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


Author for correspondence:
Paul J. Schulte
Tel: +1 702 895 3300


  • 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.


Water flow along the xylem of tracheid-bearing conifers depends on flow through inter-tracheid bordered pits. Bordered pits include an extension of the secondary cell wall which arches over the pit, leaving a pit canal on either side, which is considerably narrower than the width of the pit membrane (see, for example, Zimmermann, 1983). A nonconducting structure, called a torus, located at the center of the pit membrane, is found widely among gymnosperms, although not in all taxa (Bauch et al., 1972). The borders forming the pit canals, together with the torus, appear to act as a valve that may prevent the spread of embolism between tracheids by displacement of the torus against the pit borders (Domec et al., 2006). The torus is held in place by a flexible margo formed from microfibril strands, and Delzon et al. (2010) have suggested that cavitation resistance depends on margo flexibility in addition to the degree of overlap between the torus and adjacent pit borders. However, Hacke et al. (2004) have suggested that a highly flexible margo may allow the torus to slip from a location blocking the pit, thus allowing an embolism to spread between tracheids. In addition to the torus and margo structure acting as a valve that may block the spread of embolism, the open spaces between these margo strands form pores through which water flows in crossing the pit membrane. The pores in pit membranes of conifers are readily observed by scanning electron microscopy (SEM), whereas most angiosperm pit membranes have much smaller pores that have only recently begun to be observed directly. As a result of the more porous nature of conifer pit membranes, they appear to have considerably lower resistivity to flow relative to angiosperm pit membranes (3.3–43-fold, Hacke et al., 2004; nearly 60-fold, Pittermann et al., 2005). Broadly speaking, the pits account for at least 50% of the total xylem resistance among tracheid-bearing species (Choat et al., 2008). All of these pit components, including the pit canals formed by the pit borders, the torus obstructing flow and the pores of the margo region of the pit membrane, contribute to the overall flow resistance from tracheid to tracheid through the pits.

Detailed studies of the structure of conifer pits and pit membranes date back to at least Bailey (1913), with attempts to estimate membrane pore sizes by forcing solutions with particles through the xylem, or utilizing gases and fluids for this purpose (Petty & Preston, 1969). More recent studies have combined measurements of pit dimensions and pit membrane pores from SEM with either experimental methods or analytical approaches to estimate the contribution of various pit components to the flow resistance (Choat et al., 2008). Estimates are available in the literature for pit membrane resistivity, but these have not typically been obtained from direct measurements, but as an unknown component of a model to be determined by solution of the model (e.g. Schulte & Gibson, 1988). Models have also been developed with approximations of pit membrane structure (Lancashire & Ennos, 2002; Hacke et al., 2004), but the physical structure of these models does not appear to closely match real pit membranes that have been observed. Valli et al. (2002) have developed a model based on the fluid dynamics of a pit with detailed structure for the pit canal and torus, although the pit membrane had to be treated as a porous medium without the specific structural details of actual pit membranes.

Engineering studies of fluid flow have long utilized computational approaches to solve the flow equations. Recent advances in the technology for the development of microscale mechanical or chemical devices have given rise to fields such as microfluidics and even nanofluidics (Abgrall & Nguyen, 2009; Colin, 2010). Aspects of the background and approaches in these fields should be useful for studying flow in the similar sized subcellular components of conducting cells in the xylem of plants. Therefore, the purpose of this study was to develop models of individual pits of conifer tracheids having borders and a torus–margo structure based on SEM images showing details of these features.


Theoretical background

A general description of fluid flow can be found in the Navier–Stokes equation, based on Newton’s Second Law, combining terms describing the acceleration of the fluid and forces acting on the fluid (Munson et al., 1990). Such forces include pressure, shear forces within the fluid and perhaps even gravitational forces. An expression of this equation for steady-state flow (constant velocity with respect to time) and neglecting gravity is:

image(Eqn 1)

(V, velocity vector (m s−1); p, pressure (Pa); μ, fluid dynamic viscosity (Pa s); ρ, fluid density (kg m−3)). A continuity expression is also needed, here assuming an incompressible fluid with constant density:

image(Eqn 2)

In both equations, the symbol ∇ refers to a partial gradient with respect to the spatial variables (x, y, z for a three-dimensional model). One solution of the Navier–Stokes equation for a special case of steady-state laminar flow in a circular tube has appeared in the plant literature as the Hagen–Poiseuille equation (see Munson et al. (1990) for derivation).

