Simulation of carbon allocation and organ growth variability in apple tree by connecting architectural and source–sink models

Abstract

INTRODUCTION

Understanding and modelling how plants allocate carbohydrates, coming from photosynthesis or reserves, to the different organs is of major interest since carbohydrate and biomass allocation between organs determines many growth processes (Poorter et al., 2012), including fruit size and quality (Martre et al., 2011). Carbon allocation models range from empirically-based compartment models (e.g. Goudriaan and Van Laar, 1994; Dufrêne et al., 2005) to mechanistic models of phloem transport (e.g. Minchin and Lacointe, 2005; Jensen et al., 2012; Sellier and Harrington, 2014). The first approach appears too simple to deal with plants displaying a large number of organs organized into a complex architecture. Conversely, the second requires complex calibration and has been only calibrated so far on simple plant architectures without any branching [two sources, three sinks (Thorpe et al., 2011)].

For trees, several approaches have been proposed to simulate carbon allocation within complex architectures. These approaches differ mainly in the ways they consider the effects of distances between sources and sinks (Lacointe, 2000). Based on the common assimilate pool theory first developed for annual plants (Heuvelink, 1995), some modelling approaches have suggested that these distances can be ignored and sink strength should be considered instead (i.e. the ability of an organ to attract assimilates; Ho, 1988) as the only driving force for carbon allocation (Mathieu et al., 2009; Wang et al., 2012). Nevertheless, these approaches neglect many results showing the influence of local assimilate availability on organ growth in different tree or liana species [peach, Prunus persica (Génard, 1992); kiwi fruit, Actinidia deliciosa (Piller et al., 1998); grapevine, Vitis vinifera (Pallas et al., 2010)]. Moreover, a large amount of variability in fruit growth and quality within fruit trees that can be related to local carbon availability and fruit position in the branching system is often observed (Jackson et al., 1971; Pavel and DeJong, 1993; Lauri and Lespinasse, 2001), emphasizing the necessity to take into account distance effects when modelling assimilate allocation.

To deal with this distance effect, models based on an electric circuit analogy and including a parameter to model the resistance to mass flow in the phloem have been developed for the peach tree (Allen et al., 2005) and kiwifruit vine (Cieslak et al., 2011). This approach requires time-consuming simulations and complex calibrations. By contrast, other models have taken into account distance effects using empirical coefficients that can be directly calibrated on the observed organ growth variability (Balandier et al., 2000).

Moreover, no clear consensus exists so far on the most appropriate topological scale for the computation of carbon allocation processes. In functional structural plant models, carbon allocation can be simulated at the organ scale (e.g. Allen et al., 2005), at the shoot scale or at the compartment scale [e.g. in QualiTree (Lescourret et al., 2011)]. However, considering higher scales than organs facilitates data representation and collection and reduces the computational time of the model (Ong et al., 2014). In this context, QualiTree is a ‘source–sink’ plant model that integrates an empirical function to simulate the impact of distances on assimilate allocation in a simplified representation of plant architecture (Lescourret et al., 2011). More precisely, QualiTree is dedicated to the peach tree and considers the plant as a set of compartments (roots, trunks) and production units corresponding to 1-year-old stems and all the organs (leafy shoots and fruits) they bear. For each entity (compartment or production unit), current growth results from the balance between assimilate supply and demand. Assimilate demand is related to organ sink strengths and includes growth and maintenance respiration. Assimilate demand for growth depends on organ relative growth rates and thermal time. Assimilate supply results from (1) a simplified light interception model based on a turbid medium hypothesis allowing the computation of photosynthesis activity (Mirás-Avalos et al., 2011); (2) carbohydrate reserve mobilization; and (3) reallocation of carbohydrates based on distances between entities. QualiTree also integrates a fruit model allowing the estimation of several quality traits (Lescourret and Génard, 2005). The model has been calibrated for peach varieties and has revealed its ability to simulate the impacts of crop load and water stress on fruit weight and quality (Mirás-Avalos et al., 2011, 2013). Nevertheless, this model has some limitations that need to be overcome to extend its domain of application. First, the model uses a simplified representation of plant functioning and architecture, grouping organs by production units in which each organ type (fruits or leafy shoot) displays the same growth, thus limiting the capacity of the model to represent within-tree growth variability. Second, tree architectures are inputs and in previous studies (Mirás-Avalos et al., 2011) they were based on observed 3D tree architectures. Nevertheless, describing 3D tree architecture, especially with digitizing methods that make it possible to record 3D coordinates (Sinoquet et al., 1997), is time-consuming and limits the ability of the model to simulate organ growth on various tree architectures.

One way to overcome these limitations is to use architectural plant models that allow the simulation over years of plant architecture based on a description at the organ level (Vos et al., 2010; DeJong et al., 2011). Indeed, simulation outputs of these models provide an exhaustive description of organ and shoot dimensions and characteristics within tree architecture that can be used as the input of QualiTree. Among these models, MappleT simulates apple tree (Malus × domestica) development over years using Markovian statistical models to simulate terminal bud fates and branching patterns together with a biomechanical model to simulate branch orientation and bending (Costes et al., 2008). The model has been calibrated for the ‘Fuji’ cultivar and has been used to analyse the impact of organ geometry and plant topology on light interception efficiency (Da Silva et al., 2014a). In MappleT, the within-tree variability in shoot growth is driven by stochastic rules depending on the cultivar and tree age without accounting for underlying physiological processes. In this context, connecting the carbon allocation of QualiTree with architectures simulated by MappleT could allow simulation of both the within-tree variability of shoot and fruit growth and its dependency on environmental conditions and microclimate within tree architecture.

