SEARCH

SEARCH BY CITATION

Keywords:

  • functional–structural model;
  • nested radiosity;
  • plant architecture;
  • red:far-red ratio;
  • shade avoidance;
  • tillering;
  • virtual plant;
  • wheat (Triticum aestivum)

Summary

  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and Methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References
  9. Appendix: Derivation of
  • • 
    The outgrowth of tiller buds in Poaceae is influenced by the ratio of red to far-red light irradiance (R:FR). At each point in the plant canopy, R:FR is affected by light scattered by surrounding plant tissues. This paper presents a three-dimensional virtual plant modelling approach to simulate local effects of R:FR on tillering in spring wheat (Triticum aestivum).
  • • 
    R:FR dependence of bud outgrowth was implemented in a wheat model, using three hypothetical responses of bud extension to R:FR (unit step, curvilinear and linear response). Bud break occurred when a threshold bud length was reached. Simulations were performed for three plant population densities.
  • • 
    In accordance with experimental observations, fewer tillers per plant were simulated for higher plant population densities. The linear and curvilinear responses caused a delay in the increase in tiller number compared with experimental data. The unit step response approached experimental results best. It is suggested that a model based on relatively simple relations can be used to simulate degree of tillering.
  • • 
    This study has shown that the virtual plant approach is a promising tool with which to address crop morphological and ecological research questions where the determining factors act at the level of the individual plant organ.

Introduction

  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and Methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References
  9. Appendix: Derivation of

Tillering is an important property in crops (cereals and grasses) of the Poaceae family. It enables these crops to cope with intraspecific competition by optimizing the formation of ear-bearing shoots in relation to the resources available (Darwinkel, 1978). Next to nitrogen availability (Davies, 1971; Longnecker et al., 1993), light plays an important role in the regulation of tillering. The number of tillers per plant and the rate of tiller appearance have been related to the intensity of photosynthetically active radiation (PAR), for example, in wheat (Triticum aestivum; Bos, 1999) and perennial ryegrass (Lolium perenne; Bahmani et al., 2000), especially in nearly closed canopies (Simon & Lemaire, 1987). Also, a lower ratio between the intensities of red and far-red light (R:FR) has been related to reduced tillering in Poaceae, for example, in Italian ryegrass (Lolium multiflorum) and dallisgrass (Paspalum dilatatum) (Casal et al., 1986), barley (Hordeum vulgare; Skinner & Simmons, 1993) and weeping lovegrass (Eragrostis curvula; Wan & Sosebee, 1998). Within a canopy of plants, R:FR declines because of differential reflection and transmission of red and far-red light by green plant parts (Holmes & Smith, 1977b). R:FR is perceived by phytochrome photoreceptors located in vertically orientated plant organs such as elongating leaves, sheaths of mature leaves, and internodes (the ‘sites of perception’) (Cordukes & Fisher, 1974; Morgan et al., 1980; Skinner & Simmons, 1993). Changes in R:FR precede mutual shading, consequently serving as a warning signal for competition for light that is likely to occur in the near future (Ballaréet al., 1987; Smith, 2000; Franklin & Whitelam, 2005).

An experimental study (Evers et al., 2006) on the light conditions inside canopies of spring wheat plants showed that tiller buds remain dormant when the fraction of PAR intercepted by the canopy exceeds 0.4. That threshold value was fairly stable over a wide range of intensities of incoming light and plant population densities. In the same study, observations were also carried out on R:FR values within the canopy at soil level at the time of cessation of tiller appearance. Similar to PAR intercepted, it was observed that tiller buds remained dormant at specific values of R:FR, independent of plant population density, but these values depended on light intensity (0.32 in full light and 0.51 in 75% shade). These findings on cessation of tillering in specific light conditions within the canopy corroborated results of Sparkes et al. (2006) for wheat, and confirmed suggestions by Simon & Lemaire (1987) for perennial ryegrass and Italian ryegrass, and by Lafarge & Hammer (2002) for sorghum (Sorghum bicolor).

