Journal of Experimental Botany, Vol. 53, No. 373, pp. 1485-1493,
June 2002
© 2002 Oxford University Press
Original Papers |
Modelling the hydrodynamic resistance of bordered pits
School of Biological Sciences, University of Manchester, 3.614 Stopford Building, Oxford Road, Manchester M13 9PT, UK
Received 2 August 2001; Accepted 20 February 2002
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Abstract |
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Previous studies of the hydrodynamics of plant stems have shown that resistance to flow through bordered pits on the side walls of tracheids makes up a significant proportion of their total resistance, and that this proportion increases with tracheid diameter. This suggests a possible reason why tracheids with a diameter above around 100 µm have failed to evolve. This possibility has been investigated by obtaining an estimate for the resistance of a single pit, and incorporating it into analytical models of tracheid resistance and wood resistivity. The hydrodynamic resistance of the bordered pits of Tsuga canadensis was investigated using large-scale physical models. The importance of individual components of the pit were investigated by comparing the resistance of models with different pore sizes in their pit membrane, and with or without the torus and border. The estimate for the resistance of a real bordered pit was 1.70x1015 Pa s m-3. Resistance of pits varied with morphology as might be predicted; the resistance was inversely proportional to the pore size to the power of 0.715; removing the torus reduced resistance by 28%, while removal of the torus and border together reduced it by 72%. It was estimated that in a ‘typical tracheid’ pit resistance should account for 29% of the total. Incorporating the results into the model for the resistivity of wood showed that resistivity should fall as tracheid diameter increases. However, to minimize resistance wider tracheids would also need to be proportionally much longer. It is suggested that the diameter of tracheids in conifers is limited by upper limits to cell length or cell volume. This limitation is avoided by angiosperms because they can digest away the ends of their cells to produce long, wide vessels composed of many short cells.
Key words: Bordered pit, flow, models, tracheids.
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Introduction |
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The hydraulic resistance of xylem in the stems and branches of plants is an important physical characteristic that determines the rate at which water can be supplied to their leaves (Zimmermann, 1983
). Xylem with high hydrodynamic resistance would limit the leaf area that a plant can supply with water (Bond, 1989) or mean that it required more conducting tissue to perform this task (Becker et al., 1999).
One way in which plants could reduce their stem resistance is by developing xylem elements with wider lumens. This has been achieved in angiosperms by stacking together columns of short, wide, open-ended cells to produce long xylem vessels (Bailey and Tupper, 1918). It is therefore, at first sight, puzzling why the vast majority of gymnosperms still rely on relatively narrow tracheids for water transport.
Many studies have shown that the hydrodynamic resistance of plant axes can be fairly closely modelled by regarding xylem as a number of closely packed cylindrical tubes with diameters equal to those of the lumens of the tracheids and vessels (Zimmermann, 1971; Calkin et al., 1985, 1986; Schulte et al., 1987; Schulte and Gibson, 1988). In such analyses the cells are assumed to be ideal pipes, each of which has a resistance, R, equal to
 | (1) |
). First, the narrowness of the ends increases the pipe resistance there. Second, water must travel laterally between adjacent cells through bordered pits in their side walls if it is to travel up the plant. Pits (Figs 1, 2) therefore add another resistance component in series with the pipe resistance of the cell lumen (Calkin et al., 1986; Schulte and Gibson, 1988).
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All the available evidence suggests that pit resistance can be significant. In a range of gymnosperms and angiosperms it has been estimated to account for between 12–70% of the total resistance (Schulte and Gibson, 1988
). Furthermore, these authors found that the relative importance of pit resistance increased with tracheid diameter. These results suggest that increased pit resistance might be the factor that prevents wider tracheids from evolving; above a critical diameter it would make them less efficient hydraulically rather than more efficient. Up until now it has not been possible to test this possibility by modelling the hydrodynamic resistance of tracheids. Two things are needed: a reliable figure for the hydraulic resistance of a single bordered pit; and an analytical model of idealized tracheids that includes the resistance of the pits. This paper tackles both these omissions and, in particular, describes the first attempt to quantify the hydrodynamic resistance of a single bordered pit.
