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

Tree Physiology 30, 901–913 doi:10.1093/treephys/tpq034 Xylem anisotropy and water transport—a model for the double sawcut experiment PAUL J. SCHULTE1,2 and DAVID G. COSTA3 School of Life Sciences, University of Nevada, Las Vegas, NV 89154-4004, USA 2 Corresponding author () 3 Department of Mathematical Sciences, University of Nevada, Las Vegas, NV 89154-4020, USA Received December 7, 2009; accepted March 31, 2010; published online May 14, 2010 Summary 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. Keywords: anisotropic, conductivity, tensor, tracheid. Introduction Early studies of water flow through stems of large woody plants involved the creation of overlapping sawcuts to the stems of these trees in an attempt to understand the pathways for flow. For such studies utilizing either injected stains (Greenidge 1955) or radioactive tracers (Postlethwait and Rogers 1958), the cuts would presumably sever all direct vertical connections between roots and leaves, but nonetheless the leaves did not wilt. This result may have been thought initially to be evidence against the prevailing cohesion–tension theory of water ascent. However, careful experiments on the effects of distance between the cuts showed that cuts closer than some minimum separation associated with the length of individual vessels (occurring in the angiosperm species studied) did lead to wilting of the foliage because air bubbles were introduced, blocking flow in all vessels between the cuts (MacKay and Weatherly 1973). Thus, it became clear that flow around the cuts was made possible by lateral transfer between water-filled conduits of the xylem, although the conductivity to flow may be lower laterally than axially along the stem. Such differences in lateral and axial conductivity also make sense for plants with xylem conduits like tracheids based on their anatomical characteristics. These cells are long and narrow (of the order of a few millimeters in length and tens of micrometers in diameter). Tracheids are also imperforate, and flow from cell to cell depends on pits in the cell walls. Roughly speaking, passage through one set of pits allows flow through the length of one cell. But for lateral flow perpendicular to the axis of the cell, flow through the pits of one cell only achieves a distance of one cell diameter. In addition among conifers with tracheids, pits are far more common on radial walls of the cells than on tangential walls (Siau 1971, Panshin and de Zeeuw 1980; radial walls are those oriented along a radial line out from the stem center). This consideration would suggest that lateral conductivity might be greater in the tangential direction than in the radial direction. Thus, based on the anatomy of conducting cells, one would obviously expect the xylem to be an anisotropic conducting medium, where conductance to flow depends on direction. One expression of the anisotropic nature of plant xylem arises with the term sectoriality. If the lateral conducting ability across a cylindrical stem is sufficiently limited, the stem xylem is effectively isolated into sections tangentially around the stem. Numerous studies have suggested that sectoriality is variable among plants. Larson et al. (1993, 1994) have shown that long-lived individuals (>1000 years) of northern white © The Author 2010. Published by Oxford University Press. All rights reserved. For Permissions, please email: 1 902 SCHULTE AND COSTA cedar (Thuja occidentalis L.) found on limestone cliffs along the Niagara Escarpment in southern Ontario, Canada are highly sectored—individual roots are strongly connected with individual branches, and this characteristic may be important for their survival. Other studies have suggested that the xylem is well connected around the stem, and this degree of interconnectedness may be an important characteristic of plants for their adaptation to various environments (Zanne et al. 2006, Taneda and Tateno 2007). Therefore the previously cited experiments with overlapping cuts in the stem (after which at least some flow still continues), along with the more recent studies of sectoriality just described, suggest that the capability for lateral flow as well as flow along the stem is an important property of stems deserving further study. The purpose of the present work was to develop an approach for incorporating the anisotropic nature of the xylem into a transport equation for flow and also to test this model experimentally. We revisit the double sawcut experiment by measuring the hydraulic conductance of stems while a series of stepwise cuts are made into the stem and from a mathematical perspective by creating models of cylindrical stems with cuts of various depths. Although this is obviously an artificial system, the authors hope to show that it can be useful for developing an approach incorporating anisotropic conductivity that will be applicable in more complex but naturally occurring systems such as branch junctions. Theoretical development Anisotropy and the hydraulic conductivity tensor Mathematical descriptions and models have been useful tools for understanding the transport process in plants (Pickard 1981, Nobel 1991). For 1D approaches including branched 1D electrical circuit analog models, flow resistances or con- ∂P ∂P + kxy ∂x ∂y ∂P ∂P + kyy −qy = kyx ∂x ∂y −qx = kxx ð1Þ where qi is the flow (volume flux; m3 m−2 s−1 or m s−1) in the direction i, P is the pressure (MPa) or other driving force for flow in the x or y direction, and kij (...truncated)