Solid mechanics of the torus-margo in conifer intertracheid bordered pits.

Bordered pits of many conifers include a torus-margo structure acting as a valve that prevents air from spreading between tracheids, although the extent of torus deflection as a function of applied pressure is not well known. Models were developed from images of pits in roots and stems of Picea mariana (Mill.) BSP. A computational solid mechanics approach was utilised to determine the extent of torus deflection from pressure applied to the torus and margo. Torus deflection increased in nonlinear fashion with applied pressure. The average pressure required for sealing the pit was 0.894 MPa for stems and 0.644 MPa for roots, although considerable variation was apparent between individual pits. The pits of roots were wider and deeper than those of stems. For stems, the pit depth did not increase with pit width; thus the torus displacement needed to seal the pit was less than for pits from roots. The pressure required to seal the pit depends upon anatomical characteristics such as pit width and pit depth. Although the torus displacement for sealing was greater for roots because of their greater depth, the pressures leading to sealing were not significantly different between roots and stems.

Computational models evaluating the impact of sieve plates and radial water exchange on phloem pressure gradients.

The sugar conducting phloem in angiosperms is a high resistance pathway made up of sieve elements bounded by sieve plates. The high resistance generated by sieve plates may be a trade-off for promoting quick sealing in the event of injury. However, previous modeling efforts have demonstrated a wide variation in the contribution of sieve plates towards total sieve tube resistance. In the current study, we generated high resolution scanning electron microscope images of sieve plates from balsam poplar and integrated them into a mathematical model using Comsol Multiphysics software. We found that sieve plates contribute upwards of 85% towards total sieve tube resistance. Utilizing the Navier-Stokes equations, we found that oblong pores may create over 50% more resistance in comparison with round pores of the same area. Although radial water flows in phloem sieve tubes have been previously considered, their impact on alleviating pressure gradients has not been fully studied. Our novel simulations find that radial water flow can reduce pressure requirements by half in comparison with modeled sieve tubes with no radial permeability. We discuss the implication that sieve tubes may alleviate pressure requirements to overcome high resistances by regulating their membrane permeability along the entire transport pathway.

Pit membrane structure is highly variable and accounts for a major resistance to water flow through tracheid pits in stems and roots of two boreal conifer species.

The flow of xylem sap in conifers is strongly dependent on the presence of a low resistance path through bordered pits, particularly through the pores present in the margo of the pit membrane. A computational fluid dynamics approach was taken, solving the Navier-Stokes equation for models based on the geometry of pits observed in tracheids from stems and roots of Picea mariana (black spruce) and Picea glauca (white spruce). Model solutions demonstrate a close, inverse relationship between the total resistance of bordered pits and the total area of margo pores. Flow through the margo was dominated by a small number of the widest pores. Particularly for pits where the margo component of flow resistance was low relative to that of the torus, pore location near the inner edge of the margo allowed for greater flow than that occurring through similar-sized pores near the outer edge of the margo. Results indicate a surprisingly large variation in pit structure and flow characteristics. Nonetheless, pits in roots have lower resistance to flow than those in stems because the pits were wider and consisted of a margo with a larger area in pores.

Computational fluid dynamics models of conifer bordered pits show how pit structure affects flow.

• The flow of xylem sap through conifer bordered pits, particularly through the pores in the pit membrane, is not well understood, but is critical for an understanding of water transport through trees. • Models solving the Navier-Stokes equation governing fluid flow were based on the geometry of bordered pits in black spruce (Picea mariana) and scanning electron microscopy images showing details of the pores in the margo of the pit membrane. • Solutions showed that the pit canals contributed a relatively small fraction of resistance to flow, whereas the torus and margo pores formed a large fraction, which depended on the structure of the individual pit. The flow through individual pores in the margo was strongly dependent on pore area, but also on the radial location of the pore with respect to the edge of the torus. • Model results suggest that only a few per cent of the pores in the margo account for nearly half of the flow and these pores tend to be located in the inner region of the margo where their contribution will be maximized. A high density of strands in outer portions of the margo (hence narrower pores) may be more significant for mechanical support of the torus.

Xylem anisotropy and water transport--a model for the double sawcut experiment.