One method which could be used to find the hydrodynamic resistance of a single pit experimentally would be to compare the resistance of axes before and after the pit membranes had been digested using cellulases (Calkin et al., 1986). A knowledge of the number and distribution of pits would then allow the resistance of an individual pit to be estimated. However, cellulases do not have an effect in some species (Schulte and Gibson, 1988), and it is hard to be sure that no pit resistance remains even if the membranes seem to have been successfully digested. Using this technique, it was calculated that the membrane resistivity of the bordered pits of angiosperms and a cycad varied between 1.04 and 28.8x106 Pa s m-1 (Schulte and Gibson, 1988). However, these authors give no figures for the resistance of a single pit and no figures at all for conifers.
Another possible approach would be to measure the resistance of individual pits directly. However, because of their small size, such measurements would be extremely difficult to perform. Zwieniecki et al. used a microcapillary technique to measure the radial conductance of vessels in Fraxinus americana (Zwieniecki et al., 2001a), but not having counted the number of pits in the walls they were not able to calculate the hydraulic resistance of a single pit. However, doing similar experiments with the much narrower conifer tracheids would be much more difficult to perform. For such small structures, modelling is likely to prove a more practicable method. Good scanning electron micrographs of the morphology of the bordered pits of the eastern hemlock Tsuga canadensis (Zimmermann, 1983; Fig. 2
) can act as the basis for such models. One possible approach is to use computer finite element analysis (Schulte and Castle, 1993a, b) as has been used to investigate the resistance of scalariform plates in xylem vessels. The alternative approach used in this study was physically to model bordered pits using the methods developed for scalariform plates (Ellerby and Ennos, 1998). To do this, greatly scaled-up models are used while the Reynolds number is kept down to realistic values by using glycerol rather than water as the experimental fluid. Keeping the Reynolds number below 1 is important because it ensures that flow is kept laminar (Vogel, 1994). Physical modelling is cheap and easy to perform and has the advantage that models can be so readily altered that the effects of individual components of the pits can be investigated. The difficulty in either modelling approach is that bordered pits are so complex and irregular that their morphology can only be approximated.
To investigate the importance of pit resistance, two simplified analytical models for the hydrodynamic resistance of real tracheids were developed, based on consideration of how water flows through them. The first, which models the resistance of just a single tracheid, was combined with the results for pit resistance to estimate the relative contribution of lumen and pit resistance in the tracheids of a typical conifer. Finally, a more general version of this model was used to investigate how the resistivity of wood depends on the dimensions of tracheids and the distribution of pits. Together, these models helped to show how pit resistance will influence the optimal size and shape of tracheids, and so help explain why conifers retain narrow tracheids.
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Materials and methods |
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Modelling the hydrodynamic resistance of a single bordered pit
The experimental set-up:
The experimental set-up was extremely simple, and was similar to that used previously (Ellerby and Ennos, 1998
) (Fig. 3). Plastic waste pipes with an internal diameter of 32 mm were attached horizontally to the bottom outlet of a 20 l aspirator via a glass tap which had an internal diameter of 28 mm. The tap could start and stop the flow of fluid. PTFE thread tape sealed the link between the tap and the aspirator and pipe. A spirit level was used to keep the models level, and the end of the pipe was fitted with a weir over which the fluid could flow readily, but which ensured that the pipe was always completely filled with fluid. The aspirator was then filled with glycerol to a height of exactly 0.065 m above the centre of the outlet, at which point a reference mark was made.
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The resistance to flow of the pipes and the models was calculated by fully opening the tap, letting the glycerol drain through them, and measuring the volume flow rate, Q. This was determined by measuring the time taken to fill a 100 ml measuring cylinder with the glycerol. To calculate the exact volume the measuring cylinder was weighed before and after using an electronic balance, and fluid mass converted to volume by dividing by the fluid density (1261 kg m-3) of glycerol. During flow measurements the aspirator was kept topped up to the 0.065 m mark to maintain a constant pressure head. Since flow was driven by the pressure head of the glycerol in the aspirator and resisted by the combined resistances to flow of the outlet and connector and of the model, the resistance, R, is readily calculated using the equation (Vogel, 1994
)
 | (2) |
gh, where is the density of glycerol (1.261x103 kg m-3 at 20 °C), g is the gravitational field strength (9.81 N kg-1) and h is the height of the surface of the glycerol above the tube outlet (here 0.065 m).
