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Author Topic:   Do you really understand the mathematics of evolution?
Kleinman
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Posts: 386
From: United States
Joined: 10-06-2016
Member Rating: 1.1


Message 136 of 239 (877869)
06-22-2020 5:55 PM
Reply to: Message 135 by Taq
06-22-2020 5:20 PM


Re: Does competition accelerate DNA evolution?
Kleinman writes:

The environment selects out all variants that can't feed on the particular food source because of the incorrect beak shape and the remaining variants increase in frequency and the population inbreed for those phenotypes. But what that has done is reduced the diversity of the individual populations.

Taq writes:

If there were two traits under selection, wouldn't that increase the number of individuals for each of those traits, and increase the chances of two organisms mating and having those traits combined in their offspring?


Perhaps, but you would still have to get the correct variants mating and recombining to get the offspring with those traits. A dog breeder might want a short dog with short hair so he starts breeding for those traits. But those traits have to already exist in the gene pool to do that genetic transformation. But, let's say the breeder wants green dogs with 5 legs. If the alleles don't exist in the gene pool, there is no chance to get the particular variant no matter how many recombination events occur. You would still need the mutations that give extra limbs and the color you are looking for.
Kleinman writes:

Unless those alleles already exist in the population such as with Darwin's finches, you still need to get the appropriate mutations for the next evolutionary step. And you know that it takes 3e9 replication to get those variants.

Taq writes:

So after 3e9 replications we have nearly every possible beneficial point mutation in eat least one individual. What happens when they start mating and mixing those beneficial mutations together?


You should know the answer to that. In that population of 3e9, you will have one member with mutation A, one member with mutation B and the rest of the population will have different mutations or no mutations depending on the length of the genome. Here's an easy way to think of this problem. This is a random card drawing problem. You have 3 kinds of cards in the deck. You have one A card, one B card and the rest of the cards are not A or B cards, call them C cards. And there are (3e9)-2 C cards in the deck. Draw 2 cards from the deck and the vast majority of the time they will be two C cards, less often you will draw an A and a C card or a B and a C card and very rarely an A and a B card. In a previous message I showed you how to do this math.

You might want to argue that A and B will replicate giving more A and B cards for your deck but so will the C cards keeping the A and B cards at low frequency in the deck.

Recombination, unlike DNA evolution, is dependent on the frequency of variants in a population. That's why for the Darwin's finches example, the reduction in population size due to the kind of food available in the particular environment improves the probability for the particular recombination event but reduces the probability of a DNA evolution event which depends on the number of replications.


This message is a reply to:
 Message 135 by Taq, posted 06-22-2020 5:20 PM Taq has responded

Replies to this message:
 Message 137 by Straggler, posted 06-22-2020 6:41 PM Kleinman has responded
 Message 141 by Taq, posted 06-23-2020 12:27 PM Kleinman has responded

  
Straggler
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Posts: 10312
From: London England
Joined: 09-30-2006
Member Rating: 3.0


Message 137 of 239 (877874)
06-22-2020 6:41 PM
Reply to: Message 136 by Kleinman
06-22-2020 5:55 PM


Re: Does competition accelerate DNA evolution?
Can you give an example of a well specified question, answerable with a simulation, where your model and the scientifically accepted model would give different results?

The two simulations could then be tested to see which one more accurately reflects observable reality.


This message is a reply to:
 Message 136 by Kleinman, posted 06-22-2020 5:55 PM Kleinman has responded

Replies to this message:
 Message 138 by Kleinman, posted 06-22-2020 7:31 PM Straggler has responded

  
Kleinman
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Posts: 386
From: United States
Joined: 10-06-2016
Member Rating: 1.1


Message 138 of 239 (877876)
06-22-2020 7:31 PM
Reply to: Message 137 by Straggler
06-22-2020 6:41 PM


Re: Does competition accelerate DNA evolution?
Straggler writes:

Can you give an example of a well specified question, answerable with a simulation, where your model and the scientifically accepted model would give different results?
The two simulations could then be tested to see which one more accurately reflects observable reality.


