ARTICLE IN PRESS

Journal of Theoretical Biology 247 (2007) 788–803 www.elsevier.com/locate/yjtbi

The relevance of xylem network structure for hydraulic efficiency and safety

Lasse Loepfea, Jordi Martinez-Vilaltaa,Ã, Josep Pin˜ola, Maurizio Mencuccinib

aCenter for Ecological Research and Forestry Applications (CREAF), Autonomous University of Barcelona, E-08193 Bellaterra, Spain bSchool of GeoSciences, University of Edinburgh, Edinburgh, UK

Received 21 December 2006; received in revised form 21 March 2007; accepted 29 March 2007 Available online 1 April 2007

Abstract

The xylem is one of the two long distance transport tissues in , providing a low resistance pathway for water movement from roots to . Its properties determine how much water can be transported and transpired and, at the same time, the plant’s vulnerability to transport dysfunctions (the formation and propagation of emboli) associated to important stress factors, such as droughts and frost. Both maximum transport efficiency and safety against embolism have classically been attributed to the properties of individual conduits or of the pit membrane connecting them. But this approach overlooks the fact that the conduits of the xylem constitute a network. The topology of this network is likely to affect its overall transport properties, as well as the propagation of embolism through the xylem, since, according to the air-seeding hypothesis, drought-induced embolism propagates as a contact process (i.e., between neighbouring conduits). Here we present a model of the xylem that takes into account its system-level properties, including the connectivity of the xylem network. With the tools of graph theory and assuming steady state and Darcy’s flow we calculated the hydraulic conductivity of idealized wood segments at different water potentials. A Monte Carlo approach was adopted, varying the anatomical and topological properties of the segments within biologically reasonable ranges, based on data available from the literature. Our results showed that maximum hydraulic conductivity and vulnerability to embolism increase with the connectivity of the xylem network. This can be explained by the fact that connectivity determines the fraction of all the potential paths or conduits actually available for water transport and spread of embolism. It is concluded that the xylem can no longer be interpreted as the mere sum of its conduits, because the spatial arrangement of those conduits in the xylem network influences the main functional properties of this tissue. This brings new arguments into the long-standing discussion on the efficiency vs. safety trade-off in the plants’ xylem. r 2007 Elsevier Ltd. All rights reserved.

Keywords: Xylem; Efficiency vs. safety trade-off; Connectivity; Drought resistance; Hydraulic conductivity; Water transport; Embolism; Network; Model

1. Introduction determine the overall transport efficiency of the xylem are its maximum hydraulic conductivity and its vulnerability to The main function of the xylem is to provide a low- cavitation. A high maximum conductivity lowers its resistance pathway for water transport within the soil– probability to become a bottleneck in the pathway between plant–atmosphere continuum (SPAC). According to the the soil and the leaves when plenty of water is available cohesion-tension theory water ascent in plants takes place and, therefore, a limiting factor to photosynthetic capacity in a metastable state under tension. This negative pressure and plant growth (Brodribb and Feild, 2000; Stiller et al., in the xylem makes it vulnerable to cavitation, i.e., the 2003). On the other hand, when water is scarce, the expansion of gas bubbles in the conduits (Tyree and resistance to cavitation (and embolism) is crucial. A Zimmermann, 2002). The two main properties that number of studies have shown that vulnerability to embolism is related to drought tolerance (Maherali et al., ÃCorresponding author. Tel.: +34 93 581 1920; fax: +34 93 581 41 51. E-mail addresses: [email protected] (L. Loepfe), Jordi.Martinez. 2004; Martinez-Vilalta et al., 2002a; Sperry et al., 2002). [email protected] (J. Martinez-Vilalta), [email protected] (J. Pin˜ol), Some experimental data suggest that these two goals [email protected] (M. Mencuccini). (high maximum conductivity and high resistance to

