Utilizing Mathematic Principles, Essay Example

Utilizing Mathematic Principles of the Hardy-Weinberg Theorem to Understand Population Genetics

Introduction

Our understanding of population genetics is derived primarily from the genetic experiments conducted by Gregor Mendel in the early 1800’s (Reece et al. 265). His use of pure breeding pea plants to derive offspring and determine phenotypic ratios, otherwise known as the physical trait appearance of offspring, allowed us to have a greater understanding of the genotype of these same plants, which represents the genes present that contribute to the traits we see. He ultimately determined that for each gene, there are two alleles that contribute to the traits that are presented in offspring. Some of these alleles are dominant, which means we are more likely to see them expressed in offspring, while some are recessive, which means we are less likely to see them expressed in offspring. When two dominant alleles are contained within the genotype of an organism, they will always be seen in the phenotype. This is also the case when two recessive alleles are contained within the genotype of an organism. However, if the genotype of an organism has one dominant allele and one recessive allele, only the dominant allele will show in the phenotype. Although extensions to these principles have been discovered over time, these are the basic attributes that have given rise to our understanding of genetics as a whole.

The Hardy-Weinberg Principle was developed utilizing Mendel’s observations in order to gain a greater understanding of how allele and genotype frequencies shift in the presence of evolutionary pressures. This theory was also based, in part, off of observations made by Charles Darwin in his development of the natural selection theory (Haldane 480). This theory states that genetic traits will become more or less frequent in a population dependent upon their ability to confer an advantage in survival to an organism. It was compiled based on a series of observations that he had made while studying a variety of organisms in the Galapagos Islands, which determined that the diverse organisms there tended to be well-suited for their environment. He pondered upon this and ultimately decided that organisms that were unable to adapt to their surrounding environment would not survive, while those that were suited to do so would survive to pass down their traits through sexual reproduction. This concept derived from the fact that there is a natural variation among species, and some of these traits are unfavorable while others are favorable depending upon the particular environmental conditions that are present. The favorable traits will be transmitted to offspring more effectively, which will contribute to a larger presence of this gene in future populations. Likewise, undesirable traits tend to diminish over time. Furthermore, favorable traits were established over time due to genetic mutations. Genetic mutations that established a phenotypic consequence that was considered positive for the organism’s survival would be passed down and possibly further modified over time. Deleterious mutations would be similarly removed from the population.

The Hardy-Weinberg Principle combines both research that had been conducted by Mendel and Darwin in order to determine how populations evolve, or slowly change characteristics, over time (Crow 821). He found that the aspects of a population that control how genes are transmitted over time depend upon an organism’s mate choice, random genetic mutation, natural selection, genetic drift, gene flow, and meiotic drive. In the complete absence of these influences, the genes contained within a population are expected to remain constant. Therefore, he developed several formulas in which population genetics in simplistic cases could be tracked. The derivations of these formulas allowed evolutionary scientists to gain greater insight into the biologic factors that control population change.

It has been demonstrated that the Hardy-Weinberg Principle has many useful applications in the prediction of population change and to gain a greater understanding of drivers of evolution. However, it may also be possible to determine whether the Hardy-Weinberg Principle can be utilized to predict how gene frequencies will be inherited from generation to generation based on a series of assumptions regarding the population of interest. This understanding can be utilized to track how future generations will gain or lose certain characteristics. I hypothesize that consideration of the variance of traits, in addition to the frequency of these traits within a population will have a direct impact on our ability to predict which traits will continue to exist in a population after a given duration of time.

Derivation of the Hardy-Weinberg Principle

The Hardy-Weinberg Principle considers the presence of two alleles in a population to determine which alleles are more likely to be transmitted to future generations. To denote the variations of allele combinations that can exist, this principle utilizes p to represent dominant homozygotes and q to represent recessive homozygotes. Likewise, pq is utilized to denote heterozygotes with one dominant and one recessive allele. More frequently, these proportions are represented by p², q², and 2pq, respectively. According to the genetic principles revealed by Mendel, we expect 25% of the offspring of two different heterozygous populations to be 25% homozygous dominant, 25% homozygous recessive, and 50% homozygous. Therefore, the aforementioned values have been calculated to reflect this trend. In the Hardy-Weinberg Principle, these three variation of options are added together to represent the whole population. Mathematically, the formula reads p² + 2pq + q² = 1. For simplicity, this equation can also be represented as (p + q)² = 1. The solution of this equation is p + q = 1, which provides us with a better understanding of the representation of each individual allele in the population, outside of the consideration of allele pairs.

