Plant architecture and the allometry of hydraulic transport, light interception and growth

The metabolic scaling theory identifies network architecture as a major predictor of whole plant metabolism via hydraulic conductance of the xylem and the shared stomatal pathway for water loss and carbon gain. To predict hydraulic properties, this theory utilizes the West, Brown, Enquist (WBE) architectural model, which is based on principles of space-filling, biomechanical stability, and optimality of hydraulic transport and it is meant to be generally representative of plants. However, plants are highly diverse in their network architecture. Does this diversity matter or does it represent different ways of accomplishing the same task? This dissertation addresses that question by extending WBE to include architectural variation and by testing model predictions and assumptions. The model predicts the scaling exponent between hydraulic conductance and plant size. This exponent depends on the ""bottleneck"" effect, where greater hydraulic resistance in leaves and twigs steepens the exponent. The bottleneck effect was greater when xylem conduits were much larger or more abundant in the trunk than in the twigs. Observed diversity in xylem properties predicted that different functional groups had substantial overlap in hydraulic transport and its scaling. Branching architecture did not influence the bottleneck effect. However, deviating from WBE increased hydraulic conductance and biomechanical stability while requiring less tissue but reducing light interception. Branching could alter hydraulic scaling if architecture changed ontogenetically, which data suggested. MST assumes direct proportionality between sap flow and growth. This was supported in five of six tested species. However, tree species grew more per water use than shrubs, likely reflecting differential allocation. Differences between species were partially attributable to xylem anatomy and plant size. Among this variation in xylem anatomy, branching architecture, and plant stature, the dimensions of leaves and twigs also vary with thicker twigs curiously tending to support few large leaves instead of many small leaves (Corner's rule). Why do plants coordinate leaf and twig size? Corner's rule was recast as the prediction that larger twig leaf areas are composed of larger leaves. Species supported this prediction and had highly convergent scaling. A model predicted that Corner's rule exists to optimize the return on investment in leaves.

PLANT ARCHITECTURE AND THE ALLOMETRY OF HYDRAULIC TRANSPORT, LIGHT INTERCEPTION AND GROWTH by Duncan D. Smith A dissertation submitted to the faculty of The University of Utah in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Biology The University of Utah August 2015 Copyright © Duncan D. Smith 2015 All Rights Reserved The University of Utah Graduate School STATEMENT OF DISSERTATION APPROVAL The dissertation of Duncan D. Smith has been approved by the following supervisory committee members: John S. Sperry Chair April 30, 2015 Date Approved Frederick R. Adler Member April 30, 2015 Date Approved David R. Bowling Member April 30, 2015 Date Approved Brian J. Enquist Member Date Approved Thomas A. Kursar Member April 30, 2015 Date Approved and by M. Denise Dearing Chair/Dean of the Department/College/School of Biology and by David B. Kieda, Dean of The Graduate School. ABSTRACT The metabolic scaling theory identifies network architecture as a major predictor of whole plant metabolism via hydraulic conductance of the xylem and the shared stomatal pathway for water loss and carbon gain. To predict hydraulic properties, this theory utilizes the West, Brown, Enquist (WBE) architectural model, which is based on principles of space-filling, biomechanical stability, and optimality of hydraulic transport and it is meant to be generally representative of plants. However, plants are highly diverse in their network architecture. Does this diversity matter or does it represent different ways of accomplishing the same task? This dissertation addresses that question by extending WBE to include architectural variation and by testing model predictions and assumptions. The model predicts the scaling exponent between hydraulic conductance and plant size. This exponent depends on the "bottleneck" effect, where greater hydraulic resistance in leaves and twigs steepens the exponent. The bottleneck effect was greater when xylem conduits were much larger or more abundant in the trunk than in the twigs. Observed diversity in xylem properties predicted that different functional groups had substantial overlap in hydraulic transport and its scaling. Branching architecture did not influence the bottleneck effect. However, deviating from WBE increased hydraulic conductance and biomechanical stability while requiring less tissue but reducing light interception. Branching could alter hydraulic scaling if architecture changed ontogenetically, which data suggested. MST assumes direct proportionality between sapflow and growth. This was supported in five of six tested species. However, tree species grew more per water use than shrubs, likely reflecting differential allocation. Differences between species were partially attributable to xylem anatomy and plant size. Among this variation in xylem anatomy, branching architecture, and plant stature, the dimensions of leaves and twigs also vary with thicker twigs curiously tending to support few large leaves instead of many small leaves (Corner's rule). Why do plants coordinate leaf and twig size? Corner's rule was recast as the prediction that larger twig leaf areas are composed of larger leaves. Species supported this prediction and had highly convergent scaling. A model predicted that Corner's rule exists to optimize the return on investment in leaves. iv CONTENTS A B S T R A C T iii LIST OF F IG U R E S ................................................................................................................viii LIST OF TA B L E S x A C K N O W L E D G M E N T S xi C H A P T E R S 1. IN T R O D U C T IO N ........................................................................................................... 1 1.1 The chapters .................................................................................................................. 2 1.2 References...................................................................................................................... 4 2. A S PEC IES -LEV EL MODEL FO R METABOLIC SCALING IN TR E E S I. EX P LO R IN G BO UN D AR IE S TO SCALING SPACE W IT H IN AND ACROSS S P E C IE S ........................................................................ 6 2.1 Summary........................................................................................................................ 6 2.2 Introduction .................................................................................................................. 7 2.3 Model description......................................................................................................... 11 2.3.1 Branching architecture and mass a llom e try ................................................. 11 2.3.2 Xylem architecture and water use allometry (Q = k2DqB0) ....................... 12 2.4 Model results.................................................................................................................. 15 2.4.1 Size dependent water-use a llom e try ............................................................... 15 2.4.2 Influence of individual traits on water use s c a lin g ..................................... 15 2.4.3 Functional tree types in scaling space............................................................. 17 2.4.4 Intraspecific vs. interspecific scaling............................................................... 18 2.5 Discussion...................................................................................................................... 19 2.6 Acknowledgements ....................................................................................................... 22 2.7 References ....................................................................................................................... 22 S2 Supporting information ................................................................................................ 34 52.1 Size-dependent water use a llom e try ............................................................... 34 52.2 Functional tree types in scaling space............................................................. 34 52.3 Intra- vs. interspecific scaling........................................................................... 35 3. D EV IA T IO N FROM SYMMETR ICALLY SELF -SIMILAR B R A N C H IN G IN TR E E S P R ED IC T S ALTER ED H YD RA UL ICS, M EC H A N IC S , L IG H T IN T E R C E P T IO N AND METABOLIC SCALING ........................................................................................................................... 39 3.1 Summary ......................................................................................................................... 39 3.2 Introduction .................................................................................................................. 40 3.3 Methods and model description ............................................................................... ...42 3.3.1 The path fraction index for tree fo rm ................................................................42 3.3.2 Empirical path frac tio n s.......................................................................................42 3.3.3 Tree building model ..............................................................................................43 3.3.4 Hydraulic conductance of model trees .......................................................... ...46 3.3.5 Volume of model tre e s............................................................................................47 3.3.6 Mechanical stability of model tre e s ................................................................. ...47 3.3.7 Light interception of model trees .......................................................................48 3.3.8 Scaling predictions ............................................................................................. ...49 3.4 Results................................................................................................................................50 3.4.1 Measured, modeled and estimated path fractions...........................................50 3.4.2 Pf and whole tree hydraulic conductance.........................................................51 3.4.3 Pf and total stem volume.....................................................................................51 3.4.4 Pf and mechanical stability ................................................................................52 3.4.5 Light absorption and P f .......................................................................................52 3.4.6 Scaling and Pf .................................................................................................... ...52 3.5 Discussion.........................................................................................................................53 3.6 Acknowledgements ..........................................................................................................57 3.7 References ..........................................................................................................................58 S3 Supporting information ...................................................................................................69 53.1 Materials used in Pf measurements..................................................................69 53.2 WBE compatibility with L|(D) function (Eqn. 3.8)......................................69 53.3 Branch segment hydraulic conductance ...........................................................71 53.4 Partial derivation of Lcrit(DT) function (Eqn. 3.13)......................................73 4. C O O RD IN A T IO N B E TW E EN WATER TR A N S PO R T CA PA C ITY , BIOMASS G R OW TH , METABOLIC SCALING AND S PEC IE S STATURE IN CO -O C C U R R IN G SHRUB AND T R E E S P E C IE S ___ _79 4.1 Abstract ......................................................................................................................... ...79 4.2 Introduction .................................................................................................................. ...80 4.3 Materials and methods ...................................................................................................82 4.3.1 Location and species ........................................................................................... ...82 4.3.2 Whole-plant hydraulic conductance ............................................................... ...82 4.3.3 Estimating aboveground growth .........................................................................84 4.3.4 Modeling hydraulic scaling from structure ................................................... ...85 4.3.5 Statistics ...................................................................................................................87 4.4 Results ................................................................................................................................88 4.4.1 Sapflow sensor a c c u ra c y .......................................................................................88 4.4.2 M , H , and D allometries for predicting g row th .............................................88 4.4.3 Growth by conductance scaling (Eqn. 4 .1 ) ................................................... ...88 4.4.4 Water use by conductance scaling (Eqn. 4 .2 ) ..................................................89 4.4.5 Environmental v a ria tio n .......................................................................................89 4.4.6 Growth by sapflow scaling (Eqn. 4.3) and W U E ...........................................90 4.4.7 Conductance by mass scaling (Eqn. 4.4)...........................................................90 4.4.8 Sapflow by mass scaling (Eqn. 4 .5 )................................................................. ...91 4.4.9 Growth by mass scaling (Eqn. 4 .6 )................................................................. ...91 4.4.10 Modeling hydraulic conductance by mass scaling...........................................91 4.5 Discussion ..........................................................................................................................93 4.6 Acknowledgments............................................................................................................96 4.7 References ..........................................................................................................................96 vi 5. C O N V ERG EN C E IN LEAF SIZE VS TW IG LEAF A R EA SCALING: DO PLA N T S O P T IM IZ E LEAF A R EA P A R T IT IO N IN G ? .....................116 5.1 Abstract ......................................................................................................................... 