To be honest, back when I was in college and even when I first started teaching Biology, whenever I hit the Hardy Weinberg questions, I thought big whoop who cares. It’s just a couple of equations that describe a population genetics. And they only work if things don’t change and how often does that happen? It wasn’t until a colleague of mine was getting all excited about teaching Hardy Weinberg, that I found that maybe it is a big warp and maybe I should care.
You see in science, whenever you’re studying something you always want a control group, to compare to your experimental group. And if you are studying something like human evolutions, it’s really hard to get funding, to build yourself a giant second earth. Put your control humans on, compared to this earth's humans. Instead, using some basic statistical math, you can just mathematically create a control group.
It’s kind of like if a friend of yours. If you think there is something funny going on with his coin, well you don’t need to sit there and flip it 100 times and then compare that Result to 50 other coins that you also flipped 100 times. Instead you could just say, well assuming there’s nothing weird with the way he’s flipping it, or weird with the weight or shape of the coin., it should be 50-50, it’s the same thing.
I assume if you’re watching this video, you’ve done some genetic problems. And you know just how useful Punnett squares for predicting the outcome, when you have two different organisms breeding. Well the Hardy Weinberg equations are essentially Punnett squares for an entire population.
Before I go much further we’re going to spend a little bit time defining the terms of the Hardy Weinberg equations and making sure you know what those mean. Next as a statistical tool, Hardy Weinberg is based on a few assumptions. And if it’s predictions don’t much reality, then it indicates that one or more of those assumptions have been broken. The last thing we’ll do is, we’ll go through a couple of the standard Hardy Weinberg equations problem sets that you’ll see on the actual on the actual AP exam.
I’ve already mentioned one of the basic concepts of Hardy Weinberg, which is a population. A population is a group of interbreeding organisms in a particular area. So what are some of the other terms? Well there is this other concept called a gene pool. You got to understand a gene pool is an abstract idea. You can’t go swimming it. What a gene pool is just the sum total of all of the genes in a population. Let’s take a look at an example.
In this population, we have six individuals. And we’re going to be looking at the tongue rolling gene, the ability to do this. That’s a dominant trait. So we can see here that we have two people who are homozygous dominant for the tongue rolling ability. Three individuals who are heterozygous, and then one individual in our population of six, who is homozygous recessive i.e. a non tongue-roller.
So how big is our gene pool? How many copies of this gene do we have? Well we know that every individual has two copies of every gene, that’s Mendel’s first law. So two times six is 12. So we have a gene pool of six genes here. So what’s the frequency of that big R within our gene pool? Let’s take a look.
So I could write big R is one, two and I do some math I count up. And I see seven out of my 12 possible alleles is the big R. I type that into my calculator and get roughly that’s equal to 0.58. In Hardy Weinberg terms they call that the frequency of the dominant allele. And they use the variable ‘p’ to represent whatever the dominant allele is.
What if I was trying to figure out the frequency of this recessive little r, or in Hardy Weinberg terms, ‘q’? Well I could sit there and I could count out the number of the little r’s divide that by twelve. Or I can rip up my incredible math skills and I can say, "Well I know p plus q that frequency of the recessive allele that’s a grand total of one."
So I just plug in my numbers 0.58+q= 1. Subtract 0.58 from both side and I get q equals 0.42. Not so tough. Now what if I was trying to, well what does this mean? It means that the chances of me randomly spinning around and pointing at somebody, and that person happens to have at least big R is 0.58. What’s the chance of that person having two big R’s? Well, if you remember from your math classes, I will just take the probability of it happening once and multiply by the probability of that happening again. Essentially 0.58 times 0.58 or more simply p.
Hey wait a minute, let’s take a look at that p+q=1 thing, again. So p+q=1 let’s see. if I square both sides of this equations I get p² plus 2pq plus q² and 1² is 1.So p² is the chances of somebody being homozygous dominant, well that makes sense. The chance of getting a p and p. q² is the frequency of the homozygous recessive individuals. And then that leaves 2pq to be those individuals who are heterozygous. So that makes sense, p times q, why 2. Well you could get the big R from your mommy and little r from daddy, or the other way around. So there’s two ways to get a dominant and recessive and trait. So that’s why we have that 2 there, and is not just because of the math, it's because it works and it represents reality.
