The Mathematics of Linked Optimisation for Water and Nitrogen Use in a Canopy

We develop, and discuss the implementation of, a mathematical framework for inferring optimal patterns of water and nitrogen use. Our analysis is limited to a time scale of one day and a spatial scale consisting of the green canopy of one plant, and we assume that this canopy has fi xed quantities of nitrogen and water available for use in photosynthesis. The effi ciencies of water and nitrogen use, and the interactions between the two, are strongly affected by physiological and physical properties that can be modeled in different ways. The thrust of this study is therefore to discuss these properties and how they affect the effi ciencies of nitrogen and water use, and to demonstrate, qualitatively, the effects of different model assumptions on inferred optimal strategies. Preliminary simulations suggest that the linked optimisation of nitrogen and water use is particularly sensitive to the level of detail in canopy light penetration models (e.g., whether sunlit and shaded fractions are pooled or considered independently), and to assumptions regarding nitrogen and irradiance gradients within leaves (which determine how whole-leaf potential electron transport rate is calculated from leaf nitrogen content and incident irradiance).


Introduction
Over the last few decades, our understanding of the ecology of relationships between light, nitrogen, water and carbon within plant canopies has developed considerably. Much of this progress has taken place in the context of an implicit 'optimality paradigm,' which recognizes that biological patterns in nature should refl ect the tendency of natural selection to preserve strategies that make the most of limited resource supplies. In particular, synthetic theories have emerged to predict basic optimal trends for water use (Cowan 1977, Cowan and Farquhar 1977, Cowan 1982, Cowan 1986, Mäkelä et al. 1996 and nitrogen use (Field 1983, Hirose and Werger 1987, Farquhar 1989, Schieving and Poorter 1999. While extensive testing has generally corroborated the qualitative trends predicted by these singleresource theories (Farquhar et al. 1980a, Field 1983, Hirose and Werger 1987, Werger and Hirose 1991, Schieving et al. 1992, Evans 1993, Berninger et al. 1996, Hari et al. 1999, Koskela et al. 1999, Gonzales-Real and Baille 2000, Hari et al. 2000, these theories have had limited success in replacing the predominantly semi-mechanistic and semi-empirical basis of models for gas exchange at the individual scale and above. For example, the prediction that leaf nitrogen content should scale with irradiance is used to drive canopy models that are otherwise not based explicitly on optimality theory (e.g., Sellers et al. 1992, Amthor et al. 1994, Leuning et al. 1995, Lloyd et al. 1995. Furthermore, the 'optimal' profi les used in such models are often quite different from empirically observed profi les, though the assumption is retained because the resulting models can predict gas exchange accurately.
A strong and direct physiological linkage exists between the effi ciencies of nitrogen and water use (e.g., Field et al. 1983, Reich et al. 1989, Cordell et al. 1999, and this connection may be responsible for the limited success of theories that deal with only one of these two resources. By identifying 'linked optima,' that is, by simultaneously optimizing both resources -as a plant in nature must do when faced with limited supplies of water and nitrogen -it may be possible to begin to make effective, testable predictions of whole-plant gas exchange strictly on the basis of optimality theory. However, the rigorous linked optimality criteria require calculation of the sensitivity of spatially and temporally integrated gas exchange patterns to changes in resource allocation strategies, and these sensitivities can not be calculated analytically. The goals of the present work were to provide a complete conceptual and mathematical framework for the simultaneous optimization of nitrogen and water use in a canopy (including derivations of key equations that are too detailed and esoteric to present in more applied work), to identify important features for inclusion in a simulation model that will perform this optimization, to demonstrate the effects of these features, and to raise salient questions for more detailed analysis. We present the abstract mathematical solution for water and nitrogen use optimisation in a single plant in Section 2, and attempt to resolve conceptual diffi culties in its interpretation and potential ambiguities in its implementation. In Section 3 we discuss physiological and physical considerations that can strongly affect the relationships between assimilation rate and water and nitrogen use, and which can, therefore, also change the optimal nitrogen profi les that one would infer from a simulation model. We demonstrate these effects graphically by manipulating a leaf gas exchange model (presented in the Appendix) and by inferring optimal nitrogen profi les from a detailed canopy simulation model (based on the models presented by Leuning et al. (1995) and de Pury and Farquhar (1997), but not presented in detail here). Finally, in Section 4, we summarize the analyses of Sections 2 and 3, suggest applications of the mathematical framework developed in this paper, and offer some thoughts on the role of optimality theory in plant ecophysiology. In the Appendix, we apply the abstract optimal solution (from Section 2) to a general model of leaf gas exchange, to develop mathematical expressions that can be used in detailed simulation models of canopy gas exchange. In an accompanying paper (Farquhar et al. 2002), we apply the abstract solution to a more simplifi ed gas exchange model (which excludes self-shading and temperature effects) to evaluate general ecological implications of linked optimal water and nitrogen use.
We will begin by arbitrarily defi ning the term optimal, with regard to resource use strategies, as describing a strategy that maximizes the ratio of daily net CO 2 assimilation to the available quantity of resource. Many other defi nitions are possible, including reproductive success (the ultimate goal from a Darwinian perspective) and assimilation averaged over a longer time period. However, reduced carbon can be used for nearly any purpose (e.g., building seeds to reproduce, leaves to compete for light, or protective structures to ward off predators), and a single day is a convenient temporal context that subsumes much of the predictable variation in environmental factors that an individual plant faces. Therefore, we believe this defi nition of 'optimal' represents an appropriate compromise between generality and relevance, and proceed with no further discussion of the term. Givnish (1986) has discussed this issue in depth.
Our problem is formally stated as follows: Suppose a plant has a fi nite daily water supply, E t , and a fi nite amount of functional nitrogen in its leaves, N t . (These terms represent, respectively, the transpiration rate per unit leaf area, E, integrated over all leaves in the plant and over one day, and the leaf nitrogen content per unit leaf area, N, integrated over all leaves in the plant). The goal is to maximize the plant's net carbon gain in a single day, A t (the integral of leaf net CO 2 assimilation rate, A, over a day and over all leaves in the plant). Therefore, we wish to know how the control variables, which are stomatal conductance (g sc ) and leaf nitrogen content (N), should vary among leaves (and, for g sc , over the course of the day) in order to maximize A t . Formally, the solution to this problem is the set of functions, g sc (t,L) and N(L), that maximizes A t in the following expression: (1) (In Eq 1, L is cumulative leaf area index, which ordinates the 'position' of a leaf relative to the uppermost leaf layer, L t is total canopy leaf area index, and t is time. Note, however, that the subscript 't' in A t , E t , and N t denotes total rather than time.) The calculus of variations can be used to obtain the following solution (e.g., Wan 1995, p. 461-463): In Eq 3, A L is the mean of A over a single day in a layer L: (Note that Eqs 3 and 4 could also be expressed in terms of the total of A over the day in the layer L, in which case the numerical value of η would simply differ by a factor equal to the daylength.) The Lagrange multiplier lambda (λ) is a constant over the domain of E and g sc (which is time and cumulative leaf area index), and the multiplier eta (η) is a constant over the domain of N (which is cumulative leaf area index). Because A, E and N are state functions with respect to g sc and N (in other words, A, E and N can be calculated knowing the values of g sc and N only at the local point in time and space, and without knowing how g sc and N vary around that point), their derivatives with respect to ∂g sc /∂L, ∂g sc /∂t, and ∂N/∂L are zero. Therefore, the operators Λ and Η degenerate to ∂/∂E and ∂/∂N, respectively, and Eqs 2 and 3 reduce to Eqs 5 and 6, respectively: The equalities in Eqs 5 and 6 are expressed with the distinct notation of 'optimal identities,' because they represent relations that hold at, and only at, the optimum. In general, the quantities ∂A/∂E and ∂A/∂N can differ among leaves, but in an optimal plant, they are everywhere equal to 1/λ and 1/η, respectively. This is discussed in Section 2.4.