0022-5193/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.jtbi.2007.03.036 ARTICLE IN PRESS L. Loepfe et al. / Journal of Theoretical Biology 247 (2007) 788–803 789 embolism) cannot be achieved independently, implying that network) has never been explicitly considered in studies of there is a trade-off between maximum hydraulic conductiv- xylem function. ity and safety in the xylem, at least when comparing tissues Many aspects of life show network structures. A great within the same individual (Hacke et al., 2000; Lo Gullo and number of studies have been conducted for social, Salleo, 1993; Mencuccini and Comstock, 1997; Martinez- information and technological networks. But there are Vilalta et al., 2002b; Pockman and Sperry, 2000; Tyree et al., also many aspects of biology that show a network 1994). However, the principles underlying this trade-off are structure. Good examples are metabolic pathways, gene still poorly understood. expression or neural networks. In all of them network The conductivity (K) of the xylem was classically estimated topology affects its function, and there is no reason to by adding up the conductivities of the conduits found in a think that this should not be the case for the xylem. cross-section of wood, using the Hagen–Poiseuille equation Although all these networks are very different in appear- to calculate the conductivity of each conduit. This calcula- ance, they all share some very fundamental properties (see tion consistently overestimates the conductances measured Albert and Barabasi, 2002 or Newman, 2003 for reviews). experimentally on wood segments. The discrepancy (about Graph theory is often used to study complex networks, the 20–70% of measured K) is universally attributed to the average degree (or connectivity) being the single most resistance of inter-conduit pit pores (Chiu and Ewers, 1993; employed parameter to describe network topology. Lancashire and Ennos, 2002; Tyree and Zimmermann, 2002 In the xylem, connectivity (/kS) corresponds to the and literature cited therein), as sap has to cross a porous average number of different neighbour conduits to which a membrane to flow from one conduit to the next. The overall conduit is connected. For instance, the capacity of a hydraulic resistance is thus considered to be the sum of conduit to transport water will be limited by its own lumen resistance and inter-conduit resistance in series. Based resistance only if there is no constraint to its water supply on this assumption, it is possible to estimate inter-conduit elsewhere in the network. Like a motorway without any resistance by substracting the calculated lumen resistance access roads would be empty—never mind how many lanes (using the Hagen–Poiseuille equation) from the resistance it has, a xylem conduit will not conduct sap if it is not measured experimentally. By doing that, Sperry et al. (2005) connected to a conducting cluster that connects roots to concluded that inter-conduit resistance and lumen resistance leaves. Following this logic, it seems reasonable to expect are co-limiting, i.e., each is responsible for about half of the that the more conduits a conduit is connected to (i.e., total resistance of a wood segment. But Schulte et al. (1987) the greater its connectivity), the higher will be the flow showed that even after dissolving the porous membrane of through it. the inter-conduit connections, the measured conductivity was According to the air seeding hypothesis, embolism still 30% lower than the conductivity predicted by the propagates from an air-filled conduit to a functional one Hagen–Poiseuille equation. through the porous membrane that connects them, On the other hand, drought-induced embolism is depending on the diameter of the largest pore in the believed to spread between conduits as a function of the connection. The first condition for a conduit to be maximum size of the pores in the inter-conduit membrane embolised is that it is connected to an air-filled conduit. connecting them (air-seeding hypothesis; Zimmermann, This suggests that a conduit will be more vulnerable to 1983). Pit pore size has often been estimated from embolism the more connections it has, since more vulnerability curves making direct use of the capillarity connected conduits will be more likely to be connected to equation (e.g., Sperry and Tyree, 1990) and a direct an already air-filled conduit. At the tissue level we would relationship between air-seeding pressure and pit pore size expect that high connectivity would facilitate the spread of has been measured (Jarbeau et al., 1995). However, Choat emboli and therefore increase the vulnerability to drought et al. (2003) could not establish any direct correspondence induced embolism. between pit pore size and vulnerability to embolism, as Here we use graph theory to build a model of the xylem pores large enough to fit the predicted values could not be that explicitly takes into account its network structure. The detected. This suggests that this relationship is at least not model simulates water transport in a wood segment of a as straightforward as previously thought and that pit-pore under laboratory conditions. Our aim was to dimensions may not be the only characteristics that study the influence of connectivity on the hydraulic determine the spread of embolism in the xylem. properties of the xylem. Specifically, our hypotheses were: Structurally, the xylem is a network of interconnected (1) Connectivity co-limits hydraulic conductivity together conduits (Cruiziat et al., 2002; Tyree and Zimmermann with conduit size and inter-conduit resistance, and (2) 2002). This structure has been known for decades (Braun vulnerability to embolism increases with the connectivity of 1959; Burggraaf, 1972; Zimmermann, 1971) and showed the xylem network. again recently by 3D-images of X-ray computed micro- tomography (Steppe et al., 2004) and series of cross- 2. Methods sections taken with epifluorescence microscope (Kitin et al., 2004). However, to our knowledge, the way conduits Our model simulates an ideal xylem segment opened to are arranged in space (i.e., the topology of the xylem the atmosphere at both ends. An externally imposed ARTICLE IN PRESS 790 L. Loepfe et al. / Journal of Theoretical Biology 247 (2007) 788–803 pressure gradient drives the flow of water from one end to the other. The xylem is modelled as a three-dimensional set of conduits interconnected through porous membranes (Fig. 1). The conduits are created from a homogeneous 3D lattice in which the longitudinal distance between two adjacent nodes represents a vessel element (or a fraction of a tracheid), and each node has the same probability to become an end of conduit. Each conduit is randomly assigned a diameter that determines its hydraulic resis- tance. Nodes belonging to different conduits can be connected by ‘‘pit membranes’’, whose resistance adds to the resistance of the conduits. A linear system is obtained by assuming Darcy’s law and steady-state conditions. The system is solved numerically to obtain the equilibrium pressure at each node of the three- dimensional grid. This allows us to calculate the overall flow, but also the flow within every single conduit element. The conductivity of the xylem segment is defined as the overall flow divided by the (external) pressure gradient driving the flow. This approach is similar to the one used by Cochard et al. (2004) to model the hydraulic architec- ture of venation. The model is described in detail in the following sections.

2.1. The model

2.1.1. Creating the conduits The xylem was created on a regular three-dimensional grid of nodes. The three dimensions were defined as follows: x, the axial path, in the direction of the simulated flow; y, the radial dimension; and z, the tangential Fig. 2. (A) Schematic representation of the idealized xylem in 2D. Solid dimension of the simulated wood segment (Fig. 2). The lines represent conduit elements, dotted lines inter-conduit connections. distance between two points in the x (axial) axis represents The black spots are the reference points (nodes) for which pressure is the length of a conduit element. On the other hand, the calculated. Lc is conduit length and Lce the length of a conduit element separation between two adjacent nodes in the radial and (equivalent to the distance between two nodes on the x-axis). Pc is the probability to connect two nodes by an inter-conduit connection. (B) tangential dimensions has no physical meaning with regard Close-up of the connection between two adjacent conduits. Dc is conduit to water flow. The average separation between nodes in the diameter, Am the contact area of the connection, Pi and Pj are the y-andz-axis was determined by conduit density, but two pressures at the nodes i and j, Fij is the flow between them. (C) Close-up of the cell wall area in the connection between two adjacent conduits. Secondary cell wall is depicted in grey. The dotted lines represent the pit

membrane. (D) Close-up of a pit membrane (side view). Dp is the maximum pore diameter and fp the fraction of the contact area occupied by pores.

conduits that were separated by only one edge were considered to effectively touch each other if they were connected by an inter-conduit connection. The algorithm used to create the conduits was based on the inverse of the double-difference (DD) algorithm for measuring conduit-lengths with randomly distributed end walls (Skene and Balodis, 1968; Tyree and Zimmermann, 2002). Every node in the grid had a probability Pe to be the end of a conduit; starting from the downstream end of the simulated segment, nodes were connected to those im- Fig. 1. Example of a 3D representation of the xylem network according to mediately below (x-dimension) with a probability (1Pe). our model. Darker colour indicates higher flow rate. Cylinders represent These links represent the conduit elements. Any conduit conduits, lines indicate connections between conduits. element sharing a node was considered to be part of the ARTICLE IN PRESS L. Loepfe et al. / Journal of Theoretical Biology 247 (2007) 788–803 791 same conduit. No node could be associated to more than All the conduit elements in a conduit had the same two conduit elements, i.e., no bifurcations were possible. diameter. Individual conduit diameters (Dc) were sampled This algorithm leads to a negative exponential conduit from a log-normal distribution based on experimental data length distribution (Nijsse, 2004). The probability Pe was (Table 1). The resistance of a conduit element (Rce) was selected to match a given average conduit length (Lc) (see calculated using the Hagen–Poiseuille equation: Table 1). 128 m Lce All conduits were oriented in the direction of the overall Rce ¼ , (1) p D4 flow (i.e., there were no conduits perpendicular to the c direction of the flow) and were mostly straight and parallel. where m is the dynamic viscosity of water at 25 1CandLce However, a small probability was introduced so that some the length of the conduit element. It has been shown that nodes could randomly connect to nodes that were not the Hagen–Poiseuille equation provides an accurate aligned (i.e., with different y and z coordinates; Fig. 2). estimate of the actual resistance of individual conduits This introduced a slight deviation from the axial path (Sperry et al., 2005; Zwieniecki et al., 2001a). which resulted in increased tortuosity and a greater realism After connecting the conduits (see below), one or more in the representation of the xylem network (Andre´, 2002; clusters of conduits emerged that connected the two axial Zimmermann, 1983). Straight and parallel conduits would ends of the simulated wood segment. However, not all limit connectivity, as conduits can only connect through pit conduits belonged to those clusters. The conduits that were membranes with their immediate neighbours. On the other not connected to the conducting cluster were filtered out, hand, a very high tortuosity in our model would lead to a since they did not contribute to the overall flow and their great loss of conduits on the sides of the segment, reducing presence introduced instabilities in the solution of the the average conduit length. We have chosen an inter- resulting linear system. Conduits or clusters of conduits mediate value of tortuosity, which maximizes connectivity that where connected to the conducting cluster with only for a given probability to connect without affecting average one connection were retained. Although they did not conduit length. contribute to the overall flow, they could have a water

