Rebuttal To Creationists - "Since We Can't Directly Observe Evolution..."

Biologically, microevolutionary events in sexually reproducing organisms do add up and may lead to speciation.This is why we can map groups of species into nested hierarchical cladograms based on fossils, genetics, morphology and the patterns of Endogenous Retroviral Insertions in their genomes.Your proposal of mathematical rules for all of biology based on your calculations from 2 lab experiments using bacteria fails to model reality as reported by observers of evolution over the past 200 years.

I'm not averaging. I'm summing. Do you know the difference? You can't use a sexually reproducing population to model an asexually reproducing population. Do you know why? The mutation rate in E. coli is about one mutation per 1,000 replications.

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. The major conclusions are (i) the mutation rate of a wild-type E. coli strain is ∼1 × 10−3 per genome per generation;

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The size of the E. coli genome is about 5 million bases. At 3 SNP's per base that would be 15 million possible SNP's. This would then require 15 million * 1,000 generations to produce all possible mutations if we assume all mutations are equally probable and no repeat mutations (which isn't true, but we are just estimating here). That would be 15 billion replications for E. coli. If we also assume that there is just a single beneficial mutation possible in the entire genome, then it would take that many replications to get that one beneficial mutation. If there are thousands of possible beneficial mutations then it would take far fewer replications.The mutation rate for humans is 50,000 mutations per 1,000 births, not 1 like in E. coli. If we look at a 6 billion base diploid genome, that is 18 billion possible SNP's. At 50 mutations per offspring that would require 360 million births to get all of the possible SNP's like our model above. Moreover, these mutations would not be kept in lineages. They would spread through the population. Some would be lost because the mutation would be heterozygous to begin with and not everyone will have offspring in a steady population. However, a sizeable chunk of this variation can be kept because they are not limited to one lineage.At a steady population of just 100,000 humans it would take 3,600 generations for all of these mutations to occur in our simplistic model. At 25 years per generation, that would be 90,000 years. There would be many different beneficial mutations possible across the ancestral genome, and those would continually move towards fixation as 5 million new mutations are created in each generation.The reason why there are steps in both the Kishoni and Lenski experiments is because different beneficial mutations in different genes can not be combined into a single E. coli genome. That wouldn't be the case for a sexually reproducing population. In a sexually reproducing population a beneficial mutation that happens in gene A in one individual and a beneficial mutation that happens in gene B in another individual can be combined into a single genome in later generations.Do you understand this?

Kleinman:

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Populations are made up of individuals. You can't average over populations to determine what is happening with each individual.

Taq:

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I'm not averaging. I'm summing. Do you know the difference?

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Summing what? You start with the assumption that you have a population of 100,000. Is it 100,000 humans, 100,000 chimpanzees, 100,000 common ancestors, or some combination of all 3? If you don't consider each individual and what happens with each replication, you have mush.

Kleinman:

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If you think your model is correct, use it to describe what is happening in the Kishony and Lenski experiments.

Taq:

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You can't use a sexually reproducing population to model an asexually reproducing population. Do you know why?

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So you think that adaptive alleles are formed differently for asexual reproducers and sexual reproducers? Aren't these alleles both formed from DNA (sometimes RNA)? Don't the genetic sequences in asexual reproducers and sexual reproducers both have mutation rates associated? For someone trained in microbiology, you have very little understanding how drug resistance evolves.

Kleinman:

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Explain why it takes a billion replications for each adaptive mutational step in these experiments.

Taq:

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The mutation rate in E. coli is about one mutation per 1,000 replications.

