Hahn’s minimal model for the Calvin cycle

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.

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Photorespiration and RuBisCO

The enzyme at the centre of the fixation of carbon in photosynthesis (which is the ultimate source of our food) is RuBisCO (Ribulose Bisphosphate Carboxylase Oxygenase). It is sometimes called the ‘lazy enzyme’ because it works so slowly. This is unjust since the process is so difficult that RuBisCO is the only enzyme that can do it. So it is like the situation of a strong man who is the only one who can carry very heavy rocks and who is called lazy because it takes him so long to move the rocks. The name already indicates that this enzyme catalyses two different reactions with the same substrate, ribulose bisphosphate. So it might also be claimed that it does not concentrate well. Instead of spending all its time on the useful process of carboxylation it apparently wastes a lot of time on the hobby of oxygenation. Carboxylation produces two molecules of phosphoglycerate (PGA) by combining a molecule of carbon dioxide with one of ribulose bisphosphate. In the oxygenation reaction only one molecule of PGA is produced together with one of phosphoglycolate. At least superficially the latter substance is not only useless but actual causes the cell a lot of trouble getting rid of it. I have not seen a definitive explanation about what purpose this process, photorespiration, might serve. One idea is that it might act as a safety valve. A too large input of energy to the photosynthesis system might damage it and photorespiration can be used to dissipate this energy in a relatively harmless way when necessary. The relative rates of the two reactions are influenced by the concentration of carbon dioxide and for normal atmospheric concentrations the relative rate of photorespiration is quite high.

The discussion up to now has been appropriate for what is called photosynthesis. The name comes from the fact that the first molecule to be produced is the three-carbon molecule PGA. There is an alternative called photosynthesis. This also contributes to food production, even to the food I eat. My generic breakfast is cornflakes and the most economically important plant which uses photosynthesis is maize. In this process carbon dioxide is used to make malate. Malate is not directly useful but is an intermediate. It is transported to an internal compartment (bundle-sheath cells) where it releases carbon dioxide. The result is an increased concentration of carbon dioxide and RuBisCO, which is waiting there, is pushed towards doing more carboxylation at the expense of oxygenation. Since the process leads to greater crop yields under certain conditions there is a project to genetically engineer rice so as to produce a variant. There is third type of photosynthesis , CAM photosynthesis for crassulacean acid metabolism. It is used, for instance by pineapples. There are similarities with with the difference that while in carbon dioxide is harvested in one place and released in another in CAM it is harvested at one time (during the night) and released at another (during the day).

Models for photosynthesis, part 4

In previous posts in this series I introduced some models for the Calvin cycle of photosynthesis and mentioned some open questions concerning them. I have now written a paper where on the one hand I survey a number of mathematical models for the Calvin cycle and the relations between them and on the other hand I am able to provide answers to some of the open questions. One question was that of the definition of the Pettersson model. As I indicated previously this was not clear from the literature. My answer to the question is that this system should be treated as a system of DAE (differential-algebraic equations). In other words it can be written as a set of ODE coupled to a set of algebraic equations . In general it is not clear that this type of system is locally well-posed. In other words, given a pair with it is not clear whether there is a solution of the system, local in time, with and . Of course if the partial derivative of with repect to is invertible it follows by the implicit function theorem that is locally equivalent to a relation and the original system is equivalent to . Then local well-posedness is clear. The calculations in the 1988 paper of Pettersson and Ryde-Pettersson indicate that this should be true for the Pettersson model but there are details missing in the paper and I have not (yet) been able to supply these. The conservative strategy is then to stick to the DAE picture. Then we do not have a basis for studying the dynamics but at least we have a well-defined system of equations and it is meaningful to discuss its steady states.

I was able to prove that there are parameter values for which the Pettersson model has at least two distinct positive steady states. In doing this I was helped by an earlier (1987) paper of Pettersson and Ryde-Pettersson. The idea is to shut off the process of storage as starch so as to get a subnetwork. If two steady states can be obtained for this modified system we may be able to get steady states for the original system using the implicit function theorem. There are some more complications but the a key step in the proof is the one just described. So how do we get steady states for the modified system? The idea is to solve many of the equations explicitly so that the problem reduces to a single equation for one unknown, the concentration of DHAP. (When applying the implicit function theorem we have to use a system of two equations for two unknowns.) In the end we are left with a quadratic equation and we can arrange for the coefficients in that equation to have convenient properties by choosing the parameters in the dynamical system suitably. This approach can be put in a wider context using the concept of stoichiometric generators but the proof is not logically dependent on using the theory of those objects.

