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. 2008 Jan 16;101(8):1195–1206. doi: 10.1093/aob/mcm317

Does the Structure–Function Model GREENLAB Deal with Crop Phenotypic Plasticity Induced by Plant Spacing? A Case Study on Tomato

Qiaoxue Dong 1,†, Gaëtan Louarn 1,2,†,*, Yiming Wang 1, Jean-Francois Barczi 3, Philippe de Reffye 3,4
PMCID: PMC2710282  PMID: 18199575

Abstract

Background and Aims

Plant growth models able to simulate phenotypic plasticity are increasingly required because (1) they should enable better predictions of the observed variations in crop production, yield and quality, and (2) their parameters are expected to have a more robust genetic basis, with possible implications for selection of quantitative traits such as growth- and allocation-related processes. The structure–function plant model, GREENLAB, simulates resource-dependent plasticity of plant architecture. Evidence for its generality has been previously reported, but always for plants grown in a limited range of environments. This paper aims to test the model concept to its limits by using plant spacing as a means to generate a gradient of competition for light, and by using a new crop species, tomato, known to exhibit a strong photomorphogenetic response.

Methods

A greenhouse experiment was carried out with three homogeneous planting densities (plant spacing = 0·3, 0·6 and 1 m). Detailed records of plant development, plant architecture and organ growth were made throughout the growing period. Model calibration was performed for each situation using a statistical optimization procedure (multi-fitting).

Key Results and Conclusions

Obvious limitations of the present version of the model appeared to account fully for the plant plasticity induced by inter-plant competition for light. A lack of stability was identified for some model parameters at very high planting density. In particular, those parameters characterizing organ sink strengths and governing light interception proved to be environment-dependent. Remarkably, however, responses of the parameter values concerned were consistent with actual growth measurements and with previously reported results. Furthermore, modifications of total biomass production and of allocation patterns induced by the planting-density treatments were accurately simulated using the sets of optimized parameters. These results demonstrate that the overall model structure is potentially able to reproduce the observed plant plasticity and suggest that sound biologically based adaptations could overcome the present model limitations. Potential options for model improvement are proposed, and the possibility of using the kernel algorithm currently available as a fitting tool to build up more sophisticated model versions is advocated.

Key words: Functional–structural models, GREENLAB, phenotypic plasticity, planting density, competition, source–sink relationship, parameter stability, Solanum lycopersicum

INTRODUCTION

Phenotypic plasticity is a term commonly used by plant physiologists to refer to the environmentally induced diversity of phenotypes that a given genotype can generate (Bradshaw, 1965; Via et al., 1995). It is usually expressed at the modular sub-unit level (i.e. meristems, leaves, internodes, roots), triggered by local environmental conditions (de Kroon et al., 2005). So far, however, most crop growth models have described plant resource acquisition and conversion through a set of invariable rules and parameters describing uniform functioning of whole-plant compartments (Jones and Kiniry, 1986; Jones et al., 1991; Brisson et al., 2003). Although broadly widespread and, in many cases, having reached a respectable degree of acceptance, this approach shows its limits when integrating the full range of climatic scenarios (Riha et al., 1996) or when the purpose of the modelling is to identify model parameters with genetic significance for process-based multigenic traits such as growth, morphogenesis or allocation processes (Hammer et al., 2002; Yin et al., 2003; Dingkuhn et al., 2005).

Recently, plant structure–function models have emerged as potential tools for simulating plant phenotypic plasticity at the individual-organ level (Prusinkiewicz, 1998; Yan et al., 2004). Several functional approaches have been proposed to capture the observed plasticity, ranging from purely ontogenic (Evers et al., 2005) to mostly carbon-budget-driven ones (Perttunen et al., 1998; Allen et al., 2005; Dingkuhn et al., 2006; Guo et al., 2006), including combinations (Fournier and Andrieu, 1998) and semi-empiric descriptions of the causal processes (Thornby et al., 2003). These models vary greatly in terms of versatility (generic across species or not), complexity (levels of detail incorporated in the mechanistic description of plant growth) and in the data needed to complete their parameterization. The GREENLAB model was developed as a mathematical tool to simulate resource-dependent phenotypic plasticity, with a particular emphasis being placed on achieving a trade-off between simplicity (only twelve parameters are required) and generality (Yan et al., 2004; Dingkuhn et al., 2005). It simulates individual organ initiation and growth dynamics based on a whole-plant biomass acquisition equation and on a generic function controlling the time-course of individual organ demands. Contrary to bounded growth models (Fournier et al., 2003; Renton, 2004; Kahlen, 2007), it generates phenotypic differences in organ dimensions/weight through competition for assimilates around an environment-dependent common pool. The partitioning pattern amongst organs results from multiple interactions between ontogeny, biomass production and individual organ sink strengths, and not from a predetermined phenological timetable.

