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\begin{document}

\title{\centering \large \bf A model for photosynthetic oscillations in 
crassulacean acid metabolism (CAM)}

\author{
B. Blasius$\mbox{}^1$, F. Beck$\mbox{}^1$, U. L\"uttge$\mbox{}^2$\\
{\small $\mbox{}^1$ Institut f. Kernphysik, TH-Darmstadt,
Schlo{\ss}gartenstrasse 9, D-64289 Darmstadt}\\
{\small $\mbox{}^2$ Institut f. Botanik, TH-Darmstadt,
Schnittspahnstrasse 3, D-64287 Darmstadt}
}

\date{  }

\maketitle

\begin{abstract} \em
We propose a model of crassulacean acid metabolism (CAM)
describing the varying concentrations of pools of major metabolites
by a system of coupled non-linear differential equations.
The model shows regular oscillations in normal dark-night cycles
and free running endogenous oscillations in continuous light.
The effect of temperature is incorporated in a realistic way. 
It leads to the correct dependence of the oscillatory period length 
on temperature as compared to experimental observations.
\end{abstract}


\section{Introduction}
Crassulacean acid metabolism (CAM) is a metabolic adaption of plants 
to dry environments, resulting in a circadian cycle of carbon flow
\cite{win96,osm78,lut87}.
In order to reduce water loss by transpiration through the stomata,
CAM-plants take up external \co during the night when the stomata are 
open at low evaporation demand of the atmosphere.
The \co is fixed via phosphoenolpyruvate carboxylase (PEPc), 
the key enzyme of CAM, and is accumulated 
overnight in the form of malate in the vacuole of the cell(phase I of CAM).
During daytime, the stomata are closed, the malate is released
from the vacuole, becomes decarboxylated, and the resulting \co is
refixed via ribulose-bis-phosphat carboxylase/oxygenase (RubisCO) 
in the Calvin-cycle (phase III of CAM).

It is long known \cite{wil59,wil60} that the diurnal rhythm of carbon flow in
CAM is a free running endogenous circadian rhythm,
which means that the oscillations 
persist under constant environmental conditions in continuous darkness 
and \co free air as well as in a normal atmosphere in continuous light.
The endogenous oscillations are only
stable for a limited region in the space of external control parameters. 
If certain parameters,
like for instance temperature or light intensity, are raised above
a critical level, rhythmicity breaks down and an irregular unpredictable
state appears \cite{buc84,lut92,grm96}. 
The origin of these observations, as well as the dynamical nature of the
irregular oscillations, are still open problems of the physiological mechanisms 
governing CAM.

In order to study the regulation of such systems, 
the use of simplified models and their simulation by differential equations
is a most useful tool \cite{seg84,mur93}. 
From a dynamical point of view
the biological clock can be regarded as a limit cycle
attractor in the phase-space of the system.
A system-theoretic model of the CAM-process has
been established already in 1984 \cite{nun84} in terms of a set
of coupled non-linear first order
differential equations describing the time change of concentrations
in the major metabolites of CAM. The model was able to reproduce the
measured variations of metabolite levels during a day-night cycle
astonishingly well. It was refined later to simulate the experiments with 
CAM-plants under continuous light by stabilizing the
oscillations for prolonged runs of the simulation \cite{lut92}. 
This improvement resulted in regular oscillations of {\co} uptake
in day-night cycles as well as in continuous light, which were brought
about by a hysteresis switch, or 'beat oscillator', between active and
passive transport at the tonoplast of the vacuole.
The authors could also simulate the breakdown of the oscillation
in increased irradiance. However, careful investigations later showed
that this was always associated with unphysiological states in the
levels of metabolites in the simulation, mainly due to large accumulation 
of starch by increase of light intensity.

With this situation the demand for further improvement of the model was 
given. Progress towards this, including a considerable simplification 
with respect to the dynamically independent constituents, is presented 
in this paper. We demonstrate how the effects of temperature can be 
incorporated into the model as a continuous functional dependence, leading to 
the correct period lengths of oscillations vs. temperature,
whereas high or low temperatures  result in a breakdown of the rhythm.


