#### Optimal tree structure under hydraulic constraints

The link between plant design and tissue water relations can be explored by means of a simple model. All parameters and variables are defined in Table 1. Let *E*_{f} be the transpiration rate of a unit of leaf surface in the stand. If Ψ_{leaf} is the resulting leaf water potential, and (for simplicity) neglecting the effects of tree height on gravitational potential, we can then define the hydraulic resistance per unit foliage area across the soil–plant continuum (*R*^{f}_{tot}) as:

Table 1. Variables, parameters and units used in the model Variable | Definition | Units |
---|

*A*_{s} | Sapwood area (on a ground area basis) | m^{2} m^{−2} |

*D* | Vapour pressure deficit | Pa |

*D*_{0} | Empirical coefficient for response to *D* | Pa |

*E*_{f} | Transpiration per unit foliage area | m^{3} m^{−2} s^{−1} |

*f* | Reduction factor (subscript: *D*, for vapour pressure deficit; Ψ, for soil water potential) | – |

*g*_{s} | Stomatal conductance (superscript: max, maximum) | m s^{−1} |

*h* | Tree height | m |

*l* | Tissue longevity (subscript: r, fine root; s, sapwood) | year |

*L* | Fine root density | m m^{−3} |

*r* | Fine root radius | m |

*r*_{r} | Root resistivity per unit fine root biomass | MPa s kg m^{−3} |

*r*_{s} | Sapwood resistivity | MPa s m^{−2} |

*r*_{soil} | Soil resistivity (superscript: sat, saturated) | MPa s m^{−2} |

*r*_{soil} | Soil resistivity per unit fine root biomass | MPa s kg m^{−3} |

*R* | Hydraulic resistance (superscript: f, per unit projected leaf area; g, per unit ground area; subscript: root, shoot, soil, total) | MPa s m^{−1} |

*T* | Temperature | °C |

*W* | Stand biomass (subscript: f, foliage; r, fine roots; s, sapwood) | kg m^{−2} |

*z* | Rooting depth | m |

η | Viscosity of water | MPa s |

ρ | Density (subscript: s, sapwood; r, root) | kg m^{−3} |

σ | Specific leaf area | m^{2} kg^{−1} |

Ψ | Water potential (subscript: e, entry, leaf, foliage, soil) | MPa |

Ψ_{0} | Ψ at complete stomatal closure | MPa |

| Critical leaf water potential | MPa |

- (eqn 1a)

where Ψ_{soil} is soil water potential. Leaf-specific resistance can be scaled to the stand level as:

- (eqn 1b)

where *W*_{f} is stand foliage biomass, σ is specific leaf area and *R*^{g}_{tot} is stand hydraulic resistance on a ground area basis, which can be viewed as the sum in a series of three distinct components located in the shoot, fine roots and soil, respectively.

Shoot hydraulic resistance (*R*^{g}_{shoot}) is affected by the length of the hydraulic pathway, related to tree height *h*, and by the cross-sectional area *A*_{s} of conducting sapwood, again expressed on a ground area basis (Whitehead *et al*. 1984):

- (eqn 2)

where *r*_{s} is sapwood resistivity and *W*_{s} and ρ_{s} are sapwood biomass and density, respectively. According to the pipe model theory (Shinozaki *et al*. 1964), a constant sapwood cross-sectional area has been assumed in equation 2 throughout the plant, once axes of the same branching order are summed together.

Root hydraulic resistance (*R*^{g}_{root}) is mainly associated with the movement of water from the epidermis to the stele (Passioura 1988); it is therefore inversely related to the surface of fine roots and, approximately, to fine root biomass *W*_{r}:

- (eqn 3)

where *r*_{r} is resistivity per unit root biomass.

Soil hydraulic resistance (*R*^{g}_{soil}) is also influenced by the biomass of fine roots exploring the soil. From single root theory, *R*^{g}_{soil} can be approximated as (Passioura & Cowan 1968):

- (eqn 4)

where *r* is root radius, *L* is fine root density in the soil, *z* is rooting depth and *r*_{soil} is soil hydraulic resistivity. Rooting depth is assumed to increase with fine root biomass, implying a constant fine root density throughout the soil. Soil resistivity per unit root biomass (*r*^{r}_{soil}) is therefore defined as:

- (eqn 5)

where ρ_{r} is the basal density of fine roots and is assumed to be equal to sapwood density.

Soil hydraulic resistivity is a direct function of soil water potential and can be expressed as (Campbell 1985):

- (eqn 6)

where the resistivity of saturated soil (*r*^{sat}_{soil}), soil entry potential (Ψ_{e}) and the empirical coefficient *b* are all functions of soil texture. Tabulated values of soil hydraulic parameters for key textural types, computed according to Campbell (1985) can be found in Sperry *et al*. (1998). Soil saturated resistivity is lower in coarse soils. At the same time, the coarser the soil, the steeper the increase in soil resistivity with decreasing water potentials.

Experimental evidence (reviewed for *P. sylvestris* in the Results section) suggests that under natural conditions, leaf water potential does not usually exceed a critical value , which could be dictated in coniferous species by the risk of diffuse xylem embolism and tissue dieback (Tyree & Sperry 1988). The maintenance of such a functional homeostasis imposes a tight constraint on plant structure, as it requires from equations 1–4 that hydraulic resistances conform to:

- ( eqn 7)

Equation 7 is fully equivalent to the model of Whitehead *et al*. (1984) when the role of roots is disregarded (when *r*^{r} = *r*^{r}_{soil} = 0) and to the model presented by Givnish (1986) when, in contrast, stem hydraulic resistance is neglected (when *r*_{s} = 0).

