The minimal model of Hahn for the Calvin cycle

Abstract

There are many models of the Calvin cycle of photosynthesis in the literature. When investigating the dynamics of these models one strategy is to look at the simplest possible models in order to get the most detailed insights. We investigate a minimal model of the Calvin cycle introduced by Hahn while he was pursuing this strategy. In a variant of the model not including photorespiration it is shown that there exists exactly one positive steady state and that this steady state is unstable. For generic initial data either all concentrations tend to infinity at lates times or all concentrations tend to zero at late times. In a variant including photorespiration it is shown that for suitable values of the parameters of the model there exist two positive steady states, one stable and one unstable. For generic initial data either the solution tends to the stable steady state at late times or all concentrations tend to zero at late times. Thus we obtain rigorous proofs of mathematical statements which together confirm the intuitive idea proposed by Hahn that photorespiration can stabilize the operation of the Calvin cycle. In the case that the concentrations tend to infinity we derive formulae for the leading order asymptotics using the Poincaré compactification.

Analysis of a model of the Calvin cycle with diffusion of ATP

“…This is the fifth equation of the system ( 14) in [6]. Combining ( 5) and ( 9) and substituting (11) into the result gives…”

“…There is no contradiction here -both systems of equations are valid. In fact (11) is the same as equation (6.16) in [14] and ( 9) is obviously equivalent to an equation in [14]. The other relevant equations in [14] are…”

“…Thus this is an example where it can be expected that there are solutions which are oscillatory in the sense that some of the concentrations not monotone. Note for comparison that in a simpler analogue of this model due to Hahn [7] the eigenvalues of the linearization about a steady state are always real [11]. Fig.…”

“…Suppose that at a steady state all concentrations are equal to one and that c = 2. The reaction constants can be chosen as (6,11,11,2,6,1,1,17). We call this steady state P 2 .…”