Stochastic modelling of tree architecture and biomass allocation: application to teak (Tectona grandis L. f.), a tree species with polycyclic growth and leaf neoformation

Stochastic modelling of tree architecture and biomass allocation: application to teak (Tectona... Abstract Background and aims For a given genotype, the observed variability of tree forms results from the stochasticity of meristem functioning and from changing and heterogeneous environmental factors affecting biomass formation and allocation. In response to climate change, trees adapt their architecture by adjusting growth processes such as pre- and neoformation, as well as polycyclic growth. This is the case for the teak tree. The aim of this work was to adapt the plant model, GreenLab, in order to take into consideration both these processes using existing data on this tree species. Methods This work adopted GreenLab formalism based on source–sink relationships at organ level that drive biomass production and partitioning within the whole plant over time. The stochastic aspect of phytomer production can be modelled by a Bernoulli process. The teak model was designed, parameterized and analysed using the architectural data from 2- to 5-year-old teak trees in open field stands. Key results Growth and development parameters were identified, fitting the observed compound organic series with the theoretical series, using generalized least squares methods. Phytomer distributions of growth units and branching pattern varied depending on their axis category, i.e. their physiological age. These emerging properties were in accordance with the observed growth patterns and biomass allocation dynamics during a growing season marked by a short dry season. Conclusions Annual growth patterns observed on teak, including shoot pre- and neoformation and polycyclism, were reproduced by the new version of the GreenLab model. However, further updating is discussed in order to ensure better consideration of radial variation in basic specific gravity of wood. Such upgrading of the model will enable teak ideotypes to be defined for improving wood production in terms of both volume and quality. Tectona grandis, GreenLab, polycyclism, neoformation, annual growth, dry biomass, stochastic model, plant architecture INTRODUCTION For a given genotype of tree, the variability observed in tree growth results largely from the variation and heterogeneity of environmental factors (e.g. light, temperature, access to nutrients, etc.) which affect biomass formation and allocation, as well as meristem functioning (Mathieu et al., 2008). Meristem activity may include periods of extension followed by rest periods between and within annual shoots (Barthélémy and Caraglio, 2007). The rest period depends on climatic conditions, particularly the length of dry seasons for tropical species. In tropical climates, a short dry season can involve polycyclic growth, which may be an adaptive process of drought period avoidance as in some Mediterranean species (Hover et al., 2017). This is the case for the teak tree (Tectona grandis Linn f., Lamiaceae), which has a large natural range in India, Laos, Myanmar and Thailand (Tewari, 1999; Hansen et al., 2015). This species grows naturally under a wide range of environmental conditions from dry areas (500 mm) to humid areas (5000 mm; Hansen et al., 2015). Teak wood is known for its high commercial value due to its mechanical quality, durability, exceptional aesthetics and resistance to fungus and insect attacks; teak is also used all over the world (Moya et al., 2014; Quintero-Méndez and Jerez-Rico, 2017). This species has therefore been the focus of numerous studies seeking to increase and improve plantation productivity (Kokutse et al., 2004, 2006) and optimize planting designs (Leroy et al., 2009). A recent study highlighted the relationships existing between primary and secondary growth and their consequences for wood properties of teak (Tondjo et al., 2014). Teak trees exhibits rhythmic growth, with one or two rest periods depending on the number of dry seasons. The teak trees are without leaves during the main dry season from November to March. In the studied trees, annual growth was polycyclic due to a short dry season from July to September. Another feature of teak growth is that each growth unit is constituted of some pre-formed leaves present at an embryonic stage in the terminal bud, followed by several neoformed leaves not present in the terminal bud (Tondjo, 2016). Beyond the usual statistical analyses and correlative models, functional–structural plant models (FSPMs) are promising tools for studying functional links between the various structural components described above and, more generally, plant development and growth variability considering the effect of environmental factors and growth strategies (Fourcaud et al., 2008). Although FSPMs have been widely developed over the last two decades, there are relatively few dedicated to trees and, moreover, generic enough to be applied to a wide range of species. Trees with their complex structure, their height and their long life span call for a reduction of structural complexity in FSPM definitions. Some approaches are restrictively applied to static tree structures, i.e. they consider the ecophysiological processes over a short period of time in which architectural development is neglected. This is the case of ECOPHYS developed to study the response of field crop species to environmental stress (Rauscher et al., 1990), EMILION dedicated to the analysis of the carbon balance within pine trees (Bosc, 2000) or SIMWAL developed to study biomass partitioning in young walnut trees (Balandier et al., 2000). In contrast, integrating plant models over the long term, e.g. several years for trees, necessitates considering ecophysiological growth processes and architectural development simultaneously. Meristems generate the plant architecture by the production of new organs and their expansion with regards to the available assimilates and local environmental conditions, and sometimes following complex ontogenetic rules, e.g. leaf neoformation or polycyclism (Barthélémy and Caraglio, 2007; de Reffye et al., 2012). Purely architectural models were developed on these bases, often using stochastic models to simulate the dynamics of the topological structure. The generic AMAPsim model is a good example of such an approach (Barczi et al., 2008). Other stochastic models were developed for specific plant species as MAppleT model (Costes et al., 2008) or for young Eucalyptus (Diao et al., 2012). Functional–structural plant models were built to simulate tree growth and carbon allocation within a dynamic architecture. This is the case of L-PEACH (Lopez et al., 2008), which is dedicated to a single fruit tree species, and LIGNUM (Perttunen et al., 1998), which is a more generic model for forest trees. In the functional–structural tree model ALMIS (Eschenbach, 2005), the development of the structure is also based on the allocation and use of resources, i.e. assimilates and nutrients, with regards to local environmental factors. However, the architectural rules considered in this model are very limited, which is a limitation for the simulation of trees with complex organization.. The GreenLab model is a discrete dynamic model, which simulates biomass production and allocation at the organ scale, taking into account the feedback effects of internal trophic competition on plant morphology (Yan et al., 2004). The particularity of this model is that it takes the form of simple mathematical equations that can be solved and parameterized easily and in a very short computer processing time. Unlike the FSPMs mentioned above, the production equation of GreenLab does not use an explicit description of the plant structure, which is time consuming, but only developmental and allometric rules that allow the evolution of organ number and size to be estimated at each growth step. This model has been applied to several annual plants (Guo et al., 2006), but also to some tree species such as beech (Letort et al., 2008), Cecropia (Letort et al., 2012) and Mongolian Scots pine (Wang et al., 2012). However, to our knowledge, there is no plant model that simultaneously takes into account architectural variability, including the stochastic developmental processes along with carbon production and allocation, in a compact mathematical form. This study sets out to integrate stochastic growth processes, and leaf neoformation and polycyclism rules into the source–sink-based GreenLab model, using existing architectural data for teak trees. MATERIALS AND METHODS Experimental set-up The study was undertaken at the Agbavé forest station (0°45’E, 6°43’N) in south-western Togo. The relative air humidity there is close to 83 % on average. Rainfall averages stand at between 1100 and 1400 mm per year, with a peak in June. The mean maximum temperature ranges between 29 and 36 °C, and the hottest months are February and March. The mean minimum temperature ranges between 20 and 36 °C, and the coolest months are July and August. There are two dry seasons, one from November to March and the other from July to September. The studied teaks from a Togolese provenance were in two plots, located close together in order to minimize environmental effects (soil and climate) on the studied variables. The 1-year-old seedlings were planted at a density of 2500 trees ha–1 (i.e. at a spacing of 2 × 2 m). The analysed teaks were 2 and 5 years old, with 10 and six individuals respectively, and had not flowered. The average tree heights were 218.85 ± 67.54 cm for the 2-year-old trees and 727.25 ± 88.61 cm for the 5-year-old trees. The mean diameters at 1.30 m were 2.77 ± 1.09 cm for the 2-year-old trees and 7.88 ± 1.84 cm for the 5-year-old trees. The architectural analysis of teak Our study was based on the concepts developed for architectural analysis (Barthélémy and Caraglio, 2007). The phytomer, the basic unit of plant structure, includes the node, the leaf, the internode and the associated axillary buds (Barlow, 1989). A growth unit is defined as the portion of an axis which develops during an uninterrupted period of extension and can easily be identified by a zone of short internodes with scaly leaves corresponding to the bud protective organs from which it derives (for teak, see Fig. 2B and C). An annual shoot is defined as the portion of stem which extended during a year. The studied teaks showed rhythmic growth with two cessation periods: one during the long dry season and one during the short dry season. Thus this growth pattern was called polycyclism. The ‘physiological age’ concept, based from botanist observation (Barthélémy and Caraglio, 2007), is used to classify branch typologies. It also expresses the degree of differentiation of the produced phytomers; as mentioned previously, there are thus four physiological ages to consider here, corresponding to the following axis categories: the main stem, the branches, the branchlets and the twigs (Fig. 2A). The 2-year-old trees were unbranched. Branching appeared from the third year of growth. The phyllotaxy is opposite and decussate. Measurements Development measurements. The trees were described phytomer by phytomer. Phytomers were counted per unit of growth to obtain the distributions of the number of phytomers per growth unit according to the axis categories. The number of growth units was counted for each annual shoot. The rate of polycyclism was defined by the number of bicyclic annual shoots on the total number of annual shoots. The polycyclism rate was computed for each axis category. As the phyllotaxy is opposite decussate, two axillary meristems are present on each phytomer. Axillary production occurrence (bud, dead bud, branch, branchlets or twigs) was reported according to node rank, from the bottom to the top of the growth unit. Tree topology, i.e. the relative position of nodes and axes, was recorded using Multi Tree Scale formalism (Godin and Caraglio, 1998) with AMAPstudio (Griffon and de Coligny, 2014). Growth measurements. On each growth unit, we recorded the length and basal diameter according to their respective axis category. The growth unit stems were dried at 103 ° for 48 h and the dry weight was measured. For the leaf measurements, two different equations were determined to estimate the single individual leaf area (s) and the dry weight (We) values from the length (L) and width (Wi) of the leaf blade: s = 0.60 × L × Wi and We = 0.004 × (L × Wi)1.11 (Tondjo et al., 2015). Geometrical parameters such as phyllotaxy, branching angles or organ allometries were also measured. Main GreenLab model concepts The assumption of the GreenLab model is that plant architecture results from meristem functioning, organogenesis and photosynthesis, biomass production and partitioning (de Reffye et al., 2008). We only considered above-ground growth modelling in this study. The rules of tree development are modelled by a dual-scale automaton (Yan et al., 2002; Xing et al., 2003), providing information about the evolution of organ numbers. Biomass assimilation is modelled as a function of total tree leaf area at time t [eqn (1)]. It is then allocated to the primary growth of phytomers and to the secondary growth of the stem proportionately to the demand of the leaf, internode and growth ring at each cycle of development. The organ dimensions are computed from rules of allometry. The model workflow is detailed as follows (see Fig. 1): the seed gives the initial pool of dry biomass used to build organs and the plant architecture at cycle 1. At each cycle i, the biomass is partitioned among competing organs in relation to plant demand, D(i), which depends on the number of organs, No, and their respective sink functions pa for leaves, pe for internodes (or primary growth) and pc for growth ring increment (or secondary growth). Biomass production depends on the leaf hydraulic resistance r, the total leaf area S and the production leaf area Sp that is defined as the whole-tree crown projection onto the ground (Cournède et al., 2008). The biomass produced at cycle i, Q(i), is stored in the common pool of carbon reserves. Lastly, at each cycle of development, the model outputs the weights of individual organs and of the whole plant. The 3-D structure is another optional output allowing 3-D visualization of plant architecture. The biomass allocated to