Sure, let's start with something easy and well recognized, the Kishony mega-plate experiment:
EXTRA MINUTES - SUPERBBUGS (Harvard Experiment explained)
I published the math of DNA evolution which describes this experiment before Kishony ran his experiment. You can find the mathematical explanation here:
(PDF) The basic science and mathematics of random mutation and natural selection

Now, compare the solution I give with Joe Felsenstein's model of DNA evolution which is described here:
Felsenstein's model of DNA evolution
Good luck getting that model to work describing the Kishony experiment (or any real evolutionary experiment). The reason his model won't work is he is using an incorrect assumption. I do not make the assumption that he makes and have rewritten the transition matrix so you can use a Markov chain model to correctly predict DNA evolution. I'm just finishing the paper and will be submitting it for publication by the end of summer.


This message is a reply to:
 Message 137 by Straggler, posted 06-22-2020 6:41 PM Straggler has responded

Replies to this message:
 Message 139 by Straggler, posted 06-22-2020 8:12 PM Kleinman has responded
 Message 143 by Phat, posted 06-23-2020 1:59 PM Kleinman has responded

  
Straggler
Member
Posts: 10312
From: London England
Joined: 09-30-2006
Member Rating: 3.0


Message 139 of 239 (877878)
06-22-2020 8:12 PM
Reply to: Message 138 by Kleinman
06-22-2020 7:31 PM


Re: Does competition accelerate DNA evolution?
And this model of yours will effectively refute evolutionary notions of common ancestry between species on Earth?

That’s Nobel prize winning stuff.....

The discussion I saw to which you seemed to have no answer was this one.

Kleinman: Four Questions About Evolution

What happened there?


This message is a reply to:
 Message 138 by Kleinman, posted 06-22-2020 7:31 PM Kleinman has responded

Replies to this message:
 Message 140 by Kleinman, posted 06-22-2020 9:01 PM Straggler has responded

  
Kleinman
Member
Posts: 386
From: United States
Joined: 10-06-2016
Member Rating: 1.1


Message 140 of 239 (877882)
06-22-2020 9:01 PM
Reply to: Message 139 by Straggler
06-22-2020 8:12 PM


Re: Does competition accelerate DNA evolution?
Straggler writes:

And this model of yours will effectively refute evolutionary notions of common ancestry between species on Earth?
That’s Nobel prize winning stuff.....


What my model does is correctly explain the mathematics of the evolutionary trajectory. With respect to the Nobel prize, Edward Tatum already spoke about this issue in his 1958 Nobel Laureate Lecture.
The Nobel Prize in Physiology or Medicine 1958
Edward Tatum writes:

In microbiology the roles of mutation and selection in evolution are coming to be better understood through the use of bacterial cultures of mutant strains. In more immediately practical ways, mutation has proven of primary importance in the improvement of yields of important antibiotics – such as in the classic example of penicillin, the yield of which has gone up from around 40 units per ml of culture shortly after its discovery by Fleming to approximately 4,000, as the result of a long series of successive experimentally produced mutational steps. On the other side of the coin, the mutational origin of antibiotic-resistant micro-organisms is of definite medical significance. The therapeutic use of massive doses of antibiotics to reduce the numbers of bacteria which by mutation could develop resistance, is a direct consequence of the application of genetic concepts. Similarly, so is the increasing use of combined antibiotic therapy, resistance to both of which would require the simultaneous mutation of two independent characters.

As an important example of the application of these same concepts of microbial genetics to mammalian cells, we may cite the probable mutational origin of resistance to chemotherapeutic agents in leukemic cells44, and the increasing and effective simultaneous use of two or more chemotherapeutic agents in the treatment of this disease.


The only thing I have done is put mathematics to his explanation. Mine is a case of standing on the shoulders of giants.

Straggler writes:

The discussion I saw to which you seemed to have no answer was this one.

Kleinman: Four Questions About Evolution

What happened there?


I think I embarrassed Swamidass when he got an undergraduate level question on probability theory wrong.
Kleinman writes:

If you double the population size, do you double the probability of a beneficial mutation occurring?

Swamidass writes:

Yes. Doubling the population size doubles the probability of a beneficial mutation, all else being equal.

Kleinman writes:

So if the population size is X and the probability of a beneficial mutation occurring is 0.6, then if the population size is 2X the probability of a beneficial mutation occurring is 1.2?

This is a common error made by people not familiar with probability theory. You are trying to use the addition rule for complimentary events (that is the probability of the beneficial mutation occurring or not occurring).