The PAR intensity and R:FR inside the wheat canopy are largely determined by the architecture of the canopy itself, i.e. the distribution in the three-dimensional canopy of tissue (selectively) reflecting, absorbing and transmitting light. Light properties influence the pattern of tillering, which in turn determines the architecture of the canopy. The current work attempts to analyse this feedback mechanism in the tillering behaviour of spring wheat plants, using a modelling approach called virtual plant modelling (Room et al., 1996), or functional–structural plant modelling (Sievänen et al., 2000; Godin & Sinoquet, 2005; Vos et al., 2007). The virtue of virtual plant models is that each element in the three-dimensional architecture of the canopy is explicitly described. This enables simulation of processes at the level of the individual organ instead of at the level of the whole plant or crop. The primary objective of the current work was to determine whether a three-dimensional virtual plant model of spring wheat is capable of simulating tiller appearance merely based on the signalling properties of localized R:FR.

Materials and Methods

  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and Methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References
  9. Appendix: Derivation of

The model used in this study is an architectural model called ‘adel-wheat’ (Fournier et al., 2003), which has been calibrated and validated for spring wheat (Evers et al., 2005, 2007), in conjunction with a light interception model called ‘nested radiosity’ (nr) (Chelle & Andrieu, 1998; Chelle et al., 1998).

adel-wheat

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).

image

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:

  • image(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.

(A) Unit step response (Abramowitz & Stegun, 1972). In this type of response, F is either 1 or 0:

  • image(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:

  • image(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:

  • image(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.

image

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.

Simulations

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’).

Experiment

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.

Results

  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and Methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References
  9. Appendix: Derivation of

The number of simulated tillers per plant is shown in Fig. 3 (unit step response), Fig. 4 (curvilinear response) and Fig. 5 (linear response). The final number of tillers reached in each of the simulations is given in Table 1. Across all simulations, the number of tillers per plant decreased with increasing population density, increasing Qt value, and increasing influence of R:FR on bud development (through the relationship between R:FR and F). These factors similarly affected frequencies of occurrence of individual tillers (data not shown); the largest effects were found for tillers with the highest rank (i.e. tillers appearing late in plant development), whereas low-ranked tillers were less affected.

image

Figure 3. Number of tillers appeared per plant vs physiological age of the plant, for (a) 100, (b) 262 and (c) 508 plants m−2; vertical bars represent the standard error (n = 12). The unit step response of the ‘bud fate’ function F to the ratio of red to far-red light irradiance (R:FR) was used (see Fig. 2a). Symbols represent experimental data (closed circles) and simulation output using a Qt value (the threshold R:FR below which bud break is arrested) of 0.4 (squares), 0.6 (diamonds), 0.8 (triangles) or 0.9 (open circles). The line of the experimental data was extended to visualize the maximum number of tillers appeared per plant; the decline of the line as a result of senescence is not shown to avoid unnecessary complexity.

Download figure to PowerPoint

image

Figure 4. Similar to Fig. 3, but here the curvilinear response of the ‘bud fate’ function F to the ratio of red to far-red light irradiance (R:FR) was used (see Fig. 2b); simulations for a Qt value (the threshold R:FR below which bud break is arrested) of 0.9 were not performed.

Download figure to PowerPoint

image

Figure 5. Similar to Fig. 3, but here the linear response of the ‘bud fate’ function F to the ratio of red to far-red light irradiance (R:FR) was used (see Fig. 2c); simulations for a Qt value (the threshold R:FR below which bud break is arrested) of 0.9 were not performed.

Download figure to PowerPoint

Table 1.  Maximum tiller number reached per plant for Qt values (the threshold red:far-red ratio below which bud break is arrested) of 0.4, 0.6, 0.8 and 0.9
F response:Unit stepCurvilinearLinear
Density:100262508100262508100262508
  1. Values are mean (standard error); n = 12.

  2. F response, type of response of the ‘bud fate’ function F to the ratio of red to far-red light irradiance (R:FR) (see text and Fig. 2); density, plant population densities (in plants m−2) for which the simulations were performed; Exp., actual data from the experiment.

