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

TL;DR

This paper analyzes a mathematical model of the Calvin cycle incorporating ATP diffusion, revealing multiple inhomogeneous steady states and their instability, with implications for understanding photosynthesis dynamics.

Contribution

It introduces a reaction-diffusion model of the Calvin cycle with ATP diffusion and proves the existence of multiple inhomogeneous steady states.

Findings

Existence of infinitely many spatially inhomogeneous positive steady states.

All positive steady states are nonlinearly unstable.

Numerical simulations show non-monotonic concentration dynamics.

Abstract

The dynamics of a mathematical model of the Calvin cycle, which is part of photosynthesis, is analysed. Since diffusion of ATP is included in the model a system of reaction-diffusion equations is obtained. It is proved that for a suitable choice of parameters there exist spatially inhomogeneous positive steady states, in fact infinitely many of them. It is also shown that all positive steady states, homogeneous and inhomogeneous, are nonlinearly unstable. The only smooth steady state which could be stable is a trivial one, where all concentrations except that of ATP are zero. It is found that in the spatially homogeneous case there are steady states with the property that the linearization about that state has eigenvalues which are not real, indicating the presence of oscillations. Numerical simulations exhibit solutions for which the concentrations are not monotone functions of time.