assuming that there has not been sufficient time for a drunkard's walk to, by chance, having increased the frequency. �And in the examples we have been using, where the initial population was grown from a double-sensitive organism, there hasn't been sufficient time for a significant amount of drift. �Thus the *number* of individuals with resistance to the antibiotic is equal to the mutation rate to that genetic state times the population size examined for that genetic state. �At generation one of selection for the genetic state of antibiotic resistance, the *number* of individuals with the resistant genetic state has not changed. �But the *fraction* of the population with the resistant state has changed dramatically und
> >
> > er these conditions, from 10^-8 to 1.0. �After 30 generations of population doubling, almost regardless of whether or not one continues to use the selective conditions of antibiotic present, the *fraction* of the population resistant to the antibiotic remains essentially unchanged (1.0, with only back mutation providing the sensitive genetic state), but the *number* has increased.
>
> Hersheyh, you are conflating your ideas before you even properly
> understand mutation and selection. You first need to understand that
> it is not the fraction of population which determines the number of
> trials for a particular mutation; it is the absolute size of the
> subpopulation. This is why recovery of the subpopulation size must
> occur first before there is a reasonable probability that the next
> beneficial mutation in an evolutionary sequence will occur. Even if
> the fraction of the population with the first beneficial mutation is
> 1, the population size still must be large enough to do sufficient
> trials that the next beneficial mutation will occur at the proper
> locus.
Haven't I just pointed that out? If the fraction of the population having the mutant is 1.0 after selection, it takes about 30 doublings to produce a population of 10^9 individuals, essentially all of whom will have that first mutant. Are you claiming that if I start with 10 individuals with the mutant state that, after 30 doublings, the frequency of that mutant in the population will be 10^-8?
> >
> > I certainly agree that the probability of finding a double-mutant is importantly a function of the number of individuals with the first mutation selected for. �The odds of finding a double-mutant for a second mutation in the cells selected for resistance to the first antibiotic is low if you do that selection in the first few generations after selection. �But it becomes increasingly possible in larger cells precisely because the size of the population with the first genetic resistant state is now large. �It is large precisely because of the earlier selection changing the fraction of cells with the first resistant state and subsequent selective growth of that subfraction.
>
> Why does the �size� of the cell have anything to do with this?
Sorry, that should read "cell population". AKA cell numbers.
> It is
> the size of the subpopulation with the first beneficial mutation which
> drives the probabilities of the second beneficial mutation occurring.
Which is *exactly* what I am saying. And I also use the *correct* multiplication of probabilities when I do so. For the first selection step, where I select for resistance to one antibiotic and don't care whether the genetic state of the other gene is mutant or non-mutant because my selective conditions don't involve selection for or against that gene, I use only the probability of there being a mutant capable of resisting the one antibiotic to calculate the probability of one or more cells resistant to that antibiotic. And I also correctly use a probability of 1.00 (or close to it) for the frequency of that antibiotic in the third step when I select for double-mutants. Would you use 10^-8 for the third step?
> Any of the remaining population that is not on the same fitness
> trajectory only represents competitors for the resources of the
> environment and slows the growth of the subpopulation that is trying
> to amplify its first beneficial mutation. This is why in the Lenski
> experiment it takes hundreds of generations to amplify a beneficial
> mutation, not your estimate of 30 generations of clonal doubling.
> Lenski�s diverse populations are competing for the limiting resource,
> glucose.
>
It took about 200 generations per mutational step. The reason it took longer has to do with the relative fitness of the mutant compared to non-mutants. To take 30 generations, the mutant strain would have to grow twice as fast as the w.t. For a 10% increase in fitness, it takes longer for the frequency of the population having that variant (or any variant with any of the more beneficial traits) to be large enough for there to be a high probability of a second beneficial mutation in one of the cells having any of the other beneficial mutations. I certainly agree that 30 generations is roughly the minimum time to restore a population where the w.t. has all died.
> >
> > [snip]
> >
> > > Evolutionists for decades have used the Poisson distribution function
> > > in an attempt to describe the mutation and selection phenomenon. I
> > > believe this is not the correct probability distribution to use
> > > because the random mutation is not a Poisson random variable.
> >
> > The Poisson is only used as an estimate of the binomial probability distribution when certain conditions apply. �They apply in the cases I used it in. �So your real problem is that you disagree with the use of the binomial probability distribution. �You do so even though, with the exception of the division of 4, the equation you "derived" is nothing but the binomial probability distribution and is based on its assumptions.
>
> And for the conditions of the mutation and selection phenomenon, the
> Poisson distribution is not a good approximation for the binomial
> distribution. I pointed this out to explicitly why the Poisson
> distribution is not correct here.
No you haven't. You have repeatedly asserted that the Poisson distribution is incorrect, but you have never, not even once, given any reason why it is incorrect other than that you divide the mutation rate by 4 so the numbers don't match what you calculate. The problem, of course, is your stupid division by 4. I have it on good authority, one competent in probability theory, that under the conditions used, the Poisson is quite accurate.
> In the mutation and selection
> process, the number of trials is actually quite small (only 10 per
> generation using your numbers) and the probability of the beneficial
> event is much larger than zero.
Gee. And here I was under the delusion that the number 10 was the *expected mean number* of mutants when the mutation rate was 10^-8 and the population size was 10^9. If, under those conditions, the number of *trials* is 10, what is the mutation rate? I say that the mutation rate is the number of mutants observed/total number of individuals examined. That is the number of "events" seen divided by the number of "trials" in my understanding. If 10 is indeed both the *expected mean number of mutants* under these conditions *and* the *number of trials under these conditions*, doesn't that make the mutation rate equal to one? Which you then divide by 4?
> If you use my numbers the probability
> of the beneficial event is 1/4 when the trial occurs.
So the "mutation rate" is not, in your bizarro teminology, the number of mutants seen divided by the total number of individuals examined, but instead is always 1/4? Or is it 10/4? The number of trials divided by the number of different nt's in DNA? And how do you identify a "beneficial event"? And aren't you assuming that only 1 of the 4 nts can ever be a "beneficial event"? How do you identify when a 'trial' has occurred?
> If you want to
> claim that there are only three possible outcomes from a point
> mutation then the probability of the beneficial event is 1/3 from a
> single trial.
How do you identify that a trial has occurred without any knowledge of the starting genetic state or the end genetic state? Pull a number out of your *ss? I know how I calculate, empirically, the mutation frequency. How do you do it?
> Either way, the probability is much larger than zero and
> the Poisson distribution is not the correct approximation for either
> the binomial distribution or the correct probability function which
> describes the mutation and selection phenomenon in this case.
Oh, dear Dr. Dr. Genius, brilliant omnipotent mathematician, on bended knee, I pray thee to explain exactly what you think m/4 means in your equation. Specifically, how do you calculate and/or empirically determine m?
> And you
> should recall that I never said that the binomial distribution was a
> bad approximation for the mutation and selection phenomenon, I said
> the Poisson distribution was a bad approximation for the process and I
> have provided the mathematical justification for this reasoning.
No, you haven't. You have repeatedly asserted both that your equation was not a binomial distribution calculation, yet by simply replacing m/4 by p, we can easily observe that it is identical to a binomial distribution calculation (specifically calculating the probability of finding one or more mutants in a given population assuming certain conditions - in your case, assuming that you use simultaneous selection against two mutant traits that would only be present in the population at roughly the mutation frequency from w.t.). Now you are claiming that n or n*g (whichever calculates the total number of organisms tested) is NOT the number of trials that we examine for the presence or absence of the mutant state. Instead the number of trials is the mean expected number of mutants given an arbitrary mutation rate and a population size of n. Yet you never show how you can get the value of mutation rate = m/4 = 10^8/4 from that number of trials.
> But
> there are also two significant differences between the binomial
> probability function and the correct probability function for mutation
> and selection. The first is there are four possible outcomes from a
> point mutation, not two.
So? First, there are not four possible outcomes from a point mutation if you know you have had a point mutation. The only way to know you have a point mutation is to know the initial nt at that site. And that means that you cannot identify a G (assuming that is the initial nt) to G as a point mutation because there has been no *change*. For something to be a mutation, there must be a *change* in genetic state that is identified somehow. No detectable change, no mutation. Period. Second, you are assuming that only one other nt of the three produces a *mutant* state. And also assume that the other three nts are all equally likely to occur by point mutation. But the main problem is that you have *already* stated that the mutation rate is m, since m is the probability that a mutation or change, has occurred at this nt. That makes m/4 just the mutation rate divided by 4.
> The second is the binomial distribution was
> derived without consideration of the order of events. The order of
> events is crucial in the mutation and selection process as you have
> now finally acknowledged above.
>
I have repeatedly pointed out that the order of events (history) is crucial in determining the probability of a double mutant and how quickly one can obtain same. That is exactly why I point out the difference in probability of the three-step sequential process producing a double-mutant compared to your single-step simultaneous selection process. History can change the frequency of different alleles in a population. You just rant on about joint probability without even asking questions about the difference in allele frequency that can occur when selection is working. You talk as if the frequency or probability of a variant in a population is *always* equal to the mutation frequency.
> >
> > > In
> > > addition, the Poisson distribution does not properly relate population
> > > size, number of generations and mutation rate for computing the number
> > > of trials for a particular mutation.
Of course the Poisson properly discusses the total number of trials as well as the number of events/trial. It calculates probability differently than the binomial probability distribution, but, given that the number of trials is the total number of individuals examined, it certainly takes both mean population size examined per generation and number of generations into account. As pointed out, the term lambda in the Poisson exponent is the same as p*(n*g), where p is the event rate per trial and n*g is the total number of trials.
This is no different than using the two different equations for exponential growth. y = a*b^x = a*e^bx.
> > > I have derived what I believe is
> > > the correct probability function for computing the probability of two
> > > mutations A and B to occur. I�ll repeat the derivation here for you.
> >
> > > Probability of two beneficial mutations occurring (not simultaneously)
> > > at two loci as a function of population size and number of
> > > generations.
> >
> > > The following are the definition of the variables used.
> > > n -- is the total population size
> > > nA -- is the fraction of the total population size with mutation A
> > > nGA � is the number of generations for beneficial mutation A to occur
> > > nGB � is the number of generations for beneficial mutation B to occur
> > > mA -- the probability that in one organism in one generation, a
> > > mutation A will affect a specific locus in the genome
> >
> > Ordinarily this would be considered the "mutation rate." �But the Dr. Dr. does not
> > ...
> >
> > read more �- Hide quoted text -
In fact, I have repeatedly pointed you to math sites, and directly quoted from it, the conditions under which the Poisson distribution is a good estimator of the binomial probability distribution, which is the equation you "derived". But, then, you seem to think the number of trials in your equation is the same as the number of mutants expected rather than the number of individuals examined for the mutant state. But, then, you also think you can calculate the mutation frequency without being able to identify a mutant or distinguish it from a non-mutant. You seem not to understand that the word mutant means changed.
