For a complete list of my publications, follow this link.
For data sets generated in the course of my research program, follow this link.
The work of my group lies at the interface between chemistry, biology and mathematics, with occasional excursions into physics. We use a combination of mathematical reasoning and computer simulation to try to understand a variety of phenomena, with a particular emphasis on biological and chemical systems. Along the way, we have the opportunity to study mathematical problems such as the relationship between delay and ordinary differential equations. To give a flavor of what we do, here is a commented list of some of our recent publications:
I have long been interested in biochemical dynamics, and particularly in competitive dynamics. In this paper, we study the classical competitive inhibition mechanism in an open system, but with a slight twist: We consider a stochastic model which describes the kinetics when there are few molecules of a given type. In a living cell for instance, there are often just a few hundred or perhaps a few thousand copies of a particular enzyme. We find that this very simple mechanism (and a number of closely related mechanisms which we don't specifically study here) can then produce sustained oscillations of respectable amplitude. This is a bit of a surprise since the usual mass-action (ordinary differential equation) model always has a stable equilibrium point. Moreover, we predict that there is an optimum volume at which these oscillations are most observable. For the particular parameters we studied here, this optimum volume coincides with the volumes of bacteria or of eukaryotic organelles, which opens up interesting new possibilities for studying cellular rhythms.
Transcription is the first step in gene expression in which DNA is copied (transcribed) into RNA. Cells typically have just one or two copies of any given gene. We therefore can't use classical mass-action kinetics to describe processes like transcription since the "concentration" of a gene isn't a continuous variable. In this paper, we developed and studied a stochastic model of gene transcription. A stochastic model is one in which we consider the fact that reactive events (e.g. collisions of reactants) are random variables. We were able to derive various statistics from our model (e.g. mean transcription time). We also had the opportunity to compare two different treatments of our stochastic model and studied the conditions under which they were and were not equivalent. In addition to the insights gained into transcription kinetics, this kind of work is very helpful in understanding the kinds of approximations we can make in treating complex systems like this one.
When there are only a few molecules of a particular reactant in a system, the randomness of reactive events becomes important, leading to a stochastic description of the reaction kinetics. Genetic control systems are examples of systems for which it is necessary to adopt a stochastic description since genes are generally present in cells in just two copies. The stochastic treatment of transcription (synthesizing RNA from a DNA template) is however highly challenging because it involves (in a typical case) hundreds of elementary reactions (stepwise addition of nucleosides to the transcript). There have recently been quite a few papers which have proposed methods for accelerating the stochastic simulation of genetic control systems by replacing the hundreds of synthesis steps required to make an RNA transcript by a single step with delayed output. The question left unaddressed by all these papers is whether these new algorithms generate time series which are statistically identical (or at least similar) to the time series generated by stochastic simulation of a detailed model. Using our transcription model, we were able to answer this question with a qualified yes.
Model reduction is one of the most active threads of my research program. The challenge is to faithfully reproduce the behavior of a system on time scales of experimental interest. Slow invariant manifolds are surfaces on which a system evolves after the decay of transients. Calculating these surfaces thus gives us a reduction of a model, at least to the extent that the equation of the manifold is accurately computed. The ILDM (intrinsic low-dimensional manifold) is an approximation to the slow invariant manifold developed by Maas and Pope which has proven extremely popular in applications. Our paper describes a method which is mathematically equivalent to the ILDM but which may be faster.
Marc R. Roussel and Rui Zhu (2006) Exactly reduced chemical master equations. In Alexander N. Gorban, Nikolaos K. Kazantzis, Ioannis G. Kevrekidis, Hans Christian Öttinger and Constantinos Theodoropoulos (Eds.), Model Reduction and Coarse-Graining Approaches for Multiscale Phenomena, Springer, Berlin, pp 295-315 (invited paper).