The case of the apple tree is especially relevant from a biological point of view as a means of dealing with the question of within-tree growth variability. Indeed, large variability in fruit weight (Reyes et al., 2016) and shoot length (Costes et al., 2003; Seleznyova et al., 2008) has been observed within tree architecture. Moreover, unlike peach, in which pruning practices tend to homogenize shoot lengths, in the apple tree pruning is generally more tailored and maintains large variability in shoot and fruit growth. Physiological determinants of this variability remain partially unknown and many hypotheses have been suggested, such as the light environment of fruits and buds (Barrit et al., 1987), tree crop load (Racksó, 2006) and variations in shoot growth potential depending on bud position within the branching structure (Renton et al., 2006). The modelling approach developed in this study, connecting a representation of tree architecture and a carbon model, should be a way to give new insights into the effects of organ potential growth rate and shoot and tree carbon availability on the observed variability at the tree scale.

Our study aimed first to adapt QualiTree to the specific case of apple and second to develop a modelling framework to couple QualiTree with architectural data provided by MappleT. The modelling approach was performed based on 4-year-old trees of the ‘Fuji’ cultivar. Model outputs were compared with digitized trees considering organ final size and growth dynamics as well as their within-tree variability. The following questions were addressed: (1) is the simplified carbon allocation model and architecture used in QualiTree adequate to deal with the observed organ growth variability? (2) to what extent does carbon allocation determine organ growth variability and shoot polymorphism? (3) what is the impact of distances between organs on carbon allocation? (4) is the modelling approach adequate to deal with agronomical topics such as varying crop loads?

MATERIALS AND METHODS

Connection of MappleT and QualiTree models

In this study, four ‘Fuji’ trees were simulated with MappleT from planting until full bloom (15 April in this study) of the fourth year of growth. Since MappleT includes stochastic processes, the simulated trees displayed contrasted architectures (different numbers and positions of shoots and fruits). The values of the different parameters were those presented in Costes et al. (2008). MappleT simulation outputs included values at full bloom for organ dimensions, tree topology and spatial coordinates of each organ. These outputs were written in the multi-scale graph (MTG) format (Godin and Caraglio, 1998), including three topological scales [metamer, growth unit (GU), tree]. Vegetative GUs were classified according to their type based on the length that each shoot can reach at the end of the growth season (short, <5 cm; medium, between 5 and 20 cm; long, >20 cm). MappleT also includes a fruit set probability and assumes that each reproductive GU cannot bear more than one fruit.

In QualiTree, plant architecture is described in a dedicated Mysql database and is simplified compared with MappleT. It is based on a combined description at a production unit scale (PU) and a global compartment scale (Fig. 1). This latter includes three compartments: wood older than 2 years (old wood); roots older than 1 year (old root); and new roots developed during the current growth cycle. Each PU gathers all the organs developed from a 1-year-old shoot and includes three entities (leafy shoot, stem and fruits). The leafy shoot entity includes all annual shoots, the stem corresponds to all 1-year-old stems and the fruit entity includes all the fruits growing within each PU.

Fig. 1.

Schematic representation of the procedure used to convert multiscale tree graphs (MTGs) generated by MappleT into QualiTree architecture in a simplified example. (A) MTG generated by MappleT and represented at the growth unit (GU) scale. (B) Conversion of the MTG into QualiTree architecture described at the production unit scale. S, M, L and I refer to short, medium, long and inflorescence GUs, respectively. (C) Schematic representation of the plant in MappleT. (D) Schematic representation of the plant in QualiTree. The numbers in C refer to the GU label.

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A procedure implemented in Python was developed to convert MappleT architectures and related outputs at full bloom into appropriate formats to input them into the Mysql database of QualiTree. This procedure allows gathering all the annual shoots simulated by MappleT into PUs and estimating initial dry weights of all entities and compartments (old wood, old roots, new roots and PUs). Initial leaf dry weights were estimated as the product of leaf area provided by MappleT and specific leaf weight (SLW; see Supplementary Data Table S1). Initial internode dry weights of leafy shoots, stems and old wood were estimated based on internode length and diameter simulated by MappleT assuming a cylinder shape and a constant internode density (dens). Since the root compartment is not simulated by MappleT, the initial weight of the old root compartment was estimated as the product of old wood dry weight and shoot/root ratio (SReq). The initial dry weight of the new root compartment was estimated as the product of annual shoot dry weight and SReq.

In the previous version of QualiTree (Lescourret et al., 2011), all the leafy shoots were identical within each PU. Since shoot polymorphism exists in the apple tree (Costes et al., 2003), we integrated the three types of leafy shoots provided by MAppleT in each PU. As a consequence, in this new version of QualiTree each PU was described by the number and type of leafy shoots it contains (Fig. 1).

Description of QualiTree

QualiTree has been fully described by Lescourret et al. (2011) and Mirás-Avalos et al. (2011). In the following section, only the main hypotheses of the model and the changes made for the apple tree are presented (Fig. 2).

Fig. 2.

Schematic representation of steps in the simulation of carbon balance in QualiTree for a 1-d period. In the first step the amount of available carbohydrate for each compartment (production units, old wood, young and old roots), including photosynthesis and carbohydrate reserve, is evaluated. In the second step this carbohydrate supply is allocated to each compartment to satisfy the maintenance respiration and growth of leafy shoots. In the third step the remaining carbon supply is computed at the compartment scale for (1) the growth of compartments other than leafy shoots and (2) the part of respiration and leafy shoot growth that was not satisfied in step 2. In step 4 the remaining carbon supply of each entity is allocated among all the entities of the tree depending on their distances [eqn (6)]. In step 5, organ growth and reserve storage are simulated.

