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Author | Topic: Do you really understand the mathematics of evolution? | ||||||||||||||||||||||||
Kleinman Member (Idle past 365 days) Posts: 2142 From: United States Joined: |
Kleinman writes:
Then, explain to us how epistasis affects the mathematics of either the Kishony or Lenski experiments. And show your math if you think you understand the mathematics of evolution.
You are conflating two concepts, the creation of new alleles (DNA evolution) and the expression of any one allele (epistasis).Taq writes: I never confused those. Epistasis is a description of how mutations and alleles interact with one another to give rise to a phenotype.Kleinman writes:
And the more complex the fitness landscape, the more complex any evolutionary trajectory. And if you want to see a bottleneck in a population, have a big change in the environment, that is if the population is not driven to extinction. Big environmental changes are far more difficult for populations to adapt to. Even the Kishony experiment demonstrates this. If the step change in antibiotic concentration is too large, the experiment does not work. That's because 2 or more mutations are needed to give the necessary improvement in fitness. DNA evolution works most efficiently when only a single mutation gives improved fitness in the given environment. The reason for this is the multiplication rule which requires exponentially more replications for that evolutionary process to have a reasonable probability of occurring.
But even if they did, why would you think that it would take fewer than a billion replications for each evolutionary step to evolve and adapt to the antibiotic selection pressure?Taq writes: That's not the point. What we are saying is that big changes in environment can lead to more evolution. This is due to fitness landscapes.Kleinman writes:
I didn't say the beneficial mutations have to occur at the same time. But the multiplication rule still applies no matter when the beneficial mutations occur. What improves the probability of this happening is the ability of the particular variant's ability to replicate. The key point that you are missing here is that all these beneficial mutation somehow have to end up in some common lineage for this lineage to be adapted to all these selection pressures.Taq writes: However, they don't have to be acquired all at the same time. In fact, different lineages can take different pathways, such as one lineage adapting to temperature first, followed by salt concentration. A different lineage could adapt to salt concentration first, then temperature. There could also be multiple different mutational pathways for each environmental challenge, such as bats and birds having different adaptations for flight. Let's use your slot-machine analogy. Let's say there are two possible jackpots, a super jackpot, and a big jackpot. And let the probability of winning the super jackpot is one in ten million and the probability of winning the big jackpot is one in a million with each pull of the arm. Do you understand how to compute the probability of winning both of those jackpots as a function of the number of pulls on the arm? It's essentially the same math as the Kishony experiment if he were to use two drugs. And you are still having difficulty understanding that every evolutionary pathway must obey this math. Kishony's bacteria may have multiple different sets of mutations that can give resistance to the drug used but it still is going to take a billion replications for each evolutionary step on each of these trajectories. These are the mathematical facts of life of DNA evolution.
Kleinman writes:
You don't need to do that. Simply take the bacteria that has already adapted to ciprofloxacin and use those as the starter population for the trimethoprim experiment. All you are demonstrating is that DNA evolution works most efficiently a single selection pressure at a time. That's how MRSA was created except in the clinical environment. When one drug failed, go on to the next drug until that drug fails. If you are going to gamble with microbes, you should understand the rules of the game.
But when Kishony runs his experiment with two drugs, only when some member of the population has a beneficial mutation for both ciprofloxacin and for trimethoprim will that member be able to grow in the next higher drug concentration region.Taq writes: If the experiment were set up with different regions of the plate having different antibiotics then you could have lineages adapting to one drug and then the other.Kleinman writes:
The limited carrying capacity environment in the Lenski forces his populations to compete for the limited glucose. If Lenski were to run his experiment in 100ml of solution instead of 10ml, the more fit variant will achieve enough replications to have a reasonable probability of another beneficial mutation occurring before the less fit variants are driven to extinction, fixation is not necessary for a beneficial mutation to occur in this case. What you should try to understand is that the human body has a huge carrying capacity for microbes. The population size of these microbes easily achieves the numbers necessary for drug-resistant variants to occur. In other words, the Kishony experiment would not work in a standard-sized petri dish, you need very large populations for DNA evolution to work.
And fixation is not required for DNA evolution to occur.Taq writes: Then why did you say the following in a previous post? "What this does is it forces his populations to compete for the limited resources and the more fit variant must drive the less fit variant to extinction in order to accumulate the necessary replications for improving fitness."Kleinman writes:
Not so. Lenski has run experiments using thermal stress for his selection condition. You are speculating.Taq writes: You are also speculating when you say that adaptations to temperature and salt concentrations must be different mutations.https://courses.pbsci.ucsc.edu/...i_&_Bennett_AmNat_1993.pdf
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Kleinman Member (Idle past 365 days) Posts: 2142 From: United States Joined: |
Kleinman writes:
The reason this is the case is that the carrying capacity in the drug-free region is not large enough to accommodate both the wild-type and the drug-resistant variants multiplication. I tried to explain this in an earlier post but I'm just not getting through to you. That's why when Kishony runs his experiment with ciprofloxacin, any variant with a mutation that might be beneficial for trimethoprim (or any other class of antibiotics for that matter) cannot successfully evolve on his plate.Taq writes: I don't see why this would be the case. If resistance to cipro is gained by a single substitution mutation then it wouldn't matter if the founding population also has a single substitution for trimethoprim resistance. It is just a matter of having enough divisions to get that cipro mutation. If you took a cipro resistant colony and grew it up in large numbers on trimethoprim plates you would have the same chances of getting that mutant as you would with a cipro sensitive clone. Try and understand it this way. Imagine you have a vast, unlimited size petri dish with infinite carrying capacity and no antibiotics. You start with a single wild-type bacterium that is sensitive to all antibiotics. This bacterium doubles, the resultant bacteria double, and on and on with members accumulating mutations at the frequency of the mutation rate. After 30 doublings, you will have a billion bacteria and for a mutation rate of e-9, you will have on average, some member with a mutation at every site in the genome. That means there will be about 1 member with a beneficial mutation for ciprofloxacin, another member with a beneficial mutation for trimethoprim but the rest of the population will have mutations at other sites or no mutations at all. Then this population continues to double for 30 more generations. You will then have a billion members with the ciprofloxacin mutation (plus any other mutations accumulated including one for the trimethoprim), a billion members with the trimethoprim mutation (plus any other mutations including one for the ciprofloxacin), and for the remaining population of a billion minus the ciprofloxacin and trimethoprim variants, you will have a quintillion members (minus a few members along the way that get either the ciprofloxacin or trimethoprim mutations). The bands in the Kishony experiment give a region where the drug-resistant variants can grow without having to compete for the depleted resources in the drug-free region. Here's a Venn Diagram of the system:
Kleinman writes:
That answers that, you don't understand the mathematics of evolution so you snip out the part highlighted in red.