Interestingly, this model formalism has proved to be able to represent evolutions of biomass production and allocation for a wide variety of plant architectures, ranging from simple, single-stem plants (Guo et al., 2006, for maize) to branching annuals (Kang et al., 2006, for chrysanthemum) or complex trees (Guo et al., 2007, for pine tree). Furthermore, inter-seasonal parameter stability (Ma et al., 2007) has demonstrated the capacity of GREENLAB to explain a major part of the inter-annual phenotypic variance (namely variations in total biomass) and thus to provide relevant genotypic parameters to characterize growth and biomass allocation under defined growing conditions (location, planting density, irrigation schedules). The present study aims to evaluate the generalization of these modelling concepts to a broader range of environmental conditions (i.e. inducing more accentuated phenotypic plasticity). To do so, we investigated the consequences of planting density, a possible management option to compensate for annual fluctuation in light availability through modulation of inter-plant competition (Rodriguez and Lambeth, 1975; Papadopoulos and Pararajasingham, 1997) on parameter stability of the greenhouse tomato, a crop exhibiting strong morphological adaptations to variations of incoming light (Heuvelink, 1996). Such an assessment step is essential to legitimize the practical use of any model for optimizing crop management.

As quantitative characterizations of morphogenetic responses have been scarce in tomato (Papadopoulos and Ormrod, 1991; Heuvelink, 1995), a first step in our approach was to quantify the effects of plant spacing on development, growth and allocation processes in the tomato cultivar studied. Then, assuming that the plant response was entirely mediated through light intensity effects, we checked the ability of the model to reproduce the observed resource-dependent plasticity (above- and below-ground), and we examined the accuracy of model fits and the stability of the optimized crop parameters in the various situations studied. A critical appraisal of model behaviour was finally carried out.

MATERIALS AND METHODS

Experimental design

An experiment was carried out during spring 2006 (sowing date 15 March) on tomato (Solanum lycopersicum L., ‘ZhongZa 9’) at the Chinese Academy of Agricultural Science, Beijing, China (39·55 N, 116·25E). Plants were grown in a solar greenhouse (i.e. plastic tunnels traditionally used in China; Heuvelink, 2005) in regularly spaced 7-L pots. Three density treatments were applied: high density (HD; 0·3 × 0·3 m = 11·1 plants m−2), intermediate density (MD, 0·6 × 0·6 m = 2·8 plants m−2) and low density (LD; 1 × 1 m = 1 plant m−2). Lateral branches were systematically removed after emergence to keep only the main sympodium. Individual tomato plants were trellised vertically and organized in plots of 30 pots surrounded by two rows of guard plants at the same density. Irrigation and fertilization were similar for all pots at a given density and adjusted to the evaporative demand in their immediate vicinity.

Air temperature and relative humidity were measured on the experimental site with a capacitive hygrometer placed in a standard radiation shield, at a height of 2 m from the soil. Additional temperature probes were placed in the canopy at each density. Photosynthetic photon flux density (PPFD) above the canopies were also measured with PPFD sensors (LI-190SB connected to a LI-1400 portable datalogger; LI-COR, Lincoln, NB, USA). Data were stored in a datalogger (Galileo 32, Eldar Shany Agricultural Control, Beijing), with measurements taken every 30 s and an average calculated over 10-min periods. Daily average air temperatures ranged from 19·1 °C to 28·6 °C, while average maximum PPFD and VPD were 18·26 mol m−2 d−1 and 1·9 kPa, respectively. Resulting daily potential evaporations were calculated for each density and ranged between 0·7 mm and 9·5 mm (mean values over the growing period: 4·73 mm d−1 at LD; 4·21 mm d−1 at HD).

Model description

The GREENLAB model was described in detail by Yan et al. (2004), with some adaptations being introduced by Guo et al. (2006) to link matter production to potential evapotranspiration (PET) in the absence of a soil-water limitation. Only the main principles and specific options used here for tomato are presented.

The model dynamically represents the production, dry matter allocation and morphogenesis of a plant, based on a small set of recurrent mathematical equations and generic metamorphic rules. It is executed at time steps corresponding to organogenetic growth cycles (GC), equal to the thermal time (degree days, °Cd) needed to generate a new metamer. Plant topology is generated by an automaton (Yan et al., 2004) providing compartments (organs) that represent sinks for biomass in the course of their development. Biomass acquisition is simulated by applying atmospheric, evaporative demand (PET) to the exposed green leaf area, and by linearly converting the resulting transpiration rates into dry biomass assimilation using an optimized value for transpiration efficiency (Howel and Musick, 1984). Three dimensional (3-D) architectural representations can be generated, but this step remains optional as the major modelling concepts of GREENLAB are fundamentally process-based and independent of organ geometry.