Table 1 Nomenclature and values of the input parameters used

Name Description Values Values Source Experiment 1 Experiment 2

Conduits

Dc Average conduit diameter (lm) 34 10–50 Wheeler et al. (2005) Dc_cv CV of the lognormal distribution of conduit 0.38 0.38 Average of 13 diferent species diameters (mm) (Martinez-Vilata, unpublished data)

Pe Probability of a node to be a conduit end point 0.925 0.5–0.99 Ajusted to fit Lc Lc Average conduit length (mm) 21 10–50 Wheeler et al. (2005) z Deviation of the axial path of the conduits 0.25 0.25 Estimated Inter-conduit connections

Dp Average maximum pit pore size (nm) 50 10–100 Estimated from literature Dp_cv CV of the lognormal distribution to obtain the 0.5 0.5 Estimated from SEM pictures max pit pore size of each connection (nm) fp Fraction of the contact area occupied by pores 0.35 0.1–0.5 Estimated fa Fraction of the total conduit area in contact 0.0945 0.0945 Wheeler et al. (2005) with another conduit Connectivity

Pc Probability that two neighbour nodes are 0.01–0.4 0.01–0.4 Estimated connected through a porous membrane Dimension of simulated segment 2 As Area of the segment (mm ) 2 2 Optimized to reduce computational time without influencing results

Ls Length of the segment (mm) 200 200 Optimized to reduce computational time without influencing results

Lce Length of a conduit element (or axial distance 2 2 Optimized to reduce computational between two nodes) (mm) time without influencing results 2 dc Conduit density in a cross section (mm ) 176.5 176.5 (U. Hacke, personal communication) General constants g Surface tension of water at 25 1C (Pa) 0.072 103 0.072 103 — m Viscosity of water at 25 1C (Pa*s) 1.002 103 1.002 103 —

The parameters that were varied in Experiment 2 are marked bold. ARTICLE IN PRESS 792 L. Loepfe et al. / Journal of Theoretical Biology 247 (2007) 788–803 storage function and their presence affects the calculation 2.1.3. Calculating maximum hydraulic conductivity of the theoretical hydraulic conductivity of a cross-section We assumed steady state for each node, i.e., that the of the segment. system was in equilibrium. The flow between any two nodes depended on the pressure difference between them and the resistance of the edge (conduit element or inter- 2.1.2. Inter-conduit connections conduit connection) linking them, according to Darcy’s Inter-conduit pit connections were simulated as links law. Combining these two basic assumptions we obtained a between adjacent nodes of the same x-axis level belonging linear equation for each node i: to different conduits. Each node had a probability Pc to be X connected with an adjacent node. This probability deter- DPi;j mined, together with the conduit length distribution, the i ¼ 0, (6) Ri;j connectivity (/kS) of the network. When the value of conduit length was fixed, /kS was a logarithmic function where DPi,j and Ri,j are the pressure differences and of Pc. Each connection was characterized by three proper- resistances, respectively, of each edge i connected to a ties: number and average diameter of pores, that deter- given node j. Combining all nodes in the simulated segment mined hydraulic resistance (Rm), and maximum pit pore leads to a linear system: size (Dp), that determined the likelihood of emboli spreading between conduits. K P ¼ F. (7) The resistance of a given connection (Rm) was calculated Being K the symmetric sparse matrix of conductances assuming that the porous membrane is an infinitely thin between any two nodes, P the vector of pressures at each plate with perfectly circular pores, as (Sperry and Hacke, node and F the vector of flows. When two nodes are not 2004; Vogel, 1994) connected the conductance between them is zero (the 24 m resistance is infinite). The vector F equals zero for all the ¼ Rm 3 , (2) entrances but those associated to the nodes connected to D np e either axial end of the system. This linear system was solved where De is the equivalent pore size and np the number of numerically using the conjugate gradient algorithm (Weis- pores in the connection. De represents the diameter that sein, 2002) in order to obtain the pressure at each node. pores would have if the total resistance of a connection was This algorithm was implemented from the ‘‘mtj’’ JAVA- spread between an equal number of pores of identical library (Heimsund, http://rs.cipr.uib.no/mtj/). The ‘‘mtj’’ diameter, and was estimated from the maximum pore size library includes an algorithm to detect if the solution does of the connection (Dp)as(Sperry and Hacke, 2004) not converge. We added two additional criteria to ensure that the steady-state solution was found: the iterative De ¼ 0:63 Dp. (3) algorithm was not exited until (1) the difference of the The maximum pore size in a connection (Dp) was overall flow (Ft) between two consecutive iterations was 15 3 1 sampled from a global log-normal distribution of known less than 10 m s , and (2) the relative change in Ft was mean and standard deviation (Table 1). o0.01%. The overall flow (Ft) was calculated as the sum of To calculate np we assumed that only a fraction of the the differences of incoming and outgoing flow in each node contact area (Am) was actually occupied by pores (fp), as in connected to the proximal end of the system. The a real conduit part of the surface is occupied by the consistency of the solution was assessed by checking that secondary cell wall and the fibres of the pit membrane: the incoming flow at the proximal end of the segment was equal to the outgoing flow at the distal end (calculated in f p Am np ¼ . (4) the same way). Surface specific conductivity (K ) was 1 p D2 s 4 e calculated as

The contact area between conduits (Am) was calculated F t Ls assuming that a constant fraction (fa) of the total conduit Ks ¼ . (8) DPo As surface area (Ac) was in contact with other conduits

