1. Hardy-Weinberg Equation begins

2. p Hi, I’m p and I represent the frequency of the Dominant allele

3. If you don’t remember the genetic terminology, here is a little reminder. Individuals like humans, are diploid which means that they have two copies of each chromosome. Even though humans have a total of 46 chromosomes, there are actually 23 chromosome pairs. Each chromosome has many alleles that represent genetics for a particular trait. Ex blonde hair, blue eyes Since you get two copies of each chromosome, all the alleles are also paired. When they are paired they determine how you look– your phenotype. You can get combinations as follows: TWO DOMINANT ALLELES = DOMINANT PHENOTYPE A DOMINANT AND A RECESSIVE = DOMINANT PHENOTYPE TWO RECESSIVE ALLELES= RECESSIVE PHENOTYPE

4. In a population, there are many people with many different traits, but if only one trait is considered, it becomes much more simple. Given one trait, in a simple model, there are only two possible alleles, either dominant or recessive Previously, you learned about punnett squares and how to calculate probabilities for phenotypes given a cross between two individuals. Ex Brown hair is dominant -B --- Blonde hair is recessive =b Given the cross Bb x Bb determine the punnett square Bb B b BBBb bb Here the phenotypes are ¾ brown hair and ¼ blonde hair. RECALL THE MENDELIAN 3:1 RATIO

5. Now consider that rather than this being one cross it represents the crosses for a large number of individuals in the population. The people with homozygous brown hair are BB IN THIS POPULATION: The people with heterozygous brown hair are Bb The people with homozygous blonde hair are bb From the previous punnett square, you might say that 75 are brown hair and 25 are blonde hair. This is the 3:1 ratio and 75+25 = 100 total population. p Hi I’mI’m back to substitute in for the dominant allele q Meet my partnerHe is here to substitute in for the recessive allele pq p q pp pq qq = pp =pq =qq Given that 75 people are dominant brown hair pp+pq+pq=75 REMEMBER this includes both heterozygotes and homozygotes Given that 25 people are recessive blonde hair qq=25

6. A DINNER SCIENCE DISCUSSION 1908 Reginald Punnett -- inventor or punnett square analysis for genetics Godfrey Hardy – eminent mathematician I say Godfrey old boy, do you think there could be a way of determining the occurrence of certain alleles in the population utilizing my punnett squares ? In addition, could it help determine if alleles change in a population? Godfrey wrote the following on his napkin. p + q = 1 P 2 + 2pq + q 2 =1 This means that together the total brown and blonde alleles are 100% This is the combination of the algebra from the previous slide where pp+pq+pq+qq =1 and 1 is 100% REMEMBER pp is mathematically pxp or p 2 There are 2 pxq values so this is shortened to 2pq And finally qq the recessive allele is mathematically q2

7. The Birth of the Hardy-Weinberg Principle The allele frequencies of a population will not change from generation to generation, as long as the following conditions are met: the population is very large mating opportunities are equal no mutations occur no migration occurs no natural selection occurs—all individuals have an equal chance of reproductive success p + q = 1 Do you remember the equations yet? Here is a reminder below P 2 + 2pq + q 2 =1

8. Given the hair colour scenario, populations essentially will not change in their genetic composition if they obey Hardy-Weinberg In this picture, the percentage of blonde haired and brown haired individuals will remain constant even though people will freely marry (called random mating) and have children within this population. In addition, the population may expand, but the percentages will remain the same. The reason for this is the genetics. A specific gene is found in a certain number of people. This is called the gene frequency p q Hi I’m And I’m here to help you. I actually represent the gene frequency of the Dominant allele. I, however am not as useful as my friend q for solving Hardy Weinberg problems Hi, I’m And you should always try to determine my value first. Even if they give you p, use the equation p+q=1 to determine me. You will know when you get me in a question because it will say the word fre q ency