Domains of Differentiation
Eqs 5 and 6 are strictly ambiguous, because partial derivatives can be used to express relative changes in two variables when any arbitrary subset of the other variables in the system remain constant. We must therefore be careful to identify which variables can change, and which cannot, as E and N vary in Eqs 5 and 6. These derivatives express gradients in A along domains of 'candidate' values of E (or more precisely, g sc ) and of N. Neither Eqs 5 nor 6 allow any externally imposed environmental variables to vary. (If we believe that leaf gas exchange can signifi cantly affect the atmospheric conditions within the canopy, this assumption must be revised; however, we proceed on the assumption that the ambient humidity, CO 2 partial pressure, air temperature, wind speed, incident irradiance, and absorbed radiation are independent of leaf gas exchange per se.) In Eq 5, we can safely assume that N will not differ among candidate values of g sc at the same point in (t, L) space; however, the leaf temperature, intercellular CO 2 partial pressure, and evaporative gradient may change with E, and therefore must be allowed to vary among candidate values of g sc . For Eq 6, the 'variables' whose constancy or variability we must evaluate are in fact functions of time over a single day, because the optimization domain of N(L) and A subsumes a whole day. In this case, the daily timecourse of stomatal conductance g sc (and thus transpiration rate, E, and all variables that can change with E) may respond to changes in photosynthetic capacity among candidate N values.
This distinction can be confusing, because it is reasonable to ask, When would such variation of g sc with N actually be observed in an optimal canopy? There is no clear answer to that question, because it involves an open physiological problem: how do these allocation controls oper-ate in real plants experiencing changing conditions? Perhaps g sc would be observed to vary with N as nitrogen is reallocated among leaves in response to an externally imposed change in the light environment (for example, if a forest gap opened up nearby due to logging needed to supply optimality theorists with paper on which to print their many equations). We claim no novel insight to answer this question. Our mathematics do not describe the mechanisms or physiological processes by which plants optimise resource allocation; rather, we describe properties of gas exchange in plants that have optimised the use of fi nite resources (by unknown means that do not concern us). Investigators can use the expressions that we provide to infer optimal resource use strategies from abstract models, and to evaluate the proximity of observed states of real plants to those theoretical optima. It is therefore critical to identify precisely the equations that characterize the optima.
These equations specify that the marginal gain in net assimilation rate associated with an increase in either N or E should be uniform over all possible domains. If we assume that the smallest time domain on which N can change is one day, then the marginal gain of assimilation rate with N must allow for the potential response of stomata to the changes in photosynthetic capacity resulting from a change in N. On the other hand, the marginal gain of assimilation with E occurs on a time domain vastly shorter than the rate at which N can respond to anything, so N must be considered a constant when calculating the marginal gain of A with E.
To summarize, both Eqs 5 and 6 assume that external environmental conditions are independent (so their timecourses are constant); Eq 5 allows all physiological variables except N to vary as E varies at a single point in (t, L) space; and Eq 6 allows the timecourses of all physiological variables to vary as N varies at a single point in (L) space. We will denote these refi nements by enclosing the partial derivatives in parentheses and adding subscripts with the names of the variables to be considered constants: Buckley, Miller and Farquhar The Mathematics of Linked Optimisation for Water and Nitrogen Use in a Canopy In these equations, the subscript 'env' refers to all externally imposed environmental variables that are not directly affected by transpiration or assimilation.

Domains of Differentiation: Reprise
Note that the 'optimality criteria' in Eqs 7 and 8 are not explicitly linked, in the sense that optimal water use is defi ned by Eq 7 alone, and optimal nitrogen use is defi ned by Eq 8 alone. In the preceding section we argued, on the physiological basis that nitrogen and water use 'decisions' are made on very different time scales, that the optimally invariant derivative of A with respect to N must allow conductance (and therefore E) to vary. However, the interpretation and analytical calculation of η is complicated greatly by the implicit freedom of E (see Eqs A55 to A60 in the Appendix). It turns out that these complications can be ignored when water and nitrogen are being co-optimized.
(The integral of time in the fi rst expression has been replaced with t dl , the daylength.) The terms in the integrand of Eq 9 may vary with position in the canopy (L), because nothing in the preceding derivation constrains them to invariance among layers. Suppose a small amount of nitrogen, say δN, is moved from one layer (L = L 1 ) to another layer (L = L 2 ) in a canopy where nitrogen and water are used optimally. Using subscripts to denote layers, δN 1 = -δN 2 . This leads to Applying Eq 10 to 9, rearranging, eliminating t dl , and using δN ⋅ dE/dN = δE, we have Because water use is assumed optimal, ∂A/∂E is constant (and equal to 1/λ) and can be pulled out of the integrals on the right-hand side of Eq 11. However, the integrals that then remain, ∫δE 2 dt and ∫δE 1 dt, must be equal and opposite to satisfy the constraint on daily water use (E t ). This causes the right side of Eq 11 to vanish, forcing the left side equal to zero. This has two major implications: (The second result is obtained by removing 1/λ from the integrals on the right side of Eq 9, and decomposing δE back into δN ⋅ dE/dN.) Since layers 1 and 2 are arbitrary, the equalities in Eqs 12 and 13 specify an invariance among all canopy layers. Bringing 1/t dl back into the equations and expressing the integrals as daily average values (indicated by overbars), where nu and zeta (ν and ζ) are constants among layers in a canopy with optimal nitrogen use, in the same sense that η is a constant. Eq 15 is a very important result, because it formally expresses the intuitive notion that stomatal conductance should track photosynthetic capacity (Wong et al. 1979). Informally, this equation might be translated to say, 'If a leaf somehow acquires a bit more nitrogen, then stomatal conductance should also increase a bit, so that the increase in leaf nitrogen content is matched by an increase in transpiration rate.' Alternately, 'if nitrogen is moved from one leaf to another, water [i.e., transpiration rate] should follow.' Furthermore, Eq 15 shows that transpiration rate, rather than conductance per se, should track leaf nitrogen content, rather than photosynthetic capacity per se. Eqs 14 and 15 form a corollary of Eq 8 that applies when total canopy transpiration rate is constant and water use is optimal. The condition in Eq 14 -invariance of ν among canopy layers -may be used interchangeably with invariance of η as a criterion for optimal nitrogen use in such a canopy. In summary, we have shown that the general criterion for optimal nitrogen use is invariance (among canopy layers) in 1/η, the total derivative of A with respect to N (Eq 8). When a fi nite water supply is also being used optimally, invariance in 1/ν, the partial derivative of A with respect to N at constant E, suffi ces to guarantee optimal nitrogen use.

