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Author | Topic: Do you really understand the mathematics of evolution? | |||||||||||||||||||||||
Taq Member Posts: 10038 Joined: Member Rating: 5.3 |
In the single selection pressure case, the evolutionary steps are independent. Each step is a new binomial probability problem independent of the previous step and a new sample space occurs for each step. You can see this in the Kishony experiment, mutations A1 and A2 occur in separate drug-concentration regions and these regions correspond to the mathematical sample spaces. I was under the impression that you were combining the antibiotics into one region which would require adaptations to both drugs in order to adapt to the single new region. Is this not the case? This would mean that even if we get a mutation for resistance against one of the drugs, that mutation won't be selected for.
You can do that but then you are not talking about evolution. You are talking about migration. How is the genetic variation within a population not a factor in evolution?
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Kleinman Member (Idle past 356 days) Posts: 2142 From: United States Joined: |
Kleinman writes:
Your impression is correct and the same mathematics for DNA evolution to 2 drugs applies for the case if a single drug is used but the step increase between regions requires 2 mutations for adaptation. And you are correct, in either case, if a member of the population has only one of the two mutations necessary for growth in the next higher concentration region when two correct mutations are required, that member with only a single mutation would be selected out.
In the single selection pressure case, the evolutionary steps are independent. Each step is a new binomial probability problem independent of the previous step and a new sample space occurs for each step. You can see this in the Kishony experiment, mutations A1 and A2 occur in separate drug-concentration regions and these regions correspond to the mathematical sample spaces.Taq writes: I was under the impression that you were combining the antibiotics into one region which would require adaptations to both drugs in order to adapt to the single new region. Is this not the case? This would mean that even if we get a mutation for resistance against one of the drugs, that mutation won't be selected for.Kleinman writes:
If you want to talk about the genetic variation of the entire world-wide population of e coli, you have to do that in the context of the different environments that the different populations are evolving. I don't know how large that world-wide population is but for the sake of discussion, assume it is 1e20. And let's assume that these bacteria are growing in a vast idealized environment such as Kishony's drug-free region. Then, using the mathematics I've presented, you can calculate the diversity of that population if it was a vast single colony as a function of the mutation rate. But that vast idealized environment doesn't exist. In the real world, there are many selection pressures such as thermal stress, starvation, toxins, dehydration, competition from other replicators, predation, etc. These selection pressures reduce the diversity of populations, reducing or eliminating those variants that don't have sufficient reproductive fitness. You can do that but then you are not talking about evolution. You are talking about migration.Taq writes: How is the genetic variation within a population not a factor in evolution? So, bringing a drug-resistant variant of e coli will change the behavior of the Kishony experiment for that particular drug but it doesn't change the fundamental physics and mathematics of DNA evolution.
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Taq Member Posts: 10038 Joined: Member Rating: 5.3 |
Kleinman writes: If you want to talk about the genetic variation of the entire world-wide population of e coli, you have to do that in the context of the different environments that the different populations are evolving. We could talk about the population of E. coli in a single person's gut, if we wanted to. Would you agree that the genetic variation of the E. coli population in your gut is probably greater than that used in the Kishony experiment? Doesn't the starting genetic variation of a population affect how that population evolves, and how the math of population genetics applies to it?
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Kleinman Member (Idle past 356 days) Posts: 2142 From: United States Joined: |
Kleinman writes:
You are correct, the carrying capacity of our gut is greater than the Kishony experiment. What that means is that there is a high probability of drug-resistant variants already in that environment. The way to correctly address this was described by Edward Tatum in his 1958 Nobel Laureate lecture: If you want to talk about the genetic variation of the entire world-wide population of e coli, you have to do that in the context of the different environments that the different populations are evolving.Taq writes: We could talk about the population of E. coli in a single person's gut, if we wanted to. Would you agree that the genetic variation of the E. coli population in your gut is probably greater than that used in the Kishony experiment? Doesn't the starting genetic variation of a population affect how that population evolves, and how the math of population genetics applies to it?Edward Tatum – Nobel Lecture - NobelPrize.org Edward Tatum writes:
For many years, the standard of care taught in medical schools has been the use of single-drug therapy for the treatment of infectious diseases. In most cases, this works ok if the patient being treated has a good functioning immune system which removes any of the resistant variants that the antibiotic does not. But, if the patient has a poorly function immune system, single-drug therapy, especially at low doses is the formula for selecting for drug-resistant variants and treatment failure.
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.