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Light interception model. In QualiTree, light interception is estimated at the PU level using a turbid medium hypothesis inside a shape including 80 % of total leaf area (for further details of the light model see Mirás-Avalos et al., 2011). Since the canopy shape defined as two imbricate ellipsoids for peach trees was not suitable for apple trees, we modified it to allow the usage of user-defined ellipsoids (Supplementary Data Fig. S1). The parameters of these ellipsoids were calculated with PlantGL under OpenAlea (Pradal et al., 2008). The parameters of these ellipsoids were then included in QualiTree to define the encompassing shape in which light interception is simulated. The incident photosynthetically active radiation per unit leaf area (PAR_i, μmol photon m⁻² s⁻¹) was computed for each production unit at an hourly time step depending on the sun’s position, radiation intensity and leaf area density within the ellipsoid.

For each entity (old and new wood, old and new roots and PUs), reserve biomass is assumed to be partially mobilized to sustain growth and maintenance respiration. A constant parameter (R_m) for each entity determines the proportion of reserve biomass that is mobilized.

Carbohydrate demand for growth and respiration. For each entity, maintenance respiration is computed daily as the product of (1) its dry weight (gDM), (2) the value of a Q10 law with one dimensionless parameter (henceforward termed Q10) to deal with the temperature effect on respiration, and (3) a respiration rate parameter (MRR, gC gDM⁻¹ s⁻¹). Moreover, a growth respiration coefficient (GRC_i, gC gDM⁻¹) is taken into account and represents the amount of carbon per unit dry mass allocated to each entity that is not converted into biomass.

Daily carbohydrate demand for leafy shoot growth (Supplementary Data Fig. S2) is simulated based on an initial relative growth rate parameter (RGR_ini,i, Cd⁻¹) depending on shoot type (i for short, medium and long shoots). The equation used for leafy growth demand [D_ls,i(t), gC] is:

$Dls,i(t)=(CCls+GRCls)×RGRini,ls,i×Im⁡(t)−1×ΔDD(t)×SBls,i(t)×f(t)×g(t)$

(3)

where CC_ls (gC gDM⁻¹) is a parameter representing the proportion of carbon in structural biomass, SB_ls,i is shoot structural dry weight (gDM), Im (dimensionless) is the ratio between the current shoot:root ratio and the shoot root ratio at equilibrium (SR_eq), ΔDD is the daily increment in thermal time (°Cd), g is a function ranging from 0 to 1 and simulating the decrease in shoot demand with thermal time (Fig. S2). The g function is defined as follows:

(4)

where dd_end,i (°Cd) is a parameter representing the thermal time at the end of growth for each shoot type i. ΔDD is computed using a single sine method (Lescourret et al., 1998) with two parameters, a lower (T_min) and an upper temperature (T_max). f is a function ranging from 0 to 1 and simulating the decrease in shoot demand when current shoot biomass increases (Fig. S2), as observed in previous studies (Grossman and DeJong, 1995). The maximal weight of each leafy shoot (SB_max,i, gDM) is a parameter that depends on the type of shoot (i). f is defined as follows:

(5)

For old root, old wood and stems, a constant RGR (RGR_i, Cd⁻¹) throughout the growth season is used to evaluate their daily growth demand. For new roots, the computation of growth demand includes an initial RGR value (RGR_ini,roots, Cd⁻¹) and takes into account the shoot:root ratio imbalance (Im). In the previous version for peach, an initial RGR value and a decrease in RGR with time was used to simulate fruit growth demand. In the case of the apple tree, this decrease in fruit growth rate was not observed (Lakso et al., 1995). Similarly, our observed data showed an almost constant rate of fruit weight growth when expressed in thermal time units. Thus, an absolute growth rate (GR_f, gDM Cd⁻¹) was considered to estimate the daily demand of each fruit.

Biomass allocation phase between organs. A first phase of carbon allocation to sustain maintenance respiration of each entity and leafy shoot growth of PUs is simulated in QualiTree. If carbon supply is enough to satisfy maintenance respiration and leafy shoot growth, a remaining supply is computed for each entity. Conversely, if the carbon supply is not enough a remaining demand is computed.

After carbon exchanges, the first priority in carbon allocation is assumed to be the remaining part of maintenance respiration, whereas growth has second priority and carbon storage only occurs if growth demand is fully satisfied. At the PU scale, the remaining part of leafy shoot growth has first priority, whereas fruit and stem growth has the second priority. The amount of carbon allocated to fruits and stems is considered to be proportional to their own demand. Moreover, carbohydrate allocation among the different types of leafy shoot within each PU is also considered proportional to their own demand. After this last phase of carbon allocation, leaf area expansion is modelled for each leafy shoot as the product of a constant specific leaf weight (SLW, g m⁻²) and leaf structural weight, assuming a constant proportion of leaf dry weight in structural leafy shoot dry weight (r₁).

Parameter estimation

Parameter estimation was performed using literature data, analysing data from experiments or using directly one of the simulated trees. Values of about 40 parameters were newly estimated for apple and values for peach were used when no data were available on apple (around ten parameters; Table S1). Variables related to maintenance respiration (MRR), growth respiration demand (GR) and organ carbon concentration were estimated from the literature, using mainly Lakso (1994) and Walton et al.(1999). Parameters related to the part of reserve that can be used by each entity at each time step (Rm) were peach values. New analyses of data from Benzing (1999) and Massonnet (2004) (Table S1) were performed to estimate the values of parameters related to photosynthesis activities (p₁, p₄) as well as the values for old roots, new roots and stem RGR. The value for fruit growth rate (GR_f) was the same as that in MappleT (Costes et al., 2008).