Qt value
0.418.00 (0.00)18.00 (0.00)17.92 (0.08)17.33 (0.62)13.25 (0.75)9.33 (1.16)13.67 (0.77)6.17 (0.63)2.58 (0.35)
0.618.00 (0.00)17.58 (0.19)13.24 (0.98)14.58 (0.83) 6.17 (0.81)4.33 (1.01) 6.50 (0.61)2.08 (0.51)0.00 (0.00)
0.817.42 (0.58)11.17 (1.46) 6.83 (0.79) 7.17 (1.12) 0.58 (0.31)0.00 (0.00) 2.00 (0.68)0.00 (0.00)0.00 (0.00)
0.912.58 (1.04) 5.58 (1.41) 1.33 (0.36)
Exp. 8.90 (0.26) 5.65 (0.26) 3.65 (0.22) 8.90 (0.26) 5.65 (0.26)3.65 (0.22) 8.90 (0.26)5.65 (0.26)3.65 (0.22)

Using the unit step response, the timing of the increase in simulated tiller number per plant was close to the experimental data until a physiological age of c. 6–7 phyllochrons (100 plants m−2) or 5 phyllochrons (262 and 508 plants m−2), for any value of Qt (Fig. 3a–c). Beyond that point, a divergence in the dynamics of tillering among the various simulations could be observed. Simulations using Qt values of 0.4, 0.6 and 0.8 eventually yielded an average number of tillers per plant higher than that observed experimentally (Table 1). This was also the case for a Qt value of 0.9, but only at 100 plants m−2. At 262 plants m−2, the simulated final number of tillers per plant was similar to the experimental data, and at 508 plants m−2 the final number of tillers per plant was considerably lower than that observed experimentally (Table 1).

Compared with the situation using the unit step response, the use of the curvilinear response of F to R:FR resulted in a slight delay in the physiological age at which the increase in the number of tillers per plant started (Fig. 4a–c). However, among Qt values, the differences in the simulated time–courses of tiller number occurred earlier than in the case of the unit step response, starting at a number of tillers per plant close to zero. The number of tillers produced was generally lower than for the unit step response, resulting in fewer tillers being finally produced (Table 1). For the simulated population densities of 262 and 508 plants m−2, a Qt value of 0.6 resulted in a final tiller number similar to that in the experimental data.

The linear response caused an even greater delay in the physiological age at which the increase in tiller number per plant started (Fig. 5a–c). Beyond a physiological age of c. 4.5 phyllochrons, tillers started to appear and immediately differences in the simulated courses of tiller number per plant among Qt values were visible. In all but two of the simulations, the number of tillers per plant stayed below the experimentally observed numbers. The only exceptions were the simulations at 100 and 262 plants m−2 and a Qt value of 0.4, in which the final number of tillers was higher than the number of tillers produced in the experiment (Table 1).

When plant population densities were compared directly, it appeared that there was initially no difference in the number of tillers per plant among plant population densities until a certain moment in development. Figure 6 provides a clear example in the case of a Qt value of 0.8 and the unit step response of bud development to R:FR. No difference was observed among plant population densities until a physiological age of c. 4.5 phyllochrons; beyond this physiological age, the time–courses of the simulated number of tillers started to diverge among plant population densities. In most other simulations, similar patterns could be observed. A visualized example of the simulations is provided in Fig. 7 for two plant population densities (100 and 262 plants m−2). The figure illustrates the plastic response of the wheat plants to plant population density at four ‘snapshots’ during development.

image

Figure 6. Simulated number of tillers appeared per plant vs physiological age of the plant, for 100 plants m−2 (squares), 262 plants m−2 (diamonds), and 508 plants m−2 (triangles); vertical bars represent the standard error (n = 12). The Qt value (the threshold R:FR below which bud break is arrested) of the simulations was 0.8, and the unit step relationship between bud development and R:FR was used.

Download figure to PowerPoint

image

Figure 7. Example of the visual output at four moments in thermal time, from simulation runs at (a) 100 and (b) 262 plants m−2.

Download figure to PowerPoint

Discussion

  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and Methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References
  9. Appendix: Derivation of

The relationship among population density, R:FR and bud development

Overall, the simulated effects of plant population density on the degree of tillering agreed with the experimental observations. When plant population density was increased, R:FR dropped below Qt earlier during canopy development than at lower plant population densities, resulting in a reduction of tiller bud outgrowth. Also, when Qt was increased, the number of tillers per plant was generally lower as Qt was exceeded at earlier stages of canopy development.