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Taq Member Posts: 10038 Joined: Member Rating: 5.3 |
Kleinman writes: You are correct, the carrying capacity of our gut is greater than the Kishony experiment. What that means is that there is a high probability of drug-resistant variants already in that environment. We also have to be careful not to fall victim to the Sharpshooter fallacy. It is entirely possible for a new mutation to be neutral in one genetic background and beneficial in another genetic background. This beneficial phenotype would be dependent on two mutations. So how many neutral mutations can become beneficial mutations when combined with new mutations? I don't think we can really know this number for any genome. If there are millions of possible beneficial interactions, then it isn't surprising that a beneficial phenotype emerges that requires two mutations, one of which is neutral all by itself. It would be incorrect to draw a bulls eye around this phenotype and then claim that it is highly improbable that such a trait emerged.
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Kleinman Member (Idle past 356 days) Posts: 2142 From: United States Joined: |
Kleinman writes:
What you are missing is that with a mutation rate of 1e-9 that with 3e9 replications that you have shot at every target possible and hit each target on average once somewhere in the population. And what that means, for example, in the Kishony experiment that you are going to have some member of that population with a beneficial mutation for the Ciprofloxacin environment and a different member of the population with a beneficial mutation for trimethoprim. What determines whether either of these mutations are beneficial is the environment in which that variant is trying to grow.
You are correct, the carrying capacity of our gut is greater than the Kishony experiment. What that means is that there is a high probability of drug-resistant variants already in that environment.Taq writes: We also have to be careful not to fall victim to the Sharpshooter fallacy. It is entirely possible for a new mutation to be neutral in one genetic background and beneficial in another genetic background. This beneficial phenotype would be dependent on two mutations.Taq writes:
Sure you can know. Again, using the Kishony experiment as the example, we know that in that population of 3e9 that there about 13.8 million variants. The vast majority of those variants cannot grow in the regions where there are drugs. So the vast majority of mutations will be neutral or detrimental. To determine how many beneficial mutations there are would require genetic sequencing. Here is a paper where describes this: So how many neutral mutations can become beneficial mutations when combined with new mutations? I don't think we can really know this number for any genome. If there are millions of possible beneficial interactions, then it isn't surprising that a beneficial phenotype emerges that requires two mutations, one of which is neutral all by itself. It would be incorrect to draw a bulls eye around this phenotype and then claim that it is highly improbable that such a trait emerged.https://www.brown.edu/...Publications/Weinreich_etal2006.pdf Note that Weinreich makes an error in the fundamental mathematics of DNA evolution (which was not recognized by the peer-reviewers) by stating that each step on the evolutionary trajectory requires fixation Weinreich writes:
The Kishony experiment shows that claim is incorrect.
Thus, the relative probability of realizing any particular mutational trajectory is the product of the relative probabilities of its constituent mutations, because under our assumption the choice of each subsequent fixation is statistically independent of all previous fixations (12).
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Taq Member Posts: 10038 Joined: Member Rating: 5.3 |
Kleinman writes: What you are missing is that with a mutation rate of 1e-9 that with 3e9 replications that you have shot at every target possible and hit each target on average once somewhere in the population. And what that means, for example, in the Kishony experiment that you are going to have some member of that population with a beneficial mutation for the Ciprofloxacin environment and a different member of the population with a beneficial mutation for trimethoprim. What determines whether either of these mutations are beneficial is the environment in which that variant is trying to grow. There is a universe that exists outside of the Kishony experiment.
Sure you can know. Again, using the Kishony experiment as the example, we know that in that population of 3e9 that there about 13.8 million variants. The vast majority of those variants cannot grow in the regions where there are drugs. Antibiotic resistance isn't the only beneficial adaptation that exists in the universe. I would also suspect that there are examples of antibiotic resistance where there are multiple mutated bases that can give rise to the same phenotype. I would also agree that a mutation doesn't have to reach fixation in order to interact with new mutations.
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Kleinman Member (Idle past 356 days) Posts: 2142 From: United States Joined: |
Kleinman writes:
Do you think that DNA evolution works differently outside of the Kishony experiment? Outside of the Kishony experiment, there a multiple simultaneous selection pressures. Do you think that DNA evolution works more efficiently under those circumstances? Even in the simple Kishony experiment, it takes exponentially more replications for adaptation to occur if 2 drugs are used and likewise if the increase in drug concentration is too large between bands, the population is facing the same mathematical constraints. So what, elsewhere in the universe changes these mathematical facts of life?
What you are missing is that with a mutation rate of 1e-9 that with 3e9 replications that you have shot at every target possible and hit each target on average once somewhere in the population. And what that means, for example, in the Kishony experiment that you are going to have some member of that population with a beneficial mutation for the Ciprofloxacin environment and a different member of the population with a beneficial mutation for trimethoprim. What determines whether either of these mutations are beneficial is the environment in which that variant is trying to grow.Taq writes: There is a universe that exists outside of the Kishony experiment.Kleinman writes:
It is quite likely that in that 13.8 million variants in the Kishony experiment, there is a variant that would have improved fitness for the Lenski experiment. And sure, you can have different variants that achieve resistance to a given selection pressure. These different variants will have the same phenotype but each of these variants will have taken their own particular evolutionary trajectory to achieve their particular genotype. That is what the Weinreich paper is all about. And each step for each of these different evolutionary trajectories will require about 3e9 replications.