For leafy growth demand, SB_max and dd_end could lead to the same behaviour in terms of decrease in RGR with time. As a consequence and in order to limit over-parameterization of the model, we only characterized differences between shoot types based on SB_max and used the same parameter value for dd_end whatever the shoot type (Fig. S2). SB_max was estimated as the weight corresponding to 5 cm for short shoots, 20 cm for medium shoots or 40 cm for long shoots, using allometric relationships established by Massonnet (2004). We set dd_end to 480  °Cd, corresponding to the mean thermal time at which the longest shoots stopped producing new leaves (Massonnet, 2004). RGR_iwas estimated directly on data collected by Massonnet (2004) in 2002, which were then also partially used for model validation (Experiment 2; see below). The same RGRi values were considered for the three types of shoot because no clear difference was observed in the experimental data.

Direct calibration of α [eqn (5)], which drives the impact of distances between entities on carbohydrate allocation, was performed on one of the simulated ‘Fuji’ trees (‘Fuji’ 2; Fig. S1). For this, a set of simulations was performed with the α parameter ranging from 0 (no distance effect) to 0·15 (high distance effect), with a step of 0·005. These simulations were first used to estimate the impact of α on mean fruit weight and its variation within trees. Then, for each value of α, the coefficient of variation (CV) of fruit weight within two digitized 4-year-old ‘Fuji’ trees (‘Fuji’ 42 and ‘Fuji’ 47) was compared (see Model simulation and validation section for further description of these trees) with the simulated CV for the ‘Fuji’ 2 tree. As the lowest difference between the simulated and observed CV was observed with α equal to 0·035, this value was used for simulating carbon allocation for the three other trees.

Model simulation and validation

Simulations with QualiTree were performed using the four trees simulated with MappleT from 15 April (full bloom) to 15 September (harvest date), using meteorological data (temperature and radiation) recorded in 1998 at the INRA experimental station, near Montpellier. During this period mean daily temperature ranged from 9·3 to 29·1  °C, with an average value of 20·4  °C, and daily photosynthetically active radiation ranged from 2·9 to 57·1 mol m⁻² d⁻¹ with an average value of 41·6 mol m⁻² d⁻¹.

Simulations were compared with data collected in Experiments 1, 2 and 3, carried out in 1998 (Costes et al., 1999), 2002 and 2003 (Massonnet, 2004), respectively, on two ‘Fuji’ trees (‘Fuji’ 42 and ‘Fuji’ 47) planted in 1995. Trees were not pruned but were irrigated and fertilized to avoid any water or mineral deficiency. Thinning was controlled in order to keep only one fruit per inflorescence to avoid competition between fruits within inflorescences.

In 1998 (Experiment 1), two 4-year-old trees were digitized at the end of the growth season. The dataset included the numbers of different types of shoots and their lengths, as well as the fresh weight and water content of each fruit at harvest. Simulation outputs were compared with observed data based on (1) total plant leaf area, (2) total fruit number and total fruit weight at harvest, (3) numbers of short, medium and large shoots, (4) medium leaf area of medium and long shoots, (5) mean fruit dry weight and fruit dry weight distribution at harvest (15 September for simulations) and (6) crop load expressed as fruit number per unit leaf area. Leaf area was estimated from an allometric relationship between shoot length (L) and leaf area for long and medium shoots (La) using the empirical linear function determined by Massonnet (2004) (La = 10·2 × L + 199, with L in cm and La in cm²). For short shoots, the allometric relationship was not accurate enough to correctly estimate leaf area from length, and was thus assumed to be equal to the mean observed values for this type of shoot (60 cm², estimated from Massonnet, 2004). For individual fruit dry weights, Shapiro tests were performed on simulated and observed data to test the normality of distributions, Bartlett tests were performed to compare simulated and observed variances and Student’s t-tests were performed to compare mean simulated and observed values.

The two other experiments (Experiments 2 and 3) were used to assess the accuracy of model simulation for the timing of shoot leaf area expansion. In these experiments trees were 8 and 9 years old. Shoot length was recorded every week on medium and long shoots from bud burst until the end of the experiment. Simulated timings of leaf area expansion were compared with observations for medium and long shoots. Simulated and observed data on leaf area expansion were compared based on the root mean square error (RMSE).

Finally, the impacts of crop load on mean fruit dry weight and its variability, plant leaf area, reserve content and photosynthesis activity were assessed to evaluate the ability of the model to deal with an agronomic scenario. Ten crop load levels were simulated by randomly adding or removing fruits on one of the simulated trees (‘Fuji’ 2; Fig. S1).

RESULTS

Simulation outputs at the tree and compartment scales

Biomass and number of entities associated with each compartment or PU within the four 4-year-old ‘Fuji’ trees simulated by MappleT were estimated at the beginning of the growth season, grouping organs as in the QualiTree database (Table 1). The whole-tree dry weight was highly variable [900·8 g (‘Fuji’ 0) to 2176·1 g (‘Fuji’ 3)], as were shoot number, shoot type composition and stem weight among PUs. The number of shoots per PU ranged from 1 (42 % of PUs) to more than 10 (9 %), with a mean value of 2·8. The mean number of fruits per PU was equal to 0·22 and a large variability in this variable was simulated, with more than 70 % of PUs without fruit, 25 % bearing one fruit and 5 % bearing more than one fruit.

Mean values of shoot and fruit number simulated by MappleT were close to observations in Experiment 1 (Fig. 3; simulated, 946 and 66; observed, 813 and 59, for shoot and fruit number, respectively). Similarly, mean values of total leaf area and total fruit dry weight at harvest simulated by the modelling approach combining MappleT and QualiTree simulations were close to observations (simulated, 16·0 m² and 3·23 kg; observed, 13·1 m² and 2·68 kg, for total leaf area and total fruit dry weight, respectively).