Then, explain to us how epistasis affects the mathematics of either the Kishony or Lenski experiments. And show your math if you think you understand the mathematics of evolution.Taq writes: The genetic background of the population will determine the probability of arriving at a beneficial mutation.Kleinman writes:
What you refuse to accept is that the steps in every evolutionary trajectory are joint by the multiplication rule. And if it takes multiple mutations to improve fitness in an evolutionary step, you have multiple instances of the multiplication rule acting simultaneously. That's why the Kishony experiment won't work on his large petri when two drugs are used or when the step increase in drug-concentration is so large that it requires two or more mutations for that step.
And the more complex the fitness landscape, the more complex any evolutionary trajectory.Taq writes: A complex landscape can afford multiple evolutionary pathways towards increased fitness, but each pathway is as simple as it would be in a simple landscape. If you want to define complexity as the number of possible evolutionary pathways then I would agree.Kleinman writes:
You can't explain mathematical behavior of the Kishony or Lenski experiments so now you start storytelling. Why does it take a billion replications for every evolutionary step in the Kishony experiment? Please show your math.
Big environmental changes are far more difficult for populations to adapt to.Taq writes: That depends on a lot of factors. When the first limbed fish started pushing themselves onto land they had very little competition from other land animals. However, a poorly adapted fish will have a very tough time moving onto land now since there are a lot of well adapted tetrapods filling those niches. The K/T meteor impact produced a massive change in environment, and what resulted was the radiation of mammals and birds, one of the biggest surges in evolutionary change in the fossil record.Kleinman writes:
I havn't ignored that at all. It doesn't matter which evolutionary trajectory that a lineage takes. It doesn't change the math.
DNA evolution works most efficiently when only a single mutation gives improved fitness in the given environment. The reason for this is the multiplication rule which requires exponentially more replications for that evolutionary process to have a reasonable probability of occurring.Taq writes: What you seem to be ignoring is that there is often more than one mutation that is beneficial, and evolution can search for those solutions in parallel. It's not as if it has to be one mutation at a time. Epistasis also demonstrates that neutral mutations can become beneficial at a future time point as they interact with new mutations and new environments.Kleinman writes:
That math will work if the mutation rate for your first case is 5e-8 and 5e-7 for you second case. For the Kishony and Lenski experiments, the beneficial mutation closer to about e-9. But the mutation rate is not the major driving factory in DNA evolution, it is the number of selection pressures the population is adapting to because each selection pressure introduces another instance of the multiplication rule in the evolutionary trajectory. This is why despite its very high mutation rate, hiv cannot evolve efficiently to 3 drug therapy. Let's use your slot-machine analogy. Let's say there are two possible jackpots, a super jackpot, and a big jackpot. And let the probability of winning the super jackpot is one in ten million and the probability of winning the big jackpot is one in a million with each pull of the arm. Do you understand how to compute the probability of winning both of those jackpots as a function of the number of pulls on the arm?Taq writes: Given the right population dynamics and environments, it is pretty simple. With an initial increase of 20 million you are almost guaranteed to get the first mutation. The winner of that jackpot starts dividing until it has 2 million offspring which nearly guarantees a winner for the next jackpot.Just a moment... Kleinman writes:
Is that so? You are obviously not aware that single-drug therapy has been the standard of care for infectious diseases for years. That's why MRSA is so common.
You don't need to do that. Simply take the bacteria that have already adapted to ciprofloxacin and use those as the starter population for the trimethoprim experiment. All you are demonstrating is that DNA evolution works most efficiently a single selection pressure at a time. That's how MRSA was created except in the clinical environment.Taq writes: Those types of scenarios rarely exist outside of labs. Outside of labs, species exist in a diverse ecology, and rarely do they live in a tightly controlled environment that only differs by one variable.Kleinman writes:
What is your logic and reasoning to think that beneficial mutations for one selection pressure would be the same for a totally different selection pressure?
Not so. Lenski has run experiments using thermal stress for his selection condition.Taq writes: That's one lineage of one species of bacteria. It can't be generalized to all bacteria or all life.