What Exactly Do Lambda and Eta (λ and η) Mean?
This subsection presents an informal synopsis of the problem and its solution (which were presented formally and mathematically in Section 2.1), with the hope of clarifying its solution. The problem is this: An individual plant has a fi nite, fi xed supply of each resource. The water supply is symbolized by E t (meaning total transpiration rate; note that this subscript t denotes total, rather than time) and the nitrogen supply by N t (total nitrogen content). The solution is this: if a small additional amount of either resource becomes available, it should produce the same increase in carbon gain wherever it is invested. The small increase in transpiration rate that causes a given increase in assimilation rate (the 'marginal cost' of carbon in units of water) is represented by the Greek symbol lambda (λ), and the marginal cost of carbon in units of nitrogen is denoted by eta (η).
Three key points help in interpreting these quantities. First, the marginal gains (∂A/∂E and ∂A/∂N) must be distinguished from their numerical values in an optimal plant (λ and η). The marginal gains can be calculated for any leaf in any plant, whether that plant has optimal resource use strategies or not. On the other hand, λ and ν are abstract constructs that can not be measured; they can only be inferred, mathematically, from a model of plant functioning. To clarify this distinction, Eqs 7 and 8 use a special notation -the 'identity' symbol, ≡, with the letters 'opt' written over the symbol -to represent a defi nite equality that holds at (and only at) the optimum, and which therefore defi nes the optimum. This notation may be translated in English to "... is, in an optimal plant, equal to ...".
Second, λ is not a unique function of E t , nor is η a unique function of N t . Both λ and η are complex functions of E t , N t , and the surrounding environmental conditions. It is quite easy to become confused about whether λ and η are constants or variables, so we wish to emphasize the following point: λ and η are constant within a single plant that has fi xed supplies of water and nitrogen and that uses these supplies optimally, but λ and η may vary among plants, and they may vary for a single plant on time scales long enough that E t and N t can change.
Third, if we imagine the plant comparing 'candidate' resource-use strategies in order to fi nd the best one, all of the candidates have the same values of E t and N t , but not necessarily (and not generally) the same values of λ and η. In fact, only one of these candidate strategies (the optimal one) can even be described by a unique pair of values for λ and η, because, by defi nition, one or both marginal gains varies within the plant in all suboptimal strategies.

Gas-Exchange Model
Features that Can Affect Calculated Optimal Strategies

Overview: Interactions between the Economies of Water and Nitrogen Use
How are the optimization of canopy nitrogen and water use linked, that is, why do we suppose that these two processes are fundamentally interdependent? First we must carefully defi ne the scope of this question: we are not asking questions about the effi ciency of resource acquisition or transport, nor of assimilate partitioning. Those issues require treatment of long-term feedbacks between the allocation of what we may call 'primary resources' (water and nitrogen in this case) and allocation of the secondary resource (fi xed carbon) that is procured by the allocation of primary resources. We wish to determine how fi xed supplies of nitrogen and water should be used by the leaves of an individual plant (one 'canopy'), where the goal is to maximize the quantity of carbon gained by using these resources. The question is implicitly confi ned to a time scale on which the resource supplies do not change. For example, the effects of periodic drought or changing litter quality are categorically excluded from this analysis. The 'linkage' question then becomes, How do nitrogen use and water use interact in leaves? To conceptualize the mutual infl uences between nitrogen and water, it is helpful to think of them as currencies used to purchase two commodities (light and CO 2 ) which are then combined in the leaf to yield a product (carbohydrate). However, there is an interesting twist to this analogy. The 'price' of one of the commodities (CO 2 ) decreases as the rate of production increases, making the two currencies somewhat interchangeable: by investing more nitrogen in photosynthetic machinery, the leaf can decrease the cost of carbon in units of water (mechanistically, greater photosynthetic demand draws down c i , the CO 2 partial pressure within the leaf; this increases the diffusion gradient, requiring lower stomatal conductance, and therefore less water use, for a given photosynthetic rate). Conversely, by investing more water, i.e., by opening stomata, the leaf can decrease the cost of carbon in units of nitrogen (higher stomatal conductance increases c i , requiring less nitrogen to sustain a given photosynthetic rate). This means that a plant can, to some extent, substitute the more abundant of these two resources for the less abundant one. This leads to the interesting hypothesis, borne out by some evidence (Field et al. 1983, Reich et al. 1989, Cordell et al. 1999) that nitrogen use effi ciency (NUE, the ratio of CO 2 assimilation rate, A, to nitrogen content, N, or nitrogen fl ux, j N ) and water use effi ciency (WUE, the ratio of A to transpiration rate, E) should be inversely related when the comparison is performed among, rather than within, individual plants.
These nitrogen-water interactions are implicit in accepted small-scale biophysical and biochemical models of leaf gas exchange. Carbon dioxide demand is controlled by nitrogen via investment in photosynthetic enzymes (Evans 1983, Field and Mooney 1986), CO 2 supply is controlled by water via stomatal conductance (Farquhar and Sharkey 1982), and the two controls interact via c i . What remains to be determined, however, is the role of nitrogen-water interactions per se -particularly substitution -in the optimal use of these two resources. Because this optimality problem is defi ned by fi nite supplies of both resources at the whole-plant scale, nitrogenwater interactions and substitution may complicate attempts to extend the optimal control of gas exchange to whole plants (Field 1991).
In this section, we discuss several major features of gas exchange modeling as they relate to the effi ciency of nitrogen and water use. Our goal is to identify features that can have a large infl uence on predicted optimal strategies for nitrogen and water use.

