## 2. Model Development

The model has two state variables: structural dry matter,

$S$, in g[SDM] m

^{−2}[ground] and non-structural dry matter,

$C$ in g[CHO] m

^{−2}[ground] [

10].

Figure 1 is a summary flow diagram showing the state variables, rate variables, and driving variables (light, temperature, CO

_{2}) of the model. The terms “carbohydrate” (CHO) and “non-structural carbohydrate” (NSC) are considered synonyms, and dry matter (DM) stands for the combined nonstructural, and structural dry matter.

The differential rate equations are

and

where

$s$ is structural growth rate and

$c$ is the rate of change of

$C$. Furthermore,

$p$ is potential gross photosynthesis rate,

$i$ is inhibited photosynthesis,

$m$ is maintenance respiration, and

$g$ is growth respiration, all in g[DM] m

^{−2}[ground] h

^{−1}. The parameters mentioned here, and used later in the model, are defined in the Abbreviations, normalized per gram [SDM].

Photosynthesis is the only process that increases

**C**. I assume that potential photosynthesis per unit lit leaf area,

$p$, is a product of three functions, one of light,

$L$ (in μmol[PAR] m

^{−2}[leaf] s

^{−1}), one of temperature,

$T$ (in °C), and one of carbon dioxide concentration, C

_{a} (in μmol[CO

_{2}] mol

^{−1}[air]):

where

P_{max}, in g[DM] m

^{−2}[leaf] h

^{−1}, is maximum photosynthetic rate and

$f$,

$h$ and

$k$ are dimensionless correction terms, all positive and smaller than one. This is similar to the version of Gent and Seginer [

9], with an extra term for CO

_{2}. It fits the data of single leaf photosynthesis of tomato under various conditions of light, temperature and CO

_{2} (data not shown). Photosynthesis on a leaf area basis expressed in μmol[CO

_{2}] m

^{−2}[leaf] s

^{−1} is converted using the constant 10

^{−6} × 30 × 3600 into

p in g[DM] m

^{−2}[leaf] h

^{−1}. Photosynthesis per unit ground,

$p$, is obtained from

$p$ by multiplying by the sun-lit leaf area index,

**Λ** (in m

^{2}[leaf] m

^{−2}[ground]), namely

**Λ** itself may be calculated for small plants with all leaves illuminated from a measured value of the structural leaf area ratio,

$\lambda $, (in m

^{2}[leaf] kg

^{−1}[SDM]), and

$S$:

where I assume a constant leaf area ratio for seedlings.

Hence, Equation (4) becomes

I assume that photosynthesis is inhibited in proportion to the NSC available at dawn, as in the steady-state model [

9], namely

where

${C}_{d}$ is NSC at dawn, and

**β** (m

^{2}[ground] g

^{−1}[NSC]) is a positive constant smaller than

$1/{C}_{d}$. Consequently, actual photosynthesis,

$p-i$, is

Metabolism leading to growth is limited by temperature as well as by NSC. Let

$n\left\{T\right\}$ be the maximum rate of metabolism at a given temperature, both conversion to structure and all respiration. I assume this limit increases exponentially with temperature over the entire range

where

${n}_{0}$ (in g[DM] m

^{−2}[ground] h

^{−1}) is metabolism of a crop of size

$\lambda S$ = 1 and at 0 °C.

Maintenance respiration is assumed to increase exponentially, except with a temperature response,

θ, that may differ from

ζ,

Structural growth,

$s$, is proportional to growth respiration,

$g$, via the conversion efficiency,

**ε**, from carbon in photosynthate to carbon in structure,

At high temperature, when

**n**{

T} is much greater than

**m**{

T}, growth is limited by

**α**,

Here,

**α**, in h

^{−1}, the response of growth to NSC, appears to be dependent on photoperiod but independent of temperature. In the light,

**α**_{L} may have a higher value than in darkness

**α**_{D}. As long as NSC is not completely depleted, growth continues to utilize NSC.

If NSC is completely depleted while

$p<m$, maintenance respiration is supported by sacrificing structural material. In that case

To describe the variation in NSC over the diurnal cycle,

**S** is normalized to 1 g[SDM] after each iteration. This requires subtracting a term in

**Cs**, corresponding to the NSC lost in this renormalization

Given the states

$S$ and

$C$ of the crop at any point in time, and while

$C>0$, the terms of Equation (2) may be calculated from Equations (8), (10), and (14),

The rates $p$ and $m$ are unaffected by the level of $C$ in Equation (2). Once the rates $c$ and $s$ are known, the next-step values of the states $S$ and **C** may be calculated, and then the next set of rates and so on.

The parameters for NSC and respiration are expressed in g [CHO] g

^{−1}[SDM] h

^{−1} (

Table 1). The model was simulated in the VENSIM simulation language (Ventana Systems, Harvard, MA, USA), starting with an arbitrary initial state. Five days of periodic weather resulted in a transition to an almost periodic trajectory of

**S** and

**C**. Relative growth rate on the fifth day, RGR, was calculated from the sum of growth increments over each hour divided by the elapsed time.

## 3. Parameter Estimation

Altogether, the dynamic model has 14 parameters:

**P**_{max},

**γ**,

**κ**,

**τ**,

**δ**,

$\lambda $,

**β**,

${n}_{0}$,

**ζ**,

${m}_{0}$,

**θ**,

**ε**,

**α**_{L}, and

**α**_{D} (

Table 1). As much as possible, the parameter values used were equivalent to those for tomato in the steady-state model [

9]. Values for

P_{max},

γ,

δ,

τ, and

κ in Equation (3) were determined by least squares fit to data from measurement of gross photosynthesis of tomato leaves [

9], except data were measured at CO

_{2} concentrations of 200, 400, and 1600 μmol[CO

_{2}] mol

^{−1}[air] (data not shown). The regression between measured and observed data was R

^{2} = 0.94. A

P_{max} value was found which optimized the prediction of growth rate and NSC over all environments. Leaf area ratio, usually reported per g [DM], was transformed to

**λ**, by subtracting NSC from the actual DM. The parameter

**β** was chosen to optimize the fit between observed and predicted NSC, when it accumulated at cool temperatures. Maintenance respiration,

**m**_{0}, and its temperature dependence,

θ, were determined from observations of dark respiration of tomato leaves at various temperatures [

9]. This gave a

Q_{10} = 2. The values of parameters describing metabolism,

**n**_{0} and ζ, were determined by fitting growth data at temperatures cooler than 20 °C. The resulting

Q_{10} = 3, was different from

θ for maintenance. A value of 0.75 is used for

**ε**, the parameter related to growth efficiency, based on the analysis of Penning de Vries et al. [

32].

The parameter

**α**, the response of growth to NSC, is a new element compared to the steady-state model [

9]. Growth and diurnal variation in NSC were sensitive to this parameter at warm temperatures when

**n**{

T} did not limit metabolism. A lower limit to

**α**_{D} was set by the growth rates observed under this condition. An upper limit was set by the minimum NSC content observed, or by the magnitude of the diurnal variation of NSC. The dependence of

**α**_{L} in the light was given by the losses of NSC over the light period.