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Kleinman Member (Idle past 365 days) Posts: 2142 From: United States Joined: |
Kleinman writes:
In this discussion, we are talking about two evolutionary processes. The first evolutionary process is what Darwin calls the "struggle for existence" or what we usually call competition. And the second evolutionary process is what Darwin calls "adaptation" or what we are calling DNA evolution. The Lenski and Kishony experiments are both DNA evolution processes with the important distinction that the Lenski experiment is being carried out in a highly competitive environment. So, do you really understand the mathematics of evolution, in particular, DNA evolution?mike the wiz writes: Can't say I do, but what type of, "evolution" do you mean. For example do you think the bacteria in these experiments have shown descent by modification in undergoing anatomical changes at the macro level proving that all of the designs of anatomy into the millions, can come about this way? The only clear gross anatomical changes that is occurring in these experiments is that Lenski's bacteria are getting larger.
mike the wiz writes:
What the Lenski and Kishony experiments show is that every evolutionary transitional step takes a billion replications and that just for a single selection pressure environment. And the reason for this is the multiplication rule of probabilities. I know the maths that count; bacteria + bacteria = bacteria. How about the maths of the transitional species for macro evolution? How many would have to have existed in the past? That would be an interesting study for someone like you. Do you know why? Because the disparity of animal phyla in the Cambrian precedes diversity but Darwin's tree would predict the opposite. A common misconception is the notion that a series of microevolutionary changes can add up to a macroevolutionary change. This is mathematically incorrect. Microevolutionary changes don't add, they are linked by the multiplication rule of probabilities because each of these mutations are random events.
mike the wiz writes:
Your basic idea is correct. If I understand your argument correctly, you are saying there should be far more transitional fossils in the fossil record for the theory of evolution to be correct. To get a sense of the numbers, there have been about 50 T rex skeletons discovered and it is highly unlikely that there were billions of T rexes (pinnacle predator). So if it takes a billion replications (under the best of circumstances) for each evolutionary step, where are all these vast numbers of transitional fossils of reptiles transforming into birds and fish transforming into mammals when there are 50 fossils of a single pinnacle predator which most likely existed in very small numbers compared to those replicators lower in the food pyramid? From this link: That's because to get to the level of phyla FROM something like your bacteria as an example, would take a very long time indeed, and there would have to be a lot of evolutionary diversity before reaching the phyla level, because to evolve differences at the phyla level would take hundreds of millions of years. So mathematically since you know maths and I don't, you are my go-to man, in explaining to me what is the number of transitional species that would have had to exist across all of evolutionary time COMPARED to the ones they propose they have found? I am going to say my guess leads me to below 5%. Logically that means it would be slothful induction fallacy to infer macro evolution is true mathematically, when most of the evidence is conspicuously absent. Why below 5%? That's easy to figure out even for a non-math guy like me. Because if they PROPOSE they've found about 300 transitionals on their transitional list, then even if only 3000 transitionals existed over the full span of evolutionary history, that would only be 10%. But to say 3000 transitionals existed over the full course of the history of life on earth would be LOGICALLY PREPOSTEROUS. Ergo, there would likely have been hundreds of thousands, ergo the percentage of transitionals they have found has to be tiny ergo evolution truly is limited to turning bacteria into bacteria, according to logical reasoning I DO KNOW.How many T-Rex were there in the dino era? - Quora "The food pyramid of most ecosystems is accurately described relative species abundance as a power of ten, decreasing from, e.g., 10,000 primary producers (plants), to 1000 herbivores to 100 secondary carnivores, to 10 top predators, and finally 1 pinnacle predator."
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Kleinman Member (Idle past 365 days) Posts: 2142 From: United States Joined: |
Kleinman writes:
If you understood the mathematics of evolution, you would understand how much larger the region needs to be for the experiment to work.
The reason this is the case is that the carrying capacity in the drug-free region is not large enough to accommodate both the wild-type and the drug-resistant variants multiplication.Taq writes: Then make the region larger.Kleinman writes:
How much simpler of an evolutionary experiment can you make than either the Kishony or Leski experiments? If you don't know the exact values for the mutation rates and population sizes, write out the mathematics with those numbers as variables. If you can't do it, I'll show you how to do the math.
That answers that, you don't understand the mathematics of evolution so you snip out the part highlighted in red.Taq writes: If you understood the mathematics you would know that you can't ask a vague question and expect a specific mathematical answer. It's a bit like asking for the probability of a raffle without supplying the number of players, number of tickets, and so forth. The number of beneficial mutations is going to differ between genomes and between environments. You can't ask for a calculation without supplying those numbers.Kleinman writes:
I've never said that the all the steps have to taken all at once. Why don't you try to understand what it takes to make a single evolutionary step? The math isn't that difficult.
What you refuse to accept is that the steps in every evolutionary trajectory are joint by the multiplication rule.Taq writes: I do refuse to accept statements that are false. The steps can be taken one at a time, and don't need to be taken all at once. In a fitness landscape, you don't have to go from the bottom of the hill and reach the top in one giant leap.Kleinman writes:
If I was going to ask you about the mathematical behavior of airplanes and flight, I would start with Newton's 2nd law and ask how you would derive Bernoulli's equation. But we are not talking about flight, we are talking about DNA evolution and competition. So, derive for us the mathematics which describes a lineage accumulating a specific set of mutations allowing adaptation to a selection pressure. When you do that, you will understand the mathematics of the Kishony experiment (and the Lenski experiment as well).
You can't explain mathematical behavior of the Kishony or Lenski experiments so now you start storytelling.Taq writes: That's like asking for the mathematical behavior for airplanes. You have to ask specific questions, such as the equations for lift on the wings of a specific plane, or the thermodynamics of a specific engine. You ask such vague questions and fail to understand that they can't lead to specific mathematical answers.Kleinman writes:
You are wrong Taq. The mathematics for any step on an evolutionary trajectory is the same regardless of the selection pressure. The same equation that describes a step of DNA evolution for the Kishony experiment applies to the Lenski experiment.