The use of the Navier–Stokes equation assumes that the fluid can be treated as a continuous medium that is indivisible with properties such as pressure, which vary continuously in space within the fluid. Further, it is usually assumed that walls present in the model give rise to a no-slip condition, such that fluid velocity is zero at the wall. Although these assumptions have been well supported in many decades of engineering studies, one may question whether they are appropriate at the spatial scales encountered when water passes through pores in the conifer pit membrane. As the size of the physical structures associated with fluid flow approaches the size of the molecules or the distances over which molecules move before collisions occur, the continuum assumption breaks down and the fluid may have to be treated as being composed of individual particles that interact through collisions with each other or the boundaries of the structures (molecular dynamics or free molecular flow). An important dimensionless quantity in fluid mechanics at these microscales is the Knudsen number (Kn), typically expressed for gases as the ratio of the mean molecular free path distance to a characteristic length of the structures through which flow is occurring. The mean free path is well defined for gases and, generally, for Kn > 10, free molecular flow must be considered. However, for Kn < 0.001 or Kn < 0.01 (depending on the author), the continuum approach appears to be accepted. For liquids, intermolecular forces, such as the hydrogen bonds between water molecules, complicate the concept of the molecular free path and the calculation of Kn. Molecular spacing may be the relevant quantity for liquids, suggesting a value for water of c. 0.3 nm (Sharp et al., 2005), leading Abgrall & Nguyen (2009) to conclude that flow in nanochannels with a characteristic length of more than 10 nm can be adequately described as a continuum by the Navier–Stokes equation. For dimensions in between those that clearly involve free molecular and continuum flow, a continuum approach with a slip flow condition applied to the boundaries can be considered. Typically, for a Navier–Stokes application, a no-slip condition is applied to all boundaries, such that the flow velocity parallel to the boundary is zero (flow perpendicular to the boundary is also zero to reflect a nonleaking wall). If boundaries are treated with a slip condition, shear forces do not develop at the wall and, in principle, flow could occur without resistance or loss of pressure. For the models developed in this study, solutions were also obtained with the slip condition applied to the walls of margo pores in order to determine what affect this condition would have on the overall pressure drop through the pit.

Model development

The basic conceptual model included a single pit located in the walls between two rectangular tracheids (Fig. 1). Preliminary models with this full structure showed that almost all the pressure and flow changes occurred near the pit and, because the models were intended to focus on the pits, the models presented here include a cylindrical region before and after the pit extending about halfway across the tracheid. The shapes of the pit borders (Fig. 2) were based on images of Picea mariana (Mill.) BSP. pits in Hacke & Jansen (2009). Models including the full pit membrane were based on an SEM image of a pit from a sun-developed tree and a shade-developed tree supplied by Uwe Hacke and Amanda Schoonmaker (University of Alberta, Edmonton, AB, Canada). Although the models reflect the differences found in basic pit structure between these growth habits (see Schoonmaker et al., 2010), the present purpose was to explore the ability to create realistic models of this structure and to determine the contribution of various pit components to the total resistance of the pit. The pit margo components, including the torus and pit membrane pores, were developed as two-dimensional drawings by importing the pit membrane SEM into a scaled AutoCAD drawing and then manually creating polygons for the walls of each pore. Measurements of the margo strands suggested a pore depth of 0.05 μm, and so this pore depth was created for each pore by a process of extruding the pore polygons 0.05 μm along the axis of the model. Additional details of the geometry of the model are given in Supporting Information Figs S1–S6.

Figure 1.

Conceptual diagram of a bordered pit located in the walls between two adjacent tracheids. The inset on the right shows a cut-away view with details of the pit border shapes, the torus and many of the margo pores.

