Calculus and the

Hardy-Weinberg Principle Population Genetics Population genetics is the study of change in genetic compositions of biological populations. How does the Hardy-Weinberg

Principle Relate? Example: A population geneticist would study the ratio of white and brown fur in a rat population. The Hardy-Weinberg Principle is an ideal model of the allelic frequencies of a population. This would exclude phenomena such as genetic drift, mutation, selection, or anything non-random. What does this model indicate? With all of its assumptions in place, the Hardy-Weinberg principle states that allele and genotype frequencies of a population will be unchanging throughout time. Hardy-Weinberg: Explained The Hardy-Weinberg principle is best explained through application and variables. Using p's and q's To explain this principle, the alleles named A and B will be used. The frequency of allele A is referred to as "p" and the frequency of allele B is referred to as "q". So, the allele frequencies would be graphed as such for this population: The Mathematics Behind the Principle All of the frequencies given in the chart (q^2, p^2, pq, qp) represent different percents of the infinite population. Therefore, when all of these frequencies are added up they must equal 1. This can be shown through the equation : q^2 + p^2 + 2pq = 1 Situations with More than Two Alleles The equation previously shown only applies if the only the two alleles A and B are being considered. If an extra allele "r" is included in the situation. *Note This equation can also be found by expanding (p+q)^2 into a binomial. Mathematical Explanation Similar to the diploid situation, this triploid situation can be expressed by expanding (p+q+r)^2. When multiplied out, this expands to: p^2 + q^2 + r^2 + 2pq + 2pr + 2rq = 1 What does this indicate? The Hardy Weinberg Principle, as modeled by the previous equations, is an ideal principle. That being said, it does show some important trends in populations and allows us to examine the basic processes in population genetics. Evolutionary Implications One of the most valuable aspects of this principle is what it shows in the long term for a population. Going against the idealistic basis of this idea, the Hardy-Weinberg principle shows that as time goes on, the entire population would eventually become homozygous for the most advantageous allele for each allele. This is only true if the principles shown in Hardy-Weinberg are then released into a situation with non-random variables. This can be shown graphically. Evolutionary Implications cont. In order to support this idea, it must first be understood that the rare alleles that can be formed into advantageous traits are most commonly found in heterozygotes. This graph exemplifies that because the frequency line of q^2 (homozygote) is substantially less than the value of 2pq (heterozygote) when the graph is approaching zero. This indicates that 2pq would then be more likely to hold the more rare allele, as its frequency is shown to be higher for the red line by the legend on the left. The same is held true for p^2. When p is approaching zero, the line of 2pq (heterozygote) shows a much higher frequency than the line of p^2 (homozygote). Through this concept, the principle shows a dominance among alleles that strongly indicates the process of natural selection. Thus, the general equation for the alleles p1...pi is: Non-diploid Situations Also called polyploidy, there is a general equation for situations when the organism has more than two chromosomes for each gene. It is as follows: (p + q)^c (c being the number of chromosomal copies) Complete Generalization of the Principle The process of combining multiple allele and polyploidy situations to create a general equation is fairly simple. The multiple allele general equation is raised to the "c" power to also involve the chromosome number. Therefore, our equation is: Multinomial Expansion This equation is further generalized by using multinomial expansion, which generalizes a multinomial equation with integer values of n. The general equation for a multinomial expansion is: Multinomial Expansion of General Hardy-Weinberg Equation Using the variables in the previous equation, the fully generalized equation of multiple alleles and polyploidy, our equation is: Further Application The abstract idealism of the Hardy-Weinberg principle may seem fairly useless. However, having the principle's equation, researchers are able to use the principle to look at an actual population's genotypic frequencies and determine whether or not the differences between the actual population and the Hardy-Weinberg model are due to chance. For each of the possible causes of deviation there is an equation to assign a number to the situation, such as the inbreeding coefficient, which is: simplified as: 1 - (observed frequency of heterozygotes/hardy-weinberg expecations) Conclusion When over-arching concepts such as population genetics and the Hardy-Weinberg principle are discussed, it is difficult to define and utilize specific concepts. Here lies the correlation with genetics and mathematics; without mathematics genetics would be much more difficult to conceptualize. Necessary Vocabulary Allele- a gene located at a specific location on a specific chromosome

Heterozygous- having two different alleles for one gene

Homozygous- having two of the same alleles for one gene By: Carly Phillips

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# Hardy-Weinberg Principle

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