Techniques for computing slow invariant manifolds were originally developed for systems in which the change in time of the concentrations is described by ordinary differential equations (well-mixed macroscopic chemical systems). However, when you have just a few molecules of a particular sort in a system (e.g. two copies of a gene in a typical eukaryotic cell), you can't describe the time course of the concentration of these molecules using differential equations. Rather, you have to adopt a probabilistic description. Paradoxically, although they describe very small systems, these probabilistic descriptions lead to extremely large sets of differential equations. In the first of these papers, we show that you can produce a reduced probabilistic description using variations on the methods developed for conventional chemical systems. In the second paper, we refine these methods and show how we can compute a physically sensible initial condition for the reduced model. The latter is generally an important problem in the study of reduced models since it's not always obvious where to start the reduced model to get results consistent with those of the full model.
Maya Mincheva and Marc R. Roussel (2007) Graph-theoretic methods for the analysis of chemical and biochemical networks. I. Multistability and oscillations in ordinary differential equation models. J. Math. Biol. 55, 61-86.
Maya Mincheva and Marc R. Roussel (2007) Graph-theoretic methods for the analysis of chemical and biochemical networks. II. Oscillations in networks with delays. J. Math. Biol. 55, 87-104.
Maya Mincheva and Marc R. Roussel (2006) A graph-theoretic method for detecting potential Turing bifurcations. J. Chem. Phys. 125, 204102:1-8.
In the emerging science of systems biology, people are trying to understand how living cells work by elucidating all the interactions (reactions, binding, etc.) which make up a cell's biochemistry. The trouble is that we often don't know the kinetic constants associated with these interactions, so mathematical treatments which require these constants (e.g. bifurcation analysis) must use educated guesses. The alternative is to carry out qualitative analyses which don't depend on the values of the kinetic constants. In these papers, we show how graph-theoretical analyses of biochemical pathways can determine whether or not various types of behavior can occur. The data required for these analyses are precisely of the sort supplied by current systems biology databases, namely the connectivity of the reaction network, but not the values of the kinetic constants. Paper I cleans up some loose ends left by Ivanova in her development of a method for ordinary differential equations. Paper II extends these methods to systems with delays. Biochemical systems can't respond instantly to signals or to changes in conditions since it takes time to transcribe and translate genes. Accordingly, any model which has a genetic regulatory component should in general also contain delayed terms. The extension of Ivanova's approach to systems with delays turns out to be remarkably simple. Our third paper deals with the potential for Turing bifurcations in reaction-diffusion systems. Turing bifurcations are associated with pattern formation in spatially extended systems. They are thought to be responsible for at least some developmental events, e.g. the formation of animal coat patterns (spots, stripes, etc.).
During my last sabbatical, I visited Professor David Lloyd at Cardiff University. We performed a series of yeast fermentation experiments which we monitored by membrane-inlet mass spectrometry. Because of the rapid sampling which is made possible by this technique, we were able to observe a fast metabolic rhythm with a period of about four minutes which apparently had not been seen before in yeast cultures.
Potential graduate school applicants should note that we do not carry out any experiments in Lethbridge. However, thanks to collaborations such as this one, we sometimes have interesting data sets to analyze. Moreover, the possibility would exist for a graduate student to go to Cardiff (or to other centres where I have collaborations) to carry out experiments.
This paper arose from a collaboration with Dr Andrei Igamberdiev of Memorial University. He came to me a few years ago with some very intriguing data from a plant physiology experiment showing high-frequency irregular oscillations after transfer of leaves to a low-CO2 environment. These oscillations appear to be stochastic. It may be possible to learn something about the connections between various subcellular pools from these experiments, although further research will be required to solidify this work.
In this paper, Andrei and I continue to explore factors that affect the dynamics of photosynthesis. Here, we focus on a couple of issues. The first is whether Rubisco can be modeled as a Michaelis-Menten enzyme in vivo. Two different lines of evidence lead us to conclude that the usual Michaelis-Menten-type rate expressions are invalid for Rubisco, one based on the extremely high concentration of Rubisco in chloroplasts, and the other based on the algebraic structure of the steady-state problem for this enzyme. We then consider the interesting role of carbonic anhydrase in coupling together photosystem II and the Calvin cycle. Our work here could best be characterized as preliminary. There is much left to do on both problems.