Figure 2.

Sun and shade models showing a cross-section through the pit canal with associated borders and torus. Dimensions are shown between various points with units of micrometers.

Fluid flow models were developed using Comsol Multiphysics software (Comsol, Inc., Los Angeles, CA, USA), a general program for the solution of partial differential equations (here, the Navier–Stokes and continuity equations), including the development of the model geometry and its finite element mesh (Figs S1–S6). Detailed regions of the model, such as the pit borders and pit membrane pores, were drawn in AutoCAD and imported into the model within the Comsol software. Simulations utilized a Dell T7500 64-bit workstation with two quad-core processors and 60 GB of memory. Solution requirements varied from 10 to 75 min and 10 to 56 GB of memory.

All wall faces were set as no-slip boundaries, except for the inlet and outlet faces of the model. The outlet boundary was set as a pressure condition with a pressure of zero. It should be noted that this boundary pressure is somewhat arbitrary: a negative pressure could have been used that would have led to negative pressures throughout the model; however, flow velocities and pressure gradients would have remained unchanged. The inlet boundary was set as a constant velocity of 0.1 mm s−1 normal to the inlet. Model solutions then give velocities at all points within the model and the pressures required to support these velocities. The average inlet pressure (minus the zero pressure at the outlet) divided by the total volume flow through the model expresses a resistance with units of Pa s m−3. It should be noted that, at these low flow velocities, the chosen inlet velocity is somewhat arbitrary because the resulting pressure gradients are directly proportional to the inlet velocity (verified in preliminary simulations with inlet velocities spanning two orders of magnitude). Thus, the calculated resistances will be independent of the inlet velocity. An estimate of the relative proportions contributed by each pit component to the total flow resistance of the border pit with torus and margo can be produced by assuming that the pit canals, the regions between the torus and the pit borders and the margo pores are, roughly speaking, in series. Thus, for example, the effect of the margo itself could be expressed by subtracting the resistance of the model with only pit canals and torus from the full model with margo, canals and torus. Further, the margo resistance can be multiplied by its surface area to produce a resistivity that may represent a more intrinsic property of the margo region of the pit membrane. The side walls of the inlet and outlet regions of the model (inlet and outlet cylinders extending halfway across the tracheid; see Figs S1, S3 for more details) do not correspond to a physical element of actual tracheids and were therefore set to a slip condition. This means that they do not generate shear forces in the fluid, more closely simulating a large inlet and outlet region in which the pressure and velocity changes only occur as a result of the effects of the pit borders and adjacent walls (which have no-slip conditions). The fluid density was 998 kg m−3 with a dynamic viscosity of 1.002 × 10-3 Pa s, corresponding to water at 20°C.


Pit borders and torus

Initial models focused on the effects of the pit borders and torus on flow through the pit (Fig. 3). Without the torus, the spaces between the adjacent borders on the same side of the canal contributed little to flow. The presence of a torus not surprisingly diverts flow around the torus and into the region between the pit borders. The pit canal between the ends of the borders is a region in which flow is focused narrowly, leading to the highest velocities. Although the margo is not present in these models, it is clear that flow occurs preferentially near the edge of the torus as opposed to the narrow confined spaces at the edges of the pits. Integrating the velocity of flow over this area (in the plane of the torus) indicates that more than 99% of the flow volume occurs through the inner half of the region (towards the edge of the torus) for both the sun and shade models.

Figure 3.

Velocity of flow (mm s−1) through the pits with and without a torus present, but no margo. Results from the sun model without a torus (a) and with a torus (b). Results from the shade model without a torus (c) and with a torus (d). The white arrows represent velocity vectors whose length is proportional to velocity for a location at the base of the arrow.