the wood rings is computed according to two balanced modes: the classical pipe model usual for trees (Shinozaki et al., 1964), and a common pool distribution usual for smaller plants. The production equation is expressed as follows: Fig. 1. View largeDownload slide Morphology of 5-year-old teak trees. (A) Architecture with the four axis categories or physiological ages. (B) Bicyclic annual shoot: intra-annual growth rest phase and delimitation of two successive growth units (GU1 and GU2). (C) Monocyclic annual shoot: inter-annual growth rest phase and delimitation of two successive annual shoots (AS n – 1, AS n). Fig. 1. View largeDownload slide Morphology of 5-year-old teak trees. (A) Architecture with the four axis categories or physiological ages. (B) Bicyclic annual shoot: intra-annual growth rest phase and delimitation of two successive growth units (GU1 and GU2). (C) Monocyclic annual shoot: inter-annual growth rest phase and delimitation of two successive annual shoots (AS n – 1, AS n). Fig. 2. View largeDownload slide Schematic representation of a simulation of the GreenLab model: development and growth simulation cycle, the model outputs, field measurements, the inverse method and the hidden parameters computed from the model. *Specific implementations made in the teak model. Fig. 2. View largeDownload slide Schematic representation of a simulation of the GreenLab model: development and growth simulation cycle, the model outputs, field measurements, the inverse method and the hidden parameters computed from the model. *Specific implementations made in the teak model. Q(i)=E⋅Spr⋅(1−exp(−kε.Sp⋅∑m=(i−ta)+1max(i,ta)∑φ=1mxφNaφ(m)⋅∑j=itpaφ(j−m+1)⋅Q(j−1)D(j))) (1) where E stands for the environmental conditions (set to 1 in this work), ta is the leaf expansion duration, φ is the current physiological age (from 1 to maxφ equal to 4 in the case of teak) and Nφa (m) stands for the number of leaves of physiological age φ, emitted at age m. The ε factor, assumed to be constant, is the specific leaf mass (i.e. the ratio of leaf dry mass and leaf area). k is the light extinction coefficient of the Beer–Lambert law (de Wit, 1965). Plant demand, D(i), is defined by the sum of all organ o requests, obtained from the product of the number of organs No and their sink function po, for all physiological ages: D(i)=∑o,φ(∑j=1iNoφ(i−j+1)⋅poφ(j)) (2) Improvement of the GreenLab model for the case of teak tree: stochastic development, neoformation and polycyclic growth Teak shows an alternation between rest and extension periods of meristems in shoot construction. The meristem functioning can be treated as renewal processes. The number of phytomers on an axis shows variability, and can be modelled by a binomial distribution (de Reffye et al., 2012). Successive phytomer occurrences are considered as independent events, with a probability of occurring at each cycle of development. With rhythmic development, as in teak, two laws are associated with the duration of meristem extension and with the duration of meristem rest between two successive growth units, respectively. At each cycle of development (CD), the meristem does or does not produce a phytomer, with a growth probability according to a typical Bernoulli process (de Reffye et al., 2012). The meristem growth probability is denoted b. At a given CD, a failed Bernoulli process test leads to a void entity, meaning that no phytomer is generated. The distribution of the number of phytomers in a pre-formed part of a growth unit follows a positive binomial law. The distribution of the number of phytomers in a neoformed part of a growth unit follows a negative binomial law (de Reffye et al., 1991). Pre-formed and neoformed phytomers expand immediately until meristem breakdown. Thus, the total number of phytomers in the growth unit is a convolution of both laws (Guédon et al., 2006). The probability of neoformation occurrence is defined as a single ratio denoted P. For each axis category, the distributions of the number of phytomers of a growth unit followed a convolution between a positive binomial law (k1, b1) and a negative binomial law (k2, b2), which is the meristem activity cessation law. These laws drive the appearance of the new leaves and as a consequence the production of leaf area. By knowing, for each physiological age, the development probabilities b, branching probabilities a and neoformation probability P, it is possible to compute a theoretical chronological structure. The chronological structure is a conceptual frame, composed of the full structure with an explicit time representation, containing all entities with their probability of occurring, while the topological structure corresponds to the observable structure. The chronological structure is used to compute the organic series (Kang et al., 2016) as sets of consecutive organs on the virtual plant axis. Two embedded time scales were taken into account in the model: (1) the intra-annual time periods corresponding to the appearance of successive growth units (maximum of two growth units per year in teak); and (2) the phyllochron (i.e. the time between the appearance of two successive phytomers within a given growth unit), herein named the cycle of development (CD). Model calibration The key step for model applications is its calibration on real plants, by estimating the specific parameter from experimental data. Some model parameters can be measured, while some cannot be directly assessed from experimental observations. The parameter identification process requires three steps: (1) the analysis of plant development that gives the number of produced phytomers and the topological rules within the whole tree; (2) the building of target data with the organ dry weights; and, lastly, (3) fitting of the functional hidden parameters. The measured parameters are distributions of the number of phytomers per growth unit which defined the developmental parameters (in particular b and P probabilities). The branching probability (a) results also apply to the morphological observations. For the branching process, we consider that there is a potential number of axillary branches on each phytomer (usually one or two). Branching may occur with a branching probability, a, expressed as a single ratio between the number of branched nodes and the total number of nodes. The probability that the meristem never develops a branch is thus 1 – a. At each node, there are two axillary buds, the possibilities are: two branches (1,1), one branch (1,0) or zero branches (0,0). A coupling parameter was added, tuning the branching occurrence for the (1,1) and (1,0) cases. This coupling reflects the fact that, if an axillary bud develops on one node, the second bud also shows a significant chance of developing. When the plant architecture is stochastic, a key issue of the inverse method is defining the target data for fitting. The target data were the mean dry weights of leaves or internodes per growth unit called ‘compound organic series’. This type of target contains information on the source and sink dynamics, and consequently allows the source–sink function. Fitting one or more growth stages is called ‘single’ or ‘multi-’ fitting, respectively (Guo et al., 2006). Multi-fitting requires the organic series at several growth stages. For this study, two stages of observations were taken into account: 2-year-old trees that were unbranched and 5-year-old trees that were branched, with the structure of a young tree composed of four axis categories or ‘meristem physiological ages’. The hidden parameters have to be estimated by model inversion (Zhan et al., 2003). These are the sink functions (po) for leaves, internodes and growth ring, leaf hydraulic resistance (r) and the plant production area (Sp). The hidden parameters for the teak model are specified in Table 2. The value of the leaf specific mass ε (the ratio between dry weight and leaf area) was 0.015 g cm–2. The light extinction coefficient of the Beer–Lambert law, k, was set to k = 1. The allometry between the internode length and volume was fixed to an average value, with the same value on all internodes for a given growth unit. The seed weight or initial biomass was fixed at 1 g. Plant demand for secondary growth is proportional to the leaf number of the tree. Internode allocations to secondary growth are based on Pressler’s law with a variation coefficient λ (according to the pipe model assumptions as the surface section conservation; Kang et al., 2002). However, according to the young stages of the trees, we also consider a common pool ring distribution model. The balance between those two allocation models is defined by a coefficient λ, to be fitted with the diameter field measurements with the other functional parameters. For herbaceous species, the production leaf area (Sp) is a constant over their development, while for tree species, Sp varies during tree development. The surface of the crown is large at the beginning of growth and decreases before increasing with the development of the crown. A U-shaped function was thus used to model the evolution of Sp. Sp(x)=Smax(1−SF(x)−SminSmax−Smin)+Smin(SF(x)−SminSmax−Smin) (3) where: SF(x)=Smin(1−B(x)−BminBmax−Bmin)+Smax(B(x)−BminBmax−Bmin) (4) and: B(x)=((x−0.5)T)BaSp1(1−x−0.5T)BbSp1T (5) SF corresponds to the leaf area. Smax (maximum area) and Smin (minimum area) are fixed. The Beta law B(x) in a [1, T] interval has a minimum Bmin and a maximum Bmax depending on BaSp and BbSp. These parameters were computed from the generalized least squares inverse method which optimizes the fitting of mean dry weight of organs per growth unit simultaneously with the other source–sink parameters. Model simulation and output The rules of tree development are modelled by a dual-scale automaton (Yan et al., 2002; Xing et al., 2003). This stochastic automaton was used to simulate the tree structure. It simulated organogenesis through microstates (within a growth unit), macrostates (from one growth unit to the next on the axis) and the jump relationships between them (de Reffye et al., 2008). The different states of the automaton corresponded to the physiological age classes, and the transitions of the automaton corresponded to bud differentiation (Cournède et al., 2006). In our case, the jumps were conditional; they occurred according to the development probabilities given by the binomial laws for both micro- and macrostates. Specific individual teak trees were generated from model simulations. However, the visual aspect of the 3-D plants contained less information than the fitting of the organic series. In the 3-D tree structure construction, the internode length and growth ring were estimated from the simulated fresh internode volume. This volume is computed from the ratio between the available biomass and the basic specific gravity (Kollmann and Côté, 1968). Simulations were carried out with the GreenLab Matlab environment. Sixteen different individual teak trees were generated, at 5 years old, showing random properties related to the stochastic development laws (the negative binomial laws for pre-formed and neoformed leaves, polycyclism probability and branching probability). For each tree, side views from the 3-D output were generated, with leaves. The number of growth units and phytomers, and the respective internode volume and leaf areas were reported according to the phytomer physiological age and growth unit. RESULTS Modelling plant development The teak development pattern showed the elongation of pre-formed phytomers to be seasonal and synchronous at the start of elongation followed immediately by the elongation of neoformed phytomers. The cessation of pre-formed phytomer elongation was synchronous for all meristems. Neoformation followed a negative binomial law with a probability that varied depending on the rank of the growth unit and on the axis categories (see Table 1). The cessation of neoformed phytomer elongation was spread out over time. The leaf neoformation always occurred and the transition probability from pre-formation to neoformation is equal to 1. The number of phytomers of the first growth unit was higher than that of the second growth unit. The number of phytomers of the first growth unit decreased from trunk to twigs, while the number of phytomers of the second growth unit remained relatively constant. The number of pre-formed nodes was three for all growth units regardless of the axis categories (Fig. 3). Table 1. Annual growth in 5-year-old teak trees for different physiological ages: parameters of a positive binomial law for pre-formation (k1, b1) and parameters of a negative law for neoformation (k2, b2) Axis categories φ RGU NGU M V k1 b1 k2 b2 Main stem 1 1 210 9.60 13.10 3 1 7 0.49 1 2 150 5.00 6.10 3 1 3 0. 35 Branches 2 1 149 8.20 10.40 3 1 7 0.45 2 2 70 5.40 4.30 3 1 3 0.44 Branchlets 3 1 220 5.70 5.10 3 1 4 0.41 3 2 77 4.80 3.60 3 1 2 0.48 Twigs 4 1 100 4.20 1.90 3 1 3 0.30 Axis categories φ RGU NGU M V k1 b1 k2 b2 Main stem 1 1 210 9.60 13.10 3 1 7 0.49 1 2 150 5.00 6.10 3 1 3 0. 35 Branches 2 1 149 8.20 10.40 3 1 7 0.45 2 2 70 5.40 4.30 3 1 3 0.44 Branchlets 3 1 220 5.70 5.10 3 1 4 0.41 3 2 77 4.80 3.60 3 1 2 0.48 Twigs 4 1 100 4.20 1.90 3 1 3 0.30 Φ, physiological age; RGU, growth unit rank on the annual shoot, 1 for the first growth unit, 2 for the second growth unit; NGU, number of GUs of all studied trees; M, mean number of phytomers per GU; V, variance of the distribution of number of phytomers per GU. View Large Table 1. Annual growth in 5-year-old teak trees for different physiological ages: parameters of a positive binomial law for pre-formation (k1, b1) and parameters of a negative law for neoformation (k2, b2) Axis categories φ RGU NGU M V k1 b1 k2 b2 Main stem 1 1 210 9.60 13.10 3 1 7 0.49 1 2 150 5.00 6.10 3 1 3 0. 35 Branches 2 1 149 8.20 10.40 3 1 7 0.45 2 2 70 5.40 4.30 3 1 3 0.44 Branchlets 3 1 220 5.70 5.10 3 1 4 0.41 3 2 77 4.80 3.60 3 1 2 0.48 Twigs 4 1 100 4.20 1.90 3 1 3 0.30 Axis categories φ RGU NGU M V k1 b1 k2 b2 Main stem 1 1 210 9.60 13.10 3 1 7 0.49 1 2 150 5.00 6.10 3 1 3 0. 