## Models for photosynthesis

Photosynthesis is a process of central importance in biology. There is a large literature on modelling this process. One step is to identify networks of chemical reactions involved. Another is to derive mathematical models (usually systems of ODE) from these networks. Here when I say ‘model’ I mean ‘mathematical model’ and not the underlying network. In a paper by Jablonsky et. al. (BMC Systems Biology 5: 185) existing models are surveyed and a number or errors and inconsistencies in the literature are pointed out. This reminded me of the fact that a widespread problem in the biological literature is that the huge amount of data being generated these days contains very many errors. Here I want to discuss some issues related to this, concentrating on models for the Calvin cycle of photosynthesis and, in particular, on what I will call the Poolman model.

A point which might seem obvious and trivial to the mathematician is that a description of a mathematical model (I consider here only ODE models) should contain a clear answer to the following two questions. 1) What are the unknowns? 2) What are the evolution equations? One source of ambiguity involved in the first question is the impossibility of modelling everything. It is usually unreasonable to model a whole organism although this has been tried for some simple ones. Even if it were possible, the organism is in interaction with other organisms and its environment and these things cannot also be included. In practise it is necessary to fix a boundary of the system we want to consider and cut there. One way of handling the substances outside the cut in a mathematical model is to set their concentrations to constant values, thus implicitly assuming that to a good approximation these are not affected by the dynamics within the system. Let us call these external species and the substances whose dynamics is included in the model internal species. Thus part of answering question 1) is to decide on which species are to be treated as internal. In this post I will confine myself to discussing question 1), leaving question 2) for a later date.

Suppose we want to answer question 1) for a model in the literature. What are potential difficulties? In biological papers the equations (and even the full list of unknowns) are often banished to the supplementary material. In addition to being less easy to access and often less easy to read (due to typographical inferiority) than the main text I have the feeling that this supplementary material is often subjected to less scrutiny by the referees and by the authors, so that errors or incompleteness can occur more easily. Sometimes this information is only contained in some files intended to be read by a computer rather than a human being and it may be necessary to have, or be able to use, special software in order to read them in any reasonable way. Most of these difficulties are not absolute in nature. It is just that the mathematician embarking on such a process should ideally be aware of some of the challenges awaiting him in advance.

How does this look in the case of the Poolman model? It was first published in a journal in a paper of Poolman, Fell and Thomas (J. Exp. Botany, 51, 319). The reaction network is specified by Fig. 1 of the paper. This makes most of the unknowns clear but leaves the following cases where something more needs to be said. Firstly, it is natural to take the concentration of ADP to be defined implicitly through the concentration of ATP and the conservation of the total amount of adenosine phosphates. Secondly, it is explictly stated that the concentrations of NADP and NADPH are taken to be constant so that these are clearly external species. Presumably the concentration of inorganic phosphate in the stroma is also taken to be constant, so that this is also an external variable, although I did not find an explicit statement to this effect in the paper. The one remaining possible ambiguity involves starch – is it an internal or an external species in this model? I was not able to find anything directly addressing this point in the paper. On the other hand the paper does refer to the thesis of Poolman and some internet resources for further information. In the main body of the thesis I found no explicit resolution of the question of external phosphate but there it does seem that this quantity is treated as an external parameter. The question of starch is particularly important since this is a major change in the Poolman model compared to the earlier Pettersson model on which it is based and since Jablonsky et. al. claim that there is an error in the equation describing this step. It is stated in the thesis that ‘a meaningful concentration cannot be assigned to’ … ‘the starch substrate’ which seems to support my impression that the concentration of starch is an external species. Finally a clear answer confirming my suppositions above can be found in Appendix A of the thesis which describes the computer implementation. There we find a list of variables and constants and the latter are distinguished by being preceded by a dollar sign. So is there an error in the equation for starch degradation used in the Poolman model? My impression is that there is not, in the sense that the desired assumptions have been implemented successfully. The fact that Jablonsky et. al. get the absurd result of negative starch concentrations is because they compute an evolution for starch which is an external variable in the Poolman model. What could be criticised in the Poolman model is that the amount of starch in the chloroplast varies a lot over the course of the day. Thus a model with starch as an external variable could only be expected to give a good approximation to reality on timescales much shorter than one day.