Archive for March, 2018

The Higgins-Selkov oscillator, part 2

March 29, 2018

Some time ago I wrote about a mathematical model for glycolysis, the Higgins-Selkov oscillator. In fact I now prefer to call it the Selkov oscillator since it is a system of equations written down by Selkov, although it was obtained by modifying a previous model of Higgins. At that time I mentioned some of the difficulties involved in understanding the global properties of solutions of this system. I mentioned that I had checked that this system exhibits a supercritical Hopf bifurcation, thus proving that it has stable periodic solutions for certain values of the parameters. In the hope of getting more insights into this system I decided to make it the theme of a master’s thesis. Pia Brechmann, the student who got this subject, obtained a number of interesting results. After she submitted her thesis she and I decided to carry this a bit further so as to get a picture of the subject which was as comprehensive as possible. We have now written a paper on this subject. I cannot say that we were able to answer all the questions we would have liked to but at least we answered some of them. Here I will mention some of the things which are known and some which are not.

When written in dimensionless form the system contains two parameters \alpha (a positive real number) and \gamma (an integer which is at least two). I already mentioned a result about a Hopf bifurcation and this was originally only proved for \gamma=2. In the paper we showed that there is a supercritical Hopf bifurcation for any value of \gamma. I also previously mentioned doing a PoincarĂ© compactification of the system and blowing up new stationary solutions which appear in this process and are too degenerate to analyse directly. The discussion of using blow-ups in polar coordinates previously given actually concerned the case \gamma=2 and does not seem practical for higher values of \gamma. It turns out that the technique of quasihomogeneous directional blow-ups, explained in the book ‘Qualitative theory of planar differential systems’ by Dumortier et al. can be used to treat the general case. This type of blow-up has the advantage that the transformations are given by monomials rather than trigonometric functions and that there is a systematic method for choosing good values of the exponents in the monomials.

We discovered a paper by d’Onofrio (J. Math. Chem. 48, 339) on the Selkov oscillator where he obtains interesting results on the uniqueness and stability of periodic solutions. It was suggested by Selkov on the basis of numerical calculations that for large \alpha there exist solutions with oscillations which grow arbitrarily large at late times (let us call these unbounded oscillations). We were not able to decide on a rigorous level whether such solutions exist or not. They cannot exist when a periodic solution does. When they exist they have to do with a heteroclinic cycle in the compactification. We showed that when a cycle of this kind exists it is asymptotically stable and in that case solutions with unbounded oscillations exist. However we were not able to decide whether a heteroclinic cycle at infinity ever exists for this system. What we did prove is that for all values of the parameters there exist unbounded solutions which are eventually monotone.

We also proved that when the steady state is stable each bounded solution converges to it and that when there exists a periodic solution it is unique and each bounded solution except the steady state converges to that. I find it remarkable that such an apparently harmless two-dimensional dynamical system is so resistent to a complete rigorous analysis.


Lotka’s system

March 11, 2018

The system of ODE \dot x=a-bxy, \dot y=bxy-cy was considered in 1910 by Lotka as a model for oscillatory chemical reactions (J. Phys. Chem. 14, 271). It exhibits damped oscillations but no sustained oscillations, i.e. no periodic solutions. It should not be confused with the famous Lotka-Volterra system for predator-prey interactions which was first written down by Lotka in 1920 (PNAS 6, 410) and which does have periodic solutions for all positive initial data. That the Lotka system has no periodic solutions follows from the fact that x^{-1}y^{-1} is a Dulac function. In other words, if we multiply the vector field defined by the right hand sides of the equations by the positive function x^{-1}y^{-1} the result is a vector field with negative divergence. This change of vector field preserves periodic orbits and it follows from the divergence theorem that the rescaled vector field has no periodic orbits. My attention was drawn to this system by the paper of Selkov on his model for glycolysis (Eur. J. Biochem. 4, 79). In his model there is a parameter \gamma which is assumed greater than one. He remarks that if this parameter is set equal to one the system of Lotka is obtained. Selkov obtains his system as a limit of a two-dimensional system with more complicated non-linearities. If the parameter \gamma is set to one in that system equations are obtained which are related to Higgins’ model of glycolysis. Selkov remarks that this last system admits a Dulac function and hence no sustained oscillations and this is his argument for discarding Higgins’ model and replacing it by his own. The equations are \dot x=a-b\frac{xy}{1+y+xy} and \dot y=b\frac{xy}{1+y+xy}-cy. (To obtain the limit already mentioned it is first necessary to do a suitable rescaling of the variables.) In this case the Dulac function is \frac{1+y+xy}{xy}.

