 So are there any questions from the last lectures? So we're in the middle of population genetics. I hope to finish it today. So no more sort of the math-y part of evolutionary biology. Not that you'll see any more math today, but all you'll see is in terms of equations is the Hardy-Weinberg, and that's easy. OK, so there have been a couple of questions about notes that have been posted. I believe I haven't put anything on B-space. B-space seems to become much more popular this year than it was in the past. And so I'm certainly happy to post things there. It's not difficult, but they are being posted on the course website. So next each lecture should be some links to my notes, a PDF of my notes, this bit here, and also, whenever I do have them, PDF of my keynote presentations or PowerPoint or whatever you want to call them. So you should be able to find them there. I've gotten a couple of emails about where they are. They're online. OK. So we were talking about population genetics, and we discussed how random mating by itself doesn't change allele frequencies, so what does? And the forces that do change allele frequencies are nicely summarized right here. Those are mutations, selection, genetic drift, and migration. We talked about mutations. Mutations, the ultimate source of variability in populations, it changes one allele into another allele. That's what mutation is. And I just started to talk about selection, so I want to get through the last three forces that cause changes in allele frequencies today. And to understand selection, we have an idea called fitness. So the fitness of a genotype is assigned to individuals of different genotypes. If the genotypes differ, if individuals with different genotypes differ in their survival and or reproduction, then natural selection can see those differences. So natural selection is this process that favors individuals that are more fit, that the ones that have the more favorable genotypes. And those alleles will tend to increase in a population through time through the action of natural selection. Now there's a couple of different examples of different kinds of natural selection that I want to talk about. It's very nice. One is directional selection. And this is the scenario I want you to consider. So here's our little population. It's made up of everybody's homozygous for the little allele. So we imagine that a mutation hits one of these individuals. So remember mutations land on one chromosome initially, on one chromosome and one individual. It changes that little a to be a big allele. Now the scenario is this. Let's imagine that this mutation, this big allele, is a beneficial mutation. That is to say, it's a good mutation. If you have this allele, you have a fitness advantage over your other individuals and the populations that don't have it. What will happen, what natural selection does, it tends to increase the frequency of that big allele. So over time, you would see more and more individuals have the big allele. So this might be a snapshot of the population at some intermediate point when the population is polymorphic. That is to say that there's different forms of the allele in the population. Eventually what will happen under the influence of natural selection is everybody has the big allele. So eventually everybody has the big allele. And this process of going from a mutation of very low frequency. Initially what's the frequency of an allele in a population? Can anybody tell me? Let's imagine the population size is n. That's the number of individuals in the population. What's the initial frequency of a brand new mutation? Well, it's going to be 1 over n. But how many chromosomes are in the population? So it's 1 over 2n. This is the number of chromosomes, this number of alleles in the population. Initially only one of them is of the mutant form. So all mutations start off at this very low frequency. So this process of going from a frequency of 1 over 2n to 1, that's called the fixate. The allele, once it reaches a frequency of 1, the allele is said to be fixed. It's said to be fixed in the population when the allele frequency reaches 1. And this entire process of mutation plus fixation, that's called the substitution. So if we imagine looking at a DNA sequence, say, initially you have a mutation that changes maybe a G to a C. If that C eventually reaches a frequency of 1, then it becomes fixed in the population. And then an entire process of going from mutation and then being fixed is called a substitution. Not a mutation, it's called a substitution. Mutation is the initial change in one individual's chromosome. The substitution is the process of going from that initial low frequency to high frequency. So anyways, they just want to introduce some terminology, and this is what directional selection tends to do. This is the case where the AA individuals are the best, the most fit individuals. These guys are intermediate in fitness, and these guys have the lowest fitness. When you have that situation, the natural selection will favor these two genotypes, and the big allele will increase in frequency in the population. And there's a lot of math that can describe exactly how that transition occurs. So some examples. So in humans, there's a number of interesting examples. One involves the ability to digest lactose. So it turns out almost all mammals are lactose intolerant as adults. Obviously, all mammals, when they're young, when they're first born, they can digest lactose or they wouldn't be alive. But through time, as all different mammal species mature, they lose this ability to digest