the Hardy-WeinbergEquilibrium-Some0 Helpful Suggestions MaryT. Ortiz LorettaTaras Anthea M. Stavroulakis

TN college General Biology curricula, the topic of Hardy-Weinberg equilibrium. Our experiences in classical genetics is typically followed by consider- teaching Mendelian probabilities and the Hardy- ation of population and evolutionary genetics. Weinberg model confirm that students enrolled in After fundamental genetic principles are studied, GeneralBiology who have not attainedthe established students are introduced to the Hardy-Weinbergequi- level of proficiency in mathematics have difficulty librium as a continuum of Mendelian Genetics in a participating in class discussions on this topic and population context. Population genetics provides a solving homework problems and examination Downloaded from http://online.ucpress.edu/abt/article-pdf/62/1/20/49339/4450821.pdf by guest on 28 September 2021 logical segue to the study of evolution. Populations questions. adhering to the Hardy-Weinbergequilibrium provide Mendel's laws provide the foundation for discus- a reference for studying evolutionary changes sions of evolutionary and population genetics (Dob- (Journet1986). With this as a startingpoint, Darwin's zhansky et al. 1997).The Hardy-Weinbergequilibrium observations can be justified and quantified, as well is a continuationof Mendel's principles, which show as expanded upon, to include additional aspects of that gene frequenciesunder certainconditions remain population and evolutionary genetics. constant from one generation to the next, thereby The Hardy-Weinbergequilibrium is an algebraic maintaining the allelic composition of the gene pool mathematical tool for predicting allele frequencies, of the population. During our presentation of the phenotypes and genotypes in populations (Cum- Hardy-Weinbergequilibrium, we link Punnettsquares mings 1997; Lewis 1997; Mader 1996;Postlethwait & and probabilitycalculations to previous discussions of Hopson 1995).Mertens (1992)proposed strategies for classicalinheritance. We enrich our students' learning introducingthe Hardy-Weinbergequilibrium to make experiences by extending basic principles of inheri- it meaningful and more useful to both students and tance, and providing challenging mathematicalappli- teachers. It has been our experience that students cations. Instructors can assist students through in- have difficulty with Hardy-Weinbergproblem solv- class practiceproblems. This allows subsequent focus ing. Since students often possess a genuine fear of on Hardy-Weinbergtheory and applications rather mathematics, reinforcement of basic mathematical than laboring over arithmetic operations. principles when working through Hardy-Weinberg We present Hardy-Weinbergtheory as a mathemat- problems will provide a less threatening experience, ical model demonstratinghow allelic and genotypic thereby enabling students to successfully complete frequencies remain constant in a large population problems and concentrate on understanding genetic under specified conditions. Named after its discover- concepts as they relate to population dynamics (Flan- ers, Godfrey H. Hardy, a British mathematician,and nery 1995; Journet 1986). Wilhelm Weinberg, a German physician, the Hardy- Various methods, some quite detailed, for present- Weinberg equilibrium requires the following condi- ing the Hardy-Weinbergequilibrium have appeared tions (Campbellet al. 1997; Cummings 1997;Journet in journals and textbooks (Cummings 1997; Journet 1986; Lewis 1997; Mader 1996; Postlethwait & Hop- 1986;Lewis 1997;Mader 1996;Postlethwait & Hopson son 1995): 1995). The Hardy-Weinbergmodel is presented in a 1. No gene flow: Thereis no migrationof individuals variety of ways in General Biology textbooks. In this into or out of the population. paper we offer suggestions for instructors to assist 2. No mutations: Alleles will not change from one with their presentation of this topic. Regardless of generation to the next. the approach, our suggestions can assist instructors 3. No selection:No selective force favors one pheno- by providing a unified method for explaining the type over another. 4. No genetic drift: The population is large; random fluctuations are considered negligible. MaryT. Ortiz, Ph.D., Loretta Taras and AntheaM. Stavroulakis, 5. Random mating: Individuals pair by chance, not Ph.D., are Associate Professors in the Biology Department according to their phenotypes. at Kingsborough Community College, 2001 Oriental Blvd., hold true for a Brooklyn,NY 11235. If the above assumptions given population, and:

20 THEAMERICAN BIOLOGY TEACHER, VOLUME 62, NO. 1, JANUARY2000 p = the frequencyof the dominantallele in a population Information ascertained from the answers to these q = the frequencyof the recessiveallele in a population, questions allows students to learn additional aspects about a given population, such as knowing the per- then: p + q = 1 (Equation 1) centage of carriers of a particular genetic disease. and p2 + 2pq + q2 = 1 (Equation 2) During our classroom presentation of the Hardy- Weinberg model, it is apparent that a number of Equation 1 is called the Gene Pool Equation, and students are unsure of the mathematicalcalculations. Equation2 is called the Genotype Equation.In Equa- This hesitation with mathematicsinterferes with their tion 1 the sum of the frequencies of all of the alleles ability to solve problems and analyze results. We in a population must equal one (indicating 100%). are frequently asked to provide additional practice For example, if p = 0.2, then q must equal 0.8. In problems. GeneralBiology textbooks usually provide Equation 2 the sum of the frequencies of individuals a few example problems. The instructor may find with each genotype must add up to the entire popula- herself/himself searching for additional problems in tion, or one (100%) (Cummings 1997; Lewis 1997; which the numbers work out without requiring use Mader 1993; Postlethwait & Hopson 1995). of a calculator(Lewis 1997;Mader 1993;Postlethwait Consider the following: Pointy ears are dominant & Hopson 1995). over round ears in an alien population. Let E and

Wouldn't it be convenient for instructors to have Downloaded from http://online.ucpress.edu/abt/article-pdf/62/1/20/49339/4450821.pdf by guest on 28 September 2021 e representthe dominant and recessive alleles, respec- a table containing p, q, p2, 2pq and q2 values to tively. In a population of 1,000 aliens, 800 have pointy facilitatequick, on-the-spotconstruction of examples? ears (p = 0.8; q = 0.2). The threepossible genotypes If this table were readily available, problems could in a cross between two heterozygotes are EE, Ee and be generated quickly and abundantly. Table 1 lists ee. We can demonstrate this using a Punnett square: = - p, q, p2, 2pq and q2 values for p and q 0.1 0.9 E e in increments of 0.1. More extensive tables could be constructedusing p and q values with more significant E EE Ee digits, such as 0.11, 0.12, 0.13, etc. Use of the p and q values in Table 1 to generate problems allows instructors to focus their lesson on the significance e Ee ee of the Hardy-Weinbergequilibrium instead of arith- metic. For example, in a population meeting the the if Using the values for E and e, p = 0.8 and q = 0.2, requirementsof Hardy-Weinbergequilibrium, respectively, we can use a Punnett square to calculate p = 0.7, find the frequency of: the predicted frequencies for each genotype: a. the recessive allele b. homozygous dominants q c. heterozygotes (carriers) p p2 = 0.64 or 64% d. homozygous recessives = 2 pq pq 0.16 or 16% ppp=p 0.04 or 4% e. dominant phenotypes q2= in the population. Using Table 1, the answers to a through e are 0.3, 0.49, 0.42, 0.09 and 0.91, respec- tively. That is, 3% of the population in question q pq qq=q2 contain the recessive allele, 49% are homozygous dominant, 42% are heterozygous (carriers),9% are Therefore, using Equation 2: homozygous recessive, and 91%display the dominant phenotype. 0.64 + (2 x 0.16) + 0.04 = 1 a states that the frequency of 64% + 32% + 4% = 100% Suppose problem sickle-cell anemia, an autosomal recessive condition Once the components of the equation are deter- is 20% in a given population. We have found that mined, students can be challenged with a variety of a quick review of conversions between percents and questions, as well as formulate their own practice decimals helps guide students through the mechanics problems, including: of the problem. Conversion of 20% to a decimal for 1. What percent of the population would you expect use in Equations 1 and 2 requires dividing by 100 to be homozygous dominant? (Answer: 64%) or moving the decimal point two places to the left 2. What percent of the population would you expect to obtain 0.2. Likewise, changing a decimal to a to be homozygous recessive? (Answer: 4%) percent requires multiplying by 100, or moving the 3. What percent of the population would you expect decimal point two places to the right. to be heterozygous? (Answer: 32%) Another rule of mathematics is often overlooked 4. What percent of the population would you expect by students: When multiplying factors containing to show the dominant phenotype? (Answer: 96%) decimals, the number of decimal places in the product