Complex Light Environment
The allocation of nitrogen in proportion to available light (photosynthetic photon fl ux density, irradiance) is a rather hazy problem, because, whereas the irradiance incident on a leaf can change on arbitrarily short time scales (e.g., Hari et al. 1984, Smolander 1984, Korpilahti 1988, Pearcy 1990, leaf nitrogen content probably does not change on time scales shorter than a single day. Light intensity at the top of a canopy varies through a day as the Earth rotates. In addition, upper canopy leaves may intercept beam light that would otherwise reach lower leaves, and as the sun tracks across the sky and shifts the position of these shadows and their complementary sunfl ecks, the canopy creates a temporal distribution of irradiance that is non-uniform on very short time scales for any given leaf. There will generally be at least two modes in this distribution (one for beam light, and one for diffuse and scattered light), and in reality many distinct modes will usually exist (including many penumbra, caused by the sun's angular width, by diffraction, and by interference among other penumbra; Oker-Blom (1986)); for simplicity we will compare unimodal and bimodal distributions, because they represent extreme cases.
The optimal leaf nitrogen content (N) for a bimodal irradiance distribution will not generally appear optimal if evaluated at either of the two modal irradiances, because the optimal value represents a compromise between the two extremes. In particular, if the leaf were always at the 'sunlit' irradiance, the optimal N would be higher than for a leaf that spends time at both irradiances (de Pury and Farquhar 1997). This is illustrated diagrammatically in Fig. 1, which shows that leaves near the bottom of the canopy (where the leaves spend very little time at the higher, sunlit irradiance) have a 'nitrogen defi cit' while they are sunlit, whereas leaves near the top of the canopy (which spend very little time at the 'shaded' irradiance) have an excess of nitrogen while they are shaded. Ironically, this results in an optimal nitrogen distribution wherein upper canopy leaves are more light-limited than lower canopy leaves when both are sunlit, because sunlit lower-canopy leaves are receiving far more light than they can possibly use. The signifi cance of the sunlit/shaded dichotomy decreases as diffuse light comes to dominate (Fig. 1b), as on hazy or cloudy days when the difference between sunlit and shaded irradiance is smaller. In the extreme allocation. Near the top of the canopy, most leaf area is sunlit, but the opposite is true deeper in the canopy (dashed line in the panel at far left). Because the optimal leaf nitrogen content tracks incident irradiance (to some degree), the optimal N is 'pulled' toward the sunlit irradiance at the top of the canopy, but to the shaded irradiance near the bottom of the canopy. As a result, deep-canopy leaves are relatively more nitrogen-limited, when sunlit, than top-canopy leaves, whereas top-canopy leaves have an excess of nitrogen during the brief periods that they are shaded. This effect is larger in sunny conditions (a) than it is in overcast conditions (b), because the difference between the sunlit and shaded irradiances is much smaller in overcast conditions. of a uniformly overcast sky (UOC) there is no beam light and the dichotomy disappears. A multi-modal light environment also complicates prediction of optimal stomatal conductance. If sunfl ecks are suffi ciently short, it is unlikely that stomata can respond fast enough to optimize gas exchange both for sunfl ecks and for 'shadefl ecks' (continuous periods of time spent at the lower irradiance). Furthermore, models used for light penetration through canopies often make unrealistic assumptions about leaf morphology and leaf angle distributions. Deviations from these assumptions may have signifi cant implications under certain conditions. For example, penumbral effects can dominate the character of light transmission within canopies with small or narrow leaves like pine needles (Oker-Blom 1986), essentially eliminating the beam fraction at some depth in the canopy. Other aspects of canopy light penetration, such as spacing between trees and crown shape, may substantially affect the optimal profi le of nitrogen when expressed as a function of cumulative leaf area index. If one wishes to generate nitrogen profi les that are directly, numerically testable, it may be necessary to use a more detailed and accurate treatment of canopy light penetration. Fig. 2 demonstrates how inferred optimal N profi les might depend on whether the same total irradiance is distributed over one or two irradiance fractions. The profi les in Fig. 2 were calculated for the same values of N t and E t , but inferred from a simulation in which separate sunlit and shaded irradiances were used (a, solid line), or the two irradiances were combined into a single irradiance in each layer (b, dashed line); the dotted horizontal lines represent the bottom of each canopy, and therefore their total LAI's (L t ). The combined irradiance in (b) was equal to the sum of the sunlit and shaded values, weighted by the proportion of leaf area in each light fraction. Resource supplies were the same for both profi les (N t = 211 mmol N m -2 ground and E t = 511 mol H 2 O m -2 ground ), so the areas under each profi le are also the same (and equal to N t ). (Simulations used the canopy light penetration model of de Pury and Farquhar (1997), with latitude = -20°, longitude = 152°, day = 01 Jan, atmospheric transmissivity = 0.8, daylength = 13.2 h; air temperature (T a ) varied sinusoidally between 20°C at sunrise to 30°C at solar noon.) one profi le was inferred using a light model that distinguished between 'sunlit' and 'shaded' fractions of each canopy layer (de Pury and Farquhar 1997). For the other profi le, the two light fractions were combined into a single value for each canopy layer, equal to a weighted average of the sunlit and shaded irradiances (the weights were the proportions of leaf area occupied by each light fraction in the sunlit/shaded model).
The total absorbed irradiance of each layer was therefore identical for the two profi les. As Fig. 2 shows, when the same total irradiance is distributed between two irradiance fractions rather than one, the optimal profi le has more nitrogen near the top of the canopy (and therefore less total leaf area as well).