milk as adults. And the reason they do this is there's a gene called the LPH gene. This is for lactase-fluorazine hydrolase. This gene is turned off in adults. Now, cats can actually digest milk as adults. Basically, LPH is active or is expressed in adults and cats. So basically, there's mutations in cats that cause LPH to be expressed as an adult. It's never turned off. Now, you're probably aware that there's some populations of humans that can digest milk. I can do it as an adult. And probably, if you can raise your hand, can you digest milk? And you'll probably know it if you can't because you're flatulent if you drink too much milk. There's other things you can tell too. So there's some populations of humans that actually can digest milk as adults. It turns out there's two. There's northern Europeans and some populations in Africa. And interestingly, this ability to digest milk as an adult is associated with a pastoral lifestyle. That is to say, you raise cattle, drink their milk, eat cheese. It's associated with that. So it's very recent that people have actually discovered the mutations that cause LPH to be constituently active as an adult. Let me show you that. Bring the screen down. These are the only two slides I have for this lecture. This is work that just came out a few years ago. And this looks a little bit complicated. So I'll tell you more or less what this is a figure from a paper that was published in Nature, Nature Genetics a few years ago. But basically, this is a representation of part of the second chromosome. And then these little hatches are positions where they found variation in humans. That is to say, there's a polymorphism. Some individuals might have a nucleotide T, and others might have a C, or some individuals might have the nucleotide A, and others might have a G at that position. So these are positions where they're known to be polymorphisms. And this is in this LPH gene region. And what they've done is they've sort of blown up certain regions of this gene. And they found that there were certain, so I should also tell you that there's a buzzword that you might want to know, which is called SNP. I think I mentioned it earlier, SNP, for single nucleotide polymorphism. You might actually see that in newspapers. It's a term that's thrown around enough now that I've seen it in newspapers. This basically SNP is just a place in the human genome where there's variation, a known variation among individuals. Anyways, they just blown up a certain region of this gene, a certain non-coding part of this gene. And they found that there were four mutations, four SNPs, that happened to be associated with the gene being active as adults. And here they are. The four that they found, there was a GC change here, a CT polymorphism, a TG polymorphism, a CT, and a CG. Let's see, which one is, this one right here, this one that they label 13, 9, 10, this is basically the position along the sequence in the study. This is associated with the ability to digest milk in Europeans. And these other three mutations, one, two, three, these three are associated with the ability to digest milk in lactose tolerant, as they say, in adults in these African populations. So what this means is that the ability to digest milk in humans has evolved at least twice independently, maybe more, right? And you can imagine the natural selection would favor this ability to digest milk as adults, meaning this is a huge food source for some populations of humans, cheese, milk, for instance. And they can also, using methods that I'm not gonna describe in this class, estimate when these mutations probably first occurred about 7,000, 8,000 years ago was the guest that the authors put in the paper, which is coincident with when these pastoral lifestyles were adopted. So this is an example in humans of a recent event of natural selection. And here's just to give you an idea in these African populations. This plot over here shows you these three different countries that they surveyed. And basically, they're looking at LP, what is this, lactose tolerance and intolerance, I believe. So this is just plotting the frequency of these different traits. And over here, they're showing you the frequencies of these different mutations, these different snips in the different populations, right? The main, and you're not really responsible for either of these slides. I just wanted to give you an idea of a recent event of natural selection. If you basically know the story that there's been multiple events, convergent evolution of this trait in two different human populations, I'd be quite happy. And convergent evolution of traits independent, so convergent evolution is the independent evolution of a trait that's usually taken as strong evidence that natural selection caused that evolution, okay? If you have, for instance, fusive form forms in organisms that swim through the water, that is like a kind of a torpedo shape, you can imagine that's evolved independently in mammals and whales and dolphins and seals and also in a fish, right? And you can imagine that would be a trait that'd be very strongly selected for. So just to give you an example of some convergent evolution. All right, so let's put this back up, length the screen, screen up. That's it for this. There's one more time I'll have to bring the screen down. And these past episodes of natural selection, I mean, this is an example of a trait that you might think of as a benefit. I'm glad I'm not flatchy after drinking milk, frankly. There's other examples where past events of natural selection seem to be associated with