Having got some information about the Pettersson model we may ask what happens when we go over to the Poolman model. The Poolman model is a system of ODE from the start and so we do not have any conceptual problems in that case. The method of construction of steady states can be adapted rather easily so as to apply to the system of DAE related to the Poolman model (let us call it the reduced Poolman model since it can be expressed as a singular limit of the Poolman model). The result is that there are parameter values for which the reduced Poolman model has at least three steady states. Whether the Poolman model itself can have three steady states is not yet clear since it is not clear whether the transverse eigenvalues (in the sense of GSPT) are all non-zero.

By analogy with known facts the following intuitive picture can be developed. Note, however, that this intuition has not yet been confirmed by proofs. In the picture one of the positive steady states of the Pettersson model is stable and the other unstable. Steady states on the boundary where some concentrations are zero are stable. Under the perturbation from the Pettersson model to the reduced Poolman model an additional stable positive steady state bifurcates from the boundary and joins the other two. This picture may be an oversimplification but I hope that it contains some grain of truth.

Flying to Copenhagen without a carpet

This semester I have a sabbatical and I am profiting from it by travelling more than I usually do. At the moment I am visiting the group of Carsten Wiuf and Elisenda Feliu at the University of Copenhagen for two weeks. The visit here also gives me the opportunity to discuss with people at the Niels Bohr Institute. Note that the authors of the paper I quoted in the post on NFB were at the NBI when they wrote it and in particular Mogens Jensen is still there now. I gave a talk on some of my work on the Calvin cycle at NBI today. Afterwards I talked to Mogens and one of his collaborators and found out that he is still very active in modelling this system.

I was thinking about my previous visits to Copenhagen and, in particular, that the first one was on a flying carpet. The background to this is that when I was seven years old I wrote a story in school with the title ‘The Magic Carpet’. I do not have the text any more but I know it appeared in the School Magazine that year. In my own version there was also a picture which I will say more about later. But first something about the story, of which I was the hero. I bought the carpet in Peshawar and used it to visit places in the world I was interested in. For some reason I no longer know I had a great wish at that time to visit Copenhagen. Perhaps it was due to coming into contact with stories of Hans Christian Andersen. In any case it is clear that having the chance this was one of the first places I visited using the magic carpet. The picture which I drew showed something closer to home. There I can be seen sitting on the carpet, wearing the blue jersey which was my favourite at that time, while the carpet bent upwards so as to just pass over the tip of the spire of St. Magnus Cathedral in Kirkwall. In the story it was also related that one of the effects of my journey was a newspaper article reporting a case of ‘mass hallucination’. I think my teachers were impressed at my using this phrase at my age. They might have been less impressed if they had known my source for this, which was a Bugs Bunny cartoon.

During my next visit to Copenhagen in 2008 (here I am not counting changing planes there on the way to Stockholm, which I did a few times) I was at a conference at the Niels Bohr Institute in my old research field of mathematical relativity and I gave a talk in that area. Little did I think I would return there years later and talk about something completely different. I remember that there was a photo in the main lecture room where many of the founders of quantum mechanics are sitting in the first row. From my own point of view I am happy that another person who can be seen there is Max Delbrück, a shining example of a switch from physics to biology. My next visit to Copenhagen was for the conference which I wrote about in a previous post. It was at the University. Since that a lot has happened with chemical reaction network theory and with my understanding of it. The lecture course I gave means that some of the points I mentioned in my post at that time are things I have since come to understand in some depth. I look forward to working on projects in that area with people here in the coming days.

Stability of steady states in models of the Calvin cycle

I have just written a paper with Stefan Disselnkötter on stationary solutions of models for the Calvin cycle and their stability. There we concentrate on the simplest models for this biological system. There were already some analytical results available on the number of positive stationary solutions (let us call them steady states for short), with the result that this number is zero, one or two in various circumstances. We were able to extend these results, in particular showing that in a model of Zhu et. al. there can be two steady states or, in exceptional cases, a continuum of steady states. This is at first sight surprising since those authors stated that there is at most one steady state. However they impose the condition that the steady states should be ‘physiologically feasible’. In fact for their investigations, which are done by means of computer calculations, they assume among other things that certain Michaelis constants which occur as parameters in the system have specific numerical values. This assumption is biologically motivated but at the moment I do not understand how the numbers they give follow from the references they quote. In any case, if these values are assumed our work gives an analytical proof that there is at most one steady state.