Fig. 3.

Total shoot number (A), total leaf area (B), fruit number (C) and total fruit dry weight (D) for observed (‘Fuji’ 42 and 47, dark bars) and simulated trees (‘Fuji’ 0, 1, 2 and 3, white bars). Total leaf area and total fruit dry weight were estimated at harvest. Observed trees correspond to two 4-year-old ‘Fuji’ trees digitized in 1998 (Experiment 1).

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Different phases of carbohydrate allocation among organs were simulated during the growth season by QualiTree (Fig. 4A, B). The first phase, from mid-April to the end of May, i.e. during leaf area establishment, was characterized by a strong allocation of carbon to leafy shoots. The date of the end of this first phase corresponds to dd_end [480  °Cd, eqn (4)], revealing that a large majority of shoots did not reach their potential final weight (SB_max) before this date [eqn (3)]. During this first phase, the average carbohydrate demand of leafy shoot compartment was equal to 21.0 gC d⁻¹, with inter-day fluctuations corresponding to variations in daily thermal time. During this period, the demand for root growth also reached its maximal value (20.8 gC d⁻¹) and the beginning of fruit growth was observed. This phase was followed by a phase of maximal fruit demand, with an average value of 18.1 gC d⁻¹ between the end of May and the end of August. This phase of high fruit demand was concomitant with the increase in the demand of the above-ground woody compartment. This increase results from an increase in the current weight of the woody compartment because its demand depends on its weight and on a constant RGR. From the beginning of September, fruit demand for growth tended to decrease. This decrease results from a decrease in daily temperature since, in the model, daily fruit growth is directly associated with daily thermal time. Concomitantly, the above-ground woody compartment demand for growth continued to increase due to an increase in its current weight.

Fig. 4.

Simulation outputs of QualiTree for the simulated ‘Fuji’ 2 tree. (A) Evolution of roots, above-ground woody parts, fruits and leafy shoots biomass during the growth season (15 April to 15 September). (B) Daily carbon growth demand of the compartments. (C) Whole-tree photosynthesis.

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Whole-tree photosynthesis (Fig. 4C) increased at the beginning of the growth season due to the combined increase in tree light interception efficiency and daily incident solar radiation. Thereafter, photosynthesis reached almost 40 gC d⁻¹ (2·2 gC m⁻² d⁻¹) until the end of the experiment, with inter-day variation resulting from variation in daily incident light radiation (data not shown).

Simulation of leafy shoot growth and variability

The proportions of short, medium and long shoots within trees, simulated with MappleT, were quite accurate except for a slight overestimation of short shoots and underestimation of medium shoots (Fig. 5A, B). Individual shoot leaf areas were simulated by QualiTree and resulted in mean leaf area of medium shoots close to observed values in Experiment 1 (+7·0 %), whereas that of long shoots was overestimated (+24·4 %). At the end of the season, a large proportion of shoots reached a weight close to their potential final weight of 2·5, 6 and 11 g, corresponding to leaf areas of 140, 350 and 680 cm² for short, medium and long shoots, respectively.

Fig. 5.

Simulated (lines) distributions of final leaf area for the three types of shoot (short, A; medium, B; long, C) and observed (bars) distributions for medium and long shoots at harvest (15 September). Simulations correspond to the ‘Fuji’ 2 tree. Observed values were collected in 1998 on two digitized 4-year-old ‘Fuji’ trees (42 and 47; Experiment 1). For long and medium shoots, mean simulated and observed leaf areas are mentioned on the left side of the figure. Shoot type frequencies are also represented on the left side of the figure and were computed for each type of shoot as the ratio of the number of shoots of the considered type to the total number of shoots in the tree.

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The dynamics of leaf area development were compared between simulations and observations for medium and long shoots (Fig. 6). If the entire period was considered, mean RMSE in Experiments 2 and 3 was equal to 54·8 and 58·2 cm², respectively, for long shoots and 33·4 and 21·7 cm² for medium shoots (Fig. 6). At the end of the growing season the model simulated a remobilization of structural carbohydrates of the shoots for which incident light radiation was too low to sustain their maintenance respiration. This led to a slight decrease in shoot leaf area because in QualiTree leaf area is computed directly from structural biomass using a constant allometric relationship. When expressed as relative leaf area (Fig. 6), the rate of leaf area development was accurately simulated for long shoots, whereas it tended to be slightly underestimated for medium shoots.

Fig. 6.

Observed (points) and simulated (lines) evolution of shoot leaf area for long, medium and short shoots. Simulated values correspond to the simulated ‘Fuji’ 2 cultivar (4-year-old). Observed data were collected in Experiments 2 and 3 on 8- and 9-year-old ‘Fuji’ trees on medium and long shoots. Bars represent the standard error of observed values and dashed lines represent the standard error of simulated values. The inset graph represents changes in relative leaf area (ratio of the current leaf area to the final leaf area) for this same dataset.

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Simulations of fruit growth and variability

Mean fruit dry weights were close to observations in Experiment 1 (mean observed, 45·2 g; mean simulated, 42·9 g; Table 2). The differences in mean fruit dry weight between simulated trees (from 36·0 to 49·1 g) mainly resulted from variations in their crop load since trees with lower crop loads displayed bigger fruits and trees with higher crop loads displayed smaller fruits. The simulated fruit weight variability within trees, estimated from standard deviation values at harvest, was also close to observations in Experiment 1 (Table 2, Fig. 7). Student’s t-tests and Bartlett tests showed no differences between observed and simulated trees in terms of mean and variance, i.e. if the average simulated fruit distribution on all trees together was considered. Considering each individual simulated tree, ‘Fuji’ 2 and 3 did not display any difference in mean and variance when compared with observed trees. Fruit weights of ‘Fuji’ 0 and 1 were not normally distributed, probably due to over-representation of fruit displaying high fruit weights (>50 g).

