Archive for December, 2018

Mathematics enters the canon of molecular biology

December 18, 2018

On the shelf in my office I have a heavy red tome. It is the fifth edition of ‘Molecular Biology of the Cell’ by Alberts et al., a standard work. While reading an editorial by Arup Chakraborty I learned that the sixth edition of the book contains a section on mathematical models. Its title is ‘Mathematical Analysis of Cell Functions’ and it is almost 20 pages long. It is perhaps not very deep mathematically but I am nevertheless delighted that it is there. I find it very important that a textbook of central importance in cell biology takes time to discuss mathematics in that context and presents arguments why mathematics is valuable for biology. I wonder how much of what is written there would be understood by a mathematician with no background in molecular biology. Of course that is not the intended audience of the book and it is just my idle curiosity that makes me ask the question.

Let me list some of the main themes treated in the section. I give them as informal statements. (1) Negative feedback can stabilize a steady state. (2) Negative feedback plus delay can give rise to sustained oscillations. (3) Positive feedback plus cooperativity can give rise to multistability. (4) An incoherent feed-forward loop can give rise to a transient response to a signal. (5) A coherent feed-forward loop can give rise to a delay. I should emphasize that in all cases mentioned here the models involved use ODE. The delay in (2) does not have to do with a delay equation but just with a sufficiently large number of steps which take place one after another.

I would like to make some connections between the points (1)-(5) above and mathematical theorems. I will start with (2) since that seems the easiest case. I will replace (2) with another statement which is even easier: (2a) negative feedback is a necessary condition for the existence of an attracting periodic solution. This is related to ideas of Rene Thomas which I disussed in a previous post. Suppose we have a system of ODE \dot x=f(x) where the signs of the elements of the linearization Df(x) are independent of x. Here a ‘sign’ may be positive, negative or zero. As explained elsewhere a system of this kind defines a graph called the species graph or influence graph. A feedback loop is an oriented cycle in this graph and its sign is the product of the signs of the edges making up the cycle. The system exhibits negative feedback if its influence graph contains a negative feedback loop. The claim is then that there exist no attracting periodic solutions. Before I go further I should acknowledge my debt to Frederic Beck, who wrote his MSc under my supervision on the subject of feedback. The discussion which follows has benefitted a lot from his exposition of this subject. A system without negative feedback loops as defined above is called coherent. It was proved by Angeli, Hirsch and Sontag (J. Diff. Eq. 246, 3058) that by reversing the signs of some of the variables the system can be made quasicooperative. This means that along any feedback loop all signs are equal. Note that this does not mean that signs need be equal along an unoriented cycle. This can be illustrated by the case of an incoherent feedforward loop. It follows from the results of Angeli et al. that in a coherent system there exist no attracting periodic solutions. This result can be strengthened by replacing ‘attracting’ by ‘stable’. This has been proved by Richard and Comet (J. Math. Biol 63, 593). They also claim that if the condition that the signs of the entries in the linearization are spatially constant is dropped then the analogous statement is false. Their proof of this latter statement contained an error, as noticed by Frederic Beck. They repaired it in an erratum (J. Math. Biol. 70, 957).

This discussion shows that a lot is understood about negative feedback as a necessary condition for periodic solutions although there may still be more to be discovered on that subject. I think that much less is understood about the possibility of negative feedback being a sufficient condition for periodic solutions. As a concrete example let us consider the Selkov oscillator. It does contain a negative feedback loop consisting of one positive and one negative edge. For some parameter values it exhibits periodic solutions but for others there are only damped oscillations. Can the assumption of a negative feedback loop be supplemented in some way by an assumption of delay to give a sufficient condition? What should that assumption of delay be?

Hahn’s minimal model for the Calvin cycle

December 4, 2018

As has been discussed elsewhere in this blog there are many mathematical models for the Calvin cycle of photosynthesis in the literature and it is of interest to try to understand the relations between these models. One extreme case which may be considered is to look at the simplest possible models of this system which still capture some of the essential biological features. In 1984 Brian Hahn published a model of the Calvin cycle with 19 chemical species. Like many of the models in the literature this does not include a description of photorespiration. In 1987 he extended this to a model which does include photorespiration and has 31 chemical species. Later (1991) Hahn followed the strategy indicated above, looking for simplified models. He found one two-dimensional model and one three-dimensional model. Like many models in the literature the three-dimensional model contains the fifth power of the concentration of GAP (glyceraldehyde phosphate). This arises because elementary reactions have been lumped together to give an effective reaction where five molecules of the three-carbon molecule GAP go in and three molecules of a five-carbon sugar come out. Assuming mass action kinetics for this effective reaction then gives the fifth power. In an attempt to simplify the analysis yet further Hahn replaced the fifth power by the second power in his two-dimensional model.

Sadly, Brian Hahn is no longer around to study models of photosynthesis. He was a mathematics professor at the University of Cape Town. As head of department he was asked to give an opinion on whether the position of one member of the department should be extended. He found he could not make a positive recommendation and the matter was referred to a committee. At the meeting of this committee the candidate became very aggressive and, at least partly as a consequence of this, it was decided that his position should not be extended. Some time later, in the mathematics building, the candidate beat Hahn to death. At his trial it was decided that he was mentally ill and therefore could not be convicted of murder. After less than two years he was released from confinement. From what I have read it seems that the public discussion of this case has been influenced by issues of racism: Hahn was white and the man who killed him was black.

Let me return to photosynthesis. I thought that the small models of Hahn should be subjected to detailed scrutiny. This became the project of my PhD student Hussein Obeid. Now we have written a paper on the two-dimensional model of Hahn (with quadratic nonlinearity) and the related model with the fifth power. These can be studied with and without photorespiration. Without photorespiration it turns out that there are solutions which tend to infinity at late times. These are similar to solutions previously discovered in another model of the Calvin cycle by Juan Velazquez and myself, which we called runaway solutions. The difference is that in the two-dimensional model we can see how the runaway solutions fit into the global dynamics of the system. In this case there is a unique positive steady state which is unstable. The one-dimensional stable manifold of this point divides the space into two regions. In one region all solutions tend to infinity and have the same late-time asymptotics. In the other region all solutions tend to zero. The existence of these solutions is related to the phenomenon of overload breakdown studied by Dorothea Möhring and myself in another paper. When photorespiration is added a stable steady state comes in from infinity. Convergence to the stable steady state replaces the runaway asymptotics and all solutions are bounded.

We were able to show that the dynamics for the system with the fifth power is qualitatively similar to that with the second power. The only differences are that it not possible to obtain such detailed information about the parameter ranges for which certain qualitative behaviour occurs and that it is not ruled out that non-hyperbolic behaviour might happen in exceptional cases. Hahn derives his two-dimensional model by an informal reduction of the three-dimensional one. We showed how this can be made rigorous using geometric singular perturbation theory. This allows some limited information to be obtained about the dynamics of solutions of the three-dimensional system. We hope that this paper will serve as a good foundation for later investigations of the the three- and higher dimensional models of Hahn in the future.