Abstract
Chromosomal crossover is a biological mechanism to combine parental traits. It is perhaps the first mechanism ever taught in any introductory biology class. The formulation of crossover, and resulting recombination, came about 100 years after Mendel's famous experiments. To a great extent, this formulation is consistent with the basic genetic findings of Mendel. More importantly, it provides a mathematical insight for his two laws (and corrects them). From a mathematical perspective, and while it retains similarities, genetic recombination guarantees diversity so that we do not rapidly converge to the same being. It is this diversity that made the study of biology possible. In particular, the problem of genetic mapping and linkage—one of the first efforts towards a computational approach to biology—relies heavily on the mathematical foundation of crossover and recombination. Nevertheless, as students we often overlook the mathematics of these phenomena. Emphasizing the mathematical aspect of Mendel's laws through crossover and recombination will prepare the students to make an early realization that biology, in addition to being experimental, IS a computational science. This can serve as a first step towards a broader curricular transformation in teaching biological sciences. I will show that a simple and modern treatment of Mendel's laws using a Markov chain will make this step possible, and it will only require basic college-level probability and calculus. My personal teaching experience confirms that students WANT to know Markov chains because they hear about them from bioinformaticists all the time. This entire exposition is based on three homework problems that I designed for a course in computational biology. A typical reader is, therefore, an instructional staff member or a student in a computational field (e.g., computer science, mathematics, statistics, computational biology, bioinformatics). However, other students may easily follow by omitting the mathematically more elaborate parts. I kept those as separate sections in the exposition.
Highlights
Mendel and High School Biology Sexually reproducing organisms generally combine heritable traits from two parents
While mutations could occur during meiosis, most of the variation arises from the combinations of parental traits
How do these parental traits combine? The dominant theory was that some sort of blending or averaging took place
Summary
Mendel and High School Biology Sexually reproducing organisms generally combine heritable traits from two parents. While mutations could occur during meiosis, most of the variation arises from the combinations of parental traits. The dominant theory was that some sort of blending or averaging took place. Such a mode of inheritance would result in an average of all ancestors after only a modest number of generations (imagine repeatedly mixing colors). With traits taking real values in 1⁄21,10, is used on one population, and the model described in the section ‘‘A Simple Model’’, with elements (later called alleles) taking discrete values in f0,1g, is used on another. In both cases, a population size of 100 is kept constant for the entire duration of the simulation (100 time steps). Mendel formulated the concept of a gene (unit of inheritance), and hypothesized that inheritance is governed by the following two laws of genetics: 1. Segregation: Each sexually reproducing organism has two alleles (copies) for each gene, one inherited from each parent; and in turn will contribute, with equal probability (1=2), only one of these two alleles
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