Initially the simulated tiller appearance rate was the same for all plant population densities, but at some stage the rate slowed down earlier the higher the plant population density (Fig. 6). This simulated divergence among plant population densities in the time–course of tillering was similar to the divergence observed in the experiment described by Evers et al. (2006), and to observations reported by van Oosterom et al. (2001) on pearl millet (Pennisetum glaucum) and Lafarge & Hammer (2002) on sorghum. Increasing the population density of the simulated plants apparently did not influence the R:FR perceived by the plants to such an extent that bud break of the first c. four buds was affected. This corroborates the findings of Kirby & Faris (1972), who observed in barley that development of the first four tiller buds into tillers was not affected by population density (ranging from 50 to 1600 plants m−2). The authors suggested an ‘on or off action’ type of tiller bud outgrowth as a basis for the observed patterns in tillering, instead of a gradual transition through modulation of early tiller growth. The on/off suggestion is in line with the views of Williams & Langer (1975) and Williams & Metcalf (1975), who observed an abrupt increase in bud extension rate after a lag period. The authors ascribed this transition to the physical constraint exerted by the sheath tissues that cover the bud, which either could or could not be overcome in time, resulting in on/off behaviour of tiller bud outgrowth. Such observations support the implementation of a unit step response of tiller outgrowth in the current work: tiller bud outgrowth either occurs or does not occur, but there is no modulation of growth rate.

Using the unit step response, a value of Qt of 0.8–0.9 was necessary to simulate a tillering rate and a final number of tillers comparable to those found experimentally (Fig. 3). This was higher than the value range derived from midday measurements in an experiment (0.35–0.40) (Evers et al., 2006). This may have been caused by the fact that the R:FR value of the light sources used in the model was set to a constant (1.2). By contrast, in natural conditions the value changes from c. 0.6 at dawn to peak at 1.2 for c. 6 h during the day and declines to c. 0.6 just before sunset (Holmes & Smith, 1977a). This difference could cause more tiller buds to grow out in the simulations than in reality, as in the latter case R:FR is below Qt during a large portion of the day. Nevertheless, the observation by itself that the use of an R:FR threshold value can give rise to tillering behaviour comparable to that found in field observations confirms independent findings (Sparkes et al., 2006). The nature of the relationship between bud break and R:FR (unit step, linear, or possible types of nonlinearity) remains uncertain. A curvilinear type of response would be consistent with the shape of the relation between phytochrome photoequilibrium (the fraction of phytochrome molecules in the far-red isoform) and R:FR (Smith & Holmes, 1977). It should be noted that responses other than the unit step response resulted in a delay in tiller appearance (Figs 4, 5) not observed in reality. This is in support of the unit step response, that is an absence of modulation of bud growth rate. However, to draw more definite conclusions regarding the relation between bud growth and R:FR, more detailed analysis of bud initiation rate and bud growth is necessary.

Mechanisms

Usually, shade avoidance responses such as preclusion of bud outgrowth are explained in terms of R:FR or blue light perceived by photoreceptors (reviewed in Franklin & Whitelam, 2005). In this respect, the spatial and temporal mechanism of integration of the R:FR signal is important. The perception of R:FR occurs through phytochromes in tissues at the exterior of the plant. The various exterior surfaces of the plant perceive different R:FR ratios (Chelle et al., 2007), depending on their position and orientation in the structure. Signals that are produced in different regions in the plant are apparently transported, and spatially and temporally integrated. When strong enough the signals delivered in the buds will result in inhibition of bud growth, the default being outgrowth of buds. Important aspects at the bud level include whether there is a qualitative or a quantitative response to the strength of the signal, whether or not there is a minimum duration of exposure needed, and whether or not the sensitivity of the bud to the signal changes with time.

