Paint By Numbers: An Alternative Way of Working Through Hardy-Weinberg Equilibrium Problems

For many Physical Anthropology and Biology students, figuring out Hardy-Weinberg Equilibrium problems is both challenging and dreadful.  Even I, as a student, had difficulties completing these problems, but I have learned some tricks that have assisted me in both completing these problems and teaching the concepts to my students.  I am sharing these tricks now so that students and educators can better handle this tricky subject.

The Hardy Weinberg Equilibrium (aka principle, theorem, model, or law) is a concept that was obtained by scholars Godfrey Hardy and Wilhelm Weinberg, and it is used to determine if genetic equilibrium, or lack of evolutionary change, has occurred in a population (see this post for explanations on the aforementioned terms).  In order for genetic equilibrium to occur, a set of conditions must be met, and these conditions include:

•       Mutations must not be taking place.

•       The population must be infinitely large.

•       Individuals from neighboring populations must not introduce alleles into the population.

•       Mating must take place at random and be equally fertile.

•       Natural selection must not be occurring.

Although these conditions may be present in any given population, it may be difficult to determine with absolute certainty if all of them are taking place equally, producing genetic equilibrium.  This is where the Hardy-Weinberg Equilibrium equation comes in handy.  This equation utilizes a set of variables that represent certain aspects (in proportions) of the population and the answer of the calculated variables will demonstrate if genetic equilibrium is present. 

The Hardy-Weinberg Equilibrium equation comes in two parts and are as follows:

P + Q = 1

P² + 2PQ + Q² = 1

At this point, people start to get cross-eyed, butterflies enter the stomach, some start to feel nauseated, etc.  Step one for understanding this problem is NOT TO PANIC!  As mentioned earlier, the Hardy-Weinberg Equilibrium equation is a set of variables that represent certain aspects (in proportions” of the population.  Each of these variables (from P to Q²) represents a specific aspect of the population, and this is the breakdown of those variables and what they represent:

• —  P denotes the dominant allele
• —  Q denotes the recessive allele
• —  p² = percentage of homozygous dominant individual
• —  q² = percentage of homozygous recessive individual
• —  2pq = percentage of heterozygous individuals 

(For an explanation of these terms, please go to this previous blog post.)

Figure 1: Paint By Numbers of My Little Ponies.  It is sometimes easier to think of Hardy-Weinberg Equilibrium Equations as Paint By Numbers Kits than as math problems.  Source: Google Images

 

Now if you are still feeling anxious because you are looking back at the full problem and have math anxiety, stop thinking of the Hardy-Weinberg Equilibrium equation as a math problem.  Instead, think of it as a paint by numbers kit (Figure 1).  Each of those variables (from P to Q²) is just a different color that needs to be plugged into your paint by number kit, which in this case is your equation.  Please note that I have chosen the theme of My Little Ponies as a means of relaxing tensions that you may be feeling.  My Little Ponies was and is a popular children’s cartoon show, and much like children’s cartoons the Hardy-Weinberg Equilibrium equation does not have to be difficult or intimidating.  It can be fun and enlightening if you so chose to think of it that way, which I recommend because you will find yourself less scared of it if you chose to think that way.

Hardy-Weinberg Equilibrium equation problems are often presented as a lengthy word problem, such as the following:

Figure 2: Apple Family (My Little Ponies); Google Images

 

In the land of the My Little Ponies, there exists a family of ponies known as the Apple Family (Figure 2).  This family has a unique set of genetic traits and is one of the few My Little Ponies that can and do produce ponies with green mane, which is a homozygous recessive trait.  In this family photo, we see eleven ponies with green mane.  The current population size of the entire Apple family is 100 ponies. 

Now this word problem can be very intimidating for a variety of reasons.  There is a lot of information present here, but this information is not here to confuse you.  It is here to assist you.  If you look closely in this problem, you have already been provided the information that gives us the Q² variable:

green mane is a homozygous recessive trait...eleven ponies (have) green mane

We now know that 11 out of 100 ponies exhibit green mane, but we need to turn this number into a proportion (fraction or decimal).  To do this, we take 11 and divide it by 100 (11/100) and we get .11 as our answer.  This .11 now represents the Q² variable.  We have one part of the Hardy-Weinberg Equilibrium equation completed, and believe it or not, it is now much easier to figure out the rest of the variables.

How?  We can figure out Q from Q² by getting the square root of Q², which in this case is .11.  Once you have calculated the square root of .11, you will have Q.  In this case, Q is .33 (rounded to the nearest 100^th place).  Now if you look back at both parts of the Hardy-Weinberg Equilibrium equation, you will see that P + Q = 1.  You have Q, so you can figure P.  You do this by subtracting Q (.33) from 1, which when resolved provides you with .67 (P).  You can check to make sure you completed this correctly by adding P (.67) and Q (.33).  If the resolution equals one then you have calculated P and Q correctly.  If not, then you need to go back to the previous steps and figure out where you may have gone wrong.  In this case, .67 + .33 = 1, so the equation has been completed correctly.

So far you have figured out three variables: Q², Q, and P, but you still need to figure out P² and 2PQ.  Now that you have the Q², Q, and P variables represented you can easily come up with P² and 2PQ.  You figure out P² by multiplying P (.67) by itself.  In other words, .67 x .67, which when calculated gives you .45 when rounded to the nearest 100^th place.  So now you know that .45 represents P². 

Figure 3: PEMDAS (Unedited: www.mathgoodies.com)

Figuring out 2PQ is a little trickier because you have to remember the acronym PEMDAS.  PEMDAS refers to the order of operations in mathematics, and this acronym represents Parentheses, Exponents, Multiplication, Division, Addition, Subtraction (Figure 3).  You would set up 2PQ as follows:

2 (.67 x .33)

And following PEMDAS, you would complete the equation in the parentheses (.67  x .33) first, producing an answer of (.22) [again, rounded to the nearest 100^th] and then multiply that answer by 2:

2 x (.22) = .44 = 2PQ

Now that everything is complete, this is what each of your variables is:

P= .67

Q= .33

P²= .45

Q²= .11

2PQ= .44

Once you have this information, you can solve for Hardy-Weinberg Equilibrium, as well as figure out the number of homozygous dominant and recessive individuals as well as heterozygous individuals in a population.  You can learn all sorts of additional information about the population from this information, too.  Hopefully, though, the biggest thing you have learned is not to be intimidated by Hardy-Weinberg Equilibrium equations and how best to calculate them.  All with the help of some My Little Ponies.