Book review: An introduction to mathematical biology
Posted by Development Book Reviews, on 15 December 2011
This book review originally appeared in Development. Lance Davidson reviews “Mathematical Models of Biological Systems” (by Hugo van den Berg).
Book info:
Mathematical Models of Biological Systems By Hugo van den Berg Oxford University Press (2010) 256 pages ISBN 978-0-19-958218-1 (paperback), 978-0-19-958219-8 (hardback) £27.50/$49.50 (paperback), £65/$117 (hardback)
One of the key goals of modern cell and developmental biology is to expose the underlying principles that drive cell differentiation and to elucidate how organisms construct functional multicellular structures. Thanks to advances in sequencing, high throughput screens and sophisticated imaging technologies, these fields are now awash with quantitative descriptions of gene transcription, cell signaling and cell mechanics. However, extracting key principles from the flood of new data is a major challenge for researchers and a central obstacle to fundamental progress in cell and developmental biology. The tools required to interpret this vast amount of biological data and to test hypotheses based on these studies can be found in quantitative analysis and mathematical modeling. With the book Mathematical Models of Biological Systems, Hugo van den Berg aims to contribute to the training of a new generation of biologists and mathematicians and to provide them with an introduction to the methods that are now available to quantitatively analyze biological data.
Like many quantitative biologists, my first exposure to mathematical modeling was not in the context of cell biology or developmental biology, but came through examples from physical chemistry, physiology and population ecology. In these fields, simple problems can be formulated using ordinary differential equations (ODEs) with complete statements of the state variables, such as initial conditions. As students, we learned to write ‘word-models’ and to translate these into sets of ODEs. Word models are narrative passages intended to translate the details of a biological problem such that biologists and mathematicians alike can understand the problem in a way that allows equations to be written which capture those details. For instance, we can distil the interactions between predators and prey by stating the rules that govern their populations. Rules that govern the population of prey might include sources of population growth, such as birth or migration, and losses to the population due to predation or disease. The precise statement of these rules should be complete enough to govern the mathematical formulation of the model. Given a well-defined word model, the mathematical biologist can then write a series of ODEs; for example, with variables that represent the number of predators and prey and equations to describe how populations of predators and prey change. As students, we sometimes discovered that there were closed form solutions of these ODEs, in which changes in variables can be predicted explicitly by equations. But more often we found that we could only evaluate the general dynamic behavior of the variables; for instance, whether populations of predators and prey are stable or not. The insights and training that these model-building exercises gave us were instrumental in becoming fluent in the basic skills of mathematical modeling. The processes of formulating a model and relating fundamental principles to the mathematics and experimental outcomes were often more informative than the solution itself. However, after marveling at the awesome power of ODEs, we soon realized that the solution of some, or indeed most sets of, ODEs was intractable, that there was no way to capture relevant details of complex biology with continuous variables, or that model predictions could not be tested experimentally. As such, the tool kit of ODEs used to learn the skills of mathematical modeling is less useful for developing the quantitative models that are needed to describe problems in cell and developmental biology.