35 Branches 2 1 149 8.20 10.40 3 1 7 0.45 2 2 70 5.40 4.30 3 1 3 0.44 Branchlets 3 1 220 5.70 5.10 3 1 4 0.41 3 2 77 4.80 3.60 3 1 2 0.48 Twigs 4 1 100 4.20 1.90 3 1 3 0.30 Φ, physiological age; RGU, growth unit rank on the annual shoot, 1 for the first growth unit, 2 for the second growth unit; NGU, number of GUs of all studied trees; M, mean number of phytomers per GU; V, variance of the distribution of number of phytomers per GU. View Large Fig. 3. View largeDownload slide Distribution of the number of phytomers of the first and second growth units of bicyclic annual shoots for the main stem (A and B), for branches (C and D), for branchlets (E and F) and for twigs (G) for 5-year-old trees. Observed values (circles) and theoretical values (lines). Fig. 3. View largeDownload slide Distribution of the number of phytomers of the first and second growth units of bicyclic annual shoots for the main stem (A and B), for branches (C and D), for branchlets (E and F) and for twigs (G) for 5-year-old trees. Observed values (circles) and theoretical values (lines). At the beginning of development, the first and second growth units were synchronous. The leaves were photosynthetically active over 36 CDs. The rest period was estimated at one CD. The polycyclism rate was 70 % for the main stem, 50 % for branches, 35 % for branchlets and 0 % for twigs. Thus, the polycyclism rate decreased with the physiological age of the meristem. A phytomer generally bore two branches of the same category. Only the first growth unit was branched, with a probability of 0.054 for the trunk, 0.091 for the branches and 0.055 for the branchlets. The twigs were unbranched. The pattern of tree development is shown in Fig. 4. The chronological structure shows that the growth of teak is rhythmic, with a large number of cycles of development in rest activity between each growth unit. The polycyclic growth is mostly frequent for the main stem and branches. The branching is rhythmic and differed from 1 year with the occurrence of one or two branches per node. Only the first growth unit is branched. The main stem bears all axis categories (Fig. 4A). The topological structure corresponds to the realized phytomers (Fig. 4B). Fig. 4. View largeDownload slide Chronological structure with growth and rest periods of the meristems (A) and topological structure show observable structure (B) for the 5-year-old teak tree. The stochastic constructions are simulated by the botanical automaton whose rules are derived from observations of the behaviour of the different axes: main stem (blue), branches (green), branchlets (red) and twigs (yellow). In (A), the rest cycles of development, which are numerous, are shown in white. In (B), the rest cycles of development are not present. Fig. 4. View largeDownload slide Chronological structure with growth and rest periods of the meristems (A) and topological structure show observable structure (B) for the 5-year-old teak tree. The stochastic constructions are simulated by the botanical automaton whose rules are derived from observations of the behaviour of the different axes: main stem (blue), branches (green), branchlets (red) and twigs (yellow). In (A), the rest cycles of development, which are numerous, are shown in white. In (B), the rest cycles of development are not present. Modelling biomass production The U-shaped curve describing the evolution of Sp depended on two parameters, BaSp and BbSp [see eqn (5)], computed with the other source–sink parameters (Table 2). Over the first years of growth, Sp decreased until it reached a minimal value. Then Sp increased exponentially from the third year of development until it reached a constant value (Fig. 5). Table 2. Hidden parameters of 5-year-old teak trees computed from multi-stage fitting Q0 Initial biomass 1 g r Leaf hydraulic resistance 50 Spmax U Sp curve maximum production leaf area 15 774 cm2 Spmin U Sp curve minimum production leaf area 1000 cm2 BaSp U Sp curve Beta law first coefficient 1.19 BbSp U Sp curve Beta law first coefficient 2.33 λ Growth ring model coefficient 0.17 p1a (main stem) Leaf sink strength (physiological age 1) 1 p2a (branches) Leaf sink strength (physiological age 2) 0.78 p3a (branchlets) Leaf sink strength (physiological age 3) 0.62 p4a (twigs) Leaf sink strength (physiological age 4) 0.59 p1e (main stem) Internode sink strength (physiological age 1) 7.2 p2e (branches) Internode sink strength (physiological age 2) 4.2 p3e (branchlets) Internode sink strength (physiological age 3) 5.5 p4e (twigs) Internode sink strength (physiological age 4) 1.5 p1c (main stem) Growth ring sink strength (for all physiological ages) 1.7 vp1c (main stem) Growth ring sink variation coefficient (physiological age 1) 1 vp2,3,4c (branches, branchlets and twigs) Growth ring sink variation coefficient (physiological age 2, 3 and 4) 0.7 Q0 Initial biomass 1 g r Leaf hydraulic resistance 50 Spmax U Sp curve maximum production leaf area 15 774 cm2 Spmin U Sp curve minimum production leaf area 1000 cm2 BaSp U Sp curve Beta law first coefficient 1.19 BbSp U Sp curve Beta law first coefficient 2.33 λ Growth ring model coefficient 0.17 p1a (main stem) Leaf sink strength (physiological age 1) 1 p2a (branches) Leaf sink strength (physiological age 2) 0.78 p3a (branchlets) Leaf sink strength (physiological age 3) 0.62 p4a (twigs) Leaf sink strength (physiological age 4) 0.59 p1e (main stem) Internode sink strength (physiological age 1) 7.2 p2e (branches) Internode sink strength (physiological age 2) 4.2 p3e (branchlets) Internode sink strength (physiological age 3) 5.5 p4e (twigs) Internode sink strength (physiological age 4) 1.5 p1c (main stem) Growth ring sink strength (for all physiological ages) 1.7 vp1c (main stem) Growth ring sink variation coefficient (physiological age 1) 1 vp2,3,4c (branches, branchlets and twigs) Growth ring sink variation coefficient (physiological age 2, 3 and 4) 0.7 The leaf areas are fixed Smin = 1000 cm2 and Smax = 15 000 cm2. λ is a coefficient which corresponds to the ratio of ‘pipe model’ and ‘common pool distribution’. View Large Table 2. Hidden parameters of 5-year-old teak trees computed from multi-stage fitting Q0 Initial biomass 1 g r Leaf hydraulic resistance 50 Spmax U Sp curve maximum production leaf area 15 774 cm2 Spmin U Sp curve minimum production leaf area 1000 cm2 BaSp U Sp curve Beta law first coefficient 1.19 BbSp U Sp curve Beta law first coefficient 2.33 λ Growth ring model coefficient 0.17 p1a (main stem) Leaf sink strength (physiological age 1) 1 p2a (branches) Leaf sink strength (physiological age 2) 0.78 p3a (branchlets) Leaf sink strength (physiological age 3) 0.62 p4a (twigs) Leaf sink strength (physiological age 4) 0.59 p1e (main stem) Internode sink strength (physiological age 1) 7.2 p2e (branches) Internode sink strength (physiological age 2) 4.2 p3e (branchlets) Internode sink strength (physiological age 3) 5.5 p4e (twigs) Internode sink strength (physiological age 4) 1.5 p1c (main stem) Growth ring sink strength (for all physiological ages) 1.7 vp1c (main stem) Growth ring sink variation coefficient (physiological age 1) 1 vp2,3,4c (branches, branchlets and twigs) Growth ring sink variation coefficient (physiological age 2, 3 and 4) 0.7 Q0 Initial biomass 1 g r Leaf hydraulic resistance 50 Spmax U Sp curve maximum production leaf area 15 774 cm2 Spmin U Sp curve minimum production leaf area 1000 cm2 BaSp U Sp curve Beta law first coefficient 1.19 BbSp U Sp curve Beta law first coefficient 2.33 λ Growth ring model coefficient 0.17 p1a (main stem) Leaf sink strength (physiological age 1) 1 p2a (branches) Leaf sink strength (physiological age 2) 0.78 p3a (branchlets) Leaf sink strength (physiological age 3) 0.62 p4a (twigs) Leaf sink strength (physiological age 4) 0.59 p1e (main stem) Internode sink strength (physiological age 1) 7.2 p2e (branches) Internode sink strength (physiological age 2) 4.2 p3e (branchlets) Internode sink strength (physiological age 3) 5.5 p4e (twigs) Internode sink strength (physiological age 4) 1.5 p1c (main stem) Growth ring sink strength (for all physiological ages) 1.7 vp1c (main stem) Growth ring sink variation coefficient (physiological age 1) 1 vp2,3,4c (branches, branchlets and twigs) Growth ring sink variation coefficient (physiological age 2, 3 and 4) 0.7 The leaf areas are fixed Smin = 1000 cm2 and Smax = 15 000 cm2. λ is a coefficient which corresponds to the ratio of ‘pipe model’ and ‘common pool distribution’. View Large Fig. 5. View largeDownload slide Theoretical U-shaped curve for the production leaf area (Sp, area) according to the cycle of development (CD) in the 5-year-old teak trees. One year corresponds to 37 CDs. Fig. 5. View largeDownload slide Theoretical U-shaped curve for the production leaf area (Sp, area) according to the cycle of development (CD) in the 5-year-old teak trees. One year corresponds to 37 CDs. Measurement data were compared with the fitted parameters at the tree scale in order to evaluate the model (Fig. 6). The fitted values are consistent with measured dry biomass for the main stem of 2-year-old trees, as well as for the first growth units of each annual shoots of all axes. However, they are less consistent regarding the dry biomass of the second growth units for all axis categories of the 5-year-old trees. This can be explained by the low number of second growth units found within the trees that reduced the number of samples in this category (Table 1). Fig. 6. View largeDownload slide Comparisons between measured and fitted results at tree scale for teak. Annual internode dry mass per axis categories for 2- and 5-year-old trees: first growth unit (A) and second growth unit (B); circles denote the measured results and the black curve shows the fitted results. Fig. 6. View largeDownload slide Comparisons between measured and fitted results at tree scale for teak. Annual internode dry mass per axis categories for 2- and 5-year-old trees: first growth unit (A) and second growth unit (B); circles denote the measured results and the black curve shows the fitted results. The values of the organ sink strength are relative to the values of the sink strength of the main stem leaf set to 1. The leaf sink strength decreased from the branches to the twigs (Table 2). The sink strength of the internode tended to decrease from the trunk to the twigs, except for the branchlets, where it was higher than for the branches. For the leaf, internode and growth ring compartments, the biomass null value corresponded to the rest period: teak loses its leaves during the dry season. At the start of growth, a peak in biomass production was seen at the first cycle of development corresponding to the previous year’s carbon reserves (Fig. 7). Generally, biomass production increased from the third year with the appearance of the first branch. The biomass produced by the leaves was low at the beginning of their elongation and increased with crown development over the year when the leaf area increases. The decreased biomass over the period of maximal production correspond to the intra-annual growth cessation between the successive growth units. The biomass was also higher for the first growth flush than for the second one (Fig. 7A). The biomass allocated to internodes (i.e. primary growth) was low and relatively constant over the three growth years on all axes (Fig. 7B). The biomass allocated to growth rings (i.e. secondary growth) was higher than that allocated to the leaves and to the internodes, increased over the first three growth years and remained constant afterwards (Fig. 7C). It is interesting to note that the wood ring biomass allocation was distributed from the common pool (with λ = 0.17, Table 2), and not the pipe model (λ close to 1), perhaps related to the fact that the plants are still young and of small height. Fig. 7. View largeDownload slide Dry biomass allocation for each organ compartment: leaves (A), internodes (B, i.e. primary growth) and growth rings (C, i.e. secondary growth), according to the cycle of development (CD) during the first 5 years of growth. One year correspond to 37 CDs which correspond to 36 CDs of the active period with 1 CD for the rest period. Fig. 7. View largeDownload slide Dry biomass allocation for each organ compartment: leaves (A), internodes (B, i.e. primary growth) and growth rings (C, i.e. secondary growth), according to the cycle of development (CD) during the first 5 years of growth. One year correspond to 37 CDs which correspond to 36 CDs of the active period with 1 CD for the rest period. A strong increase in biomass supply (Q) was observed when branching occurred in the third year of growth. The minimum biomass production value corresponded to the reserves brought into play during the dry season (Fig. 8A). The plant demand trends showed two minimum values per year: the first corresponded to the annual shoot elongation rest period during the long dry season, and the second corresponded to the intra-annual shoot rest period during the short dry season (Fig. 8B). The supply to demand ratio (Q/D) decreased with the increase in plant demand D resulting from tree crown development. The Q/D was higher in the second growth year since plant demand still remained low while the biomass supply was relatively high (Fig. 8C). Fig. 8. View largeDownload slide Variation in total plant dry biomass supply or Q (A), in plant dry biomass demand or D (B), and in the supply to demand ratio or Q/D (C), depending on the number of cycles of development over the first 6 years growth of teak. Fig. 8. View largeDownload slide Variation in total plant dry biomass supply or Q (A), in plant dry biomass demand or D (B), and in the supply to demand ratio or Q/D (C), depending on the number of cycles of development over the first 6 years growth of teak. Simulation results Using the calibrated model, the tree growth and development processes were simulated for the 2-year-old teaks (Fig. 9). For the 16 virtually simulated 5-year-old teaks (Fig. 10), it could be seen from a qualitative point of view that the observed variability of the tree structures resulted of the stochastic behaviour of the model. Significant leaf area changes were also found between the simulated individuals, especially when comparing structures with a low amount of branching and those which were highly branched. From a quantitative point of view, from these virtual trees, we extracted the number of phytomers of the first and second growth units for the main stem as an example (Fig. 11). The distribution of the phytomer number per growth unit was similar to the observed numbers (Fig. 3). In order to illustrate the structural variabilty and the relationship between the structure and the organ dimensions, we selected three specific virtual individuals corresponding to medial