While there are quite a lot of results in the literature on the number of steady states in systems of ODE modelling biochemical systems there is much less on the question of the stability of these steady states. It was a central motivation of our work to make some progress in this direction for the specific models of the Calvin cycle and to develop some ideas to approaching this type of question more generally. One key idea is that if it can be shown that there is bifurcation with a one-dimensional centre manifold this can be very helpful in getting information on the stability of steady states which arise in the bifurcation. Given enough information on a sufficient number of derivatives at the bifurcation point this is a standard fact. What is interesting and perhaps less well known is that it may be possible to get conclusions without having such detailed control. One type of situation occurring in our paper is one where a stable solution and a saddle arise. This is roughly the situation of a fold bifurcation but we do not prove that it is generic. Doing so would presumably involve heavy calculations.

The centre manifold calculation only controls one eigenvalue and the other important input in order to see that there is a stable steady state for at least some choice of the parameters is to prove that the remaining eigenvalues have negative real parts. This is done by considering a limiting case where the linearization simplifies and then choosing parameters close to those of the limiting case. The arguments in this paper show how wise it can be to work with the rates of the reactions as long as possible, without using species concentrations. This kind of approach is popular with many people – it has just taken me a long time to get the point.

Models for photosynthesis, part 3

Here I continue the discussion of models for photosynthesis in two previous posts. There I described the Pettersson and Poolman models and indicated the possibility of introducing variants of these which use exclusively mass action kinetics. I call these the Pettersson-MA and Poolman-MA models. I was interested in obtaining information about the qualitative behaviour of solutions of these ODE systems. This gave rise to the MSc project of Dorothea Möhring which she recently completed successfully. Now we have extended this work a little further and have written up the results in a paper which has just been uploaded to ArXiv. The central issue is that of overload breakdown which is related to the mathematical notion of persistence. We would like to know under what circumstances a positive solution can have -limit points where some concentrations vanish and, if so, which concentrations vanish in that case. It seems that there was almost no information on the latter point in the literature so that the question of what exactly overload breakdown is remained a bit nebulous. The general idea is that the Pettersson model should have a stronger tendency to undergo overload breakdown while the Poolman model should have a stronger tendency to avoid it. The Pettersson-MA and Poolman-MA models represent a simpler context to work in to start with.

For the Pettersson-MA model we were able to identify a regime in which overload breakdown takes place. This is where the initial concentrations of all sugar phosphates and inorganic phosphate in the chloroplast are sufficiently small. In that case the concentrations of all sugar phosphates tend to zero at late times with two exceptions. The concentrations of xylulose-4-phosphate and sedoheptulose-7-phosphate do not tend to zero. These results are obtained by linearizing the system around a simple stationary solution on the boundary and applying the centre manifold theorem. Another result is that if the reaction constants satisfy a certain inequality a positive solution can have no positive -limit points. In particular, there are no positive stationary solutions in that case. This is proved using a Lyapunov function related to the total number of carbon atoms. In the case of the Poolman-MA model it was shown that the stationary point which was stable in the Pettersson case becomes unstable. Moreover, a quantitative lower bound for concentration of sugar phosphates at late times in obtained.These results fit well with the intuitive picture of what should happen. Some of the results on the Poolman-MA model can be extended to analogous ones for the original Poolman model. On the other hand the task of giving a full rigorous definition of the Pettersson model was postponed for later work. The direction in which this could go has been sketched in a previous post.

There remains a lot to be done. It is possible to define a kind of hybrid model by setting in the Poolman model. It would be desirable to completely clarify the definition of the Pettersson model and then, perhaps, to show that it can be obtained as a well-behaved limiting system of the hybrid system in the sense of geometric singular perturbation theory. This might allow the dynamical properties of solutions of the different systems to be related to each other. The only result on stationary solutions obtained so far is a non-existence theorem. It would be of great interest to have positive results on the existence, multiplicity and stability of stationary solutions. A related question is that of classifying possible -limit points of positive solutions where some of the concentrations are zero. This was done in part in the paper but what was not settled is whether potential -limit points with positive concentrations of the hexose phosphates can actually occur. Finally, there are a lot of other models for the Calvin cycle on the market and it would be interesting to see to what extent they are susceptible to methods similar to those used in our paper.