116 5.2 Introduction .................................................................................................................. 116 5.3 Materials and m e th o d s ............................................................................................... 118 5.3.1 Location and species...........................................................................................118 5.3.2 Twig a rch ite c tu re ............................................................................................... 119 5.3.3 Economics model ................................................................................................119 5.4 Results.............................................................................................................................122 5.4.1 Twig a rch ite c tu re ............................................................................................... 122 5.4.2 Economics model ................................................................................................123 5.5 Discussion......................................................................................................................124 5.6 Acknowledgements .......................................................................................................127 5.7 References......................................................................................................................127 S4 Supporting Information............................................................................................... 108 vii LIST OF FIGURES 2.1 Elements of metabolic scaling theory........................................................................... 29 2.2 Size-dependent variation in the q exponent (Q a DqB0) for tree water flow rate (Q) and trunk diameter (DB0) in modeled tr e e s ...................................................... 30 2.3 Effect of individual hydraulic traits on tree sapflow rate (Qref , at trunk diameter DB0 = 72 cm), and the Q by DqB0 scaling exponent, q, for medium­sized trees ......................................................................................................................... 31 2.4 Scaling space showing tree water transport rate (Qref , at trunk DB0 = 72 cm) and the scaling exponent q (Q a DqB0) for four functional types ....................... 32 2.5 Interspecific vs. intraspecific metabolic scaling exponents (c q )............................ 33 52.1 Size-dependent deviations from power-law scaling of modeled tree geometry . . 37 52.2 Interspecific simulations of the relationship between tree water transport rate (Q) and trunk diameter (DB0) .................................................................................... 38 3.1 Tree shape in relation to tree path fraction............................................................... 63 3.2 Two trees with 10 twigs each illustrate several model p ro p e rtie s ....................... 64 3.3 Modeled range of possible Pf values (shaded) across a large range of twig numbers and basal diameters ......................................................................................... 65 3.4 Predicted tree functions vs path fraction.................................................................... 66 3.5 Summary of SMA scaling exponents with 95% confidence intervals for the three Pf ontogenies we considered ............................................................................... 67 3.6 Volume normalized light interception for the same trees shown in Fig. 3.4d . . . 68 53.1 Measured and modeled crown area s c a lin g ............................................................... 75 53.2 Three sample 1024-twig trees formed when more than one asymmetry parameter, u, is used in each tree .................................................................................................... 76 53.3 Illustration of the equations used to determine tree heights and path lengths . 77 53.4 Metabolic scaling (i.e., K a V cq) for trees that follow the crown area scaling measured by Olson et al. (2009; i.e., scenario three, S3) 78 4.1 Scaling of aboveground growth and midday sapflow with whole plant hydraulic conductance ....................................................................................................................... 103 4.2 Relationships between aboveground growth and midday sapflow....................... 104 4.3 Hydraulic conductance vs aboveground dry mass relationships ..........................105 4.4 Scaling of midday sapflow and aboveground growth vs aboveground mass . . . . 106 4.5 Relative growth rate of aboveground tissue vs current aboveground dry mass . 107 54.1 Distinction between common and interspecific scaling ..........................................108 54.2 Average sapflows measured simultaneously by potometer and sapflow sensor . 109 54.3 Height by diameter relationships for the six study sp e c ie s ...................................110 54.4 Dry stem mass scaling relationships obtained from branches (open) and entire individuals (black)........................................................................................................... 111 54.5 Dry leaf mass scaling relationships obtained from branches (open) and entire individuals (black)........................................................................................................... 112 54.6 Conduit taper shown by area-weighted mean conduit diameter in multiple stems per species .............................................................................................................. 113 54.7 Conduit packing as shown by conduits per xylem area as a function of hydraulic-weighted conduit diameter ............................................................................................. 114 54.8 Sapwood area data determined from dye uptake is cross-sections (open) and cores (black).......................................................................................................................115 5.1 Key components of the light interception model illustrated by an open canopy with low self-shading (top row) and a denser canopy with higher self-shading . 131 5.2 Mean vs twig leaf area (A vs At) relationships........................................................132 5.3 Mean leaf area and twig leaf area were well correlated with twig diameter and length within species ....................................................................................................... 133 5.4 Example of optimal A, n and At selection and A vs At scaling ..........................134 5.5 Self-shading vs canopy openness scenarios (A, C, E) next to corresponding A vs At scaling exponents (B, D, F) .............................................................................135 ix LIST OF TABLES 2.1 Major symbols and definitions...................................................................................... 26 2.2 Model inputs and outputs in order of appearance in text ..................................... 27 2.3 Model inputs used to define the hydraulic scaling of four tree types (Fig.2.4) . 28 3.1 Symbol definitions and modifiers from the main text in order of appearance . . 61 3.2 Empirical Pf measurements from trees and sh ru b s ................................................. 62 4.1 Species properties and OLS regressions used for mass predic tion....................... 100 4.2 Whole-tree (Q and K ) and total aboveground (G and M ) SMA scaling rela­tionships ..............................................................................................................................101 4.3 OLS parameters for model input equations...............................................................102 5.1 SMA regressions between all measured twig structure parameters in each species and results from common exponent te s ts ......................................................130 ACKNOWLEDGMENTS Above all others, I would like to thank Dr. John Sperry who introduced me to research as an undergraduate and was the first person to encourage me to pursue a PhD. Throughout my graduate career, he was my biggest supporter and provided invaluable assistance and advice that made the contents herein possible. I would also like to thank my committee, Fred Adler, Dave Bowling, Brian Enquist and Tom Kursar, for their insight and for keeping my view broad and asking tough questions when needed. A portion of this dissertation was a product of or was inspired by collaboration with Brian Enquist, Lisa Bentley, Van Savage and Peter Reich. I am thankful to them for including me in their meetings and research. Thank you to the Biology Department for funding much of my time as a student through teaching assistantships and to NSF for funding the remainder. Thank you to Shannon Nielsen for help with all administrative details. CHAPTER 1 INTRODUCTION Land plants are modular and indeterminate organisms. Variation in modules and how they are arranged has led to great architectural diversity (Halle et al. 1978) as lineages adapted to fill different niches. Yet among this diversity, plant networks function to address similar requirements: they physically support the arrangement of photosynthetic and reproductive tissues in space and they provide transport between spatially separated resources. Water and nutrients must move from the soil to sites of photosynthesis where light and CO2 are available and the carbon and energy captured in the form of photosynthate must move from these sites to growing and respiring tissues. The transport of water is especially important because water is often lost to the atmosphere when plants open their stomata to take in CO2. If this water is not replaced, stomata will shut to avoid water stress but consequently reduce carbon uptake. Thus, there is a direct link between water transport and carbon fixation (Hubbard et al. 2001). Water transport is a physical process dependent on the architecture network. To a large extent, the dimensions and numbers of functional xylem conduits determine a plant's hydraulic conductance. Hydraulic conductance and the pressure difference between soil and leaf determine the rate of sapflow. Therefore, plant architecture, through its effect on water transport, is a major determinant of carbon gain. The importance of hydraulic architecture and its link to carbon has long been recognized (Yang & Tyree 1993), though increased attention followed introduction of the model by West, Brown & Enquist (WBE; Enquist et al. 1998; West et al. 1999) as part of metabolic scaling theory. The WBE model simplifies and generalizes plant architecture as symmetri­cally branching networks of tapering stems plumbed with tapering xylem conduits (further details in Chapter 2). This model predicts hydraulic conductance and how it scales with network size and then assumes how hydraulic conductance scales with sapflow and growth. The scaling of hydraulic conductance is particularly important because plant growth leads to longer path lengths, which can reduce the hydraulic conductance supplying each leaf. The scaling exponent indicates how effective plants are at overcoming the consequence of longer 2 path lengths. The model predicts the maximum exponent (full compensation for longer paths), which agrees with observations across species (Niklas & Enquist 2001). However, while the model may predict the general trend across species, it does not address individual variation within and between species. Are there consequences to the architectural diversity of plants or does this diversity represent different ways of achieving the same result? This dissertation seeks to determine the impact of architectural variation on plant function, including: hydraulic transport and its scaling with plant size and growth; biomechanics and safety from buckling; and light interception at the whole plant and individual twig levels. These questions are addressed through a combination of empirical measurements and modelling, including extensions to the WBE model. 1.1 The chapters Chapter 2 extends a version of the WBE model recently updated by Savage et al. (2010) to more explicitly define the hydraulic network by including leaves, roots and non­conducting tissues: pith; heartwood and bark. This extended version is used to ask how variation in hydraulic architecture may affect whole plant physiology and, in effect, how broadly species may differ. The model is assessed using: (1) the sapflow (Q) by trunk diameter (DB0) scaling exponent, q (i.e. Q a DqB0) and (2) the sapflow rate at a reference trunk diameter, Qref . It is emphasized that the value of q depends largely on the "bottleneck effect" . That is, q is maximized when the greatest hydraulic impediment is at the end of the transport path (i.e., in the twigs or leaves). A sensitivity analysis was conducted to see how model parameters affect q and Qref . Then, "species" in different functional groups were created by combining model parameters from the literature to predict the "scaling space" of q and Qref for each functional group. Considerable scaling space overlap existed between groups with contrasting anatomy such as conifers and diffuse-porous tropical angiosperms. Results from the scaling space were used to show that intraspecific q may differ from interspecific q if larger-statured species (e.g., trees) are systematically different from smaller-statured species (e.g, shrubs) This prediction was revisited with Chapter 4. Chapter 2 represents a major stride from using the model as a general predictor of scaling exponents to a parameterizable model to predict Q and q in actual plants. This chapter was published in Functional Ecology (Sperry, Smith, Savage, Enquist, McCulloh, Reich, Bentley & von Allmen 2012). Chapter 3 extends the model further by relaxing the external branching architecture. Empirical measurements and evidence of apical control via hormones (Cline 1997) suggest 3 that symmetric branching used in the WBE model is not representative of most plants. Therefore, the model was used to ask how branching architecture affects hydraulic conduc­tance, stem biomass, mechanical stability and light interception. Notably, these last two had not been previously assessed in the WBE framework although both were invoked as principles of architecture. Perfect symmetry will make all trunk-to-twig path lengths the same. Whereas empirical measurements and initial modeling showed that any deviation from symmetry led to a nonuniform distribution of path lengths. This observation prompted the "path fraction" to quantify branching