> Evolutionists always think that anyone who claims they are
> wrong are arrogant and ignorant but it is not my teaching which has
> led to the occurrence of multidrug resistant microbes, multiherbicide
> resistant weeds, multipesticide resistant insects and less than
> durable cancer treatments.
Nor is it mine. But your equation is still just so much GIGO bullshit.
> > > The correct probability
> > > function for random recombination does not require that you consider
> > > selection. The affects of selection on random recombination only
> > > affects the probabilities implicitly by altering the number of members
> > > with each of the particular alleles.
> >
> > The probability of *recombination*, in random sexually reproducing eucaryotes (organisms that go
> > through a meiotic cycle), is determined by the *frequencies* of the alleles in the parent generation
> > and is described by the logic of the Punnet Square, which makes it about as old as modern
> > genetics. �Note that I said frequencies in the parent generation and not the *frequency of mutation*
> > from w.t. to mutant allele. �The *frequency* of alleles in any particular generation is a function of
> > its history, specifically the selective pressures that have historically acted upon different
> > phenotypes produced by the genotypes and/or the chance history of neutral drift. �Again, it is the
> > *relative frequency* of different alleles in a randomly sexually reproducing population that
> > determines the probability of recombination, NOT the absolute number of members NOR the
> > mutation probability.
>
> The Punnett square is only useful for predicting the results of a
> breeding experiment.
The Hardy-Weinberg equation says otherwise. The H-W involves the consequencess of random sexual reproduction from generation to generation.
> It is not useful for predicting the results of
> random recombination. To do that prediction, you have to apply the
> principles of probability theory correctly and you have already shown
> that you can�t do that.
Which is exactly what H-W does.
> You do have some idea of what the variables
> are including the frequencies of the alleles. Since no evolutionist
> has derived the correct probability function that would describe
> random recombination here, I�ll do the derivation for you in a week or
> two. Then you can say how amateurish the derivation is and whine about
> some feature of the calculation before you finally admit it�s correct.
>
At best, you will come up with the H-W. At worst you will probably assume that bacteria and viruses engage in sexual reproduction every generation.
> >
> > In procaryotes and viruses, recombination is a rarer event, as these organisms generally reproduce
> > clonally, and one has to define what mechanism of genetic exchange one is talking about:
> > exchange resulting from double infection of different viruses, transmission via plasmids and other
> > extra-chromosomal agents, transformation, or by viral transduction.
>
> Do you think it is the rarity of recombination for HIV that prevents
> the lateral transmission of beneficial alleles?
Yes.
> Why don�t you present
> the mathematics to substantiate your claim?
Because I would need to know values for some variables that cannot be simply arbitrarily assumed. Like the frequency of double infection of a cell in natural HIV infections (rather than in cell cultures).
> > > >> What happens to the probabilities of the random
> > > >> recombination of A and B if only one of the two all alleles amplify?
> >
> > So, are A and B different alleles of the same gene or are they different genes? �Do you know the
> > difference? �And why it makes a difference?
>
> Let A be a resistance allele to a protease inhibitor and B be a
> resistance allele to a reverse transcriptase inhibitor. Isn�t that
> which is required to accelerate the evolutionary process by
> recombination?
So, you *are* talking about variant alleles of different, unlinked genes, then. Right?
Yes. But, again, the frequency of recombination in HIV in natural infections is not nearly as easy to determine as the frequency of recombination in humans and other sexually reproducing organisms.
> > > >Not relevant.
> > > John, you shouldn�t be making this argument until you derive the
> > > probability function for random recombination. I�m not going to give
> > > the derivation of that probability function now but consider this.
> > > What if in your population every member has allele A except the member
> > > which has allele B, that is A has a frequency close to 1 in the
> > > population? That member with allele B that is B has a frequency very
> > > close to 0. What is the chance that a member with allele A will meet
> > > and recombine with the member with allele B?
> >
> > This again shows your confusion about the difference between the term 'allele' and the term 'gene'
> > or your confusion between the 'probability of mutation' and the 'probability of recombination'. �You
> > treat A and B here as if they were alternate alleles of the same gene. �And given your confusion
> > between a 'gene locus' and a 'nt site', you are probably treating the two, �A and B, as different nts at
> > the same nt site. �If that is what you mean, then recombination will not produce anything but the
> > same two alleles (unless the two alleles contain *different* mutations at *different* nt sites from
> > each other rather, in which case a rare recombination between those two independent mutations
> > will produce a double-mutant and a non-mutant).
>
> What makes you think I�m confused about what a gene and an allele are?
Because of your awkward and unclear use of the terms.
> You are the one who has been using the wrong probability distribution
> for years without ever going through the derivation of the equation
> and determining when the equation is valid to use. Why should I
> believe that you understand what a gene and an allele is?
Because I am also right about the use of the Poisson. And the binomial probability distribution. And the Luria-Delbruck distribution. And you have been wrong about each and every one of them. In fact, you have been wrong even about the meaning of terms like "trial" and "event", wrong about the meaning of p^n as opposed to p*n, wrong about what the word "mutation" means, wrong about what "beneficial" means. I think that almost covers everything you have been wrong about.
> > Ordinarily, recombination refers to recombination between *gene loci*, not recombination within a
> > gene, and recombination between alleles that differ by having different nt's at a single nt site is
> > impossible because a nt site is the limit for recombination.
>
> The mathematics I will present will give the probability function for
> recombination between gene loci.
Unlinked gene loci? For sexually reproducing organisms or for bacteria or for the various kinds of viruses? There is no "the" probability function for recombination between gene loci.
> > So tell us what you mean when you talk about *recombination*. �Are you concerned with eucaryotic
> > recombination, which occurs each and every generation? �Or are you concerned with some type of
> > gene exchange that goes on in procaryotes or viruses? �When you use the term 'allele' in your
> > discussion above, are you talking about different forms of a specific gene locus (the actual
> > definition) or are you talking about differences in nt's at a single nt site or are you talking about
> > two *different* unlinked genetic loci, each of which has alternate alleles? �Or do you not understand
> > what I am talking about?
>
> The equation I will derive for you will include multiple
> recombinatorial events, not just a single recombination event. I�ll
> give you the equation which shows how many recombinatorial trials must
> occur before you get a reasonable probability that alleles A and B
> will recombine on a single member.
>
> >
> > [snip stupidity about the Poisson, which is irrelevant]
>
> You�re the one who so stupidly uses the Poisson distribution. Do you
> want me to give you the name of the lower division mathematics text I
> used when I was a freshman in college that gives the derivation of the
> Poisson equation?
Giving me the name of a text doesn't mean you know how and when to use it.
> > > >It's really quite simple. Given various simple assumptions, such as
> > > >independent assortment, panmixis, a constant population, and frequencies
> > > >p and q for the two alleles, the expected frequency of AB individuals is
> > > >just pq. As p and q increase, pq increases. We have already specified
> > > >that p and q are increasing. If AB phenotypes are favored over A, B, and
> > > >"wild type" phenotypes, p and q will increase faster than they would in
> > > >the absence of that advantage.
> >
> > John is possibly falsely thinking that you were talking about eucaryotic recombination involving
> > organisms with a meiotic cycle and also are thinking of A and B as alternate alleles of the same
> > gene. �I think you are probably thinking of A and B as different *genes* rather than alleles of the
> > same gene. �But it is hard to tell what you mean since you have *repeatedly* refused to clarify what
> > you mean. �Probably because you don't understand the criticisms, in this case don't know the
> > difference between "allele" and "gene locus" and "nt site".
>
> If I confused John on this point, I apologize. It should not be that
> confusing. You evolutionists are claiming that recombination will
> somehow accelerate the mutation and selection process by combining a
> beneficial mutation in an allele of one gene with a beneficial
> mutation in another allele of a different gene giving our double
> mutant. I�m going to derive for you the mathematics which describes
> this process and show you why you are wrong with your claim.
The above shows how awkward your use of "allele" and "gene" is. Geneticists would claim that recombination can produce new combination of alleles in sexually reproducing individuals that would otherwise only occur by sequential mutation. They would never describe a mutation as "beneficial" unless they specified the environment in which it is beneficial.
Specifically, they would say that if the frequency of two alleles, A and A', of gene coding for function M, and the frequency of two alleles, B and B', of an unlinked gene coding for function N are present in the population in frequencies p, for A, q, for A' (p + q = 1), r for B and s for B' (r + s = 1) and the population mates at random, the frequencies of different gametes produced will be pr A;B, ps A;B', qr A';B, and qs A';B'. Multiplying these through a Punnet square will give you the population frequencies of the various genotypes expected. Multiplying those frequencies by the population size will give you the mean expected number of each genotype.
> > > John, the only thing that the Hardy-Weinberg law gives you is that the
> > > frequency of alleles remains constant when the population is in
> > > equilibrium (selection is not acting).
Actually the assumption also is that neutral drift is not acting or is not significant in the time-frame used.
It also tells you the expected frequencies of each genotype produced. Of course, if you are talking about alleles of a single gene, you cannot get recombination. You can only get homozygosity and heterozygosity.
> > And the H-W equilibrium only really exists when the population is of infinite size.
>
> The equation I will derive for you is based on a finite population and
> applies to equilibrium and non-equilibrium conditions.
>
> >
> > But if you consider A and B to be two different unlinked *genes*, each with certain *frequency* of
> > alternate alleles in the population, say p for A and q for A's alternate allele, a, and r for B and s for
> > b, then you can use the expansion to determine the frequencies of any kind of diploid offspring
> > assuming randomness. �The frequency, in the population, of gametes with the *haploid genotype*
> > A;B would be pr. �The frequency of A;b would be ps. �Of a;B would be qr. �[As an exercise to show
> > that the frequencies of the gametes add up to 1, which is the whole population of possibilities:
> > pr+ps+qr+qs = p(r+s)+q(r+s) = (p+q)(r+s) = (1)(1) = 1] �And of a;b would be qs. �
>
> This is not how to do the computation in general because a population
> can have A, B and C alleles where the C alleles are not A and not B
> alleles.
A gene (I assume you are now talking about different alleles of a single gene, although it is hard to tell because of your limited knowledge of genetic terminology) can have three alternate alleles, as does the ABO blood type gene. In that case, you have a gene with three frequencies that add up to 1. If the frequency of the A allele is p, the frequency of the B allele of that gene is q, and the frequency of the C allele of that same gene is r, then the Punnet Square would show frequencies of p^2 AA, Q^2 BB, R^2 CC, 2pq AB, 2pr AC, and 2qr BC.
OTOH, if you are talking about alternate alleles in different unlinked genes (A and A'; B and B'; C and C'), the argument would be an expansion of that given for two unlinked genes above. Choose one, whichever one you actually meant.
> What you need to accelerate the mutation and selection
> process is for one parent with the A allele and another parent with
> the B allele to recombine those two alleles into a descendent with
> both A and B alleles to give the more fit replicator, not A and C or B
> and C.