It doesn't matter which evolutionary trajectory that a lineage takes. It doesn't change the math.Taq writes: That's patently false. Different pathways are going to have different numbers of beneficial mutations which affects the math.Kleinman writes:
A study was done at one of the hospitals in my area. What they did is swabbed and cultured everyone's (patients, employees, visitors) nose on entry. 50% were carriers of MRSA. This coincides with the numbers in my medical practice. About 50% of the soft tissue infections I treat are community-acquired MRSA. The question you should ask is, why are hospital-acquired MRSA infection resistant to more antibiotics than community-acquired MRSA?
Is that so? You are obviously not aware that single-drug therapy has been the standard of care for infectious diseases for years. That's why MRSA is so common.Taq writes: How many people on the globe right now are taking multiple antibiotics at this very moment? How many more people are asymptomatic carriers of both MSSA and MRSA?mike the wiz writes:
mike the wiz has already admitted that he doesn't understand the mathematics of evolution. So, I'll help him with the answer. Are you familiar with Markov chains? I don't think you are so here is a link that explains the concept. I know the maths that count; bacteria + bacteria = bacteria. How about the maths of the transitional species for macro evolution?Taq writes: Since you seem to be playing the creationist name game, we would have to ask what a transition would be.Models of DNA evolution - Wikipedia This link gives a mathematical definition for a transition, they call it a transition matrix. Read about and try to learn something about the mathematics of DNA evolution.
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Kleinman Member (Idle past 365 days) Posts: 2142 From: United States Joined: |
Kleinman writes:
Fixation isn't occurring in the Kishony experiment. Write out the mathematics of DNA evolution for the Kishony experiment. It appears you don't understand the difference between competition and adaptation.
How much simpler of an evolutionary experiment can you make than either the Kishony or Leski experiments? If you don't know the exact values for the mutation rates and population sizes, write out the mathematics with those numbers as variables.Taq writes: Write out the mathematics of what? Fixation of mutations through neutral drift? What exactly?Write out the mathematics of what? Fixation of mutations through neutral drift? What exactly?Kleinman writes:
It isn't, but if you think it is, just do the mathematics for the first step for the Kishony experiment.
Why don't you try to understand what it takes to make a single evolutionary step? The math isn't that difficult.Taq writes: The math is different for different steps. That's the point.Kleinman writes:
I'll make it as easy as possible, just do the mathematics for the first beneficial mutation for the Kishony experiment.
So, derive for us the mathematics which describes a lineage accumulating a specific set of mutations allowing adaptation to a selection pressure.Taq writes: How many mutations? How strong is the selective pressure? Do you want me to just list the basic equations with no numbers in them?Kleinman writes:
It isn't. You just haven't learned how DNA evolution works yet. Try doing the math for the first beneficial mutation in the Kishony experiment. It's clear you need a hint to get started. Start with a single bacterium that doesn't have the correct mutation and starts replicating in the drug-free region. Then write out the probability equation of that particular mutation occurring at least once in N replications for a given mutation rate.
The mathematics for any step on an evolutionary trajectory is the same regardless of the selection pressure.Taq writes: Bullshit. If there are 15 possible beneficial mutations for a specific beneficial phenotype it will behave differently than if there is 1 possible beneficial mutation. This can also shift as the genetic background shifts due to epistatic interactions.Kleinman writes:
None of the employees or visitors were on antibiotics, and yes there is great variability in the immune status of the individuals. That should give you a big clue on why hospital-acquired MRSA is resistant to more antibiotics than community-acquired MRSA.
A study was done at one of the hospitals in my area. What they did is swabbed and cultured everyone's (patients, employees, visitors) nose on entry. 50% were carriers of MRSA. This coincides with the numbers in my medical practice.Taq writes: How many of those people were on antibiotics? Is the environment different between carriers? In other words, is there person to person variation with respect to their immune systems and condition of their nasal and pharyngeal environments? Is the competing flora the same in each person? What selective pressures does each person present to their flora, including MSSA and MRSA? Wouldn't you say that MRSA exists in many varying environments, with antibiotics being present in a very small percentage of them?Kleinman writes:
You touched on a key reason, the immune status of inpatients vs outpatients. The question you should ask is, why are hospital-acquired MRSA infection resistant to more antibiotics than community-acquired MRSA?Taq writes: That's a question you should answer. Different ecologies? And I noticed that you snipped out the portion about Markov Chains and the transition matrix. It's probably too much for you at this point. Try to understand the "at least on rule" from probability theory. That's a little easier way to understand DNA evolution.
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Kleinman Member (Idle past 365 days) Posts: 2142 From: United States Joined: |
Kleinman writes:
Sorry, I thought you were familiar with the concept of molecular evolution. Read this paragraph: Write out the mathematics of DNA evolution for the Kishony experiment.Taq writes: You need to be more specific.Molecular evolution - Wikipedia Kleinman writes:
I've given you enough specificity, you just don't understand molecular evolution well enough to formulate the probability equation that relates the mutation rate and number of replications of a particular variant to determine that probability. Read the above paragraph I linked to and think about it for a while.
It isn't, but if you think it is, just do the mathematics for the first step for the Kishony experiment.Taq writes: You haven't supplied enough information to do the calculations, nor have you defined the question with enough specificity.Kleinman writes:
The mutation rate is a variable, call it "mu". And for simplicity, assume the mutation rate is the same for transitions and transversion. And initially assume there is only a single beneficial mutation. Once you understand the simple case, I'll show you how to do the math for multiple beneficial mutations.
I'll make it as easy as possible, just do the mathematics for the first beneficial mutation for the Kishony experiment.Taq writes: What is the mutation rate? How many beneficial mutations are possible? What is the specific mutation, and what are the biases for transition and transversion mutations in the bacteria being used?Kleinman writes:
More than a single possible beneficial mutation changes the math only slightly. First do the math for the simpler case.