Until now, shade avoidance responses (Aphalo et al., 1999; Franklin & Whitelam, 2005) have only been simulated by Gautier et al. (2000) using the virtual plant model of clover (Trifolium repens). In their study, petiole length, internode length and branching delay were calculated using empirical relations with local PAR intensity and R:FR. The main difference with the current approach is that in the clover model the responses of the above-mentioned light-sensitive variables were continuous functions of both PAR intensity and R:FR, whereas in the current model the sensitive variable (degree of tillering) was basically only affected by R:FR in the bud stage; after bud break, tiller growth followed empirical laws which did not differ between simulations. Both approaches, however, made use of empirical functions; shade avoidance can only be an emergent property of the model when the biological processes that underlie the phenomena are understood, quantified and implemented.

Concluding remarks

The tillering behaviour of the modelled plants as described in the current study was based on observations and empirically derived relations. This study has shown that the methods presented allow the environmental influence at the local level (i.e. the organ level) on the development of plants in a canopy to be modelled, even when the complex chain of events between signal perception and bud response is reduced to the simple mechanism that was implemented, and allow hypotheses regarding plant photomorphogenesis to be tested. The current study, in which a tentative preliminary test was carried out of the type of response function involved in the response of tillering to light quality, is a first step in this direction. We are confident that further steps incorporating more complex plant physiological and morphogenetic processes will similarly benefit from a three-dimensional virtual plant modelling approach. Furthermore, the modelling approach offers the possibility to explore the roles of light quality and canopy structure in relation to designing management practices and ideotypes for breeding. The modelling will certainly give rise to questions regarding the exact functioning of physiological mechanisms in the context of a whole plant in its environment (e.g. the significance of the daily pattern in R:FR from the sky).

Break or dormancy of each tiller bud results from influences of the surrounding tissue (Shimizu-Sato & Mori, 2001). In the current study, no attempts were made to model the steps between perception of R:FR and production of signals, their movement to the bud and bud response (Ballaréet al., 1987; Barnes & Bugbee, 1991; Tomlinson & O’Connor, 2004). Also, tiller outgrowth and development require assimilates, which a young tiller receives from its parent shoot (Rawson & Hofstra, 1969), and which a tiller starts to produce itself when it has developed an adequately large leaf area. Therefore, to model tiller outgrowth as a result of carbon partitioning, source activity, sink strength and source–sink relationships (Heuvelink, 1996; Lacointe, 2000; Minchin & Lacointe, 2005) need to be addressed. However, further improvement of simulation of R:FR, for instance by taking into account the diurnal fluctuations of R:FR, may in itself significantly improve the results. Much more work needs to be done to explore the boundaries of virtual plant modelling in extending the interpretation, the testing and the extrapolation of plant physiological and ecological concepts. This is a considerable task.

The current study confirms the usability of the virtual plant approach based on l-systems (Mech & Prusinkiewicz, 1996; Mech, 1997) as a suitable platform to model the environmental influence at the plant organ level on the development of a canopy of plants, confirming the findings of previous studies (Fournier & Andrieu, 1999; Gautier et al., 2000; de Visser et al., 2004; Allen et al., 2005). The nr model (Chelle & Andrieu, 1998) in combination with the caribu interface (Chelle et al., 2004) provided a convenient method for calculating complex light properties such as R:FR. The modelling framework provides possibilities for further development, such as the incorporation of photosynthesis and carbon allocation and exploration of the consequences of different hypotheses regarding R:FR effects on the chain of events leading to the shade avoidance response. Our current focus is on the integration of such processes, and work in this direction is currently in progress.

Acknowledgements

  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and Methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References
  9. Appendix: Derivation of

The authors thank Peter van der Putten, Ans Hofman, Henriette Drenth (staff members of Crop and Weed Ecology, Wageningen University), and the staff of the experimental facilities UNIFARM, Wageningen University, for their contributions to the experiment. The C. T. de Wit Graduate School for Production Ecology and Resource Conservation of Wageningen University funded this work; the Netherlands Organisation for Scientific Research (NWO) provided additional financial support.