You seem to be assuming that the organism is haploid. Although there are sexually reproducing organisms that spend most of their lives in the haploid state and only undergo meiosis during a sexual stage, that is not the anthropocentric model of sex.
>
> > Using a Punnet Square and crossing the gametes to each other along with the appropriate
> > frequency of the gametes in the population will generate the frequencies of different genotypes in
> > the progeny population.
> >
> > But that is assuming that A and B are two different unlinked genes in a eucaryotic population.
>
> The equation I will derive for you will degenerate to the mathematics
> of the Punnett Square if you eliminate C alleles and have only two
> members with A alleles and two members with B alleles or if you are
> talking about diploid, a single member with some combination of A and
> B and another member with some combination of A and B. You can set up
> that circumstance in reality with a breeding experiment but now we are
> talking about random recombination such as would be seen with HIV
> where you have more than two possible alleles.
As I showed above, the cases you mention are merely an extension of the Punnet Square. I am quite certain that the Punnet Square can be put into an equation form. But I am most certainly NOT talking about recombination as would be seen in HIV, since I have no idea how frequent or infrequent recombination is in natural human infections. Perhaps you do?
> > > If you want to estimate the
> > > probability of two alleles randomly recombining, you need to write the
> > > probability function for that stochastic process. Once you do that,
> > > you can consider how selection will change the probabilities over
> > > generations as the frequencies and population sizes of the alleles
> > > change.
> >
> > Selection increases the relative frequency of the allele of a gene which has the selectively
> > advantageous phenotype. �It will do so each generation relative to the alternative allele. � That is the
> > very definition of selection. �So the relevant question is "What is the *frequency* of the alleles in
> > reproducing individuals is there at the generation I am looking at?" �Not, for recombination in
> > sexually reproducing eucaryotes, "How many, numbers, of allele a are present in reproducing
> > members of the population?"
>
> Of course selection changes the frequency of alleles or more correctly
> causes the amplification of the beneficial allele,
"Change in frequency" is more correct than "amplification". Amplification assumes that the population size increases beyond some point. That is not necessarily true. The rate of replication of the mutant can be lower in the selective conditions than the rate of replication of the non-mutant in non-selective conditions, such that maximum population size is lower. Again, there is a "change in frequency" and change in population size are not the same thing.
> so how does this
> affect the random recombination process? You have to derive the
> probability function which describes this stochastic process to
> understand this.
I await with bated breath your "stochastic process", oh genius one.
> > Of course, you could be talking about recombination in procaryotes or viruses, which are mostly
> > growing in a clonal fashion. �Until you choose to actually respond intelligently, it is hard to parse
> > out what you are talking about here.
> You are going to have a hard time understanding me until you
> understand the mathematics and you are ever so slowly understanding
> the mathematics.
>
I understand the GIGO that arises when someone who thinks he is a mathematical genius tries his/her hand at deriving equations. You get things like his "deriving" the idea that the number of "trials" is the same as the "mean expected number of events". That the "mutation rate" is not actually the "mutation rate", but 4 times the "mutation rate".
> >> This simplistic argument is driving me nuts.
> >> What have you in mind for N?
Yet the only thing you apparently would change is that you somehow think that if the probability of mutation and fixation per nucleotide is u, then you think the probability of 30 fixations must be calculated as u^30 or something. That would be the probability of 30 arbitrarily picked, but very specific, nts out of 3 X 10^9 all being mutated and fixed this generation. That, however, is not the question asked. The question asked is "How many of the 3X10^9 nts in the genome will have undergone mutation and final fixation this generation?" In that question, we are not looking at 30 specific nts and asking for the probability that those 30 had become mutated and fixed this generation. We are asking a different question. It is the difference between flipping 600 dice, knowing that the probability of a 6-face is 1/6 and asking whether flips 1 through 100 [or any 100 dice chosen at random before flipping them] all had a 6-face. The probability of that would be (1/6)^100. Or, instead, asking the question, h
ow many dice, out of 600 flipped and regardless of which ones, should show a 6-face (probability of that would be (1/6)*600 = 100.
> >Doesn't matter. Notice that the Ns cancel. The result is the same
> >whether the population is in the hundreds, as may have been the case
> >shortly after the human-chimp split, or in the billions, as today. The
> >equation does *not* handle quickly growing or shrinking populations, but
> >those are probably relatively few generations out of millions.
>
> I see, so our social engineer thinks that mutation and selection works
> efficiently with populations of hundreds.
There is no selection in selective neutrality. By definition. [Also, there is no crying in baseball.] Mutation occurs less frequently when the population is small, but drift is significantly faster when populations are small (chance % swings are larger). The reverse holds when populations are large. That is why the Ns cancels out.
> Mark, I like hearing a good
> evolutionist fairytale, tell us how humans got a different number of
> chromosomes from chimpanzees and tell us how both a male and female
> both got the same chromosome number when the change occurred and the
> lucky bride and groom met each other?
Organisms with different chromosome arrangements, including Robertsonian events that change chromosome number, can mate with each other to produce heterozygous, but balanced, progeny. In fact this is the case in about 0.2% of live births. Even individuals with ring chromosomes (but balanced) are not uncommon. Unbalanced rearrangements with chromosome 21 is a cause of hereditary Down's syndrome. The fertility of these heterozygous progeny (including balanced translocations and inversions) is not necessarily much worse than the w.t. (although multiple rearrangements are) and the new arrangement can spread in a local population. If the local population has a benefit over the larger population, in fact, this can drive the accumulation of multiple rearrangements to prevent gene exchange between the two populations. Aka, speciation.
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1684876/pdf/ajhg00361-0046.pdf
>
> >> If you are thinking billions, then you
> >> are thinking modern humans and Dr. Dr.'s confused picture of mutations
> >> propagating among living people like a plague is what you have.
> >Neutral mutations, by definition, cannot be plague-like, unless you are
> >counting invisible, inconsequential plagues.
>
> So Mark, tell us how these neutral mutations in one family line find
> there way into unrelated family lines?
By the same mechanism by which DNA from your mate's family finds its way into your children and their mate's DNA finds its way into your grandchildren. It's called sexual reproduction. Unless your children and grandchildren are clones, of course. Are they?
>
> >> So what should our model be?
> >> What does the bottleneck at which human and monkey separated look
> >> like?
> >> Note that fixation isn't really the issue - fortunately as absolute
> >> fixation of neutral single nucleotide mutations rarely happens. 10^8
> >> human births per year times 10^-8 mutation rate means that just when
> >> you think the last surviving rare type is about to die, a backmutation
> >> in a newborn somewhere extends the the persistence of the type.
> >> So what is the data that the supposed 35 million SNP differences
> >> between man an monkey comes from, and what does it really mean? I
> >> don't know. Even if it's a matter of comparing sequences (rather than
> >> e.g. extrapolating based on assumptions about population histories) I
> >> think it's more statistical that just counting -- humans too have many
> >> neutral differences from one another. Lots of definitional
> >> complications.
> >I'm not sure, but I'm pretty sure the number of differences does come
> >from comparing the sequences and counting. After all, both genomes have
> >been sequenced, and the comparing and counting is pretty easy to do by
> >computer.
> >As to what they all mean, that is why there are still geneticists working.
>
> Mark, you actually hit the nail on the head here. One of the main
> reasons evolutionists are fighting so hard here is job security. How
> many people would want to hear hersheyh’s babble once they understand
> he knows nothing about how mutation and selection actually works?
Arrogance and ignorance from someone who most certainly cannot even define "mutation" much less tell us how to distinguish between a "mutation" and a "non-mutation".
>
> John Harshman Sep 14, 2:24 pm
> Newsgroups: talk.origins
> From: John Harshman <jhar...@pacbell.net>
> Date: Wed, 14 Sep 2011 14:24:24 -0700
> Local: Wed, Sep 14 2011 2:24 pm
> Subject: Re: Trying to salvage something from this Re: The Theory of
> Evolution
>
> >Alan Kleinman MD PhD wrote:
> >> On Aug 11, 7:16 am, John Harshman <jhar...@pacbell.net> wrote:
> >>> Alan Kleinman MD PhD wrote:
> >>>> On Jul 13, 1:56 pm, John Harshman <jhar...@pacbell.net> wrote:
The human *absence* of 47 and 48 is because, in humans, the two chromosomes present in chimpanzees underwent a Robertsonian fusion. That is evident from the order of genes on the chromosomes.
> You are not doing a base by base comparison of
> the two genomes. Evolutionists look for stretches in the two genomes
> that have some similarity and line these stretches up.
That used to be the case. It isn't anymore. But keep in mind that there is no such thing as "the" human or "the" chimp genome. Genomes are much more fluid in structure than that, as evidenced by the length polymorphisms that are used in DNA fingerprinting. That is especially the case when it comes to retroviral and LINE and SINE elements expanding or contracting in number.
See below for the entire human and chimp genome data.
http://www.nature.com/nature/journal/v437/n7055/full/nature04072.html
See this for chromosome 21 specifically:
http://www.sciencemag.org/content/295/5552/131.full
http://www.pnas.org/content/100/14/8331.full
> Your 98.7%
> similarity is a load of evolutionist crap. Why don’t you line up
> chromosome 21 and tell us how similar these chromosomes are?
In coding sequences, the references say
"The BESs mapped with high confidence (13) were used to calculate the difference between the chimpanzee and human genomes at the nucleotide level. The number of sites in valid alignments (nucleotide sites that have PHRED quality values q ≧ 30) was 19,813,086. Out of this number, 19,568,394 sites were identical to their human counterparts for a mean percent identity of 98.77."
http://etd.lsu.edu/docs/available/etd-08302006-100849/unrestricted/Han_dis.pdf
> >> This data is presented for those areas which can be matched and the
> >> match is not close at all.
> >It isn't? 98.7% identity isn't close? What would constitute close, then?
>
> You evolutionists are cherry picking data out of the two chromosomes.
> Your claim is an evolutionist fabrication and is a totally
> untrustworthy estimate. Have you gotten your subscription to “Science”
> yet? Read the full article
> http://www.sciencemag.org/content/295/5552/131.abstract&usg=AFQjCNF6LJDM3GHffyRjXPABtkVjwnfm9w
> and tell us how similar chromosome 21 is for humans and chimpanzees.
See the quote from the same article you cited, just above. As for differences, they say:
"We identified 18 STSs that amplified products from human DNA but not from that of chimpanzee (shown as circles inFig. 2). Because we used genomic DNA isolated from three chimpanzee individuals, two males and one female, the effects of relatively larger polymorphisms among the chimpanzee genomes should be minimized. These 18 primer sets, together with the flanking STSs, were further tested with other primates including gorilla. Out of these, amplification products appeared exclusively in humans from seven primer pairs (filled green circles in Fig. 2 whose positions in human chr21 are about 7.2, 8.5, 10.0, 11.6, 11.8, 18.1, and 29.3 Mb from the centromeric end, respectively) (16,17), suggesting that these loci might correspond to insertions that are specific to the human lineage. Nonhuman primate specific deletions cannot be ruled out but seem less likely because this deletion would have had to occur in all primates but humans. The remaining 11 primer pairs fail to amplify any products from chimpanzee DNAs bu
t showed positive signals in some of the other primates, suggesting the existence of deletions or mutation sites at those positions in the chimpanzee genome."