This message is a reply to:
 Message 139 by Straggler, posted 06-22-2020 8:12 PM Straggler has responded

Replies to this message:
 Message 165 by Straggler, posted 06-26-2020 4:13 PM Kleinman has responded

  
Taq
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Posts: 8409
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Member Rating: 1.9


Message 141 of 239 (877921)
06-23-2020 12:27 PM
Reply to: Message 136 by Kleinman
06-22-2020 5:55 PM


Re: Does competition accelerate DNA evolution?
Kleinman writes:

Perhaps, but you would still have to get the correct variants mating and recombining to get the offspring with those traits.

If those traits are being selected for then that shouldn't be a problem.

In that population of 3e9, you will have one member with mutation A, one member with mutation B and the rest of the population will have different mutations or no mutations depending on the length of the genome.

If the bird genome is similar to the human genome, then each bird will have between 50 and 100 mutations specific to them.

You have one A card, one B card and the rest of the cards are not A or B cards, call them C cards

You are ignoring selection. The beneficial allele will become more common over time. This greatly increases the chances of the two beneficial mutations finding their way into a single offspring.


This message is a reply to:
 Message 136 by Kleinman, posted 06-22-2020 5:55 PM Kleinman has responded

Replies to this message:
 Message 142 by Kleinman, posted 06-23-2020 12:53 PM Taq has responded

  
Kleinman
Member
Posts: 386
From: United States
Joined: 10-06-2016
Member Rating: 1.1


Message 142 of 239 (877924)
06-23-2020 12:53 PM
Reply to: Message 141 by Taq
06-23-2020 12:27 PM


Re: Does competition accelerate DNA evolution?
Kleinman writes:

Perhaps, but you would still have to get the correct variants mating and recombining to get the offspring with those traits.

Taq writes:

If those traits are being selected for then that shouldn't be a problem.


Real environments have many selection conditions, for example, disease, starvation, predation, dehydration, thermal stress (both heat and cold), toxins, competition from other species,... With Darwin's finches, you had very rapid intense selection from a single selection pressure (starvation) so recombination improved fitness. But if you have selection at many different loci simultaneously (lots of different cards in the deck) with each variant being at low frequency, your probability of getting the correct recombination events drops off very rapidly. But if you have an empirical example that demonstrates your claim, that would be interesting.
Kleinman writes:

In that population of 3e9, you will have one member with mutation A, one member with mutation B and the rest of the population will have different mutations or no mutations depending on the length of the genome.

Taq writes:

If the bird genome is similar to the human genome, then each bird will have between 50 and 100 mutations specific to them.


Sure, lots of non-A and non-B cards.
Kleinman writes:

You have one A card, one B card, and the rest of the cards are not A or B cards, call them C cards

Taq writes:

You are ignoring selection. The beneficial allele will become more common over time. This greatly increases the chances of the two beneficial mutations finding their way into a single offspring.


I'm describing an analogous experiment to the Kishony experiment except instead of a prokaryote, using a eukaryote that replicates with meiosis. But if you think that recombination solves your problem and you don't need DNA evolution, show us how you breed dogs and get cats. You can't get any better selection than that and they are only a node apart on your clade.

This message is a reply to:
 Message 141 by Taq, posted 06-23-2020 12:27 PM Taq has responded

Replies to this message:
 Message 145 by Taq, posted 06-23-2020 3:17 PM Kleinman has responded

  
Phat
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Posts: 14132
From: Denver,Colorado USA
Joined: 12-30-2003
Member Rating: 1.2


Message 143 of 239 (877927)
06-23-2020 1:59 PM
Reply to: Message 138 by Kleinman
06-22-2020 7:31 PM


Re: Does competition accelerate DNA evolution?
Kleinman writes:

The reason his model won't work is he is using an incorrect assumption. I do not make the assumption that he makes and have rewritten the transition matrix so you can use a Markov chain model to correctly predict DNA evolution. I'm just finishing the paper and will be submitting it for publication by the end of summer.

Just a few quick comments.
  • dwise1 is actually a man named David Wise. He has argued with Biblical Creationists for years and I see that he has lumped you into that category. Now be honest with me. Do you see yourself as a man gifted somewhat in mathematics who is a proponent of Intelligent Design? If so, have you managed to get the attention of the Discovery Institute at all? My advice is to stay humble and let your opponents become arrogant rather than you yourself. If you are in fact representing Intelligent Design, the Intelligence behind that design---the same intelligence that you likely thank for giving you math skills, wants you to stay humble.