(Sperry et al., 2005). This allowed us to calculate the total Being DPo the externally applied pressure, As the cross- wall area of each conduit in contact with other conduits. sectional area of the modelled segment and Ls its length. The area of each connection for a given conduit was Additionally, we calculated the theoretical conductivity calculated dividing this area by the number of connections of the modelled segment assuming that only conduit of the conduit (see above). The average value of the two lumens contributed to the overall resistance. This was conduits that formed the connection was used as the actual achieved by making ten virtual cross-cuts at random x- connection area: positions within the segment. The resistance of each open conduit in each cross-section was then calculated using the 1 Ac f Ac f A ¼ 1 a1 þ 2 a2 (5) Hagen–Poiseuille equation, the values added up for each m1;2 2 n n m1 m2 cross-section, and those averaged to obtain a final estimate being nm the total number of connections of the conduit. per segment. In this paper, theoretical conductivity always ARTICLE IN PRESS L. Loepfe et al. / Journal of Theoretical Biology 247 (2007) 788–803 793 refers to total segment cross-sectional area, not to lumen all other parameters constant. When available, the values area. of the fixed parameters corresponded to values measured for Acer negundo (Wheeler et al., 2005). When no data were 2.1.4. Spreading embolism available, we chose a value in the middle of a biologically After calculating the maximum hydraulic conductivity of reasonable range (Table 1). Segment dimensions were the segment, ten randomly selected conduits were embo- chosen to minimize computational time without influen- lised. This mimics injuries that might be induced in the real cing the results. Therefore, for segment length (Ls) and xylem by bites or mechanical fatigue. From those segment area (As) we chose the lowest value for which a conduits, embolism could spread according to the air- further small increase did not significantly change max- seeding mechanism (Zimmermann, 1983; Sperry and Tyree, imum hydraulic conductivity (Ksmax) or the pressure at 1988). We applied a gradually increasing external positive which 50% of conductivity was lost (P50). air pressure (Pa) simulating the air-injection technique to We first carried out two simulations with extreme values establish vulnerability curves (Cochard et al., 1992; Salleo of connectivity (/kS ¼ 3.8; /kS ¼ 13.7), obtaining a et al., 1992; Sperry and Saliendra, 1994), so that embolised vulnerability curve for each value. Then we ran 170 conduits had a positive pressure, while functional conduits simulations varying Pc within a wider range (0.01–0.4), were at atmospheric pressure. The bubble pressure (Pb), calculating for each simulation the maximum hydraulic i.e., the pressure difference between two conduits needed conductivity (Ksmax), the pressure at which 50% of for air to propagate through a given connection, was conductivity was lost (P50) and the steepness of the proportional to the maximum pore size of this connection vulnerability curve (a). / S (Dp) and the surface tension of water (g), which was The connectivity of the system ( k ) is affected both by considered constant (Sperry and Tyree, 1988; Tyree and the probability to connect of each node (Pc) and the length- Zimmermann, 2002). distribution of the conduits. In order to distinguish between the direct effect of conduit length and its effect 4 g Pb ¼ . (9) due to its influence on connectivity we carried out another Dp two series of 100 simulations each. In the first series we After the initial random embolization a pressure of varied Pc in order to maintain /kS fixed, only accepting 0.1 MPa was applied, which established a pressure differ- simulations with /kS ¼ 5.070.1. In the second series Pc ence between functional and embolised conduits (DPa). had a fixed value of 0.047, the average value of Pc in the Embolism could spread through an inter-conduit connec- previous set of simulations, and therefore connectivity was tion if DPa 4 Pb. In this way, embolism propagated from a function of conduit length. conduit to conduit as long as there was a connection with The structure of our model allowed us to analyse the large enough pores. When there were no more ‘‘open flow distribution in the xylem network with great detail. In doors’’ at the edges of the embolised cluster, spreading particular, we could obtain the flow within each conduit stopped and the overall conductivity (Ks) was calculated element. We ran 200 extra simulations, again varying Pc again as described above, but considering only the within the same range as before (0.01–0.4). For each remaining functional conduits connected to the conducting simulation we counted the number of conduit elements in cluster. which the flow was zero and the conduit elements with By gradually increasing Pa we obtained conductivities at negative flow. For the conduit elements with zero flow, we different pressures and the corresponding vulnerability determined whether they belonged to a conduit within the curve was fitted with the following function (Pammenter conducting cluster or to an isolated conduit, i.e., a conduit and van der Willigen, 1998): in which flow was zero in all its conduit elements. To explore the effect of connectivity on effective path 100 PLC ¼ , (10) length, in each of the previous 200 simulations 1000 tracers þ aðPP50Þ 1 e were sent through the virtual xylem. The tracers chose their where P50 is the pressure causing a 50% loss of way at every vertex with a probability proportional to the conductivity in the xylem and a is related to the slope of flows going through each edge. Each tracer registered the the curve. number of conduit elements it passed. Then we calculated The model was written in JAVA and is therefore the average path length of the 1000 tracers and divided it platform independent. Each individual simulation took by the shortest possible path. From that we obtained the about 20 min on a Pentium 4 processor. percentage increase of pathway length corresponding to different values of connectivity. 2.2. Virtual experiments 2.2.2. Experiment 2 2.2.1. Experiment 1 The goal of the second experiment was to determine the First we studied the effect of connectivity (/kS)on importance of connectivity in relation to other parameters. maximum hydraulic conductivity and vulnerability to This experiment was based upon making a large number of embolism, by varying only Pc and keeping the value of runs of the model with different sets of parameter values, ARTICLE IN PRESS 794 L. Loepfe et al. / Journal of Theoretical Biology 247 (2007) 788–803 chosen randomly from specified parameter distributions A (Monte Carlo sampling). We ran 18 382 simulations 0.010 varying the following parameters within the ranges given in Table 1: (1) average conduit diameter (D ), (2) average c 0.008

conduit length (Lc), (3) average maximum pit pore size ) -1

(Dp), (4) fraction of contact area occupied by pores (fp), s

-1 0.006 and (5) probability to connect two vertices (Pc).

We aimed at covering most of the spectrum of the values MPa 2 measured for diffuse-porous species. For the constant 0.004 (m parameters we used the values measured for Acer negundo max

by Wheeler et al. (2005), when available. If no data were Ks 0.002 available, we chose a value in the middle of a biologically reasonable range. For segment dimensions, we used the same 0.000 values as in Experiment 1. As the number of conduits in a 0 5 10 15 20 cross-section was fixed, the total number of conduits in the segment was an inverse function of conduit length. The number of conduits in the simulated segment was 4 500 in all B cases; the average number of conduits per segment was 2338. 0 From the 18 382 simulations, 3099 were not considered, -1 because the solution of the lineal system did not converge. For each of the 15 283 remaining simulations, we -2 calculated the maximum hydraulic conductivity (Ksmax), -3 the pressure at which 50% of conductivity was lost (P50) and the steepness of the vulnerability curve (a). By doing -4 this, we detected that in another 389 simulations the (MPa) -5 50 resolution of the pressure increment we used was too P -6 coarse to accurately estimate P50 and/or a, and these simulations were therefore rejected. -7

-8 3. Results 0105 15 20 3.1. Experiment 1 C When maintaining the value of all other parameters 12 fixed, the maximum hydraulic conductivity (Ksmax), the 10 pressure at which 50% of conductivity is lost (P50) and the steepness of vulnerability curve (a) all increased with connectivity (Fig. 3). This means that higher connectivity 8 ) provides greater maximum sap flow, but also higher -1 6 vulnerability to drought-induced embolism. The effect is