One-Phase vs. Two-Phase Models of Leaf Photosynthesis
Many previous treatments of canopy nitrogen optimization used simplifi ed models for photosynthesis -often a single hyperbolic function in which both nitrogen and light directly limit photosynthesis at all values of c i , the intercellular CO 2 partial pressure (Caldwell et al. 1986, Hirose and Werger 1987, Gutschick and Wiegel 1988, Schieving et al. 1992, Schieving and Poorter 1999. However, at suffi ciently low c i , photosynthesis is limited by RuBP carboxylation, and therefore by Rubisco activity. Because Rubisco activity is limited by leaf nitrogen content but not directly by light, photosynthesis is essentially not light-limited at low values of c i . Conversely, at high c i the electron transport rate limits carbon fl ux through the Calvin cycle, and this process is limited by both nitrogen and light. Furthermore, the nitrogen dependencies of these two phases are quantitatively different, even when light is saturating (Evans 1983, Evans 1989). Fig. 3 shows how assimilation rate, A (Fig.  3a) and its marginal gain from N investment, ∂A/∂N (Fig. 3b) vary with c i for both of these phases. Also important is the relationship between c i and k, the slope of the A vs c i relationship (sometimes called 'mesophyll conductance'). The value of k plays a central role in predicting optimal stomatal conductance, because k is a critical determinant of λ (Eq A40) (Cowan and Farquhar 1977). Fig. 3c shows how k varies with c i for the two phases of the model of Farquhar et al.1980b). Two-phase models have a fundamentally different shape for k vs c i than single-phase models because the shift from one phase to the other creates an infl ection and translation (Fig. 3c), whereas the k vs. c i curve for a single-phase model has uniformly positive curvature (cf. the curve marked 'k J ' in Fig 3c). As a result, k is most sensitive to c i (and therefore to stomatal conductance) at intermediate values of c i in a two-phase model, but at low values in a single-phase model. This distinction -two regimes in the response of photosynthetic capacity to nitrogen -dramatically impacts calculations related to the economy of nitrogen use. The two-phase treatment is particularly important for accurate analysis of nitrogen-water interactions, because the transition between these two phases is parameterized by c i , which is affected by stomatal conductance and is therefore infl uenced by water use. (The model developed in the present paper uses the photosynthesis model of Farquhar et al. (1980b) to account for these two different regimes.) One potential error arises if the sunlit/shaded dichotomy is not taken into account when using a two-phase photosynthesis model. Fig. 3d shows the demand curve and the optimal supply line (defi ned by the optimal stomatal conductance) for each of two irradiances (300 and 1200 µmol photons m -2 s -1 ). Imagine for a moment that this leaf actually spends half of its time at each of these two irradiances. Its average irradiance is therefore 750, and its average assimilation rate and stomatal conductance are the average of the high-and low-irradiance values. Imagine now that we wish to estimate gas exchange by calculating A and g sc at the average irradiance of 750. Because photosynthesis in this leaf is nearly light-saturated at a irradiance 750, the demand curves at 750 and 1200 are very similar, and so are the optimal conductances. As Fig. 3d shows, the result would be an overestimation of the average values of both A and g sc by more than 25%. This example demonstrates how it can be dangerous to mix levels of detail. It may seem intuitive that switching from a single-irradiance/ single-phase model to a single-irradiance/twophase model should improve the model's accu- 1983, Vogelmann et al. 1989, Cui et al. 1991, Han et al. 1999, Vogelmann and Han 2000. It is intuitively clear that more nitrogen should be allocated per unit of light-absorbing chlorophyll at the top of the leaf than at the bottom, but it is not clear how this allocation problem is actually solved in real leaves. This issue may appear to be of interest only to sub-leaf-scale physiologists. However, it has a dramatic impact on calculated optimal N profi les, because it determines how the whole-leaf potential electron transport rate (J, Eqs A9, A13, and A17) depends on leaf N racy, but this assumption is not necessarily warranted.

Gradients of Nitrogen and Light within Leaves
Light attenuation occurs not only at the level of whole leaves, but also at a smaller scale within leaves (Terashima and Saeki 1983). Attenuation of light by chlorophyll creates a vertical gradient of irradiance within leaves (Terashima and Saeki content (N) and incident irradiance (I). Fig. 4 shows optimal N profi les inferred using two different models (described below) to predict J from N and I. One hypothesis is that N is allocated among 'layers' of chlorophyll in direct proportion to the irradiance absorbed by each layer within the leaf. This idea is an extension of the 'superleaf' hypothesis (Field 1991), which states that light and nitrogen are equally limiting everywhere in a plant. The standard model for J (Eqs A11-A3) is implicitly based on this hypothesis, because it does not distinguish between layers in the leaf. However, at low irradiances, fewer photons are absorbed every second per unit of chlorophyll (and hence per unit of N invested in light capture), even though each photon can drive the same electron fl ow per unit of N invested in electron transport. To account for the nitrogen cost of light capture, Badeck (1995) formalized an alternative model (Eqs A14-A17; Fig. 5). In contrast to the standard model, Badeck's model assumes there is no N gradient within the leaf, so chloroplasts near the sunlit surface of the leaf are more nitrogenlimited than those near the bottom of the leaf (Fig. 5).
The weight of empirical evidence suggests that, under controlled conditions, leaves allocate nitrogen internally in parallel to the light gradient (e.g., Ogren and Evans 1993, Evans 1995, Evans 1999, consistent with the superleaf-based model. However, leaves that fl icker or rotate rapidly (e.g., Populus tremuloides), or are oriented vertically, may be just as likely to receive irradiance from either surface; in such leaves, a monotonic internal N gradient might be less effi cient than no gradient at all, in which case Badeck's model would be more appropriate than the standard superleaf-based model. Finally, within-leaf light gradients may change when leaves are illuminated by diffuse irradiance rather than by collimated or 'beam' irradiance (Ustin et al. 2001), further confusing the issue. More study seems to be warranted on this topic.

Effects of Leaf Temperature
Many previous analyses of canopy nitrogen allocation have assumed leaf temperature to be constant for simplicity (e.g., Hirose and Werger 1987, Gutschick and Wiegel 1988, Werger and Hirose 1991, Schieving and Poorter 1999. However, nitrogen and water are tightly linked by thermal effects, because changes in leaf temperature (T l ) caused by transpiration (via evaporative cooling) can signifi cantly affect the performance of photosynthetic enzymes Farquhar et al. 1980b, Brooks and Farquhar 1985, Harley et al. 1992, Bernacchi et al. 2001, which in turn affects the use-effi ciency of nitrogen invested in those enzymes. The resulting effects are not easy to intuit. First, variations in T l around a baseline value can either increase or decrease photosynthetic performance, depending on whether the baseline is above or below the optimum. Second, the effect of transpiration on T l is likely to increase with depth in the canopy, because lower leaves are more sheltered from convective heat exchange with the atmosphere than are upper leaves (Roberts et al. 1990). Third, it is possible for T l effects to critically change the topology of the relationships between assimilation rate and transpiration rate (Cowan andFarquhar 1977, Buckley et al. 1999), which in turn affects the mathematics of optimization. A full iterative solution of leaf energy balance and a treatment of intracanopy aerodynamics should help to account for these effects. Optimal N profi les can be predicted with and without these details to estimate the importance of thermal effects. For example, Fig. 6 compares an optimal canopy N profi le in which T l was determined by iterative solution of an energy balance formula (Eq A19), with the  (c), respectively. In (a) and (b), some light is absorbed but not used, because the electron transport capacity is saturated in some layers; conversely, in (b) and (c) some nitrogen is not fully utilized, because the local irradiance is too low.
profi le predicted for the same resource supplies when T l was set equal to 25°C. The isothermal profi le has higher leaf N contents in each layer, and as a result, its total LAI is lower, because the N supply is more concentrated in upper layers.