What that means is that they found seven sites on chromosome 21 that were human specific, probably because of insertions (likely of retroviruses or retrotransposon-like element migration after divergence), inversions, deletions or other small chromosome rearrangements (in both genic regions, within 10kb of a coding sequence and its introns, and non-genic ones -- although only rarely in coding sequences).
http://genome.cshlp.org/content/13/3/341.full.pdf+html
>
> In this URL, they studied chromosome 21. They report “We detected
> candidate positions, including two clusters on human chromosome 21
> that suggest large, nonrandom regions of difference between the two
> genomes.” Nonrandom means these are selective differences.
You need to read the whole article rather than just the abstract. They attribute the large differences to chromosome rearrangements. And that has been amply demonstrated by the above 2011 paper. They point out that
"To date, it has commonly been thought that single-basepair changes between the human and chimpanzee genomes would underlie the majority of these postulated regulatory differences. However,
the data we present in this study demonstrate that genomic rearrangements are a significant source of DNA variation between humans and chimpanzees, as well as other nonhuman primates. These rear- rangements provide excellent start- ing points for focused studies of gene expression differences in humans and chimpanzees as part of an effort to identify the genetic differences responsible for the biological, physiological, and behavior differences between these species."
It is hardly surprising that there are mutations other than point mutations that can accumulate either by chance alone (neutral drift) or by selection.
> Perhaps you
> want to claim these differences are due to a retrotransposon, from
> where?
From retroviruses and defective retroviruses in the germ line. Including many with multiple copies and that can and have switched or added new or lost old positions within a lineage.
>
> >> Evolutionists claim that humans and
> >> chimpanzees come from a common progenitor. Now you are claiming that
> >> many of these differences are neutral which is typical evolutionist
> >> speculation.
> >Simple observation of how proteins work.
>
> More like simple minded evolutionist speculation. Why don’t you
> explain to us why humans and chimpanzees produce identical insulin
> molecules but do not produce identical preproinsulin molecules?
Easy. There has been neutral drift replacing an Ala with a Ser in the chimp lineage at position 23 and a Val for an Ala at position 13. Both are conservative amino acid change and probably functionally neutral changes. Given that human and gorilla preproinsulin are identical, the changes likely occurred in the chimp lineage rather than in the human lineage. 2 conservative and probably functionally neutral amino acid differences in a part of the 110 amino acid protein that gets clipped off and is not strongly conserved is hardly outside the expected mean number of changes per 100 aa's. Here is a figure showing a number of preproinsulins from organisms as distant as the zebrafish.
http://www.springerimages.com/Images/MedicineAndPublicHealth/1-10.1007_s11154-010-9151-3-3
>
> > Tell us which are neutral differences and which are
> > selective differences.
> Well, it seldom matters whether a protein has leucine, isoleucine, or
> phenylalanine in a particular spot.
And alanine, leucine, and serine are also conservative changes.
>
> You are not answering. The authors of the study of chromosome 21 said
> there are long non-random stretches of bases. You have claimed that
> you don’t need selection to get non-random sequences. Tell us how you
> distinguish neutral differences from selective differences in a
> genome.
The most common method is their rate of change.
>
> > And then compute the joint probability of two
> > neutral mutations being fixed in a population.
> Are you still on about that? Your joint probability is irrelevant. We
> don't care about the joint probability of some particular set of
> mutations being fixed, only about the probability that any set of
> mutations will be fixed. Different, no?
>
> John, we all know that you don’t know how to analyze a stochastic
> process and both mutation and selection and mutation without selection
> are stochastic processes.
Selection is the opposite of a stochastic process. It is directional. Mutation without selection (that is, with drift) is a completely stochastic process. Mutation is a stochastic process. Selection is not.
> The joint probability of events occurring in
> a stochastic process has been and always will be governed by the
> multiplication rule of probabilities. The accumulation of mutations
> must always occur by common descent unless you have some lateral form
> of transfer of mutations. The probability of these mutations
> accumulating will always be governed by the multiplication rule of
> probabilities.
No. The probability of a mutation or two mutations "accumulating" (i..e., increasing in frequency beyond the mutation frequency) is governed by its (or their combined) relative selective advantage. Selection is not a random process. It is directional.
>
> >>>>>> How many with other known functions? How much "junk"?
> >>>>> Almost all is junk, just as almost all the genome is junk. Non-coding,
> >>>>> functional regions are just another few percent of the genome.
> >>>> This is the type of stupidity that evolutionist perpetuate. If they
> >>>> don’t know what a portion of the genome does, it is junk.
> >>> No, that's not how it works. We recognize junk by the fact that it
> >>> evolves at the rate of mutation.
> >> Take a look at this URL: http://www.google.com/url?sa=D&q=http://www.sciencemag.org/content/295/5552/131.abstract&usg=AFQjCNF6LJDM3GHffyRjXPABtkVjwnfm9w
> >> In this URL, they studied chromosome 21. They report “We detected
> >> candidate positions, including two clusters on human chromosome 21
> >> that suggest large, nonrandom regions of difference between the two
> >> genomes.” Nonrandom means these are selective differences
> >No it doesn't.
It means there have been significant, non-point changes in sequence. As they said in the actual article. You probably only read the creationist site's explanation or the abstract (which is probably all the creationist that wrote this read.:
http://www.godandscience.org/evolution/sld072.html
> This is the second time you have made this claim. So tell us how you
> distinguish selective genetic differences from non-selective. You go
> around claiming that most of the differences between human and
> chimpanzee genomes are neutral. Tell us how you have come to this
> conclusion. Or don’t you think that explaining how you come to your
> conclusions is relevant? You have claimed that 98.7% of the human and
> chimpanzee genomes are identical yet these authors studied chromosome
> 21 and found large non-random differences between the two chromosomes.
A chromosomal rearrangement, no matter how large, is a single mutational event. Even large changes can be quite compatible with life (see Down's syndrome) and, in some cases, even without important functional import.
>
> >> and we all
> >> should know by now that selective differences take hundreds of
> >> generations per base substitution. But you claim that neutral
> >> mutations fix at the rate of a couple of hundred per generation,
> >> thousands of times faster than selection can fix a beneficial
> >> mutation.
> >Once again you confuse numbers with rates.
>
> No I haven’t John.
Yes you have. Repeatedly.
> Even Charles Brenner is using the word “plague” to
> describe what you are claiming.
>
> >>>> If they
> >>>> don’t understand how to do a mathematical computation it is junk.
> >>>> John, just because you are ignorant what a non-coding region of a
> >>>> genome does, don’t impose your ignorance on us by claiming this is
> >>>> junk. If a region of DNA has no coding function for proteins but
> >>>> remains non-random, it does so because it has stabilizing selection
> >>>> acting on those sequences.
If a region of DNA has no coding function but is more conserved than, say, pseudogene sequences that can be eliminated without consequence, it likely has a function related to its sequence. Otherwise, even if it has a function, that function is not related significantly to sequence.
> >>> True. Which has nothing to do with what I'm talking about. Stabilizing
> >>> selection makes loci evolve at less than the neutral rate. Such loci are
> >>> only a few percent of the genome. By the way, evolution isn't so fast as
> >>> to randomize sequences in 5 million years.
> >> Just what are you talking about? I guess you missed the study I posted
> >> above about the large non-random differences on chromosome 21 between
> >> humans and chimpanzees.
> >So? How is that relevant? Do you have access to the whole article? I don't.
Anyone can access these older articles by signing up with Science. A bit more spam is the cost.
>
> It is relevant because you have claimed that the human and chimpanzee
> genomes are 98.7% similar and here is data from chromosome 21 which
> shows you are wrong. Yes I have access to the whole article and I told
> you how you could get access to the article without cost. Don’t use
> ignorance as a defense to your irrational claims.
>
> > 70% of genes code for different proteins,
> ....if by "different" you mean having at least one different amino
> acid.
>
> At least one amino acid different.
That is one or two aa in 350 or so. But that varies. Small proteins tend to be more highly conserved because they tend to have a larger fraction of their aa's involved in function. Other sequences, such as those that get clipped off and discarded (like the B section of preproinsulin or fibrinogen peptide) have more changes over the same time frame.
>
> >> large stretches of non-random differences between human and chimpanzee
> >> genomes yet neutral evolution will fix all these differences a rate of
> >> a couple of hundred per generation, thousands of times faster than a
> >> single beneficial mutation can be fixed in a population. What you are
> >> talking about is mathematical irrationality.
> >I've become convinced that you know almost nothing about mathematics
> >beyond the scraps rote learning you have displayed here.
>
> John, I am not the one who has thrown out the multiplication rule of
> probabilities.
Neither have I. I just know when to use it and when not to. And I don't think that the probability of a mutation is a fixed constant.
> This is the governing rule of the joint probability of
> events for a stochastic process. And every time you claim that I know
> almost nothing about mathematics, I’m going to remind you about how
> population size affects the probabilities of events. If you continue
> in this discussion, I’m going to teach you something about the
> practice of hard mathematical science and show you why your theory of
> evolution is a mathematically irrational belief system.
That would be a first for you.
>
> >>>> And the reason it has stabilizing selection
> >>>> pressures acting on those sequences is that it has some type of
> >>>> important function on maintaining the life and reproductive capability
> >>>> of that member. The only junk in this discussion is the evolutionist
> >>>> junk science which fails to properly explain how mutation and
> >>>> selection works.
> >>> You mistake evolution at the rate of mutation for stabilizing selection,
> >>> presumably because you have a false understanding of the mutation rate.
> >>> Neutral evolution produces only a bit more than 1% difference over 5
> >>> million years, not a randomization of sequences.
> >> You will only get randomization of sequences if there is no selection
> >> acting on that sequence. Your mathematics is faulty because 5 million
> >> years only represents about 500,000 generations and you can not fix
> >> 40,000,000 differences in two divergent populations in such a short
> >> period of time. It is mathematical irrationality to believe this.
Only if one is so stupid as not to understand the effect of sexual reproduction, which allows a second mechanism for generating new combinations of alleles beyond repeated serial mutation.
> >You seem to have stopped even pretending to have an argument and are
> >just repeating your mantra regardless of what you are supposedly
> >responding to.
>
> Here’s your big opportunity to show us how my mantra is wrong. Show us
> how 40,000,000 neutral or selective mutations sweep through or spread
> like a plague through the two populations.
We have shown you the math. It is your religious belief that all mutation and selection must involve serial mutation in a clonal organism that is the problem. That, and your failure to understand the difference between "What is the probability that 100 *specified* dice (whether specified before or ex post facto without reference to what was rolled) out of 600 all come up 6-face?" and "How many 6-faces do I expect if I roll 600 dice?" The first answer, mathematically, is (1/6)^100. The second answer gives the expected mean number as (1/6)*600. You think the answer to all such questions is (1/6)^100.