Margo pores

Models were then further developed to add the components of the margo. Using the SEM images of the margo, polygons were drawn along the margo strands forming the edges of the pores (Fig. 4). Pores were added to the margo region of the models in a stepwise fashion, so as to be able to see how additional pores affected the overall model solution. In general, the larger pores were added earlier in the model development process, such that, after three or four steps, only the smallest pores were being added, although in large numbers. The most extensive models (pores shown in Fig. 4) had 1078 and 894 pores for the sun and shade models, respectively. The relationship between the total pressure drop in the model and the total area of pores present in the model suggested that the further addition of the very small pores remaining would not lead to a substantial further reduction in pressure drop (Fig. 5). The margo pores of the sun model had a smaller mean pore area relative to those of the shade model (averaging 0.02196 vs 0.02743 μm2 per pore).

Figure 4.

Scanning electron microscopy (SEM) images of a pit membrane from a sun-grown (a) and a shade-grown (b) Picea mariana tree. Lines were drawn along the margo strands to form polygons demarcating the pores. Bars, 1 μm. The sun pit was 13 μm in diameter, and the shade pit was 10 μm in diameter.

Figure 5.

Pressure drop (inlet pressure minus outlet pressure) for sun (circles) and shade (squares) models as pores were gradually added to the model, thereby increasing the total pore area. The largest pores were added in the first three or four steps (leftmost points) and therefore the later steps involved adding large numbers of small pores.

The model solutions were used to create plots of flow velocity through the margo pores (Fig. 6). The maximum velocity for an individual pore appears to depend strongly on the size of the pore. The flow velocity was integrated through individual pores and the area of each pore was also calculated. As an illustration of the significance of large pores, for the sun model, the largest single pore (located near the inner edge of the margo at the top left of Fig. 6a) accounted for 2.1% of the total pore area and 8.6% of the total flow through the entire pit membrane. The 20 largest pores with 16% of the pore area accounted for 39% of the entire flow through the pit membrane. A corresponding analysis of the shade model showed that the single largest pore (located on the left in Fig. 6b) had 1.8% of the pore area and accounted for 5.9% of the total flow. The 20 largest pores in the shade model had 20% of the total pore area and accounted for 39% of the total flow through the pit membrane.

Figure 6.

Velocity of flow (mm s−1) through the margo pores for the sun (a) and shade (b) models. Note that the velocity scales to the right of each image reflect a different maximum flow velocity. The model inlet velocity was the same in both cases (0.1 mm s−1), but, because the sun pit was wider, the total volume flow through the sun pit was 1.7-fold greater than that through the shade pit.

As a further analysis of the flow through individual pores in the margo, the pore area, location in the membrane and integrated flow through the pore for c. 120 of the largest pores in the sun and shade models were determined. The pore location was expressed as the position radially out from the pit center of the centroid for each polygon representing a pore. The volume flow through the pore was nonlinearly related to the pore area (Fig. 7). However, it is also apparent that the radial location of the pore is important, with pores near the inner edge of the margo showing higher volume flow than equivalent area pores further out from the torus (also suggested by the velocities shown in Fig. 6). Integration of the flow velocity over the margo showed that 77.2% of the flow volume for the sun model and 83.1% of the flow volume for the shade model occurred within the inner half of the margo. Model solutions also allowed for the calculation of the Reynolds number (ratio of inertial to viscous forces). This quantity was greatest within the largest pore and in the narrowest portion of the pit canal: for the sun model, 0.0086 and 0.023, respectively, and, for the shade model, 0.0032 and 0.023, respectively. Such low Reynolds numbers reflect a laminar flow regime where viscous forces dominate.

Figure 7.

Volume flow (velocity integrated over the area of the pore) through individual pores as a function of the pore area for the sun (a) and shade (b) models. The symbol size indicates the location of the centroid for each pore as being near the inner (small symbols) or outer (large symbols) edge of the margo.