Early experiments with overlapping cuts to the stems of trees demonstrated that lateral flow within the stem must be possible to allow such trees to maintain water flow to their leaves. We present a mathematical approach to considering lateral flow in stems by treating the xylem as an anisotropic medium for flow and develop an expression of its conductivity in the form of a tensor. In both 3D models of tracheid-bearing stems with cuts (incorporating this tensor analysis) and experimental stems with steadily deepening cuts, it is shown that flow can continue despite the presence of even strongly overlapping cuts through 90% of the stem. Such remaining conducting ability was, however, strongly dependent on values for radial and tangential conductivity (conductivity to lateral flow across the stem either radially with respect to the central axis or tangentially to the stem surface). Furthermore, the lateral flow around obstructing cuts was more dependent on tangential flow around the stem upstream and downstream of the cuts than on radial flow across the stem. The relative importance of tangential flow could be accounted for by a greater tangential conductivity, perhaps related to the predominance of pits on radial walls of tracheids, and the presence of non-conducting pith and early growth rings in the stems. These results demonstrate that a consideration of anisotropy in transport properties of the xylem will be important for future studies of flow in stems around naturally occurring geometric features such as branching points.

Water flow through junctions in Douglas-fir roots.

Roots are important conduits for the redistribution of water within the rooting zone. Root systems are often highly branched, and water flow between regions undoubtedly involves passage through junctions between individual roots. This study considered junctions in the roots of Douglas-fir with regard to the resistances encountered by water flow through the xylem. Flow into the root branch distally along the main root encountered much greater resistance than flow into the branch and proximally along the main root (toward the plant stem). When the main root proximal to the junction was gradually shortened, the resistance to flow in the branch root and distally along the main root increased dramatically. Thus, flow in this manner appears to depend on lateral flow within the root over many centimetres proximal to the junction and not just within the direct connection at the junction. These results suggest that the hydraulic nature of junctions is an important aspect of hydraulic redistribution of water within the soil utilizing flow through roots.

Incorporation of transfer resistance between tracheary elements into hydraulic resistance models for tapered conduits.

The model of West, Brown and Enquist (1999) shows that hydraulic resistance in trees can be independent of path length, provided that vascular conduits widen sufficiently from tree top to base. We demonstrate that this result does not depend theoretically on branching architecture or cross-sectional conductive area of the stem. Previous studies have shown that pit membrane resistance, encountered when water moves between either tracheids or vessels, accounts for up to 60% of the total resistance in stem segments. When pit membrane resistance, which is neglected by most whole-tree hydraulic models, was incorporated in hydraulic models in three different ways, the near invariance of hydraulic resistance was preserved. If relative pit resistance was independent of tracheid size or if tracheid dimensions were scaled to minimize wood resistivity, the minimum conduit taper required for path length independence equaled that in the original model of West et al. (1999). Under the most realistic model, in which relative pit resistance increased with tracheid radius, this value was doubled. Such taper is not possible within the typical size range of tracheids over the entire length of moderately tall trees, but it might be possible for vessel-bearing trees. Preliminary results indicated that although tracheid radius in the outer growth ring initially increased basipetally from the top of an 18-m tall Douglas-fir (Pseudotsuga menziesii (Mirb.) Franco), it stabilized at mid-trunk. Also, conduit taper was not constant in this species, violating a key assumption of the model of West et al. (1999), on which the invariance of hydraulic resistance depends.

Branch junctions and the flow of water through xylem in Douglas-fir and ponderosa pine stems.

Water flowing through the xylem from the roots to the leaves of most plants must pass through junctions where branches have developed from the main stem. These junctions have been studied as both flow constrictions and components of a hydraulic segmentation mechanism to protect the main axes of the plant. The hydraulic nature of the branch junction also affects the degree to which branches interact and can respond to changes in flow to other branches. The junctions from shoots of two conifer species were studied, with particular emphasis on the coupling between the downstream branches. Flow was observed qualitatively by forcing stain through the junctions and the resulting patterns showed that flow into a branch was confined to just part of the subtending xylem until a considerable distance below the junction. Junctions were studied quantitatively by measuring flow rates in a branch before and after flow was stopped in an adjacent branch and by measuring the hydraulic resistance of the components of the junction. Following flow stoppage in the adjacent branch, flow into the remaining branch increased, but considerably less than predicted based on a simple resistance analogue for the branch junction that assumes the two branches are fully coupled. The branches downstream from a junction, therefore, appear to be limited in their interconnectedness and hence in their ability to interact.