>
> >>>>>> Of the ones that are in coding areas, how many are thought to make
> >>>>>> significant "interesting" morphological differences rather than minor,
> >>>>>> possibly non-function-altering changes to a protein?
> >>>>> Again, very few. The vast majority of differences in coding regions are
> >>>>> silent, i.e. making no difference in the protein being coded for.
> >>>> Really John? Is that why over 70% of the genes in humans and
> >>>> chimpanzees code for different proteins? I can’t tell what you are
> >>>> worse at, mathematics or the interpretation of data.
> >>> This is silly. "Over 70% of the genes code for different proteins" is a
> >>> reasonable expectation for neutral evolution. Few of these differences
> >>> mean anythng.
> >> We all know about evolutionist expectations, they are mathematically
> >> irrational. But if you want to show your work and compute the joint
> >> probability of two neutral mutations being fixed in a population, that
> >> would be some interesting evolutionist folklore to hear.
> >Mantra. At least your mantra does evolve over time, though it seems to
> >be randomly so.
>
> My mantras are selective John. They are based on hard mathematical and
> empirical evidence. Even hersheyh now agrees that I’ve derived the
> correct probability function for the mutation and selection phenomenon
> except he is whining about the 4 in the denominator of the mutation
> rate. Hersheyh still hasn’t figured out that there is more than a
> single possible outcome from a mutation.
There are two, and only two, possible outcomes: mutant and not mutant. That is what makes what you 'derived' as being based on a *binary* probability. You might want to look at what "binary" means. [There are other assumptions that go into binary probability theory, such as the assumption that each trial has an equal probability of being an event, that are violated.]
> So I guess you are not going
> to derive for us the probability of two neutral mutations being fixed
> in a population.
If the probability of a point mutation per site per generation somewhere in the entire population is u*2N and the probability of that specific mutation going to fixation is 1/2N, then the probability of mutation and fixation per site per generation is u. If there are 3 X10^9 nt sites per genome, then the probability of mutation and fixation per *genome* (all 3X10^9 sites) is around 30 if u = 10^-8. I cannot specify which 30 out of the 3X10^9 sites will have had the mutation and will have had it reached fixation. But that is the expected number of fixations per generation. Since 30, the expected mean number of fixations, is significantly larger than 2, the probability of at least 2 fixations per generation is essentially 1. Now if I wanted to know the probability that the nt at position 375 and the nt at position 2.5X10^8 in the genome had both undergone mutation and fixation, that probability would be u^2. But I am not interested in that. I can figure out which specific nt's had, in fact, changed rath
er easily ex post facto. But I cannot predict which ones will change beforehand.
>
> >>>>>> I assume this is ongoing research; perhaps the answers are not yet
> >>>>>> clear.
> >>>>> Oh, no. They're quite clear. What isn't clear is the exact number and
> >>>>> identities of the comparatively few functional differences.
> >>>> John, your irrational speculations don’t form a scientific basis for
> >>>> any of your claims. You don’t know how mutation and selection works
> >>>> and you can’t explain why over 70% of the genes code for different
> >>>> proteins in humans and chimpanzees.
> >>> By "different" you merely mean -- though you probably don't know it --
> >>> that there is at least one amino acid difference, i.e. one point
> >>> mutation. Trivial.
> >> Tens of thousands of different proteins between humans and chimpanzees
> >> fixed in 500,000 generations, that’s what a mathematically irrational
> >> evolutionist would call “trivial”. Maybe these proteins diverged
> >> during the pre-split period, you know, the banana split period.
> >>> [mantra snipped]
> >> Repeat after me, reptiles transform into birds, reptiles transform
> >> into birds, reptiles transform into birds…
> >That is indeed what the data show. Care to discuss it?
>
> So John, are you now going to claim that 98.7% of the data shows that
> reptiles transform into birds?
>
[snip]
> >> >> Only if the die has a few billion faces, almost all of which have the original allele on them. (the
> >> >> rest wouldn't be equally distributed either, would they?)
> >> > Greg, the die has only four faces for a point mutation, A, C, G and T.
> >> > If you think the die has billions of faces why don t you tell us what
> >> > a few of them are?
> >The die, to properly produce the ratio of the 'event' to 'trials', which is what your dice analogy does,
> > has to have as many faces as the minimum number of 'trials' needed to produce one 'event'. In
> > dice, that would be six faces with only one of those faces being the 'event'; that is what will give
> > you the 1 to 6 ratio. [The assumption is that the die are fair.] The 'event' in our case is a
> > *mutation*, which is any change from an identifiable non-mutant genetic state to an identifiable
> > different mutant state. That does mean, given a ratio of mutant (the 'event') to trials of 10^-8 that
> > the "die" have 10^8 faces, all but one of which is labelled not-mutant and the other face being
> > labelled "mutant". When you roll such a die, the probability that it will come up "mutant" is 10^-8,
> > isn't it?
>
> Hersheyh, how did you get so confused on this topic? Here is a quote
> from �Advanced Engineering Mathematics� by Kreyszig. �The statement �E
> has the probability P(E) then means that if we perform the experiment
> very often, it is practically certain that the relative frequency f(E)
> is approximately equal to P(E)� If we are flipping a coin and do it a
> large number of times, half the time we will get heads and half the
> time we will get tails. If we are rolling a die many times, 1/6 the
> time we will get a 1, 1/6 the time we will get a 2 and so on.
That is *exactly* what I said. You seem too stupid to even understand that the 1/2 of coin flips
(frequency of heads per flips) and the 1/6 of die rolls (frequency of 6-faces per roll) are the exact
counterpart to the 10^-8 of mutation frequency (frequency of observed mutants per tested individual).
All of them are P(E) or f(E), the ratio of the number of events divided by the number of trials. That gives
us a ratio that is the probability of the event per trial.
> In the mutation and selection phenomenon, the trial is the mutation.
> Consider if the mutation rate was 0 and the DNA replicated perfectly
> each time.
A "rate", "probability", or "frequency" is a "ratio", not a "number". To have a "mutation rate", "mutation
probability", or "mutation frequency" of 0, you would still have to divide the *number* of mutants (zero)
by the number of trials. You do know the difference between a "number" and a "ratio", don't you?
> No mutations gives no trials.
No mutations means that there are no trials (individuals tested) that had a mutation. That would make
the mutation *rate* or *frequency* or *probability*, obtained by dividing zero by the number of trials,
zero as well. But it would also then be an actual ratio rather than a number. Apparently you really don't
know the difference between a ratio and a number.
If the number of "mutations" is the number of trials, how do you calculate the "mutation frequency" or
"mutation rate"? Divide the number of "mutations" by the number of "mutations"? That would always
produce a "mutation rate" of 1.0.
> The mutation rate is simply
> the frequency which the die is rolled at a particular locus and with a
> mutation rate of 10^-8, the die is not rolled at that locus very
> often.
Most intelligent people would say that the mutation rate is rate or frequency at which one observes a
mutation in a population of cells. That allows us to actually measure a mutation rate. You seem to be
pulling the number 10^-8 out of your ass. How did you identify and measure the "frequency at which a
die is rolled at a particular locus" if you have no way to identify a mutant nor whether the die has
actually been rolled? Why don't you define what you mean by mutation *rate* and come up with an
actual rate rather than the number of mutations? IOW, try again.
> That�s why you need such large populations with this tiny
> mutation rate. You need a large population like 10^9 so that at least
> a few members will roll the die at that locus. But when those members
> do roll the die, there are four possible outcomes, A, C, T or G. I
> know this sticks in the craw of evolutionists because you can�t have a
> mutation from the original base to the original base but you don�t
> know what the original base is before the point random mutation
> occurs.
And you have not told us how one identifies the cases where "the four-sided die" was rolled.
> You can only say with certainty after the mutation occurs it
> will be one of the four bases. There are not 10^8 faces on the die,
> there are four faces on the die and the die is only rolled once in
> 10^8 replications when the mutation rate is 10^-8.
How does one measure the "mutation rate"? How does one identify a "mutant"? And what does
"mutation rate" mean if it isn't an actual rate but merely the expected mean number of "mutants" if one
arbitrarily decides that the "mutation rate" and the "number of trials" are the same number?
>
> >> Nearly all would be the original base, wouldn't they? The rest (a
> >> handful) would be labeled with the other bases, in numerical proportion
> >> to their empirically-determined probability. Surely you don't imagine
> >> that a four-sided die, with a 25% chance of landing on any letter, is a
> >> good model.
> >Actually, as a creationist, he probably does think that because he thinks that genes are assembled
> > by scratch by randomly choosing each nt from equimolar pools of the four nt's. Yeah, I know that
> > is a ridiculous description of evolution, but the "747 formed by a tornado" scenario seems to be
> > stuck in the creationist brain. It is what allows them to say that a 900 nt gene has the probability
> > of forming "randomly" of 1 in 4^900. Pure GIGO.
>
> Don�t be silly hersheyh, I don�t believe in abiogenesis. The concept
> of abiogenesis is more mathematically irrational than the theory of
> evolution and the theory of evolution is really mathematically
> irrational.
Says someone who cannot distinguish between a rate and a number.
>
[snip]
That's exactly why I mocked you Alan, because of your expectation that
others work so you don't have to.
> > I wonder how an intelligent person would have handled the problem?
>
> Instead of whining, offer a different solution.
Not whining; maybe sarcasm. And the question was rhetorical. The
obvious answer had already been given to you: Just make separate
posts. If you'd like make them all subsidiary to one new master
thread, even that would be better organized than what you did although
I don't see the point. Inventing new topic headings would also be
helpful and courteous.
> And you actually made an intelligent comment when you used the term
> bottleneck but you let John Harshman bully you out of the term. If you
> are going to argue that chimpanzees and humans came from a common
> progenitor, when a population goes through a selective bottleneck, not
> only will the beneficial alleles which allowed the population to
> survive the selection pressure be amplified after the population
> recovers, so will all the neutral alleles be amplified that these
> members are carrying at all their gene loci. This idea that tens of
> millions of neutral mutations will amplify and fix in a population
> without the aid of selection is mathematically irrational evolutionist
> crap.
A lot of very smart population geneticists with great mathematical
talent have shown why the opposite is true. In a vacuum, neither you
nor any person is credible in mathematics merely because you claim to
be so without support. In your case, even worse you affirmatively
reveal yourself to lack mathematical understanding time and again. Two
specifics that are definitive and indicative, respectively:
Bill posed a very good aptitude question many months back. I think it
was to show that any loop on a terrain must include two points of
equal altitude. I was impressed that you even attempted to answer but
your attempt didn't even begin to work. You made some excuse that
Bill's problem was in a totally different domain than the mathematics
at issue in arguing population genetics and that is true. But the
point for me is that by giving a totally wrong answer and not even
realizing it, you proved that you lack the most basic mathematical
prerequisite of a sense of rigor - the ability to discern correct
mathematical thinking. Even if you were brilliant, none of your
mathematical claims would be worth listening to if you have no
conception of rigor.