Discussion and Conclusions
The purpose of this paper was to develop a theoretical and mathematical framework for studying linked optimization of canopy nitrogen allocation and water use. In this section, we summarize the analyses and conclusions of the preceding sections, we suggest questions that may be addressed by this framework, and we conclude with a brief discussion of the role of optimality theory in plant ecophysiology.

Summary
We have stated the abstract mathematical solution for optimal water and nitrogen use in an individual plant canopy. Simply, the general solution is that net carbon gain should respond by the same amount, in any leaf, to a small increase in resource investment -the 'marginal gain,' to use economic terminology, should be constant and uniform. This is true for both water and nitrogen investments, but for nitrogen, the relevant timescale should average net carbon gain over a day or more (because nitrogen cannot be translocated very rapidly). However, although the numerical values of these marginal gains (denoted λ and η for water and nitrogen, respectively) are constant within an optimal plant, they are not, in general, constant over any broader domains. For example, λ and η will differ among plants with different water and nitrogen supplies (E t and N t , respectively), different physical environments (e.g, radiation regimes, humidity, carbon dioxide supply, and air temperature), or different physiological constraints (e.g., C 3 vs C 4 photosynthetic pathways). Several factors must be considered when attempting to infer optimal resource use strategies -i.e., when attempting to translate values of λ and η into patterns of water and nitrogen use within a plant. For example, to fi nd the values of stomatal conductance (g sc ) and leaf N content (N) that produce the correct λ and η, many 'candidate' values of g sc and N must be compared. This is accomplished by calculating the appropriate marginal gain at each candidate value. However, the marginal gains are partial derivatives in which some variables must be held constant while others are allowed to vary. Candidate g sc values must be compared at the same N and incident irradiance, but anything directly affected by g sc (including transpiration rate, and therefore leaf temperature and photosynthetic parameters) must be allowed to vary among candidate g sc values. Candidate N values must allow g sc (and anything directly affected by it) to vary, because in practice, stomata respond to things affected by N (such as photosynthetic capacity) much more quickly than N itself can change. (However, further mathematical analysis shows that this latter requirement can be dropped if water use is known to be optimal.) In the Appendix, we have derived general formulas for these marginal gains, using a standard model of leaf gas exchange and energy balance.
Some physiological and physical factors can signifi cantly affect the optimal resource use strategies inferred from any particular model. These include: (a) the use of a canopy light penetration model that accounts for sunlit and shaded fractions of each canopy layer (or more generally, for more complex penumbral effects), (b) the use of a biphasic photosynthesis model, in which the economies of nitrogen and water use depend differently on intercellular CO 2 partial pressure (c i ) in each phase, (c) the model chosen for wholeleaf potential electron transport rate, on the basis of hypotheses concerning nitrogen and light gradients within leaves, and (d) the effects, on leaf temperature, of evaporative cooling and variation of convective coupling with depth in the canopy. We used preliminary simulations to demonstrate the potential effects of these factors on inferred canopy N profi les (using a complete canopy model that is not presented here). Future work will investigate these factors in depth.

Possible Applications of the Mathematical Framework
Rigorous analysis of resource-use effi ciency and interactions between resources, facilitated by the mathematical framework developed in this study, can help investigators interpret and predict characteristics of gas exchange. This is not a trivial problem because the response of carbon gain to the investment of any single resource depends on scale, and on the status of other limiting resources. Ideally, the effi ciency with which a resource is used should be measured at a scale where the resource limitation is uniquely defi ned. For example, water use effi ciency (the ratio of carbon gain to water use) is often inferred from its relationship to certain measurable small-scale quantities, such as leaf gas exchange variables or carbon isotope discrimination signals in isolated tissues (Farquhar and Richards 1984). However, because resource constraints apply at the scale of whole plants, rather than individual leaves, an arbitrarily localized proxy measurement for water-or nitrogen-use effi ciency (WUE or NUE) will not necessarily be uniquely related to the associated canopy-scale effi ciency. For example, a set of trees may be ranked in a certain order with respect to whole-tree WUE (A t /E t ), but the local WUE (A/E) measured in upper-canopy leaves may rank the trees differently. Similarly, stomatal conductance is known to be correlated with photosynthetic capacity along some domains Wong et al. 1979), but the sense of those correlations need not hold among parallel loci in different canopies.
The mathematical framework developed in this paper can be used to predict relationships between various whole-plant and individual-leafscale properties of gas exchange, in order to identify easily-measured but robust estimators of the large-scale properties of interest. This model may also be able to predict useful relationships among whole-canopy variables such as assim-ilation rate, leaf area index, and water and nitrogen supplies. Any relationships predicted for canopy-scale variables could help explain observed ecological patterns, or predict patterns not yet observed. Furthermore, relationships linking whole-plant carbon gain to whole-plant resource supplies could become powerful scaling tools, because they would allow gross primary productivity to be easily predicted from mappable data that are related to resource limitations (e.g., soil parent material, site water balance, or NDVI).