Developmental processes are driven by thermal time, the additive accumulation of mean, daily, air temperature minus a crop-specific base temperature (10 °C, as recommended for tomato by Heuvelink and Marcelis, 1989). We assume that the thermal time elapsing between the appearances of two consecutive metamers is constant. Stem development was specifically formalized in this study to mimic the sympodial pattern observed in tomato (Calvert, 1965). In the usual case of single-stem, pruned indeterminate cultivars grown in greenhouses, seven to 11 metamers are first initiated producing only leaves (each consisting of a petiole and several leaflets) and internodes before the terminal apex turns for the first time into an inflorescence. Then, a series of two to three metamers, each terminated by an inflorescence, is regularly generated by growth of the lateral bud at the axil of the last leaf initiated. This pattern was simulated using a simple three-state automaton (after the initial seed stage): one state representing metamers bearing a truss assumed to be in a lateral position, the two others representing, respectively, the fruitless metamers before and after the first inflorescence (Fig. 1). Although basically monopodial, such a formalism is consistent with the strong linear relationship observed between thermal time and leaf appearance in tomato (de Koning, 1994). Furthermore, it enables us to account for the plant-to-plant variability observed in fruit position. Practically, truss position and fruit number per truss were forced as measured for each particular tomato plant (i.e. the number of cycles in each state and the number of fruits set for each truss were given as input), thus separating the developmental from the assimilation and allocation processes.

Fig. 1.

Fig. 1.

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(A) GREENLAB automaton for tomato development and (B) resulting plant architecture at GC = 25. Three states (i.e. types of metamers) are successively covered after the initial seed (eight juvenile metamers without reproductive organs followed by series of three metamers each composed of one metamer carrying a truss and two without a truss) to dynamically generate the plant topology. Arrows represent possible transitions between states.

Growth-related processes are driven by PET (Penman–Monteith equation used according to FAO guidelines; Allen et al., 1998). The potential for biomass production (E) during the ith growth cycle, GC(i), was computed as follows:

graphic file with name mcm31709.jpg 1

where TE represents transpiration efficiency, an optimized parameter to account for the crop parameter usually employed to convert the standard evapotranspiration rate into actual crop evapotranspiration (Allen et al., 1998).

Other GREENLAB equations are as described by Guo et al. (2006). Biomass production is calculated on the basis of the potential of biomass production E(i) and a Lambert–Beer's-law-like equation to estimate light interception:

graphic file with name mcm31710.jpg 2

where Q(i) represents the dry biomass created at time step i, r1 and r2 are empirical resistance parameters, Sj is the blade area of the jth leaf, and Sp is a kind of ground projection area of the leaf surface (see later in the text). Parameter r1 sets the leaf-size effect on transpiration per unit leaf area, while r2 accounts for the effect of mutual shading of leaves according to the Lambert–Beer's law. Plant density is integrated into the model by setting Sp equal to 1/density. In this particular case, ∑j=1n(i) Sj/Sp is equal to the leaf area index (LAI).

Organs are assumed to be competing for a shared pool of incremental biomass. Dry matter allocation (Δqo) to the various sinks (i.e. organs, o, currently in expansion) is performed according to their relative demand during GC(i) and the dry matter production during the previous cycle:

graphic file with name mcm31711.jpg 3

where D(i) is the total demand during GC(i), Po and fo are the sink strength and the sink variation function, respectively, of the organ o (representing the time course of the demand), and j is the time elapsed since organ o initiation (in GC). In the present version, the sink variation function is represented with a beta function formulated as follows:

graphic file with name mcm31712.jpg 4

Each type of organ o (blade, b; petiole, p; internode, e; fruit, f) is thus defined by four endogenous parameters: a sink strength P, and three parameters ao, bo (parameters defining the time course of the demand during organ expansion) and To (duration of expansion) defining a specific beta function. Normalization constraints are defined, sink strengths being expressed relatively to leaf-blade demand (blade sink value set at 1) and sink variation functions accepting the constraint ∑j=1To fo(j) = 1. Total plant demand D(i) is simply calculated by summing individual organ demands:

graphic file with name mcm31713.jpg 5

As a result of these model concepts, GREENLAB allows us to consider resource-driven phenotypic plasticity. Indeed, organ size appears to be variable and depends on resource availability and on the number and strength of sinks that share these resources at any given time.

Model parameterization

To parameterize the automaton simulating crop development, regular measurements of leaf number were made to determine the phyllochron (thermal time elapsing between the emergences of two consecutive leaves) and the leaf life span in each situation. Truss position and fruit number per truss were also recorded at harvest for all the plants studied.