and extreme structure complexity. Individual 3 had the largest number of phytomers with simultaneously the lowest total leaf area and individual leaf area, and the lowest total wood volume (Table 3). Individual 2 had the median value for the number of phytomers, and also showed the median value for the total wood volume and the individual leaf area, while the total leaf area value was higher (Table 3). Individual 15 had the smallest number of phytomers and the highest individual leaf area. The total wood volume value was average and the total leaf area value was high (Table 3). As shown in Fig. 10, this last individual only bore one branch. Fig. 9. View largeDownload slide Three architectures of 2-year-old teak trees and photographs of one real tree. The scale bar is 2 m. Fig. 9. View largeDownload slide Three architectures of 2-year-old teak trees and photographs of one real tree. The scale bar is 2 m. Fig. 10. View largeDownload slide Ten architectures from the sixteen 5-year-old teak simulations illustrating the stochastic aspect on the plant architectures. Plant vizualization is at 180 cycles of development. Individuals 3 and 15 show the largest and smallest number of phytomers, respectively. Individual 2 is closest to the median case. The scale bar is 2 m. Fig. 10. View largeDownload slide Ten architectures from the sixteen 5-year-old teak simulations illustrating the stochastic aspect on the plant architectures. Plant vizualization is at 180 cycles of development. Individuals 3 and 15 show the largest and smallest number of phytomers, respectively. Individual 2 is closest to the median case. The scale bar is 2 m. Fig. 11. View largeDownload slide Simulations of distributions of the number of phytomers for the first t (A) and second growth unit (B) for the main stem of 5-year-old teak trees. Fig. 11. View largeDownload slide Simulations of distributions of the number of phytomers for the first t (A) and second growth unit (B) for the main stem of 5-year-old teak trees. Table 3. Mean (s.d.) of the total number of phytomers, total wood volume (cm3), total leaf area (cm2) and individual leaf area (cm2) estimated from the stochastic simulation of 16 individuals Total phytomer number Total wood volume (cm3) Total leaf area (cm2) Individual leaf area (cm2) 16 virtual trees 180 (62) 13 784 (641) 105 373 (14 855) 681 (351) Individual 3 272 12 992 74 245 272 Individual 2 199 13 933 116 324 584 Individual 15 66 13 710 112 297 1703 Total phytomer number Total wood volume (cm3) Total leaf area (cm2) Individual leaf area (cm2) 16 virtual trees 180 (62) 13 784 (641) 105 373 (14 855) 681 (351) Individual 3 272 12 992 74 245 272 Individual 2 199 13 933 116 324 584 Individual 15 66 13 710 112 297 1703 Three individuals with a particular behaviour were noted. View Large Table 3. Mean (s.d.) of the total number of phytomers, total wood volume (cm3), total leaf area (cm2) and individual leaf area (cm2) estimated from the stochastic simulation of 16 individuals Total phytomer number Total wood volume (cm3) Total leaf area (cm2) Individual leaf area (cm2) 16 virtual trees 180 (62) 13 784 (641) 105 373 (14 855) 681 (351) Individual 3 272 12 992 74 245 272 Individual 2 199 13 933 116 324 584 Individual 15 66 13 710 112 297 1703 Total phytomer number Total wood volume (cm3) Total leaf area (cm2) Individual leaf area (cm2) 16 virtual trees 180 (62) 13 784 (641) 105 373 (14 855) 681 (351) Individual 3 272 12 992 74 245 272 Individual 2 199 13 933 116 324 584 Individual 15 66 13 710 112 297 1703 Three individuals with a particular behaviour were noted. View Large DISCUSSION Growth and development of teak Annual growth of teak (Tectona grandis L. f.) is polycyclic, with each growth unit including three pre-formed leaves and a number of neoformed leaves that varied with the growth unit rank (first or second growth unit) and with the axis category. The rhythmicity is directly linked to the climatic seasonality especially regarding growth rest in the summer season. In fact, no polycyclic growth was observed for teaks growing in Indonesia where there is no short dry season (Leroy et al., 2009). In this case, the growth is continuous over the growing season. Our observations and growth data in Togo suggest that the rest period is a strategy of avoidance of the dry period. The pre-formation and neoformation of organs has been observed in numerous tree species with a rhythmic growth (Magnin et al., 2012). In teak, the number of neoformed phytomers varies between the first and the second growth unit (when it is present) of annual shoots according to axis categories, while the number of pre-formed phytomers is low and constant. The low number of pre-formed leaves seems to be linked to the large size of leaves. This relationship has been mentioned in many tropical evergreen tree species (Gill and Tomlinson, 1971), but never in a deciduous species such as teak. Evaluation of the GreenLab model on teak data The GreenLab model has been adapted to simulate more complex architectural properties, in particular neoformation and polycyclic growth, which are encountered on teak growing in Togo. These characteristics were eventually successfully reproduced by this new version of the model using stochastic functions linked to a binomial distribution. When applied to trees, the advantage of multi-fitting, i.e. fitting two differentiation stages in parallel, is notable compared with other plants showing lower growth variability. By fitting several stages in parallel, we expect to extract a set of endogenous parameters representative of the species. Simulations using the parameter values estimated from the multi-fitting process provide the dynamics of the different biomass compartments (i.e. leaf, internode and growth ring) and the biomass supply to demand ratio (Q/D) for each measured plant. The plant demand decreases during the intra-annual growth, stops and then increases with the emergence of leaves of the second growth unit. This trend confirmed a theoretical study on the rhythmic appearance in plants (Mathieu et al., 2008). Our results showed that the variation in Q/D reflected growth rhythmicity. Biomass production was very low during the early years of growth (i.e. from 36 to 72 CDs) since the plant organ dimensions were small. This corresponded to the tree establishment phase. The increase in biomass supply and biomass demand over the third growing year was linked to occurrence of branching. The Q/D peak observed in the third year of growth corresponded to high biomass supply and low biomass demand during the third year. In the following year of growth, branching complexity increased together with plant demand, and thus Q/D decreased. Such relationships between an increase in Q/D and branching expression were also observed in Cecropia sciadophylla (Letort et al., 2012). In the case of teak, the biomass allocated to the growth ring was low at the beginning of the growing season because biomass was first allocated to organ elongation, and it was higher at the end of the growing season when the leaves were fully elongated. Thus, these results corroborate the existence of interactions between the plant structure and its strategy for biomass allocation. Stochastic modelling highlighted the variability of tree behaviour within a tree population. Some less branched trees showed large leaves and a large total leaf area despite a small number of phytomers. In teak, the range of estimated individual leaf areas from the product of leaf length and width is from 200 to 1500 cm2 (Tondjo et al., 2015). On the other hand, some other trees that were the most branched had a large number of phytomers and small leaves. These results highlighting the variability of biomass partitioning to leaves or to stems might explain the variability of the architecture within individuals. Production leaf area The production leaf area (Sp) is an important parameter introduced in the GreenLab model in order to extend the notion of leaf area index (LAI; usually used in crop models) to individual trees. Sp is constant in herbaceous crops with a homogeneous leaf area distribution, and corresponds to the unit surface area of soil considered in the estimation of LAI (Guo et al., 2006). In contrast, Sp varies with the increase of the crown size when it is applied on a single tree. This evolution was modelled here as a U-shaped function. The Sp decreased during the first two growing years due to leaf self-shading, and increased from the third year of growth due to crown expansion. Sp exponentially increased during the third and fourth years of tree development, and remained constant when the crown volume reached a threshold corresponding to equilibrium. The U-shaped function allows the GreenLab model to be used on other tree species. Focus on the wood-specific gravity The basic specific gravity allows computation of the fresh stem volume from the dry mass. Such information, which is easy to measure, is a key point in the GreenLab model as it is considered to calculate wood ring width, and thus the increment in growth unit volumes, with regard to the locally available dry matter. In a previous study carried out in Togo, the variability of teak architecture and wood properties was investigated with regard to thinning practices and provenance. In particular, the authors showed that radial variation in wood properties was linked to variability in tree ontogeny (Tondjo et al., 2014). Developing an FSPM model that considers both the determinism of dry biomass formation and allocation, and tree stochastic development could be of interest for analysing and explaining the correlation between wood radial variations and tree architectural development. Up to now, in our models, the green volume of tree axes had been computed from the ratio between the biomass supply and a constant value of the basic specific gravity (Gb), although this wood property is known to vary depending on the cambial age (for Gb values in teak, see Pérez and Kanninen, 2005; Tondjo et al., 2014). A future improvement to the model will be to take into account the changes in Gb linked to the cambial age (i.e. radial variation). Given the amount of dry biomass (carbon allocation), the Gb values can be used to calculate the ring width increment for each growth unit, and thus the total volume of green wood at any growth stage. The Gb is an important indicator of timber mechanical properties, and knowledge of its variation is a critical point for wood quality optimization. Such an improvement in the GreenLab model will thus be useful for analysing the relationships existing between tree architecture and wood quality in different silvicultural practices and environmental contexts. Positioning Greenlab in the FSPM world Until now, the GreenLab application on stochastic development had been described for some species with continuous growth (Diao et al., 2012; Kang et al. 2016, Vavitsara et al., 2017). In the present study, tree development was modelled and described numerically as an alternation of extension and rest periods, taking into consideration leaf neoformation and polycyclic growth processes. This is a significant evolution that allows more complex botanical laws to be taken into account. Plant architecture appears as the result of source–sink balances and is impacted by endogenous processes and exogenous constraints due to the environment (Barthélémy and Caraglio, 2007). Consequently, GreenLab will be able to provide a better understanding of the intraspecific variability and interaction between tree architecture and its environment, and the morphological adjustments to climate changes. There are few existing tree structural–functional models that integrate detailed botanical rules in order to understand architectural plasticity. Moreover, most of them are dedicated to specific plant species, e.g. kiwifruit vine (Cieslak et al., 2011), peach trees with L-PEACH (Lopez et al., 208) or apple trees with MAppleT (Costes et al., 2008; Migault et al. 2017). The LIGNUM model (Perttunen et al., 1998) is a more generic FSPM that can be applied to different tree species. However, these FSPMs are based on an explicit description of the plant topology using computer algorithms, which is very time consuming even to simulate one single tree. This calculation cost is a major limitation to simulate forests or orchards, which necessitates using powerful and expansive computers and/or developing specific fast algorithms (Han et al., 2017). The main advantage of GreenLab is its mathematical formulation that allows direct and fast calculation. Applications to large vegetation scales have been already carried out, with satisfactory results (Feng et al., 2012, 2014). CONCLUSION The presented version of the GreenLab model integrates new development processes such as the leaf pre-/neoformation and polycyclic growth that allow simulation of complex tree structures and their architectural plasticity. The first application to teak (Tectona grandis L. f.) data collected in Togo shows promising results and opens up interesting perspectives for studying trees and forest adaptation to climate change. The model also provides quantitative key insights as a first step to proposing plant ideotypes. Application to forest trees should integrate more information on basic specific gravity and its intraspecific variability, as this trait is fully correlated to several wood quality factors, e.g. linked to mechanical performances. ACKNOWLEDGEMENTS The authors thank the owner of the experimental plots, Professor Messanvi GBEANSSOR at the University of Lomé, and the Lomé University students for their help in the teak field measurements. The authors are grateful the two anonymous reviewers for critical and valuable comments of the manuscript. This work was supported, in part, by the AUF-PCSI programme, by a grant from CIRAD AIRD and by AMAP [Botany and Computational Plant Architecture, joint research unit which associates CIRAD (UMR51), CNRS (UMR5120), INRA (UMR931), IRD (2M123) and Montpellier University (UM27)] http://amap.cirad.fr/. LITERATURE CITED Balandier P , Lacointe A , Le Roux X , Sinoquet H , Cruiziat P . 2000 . 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Beijing, China : Springer/Tsinghua University Press . 236 – 249 . © The Author(s) 2018. Published by Oxford University Press on behalf of the Annals of Botany Company. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com. This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices) http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Annals of Botany Oxford University Press