Second, I once looked up the engineering work that you repeated
boasted of, giving a solution to some partial differential equations.
It seems that your contribution was a new solution to a problem for
which several other solutions already existed. I believe you used a
standard method not particularly well suited to the problem and
obtained therefore a clumsy solution useful only for getting a PhD. Is
that a fair summary? I did study partial differential equations long
ago (Stanford, by invitation before I graduated high school; grade of
A) so I have some understanding though not much interest. Certainly
they are practically important, but to an extent one can solve
differential equations by routine, without much sense about the fabric
of mathematics, e.g. rigor. They are so far divorced from population
genetics that your excuse reasonably applies in this case; competence
with differential equations does not give your mathematical claims any
credibility.
Given a smooth contour map, prove that an arbitrary circle drawn on
the map must contain at least two points that are 180 degrees apart on
the circle and are at the same altitude.
> I was impressed that you even attempted to answer but
> your attempt didn't even begin to work. You made some excuse that
> Bill's problem was in a totally different domain than the mathematics
> at issue in arguing population genetics and that is true. But the
> point for me is that by giving a totally wrong answer and not even
> realizing it, you proved that you lack the most basic mathematical
> prerequisite of a sense of rigor - the ability to discern correct
> mathematical thinking. Even if you were brilliant, none of your
> mathematical claims would be worth listening to if you have no
> conception of rigor.
>
<snip>
Yes. And if we look at at specific nt in billions of individuals we will get a *different* nt (for a site that has an average point mutation rate of 10^-8) about once in every 10^8 individuals. Is that or is that
not the value of the mutation rate or frequency of mutation, m, in your equation? The value, m, in your
equation is the frequency at which one finds mutants in a total examined population of size n, is it not?
If you actually calculate m differently, say by dividing the number of events (mutations) seen in a total examined population of size n (what I would call the number of trials) by what you now call the number of trials (10, which I assume you calculated by multiplying 10^9 as n by 10^-8 as mutation rate), you will always get a value of 1. Which means that value m/4 is always 1/4. Are you really *sure* that you want to call the mean expected number of mutants as the number of trials?
Now think (for once) about what you would need in order to actually calculate a "mutation frequency" and what that term actually means. Because mutation is a *change* in genetic state, a *change* from one state to another, you *must* be able to identify when there has been a *change* in genetic state in order to get the ratio of mutants observed/total number of genetic states examined. That *requires* you to be able to identify both the initial genetic state and the different genetic state. Does it or does it not?
This is simple. Can you tell me how you plan to measure the mutation frequency, m, without being able
to distinguish between "mutant" and "not mutant"? Again, 10^-8 was used in our experiments not
because it is always correct, but because it is a typical or average value that was calculated empirically
by dividing the number of point mutation changes per generation over a large number of nt sites.
Unlike coin flips and dice rolls where we can generate a theoretical expectation of frequency based on
honest coins, honest dice, and random flips, there is no such theoretical mutation rate. Only measured
mutation rates or frequencies. Measured as a frequency by counting number of organisms examined
which are mutant and dividing by the number of organisms examined. If you have some other way of
measuring the mutation rate, let us know. Otherwise your m is a mystery number that you simply pull out of some place where the sun don't shine.
If you are using point mutation at a specific site as the "phenotype" you are using to determine the mutation rate, that means you need to know the non-mutant state you are starting with and must be able to distinguish it from the mutant or changed state(s). In this example, any single nt change from the original nt at that site is classified as a "mutation" BY DEFINITION OF MUTATION. That is, the mutation rate that you actually calculate is the rate of change from a specific known nt to *any* other nt. Again, if you can measure a mutation rate
> In the mutation and selection phenomenon, the trial is the mutation.
No it isn't. We don't calculate the mutation rate, m, by dividing the number of mutants we find by the number of mutants we find. That would always be 1. We calculate it by dividing the number of mutants we find by the number of cells we had to search through to find those mutants. The number of examined organisms is the number of trials.
> Consider if the mutation rate was 0 and the DNA replicated perfectly
> each time. No mutations gives no trials. The mutation rate is simply
> the frequency which the die is rolled at a particular locus and with a
> mutation rate of 10^-8, the die is not rolled at that locus very
> often. That�s why you need such large populations with this tiny
> mutation rate. You need a large population like 10^9 so that at least
> a few members will roll the die at that locus. But when those members
> do roll the die, there are four possible outcomes, A, C, T or G. I
> know this sticks in the craw of evolutionists because you can�t have a
> mutation from the original base to the original base but you don�t
> know what the original base is before the point random mutation
> occurs. You can only say with certainty after the mutation occurs it
> will be one of the four bases. There are not 10^8 faces on the die,
> there are four faces on the die and the die is only rolled once in
> 10^8 replications when the mutation rate is 10^-8.
The above is all the evidence one needs to know that you don't even understand your own equation!
The term you call m is the *mutation rate*, aka mutation *frequency*, aka mutation *probability* and is
presented as if it were a well-known constant. We have, for the sake of argument, been using the
average rate for point mutation in man, E. coli and many other organisms (but not HIV) of about 10^-8.
But if we were to actually use the equation in a real situation, we would need to know the *real*
mutation rate or mutation frequency for that particular site, not some hypothetical or average value.
That's because there are both mutational hotspots (e.g., achondroplasia is due to a mutational hot spot)
and mutational cold spots and the variance is several orders of magnitude either way.
That means you have to know how the mean mutation frequency was determined and know how to
apply that kind of measurement to a specific nt. Do you? Can you do it if you don't know the initial nt at any site nor even whether it has been changed to a different one? If so, please explain how.
This was elegantly shown by Seymour Benzer in the 1950s using classical genetic techniques on the
phage T4. This work also demonstrated that the gene was not an indivisible unit and that there could
be recombination within as well as between genes. As well as identifying the smallest unit of
recombination, which *later* work showed to be between two nts. IOW, he "discovered" the nt as the
indivisible unit of a gene without sequencing, using phenotypic effects to identify genotypic changes.
http://www.sbs.utexas.edu/genetics/genweb/images/cistmap.gif
biuforums.com/attachment.php?aid=3657 - Israel
The *real* mutation rate, m, must be determined empirically and not assumed. We have just been using an average value that, in any actual case, would have to be calculated as *number of mutants seen in n organisms divided by n organisms* where you actually and in the real world count real mutants and real organisms. And since *any* change in a nt from the original base present counts as a mutation, that would mean that dividing by 4 is meaningless because the *actual* *observed* number of mutants
includes all 3 possible different nts that could represent a *change* in genetic state from the w.t. nt at
that site.
The question is how you measure m and what you think the term m means. For that matter, what you think the term "mutation" means. If you are unhappy with my definition, namely that mutation is a
*change* (I often add the word permanent, to distinguish changes in the actual nts from short-term
modifications like methylations that can work over several generations; there are quasi-mutations) from
one genetic state to a different genetic state. That is a more inclusive (and accurate) definition than
your apparent claim that mutation can only mean a point mutation from some unknown nt to some
unknown nt and that mutations in a gene that produce the same selective phenotype has no relevance to the selection process. But feel free to give your own definition of mutation. In my definition, the
word *change* is quite important operationally. But you apparently can put your hand to your head (or
somewhere else) and devine a mutation rate out of thin air that you can apply to a nt without knowing either the starting or mutant state(s).
>
[snip]
This is a pared down repeat that demonstrates that all that the Dr. Dr. did was re-invent the wheel (well, re-invent binomial probability distribution).
[snip]
>
> > That’s why you have to use the correct
> > probability function to describe the phenomenon and the Poisson
> > distribution is not the correct equation. And I expect you still
> > haven’t studied the derivation of the Poisson distribution yet. You
> > use the equation blindly without understanding what is being
> > calculated.
>
> I understand when it can be used as a good estimate of the binomial probability distribution. And the
> binomial probability distribution is what you have "derived", despite your division of the actual
> mutation probability by 4.
> Your equation that calculates the probability that one or more individuals with mutation A will be
> present in a population of total size, n, is:
>
> P(A) = 1 - (1-(mA/4))^(n*nGA)
This *is* your equation, is it not?
> Now I am going to simplify the symbols of that equation to demonstrate that it is nothing more than the
> binomial probability distribution solved to answer the question "What is the probability that there will be
> one or more A mutants in a total examined population of n*nGA?" where n = number of trials per
> generation and nGA = number of generations or times in which n trials are conducted or examined?
>
> First, I will change n to the mean total number of individuals examined. In most binomial probability
> equations n, in standard terminology is "the total number of trials". And the total number of organisms
> examined for the presence or absence of mutation *is* the total number of trials. Thus n is a simple
> replacement for your n*nGA
>
> That makes your equation now:
>
> P(A) = 1 - (1-(mA/4))^(n*nGA)
= 1 - (1-(mA/4))^n)
>
> Now I claim that the "real" mutation probability is mA per trial and you think the "real" mutation
> probability per trial is mA/4 per trial. [The per trial clause is needed for the equation to actually work
> out as an equation. Any *frequency* or *probability* is always a division of something by something.
> For mA to be the mutation *rate*, it must be the minimum frequency of the mutant that you get when
> the mutant state is either deleterious or is neutral without enough time for there to have been
> substantial drift. I will instead call mA the mutation *frequency*. The *frequency* of a mutant can be
> any value between the mutation rate and 1.0 depending on the mutant's selective value and past history.
>
> Our discussion of mA versus (mA/4) is merely a quantitative disagreement and not a qualitative one. So
> I will coin a term I will call pA, which symbolizes the "real" mutation probability per trial for A. I would
> plug in a number mA for pA. You would plug in a number that is 1/4 that. But pA can still stand for the
> "real" probability of the 'event' (real presence of mutation A) per trial. That makes your equation now
>
> P(A) = 1 - (1-(mA/4))^(n*nGA)
= 1 - (1-(mA/4))^n =
1 - (1-pA)^n
>
> In other words, your "derived" equation can be symbolized as
>
> P(A) = 1 - (1-pA)^n
>
> where P(A) is the probability of one or more A events in n trials.
> More generally,
> P(E) = 1 - (1-pE)^n
>
> Are you following this math so far? Do you agree with it?
>
> Now, what exactly does the term (1-pE)^n mean? (1-pE) is the probability of not-E per trial. And that
> makes (1-pE)^n the probability that *every* trial of the n trials done will come up as not-E. IOW, (1-
> pE)^n is the probability of seeing exactly zero 'events' in n 'trials'. Thus, 1 - (1-pE)^n = 1 - the
> probability of seeing exactly zero events in n trials = the probability of seeing one or more events in n
> trials, P(E).
>The above was your argument and I never had any problem with that argument.