Now I’ve been flinging a bunch of terms out there at you. So let’s take a quick look and make sure that everybody is on the same page with this.
So p is the frequency of the dominant allele in the gene pool. q is the frequency of the recessive allele in the gene pool. So these are describing the abstract concept of the gene pool. P² is the frequency of homozygous dominant people or organisms in the population. So here we’re talking about actual numbers of actual creatures. 2pq is the frequency of heterozygous individuals in the population. While q² is the frequency of homozygous recessive individuals.
So that is basically how Hardy Weinberg works. Now we say this is how it should it out. Let’s take a quick look at our original gene pool and let's plug in our numbers and see if this works.
Well we have here P², here is p. So 0.58² that's p² equals, I do some quick math and I get 0.34. And if you remember, that’s roughly 1/3, that’s 2 of our 6, hey yeah that works. What about 2pq? Well 2pq equals 2 times 0.58 times 0.42. You do some math and that’s simply 0.49. Again I’m rounding a little bit interestingly. And if I look, that’s rough 1/2 and 3 out of my 6 hey it matches.
Now you see how I’m plugging in the numbers on that, I'm just skipping to the end? Ways to get points on essays on Hardy Weinberg. And they like to ask these sorts of questions. Show the math, show the equation and it’ll help increase your score. Let’s do that last one q² and again see if it matches our predictions. q² is simply 0.42². Again I do some simple math and I get 0.18. And 0.18 that’s roughly a 6 and hey 1/6 is my population marks.
So yes this population is in Hardy Weinberg genetic equilibrium. So I’d say that the population was in Hardy Weinberg equilibrium. And that means that from one generation to the next, those gene frequencies shouldn’t change. But there is some assumptions that’s based on. So let’s take a closer look at what are the assumptions of the Hardy Weinberg equations.
First, since it’s a statistical method, it has to be assumed that we’re talking a large population. So if you predict from the Hardy Weinberg equations that you should have 64% of your population is homozygous dominant. But it turns out only say 47% are, then that is an indication that you’re either talking about a small population. Or at some point in the past, the population size greatly get reduced and that caused a tweaking of the predicted outcome.
And this is kind of like if you’re doing that thing with your friend’s coin, and you’re flipping the coins and you’re only flipping three times you know. If you only flip it like I said three times, you can’t get 1.5 heads and 1.5 tails, it just doesn’t work that way.
The thing next is no migration in and out of the population. If some individuals come in, who are all homozygous dominant, then they’re going to skew the results. So that’s another thing that Hardy Weinberg assumes isn’t happening.
Similarly no mutation. You can’t have the things changing. Because if a big R changes into a little r or vice versa again that’s skews the results. Just like if you’re flipping the coin, and instead of getting head or getting tail, the heads turns into tail or a into a foot that just screws things up.
Next up, random mating. Again this is statistical method. And it’s based on the idea that these combination are coming together randomly. And so, if you’ve been able to eliminate the first three assumptions, then that might clue in that the skewing of your Hardy Weinberg predictions is due to non random mating.
The next big one is natural selection. And this like I said is a biggy. Natural selection is giving advantage or disadvantage to one allele or the other. If say having little r was a big advantage, then the one individuals who’s homozygous recessive, he’s going to have great success in providing children for the next generation. So you’ll see a great increase in the numbers of little r’s in the gene pool.
So that’s the assumptions of the Hardy Weinberg equations. And again if everything matches our assumptions, then yes you’re in equilibrium and the frequencies will stay say one generation to the next. But if they aren’t matching out predictions, then this is the nice thing about Hardy Weinberg. Because it give scientist a direction. Rather than just saying something seems wrong, instead they can say, something is wrong. It doesn’t much my numbers, now I have direction in which to go, to solve the problem. To figure out what’s the cause.
