Diurnal dynamics of phloem loading: theoretical consequences for transport efficiency and flow characteristics.

Phloem transport is the process by which plants internally distribute assimilates. The loading of assimilates near the photosynthetic source is responsible for generating enough osmotic pressure to drive sap flow towards the sink tissues where assimilates are consumed. Phloem loading is variable and subject to a diurnal cycle. It is dominated by photosynthesis during the day and by degradation of leaf starch to sugars at night. Most studies ignore the effect of the loading cycle on transport and assume that sugar flow operates at equilibrium. In this study, phloem transport was simulated for three successive days using a finite element model of time-dependent Münch-Horwitz equations. The spatial and temporal distributions of phloem pressure, sucrose concentration, sap velocity and sucrose flux were predicted for five different time variations in sucrose loading. Results showed that periodic loading induces an alternance of two distinct transport phases: one with high pressure, concentration and sucrose flux magnitudes and another with low magnitudes. In contrast, phloem water velocity remained remarkably stable. The alternating phases persisted over time and, under source-driven variation, transport did not reach steady-state conditions for the tested configuration. However, the impact of loading dynamics on transport was mitigated by pathway effects. Oscillations were not only delayed as one travelled away from the source, their amplitude was also reduced over distance. That behaviour stabilized the supply of sucrose to the sink, which continued at moderate levels during the dark cycles. This finding suggests that transport would assist night conversion of starch to sugars in the leaf to prevent carbon starvation at distant sinks in the early morning. The propagation velocity of pressure/concentration waves in phloem was predicted to vary by a factor up to 2.5 depending on the time series chosen to describe the dynamics of loading. Finally, the model predicted that up to 87% of the amount of sucrose loaded over 48 h would be unloaded under time-dependent loading, whereas only 76% would under constant-rate loading. This additional efficiency was periodic. It did not increase significantly the overall efficiency of the system but could be responsible for inducing rhythms in sink activity.

A surface model of nonlinear, non-steady-state phloem transport.

Phloem transport is the process by which carbohydrates produced by photosynthesis in the leaves get distributed in a plant. According to Münch, the osmotically generated hydrostatic phloem pressure is the force driving the long-distance transport of photoassimilates. Following Thompson and Holbrook[35]'s approach, we develop a mathematical model of coupled water-carbohydrate transport. It is first proven that the model presented here preserves the positivity. The model is then applied to simulate the flow of phloem sap for an organic tree shape, on a 3D surface and in a channel with orthotropic hydraulic properties. Those features represent an significant advance in modelling the pathway for carbohydrate transport in trees.

A mathematical framework for modelling cambial surface evolution using a level set method.

BACKGROUND AND AIMS: During their lifetime, tree stems take a series of successive nested shapes. Individual tree growth models traditionally focus on apical growth and architecture. However, cambial growth, which is distributed over a surface layer wrapping the whole organism, equally contributes to plant form and function. This study aims at providing a framework to simulate how organism shape evolves as a result of a secondary growth process that occurs at the cellular scale. METHODS: The development of the vascular cambium is modelled as an expanding surface using the level set method. The surface consists of multiple compartments following distinct expansion rules. Growth behaviour can be formulated as a mathematical function of surface state variables and independent variables to describe biological processes. KEY RESULTS: The model was coupled to an architectural model and to a forest stand model to simulate cambium dynamics and wood formation at the scale of the organism. The model is able to simulate competition between cambia, surface irregularities and local features. Predicting the shapes associated with arbitrarily complex growth functions does not add complexity to the numerical method itself. CONCLUSIONS: Despite their slenderness, it is sometimes useful to conceive of trees as expanding surfaces. The proposed mathematical framework provides a way to integrate through time and space the biological and physical mechanisms underlying cambium activity. It can be used either to test growth hypotheses or to generate detailed maps of wood internal structure.

Crown structure and wood properties: Influence on tree sway and response to high winds.

Wind can alter plant growth and cause extensive, irreversible damage in forested areas. To better understand how to mitigate the effects of wind action, we investigated the sensitivity of tree aerodynamic behavior to the material and geometrical factors characterizing the aerial system. The mechanical response of a 35-yr-old maritime pine (Pinus pinaster, Pinaceae) submitted to static and dynamic wind loads is simulated with a finite element model. The branching structure is represented in three dimensions. Factor effects are evaluated using a fractional experimental design. Results show that material properties play only a limited role in tree dynamics. In contrast, small morphological variations can produce extreme behaviors such as either very little or nearly critical dissipation of stem oscillations. Slender trees are shown to be relatively more vulnerable to stem breakage than uprooting. Dynamic loading leads to deflections and forces up to 20% higher near the base of the tree than those calculated for a static loading of similar magnitude. Effects of branch geometry on dynamic amplification are substantial yet not linear. The flexibility of the aerial system is found to be critical to reducing the resistance to the airflow and thus to minimizing the risk of failure.

A finite element model for investigating effects of aerial architecture on tree oscillations.

A finite element model was developed to study the influence of aerial architecture on the structural dynamics of trees. The model combines a complete description of the axes of the aerial architecture of the plant with numerical techniques suitable for dynamic nonlinear analyses. Trees were modeled on the basis of morphological measurements that were previously made on three 4-year-old Pinus pinaster Ait. saplings originating from even-aged stands. Calculated and measured oscillations were compared to evaluate model behavior. The computations allowed the characteristics of the fundamental mode of vibration to be estimated with satisfactory accuracy. Inclusion of a topological description of the aerial system in a mechanical model provided insight into the effect of tree architecture on tree dynamic behavior. Simplifications of the branching pattern in the model led to overestimations of the natural swaying frequency of saplings by 10 to 20%. Inadequate values of stem and root anchorage stiffness resulted in errors of 10 to 20%. Modeling results indicated that aerodynamic drag of needles is responsible for 80% of the damping in the studied trees. Additionally, damping of stem movement is reduced by one half when branch oscillations are not considered. It appears that the efficiency of the dissipative mechanisms depends directly on crown topology.

A mechanical analysis of the relationship between free oscillations of Pinus pinaster Ait. saplings and their aerial architecture.

The aim of this study was to investigate the influence of aerial architecture on the dynamic characteristics of young maritime pines (Pinus pinaster Ait.) using a mechanistic approach. For this purpose, three 4-year-old saplings with prominent differences in their branching patterns were submitted to free oscillation tests. The tests were carried out with different methods and directions of mechanical loading in order to initiate the movement of each sapling. The oscillations of the different architectural elements, i.e. stem and branches of different topological order, were measured with inclinometers and strain gauges fixed to saplings. Successive pruning of the architectural elements was carried out to evaluate their relative influence on the dynamic characteristics of the trees. The aerial systems were digitized before the mechanical tests in order to use 3D visualization techniques and to make architectural analyses of the crown structure. Two distinct modes of deformation were detected during free oscillations. The natural swaying frequency ranged from 0.6-0.8 Hz for the saplings tested at the same period of the year. The frequency variations were partly explained by the morphological differences of the experimental subjects. The motions of the axes were found to depend on their topology, i.e. the movement of the axes of a given branching order was forced by the movement of their respective bearing axis. The axes of third branching order had a significant and negative effect on the damping of the natural deformation mode. Results point out the major role played by foliage, qualitatively and quantitatively, on the damping of tree motions and on coupling the motions of the crown components.