/ S (MPa very clear at low connectivity ( k ), but tends to saturate a 4 at higher values of /kS. Note that in these simulations all the variability in /kS was due to different values of P , c 2 since all the other parameters were kept constant. The vulnerability curves of a high and a low connectivity 0 system are shown in Fig. 4. Consistent with the previous 0 5 10 15 20 paragraph, the highly connected system had a 2.2 fold higher Ksmax, but embolised earlier and faster. In both systems the percentage of intact conduits was always higher Fig. 3. The effect of average connectivity (/kS) on (A) the maximum than the percentage of conductivity remaining. At 95% loss hydraulic conductance (Ksmax), (B) the pressure at which 50% of of conductivity, 34.7% and 18.8% of the conduits were still conductivity is lost (P50) and (C) the steepness of the vulnerability curve functional, at low and high connectivity, respectively. Note (a). Results from Experiment 1. that ‘functional’ here means not being air-filled and still belonging to the conducting cluster. Even at zero Conduit length (Lc) on itself had a noticeable influence conductivity, when the conducting cluster breaks apart, on Ksmax, but the increase of Ksmax with Lc was much only 69.5% (/kS ¼ 13.7) and 48.2% of the conduits steeper in the simulations where conduit length modified (/kS ¼ 3.8) were actually air-filled. /kS and thus, longer conduits implied greater connectivity ARTICLE IN PRESS L. Loepfe et al. / Journal of Theoretical Biology 247 (2007) 788–803 795

A 0.01 0.007 0.009 0.008 0.006 )

-1 0.007 ) 0.005 s

-1 = 3.8 -1 0.006 s -1 0.004 = 13.7 0.005 Mpa 2

MPa 0.004 (m 2 0.003 0.003 (m 0.002 Ks

Ksmax 0.002 0.001 0.001 0 0 0 20 40 60 80 0 2 4 6 Lc (mm) applied pressure (MPa) -2 B 1.0 -3 0.8

Ks(=3.8) -4 0.6 Ks (=13.7)

P50 (MPa) -5 0.4 cond.(=3.8)

-6

fraction remaining cond.(=13.7) 0.2

-7 0.0 0 20 40 60 80 042 6 Lc (mm) applied pressure (MPa) Fig. 5. The effect of conduit length on (A) the maximum hydraulic Fig. 4. Comparison between vulnerability curves for a high and a low conductance (Ks ), and (B) the pressure at which 50% of conductivity is connectivity xylem: (A) absolute values of conductance and (B) percent max lost (P ). In half of the simulations P was maintained constant (solid loss of conductivity and number of conduits. 50 c circles), in the other half /kS was kept constant (open circles). Note that

the effect of conduit length on both Ksmax and P50 was much more pronounced when longer conduits also implied higher connectivity, i.e. (Fig. 5A). Similar results were obtained for vulnerability to when Pc was fixed. embolism; P50 slightly increased with Lc when /kS was kept constant, but the effect was much more pronounced All five parameters that were varied independently (Dc, when varying conduit length affected connectivity Lc, Dp, fp, Pc) affected Ksmax. Conduit diameter was the (Fig. 5B). These results imply that most of the effect of parameter most critically affecting Ksmax: all simulations conduit length on xylem transport functions is mediated by were below a power function of degree four, corresponding its effect on connectivity. to the trend line for theoretical conductivity (Kt), obtained if all the resistance was in the lumen (Fig. 6A) (note that 3.2. Experiment 2 conduit density was constant and that Ksmax and Kt were expressed per total cross-sectional area, not per total lumen Running many simulations with different sets of input area): parameters allowed us to identify the parameters to which the model was more sensitive and detect impossible 4 KsmaxoKt a Dc . (11) combinations of input and output parameters. The scatter plots in Fig. 6 relate the values of the parameters to Ksmax, Ksmax was also sensitive to changes in Lc and Pc: P50 and a. The range of resulting Ksmax values was very networks with longer and more connected conduits had 7 2 1 1 wide (7.6 10 –0.045 m MPa s ), although 98.9% of higher Ksmax (Fig. 6D/M). This effect was accentuated 4 2 1 1 the simulations had a Ksmax 4 1.2 10 (m MPa s ) when looking at connectivity, which combines the effects of 3 2 1 1 and 59.7% had a Ksmaxo6.7 10 (m MPa s ), the Lc and Pc (Fig. 6P). High conductivity could not be known range of Ksmax for all studied species (Maherali achieved with low connectivities, which means that /kS et al., 2004). can be a limiting factor to Ksmax. Pit membrane resistance ARTICLE IN PRESS 796 L. Loepfe et al. / Journal of Theoretical Biology 247 (2007) 788–803 also influenced Ksmax, as showed by the increase of Ksmax conductivity were not considered due to the difficulty in with Dp and fp, but the trend was weaker (Fig. 6G/J). accurately estimating P50 with the employed pressure Vulnerability to embolism depended on three of the five increment (see Section 2), but probably also because the parameters we explored. P50 and a were sensitive to Dp, Lc chosen range for Dp was not wide enough at the lower end. and Pc (Fig. 6E/H/N), whereas Dc and fp did not affect the P50 was most sensitive towards Dp (Fig. 6H) a small Dp spread of embolism (Fig. 6B/K). More than 99% of the assured a certain safety against embolism, while very large simulations were within the measured range for P50 (0.18 pores made impossible a high resistance to embolism. P50 to 14.1 MPa; Maherali et al., 2004), but the whole was also affected by Lc and Pc (Fig. 6E/N) and especially spectrum was not covered, as there were no simulations by the combination of Lc and Pc—that is, connectivity with P50 4 1.25 MPa. This was partially caused by the (Fig. 6Q). Connectivity was the most important parameter fact that simulations with a very steep decrease in with regard to the steepness of the vulnerability curve

Fig. 6. Scatter plots of the simulations in which multiple parameters were varied (Experiment 2). The effects of average conduit diameter (Dc, mm), average conduit length (Lc, in mm), maximum pit pore size (Dp, in nm), fraction of contact area occupied by pores (fp), probability to connect (Pc) and average connectivity (/kS) on maximum hydraulic conductivity (Ksmax), pressure at which 50% of conductivity is lost (P50) and steepness of vulnerability curve (a) are shown. ARTICLE IN PRESS L. Loepfe et al. / Journal of Theoretical Biology 247 (2007) 788–803 797

Fig. 6. (Continued)