Resistance of the unobstructed piping and tap:
To determine the resistance of the outlet tap and of the 32 mm diameter pipe itself, flow rates were first measured for the apparatus without any attached piping and then with lengths of pipe of 0.165, 0.245, 0.35, 0.43, and 0.65 m. Measurements were repeated five times for all tests. Resistance values were calculated for each situation and plotted on a graph of resistance against pipe length. The intercept and slope of this graph were equal, respectively, to the resistance, R0 of the tap and the resistance per unit length, R/L of unobstructed piping.
Calculation of viscosity:
The viscosity of glycerol varies a great deal, both with temperature and with its purity. The experiments carried out above, therefore, gave the authors the opportunity to calculate the viscosity of the fluid we were using. Rearranging terms in equation 1, viscosity can be calculated as
 | (3) |
Calculating the Reynolds number of the pipe:
These preliminary experiments also allowed the Reynolds number to be calculated for an unobstructed pipe of length 0.43 m into which the model bordered pits would subsequently be placed. The Reynolds number of the pipe is given by the expression
 | (4) |
Construction of model bordered pits:
Models of bordered pits were designed to be as large as possible. Each model would completely fill up the pipe and so would have a diameter of 32 mm, compared with a mean diameter for the real pit (Fig. 2) of 17.5 µm. Therefore, the pits were scaled up by a factor of 1830.
Each bordered pit has three main elements: the net-like pit membrane; the central torus; and the arched domes of the borders on either side. It proved impossible to model the pit membrane of Tsuga canadensis accurately as the strands of material were arranged too irregularly and in too complex a manner. Instead, therefore, it was decided to model it by using much simpler rectangular galvanized steel grids with pore sizes similar to what they would be in a scaled-up pit membrane (Fig. 2). The pores of the real membrane are irregular in shape and very variable in size. The mean aperture was therefore estimated by dividing the membrane diagram into 10 discs from the edge of the torus to the edge of the pit and measuring the width of 10 pores in each disc. This procedure ensured that areas of large (towards the middle) and small (near the torus and edge) pores were equally sampled. The mean width ±SD of pores was 0.30±0.13 µm. Consequently in the ‘realistic’ model there were gaps of 0.55 mm between the wires, which had a diameter of 0.30 mm. Two other model membranes were used to help investigate the effect of pore size on resistance: the first had gaps of 0.85 mm between wires of diameter 0.43 mm; the third had gaps of 1.50 mm between wires of diameter 0.75 mm. As closely as possible, therefore, each grid had gaps that were just under twice the width of the wires and permeability of 41–44%.
The torus of Tsuga canadensis has a mean diameter of 8.2 µm and was therefore represented in the models (Fig. 2) by a plastic disc of diameter of 15 mm diameter which was stuck on to block off the centre of the membrane. Finally, the model borders were created using a 32 mm diameter and 11 mm wide rubber tyre from a ‘die cast’ model car kit. This was cut in half along its centre line and the rough edges smoothed. Finally, the aperture was reduced to a diameter of 10 mm (Fig. 2), representing a hole of diameter 5.5 µm in the original bordered pit by fitting hose washers within the rims of the tyres.
In order to investigate the separate effects of the different components of the bordered pit, four models were made for each mesh size. Replica pits included all the components; the others lacked the border, lacked the torus, or lacked both of them. Therefore 12 models were tested in all. All tests were performed over a 2 week period in March 2001.
Measuring the resistance of model pits:
The model pits were constructed as outlined above and fitted into the centre point of the 0.43 m length of pipe, which was cut into two to allow this. The joint was sealed with a plastic coupling and the tube could then be filled with glycerol ready for tests. Once tests had been performed the apparatus could then be drained and pit models readily changed.
To determine the resistance of model pits, each pipe with implanted pit was connected to the apparatus and the flow rate was measured. The measurement was repeated five times for each model. The mean total resistance of outlet, pipe and pit was then calculated using equation 2. The resistance of the pit alone could finally be determined by subtracting the combined resistances to flow of the outlet and of the 0.43 m pipe.
Calculating the Reynolds number of the pit:
The Reynolds number of the realistic pit model was calculated by substituting the measurement of its pore width instead of tube diameter in equation 4 and inserting figures for the flow rate through it, remembering that much of the area of the lumen was blocked off by the grid and the torus, which left only 32% of lumen area.