Sure you can know. Again, using the Kishony experiment as the example, we know that in that population of 3e9 that there about 13.8 million variants. The vast majority of those variants cannot grow in the regions where there are drugs.Taq writes: Antibiotic resistance isn't the only beneficial adaptation that exists in the universe. I would also suspect that there are examples of antibiotic resistance where there are multiple mutated bases that can give rise to the same phenotype.Taq writes:
You do understand that if the different variants in a population are forced to compete in a limited carrying capacity environment will slow DNA evolution?
I would also agree that a mutation doesn't have to reach fixation in order to interact with new mutations.
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Kleinman Member (Idle past 356 days) Posts: 2142 From: United States Joined: |
Does competition and fixation accelerate DNA evolution? If it does, why do the Lenski team say this:
Just a moment... When large asexual populations adapt, competition between simultaneously segregating mutations slows the rate of adaptation and restricts the set of mutations that eventually fix.
And what is wrong with what they are saying here?
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Taq Member Posts: 10038 Joined: Member Rating: 5.3 |
Kleinman writes: Do you think that DNA evolution works differently outside of the Kishony experiment? Not all adaptations are the same.
Outside of the Kishony experiment, there a multiple simultaneous selection pressures. Do you think that DNA evolution works more efficiently under those circumstances? I think it is relatively rare for there to be only a single substitution mutation within the entire genome that will confer increased fitness in a given environment.
You do understand that if the different variants in a population are forced to compete in a limited carrying capacity environment will slow DNA evolution? I understand that just fine. Evolution is very Malthusian in that there will be winners and losers, and this is true of neutral mutations as well.
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Taq Member Posts: 10038 Joined: Member Rating: 5.3 |
Kleinman writes: Does competition and fixation accelerate DNA evolution? That needs context. If we are talking about positive or negative selection, then competition does change the rate of fixation for those mutations under selection.
If it does, why do the Lenski team say this: You tell us.
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Kleinman Member (Idle past 356 days) Posts: 2142 From: United States Joined: |
Kleinman writes:
What is the mathematical difference between different adaptations? Does DNA evolution work differently between the Kishony and Lenski experiments?
Do you think that DNA evolution works differently outside of the Kishony experiment?Taq writes: Not all adaptations are the same.Kleinman writes:
You still haven't mastered the mathematics for DNA evolution to a single selection pressure and only a single beneficial mutation that improves fitness. If you think have mastered that math, tell us how the math changes if there are two or more possible beneficial mutations which give improved fitness to a given selection pressure.
Outside of the Kishony experiment, there a multiple simultaneous selection pressures. Do you think that DNA evolution works more efficiently under those circumstances?Taq writes: I think it is relatively rare for there to be only a single substitution mutation within the entire genome that will confer increased fitness in a given environment.Kleinman writes:
Do you understand that fine enough to explain it mathematically? To make that question more specific, how do carrying capacity, selection conditions, and mutation rates affect the DNA evolution mathematical behavior of the Lenski experiment?
You do understand that if the different variants in a population are forced to compete in a limited carrying capacity environment will slow DNA evolution?Taq writes: I understand that just fine. Evolution is very Malthusian in that there will be winners and losers, and this is true of neutral mutations as well.Kleinman writes:
You are conflating two different physical phenomena. Competition and fixation and DNA evolution are distinctly different phenomena. And you as well as Lenski don't understand that. That is why you cannot put evolution in the correct mathematical context.
Does competition and fixation accelerate DNA evolution?Taq writes: That needs context. If we are talking about positive or negative selection, then competition does change the rate of fixation for those mutations under selection.Kleinman writes:
Let's put my question in the correct context by posting the entire quote:
If it does, why do the Lenski team say this:Taq writes: You tell us.Kleinman writes:
You should understand what is wrong with Lenski's statement. Start with the first five words of the sentence, "When large asexual populations adapt". Are Lenski's populations large? You have already pointed out that our gut has even larger populations. And your calculation for adaptation for the Kishony experiment requires 3e9 replications. So, now try to explain why his populations adapt so slowly. Show your math.
Does competition and fixation accelerate DNA evolution? If it does, why do the Lenski team say this:Just a moment... Lenski Team writes: When large asexual populations adapt, competition between simultaneously segregating mutations slows the rate of adaptation and restricts the set of mutations that eventually fix.
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Taq Member Posts: 10038 Joined: Member Rating: 5.3 |
Kleinman writes: What is the mathematical difference between different adaptations? The number of possible beneficial mutations would be a good start.