\section{The model}
As already explained in \cite{nun84,lut92} the model describes the
carbon-flow and its regulation during diurnal CAM-cycles on a cellular level,
and it consists of the following three basic elements:\begin{itemize}
\item Metabolic pools with time dependent concentrations $x_i(t)$ 
 (measured in mol carbon).
\item (Carbon) flows $v_j$ describing the gain and loss terms 
 of metabolites leading to changing rates of pool concentrations, 
 $\dot{x_i}$,
 given by the net flow into the pool, $ \sum_j B_{ij} v_j$.
 The coefficients $B_{ij}$ connect the flows $v_j$ to the given pool $i$. 
\item A functional dependence, $v_j = v_j(x_k,y)$,
 of the flows on the concentrations in the pools and
 on some external control parameters $y$, determined by chemical or
 photosynthetic reactions, diffusion laws and feedback loops like
 enzyme inhibition.
\end{itemize}

The dynamics is defined by a system of coupled non-linear differential 
equations,
whose dimension is given by the number of simulated carbon pools, $n$,
\beq 
\frac{d}{dt} x_i (t) = \sum_j B_{ij} v_j(x_k,y) \qquad i=1\cdots n\,. 
\eeq
It was obvious at the outset that in principal the whole complex 
regulatory network
of the CAM  process can not be mapped within a simple model. 
Such a mapping would even be not desirable in view of the usefulness 
of modelling, i.e. depicting the essential features of the process. 
Consequently, our
scope was to construct a skeleton model which comprises the 
essential features of CAM regulation, based on our present structural
understanding of this process, and starting with the minimal number of 
metabolites whose participation can not be ignored.

Runs of our simulations showed that only 4 pools 
are essential for a successful modelling of the CAM process:
The malate concentration in the cytoplasm, $x_1$, and in the vacuole, $x_2$,
the \co concentration in the cytoplasm, $x_3$, and a pool containing the 
carbohydrates accumulated via the
Calvin cycle, and represented by starch, $x_4$. Compared with the original 
model \cite{nun84} which contained 6 pools of 
metabolites, this is a considerable simplification, 
but we could show that removal of the two pools glucose-6-phosphate (G-6-P)
 and phosphoenolpyruvate (PEP) produced no essential change in the results.

Fig.1 presents a block diagram showing
how the 4 pools are connected by the flows $v_1 \cdots v_7$.
The arrows indicate our model assumptions for CAM, which are as follows:
\begin{enumerate}
\item
The cycle starts with the dark fixation of \co by carboxylation of PEP
via the the enzyme PEPc ($v_1$). 
The PEP required as acceptor for binding of  \co is gained glycolytically 
from the starch pool. 
The stoichiometric factors in the diagram and in equ.(2) account for the number
of carbon atoms involved in the reactions.
\item
The resulting malate is actively transported into the vacuole ($v_2$), 
driven by the $H^+$-transporting ATPase at the tonoplast. 
\item
During the day malate is released from the vacuole by passive transport  
($v_3$), and subsequently decarboxylated giving
\co and $C_3$ residues ($v_4$) which in contrast to the former 
versions of the model are not neglected but restored again in the starch pool.
\item
The resulting \co is photosynthetically fixed under the influence of light 
$L(t)$ via RubisCO in the Calvin cycle and to storage carbohydrates (starch) 
($v_5$).
\item
Since the system is not closed there are flows from and to the environment, 
i.e.  uptake of \co through the stomata ($v_6$) and a utilization of 
carbohyrate from the gained starch by the plant for growth and productivity 
($v_7$).
\end{enumerate}
These processes give rise to the following rate equations:

\beq
\begin{array}{llllllll}
\dot{x_1} = & +4v_1 & -v_2 & +v_3 & -4 v_4 &      &      &     \\
\dot{x_2} = &      & +v_2 & -v_3 &        &      &       &     \\
\dot{x_3} = & -v_1 &      &      & + v_4  & -v_5 & + v_6 &     \\
\dot{x_4} = & -3v_1 &      &      & + 3v_4 & +v_5 &      & -v_7 \\
\end{array}
\eeq
In the next step the functional dependence of the flows 
on the pools has to be specified (equ 3). 
Here a number of simplifying assumptions
have also been introduced: for instance
all chemical reactions are  of first order without time delays, and
no back reactions are considered \cite{nun84}.


