Stochastic Modelling of Reaction–Diffusion Processes

This textbook is an example-driven introduction to stochastic modeling in mathematical biology. In the first six chapters, the authors develop a suite of modeling approaches for chemical reaction dynamics and diffusion, highlighting key differences and relationships between stochastic simulation algorithms, stochastic differential equations (SDEs), and deterministic ODEs and PDEs. The last three chapters briefly introduce more advanced material from reaction-advection-diffusion processes, to molecular dynamics and multi-resolution modeling, all of which are very active contemporary areas of research in both theoretical and applied domains. The textbook proceeds mainly by discussing a prototypical problem (e.g. diffusion within an ion channel) and then studying it from both analytical and computational angles (both independently and together, as appropriate).

 

The book is explicitly aimed at advanced undergraduates or beginning graduate students familiar with differential-equation based modeling, and some very basic concepts from probability. Despite these modest prerequisites, it covers an enormous breadth of stochastic modeling, and manages to touch on a number of still outstanding problems in the field. The example-driven presentation is used to quickly scaffold a reader's understanding of quite sophisticated techniques by starting with toy models of relatively simple phenomena, and slowly considering more detailed aspects of the underlying physics or chemistry, while developing more sophisticated modeling approaches. The presentation is focused on models and their analysis, so that rigorous mathematical aspects (e.g. formalizing white noise, asymptotic errors, or discussing rigorous limits of PDEs) and verbose biological details are only mentioned in passing, with copious references for the interested reader.

 

Beyond learning about these topics, the reader is shown how to view modeling from both an analytical and computational perspective, and these approaches continually reinforce one another, as they have in the historical development of the methods discussed. The reader is given several opportunities in the text to explore variations of the examples themselves, or to fill in details in derivations, in addition to other exercises collected at the end of each chapter. MATLAB codes used to generate all of the Figures in the book can be found at a companion website.  As stated in the preface, the vast majority of algorithms presented can easily be written in a variety of scientific computing languages with relative ease from the given pseudocode.

 

While the book's main focus is on chemical reaction networks and molecular diffusion, the authors make it clear that the ideas presented are rich in terms of applications, and deep in terms of connection to other areas (e.g.~statistical mechanics and stochastic analysis). Examples are given in ecological modeling (locust swarming, predator-prey interactions), electrophysiology (ion channels, excitability), as well as more obvious applications to chemistry relevant in cellular and systems biology. Beyond serving as a course textbook, the book could serve as a good general introduction to the area of stochastic modeling in biology for researchers, particularly given the copious citations to more specialist texts.

---

 

Dr. Andrew Krause is a Departmental Lecturer in Applied Mathematics at the University of Oxford. His research is primarily in mathematical biology and nonlinear dynamical systems. More information about him can be found at https://www.andrewkrause.org/.