References

  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and Methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References
  9. Appendix: Derivation of
  • Abramowitz M, Stegun IA. 1972. Handbook of mathematical functions with formulas, graphs, and mathematical tables. New York, NY, USA: Dover.
  • Allen MT, Prusinkiewicz P, DeJong TM. 2005. Using 1-systems for modeling source–sink interactions, architecture and physiology of growing trees: the L-PEACH model. New Phytologist 166: 869880.
  • Aphalo PJ, Ballaré CL, Scopel AL. 1999. Plant–plant signalling, the shade-avoidance response and competition. Journal of Experimental Botany 50: 16291634.
  • Bahmani I, Hazard L, Varlet-Grancher C, Betin M, Lemaire G, Matthew C, Thom ER. 2000. Differences in tillering of long- and short-leaved perennial ryegrass genetic lines under full light and shade treatments. Crop Science 40: 10951102.
  • Ballaré CL, Sánchez RA, Scopel AL, Casal JJ, Ghersa CM. 1987. Early detection of neighbour plants by phytochrome perception of spectral changes in reflected sunlight. Plant, Cell & Environment 10: 551557.
  • Barnes C, Bugbee B. 1991. Morphological responses of wheat to changes in phytochrome photoequilibrium. Plant Physiology 97: 359365.
  • Bos HJ. 1999. Plant morphology, environment, and leaf area growth in wheat and maize. PhD thesis. Wageningen University, Wageningen, the Netherlands.
  • Casal JJ, Sánchez RA, Deregibus VA. 1986. The effect of plant density on tillering: the involvement of R/FR ratio and the proportion of radiation intercepted per plant. Environmental and Experimental Botany 26: 365371.
  • Chelle M, Andrieu B. 1998. The nested radiosity model for the distribution of light within plant canopies. Ecological Modelling 111: 7591.
  • Chelle M, Andrieu B, Bouatouch K. 1998. Nested radiosity for plant canopies. The Visual Computer 14: 109125.
  • Chelle M, Evers JB, Combes D, Varlet-Grancher C, Vos J, Andrieu B. 2007. Simulation of the three-dimensional distribution of the red:far-red ratio within crop canopies. New Phytologist. doi: 10.1111/j.1469-8137.2007.02161.x
  • Chelle M, Hanan JS, Autret H. 2004. Lighting virtual crops: the CARIBU solution for open 1-systems. In: GodinC, HananJS, KurthW, LacointeA, TakenakaA, PrusinkiewiczP, DejongT, BeveridgeC, AndrieuB, eds. 4th International Workshop on Functional-Structural Plant Models. Montpellier, France. 194.
  • Cordukes WE, Fisher JE. 1974. Effects of shading of the leaf sheath on the growth and development of the tiller stems of Kentucky bluegrass. Canadian Journal of Plant Science 54: 4753.
  • Darwinkel A. 1978. Patterns of tillering and grain production of winter wheat at a wide range of plant densities. Netherlands Journal of Agricultural Science 26: 383398.
  • Davies A. 1971. Changes in growth rate and morphology of perennial ryegrass swards at high and low nitrogen levels. Journal of Agricultural Science 77: 123134.
  • Evers JB, Vos J, Andrieu B, Struik PC. 2006. Cessation of tillering in spring wheat in relation to light interception and red: far-red ratio. Annals of Botany 97: 649658.
  • Evers JB, Vos J, Fournier C, Andrieu B, Chelle M, Struik PC. 2005. Towards a generic architectural model of tillering in Gramineae, as exemplified by spring wheat (Triticum aestivum). New Phytologist 166: 801812.
  • Evers JB, Vos J, Fournier C, Andrieu B, Chelle M, Struik PC. 2007. An architectural model of spring wheat: evaluation of the effects of population density and shading on model parameterization and performance. Ecological Modelling 200: 308320.
  • Fournier C, Andrieu B. 1999. ADEL-maize: an 1-system based model for the integration of growth processes from the organ to the canopy. Application to regulation of morphogenesis by light availability. Agronomie 19: 313327.
  • Fournier C, Andrieu B, Ljutovac S, Saint-Jean S. 2003. ADEL-wheat: a 3D architectural model of wheat development. In: HuBG, JaegerM, eds. International Symposium on Plant Growth Modeling, Simulation, Visualization, and their Applications. Beijing, China: Tsinghua University Press/Springer, 5463.
  • Franklin KA, Whitelam GC. 2005. Phytochromes and shade-avoidance responses in plants. Annals of Botany 96: 169175.
  • Gautier H, Mech R, Prusinkiewicz P, Varlet-Grancher C. 2000. 3D architectural modelling of aerial photomorphogenesis in white clover (Trifolium repens L.) using 1-systems. Annals of Botany 85: 359370.
  • Godin C, Sinoquet H. 2005. Functional-structural plant modelling. New Phytologist 166: 705708.
  • Heuvelink E. 1996. Dry matter partitioning in tomato: validation of a dynamic simulation model. Annals of Botany 77: 7180.
  • Holmes MG, Smith H. 1977a. The function of phytochrome in the natural environment. 1. Characterization of daylight for studies in photomorphogenesis and photoperiodism. Photochemistry and Photobiology 25: 533538.
  • Holmes MG, Smith H. 1977b. The function of phytochrome in the natural environment. 2. Influence of vegetation canopies on spectral energy-distribution of natural daylight. Photochemistry and Photobiology 25: 539545.
  • Jacquemoud S, Baret F. 1990. PROSPECT: a model of leaf optical properties spectra. Remote Sensing of Environment 34: 7591.