architecture. The model was extended to be compatible with WBE architecture and followed the same principles of area-preserving branching and elastic similarity (McMahon & Kronauer 1976). Measurements and estima­tions of branching architecture suggested ontogenetic trends in the path fraction, which had implications for q. This chapter was published in New Phytologist (Smith, Sperry, Enquist, Savage, McCulloh & Bentley 2014). Chapter 4 addresses two main hypotheses: 1) growth and water transport are isometri-cally related (as assumed in metabolic scaling theory) and 2) larger statured species have steeper hydraulic and metabolic scaling (as proposed in Chapter 2). The argument for the first hypothesis can be made by assuming that within species, water use efficiency is constant over time (making assimilation isometric with sapflow) and a constant fraction of assimilate is devoted to growth (making assimilation and growth isometric). However, species may differ in the how much sapflow increases with each increase in mass (i.e., the metabolic scaling exponent). By virtue of their small stature, shrub species were expected to have shallower metabolic scaling than co-occurring tree species. If both hypotheses are correct then the difference between shrubs and trees should be reflected in their xylem anatomy - shrubs having less conduit taper and/or a smaller fraction of trunk sapwood. Xylem properties from each species were used in the model presented in Chapter 3. Emphasis is placed on the importance of understanding intraspecific relationships before explaining interspecific relationships. This chapter appeared in Plant, Cell and Environment (Smith & Sperry 2014). Chapter 5 addresses how leaves might be more accurately modeled. In Chapter 2, leaf hydraulic conductance was shown to have a strong influence on sapflow and how it scales. However, the link between leaf and twig hydraulic conductance was based on a limited data set. It has long been recognized that thicker twigs support larger leaves (Corner 1949), but the lack of a mechanistic explanation for this trend has prevented its incorporation into the model. Most data testing "Corner's rule" have been interspecific, which may confound the 4 underlying reason why thicker twigs have larger leaves. In this chapter, Corner's rule was revisited with an intraspecific data set, which led to a reformulation of the rule as: twigs with larger leaf areas also have larger individual leaves. Furthermore, species increased the number of leaves in a similar manner, which suggests a common response to some external variable. A model was developed to ask if Corner's rule results from optimal light gap filling to maximize the benefit of absorbing light minus the cost of leaves. This chapter is being prepared for submission to Ecology Letters. 1.2 References Cline M.G. (1997) Concepts and terminology of apical dominance. American Journal of Botany 84, 1064-1069. Corner E.J.H. (1949) The durian theory of the origin of the modern tree. Annals of Botany 13, 368-414. Enquist B.J., Brown J.H. & West G.B. (1998) Allometric scaling of plant energetics and population density. Nature 395, 163-165. Halle F., Oldeman R.A.A. & Tomlinson P.B. (1978) Tropical trees and forests: an architec­tural analysis. Springer-Verlag, Berlin, Heidelberg, New York. Hubbard R.M., Stiller V., Ryan M.G. & Sperry J.S. (2001) Stomatal conductance and photosynthesis vary linearly with plant hydraulic conductance in ponderosa pine. Plant Cell and Environment 24, 113-121. McMahon T.A. & Kronauer R.E. (1976) Tree structures: deducing the principle of mechan­ical design. Journal of Theoretical Biology 59, 443-466. Niklas K.J. & Enquist B.J. (2001) Invariant scaling relationships for interspecific plant biomass production rates and body size. Proceedings of the National Academy of Sciences of the United States of America 98, 2922-2927. Savage V.M., Bentley L.P., Enquist B.J., Sperry J.S., Smith D.D., Reich P.B. & von Allmen E.I. (2010) Hydraulic trade-offs and space filling enable better predictions of vascular structure and function in plants. Proceedings of the National Academy of Sciences of the United States of America 107, 22722-22727. Smith D.D. & Sperry J.S. (2014) Coordination between water transport capacity, biomass growth, metabolic scaling and species stature in co-occurring shrub and tree species. Plant, Cell and Environment 37, 2679-2690. Smith D.D., Sperry J.S., Enquist B.J., Savage V.M., McCulloh K.A. & Bentley L.P. (2014) Deviation from symmetrically self-similar branching in trees predicts altered hydraulics, mechanics, light interception and metabolic scaling. New Phytologist 201, 217-229. Sperry J.S., Smith D.D., Savage V.M., Enquist B.J., McCulloh K.A., Reich P.B., Bentley L.P. & von Allmen E.I. (2012) A species-level model for metabolic scaling in trees I. boundaries to scaling space within and across species. Functional Ecology 26, 1054-1065. 5 West G.B., Brown J.H. & Enquist B.J. (1999) A general model for the structure and allometry of plant vascular systems. Nature 400, 664-667. Yang S. & Tyree M.T. (1993) Hydraulic resistance in Acer saccharum shoots and its influence on leaf water potential and transpiration. Tree Physiology 12, 231-242. CHAPTER 2 A SPECIES-LEVEL MODEL FOR METABOLIC SCALING IN TREES I. EXPLORING BOUNDARIES TO SCALING SPACE WITHIN AND ACROSS SPECIES 2.1 Summary 1. Metabolic scaling theory predicts how tree water flow rate (Q) scales with tree mass (M) and assumes identical scaling for biomass growth rate (G) with M . Analytic models have derived general scaling expectations from proposed optima in the rate of axial xylem conduit taper (taper function) and the allocation of wood space to water conduction (packing function). Recent predictions suggest G and Q scale with M to the 0.7 power with 0.75 as an upper bound. 2. We complement this a priori optimization approach with a numerical model that incorporates species-specific taper and packing functions, plus additional empirical inputs essential for predicting Q (effects of gravity, tree size, heartwood, bark, and hydraulic resistance of leaf, root and interconduit pits). Traits are analysed individ­ually, and in ensemble across tree types, to define a 2D "scaling space" of absolute Q vs. its scaling exponent with tree size. 3. All traits influenced Q and many affected its scaling with M . Constraints driving the optimization of taper or packing functions, or any other trait, can be relaxed via compensatory changes in other traits. Reprinted with permission from John Wiley & Sons. Sperry J.S., Smith D.D., Savage V.M., Enquist B.J., McCulloh K.A., Reich P.B., Bentley L.P. & von Allmen E.I. (2012) A species-level model for metabolic scaling in trees I. Exploring boundaries to scaling space within and across species. Functional Ecology 26, 1054-1065. 7 4. The scaling space of temperate trees overlapped despite diverse anatomy and winter-adaptive strategies. More conducting space in conifer wood compensated for narrow tracheids; extensive sapwood in diffuse-porous trees compensated for narrow vessels; and limited sapwood in ring-porous trees negated the effect of large vessels. Tropical trees, however, achieved the greatest Q and steepest size-scaling by pairing large vessels with extensive sapwood, a combination compatible with minimal water stress and no freezing-stress. 5. Intraspecific scaling across all types averaged Q a M0-63 (maximum = Q a M 0-71) for size-invariant root-shoot ratio. Scaling reached Q a M0-75 only if conductance increased faster in roots than in shoots with size. Interspecific scaling could reach Q a M0-75, but this may require the evolution of size-biased allometries rather than arising directly from biophysical constraints. 6. Our species-level model is more realistic than its analytical predecessors and provides a tool for interpreting the adaptive significance of functional trait diversification in relation to whole tree water use and consequent metabolic scaling. 2.2 Introduction How does tree water use scale with tree size, and how does it differ across species? Given the essential role of water, this question is fundamental to understanding the metabolic scaling of individual trees, species, forest communities and ecosystems. Predicting the answer from vascular anatomy is the subject of this study. Modeling water use from vascular properties has a long history dating at least to da Vinci's rule of area-preserving branching (Richter 1970), continuing with the Ohm's law analogy of van den Honert (1948; Richter 1973) and culminating in the concept of "hydraulic architecture" (Zimmermann 1978) represented in contemporary models (e.g., Tyree 1988; Sperry et al. 2002; Macinnis-Ng et al. 2011). At the heart of these complex models is a simple relationship for whole-tree sap flow at steady state (Q): Q = K (A P - pgH) (2.1) where K is tree hydraulic conductance, A P is soil to canopy pressure drop, and pgH is the pressure required to offset the force of gravity on the water column (p = density of water; g, acceleration of gravity; H , tree height). Canopy xylem pressure regulation (via stomatal control of Q) constrains the (AP - pgH) term, and most of the uncertainty in hydraulic 8 modeling lies in representing K , which depends mostly on the complex anatomy of the flow path from soil to leaf. Until the revolutionary approach of West, Brown and Enquist (WBE; West et al. 1997, 1999; Enquist et al. 2000), most hydraulic modeling was based on specifying what K is from empirical inputs. In contrast, the WBE model derives what the allometric scaling of K should be by assuming a universal set of optimization criteria and an intentionally minimalist representation of plant vasculature. The WBE goal is to predict universal expectations for how K , and hence Q, and all dependent metabolic processes, should scale with plant size. The focus is on predicting the power function scaling exponent (b): Y x Mb (2.2) where Y is the variable of interest (K , Q, rates of metabolism or growth) and M is plant mass. The result is a metabolic scaling theory that emphasizes the unifying consequences of selection for optimal vascular transport under overarching constraints. Savage et al. (2010) have recently extended the theory with important improvements in how it represents vascular architecture. In this study, we present a model that strikes a middle ground between the structure-to- function optimization approach of Savage et al. (and its WBE predecessors) and the descriptive-empirical approach of more complex numerical models. We add a minimal set of hydraulic inputs to the Savage et al. analytical model with the goal of predicting the actual value of K and Q rather than proportional proxies that are sufficient for predicting scaling exponents. Our species-level model turns the proportionality in Eqn. 2.2 (Y x Mb) into an equality (Y = k0M b) by specifying scaling multipliers (k0). The additional complexity requires a numerical approach, but is justified because selection for optimal vascular function should concern traits underlying the multiplier as well as the exponent. Furthermore, variation in scaling multipliers across species could influence interspecific exponents (b) independently of the intraspecific value of b. We relax any a priori optimization criteria and allow key hydraulic inputs to be empirical, so that we can predict the "scaling space" defined by variation in k0 and b across species. Figure 2.1 provides a roadmap of the Savage et al. (2010) model. The branching architecture component (Fig. 2.1, left) specifies that the tree has symmetric, self-similar branching architecture that preserves the cross-sectional area of branches across each branching junction (da Vinci's rule; Horn 2000). Hence, the tree can be represented by a column (Fig. 2.1, center). The mass allometry module predicts 9 the best-fit power-law scaling between trunk diameter (DBo; 0 denotes trunk branch rank; symbols in Table 2.1) and tree mass (M): Dbo = k \M c (2.3) where k\ is the scaling multiplier and c the scaling exponent. The value of the exponent c is derived from well tested theory that H must scale with DB2/q3 for trees to maintain a constant safety margin from buckling under their own weight ( "elastic similarity" ; McMahon 1973). An elastically similar column has a mass exponent of c = 3/8 in Eqn. 2.3 (West et al. 1997, 1999; Enquist et al. 2000; Savage et al. 2010). The water use allometry module predicts how the steady-state rate of midday xylem transport (Q) scales with trunk diameter: Q = k2DBo (2.4) with multiplier k2 and water use exponent, q. To obtain Q, the Hagen-Poiseuille equation (Zimmermann 1983) is used to calculate tree hydraulic conductance (K ) from the number and dimensions of the xylem conduits in the tree sapwood, given by the xylem architecture module (Fig. 2.1, right). The prediction of K yields Q by Eqn. 2.1, and the scaling of Q with tree size yields the water use allometry of Eqn. 2.4. Previous derivations of the water use exponent q in Eqn. 2.4 have assumed that selection for transport efficiency has driven it to its theoretical maximum of q = 2 (for the assumed xylem architecture; West et al. 1999; Enquist et al. 2000; Savage et al. 2010). At this point, the rate of whole-tree water transport depends solely on its trunk basal area and is not negatively influenced by tree height or transport distance (Q a D2B0/H 0). The fifth metabolic isometry component of the Savage et al. model is a fundamental as­sumption of metabolic scaling theory: because photosynthetic CO2 flux and transpirational water flux are both limited by stomatal diffusion, gross photosynthesis and potential iso­metric surrogates such as total respiration and growth rate (G) should scale proportionally with Q (Enquist et al. 2007a). Combining metabolic isometry with the mass and water use allometries predicts metabolic scaling: G a Q a Mcq, where the metabolic scaling exponent is the product of the exponents for mass (c; Eqn. 2.3) and water use (q; Eqn. 2.4) scaling. If c = 3/8 (from elastic similarity) and q = 2 (the theoretical Savage et al. maximum), the metabolic exponent cq = 3/4 (West et al. 1999; Enquist et al. 2000). This prediction has provoked debate, partly over the validity of metabolic isometry and partly regarding q (e.g., Meinzer et al. 2005; Reich et al. 2006; Enquist et al. 2007a; Sperry et al. 2008). 