Parameters of the integrated recurrent equation representing production- and allocation-related processes (Table 1) were identified by optimization procedures against a target file summarizing morphological observations measured on a sample of plants during the course of their development (details of the multi-fitting procedure are described by Guo et al., 2006). For each treatment, three to five plants were measured destructively at five phenological stages between emergence and slightly before harvest of the second truss (5th GC, vegetative stage 15 of the BBCH scale, Meier, 1997; 11th GC, first inflorescence visible, stage 51; 18th GC, flowering of the second truss, stage 62; 24th GC, flowering of the 4th inflorescence, stage 64; 30th GC flowering of the 6th inflorescence, stage 66). Target files consisted of organ dry weights for all the metamers of these plants. In this study, we considered the root system as a single organ and split individual leaves into petiole (the true botanical petiole and the venation system between leaflets) and leaf blade (the sum of all the leaflets). Leaf surface areas were obtained by scanning of leaf blades and subsequent image analysis.

Table 1.

GREENLAB parameters simultaneously optimized during the multi-fitting process.

Parameters Comments
Pb Blade relative sink strength; Pb is set to 1
Pp Petiole relative sink strength
Pe Internode relative sink strength
Pf Individual fruit relative sink strength
Pr Root system relative sink strength
Bp Blade sink variation parameter
Be Internode sink variation parameter
Bf Fruit sink variation parameter
Br Root system sink variation parameter
r1 Leaf resistance factor (mmPET m2 g−1)
r2 Canopy extinction coefficient derived from Beer's law
Sp Projection surface (m2); Sp is set to 1/density
TE Transpiration efficiency
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During the optimization procedure, only one parameter (Bo, eqn 6) was optimized to define the beta function for each organ type, and the two parameters ao and bo were subsequently derived from Bo by iteration using the constraints ao + bo = 5 and

graphic file with name mcm31714.jpg 6

These constraints are empirical and generally yielded good results for different organs and plants (Guo et al., 2006).

The duration of expansion of each organ type (To) was derived directly from measurements of organ expansion (elongation, increase of diameter). These non-destructive measurements were performed on a sample of six plants per treatment for a series of metamers along the main sympodium that achieved their full expansion before the end of the experiment (metamers 6, 7, 8, 14, 15 and 16).

Three-dimensional representations of plants

A minimal set of geometrical parameters was also characterized to permit 3-D representation of the simulated tomato plants. Only the above-ground part was considered. Angles for phyllotaxy, initial elevation of petiole and leaf blade curvature were determined from measurements on a sample of two plants per treatment at the 13th, 19th and 25th GC. To do this, 3-D co-ordinates of ten points along the midrib of each leaf were recorded using an electromagnetic digitizer (3Space Fastrak Long Ranger, Polhemus, Colchester, VT; Sinoquet et al., 1998). Moreover, for each situation, empirical allometric relationships were established to link petiole and internode dry weights to their lengths and diameters (Dong, 2006).

A library of normalized 3-D shapes (truncated cones, polygons and spheres to represent internodes/petioles, leaflets and fruits, respectively) was finally used to visualize the simulated architecture. Organs were scaled-up according to the simulated biomass at each metamer position.

Statistical analysis

The ANOVA/MANOVA procedure of Statistica 6·0 (Statsoft, Tulsa, OK) was used to test for significant differences between means. Further differences between group means were determined using the Newman–Keul's test. Analysis of covariance (ANCOVA) was used to compare slopes and intercepts of the different observed–simulated linear relationships.

RESULTS

Quantitative impact of plant density on dry matter production and partitioning

Dry matter production differed significantly between the different situations studied (Fig. 2). As expected, increasing plant densities resulted in an increase of biomass production on a ground-area basis, but in a reduction of single-plant dry weights. The range of plant spacing used in the study enabled characterization of two extreme boundary treatments (LD, HD) – the intermediate (MD) density displaying individual plants close to LD with a total production (g m−2) 2·5 times higher.

Fig. 2.

Fig. 2.

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Time course of (A) measured plant dry weight and (B) dry matter production per unit area, at low (LD, 1 plant m−2), medium (MD, 2·8 plants m−2) and high (HD, 11·2 plants m−2) densities. Bars indicate confidence intervals at P = 0·05. Thermal time was measured in degree-days, base 10 °C.

Results of dry matter allocation amongst plant organs are presented in Fig. 3 for the two extreme densities. The general pattern was the same in all the treatments. During early vegetative stages, nearly 70 % of total biomass was allocated to the leaves (45 % just to the leaf blades), with roots and internodes sharing the remainder. These proportions were gradually modified after fruit set of the first truss at approximately 600 °Cd, a major part of the biomass being directed to the fruits from that time on. It can be seen that growth of the root system did not continue after this stage. A significant influence of plant spacing was observed on the time course of the fraction allocated to each plant compartment. At high density (HD), plants invested proportionally more into internodes and reduced the part of the biomass allocated to roots and to leaf blades. Conversely, the fraction allocated to the fruits remained approximately the same (around 80 %) for all treatments at the end of the experiment.