(Fig. 6R). Low values of /kS impeded fast embolism Connectivity was reduced by 26.5%, with a greater relative spread and highly connected systems never showed shallow loss in initially well-connected systems. (4) Average slopes. Therefore, vulnerability to embolism was deter- conduit length was 84.0% of the initial average conduit mined by pit pore size and connectivity, with Dp being length; the reduction being greater in well connected especially important for P50 and /kS for the steepness of systems with initially long conduits. Therefore longer the vulnerability curve. conduits tended to embolise earlier. (5) Conduit diameter When 100% of conductivity was lost, on average only remained unaltered. 59.7% of the conduits were actually air-filled. When When looking at the relationship between Ksmax and P50, looking at the xylem network at the pressure at which we observed a clear dependence between them (Fig. 7). 50% of conductivity was lost (P50) we found the following Having a very high maximum hydraulic conductivity average properties: (1) 78.1% of the conduits were still excluded the possibility of being very resistant against functional. (2) Only 16.1% of the vessels were actually air- embolism and vice versa. The model allowed the combina- filled, the rest of non-functional conduits was filtered out tion of low Ksmax and high P50, but a high conductivity was because they lost contact to the conducting cluster. (3) incompatible with a high resistance to embolism. ARTICLE IN PRESS 798 L. Loepfe et al. / Journal of Theoretical Biology 247 (2007) 788–803

0 (changing Pc) clearly show that most of the effect of -2 conduit length was due to its effect on connectivity -4 (Fig. 5A). -6 Regardless of whether it was caused by short conduits, low P , or both, when the connectivity of the system was -8 c low its total hydraulic conductance was usually much lower -10 than the values predicted without considering system-level (MPa) -12 properties (Fig. 6). This is because the classical model P50 -14 based on adding up the resistances from a cross-section -16 effectively assumes that all conduits are transporting water -18 at full capacity (i.e., that there is no constraint to water -20 flow elsewhere in the network). The fact that factors other 0 0.01 0.02 0.03 0.04 0.05 than lumen and pit membrane resistances can contribute to 2 -1 -1 Ksmax (m MPa s ) the total hydraulic resistance of the xylem network is consistent with the experimental results by Schulte et al. Fig. 7. The relationship between maximum hydraulic conductivity (Ksmax) (1987), in which after dissolving the pit membranes the and the pressure at which 50% of conductivity is lost (P50). Solid circles represent simulations with connectivity greater than 5, open triangles measured Ksmax was still 30% lower than the one predicted simulations with a connectivity lower than 5. Results from Experiment 2. by the Hagen–Poiseuille equation. In the following Note that high maximum hydraulic conductivity was incompatible with paragraphs we discuss in detail the mechanisms that might high resistance to embolism and that this trade-off is unaffected if explain this effect of network connectivity on overall considering only simulations with /kS greater than five. hydraulic resistance (see also Fig. 8). Even a very long and wide conduit with large pit pores is completely useless for water transport if it is not connected 4. Discussions at least twice to the conducting cluster. Low connectivity can isolate conduits, even large groups of them, converting 4.1. Maximum conductivity them into dead ends. In our model we filtered out the conduits that had no connection at all to the conducting It is well known that the maximum hydraulic conductiv- cluster, but more for practical (calculation) reasons than ity (Ksmax) of the xylem is related to conduit diameter (Dc). out of conviction, as we are not aware of any empirical Classical models used the sum of the hydraulic diameters in data confirming either the existence or absence of them. It a cross section to estimate Ksmax, according to the is difficult to think of any biological function of Hagen–Poiseuille equation. In our model, Dc set the unconnected conduits, so if they exist it is probably due maximum conductance, corresponding to the theoretical to a building imperfection. This effect was important only conductivity (Kt). But the vast majority of our simulations for extremely low connectivities; at values of /kS greater did not reach that maximum, in agreement with experi- than 5 less than 10% of the conduits were filtered out (data mental data showing that theoretical conductivity (Kt) not shown). Our main results were unaffected if we tends to be always higher than Ksmax measured on a wood considered only simulations with /kS 4 5 (see, for segment (Chiu and Ewers, 1993; Lancashire and Ennos, instance, Figs. 6 and 7). 2002; Tyree and Zimmermann, 2002 and literature cited Groups of conduits with only one connection to the therein). Recently, Sperry et al. (2005) showed that conducting cluster do not contribute to a steady flow of sap KsmaxE0.46 Kt over several orders of magnitude of (Fig. 8, circle a), but they might have a function as water conductivities and across different wood types. reservoirs, and were kept in the model. Our results show Water has to cross a porous membrane to flow from one conduit to another, which adds resistance to the pathway. Strategies to reduce this resistance can target either the resistance of each single connection by incrementing the pit pore size or the fraction of contact area occupied by pores—or reduce the number of connections per segment length that sap has to cross by increasing conduit length (Comstock and Sperry, 2000). Accordingly, an increase in conduit length, pit pore size or fraction of contact area occupied by pores resulted in an increment of Ksmax in our model. Interestingly, the effect of conduit length on conductivity was not only through changing the number of pit membranes that water has to cross but also through a Fig. 8. The influence of low connectivity on conductance. Circle (a) shows modification of the overall connectivity of the system. The an isolated group of conduits, circle (b) an isolated part of a conduit and simulations in which we varied Lc without modifying /kS circle (c) a conduit with negative flow. ARTICLE IN PRESS L. Loepfe et al. / Journal of Theoretical Biology 247 (2007) 788–803 799

35% 60% 0.012 0.012 ) -1 ) s 0.010 -1 0.010 30% -1 s

50% -1 0.008

0.008 MPa

2 0.006 25% MPa

2 0.006 0.004 (m

40% max 0.002 (m 0.004 20%

Ks 0.000 max 0.002 0% 20% 40% 30% Ks 0.000 15% cond. elem. flow=0 0% 20% 40% 60% 10% 20% isolated conduits cond. elements flow=0

isolated conduits 5%

10% 0% 01015205 0% 0 5 10 15 20 Fig. 10. Effect of average connectivity (/kS) on the fraction of conduits elements with flow ¼ 0. Only conduit elements that belonged to a conduit within the conducting cluster were considered. The inlet shows the fraction Fig. 9. Effect of average connectivity (/kS) on the fraction of isolated of non-conducting conduit elements vs. the maximum hydraulic con- conduits. The inlet panel shows fraction of isolated conduits vs. the ductivity (Ksmax). maximum hydraulic conductivity (Ksmax).