Pit components and resistance

The significance of the pit components could be determined from the pressure drops and volume flows through the various models (Table 1). Resistance to flow was calculated by dividing the pressure drop by the volume flow rate. The models without pit borders, or with pit borders alone, showed very low pressure drops and corresponding low resistances for flow through these models. As the torus and margo components were added to the models, pressure drops and flow resistances increased. From a knowledge of the resistance for each pit component, the fraction of total flow resistance attributable to each component could be calculated (Table 2). For the pits studied here, the pit canal accounted for the smallest fraction of resistance. However, the pit torus and margo were more significant, but varied in dominance depending on the sun vs shade pit as modeled. The margo dominated for the sun pit (63%), whereas the torus component dominated (43%) for the shade pit. The flow resistance of thin membrane-like structures can be expressed as an intrinsic quantity for that structure by multiplying the resistance by the area, giving a resistivity (Table 3). Here, the resistivity is expressed in units of MPa s m−1, which is perhaps more commonly found in the literature for pit membranes. It should be noted that two forms are presented based on the total pit membrane area (margo and torus combined) or the area of the margo alone.

Table 1.   Pressure drop (Δp) along the model and calculated resistance (R) for various pit components
Δp (Pa)R (Pa s m−3)Δp (Pa)R (Pa s m−3)
  1. Models were developed for a pit without borders, with borders forming a canal, with a torus present in addition to the canal and with a margo connecting the torus to the pit edge. Resistance was calculated from the model pressure drop (pinlet – poutlet) divided by the volume flow through the model (sun model, 5.30 E–14 m3 s−1; shade model, 3.14 E–14 m3 s−1). The pressure drop and resistance for the ‘Torus alone’ condition is obtained by subtracting values in the ‘Canal’ condition from the ‘Torus and canal’ condition. The pressure drop and resistance for the ‘Margo alone’ condition is obtained by subtracting values in the ‘Torus and canal’ condition from the ‘Margo, torus and canal’ condition.

No border0.7841.485 E+131.0913.480 E+13
Canal37.967.162 E+1486.812.769 E+15
Torus and canal232.74.391 E+15292.09.302 E+15
Margo, torus and canal634.51.196 E+16473.41.508 E+16
Torus alone194.73.674 E+15205.26.535 E+15
Margo alone401.87.569 E+15181.45.777 E+15
Table 2.   Fraction of resistance (%) attributable to the three components of the bordered pit
Fraction of resistance (%)Fraction of resistance (%)
  1. The fractional resistance was calculated from the resistance of a particular pit component divided by the sum of all three component resistances.

Table 3.   Resistivity (resistance per unit area) of the pit margo
ModelResistivity (margo area) (MPa s m−1)Resistivity (membrane area) (MPa s m−1)
  1. Resistivity is calculated on a margo area basis by multiplying the margo resistance by the area of the margo (sun model, 99.5492 μm2; shade model, 58.9049 μm2), or on a total membrane area basis by multiplying the margo resistance by the area of the entire pit membrane combining the margo and torus (sun model, 132.7323 μm2; shade model, 78.5398 μm2).


Slip boundary conditions

An assessment was also made of the possible effect of setting a slip boundary condition on the side walls of the pores for the sun model. To make the process of setting the boundary condition manageable, the sun model with 115 of the largest margo pores was used for this analysis. Four versions of this model were developed with a range of depths for the margo pores from 25 to 100 nm, and solved with the usual no-slip condition or the slip condition on the edge faces of the pores. For the no-slip condition, the pressure drop through the model increases linearly with pore depth (Fig. 8). It should be noted that the relationship does not have a zero intercept: an infinitely thin pore would still produce a pressure drop because of shear forces within the water accelerating into the pore. For the slip condition, the model with the thinnest pores showed essentially the same pressure drop as obtained from the no-slip condition. As the pore depth increases, the pressure drop with the slip condition does not increase with depth as greatly as with the no-slip condition. As noted earlier, a slip condition precludes the development of shear with the boundary wall: except for the entrance effect (acceleration of fluid into the pore), the slip pore shows no pressure drop along the pore. For the pore depth of 50 nm (used generally in the models with margo pores), the slip condition leads to a 4% lower pressure drop relative to using the no-slip condition.

Figure 8.

Comparison of model pressure drop through the pit depending on the thickness of the margo. Results are shown for the no-slip (circles) and slip (squares) conditions applied to the faces at the edges of the margo pores. Models were based on the sun pit with 115 margo pores, including the largest pores observed for this pit.