    “The only way I know to drive out evil from the country is by the constructive method of filling it with good.”Calvin Coolidge
    "A lie can travel half way around the world while the truth is putting on its shoes." ~Mark Twain "
    “As the fear of God is the beginning of wisdom, so the denial of God is the height of foolishness.”-RC Sproul, Essential Truths of the Christian Faith

    - You can safely assume that you've created God in your own image when it turns out that God hates all the same people you do.
    Anne Lamott
    I Have Strong Arguments Which I Cant Say To You~CG

  • This message is a reply to:
     Message 138 by Kleinman, posted 06-22-2020 7:31 PM Kleinman has responded

    Replies to this message:
     Message 144 by Kleinman, posted 06-23-2020 2:53 PM Phat has not yet responded

      
    Kleinman
    Member
    Posts: 386
    From: United States
    Joined: 10-06-2016
    Member Rating: 1.1


    Message 144 of 239 (877937)
    06-23-2020 2:53 PM
    Reply to: Message 143 by Phat
    06-23-2020 1:59 PM


    Re: Does competition accelerate DNA evolution?
    Kleinman writes:

    The reason his model won't work is he is using an incorrect assumption. I do not make the assumption that he makes and have rewritten the transition matrix so you can use a Markov chain model to correctly predict DNA evolution. I'm just finishing the paper and will be submitting it for publication by the end of summer.

    Phat writes:

    Now be honest with me. Do you see yourself as a man gifted somewhat in mathematics who is a proponent of Intelligent Design? If so, have you managed to get the attention of the Discovery Institute at all? My advice is to stay humble and let your opponents become arrogant rather than you yourself. If you are in fact representing Intelligent Design, the Intelligence behind that design---the same intelligence that you likely thank for giving you math skills, wants you to stay humble.


    I would call myself a good applied mathematician. And I think it is a gift. What that means is that I've been able to apply mathematical principles to physical situations. That applies not only to evolution but I've also published mathematical solutions to complex thermodynamic problems. With regards to the Discovery Institute and Intelligent Design, I am not a part of that group and I am not a proponent of that theory but I do recognize some validity to their arguments. My interest in evolution is based on having a correct understanding of how this phenomenon works both from a professional aspect (I have to deal with drug-resistant infections in my medical practice) and also from a spiritual aspect (is abiogenesis and the theory of evolution true or were we created). Normally, I don't discuss Creationism because that is a faith-based subject. What I can show scientifically is that evolution didn't do it and you can see by the response of some posters that they want to turn this into a faith-based discussion. In a way, I can see why they do this because they can't demonstrate that evolution did it. I should point out that in my medical practice, MRSA infections are common-place and I use the principles I'm discussing here quite successfully to treat these infections. That's one of the many reasons why I press these points so hard.

    I suspect you are familiar with Proverbs 27:17, so I will take your comment about being humble to heart.


    This message is a reply to:
     Message 143 by Phat, posted 06-23-2020 1:59 PM Phat has not yet responded

      
    Taq
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    Posts: 8409
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    Member Rating: 1.9


    Message 145 of 239 (877941)
    06-23-2020 3:17 PM
    Reply to: Message 142 by Kleinman
    06-23-2020 12:53 PM


    Re: Does competition accelerate DNA evolution?
    Kleinman writes:

    But if you have selection at many different loci simultaneously (lots of different cards in the deck) with each variant being at low frequency, your probability of getting the correct recombination events drops off very rapidly.

    Combining two beneficial loci will result in even higher fitness and increase the presence of both alleles.

    Sure, lots of non-A and non-B cards.

    Now you pretend that mutation rate doesn't matter.

    But if you think that recombination solves your problem and you don't need DNA evolution, show us how you breed dogs and get cats.

    I never said that DNA evolution isn't needed. What I have continually pointed out is that the specific conditions and outcomes of your pet experiments are not applicable to all situations. It isn't that hard to understand.

    How do you get different breeds of dogs and cats? SEXUAL RECOMBINATION!!!!! For crying out loud. You select dogs from different populations that have 1 or 2 of the features you want. You breed those dogs together to get all the features you want. You can get black fur from one, webbed toes from another, herding instinct from another, long snout from another, curled tail from another . . . mix them altogether, select the traits you want through controlled breeding, and what you have is a new breed.