Stochastic modelling of tree architecture and biomass allocation: application to teak (Tectona grandis L. f.), a tree species with polycyclic growth and leaf neoformation

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Abstract

Abstract Background and aims For a given genotype, the observed variability of tree forms results from the stochasticity of meristem functioning and from changing and heterogeneous environmental factors affecting biomass formation and allocation. In response to climate change, trees adapt their architecture by adjusting growth processes such as pre- and neoformation, as well as polycyclic growth. This is the case for the teak tree. The aim of this work was to adapt the plant model, GreenLab, in order to take into consideration both these processes using existing data on this tree species. Methods This work adopted GreenLab formalism based on source–sink relationships at organ level that drive biomass production and partitioning within the whole plant over time. The stochastic aspect of phytomer production can be modelled by a Bernoulli process. The teak model was designed, parameterized and analysed using the architectural data from 2- to 5-year-old teak trees in open field stands. Key results Growth and development parameters were identified, fitting the observed compound organic series with the theoretical series, using generalized least squares methods. Phytomer distributions of growth units and branching pattern varied depending on their axis category, i.e. their physiological age. These emerging properties were in accordance with the observed growth patterns and biomass allocation dynamics during a growing season marked by a short dry season. Conclusions Annual growth patterns observed on teak, including shoot pre- and neoformation and polycyclism, were reproduced by the new version of the GreenLab model. However, further updating is discussed in order to ensure better consideration of radial variation in basic specific gravity of wood. Such upgrading of the model will enable teak ideotypes to be defined for improving wood production in terms of both volume and quality. Tectona grandis, GreenLab, polycyclism, neoformation, annual growth, dry biomass, stochastic model, plant architecture INTRODUCTION For a given genotype of tree, the variability observed in tree growth results largely from the variation and heterogeneity of environmental factors (e.g. light, temperature, access to nutrients, etc.) which affect biomass formation and allocation, as well as meristem functioning (Mathieu et al., 2008). Meristem activity may include periods of extension followed by rest periods between and within annual shoots (Barthélémy and Caraglio, 2007). The rest period depends on climatic conditions, particularly the length of dry seasons for tropical species. In tropical climates, a short dry season can involve polycyclic growth, which may be an adaptive process of drought period avoidance as in some Mediterranean species (Hover et al., 2017). This is the case for the teak tree (Tectona grandis Linn f., Lamiaceae), which has a large natural range in India, Laos, Myanmar and Thailand (Tewari, 1999; Hansen et al., 2015). This species grows naturally under a wide range of environmental conditions from dry areas (500 mm) to humid areas (5000 mm; Hansen et al., 2015). Teak wood is known for its high commercial value due to its mechanical quality, durability, exceptional aesthetics and resistance to fungus and insect attacks; teak is also used all over the world (Moya et al., 2014; Quintero-Méndez and Jerez-Rico, 2017). This species has therefore been the focus of numerous studies seeking to increase and improve plantation productivity (Kokutse et al., 2004, 2006) and optimize planting designs (Leroy et al., 2009). A recent study highlighted the relationships existing between primary and secondary growth and their consequences for wood properties of teak (Tondjo et al., 2014). Teak trees exhibits rhythmic growth, with one or two rest periods depending on the number of dry seasons. The teak trees are without leaves during the main dry season from November to March. In the studied trees, annual growth was polycyclic due to a short dry season from July to September. Another feature of teak growth is that each growth unit is constituted of some pre-formed leaves present at an embryonic stage in the terminal bud, followed by several neoformed leaves not present in the terminal bud (Tondjo, 2016). Beyond the usual statistical analyses and correlative models, functional–structural plant models (FSPMs) are promising tools for studying functional links between the various structural components described above and, more generally, plant development and growth variability considering the effect of environmental factors and growth strategies (Fourcaud et al., 2008). Although FSPMs have been widely developed over the last two decades, there are relatively few dedicated to trees and, moreover, generic enough to be applied to a wide range of species. Trees with their complex structure, their height and their long life span call for a reduction of structural complexity in FSPM definitions. Some approaches are restrictively applied to static tree structures, i.e. they consider the ecophysiological processes over a short period of time in which architectural development is neglected. This is the case of ECOPHYS developed to study the response of field crop species to environmental stress (Rauscher et al., 1990), EMILION dedicated to the analysis of the carbon balance within pine trees (Bosc, 2000) or SIMWAL developed to study biomass partitioning in young walnut trees (Balandier et al., 2000). In contrast, integrating plant models over the long term, e.g. several years for trees, necessitates considering ecophysiological growth processes and architectural development simultaneously. Meristems generate the plant architecture by the production of new organs and their expansion with regards to the available assimilates and local environmental conditions, and sometimes following complex ontogenetic rules, e.g. leaf neoformation or polycyclism (Barthélémy and Caraglio, 2007; de Reffye et al., 2012). Purely architectural models were developed on these bases, often using stochastic models to simulate the dynamics of the topological structure. The generic AMAPsim model is a good example of such an approach (Barczi et al., 2008). Other stochastic models were developed for specific plant species as MAppleT model (Costes et al., 2008) or for young Eucalyptus (Diao et al., 2012). Functional–structural plant models were built to simulate tree growth and carbon allocation within a dynamic architecture. This is the case of L-PEACH (Lopez et al., 2008), which is dedicated to a single fruit tree species, and LIGNUM (Perttunen et al., 1998), which is a more generic model for forest trees. In the functional–structural tree model ALMIS (Eschenbach, 2005), the development of the structure is also based on the allocation and use of resources, i.e. assimilates and nutrients, with regards to local environmental factors. However, the architectural rules considered in this model are very limited, which is a limitation for the simulation of trees with complex organization.. The GreenLab model is a discrete dynamic model, which simulates biomass production and allocation at the organ scale, taking into account the feedback effects of internal trophic competition on plant morphology (Yan et al., 2004). The particularity of this model is that it takes the form of simple mathematical equations that can be solved and parameterized easily and in a very short computer processing time. Unlike the FSPMs mentioned above, the production equation of GreenLab does not use an explicit description of the plant structure, which is time consuming, but only developmental and allometric rules that allow the evolution of organ number and size to be estimated at each growth step. This model has been applied to several annual plants (Guo et al., 2006), but also to some tree species such as beech (Letort et al., 2008), Cecropia (Letort et al., 2012) and Mongolian Scots pine (Wang et al., 2012). However, to our knowledge, there is no plant model that simultaneously takes into account architectural variability, including the stochastic developmental processes along with carbon production and allocation, in a compact mathematical form. This study sets out to integrate stochastic growth processes, and leaf neoformation and polycyclism rules into the source–sink-based GreenLab model, using existing architectural data for teak trees. MATERIALS AND METHODS Experimental set-up The study was undertaken at the Agbavé forest station (0°45’E, 6°43’N) in south-western Togo. The relative air humidity there is close to 83 % on average. Rainfall averages stand at between 1100 and 1400 mm per year, with a peak in June. The mean maximum temperature ranges between 29 and 36 °C, and the hottest months are February and March. The mean minimum temperature ranges between 20 and 36 °C, and the coolest months are July and August. There are two dry seasons, one from November to March and the other from July to September. The studied teaks from a Togolese provenance were in two plots, located close together in order to minimize environmental effects (soil and climate) on the studied variables. The 1-year-old seedlings were planted at a density of 2500 trees ha–1 (i.e. at a spacing of 2 × 2 m). The analysed teaks were 2 and 5 years old, with 10 and six individuals respectively, and had not flowered. The average tree heights were 218.85 ± 67.54 cm for the 2-year-old trees and 727.25 ± 88.61 cm for the 5-year-old trees. The mean diameters at 1.30 m were 2.77 ± 1.09 cm for the 2-year-old trees and 7.88 ± 1.84 cm for the 5-year-old trees. The architectural analysis of teak Our study was based on the concepts developed for architectural analysis (Barthélémy and Caraglio, 2007). The phytomer, the basic unit of plant structure, includes the node, the leaf, the internode and the associated axillary buds (Barlow, 1989). A growth unit is defined as the portion of an axis which develops during an uninterrupted period of extension and can easily be identified by a zone of short internodes with scaly leaves corresponding to the bud protective organs from which it derives (for teak, see Fig. 2B and C). An annual shoot is defined as the portion of stem which extended during a year. The studied teaks showed rhythmic growth with two cessation periods: one during the long dry season and one during the short dry season. Thus this growth pattern was called polycyclism. The ‘physiological age’ concept, based from botanist observation (Barthélémy and Caraglio, 2007), is used to classify branch typologies. It also expresses the degree of differentiation of the produced phytomers; as mentioned previously, there are thus four physiological ages to consider here, corresponding to the following axis categories: the main stem, the branches, the branchlets and the twigs (Fig. 2A). The 2-year-old trees were unbranched. Branching appeared from the third year of growth. The phyllotaxy is opposite and decussate. Measurements Development measurements. The trees were described phytomer by phytomer. Phytomers were counted per unit of growth to obtain the distributions of the number of phytomers per growth unit according to the axis categories. The number of growth units was counted for each annual shoot. The rate of polycyclism was defined by the number of bicyclic annual shoots on the total number of annual shoots. The polycyclism rate was computed for each axis category. As the phyllotaxy is opposite decussate, two axillary meristems are present on each phytomer. Axillary production occurrence (bud, dead bud, branch, branchlets or twigs) was reported according to node rank, from the bottom to the top of the growth unit. Tree topology, i.e. the relative position of nodes and axes, was recorded using Multi Tree Scale formalism (Godin and Caraglio, 1998) with AMAPstudio (Griffon and de Coligny, 2014). Growth measurements. On each growth unit, we recorded the length and basal diameter according to their respective axis category. The growth unit stems were dried at 103 ° for 48 h and the dry weight was measured. For the leaf measurements, two different equations were determined to estimate the single individual leaf area (s) and the dry weight (We) values from the length (L) and width (Wi) of the leaf blade: s = 0.60 × L × Wi and We = 0.004 × (L × Wi)1.11 (Tondjo et al., 2015). Geometrical parameters such as phyllotaxy, branching angles or organ allometries were also measured. Main GreenLab model concepts The assumption of the GreenLab model is that plant architecture results from meristem functioning, organogenesis and photosynthesis, biomass production and partitioning (de Reffye et al., 2008). We only considered above-ground growth modelling in this study. The rules of tree development are modelled by a dual-scale automaton (Yan et al., 2002; Xing et al., 2003), providing information about the evolution of organ numbers. Biomass assimilation is modelled as a function of total tree leaf area at time t [eqn (1)]. It is then allocated to the primary growth of phytomers and to the secondary growth of the stem proportionately to the demand of the leaf, internode and growth ring at each cycle of development. The organ dimensions are computed from rules of allometry. The model workflow is detailed as follows (see Fig. 1): the seed gives the initial pool of dry biomass used to build organs and the plant architecture at cycle 1. At each cycle i, the biomass is partitioned among competing organs in relation to plant demand, D(i), which depends on the number of organs, No, and their respective sink functions pa for leaves, pe for internodes (or primary growth) and pc for growth ring increment (or secondary growth). Biomass production depends on the leaf hydraulic resistance r, the total leaf area S and the production leaf area Sp that is defined as the whole-tree crown projection onto the ground (Cournède et al., 2008). The biomass produced at cycle i, Q(i), is stored in the common pool of carbon reserves. Lastly, at each cycle of development, the model outputs the weights of individual organs and of the whole plant. The 