Model solutions for the two sample pits from Picea mariana allowed for an assessment of the role of a variety of structural features of these bordered pits on water flow. The presence of a torus at the center of the pit acts to divert flow around the torus and into the spaces between the pit borders. Without a margo, nearly all the flow would occur near the torus edge. Of course, the torus must be supported by a margo, but, even when the margo was added to the models, over three-quarters of the flow around the torus occurred through the inner half of the margo. The size of the pores in the margo had a strong effect on the flow through each pore. As a result, only a few per cent of the pores (the largest) accounted for nearly half of the flow through the margo. A nonlinear relationship between pore size and flow is to be expected, given a predicted diameter to the third power relationship with flow for an isolated pore in an infinitely thin plate (Vogel, 2003), although the pores in a pit membrane margo are not isolated from one another. In addition, the pore location was significant: pores near the inner edge of the margo showed two- to three-fold greater flow than equivalent area pores at the outer edge. Pittermann et al. (2010) also suggested that a small number of large pores in the margo might have a great impact on the overall membrane resistance.

The pit canal accounted for 6% and 18% of the flow resistance of the pit for sun and shade models, respectively. This difference probably arose because the pit canal was wider in the sun model (4.0 μm) than in the shade model (2.6 μm). For both models, the presence of a torus and margo accounted for the majority of flow resistance through the pit. However, which component dominated appeared to depend on the structure of the pit. The sun pit was wider with more space between the torus and pit borders where they overlapped. In addition, the pores in the sun pit margo were more numerous, but narrower, than those in the shade pit. As a result, the sun model showed a greater role for the margo in total pit resistance. Domec et al. (2006) estimated that not more than one-quarter of the pit resistance could be attributed to the pit membrane in studies of Douglas-fir; however, their images of the pit membranes suggest that the pores in the margo for that species were considerably larger than those found in the present study. Choat et al. (2008) cite a contribution of < 5% for the pit canal from a variety of studies. The modeling study of Valli et al. (2002) predicted a 25% role for the pit canal, a 25% contribution by the margo and the remainder caused by the interior structure of the pit between its borders. It is likely that much of this variation in pit component contribution over many studies is the result of a correspondingly wide range in the relative dimensions of pit structures, including margo arrangement. A variety of pit border shapes can be found among conifers with bordered pits (e.g. see Hacke & Jansen, 2009; Pittermann et al., 2010). Many pits show fairly straight borders, such as applied for the present study based on P. mariana pits. Others, however, have a pronounced curvature to the pit border and even a thinning of the border near the tip adjacent to the pit canal. One might speculate that the curved borders would increase the space between the torus and the pit borders and allow lower resistance for flow around the torus, perhaps also increasing the flow through pores in the outer regions of the margo. Further studies with a set of species having a range of pit border shapes would be useful to address questions about the significance of such shapes.

The calculated margo resistances were also expressed here as the resistivity of the margo (resistance per unit area). For the sun model, with its somewhat narrower pores, the margo resistivity (margo resistance multiplied by margo area) was 0.75 MPa s m−1, compared with 0.34 MPa s m−1 for the shade model. Using the entire pit membrane area (margo and torus) as the area basis, margo resistivity values were 1.01 and 0.45 MPa s m−1 for the sun and shade pit models, respectively. These values are at the higher end of the wide range for pit membranes predicted by Hacke et al. (2004), but close to the estimate of 0.4 MPa s m−1 by Lancashire & Ennos (2002) and within the range of 0.2–10 MPa s m-1 or greater cited by Choat et al. (2008). Given the porous nature of these pit membranes relative to those of angiosperms, it is not surprising that the estimates of membrane resistivity and entire pit resistance are three to four orders of magnitude lower than those cited for some angiosperms (Choat et al., 2006). In studies of 26 woody angiosperm species, Jansen et al. (2009) found maximum pit membrane pore sizes from 10 to 225 nm, c. 20-fold smaller than the largest pores observed here, probably accounting for the differences in pit membrane resistivity between conifer and angiosperm pits.