9. Lets try a sample simple problem In a population blonde hair is recessive to brown. The gene frequency of the blonde allele is 0.2 Determine the frequency of the brown allele. TRY IT ANSWER Since you are given q=0.2 you simply utilize the simple formula p+q=1 to determine that p =0.08 Now, extending on this same question, you could be asked to determine the percentage of individuals who are actually blonde haired in the population. You may think you already know, but you are wrong if you say 0.2 because that is a gene fre q uency not a number of individuals! In order to determine the number of individuals, you must utilize the variables from the second equation. P 2 + 2pq + q 2 =1 P 2 + 2pq is the decimal percent of number of brown hair people DOMINANT q 2 is the decimal percent of the number of blonde hair people (RECESSIVE) THIS IS WHAT WE WERE ASKED FOR IN THE QUESTION! ANSWER Since q=0.2 q 2 =0.04 Now multiply by 100% and you have 4% blondes TRY IT

10. Now try this challenging? question. If the percentage of blondes was determined to be 4% what is the percentage of brunettes in our population? TRY IT ANSWER Well the answer is very simple if you think clearly and do not get caught up in the math. If 4% are blondes all the others are brunettes so it is 100%-4% = 96% Now there is the hard way of determining this answer and eventually you will have to know how to utilize this aspect of the formula so here is an alternate solution:P 2 + 2pq + q 2 =1 P 2 + 2pq = decimal percent of brunettes Substituting in the unknowns p= 0.8 q=0.2 0.64 + 2(.16) = 0.96 x100% = 96% brunettes Therefore the percentage of brunettes in the population is 96% Ok so good so far. Now lets try to interpret some of the other information, once again without getting caught up in the math. What percentage of brunettes in the population are heterozygous brown hair? TRY IT ANSWER The math is already done for you from the alternate solution above. You, of course, would have to do this if you did the shortcut. 2pq = 2(.16) = 0.32 x 100% = 32% heterozygous brunettes

11. INTERESTING FACTS Godfrey Hardy developed the equations just a little before Weinberg who also independently developed the same formula. Hardy thought the equations so simple that he hardly thought to publish and thus claim his discovery. Only at the insistence of Punnett did he do so. It was lucky he did, for even though he was an eminent mathematician, this is the only equation he is widely known for today.

12. P Hi again, I’m And I am the value given for the dominant gene frequency Now meet my big brother p2p2 Here, p 2 is used to indicate the number of DOMINANT homozygous individuals. This may also be rephrased in a deceiving way, so watch out. It could say; The frequency of homozygous black individuals—where black is the dominant allele. This is still the p 2 value. You must also be aware that it is very unusual to get p 2 initially in the word problem, but it is possible. Remember, that once you have p 2 you could obtain p by simply square rooting the p 2 value. Ex p 2 = 0.81 p = 0.9

13. q Hi I’m Remember, that as a general rule you should always start any H-W problem with me! I represent the fre q uency of the recessive allele Now meet my big brother q2q2 Now, you will know that you are given a q 2 value if the question stipulates the number of recessive individuals. Anyone who expresses the recessive phenotype must be a double recessive without exception. So you know that you have the q 2 value if the question is phrased like; The number of individuals with white fur is 16 out of a population of 100. The q 2 value would thus be 0.16. The next step is always to determine the q value.

14. Now lets do some more problems, but I don’t want you to use your calculator right now. It is important to get your brain thinking, and for you not become reliant on the machine. In order to prepare your brain for the shock of actually having to do some math calculations, lets try practising. The square root of 16 is?Try it!= 4 The square root of 9 is?Try it!= 3 The square root of 0.16 is?Try it!=0.4 The square root of 0.09 is?Try it!=0.3 The square root of 49 is?Try it!= 7 The square root of 0.49 is?Try it!= 0.7 If q 2 = 90/1000, change the value to a decimalTry it!q 2 =0.09 Now solve for q in the above lineTry it!q=0.3 If q 2 = 36/100, change the value to a decimalTry it! q 2 =.36 Now solve for q in the above lineTry it! q = 0.6 If q 2 = 160/1000, change the value to decimalTry it! q 2 = 0.16 Now solve for q in the above lineTry it! q =0.4

15. Now it’s time to analyze the word problems and determine if you can figure out the unknowns you are given. All of the following represent populations in Hardy-Weinberg equilibrium In humans, red hair is recessive to brown. The gene frequency of red hair is 0.6. What value are you given? Try it! You are given q = 0.6 In rats white fur is recessive to brown. The number of white rats is 36 out of population of 100. What value are you given? Try it! You are given q 2 =.36 In moths, red wing spot colour is dominant to white. There are 18 white spotted moths in a population of 200 moths. What are you given? Try it! You are given q 2 = 18/200 or in lowest terms q 2 = 0.09 REMEMBER, every time you get q 2 you must convert it to q to determine the other values! DON’T FORGET!!!!!!

