Enriching the Teaching of Biology with Mathematical Concepts

Abstract

Secondary school educators are told to teach more mathematics and science to our students to help them become more proficient in the two subjects. Coordination of mathematics and science teaching is recognized as another means to improve proficiency. The National Science Foundation has funded the Mathematics, Science and Technology Partnership (Grant EHR-0314910) at Hofstra University and Stony Brook University to enhance mathematical proficiency of middle school students through coordinated teaching of mathematics, science, and technology. Involvement in this project has inspired me to incorporate mathematical concepts in my own college classroom. In particular, I have used mathematical concepts to enhance the teaching of a biological course in Molecular Immunology, the study of the immune system. This course is taught to undergraduates and to perspective biology teachers at Stony Brook University. Three areas in the course have become enriched through the introduction of mathematical concepts. These areas are also covered briefly in other biology courses such as Molecular and Cellular Biology, Microbiology, and in various laboratory courses at Stony Brook University. The incorporation of mathematical concepts in these three areas is described below. Symmetries in Antibody Structures Antibodies or Immunoglobulins (Ig) are proteins that recognize foreign molecules called and facilitate their clearance (Kindt et al., 2007; Roitt et al., 2001). Antigens can be proteins, polysaccharides, lipoproteins, and glycolipids, although it is usually protein antigens that elicit the best antibody production. Line and/or rotational symmetries found in antibody structures relate to the function of the various types of antibodies and help in teaching the strength of antibody/ antigen interactions. The simplest antibodies (IgG, IgE and IgD) have a basic four-chain structure consisting of two identical chains and two identical chains covalently held together with disulfide bonds (see Figure 1). Two antigen-binding domains of the antibody are at the tops of the arms where the light and heavy chains meet as shown in Figure 1A. Distinct line symmetry can be observed in the antibody structure, which divides one pair of light and heavy chains with the other pair (see IgG in Figure 1B). The simpler antibodies are therefore bivalent, meaning they have two identical binding sites for the foreign antigens. Bivalency increases the functional affinity, called avidity, of the antibody for its target antigen and allows for clumping or agglutination of antibody/antigen complexes. These complexes induce the immune system to take further action to protect the organism from the invading foreignness (Kindt et al., 2007; Roitt et al., 2001). [FIGURE 1 OMITTED] I use the simple concept of geometric symmetry in antibody structure to expand the student's understanding of the difficult concept of avidity. Some antibodies have multiples of the basic structure and exhibit additional line and rotational symmetries. For example, the structure of IgA consists of two basic structures covalently linked to each other through a joining protein; its dimeric structure has two line symmetries, vertical and horizontal (see IgA in Fig. 1B). IgA also has a rotational symmetry of 180[degrees]. The structure of IgM (see Figure 2) is even more interesting since it has ten basic structures covalently linked to each other through a joining protein (see IgM in Figure 1B). IgM therefore has multiple line and rotational symmetries (if the joining protein is overlooked). These symmetries greatly increase the avidity of IgM for its target antigen and increase the formation of antibody/antigen complexes. Using the mathematical concept of geometric symmetry enlivens the discussion of antibody structure and enhances an understanding of the relationship between antibody structure and function (see Figure 2). Avidity is defined just as the strength of interaction between a multivalent antibody and antigen. …