Optimization as an Alternative to Traditional Reduction
It is our view that optimality can be a powerful paradigm for constructing ecological models. As an ecological principle for predicting or interpreting plant gas exchange, optimality derives its utility mainly from two features. First, identifying an optimal pattern reduces the system of interest to a small number of degrees of freedom by eliminating the infi nite complement of suboptimal patterns. Second, an optimal pattern represents a logical extreme towards which phenotypes in the natural system under study may be reasonably hypothesized to have evolved. In other words, optimization uses the closest thing biology has to a scientifi c law -natural selection -to distill a complex system into an essential kernel of arbitrary simplicity. This kernel can then be used as a basis in the truest mathematical sense. However, simplifi cation, or the elimination of complexity, is more commonly achieved in science by analytical reduction (i.e., linearisation of mathematical models, which assumes complex dynamics do not emerge from essential aspects of mechanisms), empirical reduction (elimination of noise by statistical modeling, which assumes complexity is random), or scale reduction (identifi cation of some arbitrarily small scale as the unique locus of mechanistic control, and ignoring processes that only arise, and are thus only defi ned, at larger scales). Reduction is, in some sense, the mother of all null models. Not only does it simplify; it demystifi es the process of simplifi cation so much that it is the fi rst recourse of investigators faced with a complicated system.
In other words, reduction has become an implicit paradigm. This is probably due in part to its success in other sciences (many physical systems are often well-described by a linear basis that is polluted only by random variation or by fundamentally separable systematic effects), but it is also popular because it is conceptually simple.
Consider, for example, Brownian motion. While certainly interesting as an emergent phenomenon, it can be described by a few physical laws that apply universally to each member of an enormous population of essentially identical objects. That is, the dynamics of particles are created by interactions that are external to those particles, so the two features -object and force -can be mathematically separated and thus described by a linear basis. A canopy of leaves, on the other hand, may be a collection of essentially similar objects in some respects, but an understanding of canopy properties requires consideration of the whole canopy as a single coordinated object with internal dynamics. (In the canopy, internal coordination derives from the need for leaves to share a common resource pool). Therefore, if the goal is to understand canopy-level phenomena, it is not suffi cient to divide the canopy into independent units of a defi ned scale. Those units must be integrated by metascale processes, such as the allocation of a common resource pool.
Were there no resource limitation, the canopy would have neither reason nor power to tyrannize its citizen leaves. It is the force of competition (interspecifi c, intraspecifi c, or even intrasomal) and the constraints of history (chiefl y genetics and environmental change) that ecology adds to the sometimes separable forces of physical law. This renders biological dynamics essentially nonlinear. Thus linearisation may be not only inadequate, but quite inappropriate as a tool for simplifying biological systems. Optimization, despite its own bugaboos, may be a more viable tool.
Furthermore, many physical systems are quite interesting at equilibrium, when linear models are most unassailable; at worst, their equilibrium behavior is often explained by forces in the direction of an unambiguous equilibrium state. Ecological systems, on the other hand, are fundamentally historical -they generally have no meaningful equilibrium or steady-state condition.
This may be partly because their internal dynamics are very complicated (in which case there may be many relatively unstable equilibrium), and it may also be because they are embedded in a much larger system with complex dynamics across many scales of space and time (in which case there may be no equilibrium at all). The present state of an ecological system often makes sense only in the context of a convincing story explaining how that state came to be.
One may justifi ably argue that in implementing optimization, we employ reductionist techniques. Ironically, this argument, in its logical dissolution, clarifi es our thesis. It is true that we write down simple equations based on separation of processes -we linearise the physical system at some scales. However, we also link these processes with physical models of their interactions, thereby explicitly integrating these processes. Where reduction fails is as a tool for interpreting the behavior of these integrated models. Many ecologically important phenomena will be lost or misinterpreted if we attempt to simply the behavior of an integrated system by reduction, i.e., dis-integration. Optimality offers another method of fi nding the essence of a natural system -the modal behavior from which it tends to deviate unpredictably and perhaps meaninglessly -while preserving the historical basis of all long-term biological dynamics: natural selection. Total of 60 references Table 1. General parameters and variables.

Name Units
A net CO 2 assimilation rate of leaf µmol CO 2 m -2 leaf s -1 E transpiration rate of leaf mmol H 2 O m -2 leaf s -1 N leaf functional N content mmol N m -2 leaf g sc (g sw ) stomatal conductance to CO 2 (H 2 O) mol air m -2 leaf s -1 L cumulative leaf area index m 2 leaf m -2 ground t time s A t net CO 2 assimilation rate of plant µmol CO 2 m -2 ground s -1 E t transpiration rate of plant mmol H 2 O m -2 ground s -1 N t plant functional N content mmol N m -2 ground L t whole-plant cumulative leaf area index m 2 leaf m -2 ground λ invariant value of (pE/pA) N in optimal plant mmol H 2 O µmol -1 CO 2 η invariant value of pN/pA in optimal plant mmol N µmol -1 CO 2 s ν invariant value of (pN/pA) E in optimal plant mmol N µmol -1 CO 2 s leaf (air) temperature K g bc (g bw ) boundary layer conductance to CO 2 (H 2 O) mol air m -2 leaf s -1 g tc (g tw ) total conductance to CO 2 (H 2 O) mol air m -2 leaf s -1 ω d g tc /dg tw unitless g bh boundary layer conductance to heat mol air m -2 leaf s -1 c i (c a ) intercellular (ambient) CO 2 mole fraction µmol CO 2 mol -1 air w i (w a ) intercellular (ambient) H 2 O mole fraction mmol H 2 O mol -1 air V m maximum RuBP carboxylation rate µmol CO 2 m -2 leaf s -1 J potential electron transport rate µmol em -2 leaf s -1 J m maximum potential electron transport rate µmol em -2 leaf s -1 I a useful irradiance absorbed by PSII (αFI) µmol photons m -2 leaf s -1 α leaf absorptance unitless I irradiance (photosynthetic photon fl ux density at leaf surface) µmol photons m -2 leaf s -1 F (fraction of αI absorbed by PSII)(quantum yield) unitless θ A (θ J ) colimitation factors: unitless