It isn't. You just haven't learned how DNA evolution works yet.Taq writes: The math seems pretty straightforward. If there are 15 possible beneficial mutations that can produce antibiotic resistance then resistance will emerge sooner than it would if there was 1 possible beneficial mutation.Kleinman writes:
Start with one. Once you do that math, then you can try to do the math when there are multiple possible beneficial mutations.
Start with a single bacterium that doesn't have the correct mutation and starts replicating in the drug-free region. Then write out the probability equation of that particular mutation occurring at least once in N replications for a given mutation rate.Taq writes: How many possible beneficial mutations are there?Kleinman writes:
Try doing the math for the simple lab experiments first. And in any mathematical model, you want to include only those variables that have a significant effect on the predictability of that model. If we were studying Newton's laws, I wouldn't expect you to predict the motion of a bridge or building in an earthquake. You need to start your study with the motion of a pendulum or a mass and spring. Once you master the simple cases, you can go on to the more complex cases.
None of the employees or visitors were on antibiotics, and yes there is great variability in the immune status of the individuals.
Taq writes: There is also a lot of variability in the flora of each individual, the genetic makeup of the flora competing with S. aureus, and there is still variability between strains of MRSA. It isn't like the simple lab experiments you are describing where there are far fewer variables.Kleinman writes:
You've got a point there. So do the mathematics for a single beneficial mutation with a constant mutation rate as a function of the number of replications of a particular variant. Hint: Use the "at least one rule" from probability theory.
And I noticed that you snipped out the portion about Markov Chains and the transition matrix. It's probably too much for you at this point.Taq writes: It isn't. You don't seem to understand how to ask specific questions. There's no reason to move on to the more complex stuff if you can't figure out the simple stuff.
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Kleinman Member (Idle past 365 days) Posts: 2142 From: United States Joined: |
Kleinman writes:
Then explain to us the following quote from this link (and show your math): Sorry, I thought you were familiar with the concept of molecular evolution.Taq writes: I am familiar with the concept. What you don't seem to be familiar with is the mathematics of molecular evolution. Otherwise, you would understand the specifics needed in a question.Molecular evolution - Wikipedia Wikipedia writes:
Because mutations are extremely rare, they accumulate very slowly across generations. While the number of mutations which appears in any single generation may vary, over very long time periods they will appear to accumulate at a regular pace.Taq writes:
Yes, I can answer this question. If the draw of the tiles is random and the probability of drawing any particular tile is equal (same size, same shape, etc.), then that probability is 1/N, where N is the total number of tiles in the bag. See Taq, that's how you use variables in a probability calculation.
As an example, I have a big bag full of tiles. I reach in and draw out a tile with the number 33523 on it. What is the probability that I would draw out a tile with that number on it? Can you answer that question?Kleinman writes:
OK
The mutation rate is a variable, call it "mu". And for simplicity, assume the mutation rate is the same for transitions and transversion. And initially assume there is only a single beneficial mutation.Taq writes: Thank you. That is the type of specifics needed. Vaguely asking for the math of evolution just won't do. We will also assume a 4.6 million base pair genome as is seen in the E. coli model organism.Taq writes:
That's a little different way of looking at the math but that is essentially correct. And if mu=1e-9, 1.38E7/4.6E6*mu)=3.00E+09. The reason why your estimate is 3x larger than my estimate, I'm assuming the mu is the "beneficial" mutation rate and you are assuming the "total" mutation rate where there are 3 possible incorrect mutations at the particular site. If mu is mutations per base per replication then we would multiply mu by the number of bases in the genome which is 4.6E6*mu, this represents the number of mutations per replication. If we are requiring a specific substitution, then we would need 3 substitutions at each base to produce all possible mutations, or 1.38E7 mutations. The final equation is 1.38E7/(4.6E6*mu) as the number of replications needed. Is that what you are looking for? Now, what happens to that 1 variant member of the 3e9 population with the beneficial mutation in the Kishony experiment?
Kleinman writes:
Not significantly. When we finish with the simpler model, we can discuss how multiple beneficial mutations affect the math.
More than a single possible beneficial mutation changes the math only slightly.Taq writes: But it does change it.Kleinman writes:
That's a problem for you, not for me. The purpose of mathematical modeling is to do an analysis of a physical system. For example, if you want to do the analysis of a bridge, the geometry of the bridge is a variable, the physical properties are a variable, the loading forces are variables. So if you change the material used for making the cables which have a different tensile strength, you can just change that one variable and see how that affects the stresses in the structure. The math you did above is ok but it is not as flexible as writing the full governing equations of the evolutionary process. For example, what happens if the bacteria used has a different genome length? Do you think that will change the number of replications for the beneficial mutation to occur?
If we were studying Newton's laws, I wouldn't expect you to predict the motion of a bridge or building in an earthquake. You need to start your study with the motion of a pendulum or a mass and spring. Once you master the simple cases, you can go on to the more complex cases.Taq writes: The problem is that you are asking for the math of a bridge without being specific as to what properties of the bridge you are interested it. Are you interested in the tensile strength of the cables? Are you interested in the aerodynamic profile of the bridge in the wind? Do you see the problem?
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Kleinman Member (Idle past 365 days) Posts: 2142 From: United States Joined: |
Kleinman writes:
Let's put the quote back up which I ask you to explain: Then explain to us the following quote from this link (and show your math):Taq writes: Not until you explain how "show us the mathematics of evolution" is specific enough to approach any equation or question.Molecular evolution - Wikipedia Wikipedia writes:
Taq, you have actually started to explain why mutation accumulates very slowly across generations in your previous post when your math shows that it takes 3e9 replications for the first beneficial mutation in the Kishony experiment. So, how many replications of that new variant for the next beneficial mutation to occur on that variant? Hint: Just apply your math again. And note, that is the regular pace for a single selection pressure.