Fig. 7.

Observed (bars) and simulated (lines) frequencies of fruit dry weight at harvest (15 September). Simulated values are represented for the four simulated 4-year-old ‘Fuji’ trees. Observed values were recorded on the two digitized 4-year-old ‘Fuji’ trees in 1998 (Experiment 1). The value of the α parameter [eqn (5)] was calibrated directly for the ‘Fuji’ 2 tree using observed data. This value was kept constant for the other trees. The inset graph represents simulated frequencies considering all simulated trees together.

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Impact of carbohydrate allocation coefficient between shoots and tree crop load on shoot and fruit growth

The parameter α had a strong impact on the simulation of mean fruit dry weight and its variation within trees (Fig. 8). When α was close to zero (i.e. no impact of distances on biomass partitioning), all fruits had almost the same amount of carbohydrate available for growth. In this case, fruit dry weight mainly resulted from the supply/demand ratio at the tree scale, leading to low variability in fruit dry weight (s.d. = 1·5). Conversely, when the α value was high, fruit growth mainly depended on carbohydrate availability at the shoot scale. This carbon availability was quite variable among shoots due to variations in incident radiation within the canopy (Supplementary Data Fig. S3), thus leading to large variation in fruit dry weight (s.d. = 20·1). In our case, a medium value of α (0·035) was observed to give the simulated values that best represented the observed fruit dry weight variability within trees.

Fig. 8.

(A) Relationships between fruit dry weight at the end of simulation (15 September) and the total photosynthesis/fruit number ratio of the production unit in which each fruit was located for four α values (low, 0·015; medium, 0·035; high, 0·075; very high, 0·15). Medium α corresponds to the situation represented in Fig. 7. Points represent simulation results for each individual production unit and the lines result from sigmoidal adjustments performed on simulated values. (B) Distribution of individual fruit weights for the four α values. Crosses represent the mean value of fruit dry weight for each α value. Simulations were performed with the input architecture of the ‘Fuji’ 2 tree.

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Finally, the behaviour of the model was analysed in response to crop load variations (from 1·2 to 13·3 fruits m⁻²; Fig. 9). An increase in crop load induced a small increase in shoot photosynthetic activity (Fig. 9C) resulting from eqn (1), which includes a feedback retroaction between photosynthesis and shoot reserve content (Fig. 9B). Nevertheless, despite this increase in carbohydrate supply, mean fruit dry weight decreased from 50·9 to 33·9 g when crop load values ranged from 1·2 to 13·3 fruits m⁻² (Fig. 9A). Mean fruit dry weight showed a higher sensitivity to crop load than total leaf area, values of which ranged from 15·1 to 16·4 m². The model also simulated a high impact of crop load on fruit weight variability, with an increasing standard deviation of fruit weight when crop load increased. More precisely, the impact of crop load was high on the weight of the smallest fruits in the tree. Indeed the weight of the smallest fruit was equal to 8·2 g for a crop load of 13·3 fruits m⁻² whereas it only reached 43·2 g for a crop load of 1·2 fruits m⁻². Conversely, the impact was low on the weight of the biggest fruits (around 65 g whatever the treatment).

Fig. 9.

Results of simulations with contrasted crop loads. The architecture of the ‘Fuji’ 2 tree was used for these simulations and different crop loads were simulated with random addition or removal of fruits on production units. Crop load was estimated as the ratio of fruit number to tree leaf area. (A) Mean dry fruit weight (points) and weights of the smallest and biggest fruits (dashed lines) according to crop load. (B) Final tree leaf area and mean carbohydrate reserve content in leafy shoots depending on crop load. (C) Impact of crop load (1) on the amount of intercepted radiation throughout the growing season at the tree scale [photosynthetically active radiation (PAR) divided by the number of fruits (PAR/Fruits), (2) on mean daily photosynthesis per leaf area unit (Pn efficiency).

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DISCUSSION

The model accurately simulates tree-scale variables

At the tree scale, the model simulated accurately the mean numbers of the different shoot types, plant leaf area, total fruit dry weight at harvest and crop load. These results could be considered both as a new validation of MappleT for the number of shoots (after Costes et al., 2008) and as a first validation at the plant scale of the modelling approach connecting MappleT and QualiTree for the variables related to organ biomasses and dimensions. The stochastic formalisms implemented in MappleT led to the simulation of trees displaying contrasted values for all these variables. Nevertheless, the number of observed trees was not large enough to statistically evaluate the accuracy of model predictions in term of variance. In this study, both observed and simulated crop load values (around 4 fruits m⁻² of leaf area) were lower than those observed for adult trees [around 7 fruits m⁻² of leaf area reported by Wünsche et al. (2000) in typical agronomic conditions] but may be relevant for young trees displaying large vegetative development and a low number of fruits, as observed in our study (Wünsche et al., 2000). Moreover, simulated photosynthesis values at the tree scale were close to literature data. Indeed, the simulated mean daily photosynthesis in summer months was equal to 2·9 gC m⁻² d⁻¹ (Fig. 4 for ‘Fuji’ 2) and previous studies reported slightly lower values of around 2·3 gC m⁻² d⁻¹ for this same period and for 3- and 4-year-old ‘Golden’ cultivar apple trees (Wibbe et al., 1993).

