# Mathematical Modeling - Hardy-Weinberg: Biology Lab

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• 0:02 Mathematical Models
• 2:26 The Hardy-Weinberg Theorem
• 6:05 Using the…
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Lesson Transcript
Instructor: Jennifer Szymanski

Jen has taught biology and related fields to students from Kindergarten to University. She has a Master's Degree in Physiology.

What is a scientific model? How can we create them? How can they be used in biology? In this lesson, you'll answer all of these questions as you explore how a model called the Hardy-Weinberg theorem can be used to predict how a population will change over time.

## Defining and Creating Mathematical Models

When most of us think of a model in science, we think of a tangible object - maybe a series of Styrofoam balls painted to represent the order of the planets in our solar system or a plastic representation of a plant cell, enlarged and with a key present to indicate all of its external and internal structures. These physical models are just one kind of scientific model, though.

By definition, a scientific model is any object or process that is used to simplify the real world. This means that scientific models also include things that we can't see or touch, like computer programs and mathematical equations. How do scientists create mathematical models? Although there's not one 'set' method, most scientists agree on a few basic steps.

First, you must ask a question. 'What will happen if?' or 'why does?' are good starting points from which to create a model. Next, you should refine the question until it is clear and concise. A mathematical equation can't handle too many parameters at once; it must address a specific set of variables. It often takes several attempts to refine a scientific question.

The third step is to describe the biological system qualitatively (that is, using non-mathematical or statistical terms). In the case of evolution, which is genetics based, you might describe an organism's lifecycle using terms like fertilization, birth, and migration. It's during this step that you'll begin to determine the limitations of your model. Remember, life is messy! Not all adult organisms breed successfully and not all offspring survive. These and other limitations can affect how effective your model is in simplifying a real-world process.

Next, determine a quantitative (or number based) description. What variables should you choose to represent parts of an organism's lifecycle? What types of mathematical operations can effectively represent aspects of your biological system? The next steps are to use your mathematical model to evaluate your biological system, analyze your results, and examine the results in the context of the biological system. We'll look at these steps in the context of one of biology's best-known mathematical models: the Hardy-Weinberg theorem.

## The Hardy-Weinberg Theorem

One of the most important mathematical models in biology is called the Hardy-Weinberg theorem. Named for a pair of early 20th century scientists, this theorem is a mathematical model that shows the relationship between the alleles present in a population and how that population is likely to change over time. Before we explore what that means, let's review a few of the key terms that we'll be using in this part of the lesson.

Recall that most living organisms are diploid; that is, they have two copies of each chromosome, one from each parent. On each chromosome are genes, the basic unit of heredity. Different forms of a gene are called alleles. For example, many studies suggest that the ability to digest lactose (or milk sugar) as an adult is determined by the presence of a specific allele. This allele is dominant, which means that it is expressed, or seen, in an individual. The allele that is hidden is called recessive.

If an individual receives an allele that allows them to digest lactose from their mother, but the other allele from their father, then they will be able to digest lactose. This individual is called heterozygous for this trait because it has one dominant allele and one recessive allele. If an individual receives either both dominant alleles or both recessive alleles, then they are homozygous for that trait. Homozygous and heterozygous refer to a genotype, or genetic makeup, of an individual, while the trait 'seen,' or expressed, is called phenotype.

The Hardy-Weinberg theorem uses two equations to predict how the number of alleles (or allelic frequency) might change in a population (a group of interbreeding individuals) over time. It also predicts how the genotypic frequencies of a population might change over time. How?

First, the Hardy-Weinberg theorem visualizes that all of the alleles for a generation of individuals belongs to a single large group, called a gene pool. It also makes five very important assumptions. First, it assumes that offspring always arise from random mating - no pretty feathers or pheromones to attract a mate: all mating is random. Next, it assumes that natural selection does not exist. Under this assumption, rabbits with white fur and rabbits with dark fur have an equal shot at passing those genes to the next generation, even if they both live in a snowy environment.

It also assumes that there is no addition or subtraction of alleles into the gene pool. Populations don't gain or lose members by individuals migrating to a new area or joining a herd. Hardy-Weinberg also assumes there are no mutations to DNA. In other words, alleles stay the same from generation to generation. Finally, the theorem assumes that there is no genetic drift. Essentially, this means that every allele has an equal chance of showing up in the next generation. No individuals are luckier than others, or have more offspring, or live longer lives.

You may be thinking, 'These assumptions just don't happen in nature.' That is absolutely correct. In a way, the Hardy-Weinberg theory functions as a null hypothesis, a statement that declares, 'nothing can ever affect a population's alleles.' Then, when things do affect them, we can see exactly how.

## Using the Hardy-Weinberg Theorem

The starting equation is quite simple: (p + q) = 1, where, p is equal to the frequency of the dominant allele for a trait in a population, and q represents the frequency of the recessive allele. If you remember your algebra, and you square that binomial, you get p^2 + 2pq + q^2 = 1. p^2 represents the frequency of the homozygous dominant genotype, 2pq represents the frequency of the heterozygous phenotype, and q^2 represents the frequency of the homozygous recessive phenotype.

Say, hypothetically, in a population of 1,000 rabbits, coat color is determined by a single allele. The dominant allele creates a brown coat, and the recessive allele creates a white one (we'll use capital B to represent the dominant allele, and lowercase b to represent the recessive allele, and assume that capital B is completely dominant to lowercase b).

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