\beqn
v_1 & = & c_1 \, x_4 / (1+k x_1) \\
v_2 & = & c_2 \, x_1 \nonumber \\
v_3 & = & c_3 \, T f(x_2) x_2\nonumber \\
v_4 & = & c_4 \, x_1\nonumber \\
v_5 & = & c_5 \, L(t) x_3\nonumber \\
v_6 & = & c_6 \, ( c_{ext}(t) - x_3) \frac{1}{x_3} \nonumber \\
v_7 & = & c_7 \, x_4 \nonumber
\eeqn
Flows depend only linearly on substrate concentrations,
the nonlinearity of the equations results exclusively from the regulation 
elements which are as follows:
\begin{itemize}
\item Regulation of transport of malate between cytoplasm and vacuole.
In the absence of a satisfying theory 
of the membrane transport process this is done by means of an effective
beat oscillator, or hysteresis switch \cite{lut92}. It is described in the 
model by a function $f(x_2)$ 
which switches the efflux of malate
to "on" ($f=1$) if the vacuole is filled to a certain level, 
and to "off\," ($f=f_0$) if the vacuole is emptied below a certain level (Fig.2).
This hysteresis switch is the
origin of the endogenous oscillations in the model.
As a possible mechanism switching could be caused by free soluble Ca$^{2+}$
in the vacuole depending on the pH value in the vacuole \cite{sch94}.

\item  Feedback inhibition of PEPc by malate is described in the standard 
form $1/(1 + k x_1)$.
This regulation leads to rhythmic changes in the activity of PEPc
and is essential for CAM in order to prevent refixation of
\co which is released during day by PEPc in a futile cycle rather than via 
RubisCO in the Calvin cycle. 

\item Regulation of stomatal opening and closing due to a 
stomatal resistance proportional to internal \co concentration.
Our simulations showed that this regulation is not necessary 
to obtain endogenous oscillations and can thus be neglected
if one is interested only in the endogenous mechanism. 
This is in accordance with experimental findings on plants with
stripped off epidermis \cite{klu}.
On the other hand, regulation of stomatal resistance
is of, course, important if simulated \co exchange is to
be compared with experimental observations.
\end{itemize}
The model depends on three external control parameters: external \co 
concentration  $c_{ext}$, light intensity $L(t)$, and temperature $T$.

Our simulation runs exhibit an astonishing robustness of the
results against functional changes in the structure of the model.
As a rule we find, for instance,
that the simulated time course of metabolites does not
depend on the exact form of the flows or inhibitions of equ. (3),
justifying our simple assumptions.
Remarkably, it makes no difference
whether the active transport of malate into the vacuole
works at all times, as in equ. (3), or is switched off 
by the beat oscillator during the time of efflux:
\beq v_2 = c_2 (1- f(x_1)) x_1. \eeq
From this observation we conclude that within this model we do not get 
information about a feedback of
the beat oscillator, resp. the ordering structure of the tonoplast,
upon the function of the active proton pump.