  • Kirby EJM, Faris DG. 1972. The effect of plant density on tiller growth and morphology in barley. Journal of Agricultural Science 78: 281288.
  • Lacointe A. 2000. Carbon allocation among tree organs: a review of basic processes and representation in functional-structural tree models. Annals of Forest Science 57: 521533.
  • Lafarge TA, Hammer GL. 2002. Tillering in grain Sorghum over a wide range of population densities. Modelling dynamics of tiller fertility. Annals of Botany 90: 99110.
  • Lindenmayer A. 1968a. Mathematical models for cellular interactions in development. I. Filaments with one-sided inputs. Journal of Theoretical Biology 18: 280299.
  • Lindenmayer A. 1968b. Mathematical models for cellular interactions in development. II. Simple and branching filaments with two-sided inputs. Journal of Theoretical Biology 18: 300315.
  • Ljutovac S. 2002. Coordination dans l’extension des organes aériens et conséquences pour les relations entre les dimensions finales des organes chez le blé. PhD thesis. Institut National de Recherche Agronomique Paris-Grignon, Grignon, France.
  • Longnecker N, Kirby EJM, Robson A. 1993. Leaf emergence, tiller growth, and apical development of nitrogen-deficient spring wheat. Crop Science 33: 154160.
  • Mech R. 1997. Modeling and simulation of the interaction of plants with the environment using l-systems and their extensions. PhD thesis. University of Calgary, Calgary, Canada.
  • Mech R. 2004. CPFG, version 4.0. User's manual. Calgary, Canada: University of Calgary.
  • Mech R, Prusinkiewicz P. 1996. Visual models of plants interacting with their environments. SIGGRAPH ’96. New York, NY, USA. 397410.
  • Minchin PEH, Lacointe A. 2005. New understanding on phloem physiology and possible consequences for modelling long-distance carbon transport. New Phytologist 166: 771779.
  • Morgan DC, O’Brien T, Smith H. 1980. Rapid photomodulation of stem extension in light-grown Sinapis alba L. Planta 150: 95101.
  • Van Oosterom EJ, Carberry PS, O’Leary GJ. 2001. Simulating growth, development, and yield of tillering pearl millet. II. Simulation of canopy development. Field Crops Research 72: 6791.
  • Prusinkiewicz P. 1999. A look at the visual modelling of plants using 1-systems. Agronomie 19: 211224.
  • Prusinkiewicz P, Karwowski R, Mech R, Hanan JS. 2000. l-studio/CPFG: a software system for modeling plants. In: NaglM, SchürrA, MünchM, eds. Applications of graph transformations with industrial relevance. Lecture notes in computer science 1779. Berlin, Germany: Springer, 457464.
  • Rawson HM, Hofstra G. 1969. Translocation and remobilization of 14C assimilated at different stages by each leaf of the wheat plant. Australian Journal of Biological Science 22: 321331.
  • Room PM, Hanan JS, Prusinkiewicz P. 1996. Virtual plants: new perspectives for ecologists, pathologists and agricultural scientists. Trends in Plant Science 1: 3338.
  • Shimizu-Sato S, Mori H. 2001. Control of outgrowth and dormancy in axillary buds. Plant Physiology 127: 14051413.
  • Sievänen R, Nikinmaa E, Nygren P, Ozier-Lafontaine H, Perttunen J, Hakula H. 2000. Components of functional-structural tree models. Annals of Forest Science 57: 399412.
  • Simon J-C, Lemaire G. 1987. Tillering and leaf area index in grasses in the vegetative phase. Grass and Forage Science 42: 373380.
  • Skinner RH, Simmons SR. 1993. Modulation of leaf elongation, tiller appearance and tiller senescence in spring barley by far-red light. Plant, Cell & Environment 16: 555562.
  • Smith H. 1982. Light quality, photoperception and plant strategy. Annual Review of Plant Physiology 33: 481518.
  • Smith H. 2000. Phytochromes and light signal perception by plants – an emerging synthesis. Nature 407: 585591.
  • Smith H, Holmes MG. 1977. The function of phytochrome in the natural environment. 3. Measurement and calculation of phytochrome photoequilibria. Photochemistry and Photobiology 25: 547550.
  • Sparkes DL, Holme SJ, Gaju O. 2006. Does light quality initiate tiller death in wheat? European Journal of Agronomy 24: 212217.
  • Tomlinson KW, O’Connor TG. 2004. Control of tiller recruitment in bunchgrasses: uniting physiology and ecology. Functional Ecology 18: 489496.
  • De Visser PHB, Marcelis LFM, Van Der Heijden GWAM, Angenent GC, Evers JB, Struik PC, Vos J. 2004. Incorporation of 3D plant structures in genetic and physiological models. Acta Horticulturae 654: 171178.
  • Vos J, Marcelis LFM, Evers JB. 2007. Functional-structural plant modelling in crop production – adding a dimension. In: VosJ, MarcelisLFM, De VisserPHB, StruikPC, EversJB, eds. Functional-structural plant modelling in crop production. Dordrecht, the Netherlands: Springer, 112.
  • Wan C, Sosebee RE. 1998. Tillering responses to red:far-red light ratio during different phenological stages in Eragrostis curvula. Environmental and Experimental Botany 40: 247254.
  • Williams RF, Langer RHM. 1975. Growth and development of the wheat tiller bud. II. The dynamics of tiller growth. Australian Journal of Botany 23: 745759.
  • Williams RF, Metcalf RA. 1975. Physical constraints and tiller growth in wheat. Australian Journal of Botany 23: 213223.