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That would be 5 million bases replicated 1000 times or 1 in 5 billion base replications. I use a mutation rate value of 1e-9 or approximately 1 mutation for every billion replication, a slightly higher mutation rate.Now, think about this in terms of the Kishony experiment. He starts his experiment with a drug-sensitive founder. In the first generation the population doubles to 2, next generation 4, next generation 8,... after 30 generations (doublings), the population size will be over one billion. In other words, every site in the genome will have been replicated a billion times and in those billion replications, some member of the population will have a mutation at some given site. Since there are multiple possible base substitutions, it will actually take 3-4 billion replications or about 32 doublings to get some member of the population with every possible base substitution. It is that lucky member with the base substitution at the particular site that gives some resistance to the drug being used but there is only one member at this time with that first adaptive mutation. This member with the adaptive mutation forms a new colony that again has to do a billion replications so that at least one of its members will get the next adaptive mutation. This process works the same way for asexual or clonal replicators as well as sexual replicators. The 1/(mutation rate) replications are occurring at every site in the genome, the population is doing an exhaustive search of the entire sample space, and all mutations are being sampled, beneficial, neutral, and detrimental. The probability of beneficial mutation B occurring on some member with beneficial mutation A depends on the number of replications the A variant can do. This is how adaptive alleles are formed no matter whether the replicator is asexual or sexual. Replication of a site in a genome is the random trial and the possible outcomes at that site are a mutation occurs or a mutation does not occur.

I'm summing the number of mutations that happen in the population in that generation. We can also sum the mutations that happen over generations. I don't understand why this is so hard to understand.They would be 100,000 individuals. In the last ~1.5 million years they would be humans. In the last 200,000 years they would be anatomically modern humans, i.e. H. sapiens. Prior to ~1.5 million years they would be human ancestors. They would never be chimps. No, I don't. The difference is how the mutations are passed on to offspring. Do you understand the difference between diploid and haploid genomes? Do you understand how you got half of your genome from your father and half from your mother? Do you understand how this is different from bacteria who only have one parent that they are a clone of? If you use that number for the human mutation rate then you are way off. How many possible mutations result in antibiotic resistance? That's the first question you need to answer. The difference is the mutation segregation seen in the Kishony and Lenski experiments. That doesn't happen in sexually reproducing populations except in cases where the mutations are close to one another on the same chromosome.

Kleinman:

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Summing what? You start with the assumption that you have a population of 100,000. Is it 100,000 humans, 100,000 chimpanzees, 100,000 common ancestors, or some combination of all 3? If you don't consider each individual and what happens with each replication, you have mush.

Taq:

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I'm summing the number of mutations that happen in the population in that generation. We can also sum the mutations that happen over generations. I don't understand why this is so hard to understand.

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Is the population of 100,000 exact clones of each other or are there different variants in the population with different sets of mutations?

Taq:

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They would be 100,000 individuals. In the last ~1.5 million years they would be humans. In the last 200,000 years they would be anatomically modern humans, i.e. H. sapiens. Prior to ~1.5 million years they would be human ancestors. They would never be chimps.

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Are all 100,000 individuals on the exact same evolutionary trajectory and do all their descendants over generations remain on that same evolutionary trajectory where each individual gets the same set of mutations as every other individual or do the different individuals get different sets of mutations and the population is genetically diverging?

Kleinman:

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So you think that adaptive alleles are formed differently for asexual reproducers and sexual reproducers?

Taq:

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No, I don't. The difference is how the mutations are passed on to offspring. Do you understand the difference between diploid and haploid genomes? Do you understand how you got half of your genome from your father and half from your mother? Do you understand how this is different from bacteria who only have one parent that they are a clone of?

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Here is one place where you get confused about this mathematical process. Ploidy of the cell only increases the number of replications of the chromosomes. So diploid cells would get two chromosome set replications for each individual replication. If we consider cell lines that replicate by mitosis such as cancers, that would increase the diversification rate because each of the chromosomes will get different sets of mutations with each replication. This is the reason why single drug targeted cancer therapy does not work when the cell population reaches about 1/(mutation rate) replications. There will already be mutant variants at the target site. On the other hand, with meiosis, you have parents each passing half the genome. If I understand your argument correctly, you are claiming that one parent passes beneficial alleles from their set of chromosomes and the other parent passes their beneficial alleles from their set of chromosomes. If I understand your argument correctly, then your population must be diverse and not clones. I have some questions for you about your population. How many beneficial alleles are in your population? What is the frequency of the different beneficial alleles in your population? Which members have the beneficial alleles? Are these beneficial alleles homozygous or heterozygous in each of the members? And how do you compute the probability that a descendant will get these beneficial alleles from any two parents in your population of 100,000?