16. What are you given? Continued with some tricks In a bat population in H-W equilibrium, short legs are dominant to long. The frequency of long legged individuals is 0.09. What value are you given? Try it! ANSWER You are given q 2 =0.09 Frequency is the trick as the GENE FREQUENCY would be q. The indicator of this trick is that these are individuals NOT gene frequencies A population of flower has red, white and pink flowers. Crosses reveal that this polymorphism is controlled by one locus and two alleles. WW individuals have red flowers, Ww pink and ww white flowers. If the frequency of white flowered individuals in the population is 0.09, what is the frequency of the pink flowered individuals if the population is in H-W equlibrium? Try it! ANSWER You are givenq 2 =0.09 thus q=0.3 Since p+q=1 p=0.7 From the equation p 2 +2pq+q 2 =1 the pink flowers would be the heterozygotes 2pq Substituting 2(0.7x 0.3)= 0.42 or 42% Therefore the frequency of pink flower plants is 42%

17. Now, lets try some questions that walk you through the steps. The frequency of the dominant allele in a population of 100 is 40%. What are you given? Try it! You are given p = 0.4, and you would determine q to continue with this question. The formula you would use to determine q would be? Think p+q =1 The value of q is therefore 0.6 You could then substitute these values to calculate all the unknown population frequency number utilizing this equation Think Use p 2 + 2pq + q 2 = 1—Determine how many in the population are the dominant phenotype Think Since q=0.6 therefore q 2 = 0.36 or 36%. This means that there are 36 out of 100 with the recessive phenotype. That means the rest must be dominant ---64 individuals are the dominant phenotype. You can also do it the long way and substitute into p 2 + 2pq This question is unusual because you initially are getting the value for p.

18. Lets try another question with a little less of a walk through. In cats black fur is dominant to white. If there are 84 black cats and 16 white cats, determine how many of the cats are heterozygotes? You are given two values. What do you do? Think think think This is a little tricky, first calculate the total number of cats 84+16= 100 cats Next. Always try to determine q The number of recessive phenotype cats is 16, thus q 2 is 16/100, If q 2 =.16, then q=.4 Now can you answer the original question -- How many cats are heterozygotes? Remember, the heterozygotes are black as they possess the dominant allele Think think think p+q=1 thus since q=0.4,therefore p =0.6 Use p 2 + 2pq + q 2 =1 -- The component you need from this equation is 2pq Just substitute 2(0.4x0.6) =.48 Thus 48% of the cats are heterozygous black If you needed to calculate how many of the cats are homozygous black, it would be easy at this point, just substitute Since p=0.6 p 2 = 0.36 -- 36% of the cats are homozygous black

19. Now you should feel a little more confident. Try this question Assume that a population of diploid individuals is in H-W equilibrium for a trait controlled by one locus and two alleles. If the frequency of the recessive allele is 0.8, what is the frequency of heterozygous individuals? Think think think ANSWER You are given q You just use p+q=1 to determine p=0.2 Once you have this substitute into the relavant part of p 2 +2pq+q 2 This is 2pq –means the decimal percent of the heterozygotes 2(0.2 x 0.8)= 0.32 Therefore the frequency of heterozygous individuals is 0.32 or 32% of the population are heterozygotes. In a population of diploid individuals in H-W equilibrium, the frequency of the dominant allele for a certain hereditary trait is 0.3. What percent of individuals in the next generation would be expected to be homozygous for the dominant gene? Think think think ANSWER You are given p. The representation of the HOMOZYGOUS dominant individuals is p 2 from p 2 +2pq+q 2 =1 Since p=0.3 p 2 = 0.09 or 9% homozygous individuals This question is rare because you are initially given the p value and you don’t need to calculate q. Usually q is what you need to find first!