Fig. 3.

Fig. 3.

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Evolution of measured instantaneous dry matter allocation to the different plant compartments at low (LD, 1 plant m−2) and high (HD, 11·2 plants m−2) density. Thermal time was measured in degree-days, base 10 °C.

Quantitative impact of density on fruit development and organ expansion

We investigated the consequences of density treatments on plant development in order to parameterize the GREENLAB automaton that generates plant organs. The vegetative development (Fig. 4) was not significantly affected by plant density. Phyllochrons (ANCOVA, P = 0·24) and final number of unfolded leaves (ANOVA, P = 0·4) were similar in all the treatments, confirming the stability of the rate of leaf emergence when expressed according to thermal time (de Koning, 1994; Louarn et al., 2007a). Leaf life span, on the other hand, was strongly reduced when density increased (ANCOVA, P < 0·001). Consequently, the number of photosynthetically active leaves (NPAL) at HD remained constant over the experiment (around 13 leaves), leaves at the bottom dying at the same rate that they emerged at the tip. Conversely, at LD and MD, NPAL increased continuously due to a very low rate of leaf senescence. Developmental variables involved in the yield components were affected in contrasting ways (Table 2). In common with the stability of the leaf-appearance rate, the number of trusses initiated did not differ between treatments (average length of the sympodial modulus was equal to 2·9 metamers in all the situations studied, P = 0·6). The number of fruits per truss, on the other hand, appeared significantly reduced at HD because of a poorer fruit-set percentage (P < 0·01). No significant effect was observed on flower initiation or on flower development (P = 0·09).

Fig. 4.

Fig. 4.

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Effect of plant density on leaf appearance (open symbols, solid lines) and leaf senescence (black symbols, dashed lines). Each point represents the average value of 3–5 plants. LD, MD and HD refer to low-, medium- and high-density spacing. Thermal time was measured in degree-days, base 10 °C.

Table 2.

Influence of plant spacing on flower and fruit development in tomato. LD, MD and HD refer to low-, medium- and high-density spacing, respectively

Final number of trusses Final number of fruits Flower buds per truss* Flowers per truss* Fruits per truss*
LD 8·2 19·8a 6·1 5·8 4·4a
MD 8·2 19·1a 6·0 5·8 4·2a
HD 8·3 14·4b 5·7 5·1 2·8b
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* Calculations made on the first four trusses (fruit set was achieved at the end of the experiment).

Different letters represent significant differences at P = 0·05 (Newman–Keul's test).

To complement these data, a detailed analysis of organ expansion was carried out to provide the To parameters required to determine the temporal pattern of sinks actually competing at any given time for the common pool of biomass (Table 3). Significant differences were found for expansion of the fruits and the internodes (LD and MD vs HD), while none were observed for leaves and petioles (deduced from leaf measurements). Increasing densities tended to reduce the time needed to complete internode expansion and to increase the duration of expansion for fruit. For all the organ types, no significant differences were observed for the duration of expansion of metamers in positions 6–7–8 and 14–15–16. All these observations were consistent with the measurements made to characterize the final dimensions of organs (Table 3). Leaf and petiole lengths were not affected by the density treatments. Conversely, internode length/diameter and fruit diameter were strongly modified. Internode elongation was significantly enhanced by increasing densities, both for internodes with, and also without, trusses. Fruit growth, on the other hand, was clearly reduced; fruit at HD averaging only 45 % of the final volume of those at LD. For a given density, parameters To were thus set constant for each organ type and for all the metamer positions, the duration of expansion in GCs being directly deduced from the measured phyllochron.

Table 3.

Influence of plant spacing on duration of expansion and final dimensions of tomato organs. LD, MD and HD refer to low-, medium- and high-density spacing, respectively

Duration of expansion (°Cd)*
 Final size (mm)
Internode (length) Internode (diameter) Leaf Tomato Internode length, T– Internode length, T+ Internode diameter, T– Internode diameter, T+ Leaf length Tomato diameter
LD 144·6 227·8a 217·6 367·3a 44·6c 93·6b 12·4a 12·1a 349·3 66·9a
MD 153·6 226·9a 216·4 370·0a 44·8c 99·3b 12·5a 12·3a 345·1 67·5a
HD 124·1 186·7b 206·2 452·8b 69·5a 132·4a 11·3b 10·6b 387·2 50·9b
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* Thermal time, measured in degree-days, base 10 °C.

T+ and T– refer to internodes with and without a truss.

Different letters represent significant differences at P = 0·05 (Newman–Keul's test).