what we think of as disease today. So for instance, there's some populations of Native Americans who have a high incidence of type II diabetes, incredibly high incidence of type II diabetes. And the hypothesis that's been used to explain this is that in the past, these populations experienced lots of starvation or near starvation. So it's selected for individuals that could maintain body fat and didn't burn calories very quickly. So it's a so-called thrifty gene hypothesis. So if you had a thrifty gene, that was to say one that made your metabolism slow, in the past, you were more likely to survive these incidents of starvation. But today in a modern environment where you can get Twinkies or Ding Dongs from the supermarket, we have no shortage of food, this particular, having this genotype is not beneficial. And so these people tend to have, like I said, a high incidence of type II diabetes. And this is one explanation for that. And some Native American populations. Okay, now next thing I want, so that's just two examples of a directional selection. Basically, this is the type of natural selection that most people think of when they think of selection, I believe. We also have examples of what we might call purifying selection. Okay, and this is an example, so this is the hypothetical example I want you to consider here. So let's imagine a population that's all homozygous for the big allele now, okay? And you have a mutation that lands on one of these chromosomes in one individual and it turns that big A allele into a little A allele, okay? In this example, the little A allele isn't advantageous as it was, the mutation is not advantageous or beneficial as it was over here, it's deleterious, this individual's sick. Okay, so what does natural selection do? It tends to remove these alleles from the population. So one question is why would you ever have these alleles floating around in a population? Well, the idea is that you have mutation constantly introduces these little A alleles into the population and natural selection is constantly removing these mutations from the population. And remember, even though the mutation rate's quite low, you have some, what, 30,000 some odd genes in your genome, so there's always gonna be, every generation, you're gonna have a couple of deleterious mutations landing on some chromosome somewhere in some individual. And so eventually what'll happen is there'll be a balance that is struck between natural selection removing these deleterious mutations and mutation introducing them and that balance of population geneticists call that mutations selection balance. I'm not gonna provide any examples of this, but this is just an example of purifying selection, natural selection is continuously trying to remove deleterious mutations from a population. And the last example of natural selection I wanna discuss is called balancing selection. And this is the example, the scenario here is that you have, once again, you have the three different genotypes, big A, big A, big A, little A, and little A, little A. In this scenario, the heterozygous individuals are the most fit, okay? So when the heterozygous individuals are the most fit, then natural selection tends to maintain both alleles in the population. And a good example of this is the sickle cell allele. This, I mean, I mentioned in the last lecture that the SS individuals, those that are homozygous for the sickle cell allele and the beta-globin gene are quite sick individuals, right? They tend to die early. But it turns out that if you're in an area that has malaria and you're heterozygous, you have, your genotype is the most fit of the three possible genotypes. You're more resistant to the plasmodium that actually causes malaria, the parasite that causes malaria, if you're AS. And in those, in malaria is a major killer in the world. I mean, it kills millions of people every year. And so, and it has in the past as well, okay? But the SS allele in many populations in equatorial regions has a frequency as high as 12%, okay, which means about roughly how many, what fraction of the individuals are gonna be homozygous for SS, about 1%, right? Point one times point one, roughly. Okay. Now the natural selection, all of these scenarios can occur quite quickly. And this is one of the major reasons why people turn to math, why mathematicians have actually have a role in evolutionary biology. As people, we can only really study populations over the course of a lifetime at most, right? Typically, the course of study is at most a decade, right? There's some examples which maybe I'll get to next week of a couple of scientists who've studied certain populations for about 40 years now, but that's very infrequent. Usually, the time that you can observe a population is about the time of a grant from the government, about four years, okay? Which means we can only see snapshots of populations. What the math tells us is that what the frequency, how rapidly changes will occur over hundreds of generations, something no human can actually study. And so, what mathematicians found out in the 1920s is that natural selection can see very small differences in the fitness effects among these different genotypes and cause the frequencies to change rapidly over geological or even over the course of even human histories, like a thousand years, okay? So the math has been able to help us predict in the future how quickly alleles should change if the environment stays constant. That's another thing I should point out is that the fitness depends critically on the current environment that the genotype is