So how will Hardy Weinberg show up on the actual exam? Before I get into it, I really recommend that you go online, and check out the official version of the AP lab that deals with Hardy Weinberg. Because that will give you a good example of the kind of questions they may show you and some of the contexts. Also in your bonus materials folder, I’ve also included a worksheet that I gave to my students.
Now because you’re getting a chance to work on it at home I’ve made the numbers pretty hard. On the AP biology test, and in fact on most teacher's test they don’t let you use a calculator. And that’s actually a good thing because that means the numbers that they’re going to give you are pretty easy. So if you don’t know them already, I’d spent a little bit of time just brushing up on your common squares and common square roots. And it’ll make spotting the right answer much faster and much easier.
So let’s take a look at one of the first kinds of questions that they’re going to show you. So here for example, if the frequency of little r, the non rolling alleles is 0.6, then what’s the frequency of the heterozygote? Or the heterozygous individuals in the population?
Well I recall little r is, that’s q. So if I know q equals 0.6, and this is one of the tricks for maxing out your score on the AP exam, the essay portion. State everything. So q equals 0.6, well then I write my equation p plus q equals 1. Again, I just got myself a point by stating one of the equations. I plug in my numbers. And again, show your work. Imagine you’re doing one of your first geometry proofs, and your teacher's sitting there with a ruler ready to get you if you skip something.
So I plug it in, so p plus 0.6 equals 1. So I solve and I get p equals 0.4. So far so good, so now p is 0.4, q is 0.6. So heterozygous individuals they’re 2pq. Plug in my numbers 2 times 0.6 times 0.4, sorry I reversed p and q. I do some math and I get 0.48. So far so good.
Now this was one of the easier questions, because they gave you what q was. What they really like to do is trick you. Let’s take a look at the standard trick question. And I can tell you it’s a standard trick question because I use it all the time on my test. 0.91 of the population can roll their tongue, which is a dominant trait. What’s the frequency of the homozygous dominant individuals?
Now people will sit there and go, "I know it’s 0.91." They’re trying to soccer you in. Because they see 0.91 they say they can roll their tongue, it’s an easy question. No it isn’t. Remember this is the potion of the population that has at least one big R. This includes both the homozygous dominants and the heterozygotes. So the trick is find the q’s, find the q² individuals.
If 100% of the population is 1, then 1 equals p², plus 2pq, plus q² right. So let me plug in. I know that P² plus 2pq is 0.91 plus q². That tells me q² is 0.09. Now I got that, now I take my square root of both sides and I get q equals 0.3. I’ve got q, now p plus q equals 1. Plug it in, go over here. And so I get p equals I do some math 0.7. Now I just say p² my homozygous dominant, is 0.7² or 0.49 and there we go. That’s it, that’s not that hard.
So the trick is, read what they’re giving you, if they say the frequency of the gene is then they’re giving you p or q. The frequency of the alleles. If they say some fraction of the population, then you start thinking they’re trying to trick me. Read very carefully. Are they saying 91% of the population do this dominant thing? They’re trying to sneak it past you. Look for the freaky ones, look for homozygous recessives. Find them, and then you can get q. Once you get q then you can get p, and then you can plug it in to figure out whatever genotype that you want. And that’s it.
Outside of the whole square root thing, none of the math is actually all that hard. And in fact, in the most recent question that involved Hardy Weinberg, they actually gave you p, and they gave you a q. So don’t get too wordy about it.
And remember, the Hardy Weinberg equation is all about describing the gene frequencies, and genotype within a population. Now, it’s based on some assumptions, things like known natural selection or a large sample size, or large population. So if all those assumptions are met, then you don’t expect to see any changes from one generation to the next. If you do see changes, or you see that the predictions don’t match the outcome, then it’s because one or more of those assumptions have been broken.
Go through the end of the chapter questions in your textbook, do some of the problems, and that will give you the practice that you need to do well in the AP exam. And if you can handle all of the questions on my worksheet, then you’re pretty good. You’ve mastered Hardy Weinberg. Just be careful I made them tough and I’ll make you actually to think.
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.