3-D structure is another optional output allowing 3-D visualization of plant architecture. The biomass allocated to the wood rings is computed according to two balanced modes: the classical pipe model usual for trees (Shinozaki et al., 1964), and a common pool distribution usual for smaller plants. The production equation is expressed as follows: Fig. 1. View largeDownload slide Morphology of 5-year-old teak trees. (A) Architecture with the four axis categories or physiological ages. (B) Bicyclic annual shoot: intra-annual growth rest phase and delimitation of two successive growth units (GU1 and GU2). (C) Monocyclic annual shoot: inter-annual growth rest phase and delimitation of two successive annual shoots (AS n – 1, AS n). Fig. 1. View largeDownload slide Morphology of 5-year-old teak trees. (A) Architecture with the four axis categories or physiological ages. (B) Bicyclic annual shoot: intra-annual growth rest phase and delimitation of two successive growth units (GU1 and GU2). (C) Monocyclic annual shoot: inter-annual growth rest phase and delimitation of two successive annual shoots (AS n – 1, AS n). Fig. 2. View largeDownload slide Schematic representation of a simulation of the GreenLab model: development and growth simulation cycle, the model outputs, field measurements, the inverse method and the hidden parameters computed from the model. *Specific implementations made in the teak model. Fig. 2. View largeDownload slide Schematic representation of a simulation of the GreenLab model: development and growth simulation cycle, the model outputs, field measurements, the inverse method and the hidden parameters computed from the model. *Specific implementations made in the teak model. Q(i)=E⋅Spr⋅(1−exp(−kε.Sp⋅∑m=(i−ta)+1max(i,ta)∑φ=1mxφNaφ(m)⋅∑j=itpaφ(j−m+1)⋅Q(j−1)D(j))) (1) where E stands for the environmental conditions (set to 1 in this work), ta is the leaf expansion duration, φ is the current physiological age (from 1 to maxφ equal to 4 in the case of teak) and Nφa (m) stands for the number of leaves of physiological age φ, emitted at age m. The ε factor, assumed to be constant, is the specific leaf mass (i.e. the ratio of leaf dry mass and leaf area). k is the light extinction coefficient of the Beer–Lambert law (de Wit, 1965). Plant demand, D(i), is defined by the sum of all organ o requests, obtained from the product of the number of organs No and their sink function po, for all physiological ages: D(i)=∑o,φ(∑j=1iNoφ(i−j+1)⋅poφ(j)) (2) Improvement of the GreenLab model for the case of teak tree: stochastic development, neoformation and polycyclic growth Teak shows an alternation between rest and extension periods of meristems in shoot construction. The meristem functioning can be treated as renewal processes. The number of phytomers on an axis shows variability, and can be modelled by a binomial distribution (de Reffye et al., 2012). Successive phytomer occurrences are considered as independent events, with a probability of occurring at each cycle of development. With rhythmic development, as in teak, two laws are associated with the duration of meristem extension and with the duration of meristem rest between two successive growth units, respectively. At each cycle of development (CD), the meristem does or does not produce a phytomer, with a growth probability according to a typical Bernoulli process (de Reffye et al., 2012). The meristem growth probability is denoted b. At a given CD, a failed Bernoulli process test leads to a void entity, meaning that no phytomer is generated. The distribution of the number of phytomers in a pre-formed part of a growth unit follows a positive binomial law. The distribution of the number of phytomers in a neoformed part of a growth unit follows a negative binomial law (de Reffye et al., 1991). Pre-formed and neoformed phytomers expand immediately until meristem breakdown. Thus, the total number of phytomers in the growth unit is a convolution of both laws (Guédon et al., 2006). The probability of neoformation occurrence is defined as a single ratio denoted P. For each axis category, the distributions of the number of phytomers of a growth unit followed a convolution between a positive binomial law (k1, b1) and a negative binomial law (k2, b2), which is the meristem activity cessation law. These laws drive the appearance of the new leaves and as a consequence the production of leaf area. By knowing, for each physiological age, the development probabilities b, branching probabilities a and neoformation probability P, it is possible to compute a theoretical chronological structure. The chronological structure is a conceptual frame, composed of the full structure with an explicit time representation, containing all entities with their probability of occurring, while the topological structure corresponds to the observable structure. The chronological structure is used to compute the organic series (Kang et al., 2016) as sets of consecutive organs on the virtual plant axis. Two embedded time scales were taken into account in the model: (1) the intra-annual time periods corresponding to the appearance of successive growth units (maximum of two growth units per year in teak); and (2) the phyllochron (i.e. the time between the appearance of two successive phytomers within a given growth unit), herein named the cycle of development (CD). Model calibration The key step for model applications is its calibration on real plants, by estimating the specific parameter from experimental data. Some model parameters can be measured, while some cannot be directly assessed from experimental observations. The parameter identification process requires three steps: (1) the analysis of plant development that gives the number of produced phytomers and the topological rules within the whole tree; (2) the building of target data with the organ dry weights; and, lastly, (3) fitting of the functional hidden parameters. The measured parameters are distributions of the number of phytomers per growth unit which defined the developmental parameters (in particular b and P probabilities). The branching probability (a) results also apply to the morphological observations. For the branching process, we consider that there is a potential number of axillary branches on each phytomer (usually one or two). Branching may occur with a branching probability, a, expressed as a single ratio between the number of branched nodes and the total number of nodes. The probability that the meristem never develops a branch is thus 1 – a. At each node, there are two axillary buds, the possibilities are: two branches (1,1), one branch (1,0) or zero branches (0,0). A coupling parameter was added, tuning the branching occurrence for the (1,1) and (1,0) cases. This coupling reflects the fact that, if an axillary bud develops on one node, the second bud also shows a significant chance of developing. When the plant architecture is stochastic, a key issue of the inverse method is defining the target data for fitting. The target data were the mean dry weights of leaves or internodes per growth unit called ‘compound organic series’. This type of target contains information on the source and sink dynamics, and consequently allows the source–sink function. Fitting one or more growth stages is called ‘single’ or ‘multi-’ fitting, respectively (Guo et al., 2006). Multi-fitting requires the organic series at several growth stages. For this study, two stages of observations were taken into account: 2-year-old trees that were unbranched and 5-year-old trees that were branched, with the structure of a young tree composed of four axis categories or ‘meristem physiological ages’. The hidden parameters have to be estimated by model inversion (Zhan et al., 2003). These are the sink functions (po) for leaves, internodes and growth ring, leaf hydraulic resistance (r) and the plant production area (Sp). The hidden parameters for the teak model are specified in Table 2. The value of the leaf specific mass ε (the ratio between dry weight and leaf area) was 0.015 g cm–2. The light extinction coefficient of the Beer–Lambert law, k, was set to k = 1. The allometry between the internode length and volume was fixed to an average value, with the same value on all internodes for a given growth unit. The seed weight or initial biomass was fixed at 1 g. Plant demand for secondary growth is proportional to the leaf number of the tree. Internode allocations to secondary growth are based on Pressler’s law with a variation coefficient λ (according to the pipe model assumptions as the surface section conservation; Kang et al., 2002). However, according to the young stages of the trees, we also consider a common pool ring distribution model. The balance between those two allocation models is defined by a coefficient λ, to be fitted with the diameter field measurements with the other functional parameters. For herbaceous species, the production leaf area (Sp) is a constant over their development, while for tree species, Sp varies during tree development. The surface of the crown is large at the beginning of growth and decreases before increasing with the development of the crown. A U-shaped function was thus used to model the evolution of Sp. Sp(x)=Smax(1−SF(x)−SminSmax−Smin)+Smin(SF(x)−SminSmax−Smin) (3) where: SF(x)=Smin(1−B(x)−BminBmax−Bmin)+Smax(B(x)−BminBmax−Bmin) (4) and: B(x)=((x−0.5)T)BaSp1(1−x−0.5T)BbSp1T (5) SF corresponds to the leaf area. Smax (maximum area) and Smin (minimum area) are fixed. The Beta law B(x) in a [1, T] interval has a minimum Bmin and a maximum Bmax depending on BaSp and BbSp. These parameters were computed from the generalized least squares inverse method which optimizes the fitting of mean dry weight of organs per growth unit simultaneously with the other source–sink parameters. Model simulation and output The rules of tree development are modelled by a dual-scale automaton (Yan et al., 2002; Xing et al., 2003). This stochastic automaton was used to simulate the tree structure. It simulated organogenesis through microstates (within a growth unit), macrostates (from one growth unit to the next on the axis) and the jump relationships between them (de Reffye et al., 2008). The different states of the automaton corresponded to the physiological age classes, and the transitions of the automaton corresponded to bud differentiation (Cournède et al., 2006). In our case, the jumps were conditional; they occurred according to the development probabilities given by the binomial laws for both micro- and macrostates. Specific individual teak trees were generated from model simulations. However, the visual aspect of the 3-D plants contained less information than the fitting of the organic series. In the 3-D tree structure construction, the internode length and growth ring were estimated from the simulated fresh internode volume. This volume is computed from the ratio between the available biomass and the basic specific gravity (Kollmann and Côté, 1968). Simulations were carried out with the GreenLab Matlab environment. Sixteen different individual teak trees were generated, at 5 years old, showing random properties related to the stochastic development laws (the negative binomial laws for pre-formed and neoformed leaves, polycyclism probability and branching probability). For each tree, side views from the 3-D output were generated, with leaves. The number of growth units and phytomers, and the respective internode volume and leaf areas were reported according to the phytomer physiological age and growth unit. RESULTS Modelling plant development The teak development pattern showed the elongation of pre-formed phytomers to be seasonal and synchronous at the start of elongation followed immediately by the elongation of neoformed phytomers. The cessation of pre-formed phytomer elongation was synchronous for all meristems. Neoformation followed a negative binomial law with a probability that varied depending on the rank of the growth unit and on the axis categories (see Table 1). The cessation of neoformed phytomer elongation was spread out over time. The leaf neoformation always occurred and the transition probability from pre-formation to neoformation is equal to 1. The number of phytomers of the first growth unit was higher than that of the second growth unit. The number of phytomers of the first growth unit decreased from trunk to twigs, while the number of phytomers of the second growth unit remained relatively constant. The number of pre-formed nodes was three for all growth units regardless of the axis categories (Fig. 3). Table 1. Annual growth in 5-year-old teak trees for different physiological ages: parameters of a positive binomial law for pre-formation (k1, b1) and parameters of a negative law for neoformation (k2, b2) Axis categories φ RGU NGU M V k1 b1 k2 b2 Main stem 1 1 210 9.60 13.10 3 1 7 0.49 1 2 150 5.00 6.10 3 1 3 0. 35 Branches 2 1 149 8.20 10.40 3 1 7 0.45 2 2 70 5.40 4.30 3 1 3 0.44 Branchlets 3 1 220 5.70 5.10 3 1 4 0.41 3 2 77 4.80 3.60 3 1 2 0.48 Twigs 4 1 100 4.20 1.90 3 1 3 0.30 Axis categories φ RGU NGU M V k1 b1 k2 b2 Main stem 1 1 210 9.60 13.10 3 1 7 0.49 1 2 150 5.00 6.10 3 1 3 0. 35 Branches 2 1 149 8.20 10.40 3 1 7 0.45 2 2 70 5.40 4.30 3 1 3 0.44 Branchlets 3 1 220 5.70 5.10 3 1 4 0.41 3 2 77 4.80 3.60 3 1 2 0.48 Twigs 4 1 100 4.20 1.90 3 1 3 0.30 Φ, physiological age; RGU, growth unit rank on the annual shoot, 1 for the first growth unit, 2 for the second growth unit; NGU, number of GUs of all studied trees; M, mean number of phytomers per GU; V, variance of the distribution of number of phytomers per GU. View Large Table 1. Annual growth in 5-year-old teak trees for different physiological ages: parameters of a positive binomial law for pre-formation (k1, b1) and parameters of a negative law for neoformation (k2, b2) Axis categories φ RGU NGU M V k1 b1 k2 b2 Main stem 1 1 210 9.60 13.10 3 1 7 0.49 1 2 150 5.00 6.10 3 1 3 0. 35 Branches 2 1 149 8.20 10.40 3 1 7 0.45 2 2 70 5.40 4.30 3 1 3 0.44 Branchlets 3 1 220 5.70 5.10 3 1 4 0.41 3 2 77 4.80 3.60 3 1 2 0.48 Twigs 4 1 100 4.20 1.90 3 1 3 0.30 Axis categories φ RGU NGU M V k1 b1 k2 b2 Main stem 1 1 210 9.60 13.10 3 1 7 0.49 1 2 150 5.00 6.10 3 1 3 0. 