Because mutations are extremely rare, they accumulate very slowly across generations. While the number of mutations which appears in any single generation may vary, over very long time periods they will appear to accumulate at a regular pace.Kleinman writes:
You got me on that one. You have got to watch your assumptions.
If the draw of the tiles is random and the probability of drawing any particular tile is equal (same size, same shape, etc.), then that probability is 1/N, where N is the total number of tiles in the bag. See Taq, that's how you use variables in a probability calculation.Taq writes: You got it wrong. You are assuming that all the tiles have different numbers. They could all have the same number, for all you know.Kleinman writes:
13.8 million are the total number of possible variants (if we consider substitutions only) from the initial founder wild-type bacterium in the Kishony experiment. How many replications necessary for the variant with the first beneficial mutation expected to occur?
The reason why your estimate is 3x larger than my estimate, I'm assuming the mu is the "beneficial" mutation rate and you are assuming the "total" mutation rate where there are 3 possible incorrect mutations at the particular site.Taq writes: mu is mu. In a 4.6 million base genome there are 13.8 million possible substitution mutations, so if you are calculating the number of replications needed to produce all possible mutations, one of which is the one of interest, then you need to use the 13.8 million number.Kleinman writes:
And how many replications of that variant necessary for the next beneficial mutation to occur?
Now, what happens to that 1 variant member of the 3e9 population with the beneficial mutation in the Kishony experiment?Taq writes: It is able to expand into the region with antibiotics.Kleinman writes:
I understand this discussion is difficult for you. It doesn't bother me if I don't get all the information necessary to give an exact answer. If fact, in the world that I work in, that is the usual case. When you asked your tile question, I gave an answer and made an implicit assumption that the tiles had different numbers and you showed my implicit assumption could be wrong. So, here's a specific question. How many replications necessary for the 13.8 million variants to occur in an evolutionary step in the Kishony experiment?
That's a problem for you, not for me.Taq writes: When someone asks a vague question and expects a specific answer the problem lies with the person asking the vague question.
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Kleinman Member (Idle past 365 days) Posts: 2142 From: United States Joined: |
Kleinman writes:
I'm not asking you to write VOLUMES. I'm just asking you something very specific. How many replications for each evolutionary step in the Kishony experiment? And how do you compute that number of replications?
Let's put the quote back up which I ask you to explain:Molecular evolution - Wikipedia Taq writes: Look at the rest of the subtopics on that Wiki page. Look at the rest of the subtopics on that Wiki page.
quote: When you ask for the mathematics of molecular evolution you are asking about all of those things. In order to supply you with the math of evolution we would have to post VOLUMES of equations. Do you understand how ludicrous that is? Kleinman writes:
It only happens in a matter of days because the generation (doubling) time for bacteria is usually minutes. But if it is happening so quickly, why won't you tell us the number of replications that occur in those matter of days?
Taq, you have actually started to explain why mutation accumulates very slowly across generations in your previous post when your math shows that it takes 3e9 replications for the first beneficial mutation in the Kishony experiment.Taq writes: Slowly? It happens in a matter of days, doesn't it?Kleinman writes:
No, I'm not asking for the mathematics for an entire field, I'm asking for the mathematics for the number of replications for the Kishony experiment. How much more specific do you want me do get? How about just giving us the mathematics for the number of replications for a single evolutionary step in the Kishony experiment?
You got me on that one. You have got to watch your assumptions.Taq writes: Exactly. When you lack specifics you have to make assumptions, and that can make calculations worthless. That's why I'm asking for specifics. On top of that, you are asking for the mathematics for an entire field, not a specific situation.Kleinman writes:
The Poisson distribution is ok. Are you aware that the Poisson distribution is the limiting case of the binomial distribution as the mutation rate approaches 0 and the number of replications goes to infinity? How many replications necessary for the variant with the first beneficial mutation expected to occur?Taq writes: That's a bit like asking how many lotteries are necessary to get a winner, or how many tickets you need to buy in order to win the lottery. It varies. It possible, however improbable, that you could have 10 or 100 times more replications than the number I calculated and not get the mutation. There could be other times where you get the mutation in the first few replications. From my reading, the best way to model this is a Poisson distribution which Luria and Delbruck worked up in their papers.https://www.youtube.com/watch?v=ceOwlHnVCqo So use either distribution and tell us how many replications for a single evolutionary step in the Kishony experiment. Kleinman writes:
You are the one having difficulty answering a specific question. Why won't you tell us how man replications it takes for a beneficial mutation to occur for a single step in the Kishony experiment? You can even use the Poisson distribution for your answer even though the binomial distribution is the correct distribution. I can show you how to do that math if you continue to have difficulty.
I understand this discussion is difficult for you.Taq writes: You need to understand psychological projection.Kleinman writes:
Why? It's a very rare thing in life when you understand something exactly. Sometimes, all you can do is give a ballpark estimate. So far, you refuse to even call Uber to get a ride to the stadium. When are you going to give us the mathematics which estimates the number of replications necessary for a single evolutionary step in the Kishony experiment? You can even use the Poisson distribution to make your estimate.
It doesn't bother me if I don't get all the information necessary to give an exact answer.Taq writes: It should.Kleinman writes:
Just use the mean value for the Poisson distribution and you can easily compute the variance (and therefore the standard deviation) of that value to see what the spread of values are. Here is a short video that explains how to do it. How many replications necessary for the 13.8 million variants to occur in an evolutionary step in the Kishony experiment?Taq writes: It would take some more effort to dig through Luria and Delbruck's work and few other papers to get the probability spreads. Is this something you are really curious about, and is it worth my effort? Is the lottery analogy enough to let you know I understand the problem?https://www.youtube.com/watch?v=vrQ9JsAPTUc If you have difficulty applying this math to the Kishony experiment, I'll show you how to do it. So tell us how many replications required for a single evolutionary step in the Kishony experiment using the Poisson distribution.
