I. ! ! ! Mathematical Modeling: Hardy-Weinberg

II. Purpose ! The purpose of these labs was to reenforce the ideas of the Hardy- Weinberg Theorem. It was also to show evolution at work in a population of organisms. Evolution usually occurs through Natural Selection which will happen as an animal needs to change and adapt to the needs of a changing environment. The proportion of genotypes in a population from generation to generation can be determined using that Hardy-Weinberg Theorem. This only works though under the conditions of Mendelian segregation and recombination of alleles is at work. This theorem shows how Mendelian genetic inheritance preserves genetic variation. The proportion of allelic frequencies can be calculated by using the formulas p2+2pq+q2 and p-q=1. Genetic frequencies would change with the changes in the environment, the phenotype or genotype that is best suited for that specific environment will be the most observed at the time. When a bottleneck effect is observed the allelic frequency will be drastically changed from what it was. In addition when a founders affect is observed two populations of a species will evolve differently and become genetically different as they are no longer breeding together. III.Hypothesis ! Lab A. If you Randomly pick variable then the offspring produced in each generation will evolve to suit their changing environment. ! Lab B. If organisms are separated randomly then the offsprings that evolve from the parents will have a smaller gene pool and evolve differently from one another.IV.Materials ! Lab A. Computer, Paper, Pencil, AP Biology Investigative Lab book, Excel (spreadsheet) ! Lab B. Paper, Pencil, Ruler, Lab Packet, Notebook, Classmates V.ProcedureLab A:Step 1: Open a program that enables the creation of spreadsheets.Step 2: Complete the steps of model making1. Formulate the question.2. Determine the basic ingredients.3. Quantitatively describe the biological system.4. Quantitatively describe the biological system.5. Analyze the equations.6. Perform the checks and balances.7. Relate the results back to the question.Step 3: To quantitatively describe the biological system you need to:1. Set variable for the frequency of the alleles.

a. Allele A can be p and Allele B can be q2. P and q must add up to be 1. So after you input a value for p in D2 you can then

put =1-D2 into D33. Next it is necessary to simulate random mating and this is done by using the

function =RAND(). In E5 input the equation =IF(RAND()<=D$2,”A”, “B”)

4. Create the same formula in E5. Then Copy them both down for 16 cells.5. Place the zygote in cells G by entering the equation =CONCATENATE(E5,F5) then

copy this formula down to match the numbers in column F.6. Columns H,I,and J are used to keep track o the numbers of each zygote’s

genotype.7. In H5 input the equation =IF(G5=”AA”,1,0). Then copy it down the column8. In J5 input the equation =IF(G5=”BB”,1,0). Then copy it down the column9. In I5 input the equation =IF(G5=”AB”,1,(IF(G5=”BA”,1,0))Then copy it down the

column10. Calculate the sums of each generation by using the sum function.11. Use the genotype frequencies to calculate new allele frequencies and to

recalculate new p and q values. 12. Make a bar graph of the genotypes using the chart tool.13. Copy the chart and use different p and q values to describe the patterns of

different allele frequencies over many generations.14. Use the model before to determine the p and q values of the next generation.15. Create graphs for the following generations to observe the change.! Lab B:Step 1: Determine if you are a PTC taster and if your earlobes are attached or unattached.Step 2: Determine of homozygous recessive frequency (q2). Do this by dividing the number of students who cannot taste PTC by number of students in the class. This gives you the q2 value. ! Do the same for attached earlobes.! Fill in the first column of table 1.Step 3: Determine the frequency of the recessive allele (q) by calculating the square root of q2. Fill in the second column of table 1.Step 4: Determine the frequency of the dominant allele (p) using the formula p+q=1, that is, p=1-q and fill in the third column of table 1.Step 5: Determine the frequency of the homozygous dominant genotype (p2) by multiplying p times p and fill in the fourth column of table1.Step 6: Determine the frequency of the heterozygous genotype (2pq) by multiplying 2 times p times q and complete column five in table 1.Step 7: Select a trait for further testing.Step 8: Test the Hardy-Weinberg law1. Calculate the genotypic frequencies for the given trait.2. Once the genotype to be used is contrived you must mate randomly with your

fellow class mates for five generations and then record your results.3. Separate 6 students onto and island to create a founders effect and mate within

your population for five generations.4. Calculate the genetic frequencies of each population and how they have

become different.