As stated above, this formula considers the reproduction of two different heterozygous breeding members of the first filial generation, in which one parent is homozygous dominant and the other is homozygous recessive. In this derivation, A will be used to represent the frequency of the dominant allele, while a will be used to represent the frequency of the recessive allele. This gives that the initial frequency of each allele in the population is:

f₀(A) = p and f₀(a) = q

Since allele frequencies are obtained by combining the alleles of each parental genotype, and the expected contribution from homozygous genotypes is 1, while the expected contribution from heterozygous genotypes is ½, we can write that:

f_t(A) = f_t(AA) + ½f_t(Aa)

f_t(a) = f_t(aa) + ½f_t(Aa)

These proportions can be represented using a Punnett square, in which the proportion of each genotype is equivalent to the product of the row and column frequency alleles from the initial generation. Thus, we can anticipate that the second filial generation of two pure breeding parents for first filial generation homozygous plants will yield offspring that are genotypically 25% homozygous dominant, 25% homozygous recessive, and 50% heterozygous. This effect is displayed below in figure 1.

Offspring of the First Filial Generation (Heterozygous Parents)

Figure 1 Graphical display of the accuracy of the Hardy-Weinberg Principle using a Punnett square.

Figure 1 demonstrates that each component of the square can be added up to give rise to 100% of the offspring options available. 25% of the offspring population is homozygous dominant (AA), 25% of the population is homozygous recessive (aa), and 50% of the population is heterozygous. Since these all add up to 100%, which can be represented by 1 in terms of a fraction, the validity of the equation p² + 2pq + q² = 1 is supported. It is important to also consider that the expression p + q = 1 is valid since the simplification of p² + 2pq + q² = 1 is (p + q)²  = 1. Taking the square root of both sides of the equation yields p + q = 1.

Extensions of the Hardy-Weinberg Principle

Although the Hardy-Weinberg Principle was developed according to the understanding that there are two alternative alleles available in the population, in reality it is possible for multiple alleles to exist. However, each organism can carry a maximum of two per gene. This complicates our understanding of dominant and recessive, and necessitates a clarification of these definitions. Thus, a gene variation is considered dominant or recessive with respect to its alternate options. In cases whether three alleles are available for one gene, an allele can be considered dominant in respect to one gene and be recessive in respect to another. However, the relationship of dominance and recessiveness with respect to three alleles can be defined based on observation.

For example, situations exist in which there are three allele options for a particular gene. In this situation, the third allele can be represented by r. To determine the proportions of each allele that exists in the population, the Hardy-Weinberg Principle can be applied. To do so, we would set (p + q + r)²  = 1. When the square root of each side is taken, we see that p + q + r = 1. Therefore, each allele in the population is still expected to equal the whole of the variability for that particular gene. To determine the homozygosity and heterozygosity of each trait, the formula can be expanded to give p² + 2pq + q² + 2pr + 2qr + r² = 1. It is important to consider that this application can be extended to additional allele possibilities, simply by adding a letter representation to the common formula, and squaring the addition of all possibilities.

It is also useful to consider that the Hardy-Weinberg principle can be extended to gain a greater understanding of polyploidy, in which organisms have more than two copies of each chromosome. This is represented by the formula (p + q)^c, in which the letter c represents the number of chromosome copies present. In normal cases, the Hardy-Weinberg Principle raises the addition of allele possibilities to 2 because organisms typically have 2 copies of each chromosome. This example will demonstrate the formula that would be utilized in cases of tetraploidy, indicating that 5 chromosome copies are present. Therefore, in a system in which two alleles are present, the formula utilized to determine the allele distribution would be (p + q)⁵. Table 1 below demonstrates the expected gene frequencies for an organism with tetraploidy. This demonstrates that the Hardy-Weinberg theorem can be used in cases in which there is multiple alleles and polyploidy, in addition to these two situations combined.

Table 1The expected gene frequencies for an organism with tetraploidy.