10 Metabolic isometry is addressed in the second paper of this series (von Allmen et al. 2012). Here, we focus on the derivation of q. For q = 2 the negative effects of tree height and distance on Q must be eliminated. Height is negated if the drop in xylem pressure from soil to canopy (A P ) compensates for gravity (pgH), making the driving force (AP - pgH; Eqn. 2.1) height-invariant. However, the (AP - pgH) term often declines with height (Mencuccini 2003; Ryan et al. 2006). Transport distance can be negated by the "bottleneck effect" where high flow resistance at the end of the xylem pipeline restricts the flow rate regardless of pipeline length. A bottleneck effect is consistent with the tapering of xylem conduits from trunk to terminal twig (West et al. 1999; Enquist et al. 2000; Sperry et al. 2008). This narrowing is captured in the Savage et al. model by a "taper function" : the conduit diameter inside the terminal twigs is assumed size-invariant and conduits widen proximally as the stems themselves widen across branch ranks (Fig. 2.1, downward "axial taper" arrow). The bottleneck effect is also influenced by how the number of conduits running in parallel changes across branch ranks. The Savage et al. model uses a "packing function" (Sperry et al. 2008) to govern the number of conduits that fit in a specified portion of wood space. Consequently, as conduits become narrower towards the twigs, their number per wood area increases (Fig. 2.1, upward "conduit packing" arrow). To optimize space-filling, Savage et al. assume a universal packing function that allocates a constant fraction of wood space to transport vs. across all branch ranks. Savage et al. then solve for optimal conduit taper on the basis of an efficiency vs. safety trade-off (see also Enquist et al. 2000). Taper is increased just enough to yield q = 2 (to maximize transport efficiency), but no more. Excessive taper would continue to widen conduits proximally, but to no effect other than to compromise safety from cavitation (larger conduits tend to be more vulnerable; Hacke et al. 2006). Is the bottleneck effect enough to yield q = 2? Savage et al. recognize that not all species have identical taper and packing functions (McCulloh et al. 2010), suggesting that the space-filling and efficiency vs. safety trade-offs they invoke may have diverse context-dependent optima (Price et al. 2007). The intentional simplicity of the Savage et al. model also excluded additional variables that potentially influence the bottleneck effect such as the terminal resistance of leaves and the presence of nonconducting heartwood and bark. The Savage et al. model also considers a basic issue in the derivation of q: the water use allometry only becomes a pure power function (e.g., Eqn. 2.4) at the limit of infinite tree size (Mencuccini et al. 2007). Thus, best-fit power functions across different size ranges 11 yield different q (and c) exponents. For example, Savage et al. solve for the rate of conduit taper that is just sufficient to make q = 2 at the limit of infinite tree size, while the same taper yields only q = 1.86 for finite-sized trees. This leads to their prediction of a metabolic scaling exponent of cq = 0.70 (3/8 ■ 1.86) in trees of actual size, with cq = 0.75 as an upper bound (Savage et al. 2010). Our species model attempts to clarify some of the uncertainty in metabolic scaling theory by revisiting the derivation of the water use allometry component (Eqns. 2.1 and 2.4). New inputs of xylem architecture and function (asterisks in Fig. 2.1, see Model Description) are added to the Savage et al. framework to improve q estimation and to enable the prediction of the k2 multiplier so that actual flow rates, Q, can be estimated. We focus on how specific hydraulic traits can effect the scaling of water use. For simplicity, we do not alter the branching architecture of the Savage et al. model (2010). We apply the new model to four objectives. (i) Using the simpler Savage et al. parameterization, we quantify the effects of finite tree size and gravity (i.e., the [AP - pgH] term) on intraspecific scaling. (ii) We determine the influence of new inputs and variable taper and packing on the water use exponent (q) and multiplier (k2). (iii) We translate how interspecific variation in wood traits translates into a map of "scaling space" - defined by all possible combinations of multipliers (k2) and the exponents (q) across species. The scaling space was simulated for four major functional tree types: conifers, ring-porous- and tropical and temperate diffuse-porous-angiosperms. (iv) Ecological drivers of scaling diversity are discussed, as are the implications for three-fourth power metabolic scaling within vs. across species. The second paper tests the model against empirical measurements (von Allmen et al. 2012). 2.3 Model description The model has 17 inputs, with default values listed in Table 2.2. The model was written as a macro in Microsoft Excel using Visual Basic for Applications and is available from the senior author. 2.3.1 B r a n c h in g a r c h i t e c tu r e a n d m a s s a llom e try Trees are represented as a symmetrically self-similar structure shown in Fig. 2.1 (left). Branches at level i (counting from i = 0 at the trunk) are identical in length and diameter. Area-preservation (da Vinci's rule; Horn 2000) sets the ratio of daughter/mother branch diameter (ft) at ft = n 1/2, where n is the daughter/mother branch number ratio (Table 2.1 defines all symbols). Elastic similarity, which requires H a dB 03 (McMahon 1973), sets the 12 daughter/ mother branch length ratio (7) at 7 = n 1/3. Modeled trees converge on elastic similarity with size as observed (Niklas & Spatz 2004). Dimensions of the terminal branch rank (twigs) are assumed constant regardless of tree size. Twig diameter was set to 2 mm. Twig length was selected to yield convergence in large trees on the desired safety factor from buckling (HB/H ). The height at elastic buckling (Hb ) was calculated according to Niklas (1994). Simulated "species" had identical branch architecture inputs (default @, 7 , n, HB/H , twig diameter, twig length; Table 2.2). The mass scaling exponent (c) was obtained from the slope of log-log plots of DB0 vs. tree volume (V = nD 2B0H /4) across networks of different size. We did not specify the multiplier k1 for intraspecific scaling. However, for simulations of interspecific scaling, k1 across species was specified by assuming branch tissue density equalled wood density (Supporting information, The "taper function" describes how xylem conduit diameter (DC, ^m) increases with stem diameter (DBi mm): where p is the "taper exponent" and k3 (^m mm-p )the multiplier. The default p = 1/3 is the smallest p yielding q = 2 at the limit of infinite tree size in the Savage et al. model (2010). The choice of the minimum DC in the terminal twigs dictated k3 (default DC twig = 10 ^m, Table 2.2). When the model was run with axial taper alone (as in the Savage et al. model), DC narrows as DBi narrows, but is constant from pith to cambium at a given branch level (Fig. 2.1, downward "axial taper" arrow). When radial taper is added, DC increases from pith to cambium, starting from Dc = Dc twig and increasing with the taper function as branch diameter is incremented (in 100 ^m steps) to DBi (Fig. 2.1, enlarged cross-section). To avoid unrealistically large DC, a maximum (Table 2.2, DC max) was set. Default DC max was set to 240 ^m because this was the greatest DC in our functional type survey (Table 2.3; Supporting information, S2.2). The number of xylem conduits per xylem area (F , mm-2 ) was calculated from conduit diameter (DC, ^m) using the "packing function" (Sperry et al. 2008): S2.3). DC = k3DBi (2.5) F = k^D'C (2.6) where d is the packing exponent (a negative number) and k4 (mm 2 ^m C) the multiplier. The choice of k4 dictated the fraction of the total wood area occupied by xylem conduits 13 (Cf < 1). For square packing (one conduit per square of space), maximum F = 106D-2 and CF = [k4/ 106]D(d+2). Savage et al. assumed an optimal d= -2 (our default), such that CF is constant from twig to trunk (or pith to cambium). The default k4 (Table 2.2) was chosen to yield CF = 0.1, a typical hardwood value (McCulloh et al. 2010). If d was less negative than 2, then Cp increased from twig to trunk and vice-versa for d more negative than 2. Xylem cross-sectional area was obtained by subtracting the bark and pith area from total branch area. Pith diameter (Dp, mm) was invariant within a tree, with a default of 1 mm. The bark thickness at level i (Tpi, mm) was calculated from branch diameter (Dpi , mm) as: Tpi = k5D%i (2.7) where a is the bark exponent and k5 (mm(1_a)) the multiplier. For simplicity, we restricted the analysis of bark thickness to the two bark functions used to test the model in the companion paper (von Allmen et al. 2012). These were from a relatively thin-barked maple (Acer grandidentatum) and a thicker-barked oak (Quercus gambelii). Maple served as the default (Table 2.2). Total xylem area was divided into nonconducting heartwood and conducting sapwood. The sapwood area (Asi, mm2 ) at level i from branch diameter (Dpi , mm) is given by: Asi = k6DSi (2.8) where s is the sapwood exponent and k6 (mm(2-s)) the multiplier. The exponent s has a maximum of s = 2 to avoid sapwood area from exceeding xylem area and a minimum of s = 1 for thin sapwood of approximately constant depth. Values of k6 and s were obtained from the companion paper on oak and maple (von Allmen et al. 2012) and the sapflux literature. Sapwood functions were adjusted to have heartwood first appear at DBi = 2.2 cm and expand to reduce sapwood to varying percentages of total basal area at Dpo = 72 cm. The default percentage was 74%. This corresponded to a default sapwood depth from the cambium of 18.9 cm. Power functions for Eqns. 2.5-2.8 were chosen because of their convenience and good fit to empirical trends. The hydraulic conductance of a branch (KBi) was calculated from branch length (Li , ^m), and the number (Nc ) and diameter (Dc , ^m) of xylem conduits, using the Hagen-Poiseuille equation. Conduit number was obtained from the packing function and the sapwood area. When the model was run with radial DC taper, 14 we integrated the Hagen-Poiseuille equation from the inner sapwood boundary (x = 0) to the cambium (x = rc) to yield the KBi: x=rc KBi = C l Nc(x)n[Dc (x)]4/(128nLi)dx (2.9) x=0 where NC (x) and DC (x) are functions of the radial distance x across the sapwood according to the packing and taper functions. The integral was solved numerically by 100 ^m increments in x (smaller increments were unnecessary). The viscosity, n, was set at 0.001 Pa s for 20 °C. The dimensionless constant C is an empirical correction factor (0 < C < 1) that accounts for interconduit flow resistance. The literature yielded default correction factors of C = 0.44 (angiosperms; Hacke et al. 2006) and C = 036 (conifers; Pittermann et al. 2005). Branch K bi was multiplied by the number of branches in level i to yield the parallel conductance of rank i. Rank conductances in series gave the hydraulic conductance of the stem network. Leaf and root system conductances were extrapolated from the branch network con­ductance. Leaf conductance was given by ratio of leaf conductance per twig conductance (K l/K t) , which was assumed to be size-invariant. This ratio is not often measured, but values from Acer grandidentatum and Quercus gambelii cited in the companion paper (von Allmen et al. 2012) provided a range. A similar approach was used to incorporate root system conductance. The shoot conductance (KS, all branches plus leaves) was multiplied by the ratio of tree-shoot conductance (K /K S) to obtain the whole-tree (root plus shoot) conductance. The default K /K S ratio was 0.5 in keeping with observations from a variety of woody plants (Sperry et al. 2002). The default was size-invariance of K /K S, but we also allowed it to increase with size (Martlnez-Vilalta et al. 2007). Steady-state tree water transport rate (Q, kg hr-1 ) at midday was calculated from tree conductance using Eqn. 2.1. Default A P = 1 MPa (Mencuccini 2002), making it size-invariant as seen for Acer grandidentatum and Quercus gambelii (von Allmen et al. 2012). Thus, the (AP - pgH) driving force decreased with tree size. In an alternative "gravity compensation" scenario, the (AP - pgH) term was size-invariant. These two options cover the range of gravity responses of trees (see Discussion). The Savage et al. model and earlier models (West et al. 1997, 1999; Enquist et al. 2000) assume isometry between Q and K , thus implicitly adopting gravity compensation. Linear regressions of log-transformed Q vs. DBo data yielded the water use allometry equation: Q = k2DBo. 15 2.4 Model results 2.4.1 Size d e p e n d e n t w a te r - u s e a llom e tr y We investigated size effects using the model parameterized as in Savage et al.: no pith, sapwood or bark, no leaves or roots and no radial taper (Table 2.2 shows remaining default inputs). The only difference from Savage et al. was that we allowed for gravitational effects. Size effects had two causes: juvenile growth that was not elastically similar (Supporting information, S2.1, Fig. S2.1) and gravitational reduction in (AP - pgH) in tall trees. In combination, these created a nonlinear (in log-log space) water use allometry (Q) with trunk diameter (DB0; Fig. 2.2). In small trees, a power-law fit gave an approximate scaling exponent of Q a DqB01'12 (Fig. 2.2, grey). The exponent increased to a maximum as elastic similarity was approached in medium sized trees: Q a DB0q=1'72 (Fig. 2.2, dark grey). In large trees, the exponent decreased as gravity (pgH) subtracted an increasing portion of the pressure difference between soil and canopy (AP = 1 MPa, Table 2.2). Thus, Q scaling became flatter in tall trees: Q a DqB00'91 (Fig. 2.2, black). In the gravity compensation scenario, A P increased with H such that the (AP - pgH) term was size-invariant and Q scaling did not flatten. Instead it reached Q a DqB01'86 in large sized trees (Fig. 2.2, dash-dotted no g line) as estimated for the Savage et al. model at their optimal taper (p = 1/3). Increasing tree size towards infinity gave the q = 2 asymptote (Savage et al. 2010). 2 .4 .2 In f lu e n c e o f in d iv id u a l t r a i t s o n w a te r u se s c a lin g New variables added to the Savage et al. framework altered water use scaling. We report effects on the exponent, q, and the multiplier, k2, for medium-sized trees (2 < DB0 < 72 cm) where Q by DB0 scaling was nearly linear in log-log space (Fig. 2.2). Rather than cite k2 values, we substitute a more intuitive proxy: the rate of water transport at a reference tree size (Qref for DB0 = 72 cm). Although the effects were quantitatively complex (Fig. 2.3), the take-home message is simple. All variables influenced Qref because they either increased tree hydraulic conduc­tance (e.g., more, wider functioning conduits, higher leaf or root conductances) or reduced it (fewer, narrower functioning conduits, lower leaf or root conductances). A subset also altered q because they influenced the bottleneck effect: either increasing the difference in distal-to-proximal balance of hydraulic conductance in the shoot (greater q) or decreasing it (lower q). One variable (size-dependent K /K S) altered q independently of the bottleneck effect. 