Appendix: Derivation of Eqns 3 and 4

  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and Methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References
  9. Appendix: Derivation of

Equation 3 describes a curvilinear relation between R:FR (Q) and the fraction of unrestrained bud extension (F). It contains two parameters: Qm, the maximum R:FR, and Qt, the threshold R:FR below which tiller bud break is arrested. Equation 3 is a second-order polynomial (a parabola), which has the general form

  • y = a(x – p)(x – q)( Eqn A1)

(p and q, the x values at which y is zero; a, a nontrivial parameter determining curvature.) The x co-ordinate of the top of the parabola (m) is located halfway between p and q; therefore

  • image

Substituting q in Eqn A1 gives

  • y = a(x – p)(x – 2m + p)( Eqn A2)

As variable F is a fraction and can therefore only take values from 0 to 1, the top of the parabola is located at (m, 1). This gives

  • image

Substituting a in Eqn A2 gives

  • image

Substituting p with Qt, m with Qm, x with Q, and y with F yields Eqn 3.

Equation 4 describes a linear relation between Q and F. It contains two parameters, which are identical to those in Eqn 3. Equation 4 is a line, which has the general form

  • y = a(x – p)( Eqn A3)

(a, the slope; p, the x value at which y is zero.) Similarly to Eqn 3, the line is supposed to go through (m, 1); therefore

  • image

Substituting a in Eqn A3 gives

  • image

Substituting p with Qt, m with Qm, x with Q, and y with F yields Eqn 4.