VI. Data Collection! Lab A:

P = Frequency of A= 0.6q= Frequency of B= 0.4 Number of each GenotypeNumber of each GenotypeNumber of each Genotype

0.4 GametesGametes Zygote AA AB BBA A BB 0 0 1A A BB 0 0 1A B AB 0 1 0B A BB 0 0 1A B AB 0 1 0A B AB 0 1 0B A BB 0 0 1B A BB 0 0 1A A AB 0 1 0B A BB 0 0 1A A AA 1 0 0A B AA 1 0 0B A BA 0 1 0B A BA 0 1 0B A BA 0 1 0B A BB 0 0 1

Sums for each genotypeSums for each genotypeSums for each genotypeSums for each genotype 2 7 7

A Bnumber of each allelenumber of each allelenumber of each allele 0.125 0.4375 0.4375

p qallele frequency next generationallele frequency next generationallele frequency next generationallele frequency next generationallele frequency next generation 0.3536 0.6614 0.6614

Generation 2Generation 2p 0.3536q 0.6614

GametesGametes Zygote AA AB BBB A AB 0 1 0B A AB 0 1 0A B AB 0 1 0B A BB 0 0 1A A AB 0 1 0B B BB 0 0 1A B AB 0 1 0A A AB 0 1 0A A AB 0 1 0A A AB 0 1 0A B AB 0 1 0B A BB 0 0 1A A AB 0 1 0B A BB 0 0 1

A B AB 0 1 0B A BB 0 0 1

0 11 5

A B0 0.6875 0.3125

p q0 0.8292 0.559

Generation 3p 0q 0.559

GametesGametes Zygote AA AB BBA A AB 0 1 0A A AB 0 1 0B A BB 0 0 1B B BB 0 0 1B A BB 0 0 1A A AB 0 1 0B A BB 0 0 1A B AB 0 1 0A B AB 0 1 0A A AB 0 1 0B B BB 0 0 1A A AB 0 1 0B A BB 0 0 1B B BB 0 0 1A B AB 0 1 0B B BA 0 1 0

0 9 7

A B0 0.5625 0.4375

p q

P= 0Q= 0.66143783

0

3

5

8

10

Generation 1

Genotype Frequencies

AA AB BB

0

2

5

7

9

Generation 2

Genotype Frequencies

AA AB BB

! Lab B:Non Taster with attached ear lobes

0

2

5

7

9

Generation 3

Genotype Frequencies

AA AB BB

0

0

0

0

1

Allelic Frequencies

P Q

Taster Non-taster Attached Earlobe Free Earlobe

x x

x x

x x

x x

x x

x x

x x

x x

x x

x x

x x

x x

x x

x x

x x

7 10 aa 7 aa 10Table 1

Trait q2 q p p2 2pq

PTC Tasting 0.58 0.76 0.23 0.05 0.35

Earlobes 0.41 0.64 0.35 0.12 0.45Table 2: Present Generation

Genotypes Genotypic FrequenciesGenotypic Frequencies Number of StudentsNumber of Students

PTC Tasting Earlobes PTC Tasting Earlobes

Homozygous Recessive (q2)

0.59 0.41 10 8

Genotypes Genotypic FrequenciesGenotypic Frequencies Number of StudentsNumber of Students

Homozygous Dominant (p2)

0.06 0.13 1 2

Heterozygous (2pq)

0.45 0.46 6 7

Breeding Round 1:! ME! MATE!! Child!F1! tt! tt! ! ttF2! tt! Tt! ! TtF3! Tt! Tt! ! TTF4! TT! Tt! ! TtF5! Tt! Tt! ! Tt

Table 4: F5 GenerationGenotypes Genotypic Frequencies Number of Students

Homozygous recessive (q2)

0.05 1

Homozygous dominant (p2)

0.57 10

Heterozygous (2pq) 0.36 6Table 5: Parental Generation

Genotypes Genotypic Frequencies Number of Students

Homozygous recessive (q2)

0.05 1

Homozygous dominant (p2)

0.57 10

Heterozygous (2pq) 0.36 6

Class Results:Heterozygotes! ! Homozygous Recessive! ! Homozygous DominantTt! ! ! ! tt ! ! ! ! ! TT! ! !

12! ! ! ! 1! ! ! ! ! 4! !

Breeding Round 2:! ME! MATE!! Child 1! Child 2! Chosen Child!F1! Tt! Tt! ! Tt! ! Tt! ! Tt! !F2! TT! Tt! ! Tt! ! Tt! ! TtF3! Tt! tt! ! Tt! ! tt! ! tt!F4! tt! tt! ! tt! ! tt! ! tt! ! !F5! tt! Tt! ! Tt! ! tt! ! tt

Our portions results:Heterozygotes! ! Homozygous Recessive! ! Homozygous DominantTt! ! ! ! tt ! ! ! ! ! TT! ! !