HARDY-WEINBERGEQUILIBRIUM 21 Table 1. p, q, p2 2pq and q2 values for p and q = 0.1-0.9 in increments of 0.1. p 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 q 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

p2 0.01 0.04 0.09 0.16 0.25 0.36 0.49 0.64 0.81 2pq 0.18 0.32 0.42 0.48 0.50 0.48 0.42 0.32 0.18 q2 0.81 0.64 0.49 0.36 0.25 0.16 0.09 0.04 0.01 must equal the sum of the decimal places in each point, we distributed Tables 1 and 2 to our classes. of the factors in the problem. For example, this is Using the tables, students solved Hardy-Weinberg encounteredwhen multiplying 0.1 by 0.1. The product problems with less difficulty. Once the obstacle of is 0.01, not 0.1, as students often report erroneously. arithmeticcalculations was overcome,genetic features In addition, encouragingstudents to familiarizethem- characterizingthe population and the significance of selves with the perfect squares (n2)for n= 1 to n = 25 the numbers obtained became the focus of the lesson. will assist them in solving problems more rapidly Using these suggestions, instructors will be able to

(Table2). Confidence with decimal manipulationand present Hardy-Weinbergequilibrium problems more Downloaded from http://online.ucpress.edu/abt/article-pdf/62/1/20/49339/4450821.pdf by guest on 28 September 2021 swift recognition of perfect squares in Hardy-Wein- easily. Students will solve problems but, more impor- berg problems empower students to quickly reach tantly, understand the significance of the Hardy- solutions, thereby easing any mathematics anxiety. Weinberg model to population genetics. We propose assisting students with their under- Once students become familiar with using Table standing of the Hardy-Weinberg equilibrium by 1 to solve problems,we encouragethem to participate developing an available pool of questions by the in active and collaborativelearning by forming study instructorusing Table 1, along with reinforcementof groups to apply facts and concepts they have learned multiplicationrules for decimals, and a solid familiar- to new situations. This group process allows them ity with common perfect squares. Enrichingthe class- to contemplatethe relationshipbetween genetic prin- room with mathematicsprovides for improved scien- ciples, experimentalfindings and observations.Does tific literacy and science education, as advocated in our model have applicationto yourpopulation genet- the NationalScience Education Standards proposed by ics lesson? the National Research Council (1996). Our approach integrates mathematical applications with biological References principles, as well as promoting mathematical understanding. Campbell,N.A., Mitchell,L.G., & Reece, J.B.(1997). Biology Concepts& Connections,2nd ed. Menlo CA: We were to if Table 1 was as Park, The curious see useful Benjamin/CummingsPublishing Company. to the students as it had been for us. After an Cummings, M.R. (1997). Human Heredity Principles & introductionto the Hardy-Weinbergmodel, a sample Issues, 4th ed. Albany, NY: Wadsworth Publishing problem was completed. Students asked questions Company. and requested additional problems to solve. At this Dobzhansky, T., Ayala, F.J., Stebbins, G.L. & Valentine, J.W. (1977). Evolution.San Francisco,CA: W.H. Freeman Table 2. PerfectSquares for n 1 to n = 25. and Company. Flannery,M.C. (1995). Math matters. TheAmerican Biology n n2 n n2 Teacher,57(1), 56-59. 1 1 14 196 Journet,A.R.P. (1986).Population genetics: A fishy process. 2 4 15 225 The AmericanBiology Teacher, 48(8), 478-482. 3 9 16 256 Lewis, R. (1997). Human GeneticsConcepts & Applications, 4 16 17 289 2nd ed. Dubuque, IA: Wm. C. Brown Publishers. 5 25 18 324 Mader, S.S. (1996). Biology,5th ed. Dubuque, IA: Wm. C. 6 36 19 361 Brown Publishers. 7 49 20 400 Mertens, T.R. (1992). Introducing students to population 8 64 21 441 & the The 9 81 22 484 genetics Hardy-Weinbergprinciple. American 10 100 23 529 BiologyTeacher, 54(2), 103-107. 11 121 24 576 National Science EducationStandards. (1996). Washington, 12 144 25 625 DC: National Academy Press. 13 169 Postlethwait,J.H. & Hopson, J.L. (1995). TheNature of Life, 3rd ed. New York: McGraw Hill, Inc.

22 THEAMERICAN BIOLOGY TEACHER, VOLUME 62, NO. 1, JANUARY2000