>
> Now, if we had a way of directly determining the probability of exactly zero events in n trials, we could
> use that instead of (1-pE)^n, right? Well, *if* the above equation (your equation) is the same as the
> equation for one or more events from a binomial probability distribution, I should be able to show it.
> The mass probability function for a binomial distribution with the parameters n (total number of trials)
> and p (probability of the event per trial) is [n!/k!(n-k)!]*p^k((1-p)^(n-k)). k = the exact number of
> events being examined.
>
> http://en.wikipedia.org/wiki/Binomial_distribution
>
> Since we have shown above that your equation = 1 - the probability of seeing exactly zero events in n
> trials, that means that P(E) should = 1 - [n!/k!(n-k)!]*pE^k(1-pE)^n-k when k = 0 if we are looking at a
> binomial probability distribution. And, if I am correct that your 'derivation' is nothing but a binomial
> probability distribution, then when k = 0,
>
> 1 - [n!/k!(n-k)!]*pE^k(1-pE)^n-k = 1 - (1-pE)^n
>
> Now, since k! = 0! = 1, the [n!/k!(n-k)!] part above reduces to [n!/0!(n-0)!] = n!/n! = 1. Moreover p^0
> = 1 too.
>
> That reduces the left side of the equal sign to 1 - [1*1*(1-pE)^(n-0)] = 1 - (1-pE)^n
> That is:
>
> 1 - (1-pE)^n (your equation) = 1 - (1-pE)^n (binomial mass probability equation)
>
> The above sure looks like an equality to me. This demonstrates that what you derived in calculating the
> probability of there being one or more mutants in a population of size n is (drumroll please) nothing but
> the binomial probability distribution.
>
> Moreover, because "the binomial distribution converges towards the Poisson distribution as the number
>of trials goes to infinity while the product np remains fixed. Therefore the Poisson distribution with
> parameter λ = np can be used as an approximation to B(n, p) of the binomial distribution if n is
> sufficiently large and p is sufficiently small. According to two rules of thumb, this approximation is good > if n ≥ 20 and p ≤ 0.05, or if n ≥ 100 and np ≤ 10."
>
> That means that, assuming the above conditions are met, 1 - (1-pE)^n should also approximately equal 1 - the Poisson probability when k = 0.
>
> That is:
>
> http://en.wikipedia.org/wiki/Poisson_distribution
>
> 1 - (1-pE)^n should roughly equal [((p*n)^k)(e^(-p*n))]/k! when k = 0. I have switched the lambda in the equation (expected mean number of occurrences of E in n trials with p*n, which is the same thing).
>
> Because k! = 1 and (p*n)^k = 1 when k = 0, the Poisson in this case simplifies to e^-(p*n). Which
> means (again assuming that n ≥ 20 and p ≤ 0.05, or if n ≥ 100 and np ≤ 10, which it is in all the cases
> we have looked at) that
>
> 1-e^(np) =~ 1 - (1-pE)^n
> Remember that, by the form of this equation, n is the total number of individuals examined for mutant
> or not mutant state. 10^9 is a large number > 20. And p, in this case the mutation frequency, was
> 10^-8, which is certainly less than 0.05.
> Lately you have been making the absurd claim that n is not the number of trials. Instead, you claim that
> np (the mean number of mutants, given a population of n and mutant frequency of p) is the number of
> trials. Yet you have been accepting the idea that m, the mutation frequency (the frequency at which a
> mutation occurs), which is defined as observed # individuals that are mutants/total # of observed
> individuals (n) is 10^-8 (as it would be if you observed 10 mutants in 10^9 cells).
> Where do you divide the number of events by the number of trials if the the number of trials is 10?
> Which symbol in your equation represents the number of events per number of trials if the number of
> trials is 10?
> So, unless you can find something wrong with my math, stop pretending that your probability
> distribution is something different from the binomial probability distribution. It isn't. That you are not
> aware of that fact is your problem, not mine.
>
> At this point, let's talk about the equation you claim gives the joint probability or something. Your
> language is so muddled, it is hard to tell what you think you are saying.
> This is a direct quote from you:
>
> "And finally, the probability that mutation B will fall on a member of
the subpopulation with mutation A
> by the multiplication rule of
probabilities is:
> P(A)*P(B) = {1 - (1-(mA/4))^(n*nGA)} * {1 – ((1-(mB/4))^(nA*nGB)}
>
This is the correct probability function for two point mutations A
then mutation B occurring not
> simultaneously as a function of
population and subpopulation size and the number of generations for
> each event for given mutation rates."
>
> In actuality, the above is the multiplication of the binomial probability that there will be one or more A
> mutants in a population of total size n times the binomial probability
> there there will one or more B mutants in a population of total *size* nA, where nA is the number of
> individuals that have mutation A. Note that nowhere in this equation is there any requirement
> that the B mutations must actually occur in an *organism* that has an A mutation. Only that it occur in
> population of the same *size* as the population containing A. I strongly suspect that is not the equation
> you thought you were writing. I suspect you wanted to calculate the joint probability of *both* A and B > occurring in a single trial (that is, jointly in a single individual). That equation would be written
> (using the simplified terminology of binomial probability distributions:
>
> P(A,B) = 1 - (1-pA*pB)^n
>
> Note that this is just the general binomial probability distribution calculating the probability of one or
> more 'events' in n 'trials', P(E) = 1 - (1-pE)^n. The difference is that the 'event' is now the *joint*
> probability of both A and B being present per trial. The probability of that 'event' (the joint event) is
> pA*pB per trial, assuming that the A and B events are independent events.
>
> At this point, it is worth reminding you that pA and pB are not constants and are not always equal to the
> mutation probability to A or B from the non-mutant state. Again, the mutation probability is the
> *minimum* probability that a trial will have that mutant. *When* we directly select for cells that have
> both mutant state A and mutant state B *from* cells that were initially neither A nor B, the mutation
> probability is, in fact, a reasonable estimate of the probability that any given trial will have that mutant.
>
> However, if I first select for A mutants and grow up a population that is 100% A mutant, the mutation
> probability of A from not-A no longer holds. Instead the probability of A in that population is 1.0 per
> trial or close to it. That is, pA and pB are not constants but depend crucially on the past history of the
> organism.
>
> I would be interested in seeing how you deal with the above that shows that all you have derived is the
> binomial probability distribution. Please go through it step by step. The math isn't all that hard.
>
> [snip]
That�s the random recombination probability function you can�t derive.
It�s very simple to see your �gross over-extrapolation� and
�evolutionist bias�. You take a mathematical model of a single gene
with two neutral alleles and the random probability that one of the
two alleles being fixed and then claim that tens of millions of
neutral alleles can be randomly fixed simultaneously. John Harshman
likens it to the dealing of a bridge hand randomly from a deck of
cards. The probability of any particular bridge hand being dealt is
vanishingly small but those hands are dealt when the game is played.
But if you are going to do this analogy, what John is claiming is that
hundreds of millions or billions of card players are each dealt 30
cards randomly from a deck that contains an astronomical number of
cards and after 500,000 generations without selection, you end up with
generations that have billions of card players with tens of millions
of identical cards in their hands. Random mutation without selection
is not going to put tens of millions of identical neutral mutations
into the genomes of millions of members of a population in 500,000
generations or any number of generations you want to use in your
calculations.
>
> > >So, exactly what is wrong with the math, mathematically? �Is it the statement that the probability of
> > > a neutral mutation that has just occurred at a nt site becoming fixed in a population = 1/(2Ne),
> > > where Ne is the effective population size? �Is it the statement that the probability of a mutation at
> > > that nt site occurring being 2Ne*u, where u is the mutation rate for that site (again, assuming
> > > selective neutrality or near neutrality)? �Are you claiming that there are not 3 X 10^9 nt sites in
> > >the haploid human genome? �Are you claiming that the vast majority of those sites are selectively
> > > crucial (a statement that is contrary to evidence since the mean mutation rate for point mutation is
> > > around 10^-8), or do you agree that most mutation is selectively neutral?
> > There is nothing wrong with the mathematics. What is wrong is your
> > extrapolation of this mathematics to multiple neutral mutations
> > simultaneously being fixed in the population.
>
> So, are you saying, then, that there is nothing wrong with the calculation that the probability of fixation �per nt site is u, the mutation rate? �That your problem then comes from the multiplication of the probability of fixation per nt site by the total number of nt sites per haploid genome to get the rate of fixation per generation per haploid genome? �Exactly how is that mathematically wrong or an extrapolation? �The terms do come out to give the rate of fixation per generation per haploid genome, don't they? �It is no different from saying that if the probability of a 6 per roll is 1/6, in 600 independent rolls I would expect to see 100 6's. �Or saying, if I line up 100 coins and the odds of heads is 1/2 per coin flip, that I would expect, over all 100 coins flipped to see 50 heads. �Are those irrational mathematical extrapolations?
>
What I am saying is that your model is not a good model of reality.
You are trying to take a special case of a single gene with only two
neutral alleles and generalize this to situations where you actually
have multiple alleles at a single gene loci and furthermore claiming
that this random process will happen simultaneously at multiple
different gene loci simultaneously and with each gene loci having its
multiple neutral alleles. This is the gross over-extrapolation that
you are doing with this model. Now I�m going to help you with your
argument but I don�t believe it will be sufficient to make your theory
of evolution mathematically rational. Neutral evolution needs
selection but the selection comes from the selection of beneficial
alleles. Charles Brenner alluded to the concept. He brought up the
term �bottleneck�. When a population is subject to selection, not
only are the beneficial alleles amplified over generations but so are
any of the neutral alleles that the subpopulation with beneficial
allele happens to be carrying. That�s how you can achieve
amplification of neutral alleles.
> > You have this enormous
> > mathematical blind spot in your thinking. You somehow throw out the
> > multiplication rule of probabilities for computing the joint
> > probability of multiple independent events for every stochastic
> > process you see fit. This is not mathematically based science you are
> > practicing. This is evolutionist mathematical irrationality.
>
> Exactly where, in the above, did I "throw out the multiplication rule of probabilities" or use it incorrectly? �Are you claiming that, if the probability of heads per coin flip is 1/2, that if I flip a hundred coins I should multiply the probability of heads in each coin flip together to get (1/2)^100? �Is that what you think the *correct* use of the multiplication rule implies? �It sure seems like you are claiming that the *correct* use of the multiplication rule in the above equations would involve u^3000000000, that is, multiplying the probability of fixation per nt by itself 3 billion times. �Again, that would be like claiming that the probability of getting heads in flips in 100 coins is (1/2)^100.
>
Study the probability function that I derived for you which shows how
two mutations will show up on a single descendent. That probability
function shows how the multiplication rule comes into play and that in
order for the two mutations to have a reasonable probability of
appearing on a descendent that selection (amplification) must come
into play in order for there to be a reasonable probability that the
two events will occur. Amplification is what changes the probability
of events in the mutation and selection phenomenon and when you claim
that you can get tens of millions of neutral mutations to amplify
without the benefit of selection, your thinking becomes mathematically
irrational.