in. If the environment changes, then the fitness consequences of that allele can change as well. And a good example of that is phenylketonuria. This is a disease that's caused by the inability to break down phenylalanine, all right? And so, individuals that have this mutation, they tend to accumulate phenylalanine in their body and they suffer from mental retardation and they often die quite early. They usually do, in fact. So clearly, that mutation is deleterious in the presence of phenylalanine. If you actually give these people a diet that doesn't have that, then they're fine, okay? They can actually live normal lives. They just have to be very careful about what they eat, okay? That mutation is deleterious in the presence of phenylalanine. Is that clear? So the environment, you know, if the environment changes in some way, then the fitness advantage of some genotype might appear or disappear or might become apparent, okay? So that's all I wanted to say about selection. Now I want to turn my attention to genetic drift, okay? And genetic drift is a random, it's sort of the random effects of the change allele frequencies. So you can imagine situations where, let's do this. So imagine a situation where big A and little A, their fitnesses are equivalent. So there's no effect of having a big A or a little A mutation in some gene on your fitness, okay? And let's imagine, think of an imaginary population that's 50%, you know, has the big A floating around and then 50% of the alleles are little A. Now the Hardy-Weinberg theory, which I told you about earlier says that, well, if that's the case, random mating alone will not change those frequencies. So the big A and little A frequencies will stay at 50% forever. That's what Hardy-Weinberg says. Turns out in real populations that have finite size, the Hardy-Weinberg theory assumes that the population sizes are very large or, formally speaking, they assume that they're infinitely large. But in finite size populations, which all populations on earth happen to be, the one of these two alleles will ultimately become fixed. That is to say, one of them will become, the population will eventually become all big A or it'll eventually become all little A. And we can actually sort of demonstrate the effect of genetic drift in a very simple experiment. So I'm gonna, we're gonna demonstrate genetic drift. So I'm gonna make 10 alleles. One, two, three, four, five, six, seven, eight, ah, nine, 10. There's our 10 alleles. And let's make one, two, three, four, let's make four of them, the white alleles and six will be the, I guess they're green, right? The chalkboard colored alleles. And let's also just label these alleles. Zero, one, two, three, four, five, six, seven, eight, nine. Now I'm not gonna explain this the following statement, but one way of thinking about makings, this is our current generation. Let's think about what's gonna, what the population will look like in the next generation. One, one way of simulating genetic drift is just to randomly sample the alleles from the current generation and put them into the next generation. So now we're not thinking about alleles as being in individuals. We're just thinking about just randomly selecting these alleles. So we need to have a way of actually selecting randomly among 10 things. And so one thing you could do is you could think about dice. And so I don't know if you're familiar with dice, but most dice have six faces. And, but then we have 10 individuals. I've got a special die here that has 10 faces on it. And the faces have the numbers zero through nine on them. So I don't know if anybody, people here know where these dice come from. Okay, so you're probably, those of you who know where these dice come from kinda realize how grim my social life was in high school. But so what I'm gonna do is I'm gonna roll these dice and each roll is gonna determine which allele is sampled and put into the next population. So I'm gonna roll the die 10 times. Actually, I need some help here. I've got two of them. So you're the closest. So you, I'm gonna roll a new roll just tell me what numbers you get. I get a six. So allele six goes in. Allele seven goes in. Keep going. I got an eight. She got a six, three, and I got a three, four, okay? And we need three more rolls. Got a five, six, one more. All right, so the one allele which was copied in, that's this color six is clear. Five is clear. Four is clear. Three is colored in. Three is colored in. Six is clear. Eight is clear. Seven is clear. Was that a zero or a six? I thought it was a six too, okay? So we have one, two, three of the white alleles, right? In this generation. So one, two, three, four, five, six, seven, eight, nine, 10. Good, we rolled the die the correct number of times. So the frequency of this allele in this population was what? What was the frequency of that allele? That's 0.4, right? What was the frequency of the allele in this generation? 