No, I haven’t turned 50 yet. However, my 50th refereed paper has now appeared in print. This therefore seems like an appropriate time to look back on some of the research I have done since I started out as a graduate student at the University of Toronto. (I had two prior papers from my undergraduate work, but these were both in areas of science I didn’t pursue.) My intention here isn’t to write a scholarly review paper, so you won’t find a detailed set of citations here. My full list of publications is, in any event, available on my web site.
My M.Sc. and Ph.D. theses were both on the application of invariant manifold theory to steady-state kinetics. I was introduced to these problems by my supervisor at the University of Toronto, Simon J. Fraser. Simon is a great person to work for. He is supportive, and full of ideas, but he also lets you pursue your own ideas. I had a great time working for him, and learned an awful lot of nonlinear dynamics from him.
One way to think of the evolution of a chemical system is as a motion in a phase space, typically a space whose axes are the concentrations of the various chemical species involved, but sometimes including other relevant variables like temperature. The phase space of a chemical system is typically very high-dimensional. The reactions that transform one species into another occur on many different time scales. The net result is that we can picture the motion in phase space as involving a hierarchy of collapse processes onto surfaces of lower and lower dimension, the fastest processes being responsible for the first collapse events, followed by slower and slower processes.1 These surfaces are invariant manifolds of the differential equations, and we developed methods to compute them. Given the equation of a low-dimensional manifold, we obtain a reduced model of the motion in phase space, i.e. one involving the few variables necessary to describe motion on this manifold.
Invariant manifold theory has been a fertile area of research for me over the years. I continue to publish in this area from time to time. In fact, one of my current M.Sc. students, Blessing Okeke, is working on a set of problems in this area. Expect more work on these problems in the future!
Toward the end of my time in Toronto, my supervisor, Simon J. Fraser, allowed me to spend some time working with Carmay Lim who, at the time, was cross-appointed to several departments at the University of Toronto, and worked out of the Medical Sciences Building. This was a very productive time, and I learned a lot from Carmay, particularly about doing research efficiently.
We worked on a set of applied problems on the lignification of wood using an interesting piece of hardware called a cellular automata machine. This was a special-purpose computer built to efficiently simulate two-dimensional cellular automata. The machine was programmed in Forth, a programming language most of you have probably never heard of, with some bits written in assembly language for extra efficiency. For a geek like me, programming this machine was great fun. I think we did some useful work, too, as our work on lignification kinetics still gets cited from time to time.
I had been to the 1992 SIAM Conference on Applications of Dynamical Systems in Snowbird which, I think, was just the second of what would become a long-lived series of conferences. There, I had discovered that there was a lot of interest in delay-differential equations (DDEs), as the tools necessary to analyze these equations were being sharpened. I had thought about the possibility of applying DDEs to chemical modelling, and decided to apply to work with Michael Mackey at McGill University, who was an expert on the application of DDEs in biological modelling. McGill was a great environment, and I learned a lot from Michael and his students. The most significant outcome of my time in Montreal was a paper published in the Journal of Physical Chemistry on the use of DDEs in chemical modelling.2
I pursued this style of modelling in a handful of papers. Eventually, I got interested in the use of delays to simplify models that can’t be described by differential equations, namely stochastic systems.3 This is another one of those ideas that I have kept following down through the years.
In my next blog post, I will reflect on some of the work I have done since arriving in Lethbridge.
1Marc R. Roussel and Simon J. Fraser (1991) On the geometry of transient relaxation. J. Chem. Phys. 94, 7106–7113.
2Marc R. Roussel (1996) The use of delay-differential equations in chemical kinetics. J. Phys. Chem. 100, 8323–8330.
3Marc R. Roussel and Rui Zhu (2006) Validation of an algorithm for delay stochastic simulation of transcription and translation in prokaryotic gene expression. Phys. Biol. 3, 274–284.