adel-wheat is an architectural model of wheat, based on the open l-system principles and the cpfg language and simulation program (Lindenmayer, 1968a,b; Prusinkiewicz, 1999; Prusinkiewicz et al., 2000; Mech, 2004). In adel-wheat, the basic unit is the phytomer, which is repeated over time to simulate development of the wheat plant. From top to bottom, the wheat phytomer consists of the following plant organs: a leaf (composed of a blade and a sheath) inserted on a node, an internode, and a tiller bud. From a given initial planting pattern of seeds, the model calculates growth and development, size, shape and orientation in space of each organ in relation to thermal time. Seed orientation (i.e. orientation of the first leaf), basal angle and curvature of leaf blades, and leaf and tiller azimuth angles are stochastic elements in the model, based on distributions derived from experimental data.
Tiller simulation is similar to simulation of the main shoot. The concept of relative phytomer number (RPN; Fournier et al., 2003) enables the derivation of properties of individual tiller phytomers from those of the main stem. This is achieved by adding a specific ‘phytomer shift’ value to the actual rank number of the phytomer, yielding the RPN, that is the (fractional) main stem phytomer number from which properties are derived. In previous work (Fournier et al., 2003; Evers et al., 2005, 2007) the probability and timing of tiller appearance followed experimentally derived distributions. Here, tillering is assumed to depend on the value of R:FR perceived by the plant organs (see ‘Bud extension and bud break’). It is important to stress the strong agreement between the properties of the simulated and observed plants used in this study (Evers et al., 2007).
Simulation of light
To simulate the influence of R:FR on tillering, effects of each individual leaf and internode on light transfer within the canopy have to be estimated. This requires calculation of multiple scattering (reflection and transmission) of light by plant organs. This was done using a radiosity method dedicated to crop canopies (Chelle & Andrieu, 1998; Chelle et al., 1998). This approach, called nested radiosity (nr), calculates both the irradiance on and the energy absorbed by each virtual plant organ. To avoid border effects, nr infinitely repeats the simulated canopy. The link between nr and the l-system simulation program cpfg is an interface called caribu (Chelle et al., 2004). This interface drives the different calls from cpfg to nr in order to simulate the spectral bands requested by the l-system (here: red and far-red). In adel-wheat, a leaf is defined by a set of polygons, the co-ordinates of which determine their positions in space. Absorption, reflection, and transmission of light at the level of a module, for example a leaf blade, are calculated by summing the contributions of the polygons that comprise the module. The source of the incoming light was a virtual hemisphere, approximated by 36 point sources (Fig. 1). The R:FR of each of these sources was assumed to be 1.2 (Smith, 1982).
Figure 1. Configuration of the light sources in the virtual hemisphere: (a) a top-down view; (b) a side view. The axes represent the X, Y and Z co-ordinates of the light sources.
Download figure to PowerPoint
Modules in adel-wheat representing leaves were provided with reflection and transmission coefficients for red and far-red light. The values were taken from recent measurements on wheat by F. Baret (INRA-CSE Avignon, unpublished), which were acquired according to the methodology described by Jacquemoud & Baret (1990). Leaf reflectance used in the model was 0.05 for red and 0.38 for far-red; transmittance was 0.03 for red and 0.49 for far-red. Stem reflectances for red and far-red were set at the same values as for leaf reflectances; transmittances were zero. Soil reflectance was set to 0.04 for red and 0.06 for far-red based on measurements using a multispectral radiometer (CropScan Inc., Rochester, MN, USA).
R:FR perception in adel-wheat
Vertically oriented organs are the sites of perception for R:FR of horizontally propagating light (Cordukes & Fisher, 1974; Morgan et al., 1980; Skinner & Simmons, 1993). Therefore, it is the vertically orientated organs (usually the base of the plant) that are lit by a red light source in experiments studying photomorphogenetic responses to R:FR (e.g. Casal et al., 1986). In adel-wheat, this part of the plant is represented by a single module, including sheaths and parts of elongating blades which form a tube from which leaf tips appear and within which the ear finally develops. This tube, or pseudostem, is located at soil level until stem (internode) elongation starts. We surmise that it is this tube that perceives the R:FR of the incoming light. At each time step, the average R:FR value perceived by the pseudostem was used in the model to determine the extension rate of the tiller buds.
Bud extension and bud break
A previous study (Evers et al., 2006) showed that the probability of tiller appearance decreased earlier during canopy development the higher the plant population density. Yet, at the time when tiller appearance ceased, the value of R:FR perceived at the base of the plants was similar (0.35–0.40 when plants were grown in full light) across plant population densities. Therefore, it was hypothesized that tiller bud outgrowth was arrested when the R:FR perceived dropped below a threshold value. This hypothesis was implemented in adel-wheat by making R:FR affect the relative extension rate of tiller buds. Bud extension was represented by an exponential growth function:
- (Eqn 1)
(L, bud length (mm); L0, initial bud length (mm); RERp, the potential relative extension rate ((°Cd)−1); t, thermal time (°Cd) since the initiation of the bud; F, a dimensionless ‘bud fate’ function.) Values of L0 and RERp are 0.09 mm and 0.024°Cd−1, respectively (Ljutovac, 2002). F represents a dimensionless ‘bud fate’ function, with a value between 0 and 1, which expresses how RERp is affected by R:FR (see Eqns 2–4 below).
