Integrating Markov Chain Models and

An approach for modelling and simulating the architectural development of apple trees is presented. The approach is based on using an L-systems framework to integrate a Markov model of terminal bud fate and a number of hidden semiMarkov models of axial bud fate. Results show that these models are able to represent the branching zones observed in apple trees at node scale, simulate the sequences of annual growth units found along branches, and capture important aspects of the bearing pattern. Integrating these models gives a simulation of the development of the whole tree that can be used to evaluate the ability of local-scale models to capture global characteristics. INTRODUCTION When studying fruit trees, it is important to consider the architecture or structure of the tree, in addition to more commonly studied issues such as fruit quality and resistance to pest and disease (Laurens et al., 2000). In the context of orchard management, the structure of the tree will have a strong influence on the amount of tree maintenance required, with certain tree forms reducing the need for pruning and chemical thinning. The branching pattern of the tree is also closely related to the important issue of the regularity of fruit bearing (Lauri et al., 1997). Previous work on apple trees (Malus x domestica Borkh) has investigated intraspecific variability in apple tree architecture (Costes and Guédon 1997, 2002, Costes et al., 2003). These studies were based on a method for measuring and analysing plant architecture that combines topological and geometrical descriptions of plants at several scales simultaneously (Godin et al., 1997, 1999). Local-scale empirical models were developed for analysing branching patterns along the trunks of different apple cultivars (Costes and Guédon 1997, 2002). A succession of distinct homogenous zones along the trunk was observed and represented in a relatively empirical way by Markov models (Guédon et al., 2001). These Markov models were also applied to analysing the zones observed in shoots of different lengths in peach trees (Fournier et al., 1998). On the other hand, L-systems provide a well-established approach for modelling the development of branching systems (Prusinkiewicz and Lindenmayer, 1990). They have been used in agronomic and horticultural applications such as simulating the structural development of cotton plants and peach trees, based on functional hypotheses (Hanan and Hearn, 2002 , Allen et al., 2004). The goal of the present study is to further develop the approach used for the trunk by parameterising Markov chain models for branching along the other annual growth units (GUs) of the tree. We also aim to extend the approach to model the succession of GUs along branches. These local-scale models will then be integrated into a global simulation of the architectural development of whole apple trees, based on L-systems. The initial aim of the study was to produce a relatively accurate whole tree simulation so that comparison between global characteristics of the simulated trees and the measured trees could be used to evaluate the ability of local-scale Markovian models to capture global characteristics of the trees. We therefore used a stochastic approach rather than a detailed modelling of the underlying functional-structural mechanisms, such as carbon allocation or hormone interaction. A long term goal is to produce an accurate simulation, which could be used to predict the architectural development of different cultivars over time, and thus investigate and specify inter-cultivar differences in fruit position and regularity. We were interested in whether an approach based on Markov chain models and L-systems would be capable of producing such a simulation. MATERIALS AND METHODS The general approach, as illustrated in Figure 1, is to start by analysing a database of branching structures and thus formulating local-scale stochastic models of branching based on Markov chains. Using L-systems, these local-scale models will then be integrated into a simulation of full tree architectural development. Simulated structures can then be compared to the original structures at the global scale. Constructing the simulation involved finding ways of answering a number of questions. When is a bud transformed into a growing apex? How many internodes and axial buds will a growing apex produce before becoming a dormant terminal bud? What will be the fate of this terminal bud? What will be the fate of the axial buds? The answers to these questions are based on the system of categorisation used in the tree structure database, and the local-scale Markovian models built from this database. Encoded Database of Measured Tree Structures The database consists of recorded measurements for two six-year-old Fuji apple trees, with development over the six years deduced by using morphological markers such as leaf scars (Costes et al., 2003). From this data base, GUs of different types were coded using numerical symbols. Proleptic vegetative short, medium and long branches were represented with a 1, 5 and 2, respectively, and sylleptic GUs with a 4. Each bud was assigned a code according to its fate, so that a bud that produced a long GU was a type 2 bud, for example. Dormant buds were represented with a 0. This categorisation defined the timing for growth within the simulation. Buds of type 1,2 and 5 grow in the following year, unless they are the ‘terminal’ bud of a floral GU (the bourse shoot bud), in which case they grow immediately (Crabbé and Escobedo, 1991). Buds of type 4 (sylleptic) always grow immediately. This categorisation also enabled the extraction of sequences representing the fate of the axillary buds along a GU, or the succession of GUs (fate of terminal buds) along a growth axis. These sequences could then be analysed to produce local-scale Markovian models of terminal and axillary bud fate. Local-Scale Markovian Models of Bud Fate 1. Markovian Model for Terminal Bud Fate. A stochastic Markovian model was used to predict the fate of terminal buds and thus the sequence of GU types along an axis. The basic model formulation is illustrated graphically in Figure 2. Formally, a Markov chain consists of a set of states, a set of probabilities governing which will be the initial or starting state, and a matrix of probabilities controlling the transitions between states. (Ephraim and Merhav, 2002). In this case, the ‘states’ correspond to the possible fates of a terminal bud: a long GU, a short GU, a medium GU, a floral GU or death. The type of the first GU of an axis is given by the hidden semi-Markov models described below, except for the trunk. The fate of the terminal bud is then controlled by a matrix of transition probabilities obtained by an analysis of the observed sequences of GUs along the axes. For example, the transition probability from a long GU to a floral GU in the model is given by calculating the proportion of long GUs in the database for which the following GU was floral. (GUs not followed by another GU were not considered). Based on previous data analysis (Costes et al., 2003), the probability of death each year is 0.3 for short GUs and zero for other GUs. 2. Hidden Semi-Markov Models for Axillary Bud Fate. A preliminary analysis of the observed sequences representing the fates of the axillary buds along a GU suggested a succession of distinct homogenous zones along all the GUs in the tree, similar to that observed for the trunk. For example, we might find a zone of mostly floral GU buds, followed by a zone of mostly dormant buds, followed by a zone of mixed long and medium GU buds. This in turn suggested the use of hidden semi-Markov chain (HSMC) models to predict these sequences. Formally, a hidden semi-Markov chain consists of a number of states, a set of initial probabilities and a matrix of transition probabilities, like a Markov chain. In addition, each state has an occupancy distribution and an observation distribution associated with it. In this case, the ‘states’ correspond to distinct homogenous zones along a GU; the initial and transition probabilities control the succession of zones found along a GU; the occupancy distribution controls the length of the zones; and the observation distribution controls the fates of the axillary buds within the zone, as shown in Figure 3. In the simulation, a previously developed HSMC model was adapted for use for the trunk (Costes and Guédon, 2002), and new models were developed for the branches.