Kleinman:

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I use a mutation rate value of 1e-9 or approximately 1 mutation for every billion replication, a slightly higher mutation rate.

Taq:

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If you use that number for the human mutation rate then you are way off.

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That's OK, we can use whatever mutation rate you want to use.

Kleinman:

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It is that lucky member with the base substitution at the particular site that gives some resistance to the drug being used but there is only one member at this time with that first adaptive mutation.

Taq:

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How many possible mutations result in antibiotic resistance? That's the first question you need to answer.

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I've already done the math. If you want to try to do it yourself, the way you do it is by doing an "at least one" probability calculation. Let's say you have "m" possible beneficial mutations. What is the probability of at least one of those "m" possible beneficial mutations occurring in "n" replications? It doesn't change the number of replications much from only a single beneficial mutation. You can observe that empirically in the Kishony and Lenski experiments where it still takes about a billion replications for each adaptive step despite the fact there is more than one possible adaptive evolutionary trajectory to the environmental selection condition.

Kleinman:

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This process works the same way for asexual or clonal replicators as well as sexual replicators. The 1/(mutation rate) replications are occurring at every site in the genome, the population is doing an exhaustive search of the entire sample space, and all mutations are being sampled, beneficial, neutral, and detrimental. The probability of beneficial mutation B occurring on some member with beneficial mutation A depends on the number of replications the A variant can do. This is how adaptive alleles are formed no matter whether the replicator is asexual or sexual. Replication of a site in a genome is the random trial and the possible outcomes at that site are a mutation occurs or a mutation does not occur.

Taq:

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The difference is the mutation segregation seen in the Kishony and Lenski experiments. That doesn't happen in sexually reproducing populations except in cases where the mutations are close to one another on the same chromosome.

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What????

Let's get back to Percy's Message 175

Kleinman:

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Note that Haldane's model has been proven to be a conservation of energy process.

Percy:

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I see no such proof in your message.

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I already pointed out previously that Haldane's frequency equation was a conservation of energy process based on the principle that it takes energy to replicate. However, Flake and Grant demonstrate this mathematically in the following paper:
An Analysis of the Cost-of-Selection Concept
Here are two quotes from the paper, the first from the abstract:

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The robust nature of the cost-of-selection concept is examined in the light of a conservation principle.
  —Flake and Grant

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and the second from the paragraph titled "A CONSERVATION PRINCIPLE"

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A quantity that is dependent only on the initial and final state of a system and is invariant to variation in the detailed motion between states is often associated in the physical sciences with a potential energy function. A fixed reference state is normally specified for such a function, and in this case the natural choice for a reference state is the final stage of complete substitution. In this context then, the potential (cost) function evaluated at the initial state (frequency) corresponds to the potential energy associated with the substitution.
  —Flake and Grant

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Kleinman:

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You have to modify his model for the particular case.

Percy:

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If you believe Haldane's model should be modified for each particular case then please describe the various processes of modification and how one selects which to use according to case.

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I could have worded this a bit better. The frequency equation pnA + qna = 1 isn't modified, we use the equation to compute the number of replications of the more fit variant to fixation. In the process of doing this calculation, one also obtains the number of deaths of the less fit variants. This is done because it is the number of replications of the more fit variant that determines the probability of the next adaptive mutation occurring in this subset of the population.

Kleinman:

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For example, Lenski's experiment includes bottlenecking so you have to modify Haldane model as shown here:
Fixation and Adaptation in the Lenski E. coli Long Term Evolution Experiment

Percy:

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Please describe how Haldane's model should be modified for the Lenski case.