    This message is a reply to:
     Message 142 by Kleinman, posted 06-23-2020 12:53 PM Kleinman has responded

    Replies to this message:
     Message 146 by Kleinman, posted 06-23-2020 3:44 PM Taq has responded

      
    Kleinman
    Member
    Posts: 386
    From: United States
    Joined: 10-06-2016
    Member Rating: 1.1


    Message 146 of 239 (877946)
    06-23-2020 3:44 PM
    Reply to: Message 145 by Taq
    06-23-2020 3:17 PM


    Re: Does competition accelerate DNA evolution?
    Kleinman writes:

    But if you have selection at many different loci simultaneously (lots of different cards in the deck) with each variant being at low frequency, your probability of getting the correct recombination events drops off very rapidly.

    Taq writes:

    Combining two beneficial loci will result in even higher fitness and increase the presence of both alleles.


    Sure, if it can happen. I've shown you how to compute that probability with only two beneficial alleles. What do you think will happen if you are trying for three loci? Have you ever played poker? What do you think gives you a better chance when drawing for a flush? Do you think you have a better chance when you hold 4 hearts and you need 1 more? How about if you hold 3 hearts and you need to draw 2 hearts. Or you hold only 2 hearts and have to draw 3 hearts?
    Kleinman writes:

    Sure, lots of non-A and non-B cards.

    Taq writes:

    Now you pretend that mutation rate doesn't matter.


    When it comes to DNA evolution, mutation rate doesn't matter much, its the multiplication rule that drives the process. And I also think that most mutations are neutral or detrimental. A and B (beneficial mutations) are not very common. That's why it takes 3e9 replications for each occurrence.
    Kleinman writes:

    But if you think that recombination solves your problem and you don't need DNA evolution, show us how you breed dogs and get cats.

    Taq writes:

    I never said that DNA evolution isn't needed. What I have continually pointed out is that the specific conditions and outcomes of your pet experiments are not applicable to all situations. It isn't that hard to understand.


    Ok, take a reptile population, start a breeding program, and produce a bird.
    Taq writes:

    How do you get different breeds of dogs and cats? SEXUAL RECOMBINATION!!!!! For crying out loud. You select dogs from different populations that have 1 or 2 of the features you want. You breed those dogs together to get all the features you want. You can get black fur from one, webbed toes from another, herding instinct from another, long snout from another, curled tail from another . . . mix them altogether, select the traits you want through controlled breeding, and what you have is a new breed.


    Sure, you can get significant change in phenotype with a breeding program. But you will always be limited by the alleles available in the gene pool. And recombination with selection reduces the diversity of the members of that population. Any new alleles must be produced by mutations and that is DNA evolution. That's why you can't produce cats from dogs or birds from reptiles using a breeding program. Recombination without error does not produce new alleles, it can only change the expression of existing alleles (phenotype).

    This message is a reply to:
     Message 145 by Taq, posted 06-23-2020 3:17 PM Taq has responded

    Replies to this message:
     Message 147 by Taq, posted 06-25-2020 12:17 PM Kleinman has responded

      
    Taq
    Member
    Posts: 8409
    Joined: 03-06-2009
    Member Rating: 1.9


    Message 147 of 239 (878058)
    06-25-2020 12:17 PM
    Reply to: Message 146 by Kleinman
    06-23-2020 3:44 PM


    Re: Does competition accelerate DNA evolution?
    Kleinman writes:

    Sure, if it can happen. I've shown you how to compute that probability with only two beneficial alleles. What do you think will happen if you are trying for three loci? Have you ever played poker?

    Have you ever played bridge where you get to combine the cards from two hands?

    Until you admit that sexual recombination exists there really isn't anything more to say. Ignoring the most basic biology gets you nowhere.


    This message is a reply to:
     Message 146 by Kleinman, posted 06-23-2020 3:44 PM Kleinman has responded

    Replies to this message:
     Message 149 by Kleinman, posted 06-25-2020 12:51 PM Taq has responded

      
    AZPaul3
    Member
    Posts: 5118
    From: Phoenix
    Joined: 11-06-2006
    Member Rating: 3.1


    Message 148 of 239 (878063)
    06-25-2020 12:44 PM


    What is it a dog or a cat?

    Micro is ok but no macro.

    Miller-Urey proved you can’t get life from non-life.

    Ok, take a reptile population, start a breeding program, and produce a bird.

    Why are there still monkeys?

    --------------------------------------------

    Nothing new here, Citizens. Move along.


    Factio Republicana delenda est.
    I am antifa.