Multi-fitting results

Model parameters were determined in each situation by multi-fitting (i.e. statistical optimization) against a target file composed of around 20 plants distributed across five phenological stages. Figures 5 and 6 present the multi-fitting results from organ to plant scales. It should be noted that the simulated curves under these conditions are the result of whole-model functions (from which parameters have been optimized; Guo et al., 2006) and are not functions directly fitted to the observed biomasses. Interestingly, we found that optimized sets of parameters enabled accurate model simulations for the wide range of situations characterized. The main traits involved in the phenotypic plasticity induced by plant spacing were clearly reproduced. Observed reductions in total plant dry weights were indeed reproduced satisfactorily (Fig. 6, ANCOVA, P = 0·15). Simulated plant dry weight was approximately 50 g at HD against 150 g at LD at the end of the experiment, that is to say very close to the actual observed values. Dynamic changes in biomass allocation were also correctly fitted (Fig. 5B). In particular, the model accounted for the substantial reduction of dry matter allocated to roots and leaves when density increased (petioles are not shown at the compartment level but behaved strictly in the same way as leaf blades, Fig. 5a), together with the quantitatively less-important reduction of biomass invested in internodes. With regard to fruit production, the very strong reductions of yield observed at HD were explained partly by a lower fruit number and partly by a decrease in fruit size. The model accounted for this latter part, reproducing accurately the very contrasting average fruit dry weights observed in the different situations (Fig. 5A), but it did not explicitly consider the changes in fruit number (forced as measured in the present exercise).

Fig. 5.

Fig. 5.

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Multi-fitting results at (A) the organ level (example of five plants grown at high density, HD) and (B) compartment level. Peaks in the organ weight measurements result from internodes that are twice as long when carrying a truss. GC is time measured in growth cycles.

Fig. 6.

Fig. 6.

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Comparison between observed and simulated total biomass production for the three planting densities. LD, MD and HD refer to low-, medium- and high-density spacing, respectively.

In contrast to leaf dry weights, leaf areas displayed very similar responses in all the treatments until the beginning of leaf senescence (Fig. 7). The stagnation of plant leaf area at HD after the 20th GC seemed to be driven mainly by leaf senescence and not by a reduction of individual leaf areas (Table 3). This was achieved through a significant increase in specific leaf area (SLA, average values equal to 295, 324 and 400 cm2 g−1 for LD, MD and HD, respectively; P < 0·001) that kept individual leaf surface areas identical under the different densities. In the present study, these variations in SLA were taken into account by adding measured individual leaf surface areas in the target files used for optimization.

Fig. 7.

Fig. 7.

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Comparison between measured and fitted kinematics of plant leaf area expansion for the three planting densities. LD, MD and HD refer to low-, medium- and high-density spacing, respectively. GC is time measured in growth cycles.

The good fittings presented above do not prove that GREENLAB formalisms are right in respect of the biological mechanisms involved in the plant responses. To address this question, a close examination of the optimized values of the model parameters is required. Table 4 presents parameter values and their respective standard deviations (s.d.) and coefficients of variation (CV). Remarkably, five parameters out of 12 showed very little variation across the contrasting situations studied. This involved particularly the parameters defining the shape of the beta laws for the different organ types (Bp, Bb, and Bf, defining the time course of organ demand during its expansion for petioles, leaf blades and individual fruits, respectively) and the parameters driving plant production per unit of leaf area (TE and r1). Conversely, the r2 parameter accounting for mutual shading of leaves (decreased by 86·4 %) and the relative sink-strength values for internodes (Pe, increased by 16·4 %), individual fruits (Pf, decreased by 44·5 %) and root systems (Pr, decreased by 46·9 %) were most affected by plant spacing. These modifications were more elevated than the corresponding CVs in the case of Pe and r2 but not for Pf and Pr. Very large CVs were indeed computed for these two parameters, probably due to significant organ-to-organ (in the case of fruits) and plant-to-plant (in the case of root systems) variations in dry weights measured within a treatment. A major conclusion is thus that the present version of the model does not capture the entire resource-dependent plasticity induced by plant spacing, and more particularly at extremely high density (HD). Indeed, good multi-fittings were achieved under these conditions only at the expense of the stability of some of the model parameters.

Table 4.

Estimated parameter values, standard deviation (s.d.) and coefficient of variation (CV) with multi-fitting for the different density treatments (see Table 1 for key to parameters). Relative differences (RD) indicated for medium density (MD) and high density (HD) represent the relative difference of the parameters as compared to low density (LD)