0.3? Would you all agree that we had no bias in terms of which allele we actually sampled in the previous generations? There's no fitness consequence in terms of you're being more likely to be sampled if you're this allele rather than this allele, right? So the frequencies change from one generation to another. They evolved, right? But there was no natural selection involved. It was just random chance, okay? So this is genetic drift you can think of as sort of being hit by the bus type of process in evolution, right? There is a random factor to which alleles are gonna be ultimately fixed in a population. And what I'll do is I'm not gonna show any, once again I said I'm not gonna show any math, but I will show simulations soon that simulate genetic drift and natural selection operating together. Now there's a couple of different terms that come up with genetic drift rather. You can keep that dye if you want, it's yours. Well, let's use this one, it's clear. So there's two different ways that you often see genetic drift expressed in evolutionary biology. One is bottlenecks, and what a bottleneck is is a temporary restriction in the population size. So what I'm gonna do is make a little graph here where we have time along this axis, there's time, and then on this axis we'll have the population size in. So you might have a population that's pretty big for a while, and then for whatever reasons, the population size decreases, and then maybe later on it re-expands. So the width here represents the, I should shade it in a bit, the width here represents the population size. And this period right here, that's the bottleneck. That's the restriction in population size. What does this mean? It means that genetic drift, so I should also mention that genetic drift is more powerful, it changes the allele frequencies more extremely in small populations than it does in big populations. So when n is small, genetic drift is stronger. When n is big, genetic drift is weaker. So when the population is large during this phase, or during this phase, then sure the allele frequencies are changing under the influence of genetic drift, but they're doing so slowly. During these periods when the population experiences a bottleneck, then allele frequencies can change very rapidly. And in fact, just through the action of genetic drift alone, alleles can be more likely to be lost during this bottleneck phase than they are here. So bottlenecks often reduce, or they reduce the genetic variation in the population. In fact, that is a general thing that genetic drift does, is it reduces, it removes alleles from a population just by chance. Though genetic drift, more powerful in small populations than in large, it removes variation from a population. And it can remove variation quite quickly when during these bottleneck periods. So that's one phenomenon you'll see discussed in the evolutionary biology literature, or population of biology literature. And there's examples of organisms today that we think have experienced bottlenecks in the recent past. For instance, cheetahs. Cheetahs have very little variation, genetic variation from one individual to another. In fact, you can't do this with humans. You can't take a patches of skin from one individual and put it grafted on to another individual without that other individual most likely rejecting that graft. You can do that in cheetahs, no problem. They're so similar one individual to the next, you can actually perform skin grafts and the individual that has the skin grafted will not reject that. So the thought is that in some time in the recent past, cheetahs went through a bottleneck and that explains why they have so little variation in the population today. The other phenomenon I wanna talk about is the founder effect. And this is the idea that, let's say we have some population, some big population, and it's gonna found a new population. So that is to say maybe some individual from this population or some small number of individuals for what maybe a bird flies to a new island or maybe some individuals cross some mountain range and they found a new population over here. Now the individuals that move from this population, first of all, it's the idea is it's a small number of individuals and there are a random sample from the population at large. So if it happens to be the case that some of these individuals that migrated that founded the new population happened to have an allele that was rare in the founding population, then that allele that is rare over here can be quite common, become quite common over here. And so often you're probably familiar with these studies of small isolated groups of humans. There's a lot of small populations in Pennsylvania like the Mennonites, the Amish, that have high frequencies of some diseases. And it's the thought is that the individual, the founding populations of these individuals just happened by chance to have a particular disease. And a good example of this is, where's my notes? Here it is. It's Huntington's disease. This is in the notes, so I'll just, for brevity's sake, call it HD Huntington's disease, which is a late onset neurodegenerative disease. It's actually a very horrible disease. It's not really, I mean, it occurs late in life, so after people have normally reproduced so natural selection doesn't see it as well. Remember, natural selections operates on individuals before they reproduce, right? So Huntington's disease is still floating around in the population, but it's at very high frequency in some populations. So there's a population in Venezuela, in the town of San Luis, where Huntington's disease is at a very, very high frequency. I think 25%. And the idea here is that the founding populations in this particular town, the individuals had this particular allele that causes Huntington's disease. This is an example of the founder effect. Okay, now what I wanna do now is demonstrate, since I can't, I don't wanna do any math, I'm gonna demonstrate with the computer simulation the effects of genetic drift, what actually happens through time when you have just genetic drift going on and just natural selection. So this is, if you're interested, this is a little computer program I wrote for the Mac. I like writing computer programs as