40% that the fraction of conduits without contribution to the 35% overall flow was relevant (up to 50%) at low connectivity, but decreased exponentially with /kS (Fig. 9). These 30% effects become even more important as embolism spreads 25% in the system (see below). Note that the situation above represents only the extreme case; conduits or groups of 20% conduits can be connected twice or more to the conducting 15% cluster, but through a connection with a very high resistance. In that case they would not use all their 10% conducting capacity, as incoming flow would be limited Increase of lumen pathway 5% by a hydraulic ‘‘bottleneck’’ elsewhere in the system. Even when a conduit belongs to a conducting cluster, 0% 0105 15 20 water can flow through only the part between its first entry connection and its last exit connection. In our model, we assumed a random distribution of the connections along a Fig. 11. Increase of effective lumen pathway length caused by conduit conduit, therefore a low connectivity increased the prob- elements with negative flow as a function of average connectivity (/kS). ability of the extremes of conduits becoming dead ends (Fig. 8, circle b). This shortened the ‘‘effective’’ conduit length. Our results showed that the number of these dead 4.2. Vulnerability to embolism ends clearly depended on connectivity (Fig. 10). In systems with very low connectivity we found conduit The air-seeding mechanism (Zimmermann, 1983)is elements in which the flow went in opposite direction to the widely accepted to be the principal cause for conductivity overall flow (Fig. 8, circle c). This implies an enlargement loss in the xylem during drought. The first condition for of the effective pathway and therefore an increase in total air-seeding to occur is that there is at least one pre-existing resistance, as water molecules have to pass through more air-filled conduit, due to mechanical failure or residual conduit elements, whose resistances add up as they are from previous embolisms (Tyree and Sperry, 1989). From connected in series. This effect was important at very low this conduit air can enter a neighbour conduit through the connectivity, but decreased as a power function of /kS, pit membrane if the pressure difference between the air- and for values of /kS greater than 5 the increase of path filled and the functional conduit (DPa) is large enough (see length was less than 5% (Fig. 11). Eq. (9)). But having a connection with a pore so that In conclusion, our results clearly showed that connectiv- DPa4Pb is not sufficient condition for a conduit to ity is potentially a factor to be taken into account in embolise. The other necessary condition is to have an air- predicting Ksmax, as a lack of connectivity can become a co- filled neighbour. limiting factor together with lumen and pit membrane In vivo videos by magnetic resonance imaging (MRI) resistances. (N.M. Holbrook, personal communication) show clearly ARTICLE IN PRESS 800 L. Loepfe et al. / Journal of Theoretical Biology 247 (2007) 788–803 how embolism starts from a few conduits and propagates 4.2.1. System-level properties and embolism towards neighbour regions. Whole regions of the xylem The only mention of system-level properties related to show no embolism, while others are already almost embolism vulnerability we have found in the literature is completely embolised. This probably means that some the concept of redundancy. Redundancy is defined as the conduits in the former region are not embolised, despite likelihood to find an alternative path when an element of a having pit pores large enough for air to spread, only given path is blocked (Carlquist, 1988; Cruiziat et al., 2002; because they do not have an air-filled neighbour. Conduits Ewers, 1985). In that sense, having a high connectivity with more connections tend to have more neighbours and would diminish the effects of losing a conduit, and therefore a higher probability to be connected to an air- therefore resistance to embolism would increase with filled conduit. And more connections also mean an increase connectivity. But, according to our results, the effect of in the chance to have at least one with a pore large enough redundancy is not enough to compensate for the increment to let air pass at a given pressure. More connections not in lost conduits due to faster spreading of embolism in only imply a higher risk of being embolised, but also a highly connected systems (see above). higher potential to propagate the embolism. Therefore, Average degree (i.e., connectivity) is probably not the high connectivity should facilitate the spreading of only system-level property that affects the vulnerability to embolism through the xylem network and cause an earlier embolism of the xylem. Complex network theory often loss of conduits. studies the degree distribution; in our case the histogram of The ideas outlined in the previous paragraph are the number of connections per conduit (for a review of supported by our results, as highly connected systems complex network theory see Albert and Barabasi, 2002 or embolised earlier and faster (Fig. 6Q/R). The idea that Newman, 2003). In random graphs like the ones studied by conduits with more connections are more vulnerable to Erdo¨s and Re´nyi (1959) the degree distribution is embolism was supported by the fact that longer conduits binominal or Poisson, whereas in scale-free networks the (that had more connections) were embolised earlier. As probability to have a certain number of connections connections were set randomly, longer conduits had more follows a power law. Random and scale-free networks connections and therefore were more likely to have an air- have very different tolerance towards error (i.e., random filled neighbour. As longer conduits often contribute more failure) and directed attack (in which case the chance to be to water transport, the embolisation of a long conduit eliminated is proportional to the number of connections). implies a great loss of conductance (Lo Gullo et al., 1995). Whereas random networks are equally vulnerable to error Also, long conduits can carry the embolism to distant and attack, scale-free networks show a great stability regions of the xylem; whereas short conduits might towards error but are extremely vulnerable to attacks restrict the spreading to a local area, in accordance to (Albert et al., 2000). There are significant differences Zimmermann’s segmentation concept and reinforcing between wood types with regard to their conduit length the idea that conduits are shorter in distal regions for distribution and, presumably, also their degree distribu- safety reasons (Comstock and Sperry, 2000). Therefore, tion. The tracheids of gymnosperms are limited to the size species with fewer but longer conduits are likely to be more of a single cell and arranged in a grid-like structure. Each vulnerable to embolism than species with more but shorter tracheid within an individual plant has therefore similar conduits. size and probably similar number of connections. Their Our result that, on average, only 22% of conduits were degree distribution is comparable to the Erdo¨s-Re´nyi non-functional at P50 is in qualitative accordance with model of a random graph. Ring-porous species, in experimental data (Lo Gullo and Salleo, 1993). In their contrast, have a very unequal length and probably degree case the divergence could be explained by the fact that distribution and are more likely to resemble scale-free wider conduits embolised earlier, which indicates that networks like the Internet, although physical limitations conduit diameter is related either to conduit length or pit probably impede extremely high degrees. In the xylem, pore size. In our model we used a conservative approach, failures due to mechanical fatigue or herbivores can most not linking conduit diameter to any other parameter. The likely be regarded as errors, in the sense that they do not difference between percent loss of conduits and percent loss happen preferentially in the most connected conduits. But of conductivity (Fig. 4) can be explained by two effects (1) as the most connected conduits are the most vulnerable to decrease in overall connectivity as the conduits are be embolised by air-seeding (because they are more likely embolised, and (2) preferential loss of longer conduits, to have an air-filled neighbour; see above), the xylem has to that often contribute more to overall flow. In our model, at deal with directed attack during drought periods. The way P50 only 73% of the non-functional conduits were actually we constructed our model lead to an exponential length air-filled (data not shown), the rest was filtered out as they and degree distribution, with a much sharper decay for no longer belonged to the conducting cluster. This might systems with short average conduits. We therefore simu- have important implications for the refilling process, lated an intermediate situation between random and scale- as following this logic only a few crucial conduits have to free networks. This probably covered well the distribution be refilled to regain a substantial part of the initial spectrum of diffuse-porous species, but probably was not conductivity. representative of Gymnosperm or ring-porous species. ARTICLE IN PRESS L. Loepfe et al. / Journal of Theoretical Biology 247 (2007) 788–803 801