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There are two issues you have to take into account when evaluating the frequency/fixation equation for the Lenski experiment. One is that you have a varying population starting at 5 million at the beginning of the day and growing to 500 million at the end of the day which is then bottlenecked back down to 5 million, the nutrients are replenished and another day's growth is started. For the initial state of a fixation/adaptation cycle, I assume the number of more fit variants is 1 and the number of less fit variants is 4,999,999. The frequency equation for that state is (1/5,000,000) + (4,999,999/5,000,000) = 1I defined a fitness parameter for the more fit variant and then solved the equation numerically. Allow the population to do 6 generations of doubling and a 7th partial generation to reach a population of 500 million. Each generation the more fit variant increases in number and frequency and the less fit variants decrease in number and frequency. I didn't try to formulate an algebraic solution because of the bottlenecking at the end of a day's growth. That discontinuity introduces a nonlinearity into the math. Interestingly, Haldane's estimate of 300 generations/fixation is the same order of magnitude that my calculation shows and is similar to what Lenski's experiment generated.If this all makes sense to you and you don't have any questions, I'll go on next to show you how to do the adaptation portion of the calculation.

Raindrops are random micro-water. They can't add up to macro-water (rivers, lakes, oceans)?

Kleinman:

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Mathematically, microevolutionary events don't add up. They are random events so the joint probability of microevolutionary events occurring is computed using the multiplication rule.

ringo:

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Raindrops are random micro-water. They can't add up to macro-water (rivers, lakes, oceans)?

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ringo, you are making the same logical inconsistency that Taq makes in his attempt to model human evolutionary fitness improvement. Adaptive mutations are particular mutations, not any mutation. The correct analogy for your raindrop concept would be, what is the probability of two particular raindrops ending up in the same body of water? You might try and argue that raindrops are indistinguishable but mutations are distinguishable. And you must model DNA evolution and the accumulation of adaptive mutations in a lineage using the multiplication rule.

Further response to Percy's Message 175

Kleinman:

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There are a couple of ways to model adaptation. You can use the "at least one rule from probability theory as shown here in this paper:
The basic science and mathematics of random mutation and natural selection

Percy:

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Instead of providing a link to a paper, please describe the model adaptation process yourself and use the link as a supporting reference.

Kleinman:

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Or you can use a Markov chain random walk calculation to compute the probability of an adaptive mutation occurring as show here:
The Kishony Mega-Plate Experiment, a Markov Process

Percy:

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Again, instead of providing a link to a paper, please describe the model adaptation process yourself and use the link as a supporting reference.

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OK. I'll use the The basic science and mathematics of random mutation and natural selection reference since that is the formulation I used to compute the probabilities in the Lenski paper, Fixation and Adaptation in the Lenski E. coli Long Term Evolution ExperimentThe Haldane frequency equation gives us a way to compute the subpopulation sizes of the more and less fit variants. We know that the less fit variants will ultimately be driven to extinction so only members of the subpopulation of the more fit variant are candidates for a beneficial mutation "A" that must occur at some site in its genome. We start with the probability that a mutation will occur at that site in a single replication. (Note that this math applies to every site in the genome, not just the site(s) that is/are candidates for adaptive mutations. That is why this is an exhaustive search for every possible mutation). That probability is the mutation rate, call it "mu". But we also need to consider that the mutation that occurs may not be a beneficial mutation. It is possible that the wrong base substitution occurs. In other words, the mutation itself is a random trial with multiple possible outcomes. We can write the set of possible outcomes as follows:
P(Ad) + P(Cy) + P(Gu) + P(Th) + P(iAd) + P(iCy) + P(iGu)+ P(iTh) + P(del) + ... = 1
where the first four terms are possible base substitutions, the next four terms represent insertions of bases, the ninth term is the probability of a deletion and the ellipsis represents any other form of mutation you can imagine.
Let P(BeneficialA) represent the probability of the mutation that gives improved reproductive fitness. Note that P(BeneficialA) is some number between 0 and 1 and for most cases will have a value of about 1/3 to 1/4. Then, the probability of mutation A occurring at the particular site in a single replication is written:
P(A) = P(BeneficialA)*mu (1)
One could think of P(BeneficialA)*mu as the "beneficial mutation rate", a probability value slightly lower than the mutation rate "mu". It should be clear that the probability of the A mutation occurring in a single replication is very low.The next step is to compute the probability of that mutation A occurring at least once in "n" replications of the more fit variant. This is done using the "at least one rule". It is a very simple rule to apply and understand. I do a step-by-step derivation in this paper The basic science and mathematics of random mutation and natural selection
for this case. If there are any questions on how to derive or apply this rule I'll try and answer them. When this rule is applied to equation (1) you get the following probability equation:P(A) = 1 − (1 − P(BeneficialA)*mu)^n (2)Equation (2) is evaluated using the population size generated by the Haldane frequency/fixation equation. I plotted the results for several different fitness parameters in the Lenski fixation/adaptation paper if you are interested. Solving Haldane's frequency/fixation and equation (2) gives the correct mathematical description of the Lenski experiment. It demonstrates mathematically why competition slows the adaptation process. Competition slows the accumulations of replications the more fit variant can do by limiting the energy available to that variant. Note that the more fit variant accumulated replications (the random trial for the next adaptive mutation) most rapidly after the variant fixes in the population. If Lenski used a larger volume in his experiment (increases the carrying capacity), it might not be necessary for that variant to fix in order to achieve the necessary number of replications to give a reasonable probability of the next adaptive mutation to occur on some member of that subset. If the carrying capacity is much larger such as in the Kishony experiment, you can have multiple different lineages taking different evolutionary trajectories to adaptation in the same environment. The math for each of these lineages is the same.If there aren't any questions or comments on this math or how to apply this math to the Lenski experiment, I'll go onto more of Percy's comments, in particular, how to do the mathematics of adaptation using a Markov process random walk model.