In our model implementation, bud extension was assumed to be blind to R:FR for bud lengths smaller than a critical bud length of 2 mm (equal to c. 120°Cd since initiation) (i.e. F = 1 if L < 2, independent of R:FR perceived), as buds of length 2–4 mm were abundant (> 65%, n = 70) in a previous study with three population densities of wheat (personal data, unpublished). This indicated that the buds had a very low extension rate from a length of 2 mm onwards, possibly suggesting unrestricted extension below that length.
Furthermore, the frequency of tiller buds with a length > 4 mm was low, indicating that, when tillers buds reached that length, they usually grew out into a tiller. This was used in the implementation of bud break in the model: when a bud reached a critical length of > 4 mm (equal to c. 160°Cd since initiation), it was allowed to grow into a tiller. This ‘decision’ to grow out was irrevocable, i.e. from that point on tiller development could not be arrested in any way.
In situations where bud size is > 2 mm but < 4 mm, F is modulated by R:FR perceived through three possible types of response.
- (Eqn 2)
(Q (Quotient), R:FR; Qt, the threshold R:FR below which bud break is arrested.) When R:FR perceived by the pseudostem of the plant is above or equal to a given threshold value (Qt), F is 1; when R:FR is below Qt, F is 0 (Eqn 2). This causes bud extension to halt when R:FR perceived is below the threshold set.
(B) Curvilinear response. This type of response describes a curvilinear response of F to R:FR:
- (Eqn 3)
(Q and Qt have the same meanings as in Eqn 2; Qm, the maximum R:FR value, i.e. the R:FR of the incoming light).
(C) Linear response. This type of response describes a linear response of F to R:FR:
- (Eqn 4)
(Parameters and conditions are identical to those in Eqn 2.)
The three types of response are visualized in Fig. 2. The derivation of Eqns 3 and 4 is given in the Appendix. The three types of relationship between R:FR and F reflect three hypothetical ways in which R:FR can affect bud development. In the unit step response, bud extension rate is not affected. In the curvilinear response, bud extension rate is affected by R:FR, but its influence is small at R:FR values close to Qm and increases with decreasing R:FR. In the linear response, the effect of R:FR on bud extension is proportional to the value of R:FR. In this sequence, the three response types can be regarded as an increasing influence of R:FR on F, which is visualized in Fig. 2.
Figure 2. Three types of relationship between the ‘bud fate’ function F and the ratio of red to far-red light irradiance (R:FR); F is a value between 0 and 1 and determines the value of the relative extension rate of a tiller bud (Eqn 1). Qt is the threshold R:FR below which F is zero (here 0.4), and Qm is the R:FR of the incoming light (fixed at 1.2). The three types are (a) a unit step response, (b) a curvilinear response, and (c) a linear response. (a), (b) and (c) correspond to relationships A, B and C in the text. (c) shows the largest influence of R:FR on F, represented by the triangular area with co-ordinates (Qt, 0), (Qm, 1), (Qm, 0); in (b) the area between these co-ordinates is smaller than in (c), and in (a) this area is 0 (note that during canopy development R:FR decreases, i.e. goes from right to left on the x-axis).
Download figure to PowerPoint
Bud length was updated at every time step, and so was the value of F. Therefore, if bud growth was suppressed on one day, it was possible to continue to grow on the next day, if R:FR was sufficiently high as a result of a change in canopy structure. The ability of a bud to grow out was removed when the internode of the preceding phytomer started to extend; this is in accordance with personal observations and the literature (Williams & Langer, 1975), but does not allow for tiller re-growth after removal of neighbouring plants and the associated increase in R:FR perceived.
Each simulation consisted of 12 virtual wheat plants, split into three runs of four plants (for computational reasons). The time step of the simulation was 1 d, and thermal time started accumulating at the appearance of leaf 1. Simulations were performed using plant population densities of 100, 262 or 508 plants m−2 (corresponding to the plant population densities used in the experiment; see following section), Qt values of 0.4, 0.6 and 0.8, and the three proposed relationships between F and R:FR (Eqns 2–4). Additional simulations with a Qt value of 0.9 were performed in the case of the unit step response. In total, 90 individual simulation runs were performed, each taking up c. 3–8 h of simulation time (depending on the number of tillers formed) on a mainstream personal computer.
The simulation output was the number of tillers present per plant at each time step (a tiller was regarded as present when its first leaf had appeared). The variation in simulation output was the result of the stochastic components of the model (see ‘adel-wheat’).
To evaluate simulation output, data on tillering kinetics were taken from an outdoor experiment with spring wheat (Triticum aestivum L., cv. Minaret), performed in Wageningen, the Netherlands (51°58′N, 5°38′E) from April to June 2004. The set-up of the experiment has been described in Evers et al. (2006).
Leaf and tiller appearance was monitored every 3 or 4 d. A leaf was considered to have appeared halfway between the last moment it was absent and the first it was recorded as being present. Leaf appearance data were used to calculate the physiological age of the plants. Physiological age is a measure of plant development; its value represents the (decimal) number of main stem leaves that have appeared. Physiological age was calculated using the function resulting from linear regression of the number of leaves that have appeared vs thermal time.