Calculating the resistance of real pits:
To calculate the individual resistance, Rind of a real pit it is necessary to apply a correction factor. Since the resistance has units Pa s m-3, physical reasoning and dimensional analysis show that the resistance of an obstruction in a pipe will be proportional to the dynamic viscosity of the fluid (which has units of Pa s) and to dimensions to the power of -3 (m-3). As the model's diameter was 1830 times that of a real pit, glycerol was used instead of water as the fluid the correction factor is given by the expression
 | (5) |
Analytical modelling of the hydrodynamic resistance of individual tracheids
To work out the effect that bordered pits have on the overall resistance of xylem vessels it is necessary to know not only how much resistance each provides but also how many pits are typically present. The more pits there are in the wall the easier it is for the fluid to pass from one tracheid to the next, and so the total resistance to flow is reduced.
Consider water flowing through a cylindrical tracheid (Fig. 4). To travel up the tree, water has to enter the tracheid in its bottom half through the pits, travel up through the tracheid, and finally leave through the pits in the top half of the cell. If pits are distributed evenly up the walls, a molecule of water will, on average, have to travel half the length of the tracheid before crossing through a pit. The resistance to fluid flow is the sum of lumen and pit resistance. The lumen resistance, Rlum, to water travelling half the length of the tracheid can be found from the Hagen–Poiseuille equation and is given by the expression
 | (6) |
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Water travels in parallel through each of the pits in the half of the tracheid, so the total pit resistance, Rpit, is given by the expression
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 | (8) |
Relative resistance of lumen and pits in a ‘typical tracheid’:
The dimensions of a typical tracheid and the number of pits in its wall are needed before the relative resistance of the lumen and pits can be calculated. Unfortunately, no studies have investigated all these characteristics for any real tracheid. However, bearing in mind that the diameter of tracheids ranges from 0.01 mm to 0.08 mm and length is between 1 and 10 mm (Zimmermann, 1983; Schweingruber, 1990), the mean aspect ratio of a tracheid cell will be approximately 100:1. Bannan gives the mean tracheid length in the conifer Tsuga canadensis as 3.95 mm (Bannan, 1970), so a rough estimate a rough estimate of their internal diameter is 39.5 µm.
The number of pits in tracheids is also very variable. However, Bailey and Tupper show a ‘typical tracheid’ (Bailey and Tupper, 1918) (which is also shown as Fig. 1.1F in Zimmermann, 1983). The number of pits, on the surface shown, of this tracheid 124, was doubled to give a total number of 248 pits per ‘typical tracheid’ since pits are almost exclusively located on the radial walls.
Analytical modelling of the hydrodynamic resistivity of wood
Equation 8 gives a figure for the hydrodynamic resistance of a single tracheid. However, for a tree it is important to minimize the combined resistance of all its tracheids. This will depend both on the trunk's dimensions and on its resistivity, the resistance per unit area and unit length. Plants should minimize the resistivity of their wood in order either to increase water supply to their leaves or, if water transport limited trunk diameter, to minimize the investment in wood material, assuming that the relative cell wall volume stays constant. Fortunately, it is straightforward to come up with an analytical expression for wood resistivity simply by assuming that it is made up from large numbers of the thin-walled tracheids shown in the model of Fig. 4. Equation 8 investigated the resistance to movement of a column of water up through a single tracheid of area 
D2/4 up a distance L/2. The resistivity, r, of the wood material composed of such tracheids is therefore
 | (9) |
An expression can therefore be found which relates the resistivity of wood to the dimensions of the tracheids, and the numbers of pits. This can then be used to calculate the optimal cell shape which minimizes resistivity. The two possible cases are examined: if the number of pits on the walls of a tracheid is kept constant; and if the density of pits on the walls of the tracheid is kept constant.
Constant number of pits:
The resistivity of wood whose tracheids have a constant number of pits whatever their size or shape can be found by combining equations 8 and 9 to give
 | (10) |
Therefore, although lumen resistivity falls as tracheid diameter increases, pit resistance rises rapidly with diameter because the number of pits through which water can travel per unit area falls. Longer tracheids should also have lower pit resistance than short ones because water will have to travel through fewer pits. The optimal tracheid diameter, at which r is minimized, for a given tracheid length, L, will therefore be intermediate and is the point where the differential of r with respect to D equals zero.
 | (11) |
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 | (12) |
At this point the lumen resistivity will equal pit resistivity and the minimum overall resistivity will be equal to twice that derived from the Hagen–Poiseuille equation.
