ARTICLE IN PRESS Journal of Theoretical Biology 247 (2007) 788–803 www.elsevier.com/locate/yjtbi The relevance of xylem network structure for plant 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 plants, providing a low resistance pathway for water movement from roots to leaves. 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 vascular plant 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.