Furthermore, biomass allocation at the compartment scale was adequately simulated with a high allocation to the vegetative part at the beginning of the growth season, followed by a high allocation to fruits, and by an increasing allocation to the secondary growth of the woody parts from mid-summer, as observed in other studies (Abbot, 1984). This timing of carbon allocation among organs and compartments can be considered, at least partly, as an emergent property of the model resulting from the simulated carbon balance, even though it also results from model parameterization, in particular from the dynamics of leafy shoot growth. Indeed, for leafy shoots, the end of their development is directly driven by a thermal time parameter [dd_end; eqn (3)]. Furthermore, in the model, the amount of carbohydrate allocated to the woody part was large and represented more than 30 % of the total available carbon, which was in accordance with previous studies reporting values of around 40 % for 3- to 4-year-old apple trees displaying a low crop load (Cannel, 1985).

Fruit growth simulations suggest partial shoot autonomy for available carbon

At organ and shoot scales, the accuracy of model simulations was evaluated using observations derived from digitizing data and allowing the estimation of both mean and standard deviation values within trees. Despite its agronomical interest, within-tree fruit growth and weight variability has not been accurately estimated in many modelling studies. For instance, in L-Peach studies, in which carbon allocation between organs was simulated, simulated variability in organ weight has not been shown (e.g. Allen et al., 2005; Lopez et al., 2008) or its accuracy has not been statistically compared with observed data (Da Silva et al., 2011). For apple, fruit weight variability at harvest has been predicted based on a regression model between fruit characteristics at the beginning of the growth season and fruit dry weight at harvest, without any representation of plant architecture and functioning (Lötze and Bergh, 2004).

In QualiTree, this variability depends on local carbon availability, which results from the amount of light intercepted and carbon transport between PUs. Carbon exchange depending on the distance between sources and sinks was driven by a parameter (α) that was directly estimated on data in order to represent the observed variability. This can lead to wrong biological interpretations about the impact of distance on carbon allocation within plants, since tuning the value of α may lead to compensation for errors in the estimation of other model parameters. Nevertheless, in this study, results with high α and low α values were far enough from the observed variability (Fig. 8) to suggest that neither the common assimilate pool assumption (Heuvelink, 1995) nor a total autonomy of shoots or branches (Sprugel et al., 1991; Lacointe et al., 2004) is adequate to simulate the within-tree variability in fruit dry weight. This result is in line with previous studies showing that fruit growth and fruit set ability depend both on the available carbohydrate of its bearing shoot and on supplementary contribution of C from non-fruiting and vigorous shoots (Grappadelli et al., 1994; Walcroft et al., 2004).

It is noteworthy that the estimated α value was higher than that recorded in Peach (0·006; Mirás-Avalos et al., 2011), suggesting fewer exchanges of carbon between shoots in apple than in peach trees. This is likely to result from the sympodial growth pattern in apple that increases the number of branch connections within trees and could in turn increase the transport resistance in the vascular pathway (Fanwoua et al., 2014).

Since carbon transport between PUs depends on PU carbon supply, an accurate estimation of photosynthetic activity and light interception is necessary to assess the relevance of the C transport model. In our study, the light interception model of QualiTree previously validated on peach (Mirás-Avalos et al., 2011) was adapted to the case of apple, by allowing the definition of user-defined ellipsoids. This new shape appeared more appropriate for the inclusion of apple tree volumes since the shape implemented in QualiTree (two imbricate semi-ellipsoids) was adapted to open-vase trained trees. Even if the light interception model was not compared with more detailed light models or with measurements to be fully validated, it gave the silhouette to total leaf area ratio (STAR) values consistent with previous studies (Da Silva et al., 2014b), i.e. low STAR values for trees older than 3 years as well as a large proportion of shoots with low incident radiation (lower than 5 mol m⁻² d⁻¹ in our case; Fig. S3). In the light interception model used in this study, each shoot belonging to the same PU was assumed to be in the same light environment, which probably leads to some oversimplifications. Indeed, Da Silva et al. (2014b) have shown that the organization of laterals along 1-year shoots strongly affects the amount of intercepted radiation at the shoot scale.

Local carbon availability and hierarchical C-allocation rules are not fully adequate to simulate shoot growth polymorphism

In this modelling approach, three types of shoots corresponding to the types previously observed in apple (short, medium and long; Costes et al., 2003) were distinguished in the leafy shoot compartment of PUs. This adaptation of QualiTree was done because the simulations of shoot growth without introducing three shoot types were far away from the observed variability (data not shown). As a consequence, each kind of leafy shoot was described with a specific set of parameters. This specific set of parameters allowed the introduction of distinct potential growth values for each shoot type, based on previous observations of different numbers of preformed leaves within buds depending on their topological position in a branching system (Costes, 2003).

Nevertheless, and despite the introduction of three types of shoots, simulations of shoot leaf area tended to be of lower quality than those of fruit growth, with a tendency to overestimate final leaf area for each shoot type and underestimate intra-type variability. These discrepancies can mainly be interpreted as resulting from the formalisms used to simulate biomass allocation to leafy shoots in QualiTree, which are based on a hierarchical allocation model in which leafy shoots are satisfied before other sinks. This led to a large proportion of shoots reaching weights close to their maximal weight, which in turn reduced the intra-type variability and generated mean values for each shoot type higher than those observed. Modifying the hierarchical allocation model by modifying the compartment priorities during the growing season (Warren-Wilson, 1967) or by adding affinity factors for biomass allocation for each organ/compartment (Escobar-Guttiérrez et al., 1998; Guo et al., 2006; Kang et al., 2014) could be a possible way to introduce relative priorities among organs. The low variability presently simulated could also be explained by the fact that the different shoots of the same type display the same weight and sink strength at the beginning of the growing season. Adding some variability in the date of bud burst in MappleT according to microclimate conditions or bud topological position could be a way to refine the simulations in accordance with previous studies showing that the timing of shoot bud burst plays a key role in the establishment of shoot growth variability (Sachs, 1991; Novoplansky, 1996). Moreover, considering the non-trophic effects on shoot growth, in particular hormonal content or hydraulic proprieties, as previously done in other modelling approaches (Prusinkiewicz et al., 2009; Da Silva et al., 2011; Renton et al., 2012; Bennett et al., 2014), could also improve our model.