6! ! ! ! 2! ! ! ! ! 3! !

Table 6: F5 Generation Genotypes Genotypic Frequencies Number of Students

Homozygous recessive (q2)

0.181818 2

Homozygous dominant (p2)

0.3294 3.6

Heterozygous (2pq) 0.489 5.4

Island Results:Heterozygotes! ! Homozygous Recessive! ! Homozygous DominantTt! ! ! ! tt ! ! ! ! ! TT! ! !

1! ! ! ! 1! ! ! ! ! 4

Table 7: Island ResultsGenotypes Genotypic Frequencies Number of Students

Homozygous recessive (q2)

0.166 1

Genotypes Genotypic Frequencies Number of Students

Homozygous dominant (p2)

0.35 2

Heterozygous (2pq) 0.48 3

VII. Assigned Questions Lab A. 1. What can you change in your model? If you change something, what does the

change tell you about how alleles behave?! You can change the original allelic frequencies and this shows that everything is completely random.2. Do alleles behave the same way if you make a particular variable more

extreme? Less extreme? ! The should behave the same way in the fact that they will combine together randomly.3. Do alleles behave the same way no matter what the population size is? to

answer this question you can insert rows of data somewhere between the first row of data and the last row the copy the formulas down to fill in the space.

! They will still behave randomly but it depends on how large the population size is.4. What would happen if there were no randomness to this selection?! It would not be an accurate display of the Hardy-Weinberg equation or evolution.5. What kind of pattern of genotypes would you expect in the next generation?! The genotype that is best suited for the environment that its living in will be most present.6. In the absence of random events ( an infinitely large population), are the allele

frequencies of the original population expected to change from generation to generation?

! No because the random events are what makes evolution happen.7. How does this compare to a population that has random gamete selection but its

small! the gametes will still display genetic variation though it will have a smaller population which would make evolution occur faster.8. What happens to allele frequencies in such a population? Is it predictable ?! They become concentrated toward one side or another, yes as long as there is random mating.Lab B.

1. List the condition that must exist for the Hardy-Weinberg law to apply.! Random mating must be present.2.What evidence would you look for to indicate that a population is evolving?! The phenotypes are changing in one direction or another. 3.Assume Hardy-Weinberg equilibrium.

a. If p=0.8 for a population, calculate the values of q,p2,q2, and 2pq. Show all your working.

q=.2 p2=.64q2= .042pq= .32

b. what would be the gene pool frequencies in the next generation if evolution does not occur?

! it would continue shifting toward the Q direction.

VIII.Conclusion! The reason behind this lab was to try to use the Hardy-Weinberg equation in a real life situation. The first lab was teaching us to create a model on the computer and to show us something that is very important to the science field , as nothing can be proven unless math is behind it. A generalized error for this section of the lab is that Excel is not the easiest application to use and the lab was meant to be used with Excel. For this reason it made it slightly more difficult to concentrate on what the lab was trying to teach. The second lab though was a more hands on way to teach the same thing. Through mating with people in the class room the different theories of selection were demonstrated while given a more realistic example of speciation and what the Hardy- Weinberg equation is composed of. Some errors that might have occurred in this section of the lab is that some of the results might not have been recorded down correction and there may be variation between the genotypes that were recorded between the class. ! In this lab i learned why and how the Hardy-Weinberg equation should be used and why it is used. I also realized what is happening to the population not just that numbers are being found through the equation. It helped to illustrate what the equation means and how it should be used. Also i learned how to create a mathematical model. More importantly than that i learned how to use Microsoft Excel. I also learned through this lab that communicating with a lot of people you have to be organized with your data, as people are moving around constantly. ! Experiments that could be done to further the study might be to look at real life examples or organisms in an environment and to observe how they are changing with the environment in which they live. In addition to this bacteria could be observed to see how it becomes resistant to antibiotics and also use to Hardy-Weinberg equation to predict what might happen.! This applies to both evolution and in a small way genetics. Evolution because it shows the amount change in a population of organisms over many different generations. There are many real life applications for the skills and topics that were learned through these labs. For example mathematical modeling is used in many different fields of science, for predicting almost everything. This can also be used to predict the evolution of a species over time and also what that species might do to the environment it lives in.

Mathematical Modeling: Hardy-Weinberg Theory

A.P Biology 3rd Block

Denise Grover