The fact that the Lotka-Volterra system admits periodic solutions can be proved by exhibiting a conserved quantity. At this point I recall the well-known fact that while conserved quantities and their generalizations, the Lyapunov functions, are very useful when you have them there is no general procedure for finding them. This naturally brings up the question: if I did not know the conserved quantity for the Lotka-Volterra system how could I find it? One method is as follows. First divide the equation for \dot y by that for \dot x to get a non-autonomous equation for dy/dx, cheerfully ignoring points where \dot x=0. It then turns out that the resulting equation can be solved by the method of separation of variables and that this leads to the desired conserved quantity.

One undesirable feature of the Lotka-Volterra system is that it has a one-parameter family of periodic solutions and must therefore be suspected to be structurally unstable. In addition, if we consider a solution where predators are initially absent the prey population grows exponentially. The latter feature can be eliminated by replacing the linear growth term in the equation for the prey by a logistic one. A similar term corresponding to higher death rates at high population densities can be added in the equation for the predators but the latter modification has no essential effect. This is a Lotka-Volterra model with intraspecific competition. As discussed in the book ‘Evolutionary Games and Population Dynamics’ by Josef Hofbauer and Karl Sigmund, when this model has a positive steady state that state is globally asymptotically stable. The proof uses the fact that the expression which defines the conserved quantity in the usual Lotka-Volterra model defines a Lyapunov function in the case with intraspecific competition. This is an example of the method of obtaining conserved quantities or Lyapunov functions by perturbing those which are already known in special cases.

It follows from Poincaré-Bendixson theory that the steady states in the Lotka model and the Higgins model are globally asymptotically stable. This raises the question whether we could not find Lyapunov functions for those systems. I do not know how. The method used for Lotka-Volterra fails here because the equation for dy/dx is not separable.

Conference on mechanobiology and cell signalling in Oberwolfach

March 2, 2018

I just attended a conference in Oberwolfach in an area rather far away from my usual interests, although it was about mathematical biology. I did meet some interesting (known and unkown) people and encountered some new ideas. Here I will just discuss two talks which particularly caught my attention.

In a talk of Takashi Hiiragi I learned a number of interesting things about embryology. The specific subject was the mouse embryo but similar things should apply to the human case. On the other hand it is very far away from what happens in Drosophila, for instance. There is a stage where the first eight cells are essentially identical. More precisely there is a lot of random variation in these cells, but no systematic differences. The subsequent divisions of these cells are not temporally correlated. By the time the number of cells has reached thirty-two an important differentiation step has taken place. By that time there are some of the cells which belong to the embryo while the others will be part of the placenta. If the first eight cells are separated then each one is capable of giving rise to a complete mouse. (The speaker did seem to indicate some restriction but did not go into details.) In order to understand the development process better one of these cells is studied in isolation. The cell contains a clock and so it ‘knows’ that it is in the eight cell stage. It then develops into a group of four cells in the same way that the eight cells would normally develop into 32. Differentiation takes place. The key symmetry-breaking step takes place when one end of the cell (at the eight-cell stage) develops an area at one end where actin is absent. This polarization then influences the further motion of the cells. It is interesting that the interaction between the cells in these processes seems to have more to do with mechanical signals then with chemical ones.

There was a talk of Fredric Cohen about cholesterol. His claim was that the concentration of cholesterol as usually measured is not a useful quantity and that the quantity which should be measured is the chemical potential of cholesterol. This has to do with the fact that cholesterol is hardly soluble in water or in hydrophobic liquids. I must say that the term ‘chemical potential’ was something which was always very opaque for me. As a result of this talk I think I am beginning to see the light. The cholesterol in a cell is mainly contained in the cell membrane. However it is not simply dissolved there as single molecules. Instead most of the molecules are interacting with proteins or with other cholesterol molecules. The chemical potential has to do with how many molecules get transferred when the system is connected to a reservoir. Only those molecules which are free are available to be transferred. So the issues seem to be the relationship between the amounts of free and bound molecules and what the real significance of the concentration of free molecules is for understanding a system.