LD
 MD
 HD
Parameter Value s.d. CV ( %) Value s.d. CV ( %) RD ( %) Value s.d. CV ( %) RD ( %)
Pe 0·54 0·01 2·04 0·52 0·008 1·58 −3·1 0·62 0·013 2·09 16·4
Pp 0·52 0·01 2·05 0·51 0·008 1·61 −1·7 0·54 0·013 2·31 5·2
Pf 5·90 10·3 174·5 7·33 12·287 166·0 24·3 3·28 1·748 53·4 −44·5
Pr 6·21 9·5 153·3 6·26 8·628 137·7 0·7 3·30 9·779 296·7 −46·9
Be 0·73 0·0006 0·08 0·70 0·00052 0·07 −4·1 0·57 0·0007 0·12 −21·9
Bp 0·67 0·0006 0·09 0·66 0·00057 0·09 −1·5 0·61 0·0008 0·14 −9·0
Bb 0·70 0·0007 0·10 0·66 0·00062 0·10 −5·7 0·66 0·0009 0·13 −5·7
Bf 0·60 0·0010 0·17 0·63 0·00061 0·10 5·0 0·62 0·0007 0·12 3·3
Br 0·34 0·0001 0·02 0·30 0·00005 0·02 −11·8 0·26 0·0001 0·04 −23·5
r1 0·22 0·0007 0·32 0·21 0·0004 0·22 −4·8 0·28 0·0011 0··36 28·0
r2 5·68 1·41 24·8 2·53 0·59 23·5 −55·5 0·77 0·19 24·2 −86·4
TE 0·92 0·00036 0·04 0·91 0·00036 0·04 0·1 0·91 0·00036 0·04 0·1
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Altogether, however, the set of parameters remained very consistent between the various experimental situations. The ranking of organ sink strengths remained unchanged (root system > individual fruit > leaf blade > internode > petiole) and proved to be characteristic of tomato, showing significant differences from other plants following the Corner architectural model (Halle and Oldeman, 1970) already calibrated with GREENLAB (Guo et al., 2006, on maize). Furthermore, evolutions in values of parameters affected by density appear to be meaningful with respect to the plasticity observed in plant growth, suggesting that these parameters are not fixed but are actually environment-dependent. Parameters r1 and r2, for example, defined a set of production curves clearly integrating a regular increase in the competition for light interception as plant density increased (Fig. 8). At LD, the rate of dry matter production gradually decreased under the effect of self-shading for plants carrying over 0·2 m2 of leaf area (15 leaves, plant height = 0·75 m, plant spacing = 1 m), while optimized parameters for MD and HD defined earlier thresholds (for plants with comparable leaf areas, Fig. 8) with a reduction in dry matter production starting as early as 0·1 m2 leaf area (8–9 leaves, plant height = 0·3 m, plant spacing = 0·3 m) for plants at HD. Responses of relative sink-strength values also displayed a strong consistency with the changes observed in the dry matter allocation pattern. Reductions in Pf and Pr could be related to the decrease of biomass allocated to these organs at HD. In fact, as sink strength is actually expressed relative to the leaf blade (systematically set to 1), these reductions were observed because the decreases of dry matter allocated to these organs was proportionally more elevated than the reduction of allocation to leaf blades caused by SLA modifications. That was not the case for internodes, which maintained a better ability to attract assimilates compared with leaf blades and in the end showed an increase in relative sink strength (Pe) under HD in spite of the slight absolute reduction of dry matter invested in this compartment.

Fig. 8.

Fig. 8.

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Visualization of the impact of optimized parameters r1 and r2 on biomass production for the three planting densities. Production curves represent the rate of dry matter production Q(i) per leaf area when E(i) is set to 1 (no environmental effect). LD, MD and HD refer to low-, medium- and high-density spacing, respectively. GC is time measured in growth cycles.

DISCUSSION

The developmental and growth responses to plant spacing characterized in the tomato cultivar studied here were in agreement with previous reports made on tomato. Major changes involved the reduction of individual plant dry weight (Gorham, 1979; de Koning and de Ruiter, 1991; Heuvelink, 1995) and modifications in the allocation pattern (Scholberg et al., 2000, on a deterministic tomato cultivar). Furthermore, Heuvelink (1995) previously demonstrated that the fraction (for a given fruit load) of total dry matter allocated to the fruits remained approximately constant, irrespective of plant density. This was achieved through detrimental effects on fruit set and on flower development (Zahara and Timm, 1973; Rodriguez and Lambeth, 1975; de Koning and de Ruiter, 1991), fruit set being the more sensitive variable (Papadopoulos and Ormrod, 1991; Bertin, 1995). Such confirmation of previous findings was expected, but quantifications in the range of planting densities studied here were required as they considerably exceeded the range traditionally employed in commercial glasshouse production (between 2·5 and 4 plants m−2 depending on latitude; Heuvelink, 2005) and tested in most studies. This study was especially designed to explore the potential plasticity of tomato plants – LD, MD and HD being representative of isolated plants, approximate production densities and very high density, respectively.