16 Figure 2.3(a) shows the cumulative effect of adding new variables. Incorporating pith and bark (defaults in Table 2.2) reduced Qref by reducing xylem cross-sectional area, with thicker bark having a greater effect (Fig. 2.3a); q was not materially changed. Adding radial taper to the thin-barked default model decreased Qref further (Fig. 2.3a, radial taper) because of the narrowing of vessel diameter towards the pith; again, q changed little. Adding heartwood reduced both Qref and q (Fig. 2.3a, sapwood%). Reducing basal sapwood area from 74 to 30% (reducing sapwood thickness from 18.9 to 6.1 cm) caused q to drop from 1.72 to below 1.64 and Qref to drop by 24% (Fig. 2.3a). Heartwood decreased Qref by reducing the cross-sectional area for water conduction. Most of this reduction was in the larger branches and trunk, which decreased the bottleneck effect of the distal branches and lowered q. The 74% basal sapwood function was adopted as the default (Table 2.2) for assessing the further effect of adding leaves in Fig. 2.3b. As the K L/K T ratio was decreased from 1 to 0.01, Qref dropped by over 80% relative to the default no-leaf model (Fig. 2.3b) because leaves reduced network conductance. The reduction was at the distal end, which increased the bottleneck effect, and q increased from 1.72 to 1.92. The measured K L/K T range was relatively narrow: from 0.27 in Quercus gambelii to 0.38 in Acer grandidentatum (von Allmen et al. 2012). The intervessel resistance factor (C, Eqn. 2.9) caused a proportional change on Qref , but no change in q because it did not influence the bottleneck. The same was true for adding below-ground resistance (Table 2.2, K /K S = 0.50). However, if K /K S was allowed to increase with Db0 as estimated for Pinus sylvestris (K /K S = 0.75D0.06 from Martlnez- Vilalta et al. 2007), q rose to 2.12. This was the only input that gave q > 2. It did so not by increasing the shoot bottleneck effect, but because root conductance increased faster than shoot conductance with size. Varying taper and packing from the universal functions assumed by Savage et al. was simulated for a default K L/K T = 0.30 (Table 2.2; grey symbol in Fig. 2.3c,d). Figure 2.3c shows the effect of taper. The default taper was DC = 7.9DBi 1/3, corresponding to DC twig = 10 ^m. DC twig was held constant while varying the taper exponent, p, by adjusting the multiplier. Decreasing taper from p = 1/3 to p = 0.2 resulted in narrower conduits proximally, which reduced network conductance (51% drop in Qref ) and the bottleneck effect (decrease in q from 1.76 to 1.59). Increasing taper above p = 1/3 had the opposite effect: Qref increased by almost 5-fold and q rose to 1.93. At p > 0.6, conduit diameter in the proximal trunk and branches had to be capped at DC max = 240 ^m (Fig. 17 2.3c, asterisked points). Saturation in q and Qref occurred at p > 9 because vessels had reached the 240 ^m cap at every branch rank except the twigs. The 240 ^m cap, heartwood and gravity prevented q from saturating at q = 2. Figure 2.3d shows the influence of the packing function, F = k4DCC. The greater the value of coefficient k4, the greater the fraction of wood area devoted to water conduction (Cf ). Increasing Cf caused a proportional increase in Qref (Cf = 0.01 to 0.6) with no effect on q (Fig. 2.3d). When varying the packing exponent d, we covaried the multiplier to keep Cf constant in the terminal twigs. Increasing d (less negative), resulted in more big trunk vessels and hence increased both the relative flow rate (Qref ) and the Q by DqB0 scaling exponent. Decreasing the exponent had the opposite effect (Fig. 2.3d, d = 2.5 to 1.5). Effects of the exponent on Qref were small compared with the effect of conducting area fraction, C f . 2 .4 .3 F u n c tio n a l t r e e ty p e s in s c a lin g sp a c e While Fig. 2.3 isolates the consequences of particular traits, actual scaling integrates variation across all traits at once to create a 2D cloud of species-specific Qref by q combi­nations. We used the model to circumscribe this "scaling space" for major tree categories: ring-porous temperate, diffuse-porous temperate, diffuse-porous tropical and conifers. For each category, we estimated the range for input variables for which multispecies data were available (Table 2.3; Supporting information, S2.2); remaining inputs, including K /K S, were defaults (Table 2.2). A version of the model (available from the second author) scanned input combinations that defined the extremes of Qref and q for medium-sized trees (2 < DB0 < 72 cm). The four tree categories occupied distinct, but substantially overlapping, scaling space (Fig. 2.4). The most efficient transporters, with the greatest Qref and scaling exponent q, were the tropical diffuse-porous trees (Fig. 2.4, green DP tropical outline). Tropical trees combined largest trunk and twig vessels with extensive sapwood area. The only parameter compromising efficiency in tropical trees was a somewhat lower Cf (Table 2.3; fewer vessels per sapwood area). Temperate ring-porous angiosperms achieved the next highest Qref (Fig. 2.4, black RP outline). Although their vessel diameter is similar to tropical trees, their limited sapwood area (Table 2.3) compromised transport and contributed to their broad q range. Their broad Qref range corresponded to a wide range in Cf (Table 2.3). Ring-porous trees overlapped considerably with their chief cohabitants, the temperate diffuse-porous angiosperms (Fig. 2.4, red DP temperate outline). Although temperate diffuse-porous trees 18 have narrower vessel diameters than ring-porous trees, this was compensated by their greater sapwood area (Table 2.3). The most surprising result was the performance of the conifers (Fig. 2.4, blue conifer outline). Although conifer tracheids have by far the narrowest conduit diameter range, they compensate by having high CF (Table 2.3), owing to the double role of tracheids in water transport and mechanical support. The high Cf of conifers placed them almost entirely within the transport capacity of temperate diffuse-porous angiosperms. Assuming "metabolic isometry," the hydraulic scaling in Fig. 2.4 predicts growth rate scaling with tree mass (G a M cq). Assuming c = 0.369 for medium-sized trees (Supporting information, S2.1), the range for the G a Mcq exponent was mapped onto the four tree types in Fig. 2.4 (cq values on upper axis). The metabolic scaling exponents ranged from 0.26 to 0.71 and excluded three-fourth power scaling. 2 .4 .4 In tr a s p e c if ic vs. in te r s p e c if ic s c a lin g Each Qref by q coordinate in the scaling space of Fig. 2.4 corresponds to a unique water use allometry (Q = k2DqB0; Eqn. 2.4) of a theoretical "species." Each species also has a potentially unique mass allometry (DB0 = k1Mc) because of interspecific variation in wood density (Supporting information, S2.3). These "species" were sampled to simulate intraspecific vs. interspecific scaling of the metabolic cq exponent. Theoretical species with q < 1.5 were excluded because they are unlikely to exist (Supporting information, S2.3 and Discussion). When species were chosen at random and assumed to reach the same maximum size regardless of Qref and q, the intraspecific cq averaged 0.63 ± 0.0011 (mean ± SE, n = 1000 trees) and the interspecific cq averaged 0.66 ± 0.0018 (Fig. 2.5, "random"). Both intraspecific and interspecific exponents fell short of cq = 0.75. In an alternative "height-biased" sampling, we assumed that species from the upper right corner of scaling space in Fig. 2.4 (greater Qref and q) would have less of a hydraulic limitation on their maximum height and grow taller than species towards the lower left corner. Average intraspecific scaling was no different from the random scenario (cq = 0.63 ± 0.0020), but the interspecific cq could be significantly steeper depending on the sensitivity of species stature to their water use allometry and the size distribution of the interspecific sample. The particular case shown (see Supporting information, S2.3 for details) shows that interspecific cq can match three-fourth power scaling (cq = 0.75 ± 0.0041; Fig. 2.5; "height-biased"). 19 2.5 Discussion The model answers our opening question by providing species-specific predictions of the water use multiplier (k2) and scaling exponent (q) in the water use allometry equation: Q = k2DqB0. Greater k2 indicates greater water transport, gas exchange and growth as predicted by metabolic scaling theory and shown empirically (e.g., Hubbard et al. 2001). A larger scaling exponent q means a greater rate of increase in these presumably competitive capacities with tree size (Hammond & Niklas 2012). In general, fertile and consistently moist habitats with low threat of cavitation should favour conducting efficiency over safety. Species adapted to such habitats should cluster towards the upper right portions of Qref by q scaling space of Fig. 2.4. Conversely, arid and freezing habitats should push species to the lower left towards greater safety but lower transport capacity. The considerable overlap in scaling space between the functional tree types exemplifies how trait variation presumably arises from ecological and evolutionary circumstances, and how divergence in scaling space is minimized by compensation between traits (Marks & Lechowicz 2006). Despite the overlap, tropical trees were distinguished by reaching the greatest maximum capacity by having large vessels with a long functional lifetime (= large sapwood areas). These features are consistent with selection favouring efficiency over safety in their relatively permissive habitat where the threat of cavitation by freezing or water stress is low (McCulloh et al. 2010). The temperate diffuse- and ring-porous trees had lower peak transport capacities than the tropical trees. Accordingly, their habitat is not so permissive, certainly not in the case of freezing-induced cavitation. The adaptation to winter freezing takes different forms in ring- vs. diffuse-porous types (Sperry et al. 1994). Ring-porous trees had essentially the same range of vessel diameters and taper exponents as tropical trees, but the large vessels are sacrificed annually to cavitation by freezing. Hence, their drop in predicted transport capacity (lower Qref ) and flatter scaling (lower q) relative to tropical trees results from giving up sapwood area. Diffuse-porous temperate trees arguably adapted to freezing by having vessels narrow enough to limit the extent of cavitation (and many also reverse cavitation in spring; Hacke & Sauter 1996; Sperry et al. 1994). Their drop in transport capacity relative to tropical trees results from narrower vessels and less taper rather than less sapwood area. Both ring- and temperate diffuse-porous adaptations to freezing result in fairly similar estimated transport capacities: short-functioning and hence few, large vessels in ring-porous trees roughly equated to long-functioning and hence numerous, narrow vessels in diffuse-porous 20 trees. The conifers exhibit convergent scaling with temperate angiosperms despite very diver­gent wood structure. Their unicellular tracheids are limited in diameter for developmental and mechanical reasons compared with multicellular vessels (Pittermann et al. 2006). Hence tracheid taper functions are flatter (lower p range, Table 2.3). The greater impact of interconduit pits in conifers (lower C, Table 2.2) is because tracheids are much shorter than vessels and water encounters more interconduit walls as it flows through a given length of branch. However, these disadvantages are largely compensated for by the efficiency of the torus-margo structure of their intertracheid pitting (Pittermann et al. 2005). The narrow tracheid diameters and low taper are made up for by maximally efficient packing functions (Sperry et al. 2008). Conifer wood is a honeycomb of tracheids and consequently has a much greater conducting area (CF up to 0.42, Table 2.3) than angiosperm xylem with its vessels dispersed in a fibre-parenchyma matrix (CF < 0.37). Conifer wood partially dodges the efficiency vs. safety trade-off by increasing efficiency with conduit number rather than conduit diameter. Conifers also had packing exponents consistently less negative than d = 2 (Table 2.3; McCulloh et al. 2010), leading to a high water use exponent (q) despite their low taper exponent (p) range. Less negative d in conifers means there is a greater fraction of space devoted to water conduction in trunks vs. twigs. Anatomically, this is likely owing to a lower ratio of tracheid wall thickness: tracheid lumen diameter ( "thickness-to-span" ratio) in trunks vs. twigs. The thickness-to-span ratio in turn scales with the strength of tracheids against implosion by internal negative sap pressures (Hacke et al. 2001). Thus, low thickness-to-span in trunk tracheids corresponds with less negative sap pressures proximally and vice-versa in the distal twigs. If the model predictions are realistic, actual trees should fall within the boundaries shown in Fig. 2.4, but not necessarily fill them, because not all modeled trait combinations may have evolved. Indeed, data on intraspecific q, while limited, appear to primarily fall within the upper portion of the predicted range. Values of q much below ca. 1.5 have not been observed in trees (Enquist et al. 2000; Mencuccini 2003; Meinzer et al. 2005; Sperry et al. 2008), which is why lower values were excluded for assessing interspecific scaling. Flow rates for trees of ca. 72 cm in diameter (ca. 4 to 125 kg hr-1 ) are also consistent with the predicted Qre/ range (Enquist et al. 1998; Wullschleger et al. 1998; Meinzer et al. 2005). A review of whole-tree water use in 67 species indicated no systematic differences in daily tree water use vs. trunk diameter between the four functional types we 21 considered (Wullschleger et al. 1998), which is consistent with their extensive overlap in Qref (Fig. 2.4). The predicted similarity of temperate tree types is supported by observed parity in whole-tree hydraulic conductance between temperate conifers and temperate angiosperms (Becker et al. 1999)). Where differences have been seen between categories, they support model predictions. Tropical angiosperm trees in one extensive comparison moved more water per diameter than temperate conifers, consistent with our model results (Meinzer et al. 2005). A more direct test of the model in a diffuse- and a ring-porous species is the subject of the second paper in this series (von Allmen et al. 2012). The functional type simulations indicate that there are multiple ways to "skin the cat" when it comes to achieving a given water transport capacity and size-scaling. The Savage et al. derivation of an optimal taper (p = 1/3, Eqn. 2.5) effectively captures the consequences of conduit taper while holding other variables constant (Savage et al. 2010). In our more detailed