Towards the use of this model for multi-year simulations and simulation of agronomic scenarios

This study provides a first step in building a complete functional structural plant model (FSPM) for apple trees that simulates organ growth and variability by connecting two previously validated models, MappleT and QualiTree. Usually, model connection relies on compatible data as input and output of models. This is achieved in modelling platforms or software that allow efficient model connection through a common data format [L-system (Karwowski and Prusinkiewicz, 2004; Boudon et al., 2012); MTG (Pradal et al., 2008; Griffon and De Coligny, 2014); or the eXtended L-system language in GroImp (Hemmerling et al., 2008)]. In our case, the MTG formalism was chosen to represent the plant architectures simulated by MappleT. However, in QualiTree the inputs are recorded in a dedicated Mysql database independently of any modelling platform and plant architecture is based on an agronomical representation (PUs, compartment). This led us to develop a new adhoc procedure with Python to gather the different topological entities of the MTGs into QualiTree architecture representation.

Currently, the modelling approach presented in this study allows the simulation of apple tree growth during one growth cycle. In future investigations, retroaction loops between QualiTree and MappleT simulations should be included. To reach this objective further improvements are needed (1) to convert compartment and sub-compartment biomasses (i.e. stem shoot biomass) into dimensions (i.e. stem length) and (2) to integrate the impact of carbon allocation on bud fates in terminal and axillary position the following year. In its current version, QualiTree does not account for physiological processes that occur after harvest, such as leaf senescence and fall, or leaf carbon remobilization (Millard and Thomson, 1989). In forthcoming research, simulating tree carbon balance from harvest until the following spring will be of major importance for multi-year simulations since shoot growth in the early phase of tree development depends on reserve dynamics during the autumn–winter period (Hansen, 1971).

Finally, this study also shows that QualiTree is adequate to formalize the interactions between shoot growth demand, organ growth and photosynthesis activity under contrasted crop load conditions. First, a lower sensitivity of leaf area growth than of fruit growth to variation of crop load ratio was simulated, as observed in previous studies (Palmer et al., 1997). Second, values of mean fruit dry weight were close to observations. The decrease of 18 % in mean fruit dry weight simulated when crop load increased from 3·4 to 8·2 fruits m⁻² can be compared to the decrease of 15 % in mean fruit dry weight observed for a crop load ratio increase from 3·3 to 8·1 fruits m^–2 by Wünsche et al. (2000). Moreover, the larger difference in fruit weights between the smallest and biggest fruits when crop load increases, as previously observed in apple (Naor et al., 1997), was accurately reproduced. The model also simulated a feedback inhibition of photosynthesis by assuming that leaf assimilate production was regulated by the amount of reserve carbon in leaf as observed and simulated in previous studies (Wünsche et al., 2000; De Schepper and Steppe, 2011). This function was relevant to the simulation of feedback retroaction between sink demand at the shoot scale and photosynthesis activity under the contrasting crop load condition, even if the effect was slightly lower than expected [increase of 18 % in photosynthesis activity when crop load ratio increased from 3·3 to 8·1 in Wünsche et al. (2000); increase of 10 % in our study for the same range of crop load variation].

This ability of QualiTree to simulate crop load effects on tree growth opens the way to the use of this modelling approach to determine optimal crop load values for different contrasted architectures that could correspond to genotype characteristics and could be simulated by MappleT (Da Silva et al., 2014a). Nevertheless, before dealing with the genotypic variability and regarding the large number of parameters of QualiTree, a sensitivity analysis with global methods (FAST, Sobol, Morris, meta-modelling; Iooss and Lemaître, 2014) should be performed to define a set of parameters that display the greatest impact on model outputs.

CONCLUSIONS

In this study, QualiTree was used as a carbon allocation model to complement MappleT because its formalisms and its representation of architecture were simpler than those of other transport models. Even if some changes were performed to QualiTree to account for the shoot polymorphism in apple trees, this modelling choice appeared adequate to simulate carbon allocation at the plant and organ scales.

This study can provide useful information and background for plant modellers, horticulturists and botanists. For plant modellers, the model proves the necessity to take into account distance effects and organ growth dynamics when simulating biomass partitioning. For horticulturists, the model could be used as a tool to simulate the combined impact of crop load and architecture on fruit weight and yield. For botanists, it suggests hypotheses on putative physiological determinants of shoot polymorphism and fruit weight within-tree variability as well as a modelling approach suitable for the testing of hypotheses on the impact of carbon balance on plant growth and architecture.

ACKNOWLEDGEMENTS

The authors thank F. Lescourret for her valuable advice on QualiTree and the AFEF team technical staff for experimental data collection. This study was funded by the APMED project (European ARIMNET, 2011 call). D.DaS. received the support of the European Union, in the framework of the Marie-Curie FP7 COFUND People Programme, through the award of an AgreenSkills fellowship. The PhD scholarship of W.Y. was supported by the 948 project (2013-2016) from Chinese Ministry of Agriculture.

LITERATURE CITED