The results for simulations in normal 12h-dark 12h-light cycles
are presented in Fig.3, which demonstrates the time variations of the 
metabolic pools in our model.
The cycle at the beginning of the dark period starts with a transport
of malate into the vacuole (phase I), leading to an uptake of \co
(dark fixation). The rate of malate transport into the vacuole decreases 
with time. With the onset of light photosynthesis begins; the \co needed for 
this process is seen in the transitory morning peak of \co uptake (phase II). 
Further, with the filled vacuole the beat oscillator switches on the efflux.
As a result the malate concentration in cytoplasm becomes very high,
deactivating PEPc by feedback inhibition. Decarboxylation of malate increases
the internal \co concentration which can become even higher than the external 
\co , leading to negative values of \co uptake even during stomatal closure 
(phase III).
After emptying the vacuole efflux stops. This leads to a reactivation
of PEPc due to sinking malate concentration in the cytoplasm.
At this time both RubisCO and PEPc can fixate \co and there is a marked uptake
of \co in the transition phase of the late afternoon and evening (phase IV). 
The very large \co uptake in phase IV observed in the simulation (Fi.3) to 
some extend contradicts experimental observations where smaller rates are 
usually detected . A possible explanation of this discrepancy could be found 
by assuming rhythmic changes in the state of PEPc, which can delay
the onset of reactivation of PEPc.

The model also produces endogenous rhythms in continuous light as shown in 
Fig. 4.
In contrast to results with the former version \cite{lut92,grm96},
increase of light intensity no longer leads to a breakdown of rhymicity, 
but only to a slight increase in the oscillation frequency.
This is due to the fact that the starch content is now held in
a physiologically meaningful range, so more light only leads to an increased 
mean level of starch.
The observed loss of rhythmicity in intense light has thus to be based on a new 
influence of this control parameter, which is not yet understood.

\section{The effect of temperature}
The original version of the CAM model did not yet contain a description of 
the important influence of temperature as external control parameter
\cite{nun84,lut92}.
It was shown later by Grams et al. \cite{grm96} that temperature can be
integrated in terms of a discrete mode-switch matrix, regulating in- and 
efflux at the tonoplast.
In this form, however, continuous temperature variations can not be
incorporated. Here we develop a new mechanism for the regulations of the 
fluxes at the tonoplast. It describes the
temperature dependence of CAM in a continuous way, and allows to give an
explanation for temperature compensation.
This approach is inspired by the investigation of temperature effects on  
tonoplast fluidity \cite{klu91,grm95}.
Thus, we relate this to a change in the
permeability of the tonoplast to malate, while a possible influence of 
temperature on enzyme reactions 
\cite{car95} is not considered in this work.

In view of the results of Kluge et al. \cite{klu91}
we assume that higher temperatures lead to a higher malate
permeability of the membrane, resulting in an
efflux proportional to temperature $T$, equ.(3),
\beq v_3  = T c_3 \, f(x_2) x_2 . \eeq
 With this modification the period length of the endogenous oscillations
can be calculated from the simulations as a function of temperature (Fig.5). 
If no efflux is allowed during influx time ( $f_0 = 0$)
this leads to a smaller period length with increase of temperature 
due to the faster efflux (Fig. 5),
which seems natural, but is just opposed to the experimental findings 
\cite{wil92,grm96}. The situation can be altered if
the hysteresis switch is changed to allow a small efflux out
of the vacuole even during influx time ($f_{influx} = f_0$) (Fig.2).
Note that temperature is effecting both, the 
hysteresis switch dependend efflux as well as the permanent efflux. Since 
efflux is already quite rapid at normal temperatures (Fig. 3),
it is not very much affected by a temperature increase.
On the other hand the permanent efflux term
can considerably extend the time necessary to
fill the vacuole since the active transport has to work against the permanent 
efflux which increases with $T$. Because of this, higher temperatures cause 
slower oscillations.  If, however, temperature
is raised above a critical value, the vacuole can not be
filled any more, and rhythmicity breaks down. On the other hand, if temperature 
is lowered below a critical value the efflux becomes
so small that the vacuole can not be emptied and rhythmicity again breaks
down. Obviously the phases of the two steady states at very low
or very high temperature differ by 180$^0$, as is also observed experimentally 
\cite{gradi95}, and evidently, with this improved hysteresis switch
the dependence of period length
on temperature in the model simulations is in rather good agreement with 
experimental findings (Fig.5). 