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Wood with wider tracheids will have lower resistivity that that with narrow tracheids. However, to minimize resistivity the tracheids must be far more than proportionally longer. Tracheid length should, in fact, increase with diameter to the fourth power, and wider tracheids should consequently be relatively more elongated.
Constant pit density:
An alternative model can be produced for the case in which it is assumed that tracheids have a constant pit density on their walls whatever their dimensions, so larger tracheids will have more pits. In this case, the number of pits in the walls of a tracheid is equal to the number per unit area, na, on the radial walls times half its total surface area, DL/2, assuming that the cells are approximately circular in cross-section. The total resistance of a tracheid to water travelling half its length is therefore
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 | (15) |
It can be seen once again that though the lumen resistivity decreases as D increases, the pit resistivity will actually increase because, though there are more pits, there are still fewer per unit area. Once again wood composed of long tracheids will have less hydrodynamic resistance, and once again the optimal tracheid diameter, at which r is minimized, for a given tracheid length, L, will be intermediate at the point where the differential of r with respect to D equals zero.
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 | (018) |
 | (17) |
At this point the lumen resistivity will equal half pit resistivity and the minimum overall resistivity will be equal to three times that derived from the Hagen–Poiseuille equation.
 | (18) |
Wood with wider tracheids should once more have lower resistivity than that with short narrow tracheids. Once more, to minimize resistivity wider tracheids should be more than proportionally wider. Their optimal diameter should scale with its length to the power of 1.5; they should therefore get relatively more elongated as they get wider, though by less than if the number of pits is held constant.
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Results |
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Resistance of the unobstructed piping and tap
The resistances of the different lengths of unobstructed pipes increased almost exactly linearly with length. Regression gave the equation for resistance, R, as
 | (19) |
Viscosity of the glycerol
Inserting the value for the slope of the regression for resistance against length, 1.05x108, into equation 2, gives the value for the dynamic viscosity of the glycerol as 2.70 Pa s. This is some 80% higher than the normal textbook value at 20 °C of 1.49 Pa s, because the laboratory was rather cold as experiments were carried out in March. Extrapolation of the curve of viscosity vs temperature gives a figure for temperature of 13 °C.
Reynolds number of the pipe
The flow rate, Q, for the 0.43 m long piping was 5.79x10-6 m3 s-1. Inserting this value into equation 4 gives a Reynolds number of 0.107. This is of the same order of magnitude as flow in real tracheids and well below the value of 10 at which flow through sharp-edged orifices in pipes becomes unsteady (Vogel, 1994). Therefore, it can be confidently assumed that the flow in the piping accurately models flow through real tracheids and pits.
Resistance to flow of model bordered pits
The resistances of the model pits are shown in Fig. 5. The increases in resistance due to the presence of the different model pits were all large compared with the initial resistance of the unobstructed 0.43 m piping. The mean resistance of the ‘realistic’ pit model, R for instance, was 7.47±0.03x108 Pa s m-3. For this case the Reynolds number of the fluid flowing through each individual pore was calculated to be 9.5x10-4, well below 1, and again showing that flow must have been laminar, just as it is in real bordered pits.
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There were also clear trends in resistance with the morphology of the pits. The resistance increased as the width of the pores in the pit, w, decreased. For the complete bordered pit, regression of log data gave the following expression for the resistance. Rind=3.48x106 w–0.715 Pa s m-3.
Removing the different elements of the pit greatly reduced its resistance. Removing the torus from the ‘realistic’ pit, so allowing water to flow across the whole width of the mesh, reduced resistance to only around 72% of the value for the complete pit. Removing the border reduced the resistance to only 44% of the value for the complete pit, while removing both the border and the torus to give a simple pit reduced resistance to only 28% of its original value. The percentage reduction was even greater for the wider meshes.
Reynolds number of the pit
The Reynolds number water flow through the realistic pit model was 2.9x10-4, well below 1, so the flow through the pit was also laminar.
Resistance to flow of real pits
Combining the results for the ‘realistic’ pit with the scaling factor given in equation 5 gives the resistance of an individual real pit, Rind, as 1.70x1015 Pa s m-3.
