Hardy-Weinberg equation

now that we're familiar with the idea of allele frequency let's build on that to developed to develop the hardy let me just in a new color and actually let me do it right over here the hardy Weinberg Weinberg principle which is a really useful principle for thinking through how what allele frequencies might be or what what probability you would have if you found someone of what's that what support percentage of a population might be homozygous recessive or homozygous dominant or might be a heterozygote and it really builds on the work we've already seen with allele frequency now before we go into that we're going to make some assumptions and these are all just assumptions that get us a stable allele frequency in the population from generation to generation we're going to assume we're going to assume that there's no selection no natural selection is going or so or even unnatural selection is going on that would change the allele frequencies so it's not like people with one of the alleles or another are going to be more or less likely to reproduce and have viable offspring we're also going to assume no mutation no mutation so we're going to assume that one of these alleles can't aren't isn't from generation to generation turning to another one or turning into maybe a a different a new a new type of trait you know whether it's I don't know green eyes or whatever else it might be and we're also going to assume large populations large populations so that would definitely throw out the example that we looked at in the last video which I did just to understand the notion of allele frequencies where we said hey look one out of the four of the one out of the four of the genes in this population or one-fourth of the alleles in this population are the dominant brown well three-fourths or 75% were the recessive blue we're going to assume large population so many many many and so that's so that if you have very small populations you could imagine that depending on how these reproduce it's very easy to get to a changes in frequencies but in larger populations that helps us make the assumption that we have stable allele frequencies so once again this is also that we have stable stable allele allele frequencies now based on that we've already seen if we take the frequency of the dominant trait which we can denote with P and to that we add the frequency of the recessive trait of the recessive I should say allele let me be very careful here the frequency of the dominant allele and to that we add the frequency of the recessive allele what's that going to be well you see in this case it'll adds up to 100% or 1 and it's always going to add up to 100% or 1 because we're assuming that there's only one of two alleles in the population so you're you have a hundred percent chance of getting one of these two that whatever percentage is going to be is going to be whatever the frequency here is one hundred percent minus that is going to be whatever Q is so these two things are going to be equal to one hundred percent or equal to one and now we can start to do a little bit of interesting mathematics that'll allow us to start thinking about things like homozygotes and heterozygotes and so to do that let's square both sides so let's square both sides of this a little bit of algebra and biology class and so when you square the left hand side this is just squaring a binomial you might want to review it if this looks like Latin to you it's on there's many algebra videos on Khan Academy that go into this this is going to be P squared plus plus 2 plus 2 times PQ plus 2 times P Q plus Q squared and of course 1 squared is still going to be equal to 1 now what are each of these terms here what are each of these terms let's just think about something P squared P squared is the same thing as P times P well P is the frequency of your dominant allele so this is a frequency of your dominant allele the percentage of the allele population I guess you could say the allele frequency that is dominant and you're multiplying that times it again well another way to think about P is this is the probability if you were randomly to pick if you were to randomly pick one of these genes so if you're randomly to pick one of these four genes and out here I'm just using my oversimplified population of course for the truths that we're about to surface to be true you're going to have to assume a large population but in this one right over here one way to view P is what's the probability if I were to pick a gene at random what's the probability that it is the variant or it is that is the it is the dominant allele what is the probability that it represents the brown variant so that's one way to view to view P so the probability probability of getting of getting a let's just write it that way a capital B a dominant brown allele so what's P times P that's the probability of getting two dominant alleles or another way of thinking about it this is the probability for someone in the population to be homeles I do be a Dom homozygous dominant so it's the probability it's the probability of someone being it's the probability of someone being capital B and capital B and so what is by the same logic what is Q squared well Q squared that's just Q times Q Q is the probability of getting one recessive allele so this is the probability of getting two recessive alleles one from your mother and one from your father so this is the probability this is the probability of if you're kind of randomly born into this population of getting two recessive alleles now what is this middle term right over here well P times Q so so PQ so one way to think about it if you said what's the probability that from your mother you get the dot you know randomly you don't nothing about them or if you rent you pick a random mother and a random father what's the probability and we have to be careful here so if you're just randomly getting alleles if you're just randomly getting alleles what's the probability that from on one side you're going to get the dominant and on from the second side you're going to get the recessive so that would be PQ that would be P times Q times q so that's getting it from say from one parent and then that's from the second parent but what about the other way around from the first parent getting the you have a cube probability of getting the recessive one and you have from the second parent you have a p probability of getting the dominant one so there's two ways of becoming a heterozygote and so if you add these two probabilities what do you get these are both PQ I'm just changing the order of multiplication you sum these two you sum these two you get two PQ so this is the probability this is the probability of being a heterozygote this is the probability of being a heterozygote so this is a pretty neat result just by making a few assumptions and reasoning through this notion of allele frequency we're able to come up with this with this expression that actually is fairly powerful in in in thinking about allele frequency in a population and actually the different genotype frequencies in a population you see it all makes sense these all add up to one the probability of someone being homozygous dominant plus the probability of someone being a heterozygote plus the probability of someone being a homozygous recessive they're going to add up to 100% because someone's going to have to be one of these three things I'm going to leave you there in this video in the next video we're actually going to use this this heart this hardy-weinberg equation to actually come up with some very interesting results about a population