 Application of the Hardy-Weinberg Principle

It is possible to determine the frequency of alleles in population using the Hardy-Weinberg principle both working forward and in reverse. For example determining the phenotype of the population for simple traits or determining the genotype for complex traits gives us a greater understanding of the allele distribution within a population. For purposes of this application, we will assume that a trait in a population has 1469 organisms that are homozygous dominant, 138 organisms that are heterozygous, and 5 organisms that are homozygous recessive. Plugging this into the Hardy-Weinberg Principle, p² + 2pq + q² = 1, we can determine that 1469² + 2(138) + 5² is equal to 100 percent of the population. Since 1750 then represents 100 percent of the population, we can compute that p is equivalent to 1469/1750. This yields p = 0.83943. Thus, approximately 83.9% of the population is homozygous dominant. Next, the portion of the population that is homozygous recessive can be determined by computing 5/1750. This yields 0.00286, indicating that about approximately .3% of the population is homozygous recessive. Lastly, the heterozygous population is represented by (2 x 138)/1750, which is equivalent to 0.15771. Therefore, approximately 15.8% of the population is heterozygous. To verify these findings, it is necessary to determine whether the proportion of the calculated homozygous dominant, homozygous recessive, and heterozygous alleles are accurate. Therefore, accuracy can be proven by demonstrating that 0.83943 + 0.15771 + 0.00286 = 1, demonstrating that this calculation accounts for 100% of the population. This application solution is in accordance to the graph and explanation of the Hardy-Weinberg Principle’s balance equilibrium, which can be seen below in figure 2. It demonstrates that when p is equal to 0.839 and q is equal to 0.002, pq must equal .158. The Hardy-Weinberg Equilibrium graph can be used in a variety of situations to ensure that the proportions of each calculated allele are equivalent to 1.

Graph of Hardy-Weinberg Equilibrium

Conclusion

It can be determined from the above research that the Hardy-Weinberg Principle can be utilized to both detect current allele distribution within population and determine how these trends will continue over time. Ultimately, the variability of traits in addition to their heritability impact the probability that a particular genetic component will remain in a population over time. Although the Hardy-Weinberg Principle was initially developed to gain a greater understanding of populations in which two alternative alleles exist, the proportionality principles of the equation allow this knowledge to be applied to the understanding of genetic traits that have greater than two alleles of in organisms in which polyploidy is more common. Therefore, the Hardy-Weinberg Principle can be continued to be applied to our ever changing understanding of modern genetics, even though it was created based on a simpler understanding.

It is clear that the Hardy-Weinberg Principle can be utilized to predict how gene frequencies can be inherited from generation to generation based on a series of assumptions regarding the population of interest. This equation can be used to determine the starting proportion of alleles available in a given population and then tracked on time by studying various aspects of the population include mate choice, random genetic mutation, natural selection, genetic drift, gene flow, and meiotic drive. Trends regarding these evolutionary principles can be developed, and ultimately modelled into a predictable growth or decay equation. In this way, we can track populations over time, but it will require close study of the way these populations change. While it is difficult to study a population with exact certainty, the Hardy-Weinberg Principles provide us with excellent guidelines to be able to determine statistically significant predictions, even with partial knowledge concerning a particular population.

In conclusion, the Hardy-Weinberg Principle is a useful tool that biologists can use to gain a greater understanding of evolutionary principles in addition to the specific populations they concern. A practical application of this theory is the greater understanding of disease in human populations and the determination of how these principles can be used to erase certain diseases from existence in addition to reducing their likelihood. Ultimately, further research in this field will provide us with enhanced knowledge regarding medicine and the human condition. It is likely that these principles will continue to be developed as we discover new and exciting information regarding genetics, and it is all thanks to our scientific forefathers that we are currently able to make these great leaps in knowledge of ourselves and of our environment.

Works Cited

Crow, J. “Hardy, Weinberg and language impediments”. Genetics. 152.3 (1999): 821–5.

Haldane, J.B.S. The measurement of natural selection. Proceedings of the 9th International Congress of Genetics. 1953:480-487

Reece, Jane B., and Neil A. Campbell. “Mendel and the Gene Idea.” Campbell Biology. 9th ed. Boston: Benjamin Cummings / Pearson Education. 2011: 265.