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Algebraic and Discrete Mathematical Methods for Modern Biology

Raina Robeva, editor
Academic Press
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The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Russell Jay Hendel
, on

This is an excellent book which would be suitable as a textbook for:

  • An upper undergraduate course in Discrete Mathematics (I would, for reasons to be explained below, prefer this book over other current Discrete Mathematics books);
  • A first or second year graduate course in Discrete Mathematics; or
  • A senior seminar focused on undergraduate research.

The book is written as a collection of 14 self-contained chapters by distinct authors under the editorship of Raina Robeva, who herself has led multiple NSF-funded development projects at the interface of mathematics and biology. The 14 chapters treat frontier biological problems using algebraic and discrete mathematical methods; while the material does not require knowledge of calculus, a few chapters rely on knowledge of linear algebra.

In what follows I first bullet the features of the reviewed book that distinguish it from other Discrete Mathematics' books. I follow this with a bulleted list of features shared with other good textbooks.

  • Portrayal of mathematics as a dynamic, real-world, interactive discipline attempting to find solutions: Too often calculus, linear algebra and discrete mathematics courses portray mathematics as a closed exact discipline. The job of the student is to master results and apply them as they arise. But this is not true on the frontiers of research, where mathematicians attempt to find solutions to real-world problems, with no guarantee of success. Competing solutions must then be evaluated in terms of reasonableness, correctness and ease of use, the same types of subjective evaluations used in the non-exact sciences.

An example from the book occurs in Chapter 1. Several biological fields such as disease etiology or gene expression require the study of interactions between large numbers of objects, the purpose being to observe dominant clusters and patterns. The book presents two software programs attempting to portray the complex relationships between these large numbers of objects using discrete graphs. The software programs in turn allow multiple presentations e.g. radial, spring and circular. Thus the student is exposed to multiple solutions to data representation along with a discussion of pros and cons

  • Advanced coverage, with minimal prerequisites, of several standard Discrete Mathematics topics: For example:
    • Graph Theory: Besides presenting the traditional definitions of vertices, edges, adjacency, paths, loops, connectivity, degree, cliques, etc. the book heavily uses several other graph-theoretic concepts such as the degree distribution, the clustering coefficient, the small world network, interval graphs, boxicity, connectance, competition number and projection graphs.
    • Boolean Algebra: Besides the traditional concepts of Boolean variables, atoms, normal forms, unary-binary Boolean functions (and, or, not), the book heavily uses, attractors, canalization, stability and criticality.

These two main characteristics — advanced coverage of topics, and portrayal of mathematics as a dynamic interactive field — are the main attractive features of the reviewed book. I now list traditional features of all good Discrete Mathematics textbooks which this book has as well.

  • Coverage of standard Discrete Mathematics topics: In its study of many varied biological problems, the book covers trees, graphs, Boolean algebras, trees, dynamic discrete systems (recursions), grammars and languages, use of linear algebra and grammars and languages.
  • Exercises, projects and research questions: The chapters contain standard practice exercises to gain mastery of concepts, projects, some involving literature review, and research questions including simply stated questions on which virtually no research has been done.
  • Bibliographies: Each chapter has several dozen references. The references contain
    • Many modern papers written in the last five years.
    • Some classic vintages. I was delighted to find in discussions of Charles Darwin not only a reference to The Origin of Species (1859) but a reference to his original paper (on which the book was based) in the Journal of the Proceedings of the Linnean Society of the London Zoo (1858).
    • Several excellent online tutorials (for example tutorials on singular value decomposition and principal cumulant component analysis);
  • Variety of methods: I have already mentioned above that linear algebra, geometric, statistical, probability and language-grammar methods are used throughout the book.
  • Software packages: When something difficult can be done by software the book mentions it. Often, the book mentions several software packages and compares and contrasts them. Some of these software packages are available online, possibly for free. These software packages are not computational but typically deal with graphs, boolean functions and matrices.
  • Pictures and diagrams: The book is not big (350 pages). Nevertheless, it carefully shows graphs and matrices (instead of relying on students to run software to see them). The student has ample opportunity from the book itself to become interested and aware of issues of various approaches.
  • A variety of current topics of research: These include disease etiology, gene-expression, predator-prey relationships, cellular and gene networks, cell differentiation, Petri nets, transmission of infectious diseases, correlated gene responses, metabolic analysis, reconstructing phylogeny, RNA analysis.

Discrete Mathematics is traditionally taught as a preparatory course for computer science majors, or a standalone course to introduce proofs and logic, or as an alternative to calculus. The reviewed book shows numerous complex problems solvable by discrete methods; the book has an emphasis on software used for matrix and graph computations, thus making it very suitable for a preparatory course for computer science majors. I therefore strongly recommend it, over other classic texts, for use in teaching Discrete Mathematics.

Russell Jay Hendel, RHendel@Towson.Edu, holds a Ph.D. in theoretical mathematics and an Associateship from the Society of Actuaries. He teaches at Towson University. His interests include discrete number theory, applications of technology to education, problem writing, actuarial science and the interaction between mathematics, art and poetry.

1. Graph Theory for Systems Biology: Interval graphs, motifs, and patterns recognition
2. Food Webs and Graphs
3. Adaptation and Fitness Graphs
4. Signaling Networks: Asynchronous Boolean models
5. Dynamics of Complex Boolean Networks: Canalization, stability, and criticality
6. Steady State Analysis of Boolean Models: A dimension reduction approach
7. BioModel Engineering with Petri Nets
8. Transmission of Infectious Diseases: Data, models, and simulations
9. Disease Transmission Dynamics on Networks: Network structure versus disease dynamics
10. Predicting Correlated Responses in Quantitative Traits Under Selection: A linear algebra approach
11. Metabolic Analysis: Algebraic and geometric methods
12. Reconstructing the Phylogeny: Computational methods
13. RNA Secondary Structures: Combinatorial models and folding algorithms
14. RNA Secondary Structures: An approach through pseudoknots and fatgraphs