They would have variation just like your average sampling of mammal species. Again, mutations spread through a sexually reproducing population. It means that mutations from different lineages are combined. You keep asking how these beneficial mutations are put in the same lineage. This is how. This is going to differ based on a myriad of conditions. There is no single answer for any of those questions. Even in the Lederberg experiment there was a 1,000 fold difference in the beneficial mutation rate for two different phenotypes. The very fact that you pretend there is a beneficial mutation rate only highlights your misunderstandings of how evolution works. Really? Even with a very small population of 100,000 individuals it only took 90,000 years to get all possible SNP's with 360 million births. If there are 20 million possible beneficial mutations this would be 20 million out of 18 billion possible SNP's. This gives us a 1 in 900 chance of getting a beneficial mutation. This means we only need 18 births to get a beneficial mutation. 18 is a lot different than 360 million. Its from the Lenski paper you keep citing. It's the first sentence in the abstract.

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When large asexual populations adapt, competition between simultaneously segregating mutations slows the rate of adaptation and restricts the set of mutations that eventually fix.
Distribution of fixed beneficial mutations and the rate of adaptation in asexual populations

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Are you telling me you don't understand the paper you keep citing?

Edited by Taq, .

You will notice that their model applies to species with haploid genomes, not diploid genomes like that seen in primates.

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It is shown for a continuous haploid model that the common standard assumptions usedin calculating the cost of gene substitution, namely, large constant population size and small constant selective value, are unnecessary.

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Which doesn't apply in sexually reproducing species.

Surprised?

Good. Then I'm likely to be right. It doesn't matter which raindrops you add. The sum is the same.

A raindrop falls and runs downhill to a small stream. Small streams flow into a larger river. Larger river dumps into ocean. Same body of water.Sexual reproduction is analogous. Lineages flow into each other and form the larger body of variation that is the population.If you understood how sexual reproduction works you would be able to figure these things out. Not for sexually reproducing populations.

Kleinman:

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Is the population of 100,000 exact clones of each other or are there different variants in the population with different sets of mutations?

Taq:

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They would have variation just like your average sampling of mammal species.

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Are some of the variants more fit than other variants and are they engaged in biological evolutionary competition?

Kleinman:

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Are all 100,000 individuals on the exact same evolutionary trajectory and do all their descendants over generations remain on that same evolutionary trajectory where each individual gets the same set of mutations as every other individual or do the different individuals get different sets of mutations and the population is genetically diverging?

Taq:

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Again, mutations spread through a sexually reproducing population.

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You aren't answering my question about whether all 100,000 individuals are on the same evolutionary trajectory. But, let's get some more detail on your statement that mutations spread through a sexually reproducing population. Is it possible that a beneficial mutation is lost in a sexually reproducing population? What happens if the beneficial allele is heterozygous rather than homozygous?