model, conduit taper emerges as one of several influences on water use scaling, underscoring the likelihood that selection on any single trait (like taper) can be relaxed by compensating changes in other variables (e.g., packing function, leaf hydraulics and sapwood area). Nevertheless, the boundaries of the Qref by q scaling space were finite and relate in context-specific ways to the same space-filling and safety vs. efficiency constraints emphasized by Savage et al. The simulated scaling space excluded three-fourth intraspecific metabolic scaling. The greatest metabolic exponent was cq = 0.71 and random sampling yielded an intraspecific mean cq « 0.63. This mean is similar to the range predicted from observed water use scaling within the few tree species where it has been assessed (Mencuccini 2003; Meinzer et al. 2005). If metabolic scaling does indeed center on cq = 0.75, as has been proposed (Enquist et al. 1998, 2000; Niklas & Enquist 2001), the reason remains ambiguous based on our results. Given the focus on three-fourth power scaling, we looked for situations where it could be consistent with the model and found two of them. Intraspecifically, if hydraulic conductance increases faster in roots than in shoots with size, q can reach or exceed 2 (q > 2) and cq < 0.75. This pattern has been proposed as a mechanism to compensate for a potential hydraulic limitation on tree height (Magnani et al. 2000). However, data are limited and equivocal, with some species showing an increase in K /K S with size (Martlnez-Vilalta et al. 2007) and others not (von Allmen et al. 2012). A related explanation that applies to interspecific scaling is that species with greater inherent transport capacity (larger k2 and/or q) would have less of a hydraulic limitation to height and grow taller than species 22 with lower transport capacity. A bias for greater stature with steeper intraspecific scaling can theoretically give metabolic cq exponents of 0.75 or higher (Fig. 2.5). While both scenarios are compatible with three-fourth power scaling, neither predicts that particular exponent from a priori optimization in the WBE sense. Our species-level model is purposefully more complex and realistic than the Savage et al. analytical version. By allowing many functional traits to simultaneously vary, the numerical model reveals how a species' metabolic scaling results from interaction between complex trait interactions and covariance. A finite scaling space appears more realistic than convergence on one particular rule. This conclusion is based on a limited number of hydraulic traits, but is likely to be reinforced when additional complexities are considered. For example, the range of branching structure is more diverse than the generic WBE default (Price et al. 2007; Bentley et al. 2013), carbon allocation may not always preserve isometry between metabolic sinks and vascular supply (Reich et al. 2006; Enquist et al. 2007b), and vascular supply itself would be modulated by dynamics of cavitation and refilling. Incorporating such complexity can translate an even broader diversity of plant functional traits into whole plant performance. Such a framework could have general utility in ecology from constraining ecosystem fluxes and stocks to exploring the optimization of trait interactions. 2.6 Acknowledgements The authors were supported by National Science Foundation Advancing Theory in Biology Award 0742800. John S. Sperry, Duncan D. Smith and Erica I. von Allmen were partially funded by National Science Foundation Grant IBN-0743148. Lisa P. Bentley was also supported by NSF Postdoctoral Fellowship in Bioinformatics DBI-0905868. 2.7 References Becker P., Tyree M.T. & Tsuda M. (1999) Hydraulic conductances of angiosperm versus conifers: similar transport sufficiency at the whole-plant level. Tree Physiology 19, 445­452. Bentley L.P., Stegen J.C., Savage V.M., Smith D.D., von Allmen E.I., Sperry J.S., Reich P.B. & Enquist B.J. (2013) An empirical assessment of tree branching networks and implications for plant allometric scaling models. Ecology Letters 16, 1069-1078. 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Major symbols and definitions Symbols Definitions ASi Sapwood area, branch level i AS/A t Sapwood area/basal area for reference tree size with Db0 = 72 cm CF Fraction of wood occupied by conduit lumens (conduit lumen fraction) C Xylem hydraulic conductance/Hagen-Poiseuille conductance (endwall correction) Db0 Trunk diameter (branch rank 0) DBi Stem diameter for branch rank i DC Xylem conduit diameter Dc max Maximum allowable conduit diameter DC twig Conduit diameter in the distal-most branch rank (twigs) Dp Pith diameter F Number of conduits per wood area g acceleration of gravity G biomass growth rate of shoot H /H b tree height/Euler buckling height K tree hydraulic conductance Kl/K t Leaf hydraulic conductance/supporting twig conductance K /K S Tree conductance/shoot conductance k0, b Generalized scaling multiplier and exponent (e.g., Y = k0Mb) k1, c Mass scaling multiplier and exponent (Db0 = k1M c) k2, q Water use scaling multiplier and exponent (Q = k2DB0) k3, p Taper function multiplier and exponent (DC = k3DPi) k4, d Packing function multiplier and exponent (F = k4DC) k5, a Bark thickness function multiplier and exponent (TBi = k5DBi) k6, s Sapwood area function multiplier and exponent (ASi = k6DBi) Li Branch segment length, level i M Shoot (aboveground) mass NC Conduit number n Daughter/mother branch number ratio Q Steady-state tree water transport rate at midday Qref Q for "reference" tree size of trunk diameter Db0 = 72 cm TBi Bark thickness, branch level i V Shoot (aboveground) volume ft Daughter/mother branch diameter ratio A P Total soil to canopy water potential difference Y Daughter/mother branch length ratio n Viscosity of water p Density of water Table 2.2. Model inputs and outputs in order of appearance in text. Power functions were used for their simplicity and good fit to empirical trends Input Default n, branch number ratio 2 Y, branch length ratio 0.794 (elastic similarity for n = 2 symmetric branching) P , branch diameter ratio 0.707 (area-preserving n = 2 symmetric branching) Hb /H , mature tree safety factor 4 Terminal twig diameter 2 mm Terminal twig length 8.1 cm DC = ksDB^; taper function k3 = 7.9 ^m mm-p , p = 1/3; Dc in ^m, Db% in mm DC max 240 ^m DC twig 10 ^m F = k4DC; packing function k4 = 105 ^m-d mm-2 ; d = -2; F in mm-2 , Dc in ^m D p , pith diameter 1 mm TBi = k5DBi bark function k5 = 0.0225 mm1-"; a = 1.05; TBi and DBi in mm ASi = keDB^; sapwood function k6 = 0.905 mm2-s; s = 1.93; Asi in mm2, Db% in mm C, endwall correction factor 0.44 (angiosperms); 0.36 (conifers) Kl/K t , leaf/twig conductance 0.30 K /K s , tree/shoot conductance 0.50 AP, total pressure drop 1 MPa Output H , DB0 and V, yielding estimates of c: DB0 a Mc K and DB0 Q and DB0, yielding estimates of q: Q = k2DB0 Estimates of G a Q a Mcq B0 27 Table 2.3. Model inputs used to define the hydraulic scaling of four tree types (Fig.2.4). Ranges adapted from the literature (Supporting information, S2.2). Additional inputs were set to defaults listed in Table 2.2. The As /At trunk is the fraction of sapwood area per basal area in a tree of DBo = 72 cm that results from the inputted sapwood function. Note that the range of leaf-to-twig conductance ratio (Kl /K t ) was assumed to be the same for all categories, as were the sapwood parameters in all but the ring-porous category. Ring-porous temperate Diffuse-porous temperate Diffuse-porous tropical Conifers DC twig, ^m 21 (16.8 - 25.2) 12 (9.6 - 14.4) 21 (16.8 - 25.2) 7 (5.6 - 8.4) DC max, ^m 145 - 240 33 - 79 158 - 240 28 - 45 Taper p 0.30 - 0.59 0.14 - 0.41 0.31 - 0.61 0.20 - 0.44 Packing d -1.34 to -2.29 -1.65 to -3.27 -2.38 to -2.0 -1.69 to -1.8 Cf 0.09 - 0.37 0.07 - 0.20 0.06 - 0.12 0.37 - 0.42 K l /K t 0.20 - 0.40 0.20 - 0.40 0.20 - 0.40 0.20 - 0.40 Sapwood s 1.05 - 1.36 1.55 - 1.91 1.55 - 1.91 1.55 - 1.91 As /At trunk 0.003 - 0.017 0.34 - 0.74 0.34 - 0.74 0.34 - 0.74 28 29 1. Branching Architecture 2. Mass Allometry D B 0 = k 1 M J 5. Metabolic Isometry G a Q a Mcq 4. Water Use Allometry qB 0 Q = fc2 D? *1eafresistance 3. Xylem Architecture heartwood sapwood bark F ig u re 2.1. Elements of metabolic scaling theory. Self-similar and symmetric branching architecture (left) that is area-preserving (central column) yields trunk diameter (DB0) by mass (Mc) scaling. Xylem conduit architecture (shown in column cross-sections) yields water use (Q, flow rate) by DqB0 scaling. Combining mass and water use yields Q by M cq scaling. If growth rate (G) is isometric with Q (metabolic isometry), then the theory yields growth rate (G) by M cq scaling. Asterisked components represent novel parameters that were not explicit in the (Savage et al. 2010) model. See Table 2.1 for other symbols. 30 Trunk diameter, DB0 (m) F ig u re 2.2. Size-dependent variation in the q exponent (Q <x DqB0) for tree water flow rate (Q) and trunk diameter (DB0) in modeled trees. Small trees exhibit flat scaling because of the nonelastically similar growth of juveniles. Medium trees are steepest because they are elastically similar and have small gravity effects. The tallest trees flatten again because of larger gravity effects, unless there is gravity compensation (dash-dotted no g line). 31 100 80 60 40 20 O 0 (a) Pith, bark, sapwood Q Savage (100%*) - • pith + thin bark # (thick bark) - radial taper - s » *74% *50% *30% *basal sapwood % -co wc TO .2 TO 15 16 17 18 19 Q by DB0 exponent (q) F ig u re 2.3. Effect of individual hydraulic traits on tree sapflow rate (Qref , at trunk diameter DB0 = 72 cm), and the Q by DqB0 scaling exponent, q, for medium-sized trees. (a) Cumulative effects of adding pith and thin bark, radial taper and sapwood of decreasing percentage of basal area (at DB0 = 72 cm) to the Savage et al. model (open symbol). Thick bark shown separately. Grey 74% sapwood point is default for b. (b) Adding leaves of decreasing conductance (KL) relative to twigs (KL/K T) decreased Qref and increased exponent q. Thick line is probable range of KL/K T, grey datum is default KL/K T (0.3) for c,d. (c) Increasing the conduit diameter taper exponent, p, makes wider conduits proximally (Eqn. 2.5) and increased Qref and q. Open symbol is default. Asterisked p exponents required proximal conduits to be capped at DC max = 240 ^m. Thick line is realistic range of exponent, p. (d) Varying the packing exponent, d, (Eqn. 2.6) had a large effect on the q exponent, but little effect on Qref compared to changing the fraction of conducting wood area (C f ). Thick lines indicate realistic ranges of d and C f . 32 G by M exponent, c • q 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 Q by Dbo exponent, q F ig u re 2.4. Scaling space showing tree water transport rate (Qre/, at trunk Dbo = 72 cm) and the scaling exponent q (Q oc Dqm ) for four functional types. Considerable overlap existed between conifers (blue), temperate diffuse-porous (red), ring-porous (black, RP) and tropical trees (green). The corresponding growth rate by mass exponent (cq) is given on the upper axis. Parameter ranges defining these tree types are given in Table 2.3. 33 CT O c 0c oCP X 0 cn £Z In o w a) <n ro E .Q CD 5 080 075 070 065 060 055 3/4 scaling - 9- - 1 1 __________In1t_e_r_ _______In1t_r_a_ __ 1 + 1 ____I_n1t_e_r_ _______Int1ra_ ________ Random Height-Biased F ig u re 2.5. Interspecific vs. intraspecific metabolic scaling exponents (cq). Exponents obtained from n = 10 repetitions of the regression in Fig. S2.2 (Supporting information, S2.3). Symbols are outliers, whisker is 10th/90th percentile, box is 25th/75th percentile, line is median. Intraspecific scaling is not influenced by whether there was a random relationship between species scaling and stature, or a height-biased relationship where species with steeper and higher scaling (upper right of Fig. 2.4 scaling space) also reached greater size. The latter "height-biased" scenario greatly steepened interspecific scaling and was the only interspecific sampling scheme that could yield exponents matching three-fourth power scaling (dashed line). 34 S2 Supporting information S2.1 S iz e -d e p e n d e n t w a te r u se a llom e try Figure S2.1A shows the height (H) by trunk diameter (DB0) allometry of modeled trees, which converges on elastic similarity (H a D^O) with size. Figure S2.1B shows the consequences of this for tree volume (V, and by implication, tree mass, M ) scaling with DB0. Trees converge on the expected c = 3/8 = 0.375 value for elastic similarity. Trees of the medium size range (gray symbols), which show approximately linear scaling in log-log space (text Fig. 2.2) have c = 0.369 rather than the asymptotic c = 0.375, which is approached in very large trees. S 2.2 F u n c tio n a l t r e e ty p e s in s c a lin g sp a c e Inputs for the packing and taper functions, including the vessel lumen fraction (CF), were taken from the survey data set of McCulloh et al. (2010) with additional data from the companion paper (von Allmen et al. 2012) and Sperry et al. (2008). The taper function yielded mean twig conduit diameters (DC twig) per type for a fixed twig diameter of 2 mm (default, text Table 2.2). The DC twig was allowed to vary ± 20% of the mean value per type (text Table 2.3, DC twig range in parentheses). Because the packing and taper functions were determined on smaller trees, they were not necessarily reliable predictors of the maximum vessel diameter in major branches and trunk (Dc max). In­stead, we obtained the type-specific range of DC max for the same species from the "inside wood" data base (InsideWood, 2004-onwards and published on the Internet at http://insidewood.lib.ncsu.edu/search) and from Panshin & de Zeeuw's textbook 1970. For species in the McCulloh et al. data set that were missing from these sources, we substituted con-generic species. The maximum Dc twig and minimum Dc max within a functional type dictated the minimum taper exponent (p) for that tree type. The upper value of p for each functional type was set to the maximum from the literature, but the trunk DC was never allowed to exceed the largest DC max for that tree type (Table 2.3). The packing function was constrained to stay within the cited range of packing exponent d while not violating the literature values of CF . Sapwood functions were obtained from several sources (Bovard et al. 2004; Gebauer et al. 2008; Hultine et al. 2010; von Allmen et al. 2012). Functions for tropical and temperate diffuse-porous trees and conifers were all constrained to initiate heartwood at DBi between 2 and 6 cm and to yield basal sapwood areas within cited fractions of total basal area (As/A t ) for the reference trunk DBo = 72 cm. For lack of information to the contrary, all three of these tree types were given the same range of sapwood functions. In ring- 35 porous trees, sapwood is limited