Dynamical Properties of Models for the Calvin Cycle

Abstract

Modelling the Calvin cycle of photosynthesis leads to various systems of ordinary differential equations and reaction-diffusion equations. They differ by the choice of chemical substances included in the model, the choices of stoichiometric coefficients and chemical kinetics and whether or not diffusion is taken into account. This paper studies the long-time behaviour of solutions of several of these systems, concentrating on the ODE case. In some examples it is shown that there exist two positive stationary solutions. In several cases it is shown that there exist solutions where the concentrations of all substrates tend to zero at late times and others (runaway solutions) where the concentrations of all substrates increase without limit. In another case, where the concentration of ATP is explicitly included, runaway solutions are ruled out.

Appendices

Appendix 1: Michaelis–Menten Theory

Consider a simple reaction which converts one molecule of the species $S$ (the substrate) to one molecule of the substance $P$ (the product). With mass action kinetics this leads to the equations $\dot{x}_S=-kx_S$ and $\dot{x}_P=kx_S$. Suppose now that this reaction is catalysed by an enzyme $E$. Then there is a reaction with reaction constant $k_1$ in which the substrate combines with the enzyme to form a complex $SE$. The reaction constant for the process of dissociation of $SE$ into $S$ and $E$ will be denoted by $k_{-1}$. Finally there is the reaction in which the complex gives rise to the product with reaction constant $k_2$ while setting free the enzyme. This gives rise to the system

$$\begin{aligned} \dot{x}_S&=-k_1 x_Sx_E+k_{-1}x_{SE},\end{aligned}$$

(8.1)

$$\begin{aligned} \dot{x}_{SE}&=k_1x_Sx_E-(k_{-1}+k_2)x_{SE},\end{aligned}$$

(8.2)

$$\begin{aligned} \dot{x}_{E}&=-k_1x_Sx_E+(k_{-1}+k_2)x_{SE},\end{aligned}$$

(8.3)

$$\begin{aligned} \dot{x}_P&=k_2x_{SE}. \end{aligned}$$

(8.4)

The first three of these equations form a closed system and thus it is natural to analyse it first and use the last equation to determine the evolution of the concentration of the product afterwards, if desired. The above system is the MM-MA version of the original simple reaction. The Michaelis–Menten kinetics will now be derived on a heuristic level. Note first that the quantity $x_{SE}+x_E$ is conserved. Call it $E_0$. Substituting the relation $x_E=E_0-x_{SE}$ into the first two evolution equations gives a closed system for $x_S$ and $x_{SE}$:

$$\begin{aligned} \dot{x}_S&=-k_1 E_0x_S+(k_{-1}+k_1 x_S)x_{SE},\end{aligned}$$

(8.5)

$$\begin{aligned} \dot{x}_{SE}&=k_1E_0x_S-(k_{-1}+k_1 x_S+k_2)x_{SE}. \end{aligned}$$

(8.6)

Now introduce $\tau =\epsilon t,\,\tilde{x}_{SE}=\epsilon ^{-1}x_{SE}$ and $\tilde{E}_0=\epsilon ^{-1}E_0$ for a constant $\epsilon$. This gives

$$\begin{aligned} x'_S&=-k_1 \tilde{E}_0 x_S+(k_{-1}+k_1 x_S)\tilde{x}_{SE},\end{aligned}$$

(8.7)

$$\begin{aligned} \epsilon \tilde{x}'_{SE}&=k_1 \tilde{E}_0 x_S-(k_{-1}+k_1 x_S+k_2)\tilde{x}_{SE} \end{aligned}$$

(8.8)

where the primes denote derivative with respect to $\tau$. In the last system it is possibly to formally pass to the limit $\epsilon \rightarrow 0$, corresponding to a very small amount of enzyme. In the limit the second equation reduces to the algebraic equation

$$\begin{aligned} \tilde{x}_{SE}=\frac{k_1\tilde{E}_0 x_S}{k_{-1}+k_1 x_S+k_2}. \end{aligned}$$

(8.9)

Substituting this back into the evolution equation for $x_S$ and gives the effective Michaelis–Menten equation

$$\begin{aligned} x_S'=-\frac{k_1k_2\tilde{E}_0 x_S}{k_1x_S+k_{-1}+k_2}. \end{aligned}$$

(8.10)

It can then be computed that in this set-up $x_P'=-x_S'$.

