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.

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Reaction networks in Copenhagen

Last week I attended a workshop on reaction network theory organized by Elisenda Feliu and Carsten Wiuf. It took place in Copenhagen from 1st to 3rd July. I flew in late on the Tuesday evening and on arrival I had a pleasant feeling of being in the north just due to the amount and quality of the light. Looking at the weather information for Mainz I was glad I had got a reduction in temperature of several degrees by making this trip. A lot of comments and extra information on the talks at this conference can be found on the blog of John Baez and that of Matteo Polettini. Now, on my own slower time scale, I will write a bit about things I heard at the conference which I found particularly interesting. The topic of different time scales is very relevant to the theme of the meeting and the first talk, by Sebastian Walcher, was concerned with it. Often a dynamical system of interest can be thought of as containing a small parameter and letting this parameter tend to zero leads to a smaller system which may be easier to analyse. Information obtained in this way may be transported back to the original system. If the parameter is a ratio of time scales then the limit will be singular. The issue discussed in the talk is that of finding a suitable small parameter in a system when one is suspected. It is probably unreasonable to expect to find a completely general method but the talk presented algorithms which can contribute to solving this type of problem.

In the second talk Gheorghe Craciun presented his proof of the global attractor conjecture, which I have mentioned in a previous post. I was intrigued by one comment he made relating the concept of deficiency zero to systems in general position. Later he explained this to me and I will say something about the direction in which this goes. The concept of deficiency is central in chemical reaction network theory but I never found it very intuitive and I feel safe in claiming that I am in good company as far as that is concerned. Gheorghe’s idea is intended to improve this state of affairs by giving the deficiency a geometric interpretation. In this context it is worth mentioning that there are two definitions of deficiency on the market. I had heard this before but never looked at the details. I was reminded of it by the talk of Jeanne Marie Onana Eloundou-Mbebi in Copenhagen, where it played an important role. She was talking about absolute concentration robustness. The latter concept was also the subject of the talk of Dave Anderson, who was looking at the issue of whether the known results on ACR for deterministic reaction networks hold in some reasonable sense in the stochastic case. The answer seems to be that they do not. But now I return to the question of how the deficiency is defined. Here I use the notation for the deficiency as originally defined by Feinberg. The alternative, which can be found in Jeremy Gunawardena’s text with the title ‘Chemical reaction network theory for in silico biologists’ will be denoted by . Gunawardena, who seems to find the second definition more natural, proves that the two quantities are equal provided a certain condition holds (each linkage class contains precisely one terminal strong linkage class). This condition is, in particular, satisfied for weakly reversible networks and this is perhaps the reason that the difference in definitions is not often mentioned in the literature. In general , so that deficiency zero in the sense of the common definition implies deficiency zero in the sense of the other definition.

For a long time I knew very little about control theory. The desire to change this motivated me to give a course on the subject in the last winter semester, using the excellent textbook of Eduardo Sontag as my main source. Since that I had never taken the time to look back on what I learned in the course of doing this and this became clearer to me only now. In Copenhagen Nicolette Meshkat gave a talk on identifiability in reaction networks. I had heard her give a talk on a similar subject at the SIAM life science conference last summer and not understood much. I am sure that this was not her fault but mine. This time around things were suddenly clear. The reason is that this subject involves ideas coming from control theory and through giving the course I had learned to think in some new directions. The basic idea of identifiability is to extract information on the parameters of a dynamical system from the input-output relation.

There was another talk with a lot of control theory content by Mustafa Khammash. He had brought some machines with him to illustrate some of the ideas. These were made of Lego, driven by computers and communicating with each other via bluetooth devices. One of these was a physical realization of one of the favourite simple examples in control theory, stabilization of the inverted pendulum. Another was a robot programmed to come to rest 30 cm in front of a barrier facing it. Next he talked about an experiment coupling living cells to a computer to form a control system. The output from a population of cells was read by a combination of GFP labeling and a FACS machine. After processing the signal the resulting input was done by stimulating the cells using light. This got a lot of media attention unter the name ‘cyborg yeast’. After that he talked about a project in which programmes can be incorporated into the cells themselves using plasmids. In one of the last remarks in his talk he mentioned how cows use integral feedback to control the calcium concentration in their bodies. I think it would be nice to incorporate this into popular talks or calculus lectures in the form ‘cows can do integrals’ or ‘cows can solve differential equations’. The idea would be to have a striking example of what the abstract things done in calculus courses have to do with the real world.

My talk at the conference was on phosphorylation systems and interestingly there was another talk there, by Andreas Weber, which had a possibly very significant relation to this. I only became aware of the existence of the corresponding paper (Errami et. al., J. Comp. Phys. 291, 279) a few weeks ago and since it involves a lot of techniques I am not too familiar with and has a strong computer science component I have only had a limited opportunity to understand it. I hope to get deeper into it soon. It concerns a method of finding Hopf bifurcations.

This conference was a great chance to maintain and extend my contacts in the community working on reaction networks and get various types of inside information on the field