35 Branches 2 1 149 8.20 10.40 3 1 7 0.45 2 2 70 5.40 4.30 3 1 3 0.44 Branchlets 3 1 220 5.70 5.10 3 1 4 0.41 3 2 77 4.80 3.60 3 1 2 0.48 Twigs 4 1 100 4.20 1.90 3 1 3 0.30 Φ, physiological age; RGU, growth unit rank on the annual shoot, 1 for the first growth unit, 2 for the second growth unit; NGU, number of GUs of all studied trees; M, mean number of phytomers per GU; V, variance of the distribution of number of phytomers per GU. View Large Fig. 3. View largeDownload slide Distribution of the number of phytomers of the first and second growth units of bicyclic annual shoots for the main stem (A and B), for branches (C and D), for branchlets (E and F) and for twigs (G) for 5-year-old trees. Observed values (circles) and theoretical values (lines). Fig. 3. View largeDownload slide Distribution of the number of phytomers of the first and second growth units of bicyclic annual shoots for the main stem (A and B), for branches (C and D), for branchlets (E and F) and for twigs (G) for 5-year-old trees. Observed values (circles) and theoretical values (lines). At the beginning of development, the first and second growth units were synchronous. The leaves were photosynthetically active over 36 CDs. The rest period was estimated at one CD. The polycyclism rate was 70 % for the main stem, 50 % for branches, 35 % for branchlets and 0 % for twigs. Thus, the polycyclism rate decreased with the physiological age of the meristem. A phytomer generally bore two branches of the same category. Only the first growth unit was branched, with a probability of 0.054 for the trunk, 0.091 for the branches and 0.055 for the branchlets. The twigs were unbranched. The pattern of tree development is shown in Fig. 4. The chronological structure shows that the growth of teak is rhythmic, with a large number of cycles of development in rest activity between each growth unit. The polycyclic growth is mostly frequent for the main stem and branches. The branching is rhythmic and differed from 1 year with the occurrence of one or two branches per node. Only the first growth unit is branched. The main stem bears all axis categories (Fig. 4A). The topological structure corresponds to the realized phytomers (Fig. 4B). Fig. 4. View largeDownload slide Chronological structure with growth and rest periods of the meristems (A) and topological structure show observable structure (B) for the 5-year-old teak tree. The stochastic constructions are simulated by the botanical automaton whose rules are derived from observations of the behaviour of the different axes: main stem (blue), branches (green), branchlets (red) and twigs (yellow). In (A), the rest cycles of development, which are numerous, are shown in white. In (B), the rest cycles of development are not present. Fig. 4. View largeDownload slide Chronological structure with growth and rest periods of the meristems (A) and topological structure show observable structure (B) for the 5-year-old teak tree. The stochastic constructions are simulated by the botanical automaton whose rules are derived from observations of the behaviour of the different axes: main stem (blue), branches (green), branchlets (red) and twigs (yellow). In (A), the rest cycles of development, which are numerous, are shown in white. In (B), the rest cycles of development are not present. Modelling biomass production The U-shaped curve describing the evolution of Sp depended on two parameters, BaSp and BbSp [see eqn (5)], computed with the other source–sink parameters (Table 2). Over the first years of growth, Sp decreased until it reached a minimal value. Then Sp increased exponentially from the third year of development until it reached a constant value (Fig. 5). Table 2. Hidden parameters of 5-year-old teak trees computed from multi-stage fitting Q0 Initial biomass 1 g r Leaf hydraulic resistance 50 Spmax U Sp curve maximum production leaf area 15 774 cm2 Spmin U Sp curve minimum production leaf area 1000 cm2 BaSp U Sp curve Beta law first coefficient 1.19 BbSp U Sp curve Beta law first coefficient 2.33 λ Growth ring model coefficient 0.17 p1a (main stem) Leaf sink strength (physiological age 1) 1 p2a (branches) Leaf sink strength (physiological age 2) 0.78 p3a (branchlets) Leaf sink strength (physiological age 3) 0.62 p4a (twigs) Leaf sink strength (physiological age 4) 0.59 p1e (main stem) Internode sink strength (physiological age 1) 7.2 p2e (branches) Internode sink strength (physiological age 2) 4.2 p3e (branchlets) Internode sink strength (physiological age 3) 5.5 p4e (twigs) Internode sink strength (physiological age 4) 1.5 p1c (main stem) Growth ring sink strength (for all physiological ages) 1.7 vp1c (main stem) Growth ring sink variation coefficient (physiological age 1) 1 vp2,3,4c (branches, branchlets and twigs) Growth ring sink variation coefficient (physiological age 2, 3 and 4) 0.7 Q0 Initial biomass 1 g r Leaf hydraulic resistance 50 Spmax U Sp curve maximum production leaf area 15 774 cm2 Spmin U Sp curve minimum production leaf area 1000 cm2 BaSp U Sp curve Beta law first coefficient 1.19 BbSp U Sp curve Beta law first coefficient 2.33 λ Growth ring model coefficient 0.17 p1a (main stem) Leaf sink strength (physiological age 1) 1 p2a (branches) Leaf sink strength (physiological age 2) 0.78 p3a (branchlets) Leaf sink strength (physiological age 3) 0.62 p4a (twigs) Leaf sink strength (physiological age 4) 0.59 p1e (main stem) Internode sink strength (physiological age 1) 7.2 p2e (branches) Internode sink strength (physiological age 2) 4.2 p3e (branchlets) Internode sink strength (physiological age 3) 5.5 p4e (twigs) Internode sink strength (physiological age 4) 1.5 p1c (main stem) Growth ring sink strength (for all physiological ages) 1.7 vp1c (main stem) Growth ring sink variation coefficient (physiological age 1) 1 vp2,3,4c (branches, branchlets and twigs) Growth ring sink variation coefficient (physiological age 2, 3 and 4) 0.7 The leaf areas are fixed Smin = 1000 cm2 and Smax = 15 000 cm2. λ is a coefficient which corresponds to the ratio of ‘pipe model’ and ‘common pool distribution’. View Large Table 2. Hidden parameters of 5-year-old teak trees computed from multi-stage fitting Q0 Initial biomass 1 g r Leaf hydraulic resistance 50 Spmax U Sp curve maximum production leaf area 15 774 cm2 Spmin U Sp curve minimum production leaf area 1000 cm2 BaSp U Sp curve Beta law first coefficient 1.19 BbSp U Sp curve Beta law first coefficient 2.33 λ Growth ring model coefficient 0.17 p1a (main stem) Leaf sink strength (physiological age 1) 1 p2a (branches) Leaf sink strength (physiological age 2) 0.78 p3a (branchlets) Leaf sink strength (physiological age 3) 0.62 p4a (twigs) Leaf sink strength (physiological age 4) 0.59 p1e (main stem) Internode sink strength (physiological age 1) 7.2 p2e (branches) Internode sink strength (physiological age 2) 4.2 p3e (branchlets) Internode sink strength (physiological age 3) 5.5 p4e (twigs) Internode sink strength (physiological age 4) 1.5 p1c (main stem) Growth ring sink strength (for all physiological ages) 1.7 vp1c (main stem) Growth ring sink variation coefficient (physiological age 1) 1 vp2,3,4c (branches, branchlets and twigs) Growth ring sink variation coefficient (physiological age 2, 3 and 4) 0.7 Q0 Initial biomass 1 g r Leaf hydraulic resistance 50 Spmax U Sp curve maximum production leaf area 15 774 cm2 Spmin U Sp curve minimum production leaf area 1000 cm2 BaSp U Sp curve Beta law first coefficient 1.19 BbSp U Sp curve Beta law first coefficient 2.33 λ Growth ring model coefficient 0.17 p1a (main stem) Leaf sink strength (physiological age 1) 1 p2a (branches) Leaf sink strength (physiological age 2) 0.78 p3a (branchlets) Leaf sink strength (physiological age 3) 0.62 p4a (twigs) Leaf sink strength (physiological age 4) 0.59 p1e (main stem) Internode sink strength (physiological age 1) 7.2 p2e (branches) Internode sink strength (physiological age 2) 4.2 p3e (branchlets) Internode sink strength (physiological age 3) 5.5 p4e (twigs) Internode sink strength (physiological age 4) 1.5 p1c (main stem) Growth ring sink strength (for all physiological ages) 1.7 vp1c (main stem) Growth ring sink variation coefficient (physiological age 1) 1 vp2,3,4c (branches, branchlets and twigs) Growth ring sink variation coefficient (physiological age 2, 3 and 4) 0.7 The leaf areas are fixed Smin = 1000 cm2 and Smax = 15 000 cm2. λ is a coefficient which corresponds to the ratio of ‘pipe model’ and ‘common pool distribution’. View Large Fig. 5. View largeDownload slide Theoretical U-shaped curve for the production leaf area (Sp, area) according to the cycle of development (CD) in the 5-year-old teak trees. One year corresponds to 37 CDs. Fig. 5. View largeDownload slide Theoretical U-shaped curve for the production leaf area (Sp, area) according to the cycle of development (CD) in the 5-year-old teak trees. One year corresponds to 37 CDs. Measurement data were compared with the fitted parameters at the tree scale in order to evaluate the model (Fig. 6). The fitted values are consistent with measured dry biomass for the main stem of 2-year-old trees, as well as for the first growth units of each annual shoots of all axes. However, they are less consistent regarding the dry biomass of the second growth units for all axis categories of the 5-year-old trees. This can be explained by the low number of second growth units found within the trees that reduced the number of samples in this category (Table 1). Fig. 6. View largeDownload slide Comparisons between measured and fitted results at tree scale for teak. Annual internode dry mass per axis categories for 2- and 5-year-old trees: first growth unit (A) and second growth unit (B); circles denote the measured results and the black curve shows the fitted results. Fig. 6. View largeDownload slide Comparisons between measured and fitted results at tree scale for teak. Annual internode dry mass per axis categories for 2- and 5-year-old trees: first growth unit (A) and second growth unit (B); circles denote the measured results and the black curve shows the fitted results. The values of the organ sink strength are relative to the values of the sink strength of the main stem leaf set to 1. The leaf sink strength decreased from the branches to the twigs (Table 2). The sink strength of the internode tended to decrease from the trunk to the twigs, except for the branchlets, where it was higher than for the branches. For the leaf, internode and growth ring compartments, the biomass null value corresponded to the rest period: teak loses its leaves during the dry season. At the start of growth, a peak in biomass production was seen at the first cycle of development corresponding to the previous year’s carbon reserves (Fig. 7). Generally, biomass production increased from the third year with the appearance of the first branch. The biomass produced by the leaves was low at the beginning of their elongation and increased with crown development over the year when the leaf area increases. The decreased biomass over the period of maximal production correspond to the intra-annual growth cessation between the successive growth units. The biomass was also higher for the first growth flush than for the second one (Fig. 7A). The biomass allocated to internodes (i.e. primary growth) was low and relatively constant over the three growth years on all axes (Fig. 7B). The biomass allocated to growth rings (i.e. secondary growth) was higher than that allocated to the leaves and to the internodes, increased over the first three growth years and remained constant afterwards (Fig. 7C). It is interesting to note that the wood ring biomass allocation was distributed from the common pool (with λ = 0.17, Table 2), and not the pipe model (λ close to 1), perhaps related to the fact that the plants are still young and of small height. Fig. 7. View largeDownload slide Dry biomass allocation for each organ compartment: leaves (A), internodes (B, i.e. primary growth) and growth rings (C, i.e. secondary growth), according to the cycle of development (CD) during the first 5 years of growth. One year correspond to 37 CDs which correspond to 36 CDs of the active period with 1 CD for the rest period. Fig. 7. View largeDownload slide Dry biomass allocation for each organ compartment: leaves (A), internodes (B, i.e. primary growth) and growth rings (C, i.e. secondary growth), according to the cycle of development (CD) during the first 5 years of growth. One year correspond to 37 CDs which correspond to 36 CDs of the active period with 1 CD for the rest period. A strong increase in biomass supply (Q) was observed when branching occurred in the third year of growth. The minimum biomass production value corresponded to the reserves brought into play during the dry season (Fig. 8A). The plant demand trends showed two minimum values per year: the first corresponded to the annual shoot elongation rest period during the long dry season, and the second corresponded to the intra-annual shoot rest period during the short dry season (Fig. 8B). The supply to demand ratio (Q/D) decreased with the increase in plant demand D resulting from tree crown development. The Q/D was higher in the second growth year since plant demand still remained low while the biomass supply was relatively high (Fig. 8C). Fig. 8. View largeDownload slide Variation in total plant dry biomass supply or Q (A), in plant dry biomass demand or D (B), and in the supply to demand ratio or Q/D (C), depending on the number of cycles of development over the first 6 years growth of teak. Fig. 8. View largeDownload slide Variation in total plant dry biomass supply or Q (A), in plant dry biomass demand or D (B), and in the supply to demand ratio or Q/D (C), depending on the number of cycles of development over the first 6 years growth of teak. Simulation results Using the calibrated model, the tree growth and development processes were simulated for the 2-year-old teaks (Fig. 9). For the 16 virtually simulated 5-year-old teaks (Fig. 10), it could be seen from a qualitative point of view that the observed variability of the tree structures resulted of the stochastic behaviour of the model. Significant leaf area changes were also found between the simulated individuals, especially when comparing structures with a low amount of branching and those which were highly branched. From a quantitative point of view, from these virtual trees, we extracted the number of phytomers of the first and second growth units for the main stem as an example (Fig. 11). The distribution of the phytomer number per growth unit was similar to the observed numbers (Fig. 3). In order to illustrate the structural variabilty and the relationship