to the single current year's growth ring (Ellmore & Ewers 1986; Zimmermann 1983). Ring-porous sapwood functions were accordingly constrained to initiate heartwood beginning in the first branch rank proximal to the twigs, and to yield the much lower range of AS/A T reported for these species. In lieu of more information of the leaf-twig conductance ratio (KL/K T ), we used the same range for this ratio for each category (Table 2.3). This range was a slight expansion of the range reported for the ring-porous Quercus gambelii (0.27) and diffuse-porous Acer grandidentatum (0.38) in the companion paper (von Allmen et al. 2012). S 2 .3 I n t r a - v s. in te r s p e c if ic s c a lin g Each data set sampled from the assemblage of theoretical species in Fig. 2.4 of the text had 100 species. As noted in the text, species with q < 1.5 were excluded on the grounds they would be unlikely to exist. For each species we assumed a uniform distribution (on log scale) of 11 tree trunk diameters from 2 < DB0 < 72cm (e.g., the medium size class, text Fig. 2.2). An RMA regression through the log-transformed data (n = 1100 trees; 11 trees from 100 species) yielded the interspecific scaling exponent for q (DB0 by Q values, Fig. S2.2) and c (DB0 by M values). The 100 species sampling was repeated 10 times for the "random" and "height-biased" scenarios to obtain an average interspecific exponent (text Fig. 2.5). The intraspecific q exponent was averaged across the 100 sampled species. A random multispecies data set of Q by DB0 (n = 1100 trees) is shown in Fig. S2.2 (grey symbols). The RMA slope through such data sets gave an average interspecific q = 1.81±0.0048 (mean ± SE), which was about 6.6% greater than the intraspecific mean of q = 1.69 ± 0.0029. In the "height-biased" scenario, species with greater Qref and q (towards upper right in Fig. 2.4) were assumed to achieve greater stature than species with lower values (towards the lower left). Data sets of 100 species were randomly sampled from Fig. 2.4 of the text along an arbitrary diagonal running from lower left to upper right. The particular diagonal used for coefficients shown in text Fig. 2.5 was Qref = 0.00033e7'2q. The DB0 range for species with progressively greater q was increased to reflect their greater maximum stature. We assumed that all species achieved a DB0 up to 0.128 m, but as q increased from 1.5 to the maximum of 1.91, the maximum Dbo for that species increased to 0.724 m. The bias towards steeper scaling in the taller species can be seen in Fig. S2.2 by comparing the black "coupled" species with the gray "random" ones. The RMA interspecific regression yielded an average q = 2.039 ± 0.0107 vs. the intraspecific mean of q = 1.71 ± 0.0055. Altering the diagonal coupling between q and Qref and altering the q-dependence of species stature yielded a wide range of exponents, even exponents greater than three-fourth. If, however, 36 there was no species size bias along the diagonal, the result was equivalent to the "random" scenario (simulations not shown). The analogous process was used to generate interspecific estimates of the mass scaling exponent, c. Each species sampled for q had a common DB0 by V c scaling, where c = 0.369 (Fig. S2.1, medium size trees). This universal scaling was converted to a species-specific DB0 = k 1M c scaling by assuming that M = V ■ wood density. Wood density values were randomly assigned to each species in the multispecies data set from the 16468 values in the database of Zanne et al. (2009; http://datadryad.org/handle/10255/dryad.235; Chave et al. 2009). An RMA linear regression through log-transformed Dp0 by M values in the data set yielded the interspecific c. No size bias was assumed with mass exponent c, in keeping with the observation that wood density does not covary with the critical buckling height, Hb (McMahon 1973; Niklas 1994). For the mass scaling exponent, c, there was little difference between interspecific c (c = 0.367) vs. intraspecific c (0.369) because wood density of most species in the randomly sampled data set fell within a limited span (613 ± 176 kg m-3 ,mean ± SD) despite a fairly wide total range (80 to 1360 kg m-3 ; Zanne et al. 2009). Multiplying the q estimates by c for the same trees yielded the metabolic scaling exponent: G x Mcq. These are cited in the text (Fig. 2.5) for the random and height-biased sampling scenarios. 37 £ x ■<D 0 0 om c Trunk Diameter, DB0 (m) Tree Volume, V (m ) F ig u re S2.1. Size-dependent deviations from power-law scaling of modeled tree geometry. Trees are grouped into small (gray), medium (dark gray), and large (black) sizes to show changes in best-fit power-law scaling exponents. A. Modeled tree height (H) converges on linear scaling (in log-log space) with trunk diameter (DB0) with size as required to maintain a constant safety margin from elastic buckling. B. Size-dependent variation in the c exponent (DB0 a Vc) for trunk diameter (DB0) and tree volume (V). The same exponent applies to tree mass (M) for constant branch tissue density. Scaling converges on c = 3/8 = 0.375 as tree size increases. 38 O t f o Wc ro 0 0.01 0.1 Trunk Diameter, DB0 (m) Figure S2.2. Interspecific simulations of the relationship between tree water transport rate (Q) and trunk diameter (DB0). Grey represents 100 randomly chosen "species" (Q by Dbo allometries from text Fig. 2.4), each with 11 individuals of different trunk diameter. Although individual data points are obscured, the important point is the similarly broad Q range across all DB0 in the "random" scenario. A single RMA regression (q = slope of gray line) through all n = 1100 data points yields the interspecific exponent: Q a DqB0. Black represents species with coupling between Qref and q and species stature. Species that achieve larger DB0 also had greater Qref and q. Hence, the Q range of the tallest trees is biased upward compared to the random scenario. CHAPTER 3 DEVIATION FROM SYMMETRICALLY SELF­SIMILAR BRANCHING IN TREES PREDICTS ALTERED HYDRAULICS, MECHANICS, LIGHT INTERCEPTION AND METABOLIC SCALING 3.1 Summary • The West, Brown, Enquist (WBE) model derives symmetrically self-similar branching to predict metabolic scaling from hydraulic conductance, K, (a metabolism proxy) and tree mass (or volume, V ). The original prediction was K x V °'75. We ask whether trees differ from WBE symmetry and if it matters for plant function and scaling. We measure tree branching and model how architecture influences K, V, mechanical stability, light interception and metabolic scaling. • We quantified branching architecture by measuring the path fraction, P f : mean / maximum trunk-to-twig pathlength. WBE symmetry produces the maximum, Pf = 1.0. We explored tree morphospace using a probability-based numerical model constrained only by biomechanical principles. • Real tree Pf ranged from 0.930 (nearly symmetric) to 0.357 (very asymmetric). At each modeled tree size, a reduction in Pf led to: increased K; decreased V; increased mechanical stability; and decreased light absorption. When Pf was ontogenetically constant, strong asymmetry only slightly steepened metabolic scaling. The Pf on­togeny of real trees, however, was "U" shaped, resulting in size-dependent metabolic scaling that exceeded 0.75 in small trees before falling below 0.65. Reprinted with permission from John Wiley & Sons. Smith D.D., Sperry J.S., Enquist B.J., Savage V.M., McCulloh K.A. & Bentley L.P. (2014) Deviation from symmetrically self-similar branching in trees predicts altered hydraulics, mechanics, light interception and metabolic scaling. New Phytologist 201, 217-229. 40 • Architectural diversity appears to matter considerably for whole tree hydraulics, mechanics, photosynthesis, and potentially metabolic scaling. Optimal architectures likely exist that maximize carbon gain per structural investment. 3.2 Introduction A large and growing body of research has focused on the coordination of hydraulic transport with the metabolism of photosynthesis and growth. While empirical research on this subject is quite extensive (e.g., Brodribb 2009), a prominent component is metabolic scaling theory (MST), which stems from the original development by West, Brown, & Enquist (WBE hereafter; 1997, 1999). The theory, as it applies to plants, centers on the premise that water transport is a colimiting factor for photosynthesis. Because water transport is a largely physical process dependent in part upon transport network structure, its scaling can be predicted from relatively simple allometric models, leading to scaling predictions for all dependent metabolic processes. The WBE model is fairly simple in its design. Plant branching structure is divided into external and internal components. The external structure follows symmetrical and self-similar branching (see Fig. 3.1a, rightmost tree), which allows the structure to be easily scaled. The external structure also conforms to biomechanical principles of area preservation and safety from gravitational buckling. The internal branching structure is the network of xylem conduits within the branches. The number and dimensions of xylem conduits are linked by simple rules to the external branch network (Savage et al. 2010; Sperry et al. 2012). Central to MST are relationships described by power functions of the form y = axb where a is a scaling multiplier and b is a scaling exponent. Oftentimes, the focus is on the proportionality, y a xb. The WBE model's prominent achievement is the analytical prediction in agreement with at least some empirical observations (Niklas & Enquist 2001) that metabolic rate (B) scales with mass (M ) to the three-fourth power (i.e., B a M 3/4; symbol definitions repeated in Table 3.1). This scaling prediction may be broken into two separate components that individually relate mass and water use to the easily measured dimension of trunk diameter, D y . The stem mass (and volume, V ) is assumed to scale with dT/c . This "volume expo­nent," c, is predicted to converge on 3/8, which is supported by theoretical and empirical considerations (McMahon & Kronauer 1976; von Allmen et al. 2012). The rate of water use, Q, is assumed to scale with D q . The model predicts Q from whole tree hydraulic conductance, K, which is calculated from internal vascular allometry. If the flow-induced 41 pressure drop from soil to leaf is size invariant, then K a Q. Because water loss and CO2 uptake utilize the same stomatal pathway, carbon assimilation should have a direct relationship to Q. If a constant fraction of photosynthate goes towards growth (a proxy for B) the result is B a Q a K a D^. The product of the "hydraulic exponent," q, and c gives the "metabolic exponent," cq: B a Q a K a M cq. The WBE derivation of cq = 0.75 arises from the prediction that q converges on 2 for infinitely large trees. Thus, c = 3/8, q = 2, and cq = 0.75. Smaller values of q (0.68 to 1.91) and, hence, cq (0.25 to 0.70) are predicted for finite trees (Savage et al. 2010; Sperry et al. 2012). Since its creation, revisions have been made to the WBE model, which have dealt with altering the branching structure within the confines of perfect symmetry (Price et al. 2007) and making the internal anatomy more realistic. The anatomical modifications have included more accurate scaling of xylem conduit number (Savage et al. 2010) and the addition of leaves, roots, and nontransporting tissues (Sperry et al. 2012). These revisions have led to more accurate predictions (Price et al. 2007; von Allmen et al. 2012) but trees were still assumed to follow symmetrically self-similar branching. Real trees show average branching ratios (daughter/mother branch number, diameter, and length) that can be similar to the constants predicted by WBE's symmetric self-similarity (Bentley et al. 2013). However, the distributions are quite broad, indicating a sizable fraction of asymmetric junctions. Even a few asymmetric junctions amongst major branches could significantly alter whole-tree symmetry. We ask whether the branching architecture of real plants deviates substantially from the WBE structure. We then address the consequences of deviation with a model. We use the WBE model as a reference point and develop a novel numerical simulation method for building trees that represent the full range of tree morphospace from WBE symmetry to maximal asymmetry. Our numerical approach uses a minimum of deterministic branching rules and instead relies on probability distributions to build branch junctions and trees of varying symmetries. Our only major branching assumptions are that trees conform to the well established patterns of area-preserving branching (Horn 2000) and network-scale elastic similarity (McMahon & Kronauer 1976). We use the improved internal anatomy of Sperry et al. (2012) but hold xylem parameters constant across simulated trees so as to isolate branching effects. We use the numerical model to investigate how deviations from WBE branching affect whole tree hydraulic conductance, total stem volume, safety from gravitational buckling, and light interception. The model is also used to predict the influence of branching architecture on the scaling of tree hydraulic conductance (exponent 42 q) and volume (exponent c) with trunk diameter, and hence how hydraulic conductance and its dependent processes scale with mass (exponent cq). 3.3 Methods and model description 3 .3.1 T h e p a th fr a c t io n in d e x fo r t r e e fo rm We developed the "path fraction," P f , to quantify how much a particular branch network deviated from the WBE ideal. The path fraction is based on the pathlengths from twig tip to trunk base. We use the symbol, L|, for this pathlength where the double arrow indicates that this length spans two extremes, twig tip to trunk base. In a WBE tree, all values of are the same. In our model, deviating from WBE by removing junction symmetry adds variation to L|. We define the path fraction as Lt Pf = L (3.1) The bar in L| refers to the mean L| for the tree and the asterisk in L| (and other symbols that follow) indicates the maximum. The L| is an approximation of plant height so we will also use this symbol for height. The maximum possible Pf is 1, which occurs when L| = L| (e.g., WBE trees; see Fig. 3.1a, rightmost tree). A high Pf corresponds to a round-shaped, spreading crown while a low Pf corresponds to a narrow crown with limited spread (Fig. 3.1). The minimum Pf is made by a structure with a central axis with twigs attached alternately. This structure minimizes L| and we refer to it as the "fishbone" structure (e.g., Fig. 3.1a, leftmost tree). We use Pf as the independent branching structure variable against which we plot the functional attributes of tree hydraulic conductance, volume, mechanical stability, and light interception. 