Stability of stationary solutions in models of the Calvin cycle

Abstract

In this paper results are obtained concerning the number of positive stationary solutions in simple models of the Calvin cycle of photosynthesis and the stability of these solutions. It is proved that there are open sets of parameters in the model of Zhu et. al. [15] for which there exist two positive stationary solutions. There are never more than two isolated positive stationary solutions but under certain explicit special conditions on the parameters there is a whole continuum of positive stationary solutions. It is also shown that in the set of parameter values for which two isolated positive stationary solutions exist there is an open subset where one of the solutions is asymptotically stable and the other is unstable. In related models derived from the work of [4], for which it was known that more that one positive stationary solution exists, it is proved that there are parameter values for which one of these solutions is asymptotically stable and the other unstable. A key technical aspect of the proofs is to exploit the fact that there is a bifurcation where the centre manifold is one-dimensional.

A Calvin Bestiary

“…The paper [19] uses computer-assisted methods which are claimed to prove the assertion about the limitation on steady states. In [5] a purely analytical proof of the assertion was given under the assumptions on the Michaelis constants made in [19]. It was also proved that the assertion depends essentially on these assumptions.…”

“…Restricting consideration to spatially homogeneous solutions reduces the resulting system of reaction-diffusion equations to a system of ODE (Model 2.4.1). It turns out that Model 2.4.1 can be analysed as in the cases of Models 2.2.1 and 2.2.2, giving the existence of two steady states, one stable and one unstable [5]. Interestingly, the solutions of Model 2.4.1 are bounded although this is nontrivial to prove [17].…”

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

“…In that case there is precisely one steady state and it is degenerate. It was shown in [4] that there are parameter values for which there is one stable and one unstable steady state. It was left open whether this is true for all parameter values for which there are two steady states.…”

“…For as an endpoint of the line segment is approached V (x ATP ) → E and hence x ATP → 0. If the other parameters are held fixed we can think of x 1 and x 2 as functions of c. As c → ∞ they behave in such a way that 4 . As c increases the function F increases.…”

“…We need to check that the vector field defining the dynamical system points into the region Q a,b for suitable choices of a and b. In the case of the part of the boundary defined by L 2 it suffices to suppose that a ≤ 1 2 k7 k4 1 4 . In the case of the part of the boundary defined by u 6 it suffices to require that b ≤ k8c 2(k2+k5)a+k8 .…”