between the structure and the organ dimensions, we selected three specific virtual individuals corresponding to medial and extreme structure complexity. Individual 3 had the largest number of phytomers with simultaneously the lowest total leaf area and individual leaf area, and the lowest total wood volume (Table 3). Individual 2 had the median value for the number of phytomers, and also showed the median value for the total wood volume and the individual leaf area, while the total leaf area value was higher (Table 3). Individual 15 had the smallest number of phytomers and the highest individual leaf area. The total wood volume value was average and the total leaf area value was high (Table 3). As shown in Fig. 10, this last individual only bore one branch. Fig. 9. View largeDownload slide Three architectures of 2-year-old teak trees and photographs of one real tree. The scale bar is 2 m. Fig. 9. View largeDownload slide Three architectures of 2-year-old teak trees and photographs of one real tree. The scale bar is 2 m. Fig. 10. View largeDownload slide Ten architectures from the sixteen 5-year-old teak simulations illustrating the stochastic aspect on the plant architectures. Plant vizualization is at 180 cycles of development. Individuals 3 and 15 show the largest and smallest number of phytomers, respectively. Individual 2 is closest to the median case. The scale bar is 2 m. Fig. 10. View largeDownload slide Ten architectures from the sixteen 5-year-old teak simulations illustrating the stochastic aspect on the plant architectures. Plant vizualization is at 180 cycles of development. Individuals 3 and 15 show the largest and smallest number of phytomers, respectively. Individual 2 is closest to the median case. The scale bar is 2 m. Fig. 11. View largeDownload slide Simulations of distributions of the number of phytomers for the first t (A) and second growth unit (B) for the main stem of 5-year-old teak trees. Fig. 11. View largeDownload slide Simulations of distributions of the number of phytomers for the first t (A) and second growth unit (B) for the main stem of 5-year-old teak trees. Table 3. Mean (s.d.) of the total number of phytomers, total wood volume (cm3), total leaf area (cm2) and individual leaf area (cm2) estimated from the stochastic simulation of 16 individuals Total phytomer number Total wood volume (cm3) Total leaf area (cm2) Individual leaf area (cm2) 16 virtual trees 180 (62) 13 784 (641) 105 373 (14 855) 681 (351) Individual 3 272 12 992 74 245 272 Individual 2 199 13 933 116 324 584 Individual 15 66 13 710 112 297 1703 Total phytomer number Total wood volume (cm3) Total leaf area (cm2) Individual leaf area (cm2) 16 virtual trees 180 (62) 13 784 (641) 105 373 (14 855) 681 (351) Individual 3 272 12 992 74 245 272 Individual 2 199 13 933 116 324 584 Individual 15 66 13 710 112 297 1703 Three individuals with a particular behaviour were noted. View Large Table 3. Mean (s.d.) of the total number of phytomers, total wood volume (cm3), total leaf area (cm2) and individual leaf area (cm2) estimated from the stochastic simulation of 16 individuals Total phytomer number Total wood volume (cm3) Total leaf area (cm2) Individual leaf area (cm2) 16 virtual trees 180 (62) 13 784 (641) 105 373 (14 855) 681 (351) Individual 3 272 12 992 74 245 272 Individual 2 199 13 933 116 324 584 Individual 15 66 13 710 112 297 1703 Total phytomer number Total wood volume (cm3) Total leaf area (cm2) Individual leaf area (cm2) 16 virtual trees 180 (62) 13 784 (641) 105 373 (14 855) 681 (351) Individual 3 272 12 992 74 245 272 Individual 2 199 13 933 116 324 584 Individual 15 66 13 710 112 297 1703 Three individuals with a particular behaviour were noted. View Large DISCUSSION Growth and development of teak Annual growth of teak (Tectona grandis L. f.) is polycyclic, with each growth unit including three pre-formed leaves and a number of neoformed leaves that varied with the growth unit rank (first or second growth unit) and with the axis category. The rhythmicity is directly linked to the climatic seasonality especially regarding growth rest in the summer season. In fact, no polycyclic growth was observed for teaks growing in Indonesia where there is no short dry season (Leroy et al., 2009). In this case, the growth is continuous over the growing season. Our observations and growth data in Togo suggest that the rest period is a strategy of avoidance of the dry period. The pre-formation and neoformation of organs has been observed in numerous tree species with a rhythmic growth (Magnin et al., 2012). In teak, the number of neoformed phytomers varies between the first and the second growth unit (when it is present) of annual shoots according to axis categories, while the number of pre-formed phytomers is low and constant. The low number of pre-formed leaves seems to be linked to the large size of leaves. This relationship has been mentioned in many tropical evergreen tree species (Gill and Tomlinson, 1971), but never in a deciduous species such as teak. Evaluation of the GreenLab model on teak data The GreenLab model has been adapted to simulate more complex architectural properties, in particular neoformation and polycyclic growth, which are encountered on teak growing in Togo. These characteristics were eventually successfully reproduced by this new version of the model using stochastic functions linked to a binomial distribution. When applied to trees, the advantage of multi-fitting, i.e. fitting two differentiation stages in parallel, is notable compared with other plants showing lower growth variability. By fitting several stages in parallel, we expect to extract a set of endogenous parameters representative of the species. Simulations using the parameter values estimated from the multi-fitting process provide the dynamics of the different biomass compartments (i.e. leaf, internode and growth ring) and the biomass supply to demand ratio (Q/D) for each measured plant. The plant demand decreases during the intra-annual growth, stops and then increases with the emergence of leaves of the second growth unit. This trend confirmed a theoretical study on the rhythmic appearance in plants (Mathieu et al., 2008). Our results showed that the variation in Q/D reflected growth rhythmicity. Biomass production was very low during the early years of growth (i.e. from 36 to 72 CDs) since the plant organ dimensions were small. This corresponded to the tree establishment phase. The increase in biomass supply and biomass demand over the third growing year was linked to occurrence of branching. The Q/D peak observed in the third year of growth corresponded to high biomass supply and low biomass demand during the third year. In the following year of growth, branching complexity increased together with plant demand, and thus Q/D decreased. Such relationships between an increase in Q/D and branching expression were also observed in Cecropia sciadophylla (Letort et al., 2012). In the case of teak, the biomass allocated to the growth ring was low at the beginning of the growing season because biomass was first allocated to organ elongation, and it was higher at the end of the growing season when the leaves were fully elongated. Thus, these results corroborate the existence of interactions between the plant structure and its strategy for biomass allocation. Stochastic modelling highlighted the variability of tree behaviour within a tree population. Some less branched trees showed large leaves and a large total leaf area despite a small number of phytomers. In teak, the range of estimated individual leaf areas from the product of leaf length and width is from 200 to 1500 cm2 (Tondjo et al., 2015). On the other hand, some other trees that were the most branched had a large number of phytomers and small leaves. These results highlighting the variability of biomass partitioning to leaves or to stems might explain the variability of the architecture within individuals. Production leaf area The production leaf area (Sp) is an important parameter introduced in the GreenLab model in order to extend the notion of leaf area index (LAI; usually used in crop models) to individual trees. Sp is constant in herbaceous crops with a homogeneous leaf area distribution, and corresponds to the unit surface area of soil considered in the estimation of LAI (Guo et al., 2006). In contrast, Sp varies with the increase of the crown size when it is applied on a single tree. This evolution was modelled here as a U-shaped function. The Sp decreased during the first two growing years due to leaf self-shading, and increased from the third year of growth due to crown expansion. Sp exponentially increased during the third and fourth years of tree development, and remained constant when the crown volume reached a threshold corresponding to equilibrium. The U-shaped function allows the GreenLab model to be used on other tree species. Focus on the wood-specific gravity The basic specific gravity allows computation of the fresh stem volume from the dry mass. Such information, which is easy to measure, is a key point in the GreenLab model as it is considered to calculate wood ring width, and thus the increment in growth unit volumes, with regard to the locally available dry matter. In a previous study carried out in Togo, the variability of teak architecture and wood properties was investigated with regard to thinning practices and provenance. In particular, the authors showed that radial variation in wood properties was linked to variability in tree ontogeny (Tondjo et al., 2014). Developing an FSPM model that considers both the determinism of dry biomass formation and allocation, and tree stochastic development could be of interest for analysing and explaining the correlation between wood radial variations and tree architectural development. Up to now, in our models, the green volume of tree axes had been computed from the ratio between the biomass supply and a constant value of the basic specific gravity (Gb), although this wood property is known to vary depending on the cambial age (for Gb values in teak, see Pérez and Kanninen, 2005; Tondjo et al., 2014). A future improvement to the model will be to take into account the changes in Gb linked to the cambial age (i.e. radial variation). Given the amount of dry biomass (carbon allocation), the Gb values can be used to calculate the ring width increment for each growth unit, and thus the total volume of green wood at any growth stage. The Gb is an important indicator of timber mechanical properties, and knowledge of its variation is a critical point for wood quality optimization. Such an improvement in the GreenLab model will thus be useful for analysing the relationships existing between tree architecture and wood quality in different silvicultural practices and environmental contexts. Positioning Greenlab in the FSPM world Until now, the GreenLab application on stochastic development had been described for some species with continuous growth (Diao et al., 2012; Kang et al. 2016, Vavitsara et al., 2017). In the present study, tree development was modelled and described numerically as an alternation of extension and rest periods, taking into consideration leaf neoformation and polycyclic growth processes. This is a significant evolution that allows more complex botanical laws to be taken into account. Plant architecture appears as the result of source–sink balances and is impacted by endogenous processes and exogenous constraints due to the environment (Barthélémy and Caraglio, 2007). Consequently, GreenLab will be able to provide a better understanding of the intraspecific variability and interaction between tree architecture and its environment, and the morphological adjustments to climate changes. There are few existing tree structural–functional models that integrate detailed botanical rules in order to understand architectural plasticity. Moreover, most of them are dedicated to specific plant species, e.g. kiwifruit vine (Cieslak et al., 2011), peach trees with L-PEACH (Lopez et al., 208) or apple trees with MAppleT (Costes et al., 2008; Migault et al. 2017). The LIGNUM model (Perttunen et al., 1998) is a more generic FSPM that can be applied to different tree species. However, these FSPMs are based on an explicit description of the plant topology using computer algorithms, which is very time consuming even to simulate one single tree. This calculation cost is a major limitation to simulate forests or orchards, which necessitates using powerful and expansive computers and/or developing specific fast algorithms (Han et al., 2017). The main advantage of GreenLab is its mathematical formulation that allows direct and fast calculation. Applications to large vegetation scales have been already carried out, with satisfactory results (Feng et al., 2012, 2014). CONCLUSION The presented version of the GreenLab model integrates new development processes such as the leaf pre-/neoformation and polycyclic growth that allow simulation of complex tree structures and their architectural plasticity. The first application to teak (Tectona grandis L. f.) data collected in Togo shows promising results and opens up interesting perspectives for studying trees and forest adaptation to climate change. The model also provides quantitative key insights as a first step to proposing plant ideotypes. Application to forest trees should integrate more information on basic specific gravity and its intraspecific variability, as this trait is fully correlated to several wood quality factors, e.g. linked to mechanical performances. ACKNOWLEDGEMENTS The authors thank the owner of the experimental plots, Professor Messanvi GBEANSSOR at the University of Lomé, and the Lomé University students for their help in the teak field measurements. The authors are grateful the two anonymous reviewers for critical and valuable comments of the manuscript. This work was supported, in part, by the AUF-PCSI programme, by a grant from CIRAD AIRD and by AMAP [Botany and Computational Plant Architecture, joint research unit which associates CIRAD (UMR51), CNRS (UMR5120), INRA (UMR931), IRD (2M123) and Montpellier University (UM27)] http://amap.cirad.fr/. LITERATURE CITED Balandier P , Lacointe A , Le Roux X , Sinoquet H , Cruiziat P . 2000 . 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Annals of BotanyOxford University Press

Published: Mar 27, 2018

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