> > What I am saying is that whether the genetic differences are selective
> > or neutral makes no real difference in the mathematics of evolution.
> > Let all the genetic differences between humans and chimpanzees be
> > selective which gives the most rapid fixation of mutations. You are
> > still no where close to being able to do the mathematical accounting
> > for these differences in 500,000 generations.
>
> Show your math here: �Ooops. �All we get is your WAG.
>
I can easily show how the probability function I derived for two
mutations can be extended to any number of mutations you want. And I
have presented empirical examples which demonstrate this mathematical
behavior. You on the other hand give us an unrealistic simplification
and no empirical examples. Your mathematical skills are limited to
plugging number into the Poisson distribution (the wrong probability
distribution to describe mutation and selection) and using the Punnett
square to describe recombination. The probability function for random
recombination is not hard to derive if you know the basics of
probability theory but apparently evolutionists don�t even know the
basics of probability theory.
>
>
>
>
> > You can be as derisive
> > as possible but that will not give you any scientific or mathematical
> > evidence to support your mathematically irrational belief system and
> > in the meantime you have bungled the basic science and mathematics of
> > the mutation and selection phenomenon and harmed millions of people in
> > the process.
>
> > >> You try to take this
> > >> model and impose the results derived on John Harshman s 40,000,000
> > >> differences between human and chimpanzee genomes.
> > >Quite successfully.
> > If you want to call it a mathematically irrational extrapolation that
> > throws out the multiplication rule of probabilities for the joint
> > probabilities of multiple independent events for a stochastic process,
> > it�s a perfect fit for your mathematically irrational belief system.
>
> > >> On average, to
> > >> account for these differences requires the fixation of dozens of
> > >> neutral mutations generation after generation for hundreds of
> > >> thousands of generations.
> > >Yes. �But fixation is actually a fuzzy boundary when you have a population of 6 billion people because that size almost >guarantees new point mutation at every site. �Basically, all that is required for fixation is that the last step from Ne-1 (or >several) individuals having an originally new mutant allele that was first acquired long ago become Ne - 0 by loss of the >few individuals having the original w.t. allele. �When you look at the chimp compared to human genome and the time >available since last divergence, the amount of difference seen is that expected if most of the genome is selectively >neutral. �That is, the mathematics appears to work in the real world under the assumption that most of the nt's in the >human and chimp genome are selectively neutral (any of the 4 possibilities will have the same functional effect). �We >*know* that not all the sequence differences are due to drift (the slowest mechanism for producing a difference). �Some >(small) fraction of difference is of selective impor
ta
>
> nc
>
>
>
> > e.
> > Hersheyh, you play fast and loose with population sizes. Do you think
> > that five million years ago there was a population of 6 billion
> > progenitors?
>
> Not at all. �In fact, the best estimate is that during most of its existence, human populations were closer to 10,000 individuals. �But I was just commenting on the difficulty of defining "fixation" in a large population. �I wasn't using the number 6 billion anywhere in any equation. �Can you possibly try to read for comprehension?
>
You have just sealed the mathematical irrationality of your theory of
evolution. What happens to your mathematics of mutation and selection
when you have a population of 10^4? How do you get two beneficial
mutations to show up on a descendent? What happens to your
probabilities of a single beneficial mutation occurring? Can you
possibly do your mathematics more irrationally?
> > This is why your analysis is a crock of hot steaming bs.
> > Why don�t you try doing the analysis of the fixation of two neutral
> > mutations in a similar manner as the fixation of a single neutral
> > mutation and present the algebra to us? Oh, I forgot, all you know how
> > to do is blah, blah, blah and plug in numbers in the wrong probability
> > distribution.
>
> No probability distribution. �But if I correctly calculated the probability of fixation for a single nt site, the probability for fixation in one of two nt sites would be twice the probability of fixation for a single nt site. �I am not interested in the probability that both of the two nts have a fixation. �I am interested in how many fixation events there will be if I look at N nts. �That is equivalent to asking the probability of getting a six in six flips of the dice. �That answer is one, and that is an answer to a different question than asking the probability of getting a six in all six flips of the dice, which is the question you are asking. �The mathematical answer to the first is (1/6)(probability of a six per flip)*6(number of flips) = 1 six per six flips. �The mathematical answer to the second is (1/6)^6 = 0.0000214 the probability that 6 flips will give a 6 every time.
>
So you believe the probability of two random events occurring is
governed by the addition rule of probabilities? Let�s see what happens
when we continue with your mathematical irrationality. The probability
of fixation of two nt sites is twice the probability of a single nt
site. So the probability of fixation of three nt sites will be three
times the probability of fixation of a single nt site and the
probability of fixing N nt sites is N times the probability of
fixation of a single nt site. If you keep this up, we can have
probabilities in the millions. Hersheyh, I don�t know how you could
have done this but you started this discussion with no understanding
of probability theory and now you actually know less than nothing
about the theory.
> > >> This drift model only takes into account the
> > >> fixation of one of two alleles as you describe above, not the fixation
> > >> of dozens of neutral alleles every generation and when in reality, you
> > >> have more than two possible alleles at a single locus.
>
> It is just correctly multiplying the probability of an event per trial times the number of trials to get the expected mean number per that many trials. �Like multiplying the probability of a 6 per dice throw by the number of dice throws to get the expected mean number of 6's in that larger number of throws.
>
The joint probability of multiple independent events is not governed
by the addition rule of probabilities; it is governed by the
multiplication rule.
>
>
> > >We are talking about point mutational changes in nt's, not alleles or alternate forms of genes. �Learn
> > > the meaning of genetic terminology, why don't you -- at least before you say more ignorant things? �
> > > In most genes (say, a coding sequence for a 300 aa protein, thus 900 nt), neutral drift is 1) less
> > > likely since the protein must function and there is more constraint on nt sequences, 2) when it
> > > occurs, is more likely to be a point mutation that does not change the aa sequence encoded, 3)
> > > when an aa is changed by neutral fixation, it will tend to be similar in characteristics (e.g.,
> > >hydrophobicity) or in an unnecessary part of the protein, 4) will, given the time of divergence
> > > between chimps and humans, produce an average of somewhat less than one aa change per
> > > average
>
> ...
>
> read more �- Hide quoted text -
>
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You are on the road to making another mathematical and scientific
blunder. Do you think that combination herbicides and pesticides are
being applied to viruses and prokaryotes? These selection pressures
are being applied to eukaryotes which regularly do recombination. And
they demonstrate the same mathematical behavior as does the viral and
prokaryotic examples. Do you want me to repost these empirical
examples for you? And you have yet to provide us with the probability
function which describes random recombination. You are still working
with your century old Punnett square. Your lecture notes must be old
and frayed.
>
> > > This is
> > the mathematically and empirically irrational crap that forms the
> > basis of evolutionism.
>
> You keep claiming that evolution is mathematically irrational, yet you are the one presenting mathematical garbage and calling it a "derivation" of the "correct probability distribution".
How would you know what the correct probability distribution for the
mutation and selection phenomenon is? You use the Poisson distribution
without ever going through the derivation of the equation and I have
shown you previously why it not even a good approximation for the
phenomenon. And now you are using the addition rule of probabilities
to compute the joint probability of events. There�s enough methane gas
coming out of your brain to power a city.
>
[snip another of hersheyh�s hypothetical examples, still waiting for a
real measurable and repeatable example]
Too bad hersheyh left out the part from the scientist credited with
developing combination therapy where he (David Ho) describes how he is
using the multiplication rule of probabilities to stifle the mutation
and selection process.
I�m so disappointed, I though crocodiles and ostriches were closely
related, after all they both have gizzards and that the missing link
was crocriches.
[snip evolutionist folklore]
So is your claim now that you can transform reptiles into birds by a
breeding program? All you can do with a breeding program is change the
expression of existing genes and this only works properly with
homologous members of a population. Evolutionists gloss over the
differences in homology between different life forms. To an
evolutionist fictionalized view of reality, chromosome number is no
barrier at all to the evolutionary process. Non-homologous life forms
cross breed all the time giving fertile offspring and more fit
replicators. There is no limit to evolutionist speculations are
irrationality.
>
[snip more evolutionist speculations]
>
> > You write so much and say so little. Amplification is the requirement
> > for a population to overcome the multiplication rule of probabilities.
> > If the population can not amplify the allele, there is a very low
> > probability that the next beneficial mutation will occur at the proper
> > locus.
>
> If by "amplification" you mean that the *frequency* of particular alleles in the population changes, sure. �That is the definition of what happens during selection.
>
When are you going to learn that it is not �frequency� which drives
the mutation and selection process, it is the subpopulation size which
drives the probabilities of events occurring?
>
>
> > >With the assumption of random mating, we would expect 0.35 for the frequency of Aa, 0.0 for the frequency of AA, and >0.65 for aa in the next generation (because we are crossing Aa X aa). �Similarly, we would expect 0.15 Bb and 0.85 bb >individuals. �Since the genes are assumed to be unlinked and *using* the multiplication rule of probabilities (correctly), >then I would expect the progeny of this mating to be (0.65)*(0.85) = 0.55 �aa, bb; (0.65)*(0.15) = 0.10 aa, Bb; >(0.35)*(0..85) = 0.30 Aa, bb; and (0.35)*(0.15) = 0.05 Aa, Bb individuals. �That adds up to 1.0.
>
> > You still haven�t figured out how to write the probability function
> > for random recombination. This is reasonable since it has taken months
> > for you to get any understanding of the probability function for
> > mutation and selection.
>
> No. �I have understood what mutation does, what selection does, and what recombination does. �You, OTOH, have repeatedly and stupidly presented your messed up derivation of the binomial probability, where you divide the mutation rate by 4, all the while claiming that that stupidity is a work of genius.
>
You are wrong hersheyh and it should be obvious to you why you are
wrong. Even Inez realizes there should be at least 3 outcomes from a
point mutation. What evolutionists like you who don�t comprehend is
that when you don�t know what the base was before the mutation
occurred requires that you consider that any of the four bases must be
taken into account as possible outcomes.
[snip more of hersheyh�s inability to comprehend the mathematics of
mutation and selection]
> > Bill posed a very good aptitude question many months back. I think it
> > was to show that any loop on a terrain must include two points of
> > equal altitude.
>
> Given a smooth contour map, prove that an arbitrary circle drawn on
> the map must contain at least two points that are 180 degrees apart on
> the circle and are at the same altitude.
Thanks; I forgot the 180 degree stipulation which is an interesting
condition in that it reminds me of an Einstein anecdote. About 1946
Einstein interviewed (graduate student) Ernst Straus to be Einstein's
mathematical assistant. Asked for an example of his mathematical
discoveries, Straus mentioned this theorem: Given any closed curve in
a plane, there are three points on the curve which form an equilateral
triangle. Einstein replied that he didn't like the theorem at all
because it combined two concepts, topology and geometry, which are
inimical to one another. (But Straus did get the job.)