Relative resistance of bordered pits and lumen
The relative magnitudes of the resistance of bordered pits and lumen can be found by entering the experimental values obtained for the resistance of a real pit and the estimates for the morphological features of the ‘typical tracheid’ into equations 6, 7 and 8. These give lumen resistance, Rlum, as 3.31x1013 Pa s m-3 and pit resistance, Rpit, as 1.37x1013 Pa s m-3. Thus pit resistance would make up 29% of the total resistance of a typical tracheid.
Optimal diameters of tracheids to minimize resistivity
The optimal diameters for tracheids to reduce resistivity were calculated for both analytical models. Assuming that the number of pits is kept constant, the optimal diameter for a tracheid of length 3.95 mm is 49 µm. If the density of pits is assumed constant the optimal diameter is 53 µm.
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Discussion |
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The physical modelling of water flow through bordered pits proved a successful technique. The experiments were straightforward to carry out, while the high viscosity of the glycerine in the models also kept the Reynolds number of the flow, both in the pipe and through the pit well below 1. There were three potential problems with the apparatus. First, the model pits were placed within a pipe rather than within the side wall of a much larger pipe. The flow regime near the models would probably therefore be rather different than in real tracheids, as flow is faster towards the centre of a pipe. However, the great majority of the resistance to flow through a pit occurs during the passage of fluid through the membrane and border (Calkin et al., 1986
), so the positioning of the model is unlikely to have significantly affected its resistance. Second, the membrane of real pits is so complex that this model of it was necessarily unrealistic. Third, without temperature control, viscosity, and hence resistance might have changed over the course of the tests. Hence the estimate of the resistance of real bordered pits is subject to error. However, the straight line relationship between viscosity and pipe length which emerged from tests conducted over a period of several days, suggests that this was not a significant factor. The experiments were able to give a sensible answer for the resistance and the use of a range of different models allowed the investigation of the relative importance of the different elements in causing resistance. The modelling also allowed the relative importance of the pit resistance in a typical conifer to be estimated and the effect of tracheid morphology and pit number on the overall hydrodynamic resistance to be analysed.
How does the estimate for the resistance of bordered pits compare with those of other authors? For a range of species Schulte and Gibson calculated that the membrane resistivity of the bordered pits of angiosperms and a cycad varied between 1.04 and 28.8x106 Pa s m-1 (Schulte and Gibson, 1988). For this study's model it is possible to obtain an equivalent resistivity by multiplying the estimate for resistance, 1.70x1015 Pa s m-3, by the area of the membrane (2.39x10-10 m2). This gives a figure for the bordered pit of 0.4x106 Pa s m-1, rather lower than theirs, but of the same order of magnitude. The difference would be expected because of the additional resistance given by the borders of the pits in this model, which are typical of conifers.
Hydrodynamic theory predicts that the resistance of an individual pit should be inversely proportional to the first power of the diameter of the pores in its membrane if the membrane area is held constant. The resistance of a single pore is inversely proportional to the third power of its diameter (Vogel, 1994), but the number of pores in the membrane will be inversely proportional to the square of pore size. Therefore, since the resistances of the pit pores are in parallel, their collective resistance will be inversely proportional to the number of pores to the first power. It was found that pit resistance was in fact proportional to pore size to the lower power of -0.715. The slightly lower exponent may be due to the additional resistance of the border and torus, which would be similar whatever the pore size. It also suggests that this estimate of pit resistance would not be unduly affected by small errors in the estimated pore size; an error of 10% in pore size would only result in an error of 7% in resistance. For this reason as well, the use of a ‘mean pore size’ also seems justified.
Another finding was that the torus and border increased the resistance of the realistic bordered pit by a factor of over three compared with a simple pit. This disadvantage must be outweighed by the well-known ability of the border and torus to act as a valve that prevents the spread of embolisms. Perhaps the valve arrangement allows the pit membrane of conifers safely to have wider pores than the simple pits of angiosperms, which have to trap embolisms without the valve mechanism. Paradoxically, therefore, it might allow bordered pits with a torus to have lower hydrodynamic resistance than ones that lack this feature. The lower resistivity of the model pit membrane than that of the angiosperms studied by Schulte and Gibson (Schulte and Gibson, 1988) (which lack the torus) certainly backs this idea up.
