Patrick Roisen

**M.Ed., Stanford University**

Winner of multiple teaching awards

Patrick has been teaching AP Biology for 14 years and is the winner of multiple teaching awards.

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The **Hardy-Weinberg** Law is an equation for calculating the frequencies of different alleles and genotypes in a population in genetic equilibrium and expressed by the formula *p + q = 1* where p is the frequency of the dominant allele and q is the frequency of the recessive allele. For genotypes the equation is *p^2+pq+q^2 = 1* in which p^2 is the frequency of the homozygous dominant genotype, 2pq is the frequency of the heterozygous genotype and q^2 is the frequency of the homozygous recessive genotype.

One of the basic things in population genetics is a concept known as the Hardy-Weinberg Law. The Hardy-Weinberg Law is a collection of two equations that is used to Mathematically calculate the frequencies of alleles within the gene pool of a population and the frequency of genotypes within that population. So I need to make sure that you understand some basic ideas first population is a group of organisms within a particular area where all are in a breeding with each other. And then you need to understand that gene pool is this abstract idea, it's the collective, or a collection of all the alleles within a particular population for any trade that you're talking about.

For example let's suppose that we had this population here now the Hardy-Weinberg equation for describing that gene pool is p+q=1, now p is a variable used to represent the frequency of the dominant allele in this case I'll be talking about the trait big R for rolling your tongue. Q is the frequency of the recessive allele within that gene pool, little r in this case or the inability to roll your tongue. Alright so if our population consisted of somebody's homozygous dominant, another persons who is homozygous dominant, somebody who is heterozygous for the tongue rolling ability and then somebody who is homozygous recessive.

What is p and whit is q in my gene pool? Well I would solve this simply by counting up 1, 2, 3, 4, 5 big R alleles out of the total number in my gene pool of 1, 2, 3, 4, 5, 6, 7, 8 alleles which gives me 0.625 so p in this case is 0.625 whereas q is I could solve this in 2 ways. One I can still go 1, 2, 3 divide by 8 or I could pop into my Hardy-Weinberg equation here p+q=1 and do some subtraction. And I could say well if p is 0.625 then I know 1-0.625 is 0.375 alright that's pretty straight forward. Now it gets a little bit more complicated when we go to the equation for describing the individual population's genotypes, but don't worry because you've actually seen this before. What I did is I took my p+q=1 and I just did what you've done before in your Math classes I squared both sides you know in Math if you do one thing to one side of the equation, you can do it to the other side of the equation and things still work out. So I get p squared plus 2pq plus q squared equals 1.

What's p squared? p squared is the frequency of individual who has homozygous dominant. In my example of tongue rolling it's big R, big R, 2pq is the frequency of big R, little r or heterozygous individuals. q squared is the frequency of little r, little r or homozygous recessive individuals and again this is for a population that's in something called genetic equilibrium that's where there's no changes going on from one generation to the next in terms if these frequencies either of the genotypes or frequencies of the alleles within the gene pool. Now an example kind of a problem that you're going to face when you're studying Hard-Weinberg is one like this, for example in a population 0.16 of that population i.e. 16% of the population cannot roll their tongues. What is the frequency of big R, little r now a common thing is for kids to go "0.16 can't roll their tongues" that's q no.

Remember q is describing the gene pool just what fraction of the total DNA in the entire population is the little r? You know that everybody has 2 copies of every gene so somebody has to have 2 copies of little r in order to show non rolling. So 0.16 is little r, little r q squared so I pop that in. q squared equals 0.16 alright what do I do next? Well that's when I whip out my Math skills and I take the square root of both sides, if q squared is 0.16 then q is the square root of 0.16 or 0.4. Now I go to my first of the Hardy-Weinberg equations p+q=1 and if I plug in some numbers, let's see p winds up being p+0.4=1 p=0.6 alright so now I go up to here which one of these is heterozygous individuals big R, little r? 2pq, so 2pq equals 2 times 0.6 times 0.4 which tells me that the frequency is 0.48 sp roughly half of the population is heterozygous they can still roll their tongue but they are actually carrying that little r, that non-rolling allele but it's recessive so they don't show it.