You still haven't mastered the mathematics for DNA evolution to a single selection pressure and only a single beneficial mutation that improves fitness. If you think have mastered that math, tell us how the math changes if there are two or more possible beneficial mutations which give improved fitness to a given selection pressure. If there are more possible beneficial mutations then you need fewer divisions in order to see an increase in fitness.
Do you understand that fine enough to explain it mathematically? To make that question more specific, how do carrying capacity, selection conditions, and mutation rates affect the DNA evolution mathematical behavior of the Lenski experiment? Why don't you tell us?
Competition and fixation and DNA evolution are distinctly different phenomena. Say what? Why do we see sequence conservation within exons when we compare genomes between species, but a lack of sequence conservation in introns?
So, now try to explain why his populations adapt so slowly. Show your math. You first.
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Kleinman Member (Idle past 356 days) Posts: 2142 From: United States Joined: |
Kleinman writes:
Do the math for the simplest case, assume there are only 2 possible beneficial mutations.
What is the mathematical difference between different adaptations?Taq writes: The number of possible beneficial mutations would be a good start.Kleinman writes:
That's the kind of vague answer I would expect from nwr and AZPaul3. Do the math that predicts how many fewer divisions in order to see an increase in fitness.
You still haven't mastered the mathematics for DNA evolution to a single selection pressure and only a single beneficial mutation that improves fitness. If you think have mastered that math, tell us how the math changes if there are two or more possible beneficial mutations which give improved fitness to a given selection pressure.Taq writes: If there are more possible beneficial mutations then you need fewer divisions in order to see an increase in fitness.Kleinman writes:
Why don't you try first? I'll even give you a hint on how to do the math. Consider the case of 2 possible beneficial mutations. What is the probability of a beneficial mutation occurring at least once at those possible sites?
Do you understand that fine enough to explain it mathematically? To make that question more specific, how do carrying capacity, selection conditions, and mutation rates affect the DNA evolution mathematical behavior of the Lenski experiment?Taq writes: Why don't you tell us?Kleinman writes:
You probably won't understand this but competition and fixation is a first law of thermodynamics process and DNA evolution is a second law of thermodynamics process. And I don't know what you are seeing with introns and exons. You barely understand the basic principles of DNA evolution to single selection pressure.
Competition and fixation and DNA evolution are distinctly different phenomena.Taq writes: Say what? Why do we see sequence conservation within exons when we compare genomes between species, but a lack of sequence conservation in introns?Kleinman writes:
I've already published the math. You can find it here: So, now try to explain why his populations adapt so slowly. Show your math.Taq writes: You first.Just a moment... And you should understand why DNA evolution is slowed in a competitive environment. You have already shown that it takes 3e9 replications for a beneficial mutation to occur. So, when you have a populations such as Lenski's populations where many variants are competing for a fixed amount of glucose, that will limit the number of replications for all variants. Then, the most fit variant must drive to extinction the less fit variants in order to have sufficient resources for that most fit variant in that particular lineage to accumulate its 3e9 replications for the next beneficial mutation. Now try to do the math for two or more possible beneficial mutations and learn why this has very minimal effect on the DNA evolution process.
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Taq Member Posts: 10038 Joined: Member Rating: 5.3 |
Kleinman writes: Do the math for the simplest case, assume there are only 2 possible beneficial mutations. It would take half as many divisions.
Why don't you try first? It would be nice to see some reciprocation.
You probably won't understand this but competition and fixation is a first law of thermodynamics process and DNA evolution is a second law of thermodynamics process. All of biology boils down to thermodynamics, as do all physical processes. In a simplistic model, energy flows from the Sun to photsynthesizers to herbivores to carnivores. There is about 10% energy transfer at each trophic level. The total energy is limited in our solar system, and energy can't increase in our isolated solar system. Imperfect replicators competing for limited resources is what drives evolution.
And I don't know what you are seeing with introns and exons. You barely understand the basic principles of DNA evolution to single selection pressure. When you compare functional genes between species you will see fewer differences between exons than you will introns. I understand selective pressures just fine.
And you should understand why DNA evolution is slowed in a competitive environment. "Slower" is a relative term. What are you comparing to? Are you comparing it to a population that increases exponential towards infinity?
You have already shown that it takes 3e9 replications for a beneficial mutation to occur. So, when you have a populations such as Lenski's populations where many variants are competing for a fixed amount of glucose, that will limit the number of replications for all variants. The number of replications is the same per culture because they are observed to reach the same density, and are started from the same number of bacteria.
Then, the most fit variant must drive to extinction the less fit variants in order to have sufficient resources for that most fit variant in that particular lineage to accumulate its 3e9 replications for the next beneficial mutation. You are assuming that the mutations have to come in a specific order. If there are multiple beneficial mutations then you can have a mix of those beneficial mutations in the population simultaneously.
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