you might have guessed. And so I figured I'd just write a little program here that does this. What I'm gonna do is, so these numbers at the bottom left, don't worry about them. These are parameters of these mathematical models that allow you to predict the frequencies of alleles at different times. And they're only there here, so I can actually change things. Let's go ahead and first of all, simulate, let's go make the frequency of the big A allele 50%. So this is what we talked about with Hardy-Weinberg. So what I did is I made the big A allele frequency 5.5 and the little A allele frequency 0.5. And we all know that random mating by itself doesn't change the frequency. So if the frequency happens to be 0.5, it stays there forever. If it happened to be 0.1, then it stays there forever, right? Now let's add in, make the population finite. So now we have to tell it how many alleles are. So let's make 50 of the alleles are gonna be, let's make it 100. So you have 100 of the alleles are of the big A type and you have 100 individuals total. So there's 200 alleles in the population. So we're starting, if I did things right here, we're starting the simulation with 50% big A, 50% little A. Let's go ahead and hit the simulate button. So in this case, the big A allele one, right? It goes to fixation. This case, that's very convenient, the big A allele is lost, right? Every time I push the button here, we're gonna get a different realization of the process. It's a random process that involves rolling dice in the computer memory essentially. Just like we did here. Of course the computer does it much more quickly than we do it. But the point here is you see these fluctuations in the allele frequencies and these fluctuations are caused by nothing more than chance, right? And eventually the allele frequencies happen to fluctuate to one or they happen to fluctuate to zero and at that point either the allele is fixed or it's lost from the population. That's all the genetic drift is. Now let's go ahead and make the population size bigger. Let's make it instead of 100, make it 10,000. Now let's see what happens. Still have random fluctuations in the allele frequencies but notice those fluctuations are smaller, right? In fact over the course of the simulation I forget what I put as the upper limit but over the course of the simulation the allele frequencies don't become fixed or lost. Now eventually if I were to run the simulation long enough that would happen. One or the other would win eventually with probability one, one of those two things will happen. But over the course of the short simulation the allele frequencies changed was small enough that you didn't see one or the other become fixed. And here's another example, here's another example. Let's make the population a little bit smaller. This is, I like doing simulations like this. There's one's lost, that's very exciting. All right, but anyways you get the idea, right? Now what is the initial frequency of an allele? It's one over two N, right? So this is kind of an unfair situation. When we started the two alleles it's sort of equal footing and I'm gonna claim that about 50% when the initial frequency is 50%, about 50% of the time the big alleles want wins and 50% of the time the little alleles wins, okay? What I'm gonna do now is make the population size a lot smaller, let's make 10 individuals. And let's start the frequency of, start the frequency of the big allele to be, it's initial frequency when it first appears in a population, it always starts off as there being one individual, right? Now what happens? Well there it was lost, right? That one individual was hit by a bus or something, right? Here it was lost again. Here it was lost again, lost, lost, lost, it won, right? There is a chance that it's gonna win occasionally, right? Let's see, lost, lost, lost, lost wins, lost, lost, lost, lost, lost, lost, lost, lost, lost, lost, lost, lost, lost, lost, lost, lost, lost, win, oh, I went through the win. But you get the idea, I'm gonna claim now that about the fraction of the time that this one wins is about 1 over 2n. The probability that a particular allele ultimately is fixed is its frequency at the time you make the prediction. That's another prediction we can make, or another statement we can make about genetic drift. OK, now that's all just the two alleles, big A and little a, have equal fitness. And basically, here I'm showing the relative fitnesses of the three genotypes. And this is just a reminder to myself, frankly. But they're all the same. So I know I set things right. Now let's make one of the alleles have an advantage. Let's make it 0.05. So this is a huge fitness advantage for being big A. And let's make the population size infinite, and let's start the frequency small. So we're just going to look at what natural selection does just by itself in an infinitely large population. Here we go. It increases the frequency of the allele. Big surprise, right? So this is, once again, there's no genetic drift. It's like I had two knobs. I have the natural selection knob and I have the genetic drift knob. I turned the genetic drift knob so there was no genetic drift. And then I turned the natural selection knob from there being no fitness difference among the genotypes to there being a fitness difference. And in this case, the big A allele, homozygous big A is the most fit. And the homozygous little a is the least fit. And this is the intermediate. So this is an example of directional selection. And it does just what I told you it does. It increases the fitness. I mean, it increases the frequency of the big A allele. We can also model, I put some notes here to myself, we can also model balancing selections. I'm going to change this parameter to be minus 1 now. Now look at the relative fitnesses. This balancing selection is the case where the heterozygous individuals are the most fit. And what did I say natural selection will do in this case? It's going to maintain both of the alleles in the population or at least that's what it wants to do. So here we go. We started off at a frequency of 0.01, 1%, I don't know if I'm doing this wrong, and it goes to some intermediate frequency. And it appears to be staying there forever, right? We could also start the frequency off at some big number. 