This type of discussion is quite standard and the reason it is reproduced here is to illuminate the relations between the three types of kinetics (MA, MM-MA and MM) by an explanation of the simplest example. There is a one to one correspondence between stationary solutions of the systems MM-MA and MM, as will now be shown. If a stationary solution $(x_S,x_{SE})$ of the system MM-MA is given then the Eq. (8.9) is satisfied. Hence the Eq. (8.10) holds and a stationary solution of the system MM is obtained. Conversely, suppose that a solution $(x_S,\tilde{x}_{SE})$ is given. Then a stationary solution of the system (8.5) is obtained. Defining $x_E=E_0-x_{SE}$ for a positive constant $E_0$ completes it to a stationary solution of the system MM-MA.

Appendix 2: A Special Class of Matrices

This appendix is concerned with the algebraic properties of some matrices of a special form which appear in this paper. Let $A$ be an $n\times n$ matrix with entries $a_{ij}$. Suppose that $a_{ii}<0$ for each $i$, that $a_{ij}>0$ for $j=i-1\ \mathrm{mod}\ n$ and that $a_{ij}=0$ otherwise. Suppose further that $(-1)^{n+1}\det A>0$. The matrix $A+\lambda I$ is positive for a sufficiently large real number $\lambda$, i.e all its elements are positive. By the Perron–Frobenius theorem [19] it has a unique eigendirection spanned by a positive vector. Let $p$ be an eigenvector of this type with components $p_i$. The corresponding eigenvalue is positive. Let it be denoted by $\beta$. Another consequence of the Perron–Frobenius theorem is that all other eigenvalues of $A+\lambda I$ have modulus smaller than $\beta$. In particular the real part of any other eigenvalue is smaller than $\beta$. The vector $p$ is an eigenvector of $A$ with eigenvalue $\alpha =\beta -\lambda$ and all other eigenvalues of $A$ have real part smaller than $\alpha$.

If $A$ is a matrix of the above special form then it can be shown that the matrix $B=A^{-1}$ is a positive matrix. One way of proving this as follows. Let $x$ be a vector in $\mathbf{R}^n$ and consider the equation $Ax=y$. Inverting the matrix is equivalent to solving this equation for $x$. The equation can be written in components as

$$\begin{aligned} a_{ii}x_i+a_{i,i-1}x_{i-1}=y_i; 1\le i\le n \end{aligned}$$

(9.1)

where the indices are to be interpreted modulo $n$. Hence

$$\begin{aligned} a_{i+1,i}x_i=(-a_{i+1,i+1})x_{i+1}+y_{i+1}. \end{aligned}$$

(9.2)

Note that the coefficients in this equation are positive. By substituting these equations into each other successively with $i$ increasing from one to $n$ it is possible to obtain an equation of the form:

$$\begin{aligned} \left( \prod _ia_{i,i+1}\right) x_1=\left( \prod _i(-a_{ii})\right) x_n+\sum c_iy_i \end{aligned}$$

(9.3)

where the coefficients $c_i$ are positive. Rearranging this gives

$$\begin{aligned} (-1)^{n+1}(\det A)x_n=\sum c_iy_i. \end{aligned}$$

(9.4)

Any other $x_i$ can be determined in an analogous way. The determinant of $A$ is equal to $\prod _i a_{ii}+(-1)^{n+1}\prod _i a_{i,i+1}$. Putting these facts together gives the proof of the desired statement.

It can be concluded from the Perron-Frobenius theorem that $B$ has a unique eigendirection spanned by a positive vector. This is also an eigenvector of $A$ and so must be proportional to $p$. The corresponding eigenvalue is $\alpha ^{-1}$ and is positive. Hence $\alpha$ is positive.