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:
On the outside, most vertebrates are roughly symmetrical, but this is not the case with respect to the arrangement of the internal organs: in humans for example, your heart leans to the left, your left lung is smaller than your right, your liver is mostly on the right, and so on. Within a species, these internal asymmetries are almost invariant, with only one person in 10 000 having their organs mirrored relative to the normal arrangement. Errors in organ placement however cause significant birth defects in the heart and in other organs. So how does the developing embryo “know” which side is the left? Part of the answer to this question is that the protein Nodal specifies the left side. Nodal activates its own synthesis, as well as the synthesis of an antagonist named Lefty. The interaction of Nodal and Lefty may be responsible for the formation of a Turing pattern, or Lefty may be responsible for stopping a wave of Nodal from invading the right-hand side of the embryo. Our first paper in this area studies the dynamics of a model of Nodal and Lefty in a single cell. We find that, over very wide ranges of parameters, the system is bistable, with coexistence of left-hand high-Nodal and right-hand low-Nodal steady states. Bistability in a single-cell model would allow both wave propagation and Turing patterns in a multi-cell model we are currently developing.
Marc R. Roussel and Rui Zhu (2006) Stochastic kinetics description of a simple transcription model. Bull. Math. Biol. 68, 1681-1713.
Marc R. Roussel (2013) On the distribution of transcription times. Biomath 2, 1307247:1–14. [Invited paper associated with Biomath 2013]
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 have been able to derive various statistics from our model (e.g. mean transcription time) and to determine conditions under which the distribution of transcription times is distinctly non-Gaussian. In my second paper, I also developed an approximate form of the distribution that might be useful for fitting experimental results under some conditions.
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.
Maya Mincheva and Marc R. Roussel (2012) Turing-Hopf instability in biochemical reaction networks arising from pairs of subnetworks. Math. Biosci. 240, 1–11.
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.). Our latest paper deals with Turing-Hopf instabilities, which lead to interesting spatio-temporal behaviors (waves, spatio-temporal chaos, etc.), some of which are relevant to our understanding of cardiac function, among other things.
Our 2007 paper on rapid photosynthetic oscillations contained a hypothesis regarding the mechanism leading to the oscillations that involved the coupling between photosynthesis and photorespiration via the carbon dioxide generated by the latter process. In order to test this hypothesis, we built a mathematical model for the coupled processes. We were able to find oscillations, but not in a realistic parameter range. The analysis involved the use of our graph-theoretical methods for analyzing delayed mass-action systems, so it was a nice illustration of the power of these methods both for identifying the potential for oscillations and for identifying the parameter range where oscillations may be found.
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.
The eukaryotic nucleus is extraordinarily crowded with macromolecules. It is also highly structured. In this paper we study the influence that differences in density of macromolecules in different parts of the nucleus can have on the rate of nuclear export. Optimal search strategies for the nuclear pore typically involve a combination of factors, for example a layer in which isotropic diffusion operates along with a layer where diffusion to the membrane is accelerated at the cost of reduced lateral mobility.
I've been working on model reduction, in one way or another, since my graduate school days. It's one of those projects that just keeps going, raising new theoretical and practical questions all the time. A few years ago, my student Terry Tang and I developed a new method for model reduction which we called the FETA method, short for functional equation truncation approximation. At the time, we showed that the FETA method was equivalent to a classic construction known as the ILDM (intrinsic low-dimensional manifold) in planar (two-variable) systems. In this paper, I prove that the FETA method is equivalent to the ILDM in general. I also show that manifolds other than the slow manifold can be computed by the FETA/ILDM method. This opens up these methods to applications in a broader range of nonlinear dynamical problems than those to which they have so far been applied.
As an example of this method, I present here the two-dimensional FETA manifold, an approximation to the attracting manifold in which the Lorenz attractor lives. The red dots are points in the FETA manifold. The blue dots are the attractor, shown here for perspective.
This paper continues our development of model-reduction methods based on the iterative solution of the invariance equation. Here, we bring our iterative methods together with lumping (a.k.a. variable aggregation). In essence, we allow a user of our methods to choose variables for the reduced model that are functions of the original model variables. In one example presented in this paper, we use a function related to the system free energy as the retained variable. The result is a kinetic model for the evolution of this thermodynamic quantity.
Sensitivity analysis is an important technique in the analysis of models. It tells us which parameters have the greatest effect on specified aspects of a solution. With my colleagues Brian and Maya, we have developed a sensitivity analysis for oscillating solutions of delay-differential equations, which allows us to recover sensitivities of the period and amplitude of the solution to the parameters, including the delays.
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