0.99, say, and see what happens. It starts from the big number and goes down. So natural selection is favoring some intermediate frequency for the two alleles. Is that clear? OK. Now let's go back to the case, I want 0.5 here. Let's go back to this case where we have directional selection acting. But this time let's do a finite population. Let's not have, let's turn on both natural selection. So I basically demonstrated natural selection by itself. I demonstrated genetic drift by itself. Let's now combine the two forces, right? What's going to happen? It's very exciting now. So we're going to make the populations finite in size. Let's start the frequency off kind of intermediate. We'll make 100 individuals, 100 big alleles, and 100. So we're going to start the frequency off at 1 half. So we're giving, we're starting the frequency off at 50, 50, and notice that the population size is kind of small. It's 100 individuals. And that, in this case, remember the big A, big A alleles always favored. If you have a big A, you're always better than if you don't have one. And it increases the frequency to 1. Let's push this button a number of times. There's fix, fix, fix, fix, fix, fix, fix, fix, fix, fix, fix, fix, fix, fix, fix, fix, fix, fix. Almost lost the fix, fix, fix, fix, fix. You see a difference in the pattern? Remember when I just had natural, when I just had genetic drift going on, there was no fitness difference among the genotypes. And I did the same thing, what happened? About 50% of the time, roughly 50% of the time big A was fixed and roughly 50% of the time big A was lost, right? Now clearly we've biased the fraction of time you're lost. Every time I hit this button, big A was fixed, right? We never saw an instance where it was lost. True? Okay, and that makes sense. But now let's go ahead and start the frequency off much smaller. So let's start the frequency of the big A allel off at its frequency that it's gonna be in when it first occurs. After all, when it first occurs, it's gonna be at a frequency. There's only one example of that allel in the population. It's still better to, so the individual that mutation lands on is one lucky individual, right? Because he or she has a real fitness advantage. But let's see what happens. Will the frequency go off to one every time? No, here's one example. There's another example that was lost, right? Here's an example of an allel that's favored being lost from the population. It's not being fixed. Lost, lost, lost, lost, lost, lost, lost is fixed there. Lost, lost. Every time I push the button, another simulation's occurring. There's another example where it was fixed. This is another example. And once again, genetic drift is being hit by a bus. This is an example of that super fit individual that's not looking the right way when they cross the street and being hit by the bus, right? You know, that allel, those good genes were unfortunately lost from the population. So this is kind of a remarkable fact and one you might think about. If natural selection is definitely a powerful force and it does change these allel frequencies, but it's not perfectly efficient, right? It doesn't take every single beneficial mutation that ever occurs and drag it to fixation, right? There's this random chance element that natural selection has to fight. So generally speaking, natural selection is more likely that the alleles are gonna be less likely to be lost through genetic drift when the populations are large. But you're still fighting this fact that when these new beneficial mutations first occur, they're at low frequency. And so basically, natural selection only sort of gains traction when the frequency of this beneficial allel gains some intermediate frequency. Then it's much less likely to be lost and natural selection can do its thing, okay? Is that clear? No? Yes? Okay, good. People are nodding, that's a good sign. All right, so what else did I wanna say? All right, so five minutes to go. I don't think there's anything else I can really demonstrate with these simulations. Maybe we could try one last one, which is let's make the heterozygous individuals more fit and let's see what happens when the populations, so I'm just doing a bunch of simulations here. Oh, ah, missed it. So I'm pushing the button. You can see how often these, there. So there's an example of balancing selection with genetic drift turned on. So most of the time, once again, I had to push the button many times to get an example where the allele wasn't lost. But in this case, the allele isn't lost, but it basically gets to this intermediate frequency and then it sort of just bounces around, right? It bounces around at that frequency and it will do that for a very long time. Now what I'm gonna do next lecture is finish up my discussion of population genetics with migration and then I'll give some examples of natural selection in the wild, okay?