The third finding was the confirmation from modelling of the significant effect of the pits on the total hydrodynamic resistance of tracheids. The estimate in this study was that in a typical tracheid, pits should account for about one-third of the total resistance, in agreement with other authors who give values ranging from 12–70% (Schulte and Gibson, 1988). This study's estimate is not particularly reliable, as it was based on crude estimates of the tracheid length, diameter and pit number. Clearly, further morphological study of conifer tracheids is needed. Once that is known, this data could be plugged into equation 8 to calculate realistic estimates of relative pit resistance. Recent work has shown that the hydrodynamic resistance of the pit membranes of xylem vessels in angiosperms may be altered by altering the concentrations of ions in the perfusing water, an effect which is put down to the swelling of pectins (Zwieniecki et al., 2001b). These authors suggest that this might present a method whereby angiosperms can control water uptake. The fact that pit resistance in tracheids has been shown here to be significant suggests that this mechanism might also be used by conifers to control their water uptake.
It has consistently been found that the relative importance of pits depends on tracheid diameter: the wider the tracheid the greater the importance of the pits (Schulte et al., 1987; Schulte and Gibson, 1988). The reason why this occurs is clearly shown by the analytical model of tracheid resistance presented in this paper and summarized in equation 8. Assuming that the number of pits per tracheid remains constant, doubling the diameter of the tracheid decreases resistance by a factor of 16, but leaves pit resistance unchanged.
Finally, the results of the analysis of wood resistivity can help cast light on why conifers have failed to evolve wider and hence more efficient tracheids. As both models show, because of the component of pit resistivity, tracheids must not only get wider to reduce resistivity, they must also increase their length by an even greater factor. Length must rise with diameter to the power of 4 if it is assumed that pit number remains constant, or to the power of 1.5 if pit density is assumed to remain constant. It is possible that limitations in xylem cell length, or cell volume (as advocated by Bailey and Tupper, 1918), prevents conifers producing tracheids with diameters above about 80 µm. Tracheids with the maximum length of 10 mm reported for conifers (Schweingruber, 1990) will have an optimum diameter, calculated from equation 12 of 62 µm, or from equation 17 (and assuming a pit density of 5.06x108 m-2, the same as that assumed for the ‘typical tracheid’) of 98 µm. Alternatively, it might instead be that tracheids reach the limit of xylem cell volume. Optimal tracheids with a diameter of 62 µm or 98 µm and length 10 mm will have a volume of 0.030 mm3or 0.076 mm3. These values are similar to the volume of the widest vessel elements in lianas (Carlquist, 1975), which have diameters around 500 µm and lengths around 35 µm. The volume of a cylinder of these dimensions is 0.069 mm3. In further support of this notion is the well-known inverse correlation between vessel element diameter and length seen in angiosperms (Bailey and Tupper, 1918; Carlquist, 1988).
As well as limits to cell length and volume, the limitations to conductivity in conifer wood are, therefore, probably related to the fact that they have been unable to evolve the ability to digest away the ends of their cells to produce the scalariform or simple plates seen in the xylem vessels of angiosperms (Bailey and Tupper, 1918; Frost, 1930a, b). By doing this, angiosperms can join short, wide, vessel cells end-to-end to produce multicellular vessels that are both very wide and very long, so that the hydrodynamic resistance is greatly reduced. The only way water can travel between individual tracheids in conifers is through pits which have a hydrodynamic resistance which is several orders of magnitude greater than even the finest scalariform plates (Ellerby and Ennos, 1998). Faced with this inability, the limitation of tracheid diameter and the higher resistivity of conifer wood which has been detected by researchers (Zimmermann, 1983) is therefore inevitable.
One compensating advantage of the arrangement in conifers is that embolisms are readily trapped by the valves in the pits between the short, narrow cells, and so if an embolism does occur in conifers it does not spread. For this reason conifers can maintain what conductivity they do have for many years, even in dry or frosty climates. As so often in biology it appears that there is no single form which is optimal in all cases. The tracheids and bordered pits with torus of conifers suit them to areas with seasonal drought or frost, whereas in the wet tropics the more efficiently conducting vessels of angiosperms would allow them to supply their leaves without needing so much conducting material.
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Acknowledgments |
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We would like to thank Dr Peter Becker for reading and commenting on an earlier draft of the manuscript, and to two anonymous referees for their constructive suggestions on how the paper could be improved.
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Notes |
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1 To whom correspondence should be addressed. Fax: +44(0)161 2753938. E-mail: Roland.Ennos{at}man.ac.uk
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References |
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