Calvin on the Calvin cycle

In a previous post I mentioned Calvin’s Nobel lecture. Now I read it again and since I had learned a lot of things in the meantime I could profit from it in new ways. The subject of the lecture is the way in which Calvin and his collaborators discovered the mechanisms of the dark reactions of photosythesis. This involved years of experiments which I am not qualified to discuss. What I will do here is to describe some of the major conceptual components of this work. The first step was to discover which chemical substances are involved in the process. To make this a well-defined question it is necessary to fix a boundary between those substances to be considered and others. As their name suggests the dark reactions can take place in the dark and to start with the process was studied in the dark. It seems, however, that this did not lead to very satisfactory results and this led to a change of strategy. The dark reactions also take place in the light and the idea was to look at a steady state situation where photosynthesis is taking place in the presence of light. The dark reactions incorporate carbon dioxide into carbohydrates and the aim was to find the mechanism by which this occurs. At the end of the Second World War, when this work was done, carbon 14 had just become much more easily available due to the existence of nuclear reactors. Calvin also mentions that when doing difficult separations of compounds in his work on photosynthesis he used things he had learned when separating plutonium during the war. Given a steady state situation with ordinary carbon dioxide the radioactive form of the gas containing carbon 14 could be introduced. The radioactive carbon atoms became incorporated into some of the organic compounds in the plants used. (The principal subject of the experiment was the green alga Chlorella.) In fact the radioactive carbon atoms turned up in too many compounds – the boundary had been fixed too widely. This was improved on by looking what happened on sufficiently short time scales after the radioactive gas had been added, of the order of a few seconds. After this time the process was stopped, leading to a snapshot of the chemical concentrations. This meant that the labelled carbon had not had time to propagate too far through the system and was only found in a relatively small number of compounds. The compounds were separated by two-dimensional chromatography and those which were radioactive were located by the black spots they caused on photographic film. Calvin remarks ironically that the apparatus they were using did not label the spots with the names of the compounds giving rise to them. It was thus necessary to extract those compounds and analyse them by all sorts of techniques which I know very little about. It took about ten years. In any case, the endpoint of this process was the first major conceptual step: a set of relevant compounds had been identified. These are the carbon compounds which are involved in the reactions leading from the point where carbon dioxide enters the system and before too much of the carbon has been transferred to other systems connected to the Calvin cycle. While reading the text of the lecture I also had a modern picture of the reaction network in front of me and this was useful for understanding the significance of the elements of the story being told. From the point of view of the mathematician this step corresponds to determining the nodes of the reaction network. It remains to find out which compounds react with which others, with which stoichiometry.

In looking for the reactions one useful source of information is the following. The carbon atoms in a given substance involved in the cycle are not equivalent to each other. By suitable experiments it can be decided which are the first carbon atoms to become radioactive. For instance, a compound produced in relatively large amounts right at the beginning of the process is phosphoglyceric acid (PGA) and it is found that the carbon in the carboxyl group is the one which becomes radioactive first. The other two carbons become radioactive at a common later time. This type of information provides suggestions for possible reaction mechanisms. Another type of input is obtained by simply counting carbon atoms in potential reactions. For instance, if the three-carbon compound PGA is to be produced from a precursor by the addition of carbon dioxide then the simple arthmetic relation indicates that there might be a precursor molecule with two carbons. However this molecule was never found and it turns out that the relevant arithmetic is . The reaction produces two molecules of PGA from a precursor with five carbon atoms, ribulose bisphosphate (RuBP). Combining the information about the order in which the carbon atoms were incorporated with the arithmetic considerations allowed a large part of the network to be reconstructed. Nevertheless the nature of one key step, that in which carbon dioxide is incorporated into PGA remained unclear. Further progress required a different type of experiment.

The measurements used up to now are essentially measurements of concentrations at one time point (or very few time points). The last major step was taken using measurements of the dynamics. Here the concentrations of selected substances are determined at sufficiently many time points so as to get a picture of the time evolution of concentrations is certain circumstances. The idea is to first take measurements of PGA and RuBP in conditions of constant light. These concentrations are essentially time-independent. Then the light is switched off. It is seen that the concentration of PGA increases rapidly (it more than doubles within a minute) while that of RuBP rapidly decreases on the same time scale. This gives evidence that at steady state RuBP is being converted to PGA. This completes the picture of the reaction network. Further confirmation that the picture is correct is obtained by experiments where the amount of carbon dioxide available is suddenly reduced and the resulting transients in various concentrations monitored.