## A model for the Calvin cycle with diffusion

In the past I have spent a lot of time thinking about mathematical models for the Calvin cycle of photosynthesis. These were almost exclusively systems of ordinary differential equations. One thing which was in the back of my mind was a system involving diffusion written down in a paper by Grimbs et al. and which I will describe in more detail in what follows. Recently Burcu Gürbüz and I have been looking at the dynamics of solutions of this system in detail and we have just produced a preprint on this. The starting point is a system of five ODE with mass action kinetics written down by Grimbs et al. and which I call the MA system (for mass action). That system has only one positive steady state and on the basis of experimental data the authors of that paper were expecting more than one. To get around this discrepancy they suggested a modified system where ATP is introduced as an additional variable and the diffusion of ATP is included in the model. I call this the MAd system (for mass action with diffusion). Diffusion of the other five chemical species is not included in the model. The resulting system can be thought of as a degenerate system of six reaction reaction-diffusion equations or as a system where five ODE are coupled to one (non-degenerate) diffusion equation. The idea was that the MAd system might have more than one steady state, with an inhomogenous steady state in addition to the known homogeneous one. Experiments which measure only spatial averages would not detect the inhomogeneity. To be relevant for experiments the steady states should be stable.

For simplicity we consider the model in one space dimension. The spatial domain is then a finite closed interval and Neumann boundary conditions are imposed for the concentration of ATP. We prove that for suitable values of the parameters in the model there exist infinitely many smooth inhomogeneous steady states. It turns out that all of these are (nonlinearly) unstable. This is not a special feature of this system but in fact, as pointed out in a paper of Marciniak-Czochra et al. (J. Math. Biol. 74, 583), it is a frequent feature of systems where ODE are coupled to a diffusion equation. This can be proved using a method discussed in a previous post which allows nonlinear instability to be concluded from spectral instability. We prove the spectral instability in our example. There may also exist non-smooth inhomogeneous steady states but we did not enter into that theme in our paper. If stable inhomogeneous steady states cannot be used to explain the experimental observations what alternatives are available which still keep the same model? If experiments only measure time averages then an alternative would be limit sets other than steady states. In this context it would be interesting to know whether the system has spatially homogeneous solutions which are periodic or converge to a strange attractor. A preliminary investigation of this question in the paper did not yield definitive results. With the help of computer calculations we were able to show that it can happen that the linearization of the (spatially homogeneous) system about a steady state has non-real eigenvalues, suggesting the presence of at least damped oscillations. We proved global existence for the full system but we were not able to show whether general solutions are bounded in time, not even in $L^1$. It can be concluded that there are still many issues to be investigated concerning the long-time behaviour of solutions of this system.

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