In a previous post I wrote about glycolytic oscillations and mentioned a mathematical model for them, the Higgins-Selkov oscillator. Higgins introduced this as a chemical model while Selkov also wrote about some mathematical aspects of modelling this system. When I was preparing my course on dynamical systems I wanted to present an example where the existence of periodic solutions can be concluded by using the existence of a confined region in a two-dimensional system and Poincare-Bendixson theory. An example which is frequently treated in textbooks is the Brusselator and I wanted to do something different. I decided to try the Higgins-Selkov oscillator. Unfortunately I came up against difficulties since that model has unbounded solutions and it is hard to show that there are any bounded solutions except a stationary solution which can be calculated explicitly. For the purposes of the course I went over to considering the Schnakenberg model, a modification of the Higgins-Selkov oscillator where it is not hard to see that all solutions are bounded.

More recently I decided to try to finally find out what happens with the Higgins-Selkov oscillator itself. Reading Selkov’s paper I originally had the impression that he had proved the essential properties of the solutions. This turned out to be mistaken. One obstacle for me was that Selkov cited a theorem from a famous Russian textbook of Andronov et. al. and I did not know what the theorem was. An English translation of the book exists in the university library here but since Selkov only cites a page number I did not know how to find the theorem. I was able to get further when Jan Fuhrmann got hold of a copy of the page in question from the Russian original. This page has an easily identifiable picture on it and this allowed me to identify the corresponding page of the English translation and hence the theorem. I found that, as far as it is applicable to the oscillator problem this was something I could prove myself by a simple centre manifold argument. Thus I realized that the results quoted by Selkov only resolve some of the simpler issues in this problem.

At this stage I decided to follow the direction pointed out by Selkov on my own. The first stage, which can be used to obtain information about solutions which tend to infinity, is to do a Poincare compactification. This leads to a dynamical system on a compact subset of Euclidean space. In general it leads to the creation of new stationary points on the boundary which are not always hyperbolic. In this particular example two new stationary points are created. One of these has a one-dimensional centre manifold and it is relatively easy to determine its qualitative nature. This is what Selkov was quoting the result of Andronov et. al. for. The other new stationary solution is more problematic since the linearization of the system at that point is identically zero. More information can be obtained by transforming to polar coordinates about that point. This creates two new stationary points. One is hyperbolic and hence unproblematic. The linearization about the other is identically zero. Passing to polar coordinates about that point creates three new stationary points. One of them is hyperbolic while the other two have one-dimensional centre manifolds. The process comes to an end. When trying this kind of thing in the past I was haunted by the nightmare that the process would never stop. Is there a theorem which forbids that? In any case, in this example it is possible to proceed in this way and determine the qualitative behaviour near all points of the boundary. The problem is that this does not seem to help with the original issue. I see no way in which, even using all this information, it is possible to rule out that every solution except the stationary solution tends to infinity as tends to infinity.

Given that this appeared to be a dead end I decided to try an alternative strategy in order to at least prove that there are some parameter values for which there exists a stable periodic solution. It is possible to do this by showing that a generic supercritical Hopf bifurcation occurs and I went to the trouble of computing the Lyapunov coefficient needed to prove this. I am not sure how much future there is for the Higgins-Selkov oscillator since there are more modern and more complicated models for glycolysis on the market which have been studied more intensively from a mathematical point of view. More information about this can be found in a paper of Kosiuk and Szmolyan, SIAM J. Appl. Dyn. Sys. 10, 1307.

Finally I want to say something about the concept of feedback, something I find very confusing. Often it is said in the literature that oscillations are related to negative feedback. On the other hand the oscillations in glycolysis are often said to result from positive feedback. How can this be consistent? The most transparent definition of feedback I have seen is the one from a paper of Sontag which I discussed in the context of monotone systems. In that sense the feedback in the Higgins-Selkov oscillator is definitely negative. An increase in the concentration of the substrate leads to an increase in the rate of production of the product. An increase in the concentration of the product leads to an increase of the rate of consumption of the substrate. The combination of a positive and a negative sign gives a negative loop. The other way of talking about this seems to be related to the fact that an increase in the concentration of the product leads to an increase in the reaction rate between substrate and product. This is consistent with what was said before. The difference is what aspects of the system are being regarded as cause and effect, which can lead to a different assignment of the labels positive and negative. The problem as I see it is that feedback is frequently invoked but rarely defined, with the implicit suggestion that the definition should be obvious to anyone with an ounce of understanding. I seem to be lacking that ounce.

May 30, 2014 at 4:06 pm |

Take a look at this article. The author considers a method for definition of feedback and use as one example the Selkov model.

http://www.biomedcentral.com/1752-0509/3/90

The Selkov model represents an open monosubstrate and monoproduct reaction catalized by an allosteric enzyme which is inhibited by the substrate and activated by the product.

So, the situation described by you “an increase in the concentration of the substrate leads to an increase in the rate of production of the product” is imposible.

(see, for example, http://users.df.uba.ar/silvina/papers/support/node5.html).

There is a positive feedback “an increase in the concentration of the product leads to an increase in the velocity of the reaction (autocatalytic term xy^2)” and a negative feedback as well “an increase in the concentration of the product leads to an increase of the outflow of it and thereby to decaying of the reaction intensity” (term -wy, w is the parameter of the outflow).

P.S. I hope that Selkov model has a future because it is very simple and useful model which describes the key features of the phosphofructokinase step in glycolytic pathway. Besides, there are up-to-date models based on it (Wolf’s model, 2000, which includes diffusion terms).

I have been studied Selkov model for a while and I am Russian, so don’t hesitate to answer any questions if you need help in translation of Russian books or something else.

June 12, 2014 at 11:43 am |

Hi, Darya. Thanks for the comments and references. I have spent some time thinking about this with the following results. It seems to me that the statements I made in my post are very close to remarks in the paper of Wilhelm you cited, although these are expressed in slightly different words. He says that Goldbeter talks about positive feedback in glycolysis but that ‘a more detailed analysis reveals that a negative feedback (according to our definition) is always contained’. Wilhelm introduces a graph (or, I would say, a multigraph since there can be several edges between two given nodes) and this makes sense for any reaction system with mass action kinetics. What is the relation to the graph of Sontag which I discussed previously? In fact the graph of Sontag is not defined for all reaction networks but only for a special class. It is defined precisely when all the edges in the graph of Wilhelm connecting a given pair of nodes with the same orientation carry the same sign (all have ordinary arrowheads or all have hammerheads). In that case we can replace each multiple edge by a single edge with the same orientation and the same sign to obtain the graph of Sontag. This means in particular that in general the graph of Wilhelm contains more information than that of Sontag and so gives a richer definition of feedback loops. It just happens that in the case of the Selkov oscillator both graphs are identical.

January 10, 2016 at 5:29 pm |

Is this work on the Higgins-Selkov oscillator that you did to prepare for your course available somewhere? E.g., can we get your course notes?

January 18, 2016 at 1:00 am |

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