The fact that we could see a clear trend for increased connectivity. The influence of Lc is particularly rich vulnerability to embolism of systems with longer conduits because on top of the effect on connectivity we have to was probably caused, at least in part, by their longer tail in consider the effect on the number of inter-conduit the degree distribution and therefore higher vulnerability to membranes that water has to cross (Fig. 5). directed attack. It is clear from the previous paragraph that the correlation between Ksmax and P50 observed in our model 4.3. Efficiency vs. safety trade-off (Fig. 7) and in real plants cannot be attributed to one single parameter. Maximum hydraulic conductivity as well as There is strong empirical evidence for the existence of an vulnerability to embolism are determined by a set of efficiency vs. safety trade-off in the xylem between different variables. In our model the parameters that had an segments of an individual plant. Roots have higher influence on both efficiency and safety were conduit length, conductivities than stems or branches, but they are also pit pore size and Pc. This list would be larger if we assumed more vulnerable to embolism (Hacke et al., 2000; less independence of the inputs (for instance, if conduit Mencuccini and Comstock, 1997; Martinez-Vilalta et al., length was related to conduit diameter or pit pore size to 2002b; Lo Gullo and Salleo, 1993; Pockman and Sperry, contact area). In addition, we did not take into account 2000; Tyree et al., 1994). There is also evidence for a safety against implosion (rupture of the membrane), weaker cross-species trade-off (Hacke et al., 2006; Wheeler where the fraction of contact area occupied by pores is et al., 2005). The most comprehensive study to date presumably important (Sperry and Hacke, 2004), or concludes that there is a negative relationship between resistance against freezing-thaw induced embolism, that maximum conductivity and resistance to embolism across has been shown to depend on conduit diameter (Dc) (e.g. species, but the relationship is no longer significant when a Davis et al., 1999). This implies the existence of a large phylogenetically independent test is used (Maherali et al., number of strategies compatible with an optimal compro- 2004). mise between efficiency and safety, which may explain why The explanations given for this trade-off concentrate a clear trade-off between Ksmax and P50 is not found across mainly at the single connection and single parameter level, phylogenetically independent taxa. Each individual plant suggesting either a trade-off between pit membrane represents the realization of one possible combination of conductivity and air-seeding pressure or a positive relation these variables depending on its genetical and environ- between conduit diameter (Dc) and maximum pit pore size mental constraints. (Dp). The most straightforward idea is a direct relation between maximum pit pore size (relevant for air-seeding) 4.4. Limitations and further development and equivalent pit pore size (relevant for conductivity), theoretically explained by Sperry and Hacke (2004), Our model has to be regarded a first approach towards experimentally measured (Jarbeau et al., 1995) and also the modelling of water transport in the xylem as a system- assumed in our model. Wheeler et al. (2005) linked level process. Consequently, we kept the representation of maximum pit pore size to pit membrane area instead of the xylem structure as simple as possible, so that the effect equivalent pit pore size, as they could not find any of system-level properties can be easily isolated and relationship between area-specific membrane resistance interpreted, and the model serves as a basic structure and P50, but the relationship suggested by Wheeler et al. where additional complications can be added. Some (2005) still implies that the trade-off is between membrane relevant aspects that were not included in the present conductivity (as larger areas conduct more) and maximum model and are likely to affect plant water transport in the pit pore size. But according to our results the contribution xylem as modelled here are: (1) An explicit representation of pit membrane resistance to total resistance can be rather of the three-dimensional arrangement of different cell types small and high Ksmax were possible even with considerably in the xylem (e.g., annual rings, xylem rays, early- vs. small pores (Fig. 6G). Martinez-Vilalta et al. (2002b) latewood). In particular, the xylem may be best represented avoided the assumption that pit membrane conductivity is as a set of separate sub-networks interconnected only relevant for Ksmax by linking maximum pit pore size to through living ray cells. (2) Water transport in the xylem conduit diameter instead of pit membrane resistance. This was simulated as a purely hydraulic process, i.e., we did not would explain why the largest conduits often embolise first, consider the effects of sap ionic composition on the as shown by Lo Gullo and Salleo (1993), but there is no permeability of the pit membrane (cf. Zwieniecki et al., experimental evidence consistent with that link. 2001b). It should be noted that recent results show that the Our results also suggest that system-level properties are mechanism underlying these effects is still poorly under- likely to play a role in the efficiency vs. safety trade-off in stood (van Ieperen and van Gelder, 2006; H. Cochard and the xylem. Connectivity increased both maximum con- M. Mencuccini, unpublished results), making it very ductivity and vulnerability to embolism (Fig. 6P/Q/R) and difficult to include them in a model until more evidence is is therefore another element to be taken into account in the collected. (3) Conduits were considered to be perfect trade-off. Either long conduits or a high density of inter- capillaries. Whereas this has shown to be a reasonable conduit connections (high Pc) caused high network approximation in some cases (cf. Zwieniecki et al., 2001a), ARTICLE IN PRESS 802 L. Loepfe et al. / Journal of Theoretical Biology 247 (2007) 788–803 the impact of this assumption has to be evaluated in any Choat, B., Ball, M., Luly, J., Holtum, J., 2003. Pit membrane porosity and serious attempt to split total xylem resistance into its water stress-induced cavitation in four co-existing dry rainforest different components. species. Plant Physiol. 131, 41–48. Cochard, H., Cruiziat, P., Tyree, M.T., 1992. Use of positive pressures to establish vulnerability curves. Further support for the air-seeding 5. Conclusion hypothesis and implications for pressure–volume analysis. Plant Physiol. 100, 205–209. This is the first model of the xylem that takes into Cochard, H., Nardini, A., Coll, L., 2004. Hydraulic architecture of leaf account its three-dimensional structure and the system- blades: where is the main resistance? Plant, Cell Environ. 27, 1257–1267. level properties of the conducting network. We showed on Comstock, J., Sperry, J.S., 2000. Tansley Review No. 119. Theoretical a theoretical basis that maximum hydraulic conductivity considerations of optimal conduit length for water transport in and vulnerability to embolism depend on multiple factors, vascular plants. New Phytol. 148, 195–218. including the connectivity of the network. Connectivity Cruiziat, P., Cochard, H., Ame´glio, T., 2002. Hydraulic architecture of increases both maximum hydraulic conductivity and : main concepts and results. Ann. For. Sci. 59, 723–725. vulnerability to drought-induced embolism and is therefore Davis, S.D., Sperry, J.S., Hacke, U.G., 1999. The relationship between xylem conduit diameter and cavitation caused by freezing. Am. J. 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