Archive for November, 2013

Geometric singular perturbation theory

November 29, 2013

I have already written two posts about the Michaelis-Menten limit, one not very long ago. I found some old results on this subject and I was on the look-out for some more modern accounts. Now it seems to me that what I need is something called geometric singular perturbation theory which goes back to a paper of Fenichel (J. Diff. Eq. 31, 53). An interesting aspect of this is that it involves using purely geometric statements to help solve analytical problems. If we take the system of two equations given in my last post on this subject, we can reformulate them by introducing a new time coordinate \tau=t/\epsilon, called the fast time, and adding the parameter as a new variable with zero time derivative. This gives the equations x'=\epsilon f(x,y), y'=g(x,y) and \epsilon'=0, where the prime denotes the derivative with respect to \tau. We are interested in the situation where the equation g(x,y)=0, which follows from the equations written in terms of the original time coordinate t, is equivalent to y=h_0(x) for a smooth function h_0. The linearization of the system in \tau along the zero set of g automatically has at least two zero eigenvalues. For Fenichel’s theorem it should be assumed that it does not have any more zero (or purely imaginary) eigenvalues. Then each point on that manifold has a two-dimensional centre manifold. Fenichel proves that there exists one manifold which is a centre manifold for all of these points. This is sometimes called a slow manifold. (Sometimes the part of it for a fixed value of \epsilon is given that name.) Its intersection with the plane \epsilon=0  coincides with the zero set of g. The original equations have a singular limit as \epsilon tends to zero, because \epsilon multiplies the time derivative in one of the equations. The remarkable thing is that the restriction of the system to the slow manifold is regular. This allows statements to be made that qualitative properties of the dynamics of solutions of the system with \epsilon=0 are inherited by the system with \epsilon small but non-zero.

Due to my growing interest in this subject I invited Peter Szmolyan from Vienna,who is a leading expert in this field, to come and give a colloquium here in Mainz, which he did yesterday. One of his main themes was that in many models arising in applications the splitting into the variables x and y cannot be done globally. Instead it may be necessary to use several splittings to describe different parts of the dynamics of one solution. He discussed two examples in which these ideas are helpful for understanding the dynamics better and establishing the existence of relaxation oscillations. The first is a model of Goldbeter and Lefever (Biophys J. 12, 1302) for glycolysis. It is different from the model I mentioned in a previous post but is also an important part of the chapter of Goldbeter’s book which I discussed there. The model of Goldbeter and Lefever was further studied theoretically by Segel and Goldbeter (J. Math. Biol. 32, 147). On this basis a rigorous analysis of the dynamics including a proof of the existence of relaxation oscillations was given in a recent paper by Szmolyan and Ilona Kosiuk (SIAM J. Appl. Dyn. Sys. 10, 1307). The other main example in the talk was a system of equations due to Goldbeter which is a kind of minimal model for the cell cycle. It is discussed in chapter 9 of Goldbeter’s book.

I have the feeling that GSPT is a body of theory which could be very useful for my future work and so I will do my best to continue to educate myself on the subject.