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Kleinman Member (Idle past 365 days) Posts: 2142 From: United States Joined: |
It seems that nwr is not the only one with a vague understanding of DNA evolution, Taq doesn't do much better. So let's see if we can help.
Taq writes:
So, if the genome length is 4.6E6 and if mu = 1E-9, it would take about 218 replications of the original founder bacterium of Kishony's drug sensative lineage before we would expect to see the first mutation in one of those 218 members, somewhere it its genome. The remaining 217 members should be exact clones.
mu is mu. In a 4.6 million base genome there are 13.8 million possible substitution mutations, so if you are calculating the number of replications needed to produce all possible mutations, one of which is the one of interest, then you need to use the 13.8 million number. |
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Kleinman Member (Idle past 365 days) Posts: 2142 From: United States Joined: |
Kleinman writes:
The reason why the Poisson distribution is ok but not correct because the probability of success (the mutation rate) is very small and population sizes are very large (the number of trials). Under these circumstances, the Poisson distribution gives similar results as the binomial distribution (the correct distribution function for this probability problem).
Why won't you tell us how man replications it takes for a beneficial mutation to occur for a single step in the Kishony experiment?Taq writes: I think you already saw it: "The Poisson distribution is ok."Taq writes:
There isn't a lot of work to go through to answer the question of how many replication necessary for the Kishony experiment to get a variant to adapt to the next higher drug-concentration region. You can even use the Poisson distribution to do your calculation. I've already given you a link that explains how to do the math. If you can't do the math for this simplest of evolutionary experiments, how are you going to do the math for the more complex evolutionary experiments?
If this is a throw away argument, I really don't want to go through the effort of digging through all the formulae and doing the math. Is there a point you want to make with all of this?Kleinman writes:
Quite correct. So, do the math for the simplest case. Assume there is only one possible beneficial mutation at a single site in the genome that will give the necessary improvement in fitness for that new variant to grow in the next higher drug-concentration. Assume you start with a single wild-type bacterium in the drug-free region. What is the number of replications of that wild-type variant needed for that improved fitness variant expected to appear? Please show your math.
Why? It's a very rare thing in life when you understand something exactly. Sometimes, all you can do is give a ballpark estimate.Taq writes: You still need valid assumptions to get ballpark estimates.
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Kleinman Member (Idle past 365 days) Posts: 2142 From: United States Joined: |
Kleinman writes:
Use whatever distribution you think is appropriate. Tell us how many replications are needed to give a reasonable probability of a drug-resistant variant occurring able to grow in the next higher drug-concentration region in the Kishony experiment and show your math.
The reason why the Poisson distribution is ok but not correct because the probability of success (the mutation rate) is very small and population sizes are very large (the number of trials). Under these circumstances, the Poisson distribution gives similar results as the binomial distribution (the correct distribution function for this probability problem).Taq writes: A Poisson distribution is a binomial distribution: "A familiar nongenetics example of a binominal probability distribution is a coin flip: heads and tails are two discrete outcomes, and the probability of each is 0.5 on any single flip. One type of binomial distribution is the Poisson distribution, which expresses the probability of a given number of occurrences of an event that occurs during a fixed time period."https://www.genetics.org/content/genetics/202/2/371.full.pdf Kleinman writes:
Don't you think that correctly describing the evolution of drug-resistance is important? And this subject is important in understanding how to treat cancer, especially as targeted cancer therapies are developed. And why is it so important to you to stop this thread? What is so terrible about correctly describing the physics and mathematics of evolution? My understanding of evolution has been helped by discussions like this. If you can't do the math for this simplest of evolutionary experiments, how are you going to do the math for the more complex evolutionary experiments?Taq writes: So what is the ultimate point you are trying to make? Explain to us why it is worth our time to do the calculations? If we do the calculations, will you admit that people do understand the calculations and stop posting in the thread? I think that Darwinian evolution is qualitatively correct. What's wrong with quantitating his theory?