\section{Conclusion}
We have shown that simulation of CAM is possible using a relatively simple 
model compared to the complicated regulatory network of the real process. 
The scope of our modelling does not lie in an exact and detailed reproduction 
of all measured quantities, 
but in constructing a minimal skeleton model for a better understanding of 
the decisive regulating 
elements of the CAM cycle. Not considered in our model are, for instance, 
regulation due to seasonal effects and the aspects of water loss.
From the manifold interactions of CAM plants with the environment only three 
factors, temperature, light intensity and external \co concentration, were 
selected to influence the dynamics of the process in form of
external control parameters.
Special emphasis in our investigations
was laid on the origin and control of endogenous oscillations. The effect of 
temperature was investigated in detail. Correct dependence of
period length could only be achieved by the assumption of a permanent leaking 
of malate from the vacuole, even during influx time. This, however, seems to 
be not unplausible in view of the fluidity properties of the tonoplast. 

Besides some minor quantitative differences between our simulations and 
experimental observations, there are still important qualitative features of 
CAM which are not reproduced within the model presented here, viz. 
damping of endogenous rhythms, and the
irregular state obtained by increasing light intensity or temperature under 
continuous light above a characteristic threshold. This qualitative deficiency 
of our model is an important indicator of what still is missing.
Regarding recent experiments on the fluidity of the tonoplast \cite{klu91},
a simulation of the ordering structure in this membrane should be most 
interesting. The reduction from the originally six metabolic pools to only 
four was a decisive step, giving a strong hint that better understanding of 
the influx-efflux dynamics at the tonoplast is urgently needed.  

\section*{Acknowledgments}
We are grateful to Thorsten Grams for valuable discussions. 
This work was supported by the Sonderforschungsbereich 199
(Teilprojekt B5) of the Deutsche Forschungsgemeinschaft, Bonn, Germany.

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\newpage
\parindent0cm
\section*{Figure captions}
{\bf Fig.1}:\\
Block-diagram of the CAM model, showing how the four pools
$x_1 \cdots x_4$ (drawn as boxes)
are connected by the flows $v_1 \cdots v_7$ (arrows).
Regulatory feedback loops are indicated as dashed lines.
The model depends on the three external parameters: light intensity,
external $CO_2$ concentration and temperature (dotted lines).

\vspace*{1cm}
{\bf Fig. 2}:\\
Hysteresis switch for regulating the efflux of the vacuole.
The abscissa $x$ denotes the concentration of malate in the vacuole ($x_2$).
A cycle starts with the empty vacuole, $x_2 = x_{min}$.
Since only a small efflux is present, $f(x) = f_0$, the vacuole is filled
by the active influx. At the critical concentration, $x_2=x_{max}$, $f(x)$
switches to the value $f(x) = 1$, and fast efflux sets in.
Once the concentraion has fallen down to $x_2=x_{min}$, $f(x)$ switches
again to $f(x) = f_0$, and the cycle starts anew.

\vspace*{1cm}
{\bf Fig. 3}:\\
Results of simulation in 12h-dark, 12-light cycles.
{\bf A}: net $CO_2$ exchange, {\bf B}: malate concentration in the 
cytoplasm (solid line) and malate concentration in the vacuole (dotted line). 
Times of simulated darkness, $L(t) = 0$, are
indicated as black bars.

\vspace*{1cm}
{\bf Fig. 4}:\\
{\bf A}: simulation results, {\bf B}: experimental findings for the plant 
{\em Kalancho\"e daigremontiana} \cite{bla96} of endogenous rhythms of net 
$CO_2$ exchange in continuous illumination.

\vspace*{1cm}
{\bf Fig. 5}:\\
Period length of simulated endogenous rhythms in continuous light, $L(t) =1$,
as a function of temperature.
Plotted are the results of two different simulations with
a minimal efflux of either
$f_0 = 0.2$ (solid line), or $f_0 = 0$ (dotted line).
As a comparison the filled circles show experimental findings 
for the plant {\em Kalancho\"e daigremontiana} \cite{gradi95}.

\end{document}