Kleinman:

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There will already be mutant variants at the target site. On the other hand, with meiosis, you have parents each passing half the genome. If I understand your argument correctly, you are claiming that one parent passes beneficial alleles from their set of chromosomes and the other parent passes their beneficial alleles from their set of chromosomes. If I understand your argument correctly, then your population must be diverse and not clones.

Taq:

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It means that mutations from different lineages are combined. You keep asking how these beneficial mutations are put in the same lineage. This is how.

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What is the probability of beneficial alleles in a diverse population recombining in the same descendant?

Kleinman:

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How many beneficial alleles are in your population? What is the frequency of the different beneficial alleles in your population? Which members have the beneficial alleles? Are these beneficial alleles homozygous or heterozygous in each of the members? And how do you compute the probability that a descendant will get these beneficial alleles from any two parents in your population of 100,000?

Taq:

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This is going to differ based on a myriad of conditions. There is no single answer for any of those questions. Even in the Lederberg experiment there was a 1,000 fold difference in the beneficial mutation rate for two different phenotypes. The very fact that you pretend there is a beneficial mutation rate only highlights your misunderstandings of how evolution works.

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Tell us how you compute the probability that a descendant will get beneficial alleles by recombination for the conditions of your model.

Kleinman:

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Let's say you have "m" possible beneficial mutations. What is the probability of at least one of those "m" possible beneficial mutations occurring in "n" replications? It doesn't change the number of replications much from only a single beneficial mutation.

Taq:

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Really? Even with a very small population of 100,000 individuals it only too 90,000 years to get all possible SNP's with 360 million births. If there are 20 million possible beneficial mutations this would be 20 million out of 18 billion possible SNP's. This gives us a 1 in 900 chance of getting a beneficial mutation. This means we only need 18 births to get a beneficial mutation. 18 is a lot different than 360 million.

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How do 20 million possible beneficial mutations end up in the lineages of all humans? Recombination? Do all 20 million possible beneficial mutations give an equal improvement in fitness to your different lineages, or is there biological evolutionary competition that causes the loss of some of the less fit variants?

Kleinman:

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What????

Taq:

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Its from the Lenski paper you keep citing. It's the first sentence in the abstract.

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When large asexual populations adapt, competition between simultaneously segregating mutations slows the rate of adaptation and restricts the set of mutations that eventually fix.
Distribution of fixed beneficial mutations and the rate of adaptation in asexual populations

Taq:

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Are you telling me you don't understand the paper you keep citing?

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Try reading beyond the abstract:

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In larger populations the situation becomes more complicated, as many distinct mutations segregate simultaneously, but only one lineage can fix in the absence of recombination. Many of the mutations are therefore wasted, and a complicated process of interference arises between the mutations competing for fixation. This process is also found to a degree in sexual populations (the Hill–Robertson effect) if recombination cannot act quickly enough to place all of the competing mutations on the same genetic background (16).

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Kleinman:

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I already pointed out previously that Haldane's frequency equation was a conservation of energy process based on the principle that it takes energy to replicate. However, Flake and Grant demonstrate this mathematically in the following paper:
An Analysis of the Cost-of-Selection Concept

Taq:

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You will notice that their model applies to species with haploid genomes, not diploid genomes like that seen in primates.

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It is shown for a continuous haploid model that the common standard assumptions usedin calculating the cost of gene substitution, namely, large constant population size and small constant selective value, are unnecessary.

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OK, Haldane's frequency equation for a sex-linked diploid is:
pn^2AA + 2pnqnAa + qn^2aa = 1
Which of the variants fix in the population, AA, Aa, or aa?

Kleinman:

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This is done because it is the number of replications of the more fit variant that determines the probability of the next adaptive mutation occurring in this subset of the population.

Taq:

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Which doesn't apply in sexually reproducing species.

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Taq, you are still confused on this point. Try thinking of it this way. It's the number of replications of a particular allele that determines the probability of an adaptive mutation occurring at some site in that allele. If that allele is homozygous in that diploid then you will get two replications of that allele with every creature replication. If the allele is heterozygous at that site, you will have only one replication of each particular allele.