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Kleinman Member (Idle past 365 days) Posts: 2142 From: United States Joined: |
Kleinman writes:
Very good! I hope AZPaul3 is following this discussion because his estimate was between 1 and infinity. 3 billion is definitely in the ballpark. There are several directions this discussion can take but let's stick with the line we are on and try to give the mathematical explanation of why the Kishony experiment won't work with two drugs instead of one. And to do this, consider the 100ml to 1 liter of culture that you described with the bacterium with the beneficial mutation in that population of 3 billion. How large would the growth solution have to be if you took those 3 billion bacteria (with the one member with the resistance mutation) in order to get a variant with the first 2 beneficial mutations? In other words, how large must the population be for a variant to occur with both mutations necessary to grow in the next higher drug-concentration region with two drugs? Assume that you need only a single mutation for each drug. Tell us how many replications are needed to give a reasonable probability of a drug-resistant variant occurring able to grow in the next higher drug-concentration region in the Kishony experiment and show your math.Taq writes: Going with a per base mutation rate of 1E-9 and a genome size of 4.6E6 bases, that would be one mutation per 217 replications. If we are looking for a specific substitution mutation that would be 1 mutation out of 4.6E6 bases and 3 possible mutations per base for a total of 1.38E7 mutations. multiply the number of replications per mutation by the number of possible mutations and you get 3E9 replications. So you would need about 3 billion replications to have a reasonable chance of getting any specific substitution mutation in E. coli with that specific mutation rate and genome size. E. coli in saturated liquid culture is about 1E6 to 1E7 per ml. If we added 1 bacterium to 100 ml to 1 liter of culture and let it reach stationary phase we should get our mutant. Now what? And by the way, the size of the genome is actually not a factor. When you have a mutation rate of 1e-9 and 3 billion replications of that genome, you will have on average, every substitution possible at every site in the genome no matter how long the genome in some member of that population. Edited by Kleinman, : Correct the title
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Kleinman Member (Idle past 365 days) Posts: 2142 From: United States Joined: |
Kleinman writes:
We are not talking about two simultaneous mutations. We are talking about a second particular mutation occurring on some member of the population where a first particular mutation has already occurred. In the context of the Kishony 2 drug experiment, in your 3 billion population, you already have a variant with a mutation for the first drug. You also have a variant with a mutation for the second drug as well. How large must the total population size be for a mutation for the second drug occur on a member that already has a mutation for the first drug or for a mutation for the first drug occur on a member that already has a mutation for the second drug? This is a classical conditional probability problem. I gave you a Venn diagram earlier in the thread to help you understand the problem:
There are several directions this discussion can take but let's stick with the line we are on and try to give the mathematical explanation of why the Kishony experiment won't work with two drugs instead of one.Taq writes: I think we can all agree that needing two mutations at once is going to make adaptation nearly impossible if you start with zero variation. Is there a point beyond that?Taq writes:
We know what the variation is for the Kishony experiment, you are starting with a population size of 3e9. Of that 3e9 population, 13.8 million are variants with each of these members having one of the possible mutations but the vast majority of the starting population is still "wild-type" that is they are exact clones of the original founder bacterium. And we have only 1 member with a beneficial mutation for the first drug and only 1 member with a beneficial mutation for the second drug. Now, the 3e9 bacteria start doubling, how large must that population be for there to be a reasonable probability of getting a variant with both the A1 and A2 mutations. I'll give you a hint here to help you understand this problem. In the single drug experiment, that single drug-resistant mutant moves into the next higher drug concentration region and must achieve 3e9 replication for the next beneficial mutation to occur.
What would this look like if you start out with a certain amount of variation?Kleinman writes:
I don't agree with that. If you can't do the mathematics, the best you can have is a vague understanding of the phenomenon. We are trying to give nwr more than a vague understanding of evolution.
When you have a mutation rate of 1e-9 and 3 billion replications of that genome, you will have on average, every substitution possible at every site in the genome no matter how long the genome in some member of that population. Taq writes: I knew as much when I started, but sometimes it is easier to follow a train of thought instead of employing algebra to eliminate redundancies.Taq writes:
That's not correct but since fixation does not pertain to the Kishony experiment, it is not worth discussing this subject at this time. In the same vein, the neutral fixation rate is equal to the mutation rate no matter the size of the population. Try to figure out how large the population size has to be for a variant to occur with 2 beneficial mutations for 2 drugs to make one evolutionary step in the Kishony experiment if 2 drugs are used.
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Kleinman Member (Idle past 365 days) Posts: 2142 From: United States Joined: |
Kleinman writes:
That's correct for the single drug experiment. How does the math work for the 2 drug experiment where some variant has to appear in the drug-free region with a beneficial mutation for each of the drugs before it can grow in the next higher drug-concentration region? How large does the founding colony have to be for there to be a reasonable probability of that variant appearing?
We are talking about a second particular mutation occurring on some member of the population where a first particular mutation has already occurred.Taq writes: The same calculations would apply, starting with the first bacterium to gain resistance to a single drug. You would need another 3 billion descendants from that first resistant bacterium to get the mutation for resistance to the 2nd drug.Kleinman writes:
The math tells us what the variation is. You start with a single wild-type bacterium without any resistance alleles. When we have 3e9 replication with a mutation rate of 1e-9, that gives us on average 1 member in the population with any of the 13.8 million possible substitution mutations, The remaining members will be wild-type clones of the original founder. Of course, somewhere else in the world they may be other variants but try to understand the Kishony experiment first.
We know what the variation is for the Kishony experiment, you are starting with a population size of 3e9.
Taq writes: That doesn't tell us what the genetic variation is. A population of 3 billion that recently descended from a common ancestor is going to have less genetic variation than a sample of 3 billion individuals from a population that has been dividing for a long time period. It is similar to the genetic variation in 10 close family members vs. the genetic variation of 10 individuals randomly sampled from across the globe.Kleinman writes:
You have started to do the math but then you said you can explain your train of thought without algebra. That is vague. And your approach to the math was slightly different than my approach but your approach is correct as well. The problem with your approach becomes obvious when you try to do the math that would predict the behavior of the Kishony experiment when using two drugs instead of one. As you try to do the mathematics for the two-drug experiment,you will see why your approach doesn't work. The problem is that you are working with two linked binomial probability problems. Perhaps you can get your approach to work but I don't think so.
If you can't do the mathematics, the best you can have is a vague understanding of the phenomenon.
Taq writes: I did the math. Your only complaint seems to be that I didn't do the math the way you wanted me to.Kleinman writes:
Fixation is not applicable to the Kishony experiment. When you finally figure out the math for the two-drug Kishony model, we can discuss fixation in the Lenski experiment (except that is fixation for a haploid and that fixation is selective). This will give you practice in applying evolutionary mathematics to real, measurable, and repeatable experimental examples of evolution. Just in case you were curious.Taq writes: Just in case you were curious.
quote: So, try to figure out how large the population size has to be for a variant to occur with 2 beneficial mutations for 2 drugs to make one evolutionary step in the Kishony experiment if 2 drugs are used.
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