The functional requirements of equation 7 could not be met unless new foliage growth was always supported by an adequate quantity of sapwood and fine roots. Moreover, the reduction in hydraulic resistance needed to sustain new foliage will be achieved at minimum cost, in order to reserve as many resources as possible for foliage growth and so maximize plant height increments and, ultimately, tree survival and fitness. In particular, optimal growth under hydraulic constraints requires that the ratio of marginal hydraulic returns to marginal annual cost for carbon investment in either roots or sapwood, once discounted for tissue turnover, should be the same (Case & Fair 1989):

- (eqn 8)

- (eqn 9)

After rearranging, the balance between sapwood area and fine root biomass is predicted to be:

- (eqn 10)

where the coefficient *c* is a function of tissue characteristics and soil hydraulic resistivity:

- (eqn 11)

The balance between sapwood area and fine root biomass will therefore depend on soil textural characteristics, and will in general shift towards larger root biomass under dry conditions.

When combined with the general requirement of functional homeostasis of equation 7, equation 10 translates into hydraulic constraints, representing the optimal balance between transpiring foliage and conductive tissues under given environmental conditions:

- (eqn 12)

- (eqn 13)

The allometric balance between *W*_{f} and *A*_{s}, *W*_{r} can be seen to depend on tree height, less carbon being allocated to foliage as the stand ages, as discussed in detail by Magnani *et al*. (2000) who give a more detailed description of the mathematical development of equations 9–13.

The basic constraints captured by equations 10–13 are not altered qualitatively by including the effects of height and gravitational potential, which are equivalent to a change in critical water potential, in equation 7.

#### Functional and structural response to key environmental parameters

The effects of temperature, air humidity and soil water potential on plant function and structural design are considered here. Low temperatures dramatically increase the hydraulic resistance of the soil–plant continuum (Cochard *et al*. 2000). In the case of sapwood resistance the effect is purely physical, and is determined by the response to temperature of water viscosity, η. The relationship can be represented as:

- (eqn 14)

- (eqn 15)

Root resistance responds even more dramatically to low temperatures, as a result of the reduced fluidity of plasma membranes that water has to cross at the level of the endodermis. This is demonstrated by a re-analysis of root hydraulic conductance (= 1/*R*_{root}) data for a coniferous, two dicot and two monocot species (BassiriRad, Radin & Matsuda 1991; Fennell & Markhart 1998; Markhart *et al*. 1979; Smit-Spinks, Swanson & Markhart 1984; Fig. 2). The foliage : fine root biomass ratio is expected in the model to follow the same relationship. Moreover, in order to compensate their greater sensitivity to cold conditions, fine roots are expected to be favoured against sapwood area at low temperatures (equation 10).

The expected impact on plant structure of air humidity and soil water depends upon the response of transpiration to the plant’s environment. Leaf transpiration can be approximated in conifers by imposed transpiration, the product of stomatal conductance (*g*_{s}) by air vapour pressure deficit (Whitehead *et al*. 1984). Air and soil humidity limitations reduce stomatal conductance below its maximum value *g*_{s}^{max}:

- (eqn 16)

where the modifiers *f*_{D} and *f*_{Ψ} range in value between 0 and 1, and represent the effects of air vapour pressure deficit and soil water potential, respectively. According to Lohammar *et al*. (1980), the reduction induced by air vapour pressure deficit can be expressed as:

- (eqn 17)

where *D* is air vapour pressure deficit and *D*_{0} is the value inducing a 50% stomatal closure. The response to soil water potential can be approximated by a simple linear function (Jones 1992):

- (eqn 18)

where Ψ_{0} represents the soil water potential corresponding to complete stomatal closure. Therefore, leaf transpiration can be represented as:

- (eqn 19)

From equation 7, the hydraulic resistance per unit foliage area that can be safely maintained will decrease asymptotically as *D* increases, mirroring the response of transpiration captured by equation 19. This constraint will cause a parallel decline in the biomass of foliage supported by unit sapwood area (Fig. 1b):

- (eqn 20)

From equation 10, the balance between foliage and absorbing roots is expected to respond in a similar way to *T* and *D*.

In response to soil drought, stomatal closure effectively prevents the onset of extreme leaf water potentials, despite the marked increase in soil–plant hydraulic resistance that is often reported (Breda *et al*. 1993; Irvine *et al*. 1998). From equations 7 and 19, the maximum resistance that can be withstood by the plant under hydraulic constraints can be expressed as:

- (eqn 21)

If is more negative than Ψ_{0}, stomata will shut completely and *R*^{f}_{tot} will therefore increase in dry soil without triggering substantial xylem cavitation (Fig. 3a), in good agreement with experimental evidence (Irvine *et al*. 1998). However, because of the parallel increase in soil hydraulic resistivity (equation 6; Fig. 3a), plant resistance per unit foliage area must be reduced if the hydraulic constraints are to be met. This will shift resources from foliage to conductive tissues under dry conditions, which will be more marked in coarse soils (Fig. 3b,c). Because of the parallel changes in the balance between sapwood and fine roots (equation 10), drought will increase allocation to fine roots (Fig. 3b), as often reported under field conditions.