In this context, and in spite of the current model limitations (the parameters r2, Pe, Pf, Pr and Br varied significantly between the different spacings), accurate model simulations obtained with sets of optimized parameters tend to confirm the potential of the model structure to accommodate a large part of the plant plasticity induced by variable spacing (i.e. modifications in dry matter production and allocation). Precisely because GREENLAB was shown to be able to perform as well under extreme conditions as it did under more usual ones, this considerably strengthens the rational basis of a structural model based on linear conversion of transpiration into dry matter and on an allocation process driven by single-sink functions to represent cohorts of organs of particular types. This result is not straightforward. Previous fits relating to fresh matter production (which is likely to be more weakly correlated to transpiration) clearly showed the inability of this set of equations to reproduce fresh weight accumulations during the whole cycle (maize grain-filling period, Guo et al., 2006; tomato fruit production period, Louarn et al., 2007b) and hence argues against the simplistic suggestion that the model is over-parameterized.

However, as modelling plant plasticity refers to the ability of a model to properly respond to a range of different conditions (rather than by simply tweaking parameter values for each condition), our results also suggest that further development is required if the model is to fully fulfil this objective. These developments should first be reasoned with respect to a meaningful validity domain (e.g. commercially useful planting densities). From this stage on, developments could take different forms depending on the parameter considered. Simple response curves (i.e. reaction norms, Via et al., 1995; De Witt and Scheiner, 2004) of the parameter to the relevant environmental variable could advantageously replace currently fixed parameters. This could be the case for relative sink strengths and, notably, for internode and root sink strengths as a function of absorbed PAR (Kolb and Steiner, 1990; Lötscher and Nösberger, 1997). More in-depth modifications of the model equations could also be considered, in particular for the production equation in which r2 plays such a key role. The most obvious option would be a coupling of current allocation formalisms with a detailed radiative transfer model (Soler et al., 2003) to more completely take advantage of the 3-D virtual plants. Such a model has already been implemented with tomato (Dong, 2006) and a sensitivity analysis is in progress to evaluate the consequences on model predictions, on model parameter estimation, and on the computation time required. However, a more simple alternative could be a model integrating a production equation and taking into account both plant self-shading and neighbour-plant competition (instead of a global Sp parameter). Such an approach, defining plant-to-plant competition factors from a vertical projection of a single plant area (Bonan, 1988), has recently been presented for a stand of pine trees (Cournède et al., 2007). A calibration of this model on the present dataset could enable checking of its generality under different planting conditions.

Because regulation of fruit number (through dynamic fruit-set prediction) and regulation of leaf area (through SLA modifications) turned out to be key processes in adaptation to increasing density; these will also have to be introduced to future versions of the model. Opportunities for linking growth and development already exist in GREENLAB and could take into account feedback effects of plant growth on flower and/or fruit development (Mathieu et al., 2004). Nevertheless, thresholds defining the sink/source ratios that induce fruit abortion remain to be evaluated. What makes this modelling approach particularly valuable is the possibility of using the generic kernel algorithm to produce the relationships required to build more sophisticated dynamic models. A version as simple as the one presented in this paper could be used as a powerful tool to resolve sink and source relationships. This could be used to deal with phenomena such as leaf senescence, fruit abortion and plant-to-plant topology variations, as far as they can be properly quantified. As we show, the model already constitutes a framework suitable for analysing organ sink-strength variations under contrasting conditions, enabling insights to be gained into the biological systems studied (e.g. leaves, internodes and root growth behaviours that can be related either to direct measurements or to independent reports in the literature: Corré, 1983; Solangaarachchi and Harper, 1987; Kolb and Steiner, 1990; Ballaré et al., 1991; Ericsson, 1995). The model could also provide a tool for accessing temporal responses of the supply/demand ratio and for dealing with complex interactions, such as resource-dependent fluctuations of SLA (Gary et al., 1993), flower initiation and fruit set, which have not been properly explained so far. In the present framework, supply and demand are indeed balanced when biomass simulations are accurate, independently of individual parameter variations.

CONCLUSIONS

This case study constitutes only a first step towards developing a dynamic structure–function model that can properly account for plant adaptation to a range of environmental conditions. We have shown that the generic GREENLAB model actually fails to completely explain observed plasticity and that it must be made more complex if it is to fully account for the effects induced by plant competition for light at very high densities. We have discussed some options for further model improvement along these lines. Meanwhile, it is worth noting that the simple fitting tool presented in this paper can be used to resolve source–sink relationships so as to better explore datasets (for instance to explore supply/demand ratios) and to efficiently build up more-sophisticated dynamic models. Ultimately, such an improved model must be challenged with independent data if we are to assess its predictive capability. The improved model will find applications in selection (Dingkuhn et al., 2005), greenhouse tomato yield and quality prediction (fruit size, Guichard et al., 2001) and management practice optimization (planting design, pruning).

ACKNOWLEDGEMENTS

This study was supported by the National Natural Science Foundation of China (#60073007), and the LIAMA laboratory in Beijing, China. The authors would also like to thank students from China Agricultural University (Beijing) and Hunan Agricultural University (Changsha) involved in the plant measurements.

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