The torus–margo style of bordered pits in many conifers is thought to provide a valve effect to reduce the spread of air from one embolized tracheid to others (Zimmermann, 1983; Choat et al., 2008). Therefore, the evolution of these structures might involve tradeoffs between the functions of water flow and embolism isolation. The presence of the torus at a location in the pit that would otherwise allow for high flow obviously has a dramatic effect of increasing resistance. However, a margo with wide pores, such as is typically found in many conifers, appears to somewhat balance this increased resistance by allowing a low-resistance pathway around the torus. Of course, for the torus to act as a valve, it must be mechanically bound to the pit edges with some strength, although in a flexible manner (Delzon et al., 2010). The margo pores in the pit membranes used for the development of the present models were somewhat unevenly distributed, with larger pores clustered towards the inner edge of the margo near the torus. A particularly nice SEM image of a conifer pit membrane, published by Zimmermann (1983), appears to show a similar distribution of pores. As noted earlier, the model results presented here indicate that the inner half of the margo provides for over three-quarters of the flow. The resistance created by the torus and pit borders leads to the ‘corner’ at the edge of the pit having little significance for flow. Perhaps a high density of margo strands in the outer regions of the margo could provide high mechanical support without producing a high flow resistance through the pit because this portion of the margo is less significant for flow. Thus, the structure of the margo may, to some extent, be optimized to provide high fluid flow, while still providing the strength for mechanical support of the torus.

In principle, one could expand upon the present models to use the pressures on the torus to calculate forces that might be associated with deflection of the torus. Moving mesh methods could be employed, whereby the margo and torus structures (and their finite element mesh) are deformed as a function of the model solution (pressures at the torus faces, for example). Domec et al. (2007) have suggested that apparent changes in conductivity with a change in applied pressure may be a result of fluid flow effects on the torus–margo. Sperry & Tyree (1990) showed similar results among several conifers, whereby conductivity was dependent on flow rate. One might also be able to consider a two-phase flow regime with a liquid and air interface. Such models coupling fluid and solid mechanics and employing a moving mesh could consider how the pit membrane structure would be deformed by flow and also by forces generated at the air–water interface. It would be particularly important to ensure that biologically relevant pressures and flow velocities were used in the model because pit membrane deflection would introduce a nonlinearity, such that flow might not be directly proportional to the pressure as in the present models.

Models employing the slip condition at the margo pore side walls resulted in a small (4%) decrease in pressure drop through the pit. Given the suggestion in the engineering literature (Sharp et al., 2005; Abgrall & Nguyen, 2009) that the Navier–Stokes equation is valid for even the narrowest pores in the present models and the dominance for flow of the larger pores in the margo of these pits, the no-slip condition for pore boundaries would appear to be appropriate. This conclusion might need to be reassessed for pit membranes lacking these wide pores, particularly such as those found among angiosperm species. Interestingly, water flow in subnanometer channels, such as carbon nanotubes (0.8 nm in diameter), appears to be semifrictionless and independent of channel length (Abgrall & Nguyen, 2009), although it is not clear whether this would be relevant for even the narrow pores in angiosperm pit membranes.

The sample size of one pit per category of sun-grown and shade-grown trees in the present study is obviously not adequate to assess the statistical significance of differences between pits in these trees as a result of their developmental environment. However, the study of four boreal forest conifers by Schoonmaker et al. (2010) concluded that shade-grown trees with significantly thinner margo strands and larger pores were more vulnerable to embolism. Further, although the shade-grown trees had narrower tracheids, their conductivity was similar to sun-grown trees, possibly because of a lower pit resistivity as a result of larger margo pores, a conclusion supported by the present modeling results. Future work could consider a larger sample of pits, but also a variety of species containing pits with observable margo pores, perhaps addressing questions of an ecological context as well.


The author gratefully acknowledges Uwe Hacke and Amanda Schoonmaker at the University of Alberta, Edmonton, AB, Canada for graciously providing the two SEM images of a Picea mariana torus–margo.