20. Now for some little twists to the question. See if you can find the tricks For a trait that is controlled by two alleles at a single locus, the frequency of the dominant allele is 0.6. What is the percent of individuals that show the dominant trait? Find the trick first Think think think ANSWER You are given p, but this time you are asked to determine p 2 +2pq Can you see why? –it’s because this represents all individuals that express the dominant allele. You must start this with the STEP 1 DETERMINE q Now continue with the solution Think think think ANSWER Since p=0.6 and p+q = 1 therefore q=0.4 Substitute the values into p 2 +2pq 0.36 + 2(0.6 x 0.4)= 0.84 or 84% express the dominant trait In a spider, grey colour is dominant to white. There are 910 grey spiders in a population of 1000. How many of the grey spiders are heterozygous? Think think think ANSWER You are given p2+2pq = 910/1000 DON’T TRY TO SOLVE THIS –GO THE EASY WAY If 910 spiders are grey that means that 90 are white – this is q 2 = 90/1000 or q 2 = 0.09 and thus q= 0.3 and p=0.7 Heterozygotes are represented by 2pq so substituting 2(0.7x0.3)=0.42 or 42% of the spiders are heterozygous grey

21. Now try something even more difficult. Can you see the trick If one gamete in five carries a recessive allele, what must be the frequency of the homozygous recessive genotype in a population at H-W equilibrium? Think think think ANSWER The gamete alleles mean the same as the frequency of the recessive allele. Therefore q=1/5 Now this is the same as q= 2/10 or 0.2 Shame on you if you needed your calculator Since q=0.2 thus the frequency of the homozygous recessive is q 2 =0.04 or 4% -------------------------NOW YOU CAN USE YOUR CALCULATOR----------------------- Suppose that 1 in 400 people in a large population have a recessive disorder. Apply the Hardy–Weinberg principle to estimate the proportion of individuals who are carriers of (i.e., heterozygous for) this disorder. Think think think ANSWER You are given q 2 =1/400 thus q 2 = 0.0025 and q=.05 Substitute into p+q=1 and p=0.95 Utilizing p 2 +2pq+q 2 =1 the carriers (heterozygous) are 2pq Substitute 2(0.95x 0.05)= 0.095 Therefore 9.5% of the people are carriers.

22. Now let’s try to look at something closer to reality. If this crowd were in H-W equilibrium, we could predict the percentages of blondes in all future generations given a simple analysis. Simplified as Dark hair etc. dominant to blonde Count how many of these people are blondes “Count the total number of people n the population I count 3573 people of which 142 are blondes Determine the future percentages of Dark haired people, blonde haired people homozygous dark haired people, heterozygous dark haired people and homozygous blondes.

23. The number of blondes is actually the number of individuals with the recessive phenotype. Thus q 2 =142/3573 If q 2 = 0.03975 then q= 0.2 Utilizing p+q=1 p=0.8 Utilize p 2 +2pq+q 2 =1 to determine the percentages p 2 = 0.64 or 64% homozygous dark hair 2(0.8 x0.2)=.32 or 32% heterozygous dark hair q 2 = 0.04 or 4% (homozygous) blonde hair Also there would be 96% Dark haired people and 4% blonde haired people, not just in this picture, but in all future generations of this population in H-W equilibrium.

24. Extension: Real Project for extra credit Why don’t you try to measure one genetic trait in our Cardinal Carter Secondary School population. See if you can choose a trait that is only due to two alleles to keep it as simple as possible. (Hair and eye colour are not good choices due to multiple alleles if you remember this from your grade 11 work) Conduct an actual count -- determine p and q and predict what the percentages of the trait would be in future generations if the population were in H-W equilibrium. (It’s not really H-W equilibrium –discuss why) In the future, we can resample compare your numbers and see if the numbers are shifting or relatively stable. Hand it in to me before the final exam if you want credit

25. CONGRATULATIONS YOU HAVE COMPLETED YOUR HARDY- WEINBERG TRAINING --GOOD LUCK WITH THESE ON THE EXAM –REMEMBER TO TRY THESE ON THE QUIZZES PROVIDED ON THE WEBSITE

26. THE END