Now here's a little trick, I'm going to let you in on a secret that Biology teachers have. We love to mess with your heads, they will rarely unless your teacher is really nice, they'll rarely say 0.16 cannot roll their tongues what they'll do is they'll say 0.84 can roll their tongues. And we'll sit in the backroom giggling we got them, because a lot of kids will sit there and go 0.84 can roll their tongues and they'll assume 0.84 is big R, little r sorry big R, big R. In actuality 0.84 is this plus that. So the trick to solving Hardy-Weinberg is look for the q squared, look for homozygous recessives. So if they say 0.91% can or 0.91 can roll their tongues you know that 0.09 cannot, and that's what you're supposed to solve for okay, little trick that I'm telling you guys but don't tell to my own students I want to trick them.

Now this is all Math, this is all based on statistics and if you've ever taken any kind of statistics class you know that there's a bunch of assumptions. When I flip a coin I assume 50% of the time it's going to come up heads, 50% of the time it'll come up tails. What are the assumptions of the Hardy-Weinberg equations? First this is assuming we're describing a large population, you know that if I flip a coin once, it's not going to come up half heads, half tails. It's either going to be 100% heads or 100% tails. If I flip it 10 times it could come up 7 times heads, 3 times tails and you're going to sit there and scream that the end of the universe is coming. No, but if you flipped it a thousand times and you got 700 heads and 300 tails then you might assume either there's something weird going on with the coin or there's something weird gong with the rules of the universe.

No mutations, if you flip a coin one side heads and the other side becomes a wing that's weird this assumes that if a population isn't undergoing genetic change, one of the assumption is that there's no mutations. Another assumption is no migration, if you flip a coin and all of a sudden somebody tosses down their own coin what? That changes things so no people moving in with their tongue rolling abilities or non of our non rolling cousins leaving. You also have to assume that mating is random for that particular trait so that people aren't suddenly going can you roll your tongue alright. And no natural selection, there's no advantage or disadvantage to that particular trait whether it's tongue rolling or shape of your ear lobes whether or not you have widows peak or whether or not your thumb goes straight up like this or bends backwards.

Alright so these are the assumptions of a Hardy-Weinberg equation and I'm betting you're thinking in your head what I used to think when I first started studied this stuff. Most of the time one of these is going to be violated and if not more. So why ever use the Hardy-Weinberg equation? I used to think that even when I was teaching it until one of my colleagues explained to me exactly why this is so useful. This gives us our control group because you know in an experiment you always need to have your control group and your experimental group. Your control group is just like your experimental group except for hopefully one thing is different.

Well if you're studying say humans you cannot get the funding and the legal ability to take 500 million humans put them on another planet earth and then wait ten thousand years and see how their population genetics have changed to study their evolution. It's kind of hard and expensive, instead we can use some Math and we can say well assuming these things we shouldn't see any difference in our predictions from Hardy-Weinberg and our reality human genetics. And then if you do see differences you can say well did I have a large population I'm I studying the population of China? That's a large population, I didn't break that assumption. Are there mutations going on? Maybe, maybe not did this happen during the migration? I start figuring out, narrowing in what exactly is causing the difference between my predictions from Hardy-Weinberg and reality?

And scientists have used this to figure out all sorts of interesting things for example some forms of diabetes you would expect should be essentially gone because they're deleterious they can lead to you dying. So why is some forms of diabetes so common, scientists had calculated that if you have a certain disease and it's more prevalent in the population in a certain frequency then there's going to be some reason why it's still there. And it turns out this has led to some people suspecting that diabetes, type 2 diabetes maybe in some circumstances actually in advantage or maybe due to people having highly efficient body systems and so their metabolism is very efficient. Which is great if you're going through a famine or things like that, but if you have easy access to McDonalds and those other things then it becomes a disadvantage.

Now let's, using our assumption let's see if our original population was in Hardy-Weinberg equilibrium. Now the way I test this, let me grab a pen is I'd say okay I know what p is so what about p squared? p squared equals 0.625 squared alright so I do the Math and I discover that p squared should equal 0.39, 0.39 let's see in my actual population 1, 2 out of 4 0.5 so 0.5 my real population is homozygous dominant but I have predicted through Hardy-Weinberg only 39% of them or 0.9 should be homozygous dominant. I violated something well this is assumption number 1, this is a small population 4 individuals is not sufficient. So that's one reason why this population is not in a Hardy-Weinberg equilibrium.