3 .3 .2 Em p ir ica l p a th fra c t ion s As a test of how much real plants deviate from the WBE structure, 40 Pf measurements were made of real branch systems. Specimens came from 15 different species and included both whole individuals and branches of open-grown trees and shrubs (species and sources in Table 3.2 and Supporting information, S3.1). Species were chosen to represent a wide range of apparent architectures. Branches were obtained by a single cut just distal to a branch junction. Path fractions were obtained in two ways. For some (mostly the entire individuals), each segment between branching points was labeled and its length, diameter, and mother segment were recorded. Twig-to-base paths were then reconstructed from these data to get all L| values. For the other specimens, L| values were measured directly by 43 following stems from base to twig tips using a marked string with 10 cm precision. For this direct method, specimens were measured in spring so the measurements were made to tips that appeared to have been active the previous season. Direct Pf measurements were time-consuming, limiting the size range to trees with trunk diameters, Dt , < ca. 5 cm. To estimate the Pf of larger trees to trunk diameters over 1 m, we used the crown area vs trunk diameter data set of Olson et al. (2009; see Supporting information, Fig. S3.1) from angiosperm trees. From their published data (including all branches and trees in sheltered and salt-sprayed environments), we obtained an ordinary least squares (OLS) regression to predict vertically projected crown areas from Dt . We matched these predictions to 3D modeled trees with the same Dt and within 5% of the same crown area. The Pf from these matching model trees were used to construct a Pf ontogeny. 3 .3 .3 T re e b u i ld in g m o d e l 3 .3 .3 .1 B ra n ch in g Our tree building model was written in the R language (R Core Team 2013) and is available from the senior author upon request. The model begins by sequentially defining junctions, starting with the trunk. At each junction, the mother branch (subscript m) splits into a number of daughters (subscript d). The number of daughters is f , the furcation number. Within each tree, we randomly chose a maximum furcation, f *, and then at each junction we chose f from 2 to f *. The f * was 2, 3, or 4, which covers the range for most botanical trees. Our f selection contrasts with the WBE model, which uses a strictly constant f (n in their terminology). We assigned each branch an order or rank, R, equal to the number of twigs it ultimately supports (Katifori & Magnasco 2012). Therefore, the starting point of each tree, the trunk, has Rm = the total number of twigs on the tree. This ranking system, illustrated in Fig. 3.2, simplifies tree building because: R is a finite integer; branch ranks change at each junction; and total rank is preserved across junctions. Each combination of mother rank, Rm, and f defines possible daughter ranks, Rd. Each daughter can only take on a certain number of different ranks because the sum of Rd must equal Rm. The first selected daughter rank, Rd,i, was always the smallest and was restricted to the range, A 1 to Z 1, where A 1 = 1 and Zi = L f J (3.2) where the floor brackets indicate the integer of the ratio. For subsequent daughter ranks, Rd,i , where 1 < i < f , the Zi is given by 44 Zi = i-1 Rm - ^ Rd,j | / ( f - * + 1) j=1 (3.3) Equation 3.3 is just a variation on Eqn. 3.2 where the numerator accounts for the fact that there is "less rank" remaining to divide and the denominator indicates the "remaining rank" is being divided among fewer undefined daughters. The intermediate values of Ai (if present) are different from the first and final Ai . For 1 < i < f , the Ai = Rd,i-1 as no daughter may be smaller than its predecessor. For the final daughter in the furcation, Rd, f , the Af = Z f such that Rdj can only take on a single value that completes the mother rank. The choice of Rd in each junction determines the symmetry of that junction. We controlled this choice by using a discrete probability distribution function to select each Rd,i at random from its respective Ai to Zi range. We defined this probability distribution with a power function because changing the exponent, u, allowed us to control the degrees of symmetry or asymmetry. When u < 0, the probability, P , of any Rd is given by PRd = Ru/ £ ( 3 . 4 ) j=A When u > 0, a slightly different equation is used, Z pRd = (Z - A + Rd) u/ £ (Z - A + j ) u (3.5) j=A For a given u > 0, Eqn. 3.5 takes the probabilities from Eqn. 3.4 with -u and mirrors them over the same A to Z range. For example, comparing u = 2 to u = -2 in a junction, PRd=A when u = 2 is equal to PRd=Z when u = -2. When u < 0, asymmetrical junctions are favored while u > 0 favors symmetry. Using Eqns. 3.4 and 3.5 with a u range of -5 to 5 created trees that populated the Pf range from maximum asymmetry ( "fishbone" trees) to perfect symmetry (WBE trees). For a given tree, we fixed u at a single value. When u was varied within a tree to produce both strongly symmetric and asymmetric junctions, the generated trees were unrealistic (Supporting information, Fig. S3.2). As an illustration of the daughter selection process, consider the circled junction in Fig. 3.2 (left tree). This tree has 10 twigs total and u = -2 was selected at random from -5 to 5. First, f * = 3 was selected from 2, 3, or 4 with equal probability. The f of the first junction (the trunk; with Rm = 10) was chosen between 2 and f * with equal probability. Choosing f = 3, the rank of the smallest daughter, Rd,1, was selected next. Because Rd,1 is the smallest and all daughters must add to 10, Rd,1 must be between 1 (A1) and 3 (Z 1), as given by Eqn. 3.2. With negative u, Rd,1 = 1 will have the greatest probability 45 (PRd i = 0.735 from Eqn. 3.4) and 3 will be very unlikely (PRd 1 = 0.082). Suppose R , i = 1 is chosen. The second daughter, Rd,2 is the next smallest so it may range from 1 to 4, as given by Eqn. 3.3. Again, the minimum, 1, is most likely to be chosen. Here, Rd,2 = 2 was chosen. The final daughter has only one option, R , f = 7, resulting in a fairly asymmetrical junction. After creating this first junction, each daughter with R > 1 became a mother and junction selection continued, keeping f * = 3 and u = -2 . The right tree in Fig. 3.2 shows how u = +2 can create much more symmetrical junctions. 3 .3 .3 .2 B ra n ch d iam e te r s After assigning all ranks, branch diameters and lengths were determined. Diameters were defined using constant twig diameters and area-preservation (i.e., D^ = f=1 D^j). With R defined as the total number of supported twigs, each with constant cross-sectional area, R is proportional to the cross-sectional area of the branch. As such, diameter, D, is a function of R and twig diameter, Dt: D = DtR0'5 (3.6) This property is illustrated by the trees in Fig. 3.2 where diameters increase with R. 3 .3 .3 .3 B ra n ch leng ths Length determination is more complicated but the guiding principle is that lengths must coordinate with diameters to achieve a constant safety factor from whole tree elastic buckling from gravity. Here, we define a new pathlength, L*, where the upward arrow indicates this length is from branch base (i.e., just above its lower junction) up to twig tip. This contrasts with the double arrow in L|, which indicates trunk to twig path. The asterisk in L| signifies the maximum pathlength (i.e., to the most distant twig). Empirical data indicate that once a trunk or branch reaches a modest D, its longest supported path, L|, tends to scale as L| « aD2/3 (Niklas 1994; von Allmen et al. 2012). The exponent of 2/3 is consistent with elastic similarity (i.e., constant deflection per length; McMahon & Kronauer 1976). The critical height at elastic buckling, Lcrit, is also predicted to follow 2/3 scaling with D: Lcrit = bD2/3, where b can be explicitly calculated from tree form and wood properties (Greenhill 1881). The shared 2/3 exponent means the safety factor from buckling (Lcrit/L|.) becomes constant at larger D. This ultimately constant safety factor, s, is equal to the ratio of the scaling multipliers: s = b/a. At smaller D, however, the L* by D scaling is steeper than 2/3. McMahon & Kronauer (1976) attribute this steeper exponent to a "virtual length," lo. If the tree is represented as an elastically 46 similar doubly tapered beam, then lo is the distance from the free end of the beam (i.e., the twig tip) to the point where the beam would taper to zero at its theoretical origin. McMahon & Kronauer (1976) show that by D scaling across all D can be fit by an equation of the form: = aD2/3 - lo (3.7) As D increases, the lo term becomes comparatively negligible and the equation converges to = aD2/3 (see Supporting information, Fig. S3.3). Branch lengths were assigned from a single version of Eqn. 3.7 (Eqn. 3.8) that was applied across all trees regardless of their branching topology. The multiplier, a, was defined as a = b/s, where s = 4 and b was calculated from a WBE tree (b = 107.94 m1/3; see Mechanical stability of model trees section). The value of lo was derived from WBE trees (see Supporting information, S3.2) and plugged into Eqn. 3.7 to produce the by D equation for all modeled trees: L\(D) = ^ d 2/3 - 0.794s d 2 /3 (3.8) Equation 3.8 gives maximum length distal to each branch segment and from this, individual branch lengths (i.e., between junctions) were determined. At a given junction, the mother branch will have a certain and its daughters will have respective values. Because larger diameters support longer paths, it will be true that the daughter with the largest diameter, D *d, will be part of the mother's longest path. Therefore, the segment length of the mother, lm, is lm = L (Dm ) - L (D d ) (3.9) Twigs, which do not support daughters, have lengths equal to their L|: lt = L|(Dt). (3.10) The use of Eqn. 3.9 can be illustrated by the left tree in Fig. 3.2. The trunk (Rm = 10) supports a maximum path of = 0.58 m (using model parameters in Eqn. 3.8). Of its three daughters, only the largest daughter (R^ = 7) lies along this path. This daughter supports a maximum path of = 0.48 m. Therefore, the length of the trunk segment must be the difference: lm = 0.10 m. 3 .3 .4 H y d r a u l ic c o n d u c ta n c e o f m o d e l t re e s The hydraulic conductance, K, for each model tree was calculated from the internal network of xylem conduits. The internal anatomy is defined from the external anatomy 47 following the recent WBE revision by Sperry et al. (2012). Briefly (see Supporting infor­mation, S3.3 for details), hydraulic conductance of each stem segment is calculated from the diameter, number, and length of functional xylem conduits (Savage et al. 2010; Sperry et al. 2012). Additional hydraulic resistances come from leaves, roots and conduit endwalls (Sperry et al. 2012). Segment conductances were combined using rules of network analysis to calculate K. Sperry et al. (2012) used the external branching parameters of WBE to study the effects of variable internal anatomy. Here, we did much the opposite, using the Sperry et al. (2012) default internal parameters while studying the consequences of branching pattern and Pf on hydraulic conductance and the hydraulic exponent, q. Tree volumes were calculated to determine their sensitivity to Pf and, hence, the sensitivity of the volume exponent, c. Total stem volume, V, was the summed volume of all cylindrical branch segments. The volume of roots and leaves was not computed but assumed proportional to stem volume. If tissue density is invariant, then V becomes a proxy for stem (and plant) mass for purposes of metabolic scaling predictions. 3 .3 .6 M e ch a n ica l s ta b i l ity o f m o d e l tre e s The effect of branching structure on mechanical stability was assessed for all model trees by comparing estimated critical heights at elastic buckling (Lcrit) relative to estimated Lcrn of WBE trees (Lcrit;WBE). Typically, Lcrit is estimated by folding all branches up to make a column and assuming the tree mechanically behaves as this column (Niklas 1994). Furthermore, this column is assumed to have straight sides. To represent the full spectrum of more realistic trees, we used the alternative method of Jaouen et al. (2007), which identifies the "main stem" (i.e., the thickest trunk-to-twig path) as the tallest mechanical structure that must support itself and all attached branches. The Jaouen et al. method accounts for the important effects of branching architecture on vertical mass distribution and Lcrit. The diameter, D, of the main stem may be described as a function of height, z, using 3 .3 .5 V o lum e o f m o d e l t re e s Likewise, the stem mass of all branches supported above z may be defined by (3.11) m (3.12) 48 where Mtot is the total tree stem mass. The exponents n and m approximate the distribu­tions of support capacity (D) and support requirement (M ) in the main stem. For each tree, these exponents were calculated from Eqns. 3.11 and 3.12 by standardized major axis (SMA) regression of logged data using the SMATR package for R (http://bio.mq.edu.au/ecology/SMATR/; Warton et al. 2006). With some modifications to Eqn. 1 from Jaouen et al. (2007; see Supporting information, S3.4), we predicted Lcrit using Values for the ratio of E (Young's elastic modulus; N m-2 ) and pg (specific weight of supporting tissue; N m-3 ) for wood are approximately constant (Niklas 1994). The cv (determined numerically in R) is the first positive root of the Bessel function of the first kind with parameter v = (4n - 1)/(m - 4n + 2) (Greenhill 1881; Jaouen et al. 2007). The 2/3 value of b in Eqn. 3.8 corresponds to all the terms in front of DT in Eqn. 3.13 where m, n, cv, and Pf were from a WBE tree. When calculating Lcrit, two requirements were imposed. (1) Values of n and m are only meaningful when the data are well fit by Eqns. 3.11 and 3.12. We removed trees where fits had r2 < 0.95. (2) When v < -1, the cv becomes somewhat erratic so these trees were also removed. Less than 7% of all modeled trees were removed for poor fits to Eqns. 3.11-3.12 and only three trees in total were excluded for v < -1. 3 .3 .7 L ight in te r c e p t ion o f m o d e l t re e s The importance of light interception is implied in the WBE model through "space-filling branching" but it has not been quantified (Duursma et al. 2010). To estimate how Pf influenced light interception, we extended the model to three dimensions. For simplicity, we restricted 3D construction to trees where f * = 2 was chosen. Determining spatial structure required specification of branching angles and rotations with respect to connecting stem segments. Each branch segment was assigned an axis that runs along its length. "Branching angle" shall refer to the angle a daughter axis makes away from its mother's axis. "Rotation" refers to the rotation around its mother's axis. We adopted a set of maximally simple rules to set these angles and applied them equally acr