      
    Kleinman
    Member
    Posts: 386
    From: United States
    Joined: 10-06-2016
    Member Rating: 1.1


    Message 149 of 239 (878065)
    06-25-2020 12:51 PM
    Reply to: Message 147 by Taq
    06-25-2020 12:17 PM


    Re: Does competition accelerate DNA evolution?
    Kleinman writes:

    Sure, if it can happen. I've shown you how to compute that probability with only two beneficial alleles. What do you think will happen if you are trying for three loci? Have you ever played poker?

    Taq writes:

    Have you ever played bridge where you get to combine the cards from two hands?

    Until you admit that sexual recombination exists there really isn't anything more to say. Ignoring the most basic biology gets you nowhere.


    Sorry, I don't know how I missed this post. Yes, I've played bridge, many years ago. And do you think that combining hands gives you a grand slam every time? Sometimes you get a yarborough.

    And don't be silly, not only do I admit that sexually recombination exists, I've given you the mathematics which describes it. And you aren't going to breed cats from dogs by this process, even if you hold all the trump cards.


    This message is a reply to:
     Message 147 by Taq, posted 06-25-2020 12:17 PM Taq has responded

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    Kleinman
    Member
    Posts: 386
    From: United States
    Joined: 10-06-2016
    Member Rating: 1.1


    Message 150 of 239 (878068)
    06-25-2020 1:06 PM


    The mathematics of DNA evolution
    dwise1 writes:

    Even then, you would still be arguing through bare links which is against forum rules! You need to explain it in your own words using links and quotes from links for support, NOT as a substitute!

    Damn you sleazy creationists! You just keep getting worse and worse.


    This quote is from the "Should we teach both evolution and religion in school thread?" What dwise1 is complaining about is that I asked him to explain Felsenstein's DNA evolution model and apply it to the Kishony experiment and predict its behavior. The Felsenstein DNA model is described on the Wikipedia page:
    F81 model (Felsenstein 1981)
    That model is a variation of the original Jukes-Cantor model of DNA evolution which is described earlier on the page:
    JC69 model (Jukes and Cantor 1969)
    These are Markov Chain models which are a form of random walk algorithms.

    If you are not familiar with these kinds of models, the Wikipedia page will be totally confusing but with a little preparation, these models are very easy to understand and calculate. The biggest problem with these kinds of models is that they can take massive amounts of calculations and huge amounts of computer time to evaluate. The biggest problem with the models on this Wikipedia page is that they will not work for describing the Kishony experiment. The problem occurs with an assumption being made in formulating the transitions matrix. Before I explain where the error is though, I'm going to show you how to do the mathematics of DNA evolution using basic principles of probability theory. This mathematics is described in detail here:
    The basic science and mathematics of random mutation and natural selection
    When I wrote this paper, the Kishony experiment had not yet been performed. I derived the governing equations based on the Weinreich studies "Darwinian Evolution Can Follow. Only Very Few Mutational Paths to Fitter Proteins". But now that we have a very visual Kishony experiment, I will go through the derivation based on that experiment.

    Here is a typical video of that experiment:
    The Evolution of Bacteria on a “Mega-Plate” Petri Dish (Kishony Lab)

    To start the derivation, we will make a couple of simplifying assumptions. First, assume that there is only one beneficial mutation to the drug. Don't worry we can debate the validity of that assumption and how much it changes DNA evolution. Second, unlike in the paper, I linked to above, the population size does not remain constant over generations. In reality, the populations grow exponentially over generations in the Kishony experiment but it is not the rate at which populations grow that determine the probabilities of the beneficial mutation to occur, it is the total number of replications of the particular variant. The third assumption we make is that the mutation rate is constant.

    So, Kishony starts the experiment by inoculating "wild-type" drug-sensitive bacteria on the drug-free region of his plate. Assume he inoculates with a single drug-sensitive bacterium. The first thing we must consider if there are more than one possible mutation that can occur at the given site this is written mathematically as:

    P(−∞ < X < +∞) = P(Ad)+P(Cy)+P(Gu)+P(Th)+P(iAd)+P(iCy)+P(iGu)+P(iTh)+P(del)+...=1

    Where Ad, Cy, Gu, and Th are the possible bases that can occur the prefix "i" means insertion of the particular base, "del" means deletion of a base, and the "..." means any other possible mutation that you can imagine. One of those possible mutations is the beneficial mutation and we assign that probability to the term P(BeneficialA). Then, the probability that the beneficial mutation (call it "A") occurs in a single replication is written:

    P(A) = P(BeneficialA)μ

    Any problem?

    Edited by Kleinman, : line clipped off


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