A.1.1 CO 2 Assimilation
To develop analytical expressions for the marginal gains (∂A/∂E and ∂A/∂N; Eqs 5 and 6), we must defi ne the function A and its derivatives in a precise manner. At steady-state, ignoring diffusive resistance within the leaf, the assimilation rate A must simultaneously satisfy a 'supply' equation (A = A s ) describing CO 2 fl ux from the atmosphere to the site of carboxylation, and a 'demand' equation (A = A d ) describing the fl ux of CO 2 that is RuBP carboxylation: where A V and A J are defi ned by Eqs A8 and A9 below. The hyperbolic minimum function, minh, is the negative root of a quadratic expression given by: In Eq A3, θ is a convexity term, ranging from zero to one, that controls the degree of co-limitation between x and y when their values are close. If θ is close to unity, there is little colimitation, and the transition between x and y is very sharp. In Eq A2 θ is subscripted with 'A' to distinguish it from an analogous term, θ J , in an expression for potential electron transport rate, J; see Eq A13.
It is convenient to identify c i as a parameter linking CO 2 supply and demand (Eqs A1 and A2), and to express it as a function of all independent variables in the supply and demand equations (removing c a as a constant): Total conductance to CO 2 is a function of stomatal and boundary layer components: In some cases, we will fi nd it more convenient to differentiate Eq A6; in some cases, Eq A7 is more easily analysed. With all dependencies thus contained in a single equation, one can construct mathematical expressions for any partial derivative. In Eq A2, The maximum velocity of RuBP carboxylation, V m , is generally taken as a linear and homogeneous function of whole-leaf nitrogen content, N: A.1.1.1 Modeling Potential Electron Transport Rate As described in Section 3, the potential electron transport rate, J, can be modeled in more than one way. The most common model calculates J as the lesser of J m , the maximum potential electron transport rate, and I a , the useful irradiance absorbed by Photosystem II. Importantly, J m and I a are calculated from whole-leaf parameters: J m is assumed linearly proportional to whole-leaf N content, N, and I a is the product of incident irradiance, I, with absorptance, α, and a spectral correction factor, F: However, the leaf absorptance, α, is determined by leaf chlorophyll content, c L , which is also limited by nitrogen. Badeck (1995) developed an alternative model that calculates J by integrating the local potential electron transport rate per unit of chlorophyll, j, over all 'layers' of chlorophyll. The layers are ordinated by cumulative chlorophyll density, c, so that the value of c in the 'lowest' layer is c L (which Badeck assumed is proportional to whole-leaf N by a constant: c L = χ c N). Chlorophyll is assumed to absorb irradiance in proportion to its local intensity, so the useful irradiance absorbed by PSII declines exponentially with c with an extinction coeffi cient k c : Note that the leaf absorbance can easily be calculated from Eq A14 as However, Badeck also assumed that the amount of nitrogen allocated to electron transport per unit of chlorophyll, and therefore also j m , the maximum potential electron transport rate per unit of chlorophyll, is the same for all layers (and it can easily be shown that j m = χ j /χ c ). As a result, in any given layer, j can be limited either by nitrogen or by light: where J m and I a are defi ned by Eqs A11 and A12, with leaf absorbance α given by Eq A15. The term I* is the incident irradiance at which the light-and nitrogen-limitations are exactly balanced in the uppermost layer (i.e., j m = i a (0)). I* is proportional to the whole-leaf electron transport capacity per unit of chlorophyll, which, in Badeck's model, is constant and therefore independent of leaf N content. If the actual irradiance, I, is large enough to saturate the electron transport capacity of the lowest layer, then all layers are nitrogen-limited and J = J m ( Fig. 5a; part (a) in Eq A17). Conversely, if I is lower than I*, then all layers are light-limited ( Fig. 5c; part (c) in Eq A17). For intermediate values of I, a transition from N-to light-limitation occurs within the leaf ( Fig. 5b; part (b) in Eq A17).

A.1.2 Transpiration and Energy Balance
Transpiration (E) is the product of total conductance, g tw , and the gradient of water vapor mole fraction from the intercellular spaces (w i ) to the ambient air (w a ): Transpiration and assimilation are linked by stomatal conductance (which is to be optimized, and is thus unconstrained) and by energy balance, which constrains T l . We present an energy balance expression, which is an implicit function for T l , because we will need to differentiate it to account for thermal effects on ∂A/∂E in the next section: In Eq A19, Φ is absorbed solar radiation, T a is air temperature, σ is the Stefan-Boltzmann constant (5.67 10 -8 J m -2 K -4 s -1 ), e atm is the atmospheric transmissivity, l is the latent heat of vaporization (18.01 J mol -1 H 2 O at 25°C), and c pa is the heat capacity of air (29.25 J mol -1 air K -1 at 25°C). The term f L decreases with depth in the canopy (i.e., with increasing L), because lower leaves are less radiatively coupled to the sky. Both f L and Φ are calculated from the detailed canopy light penetration model, and formal expressions for Φ and f L , and also for ε atm and g bh , can be found in Leuning et al. (1995) and de Pury and Farquhar (1997).

A.2 Derivation of an Expression for ∂A/∂E at Constant N
Eq 5 says that a plant is making optimal use of a fi nite, fi xed water supply if the marginal gain of assimilation rate with respect to transpiration rate is uniform and constant. We proceed below to identify a formula for this marginal gain: In Eq A20, the partial derivative at constant N and constant environmental conditions has been rewritten as a total derivative for clarity in the ensuing derivation; in essence, we have redefi ned Eqs A6 and A7 as follows: The transpiration rate does not depend on N explicitly, so it is not necessary to redefi ne E for this derivation. The term ω relates the differentials of g tc and g tw , accounting for the ratio of H 2 O and CO 2 diffusivities through stomata (1.6): The partial derivatives in Eq A24 are given by: The last term, λ , is the limiting, minimum value that λ would approach if the slope of the demand curve, k, approached infi nity. The meanings of λ (and its counterpart for nitrogen, ν , defi ned by Eqs A50 and A54 below) are clarifi ed by Fig. 7. λ may be thought of as the lowest possible cost of carbon in units of water (measured at a given c i ). The actual cost, λ, is larger because, when an increase in conductance provides more CO 2 , some of that carbon never gets fi xed -instead, it stays in the intercellular spaces (where it increases c i ), because the photosynthetic apparatus has fi nite effi ciency.

A.3 Derivation of an Expression for ∂A/∂N at Constant E
Optimal nitrogen use is achieved when the sensitivity of assimilation to leaf nitrogen is uniform throughout the canopy. It is not possible for this sensitivity to be constant in time, because nitrogen can not be moved among leaves with anywhere near the speed that would be required to track diurnal changes in irradiance. Therefore, the gain function in this case is actually the daily integral, or the time-average over one day, of assimilation: We require a formula for the response of daily-averaged A to leaf N content, N, at constant transpiration rate, E. When this derivative is invariant, we symbolize its numerical value by 1/ν. Pulling the differential operator d/dN into the integral is permitted because the domain of differentiation is orthogonal to the domain of integration (i.e., dN/dt is zero). This simplifi es the problem somewhat, in that we can fi nd the 'local' derivative dA/dN for analytical purposes, and integrate it only for computational or graphical implementation. We begin by redefi ning the assimilation functions appropriate to the domain of optimization. In this case, dA/dN compares among candidate values of N at one depth (more precisely, one value of cumulative leaf area index, L) in the individual plant canopy, and the timecourse of stomatal conductance is invariant among candidate N values: Differentiating the supply and demand functions, recognizing that leaf temperature has no explicit dependence on N, and setting the results equal, we have the following: δN/ν (where ν is the sensitivity of A to N at constant E, rather than at constant c i ; Eq A54) is the gain in assimilation that would be realized if no change occurred in the supply curve (shown as a fainter straight line), that is, if stomatal conductance did not change with leaf N. δ ν N /ˆ is the 'potential' gain -the gain that would be realized if the supply curve's fi nite slope did not cause c i to decrease as N increased. Similarly, in (b), supply curves are shown before and after an increment in E, and the corresponding gain and potential gain (the latter not limited by the demand curve's fi nite slope) are δE/λ and δ λ E /ˆ, respectively.
Solving these for dc i /dN and substituting the result into the second part of the preceding equation, we have The relative responses of V and J to N (dlnW/dN in Eq A50) can be calculated from Eqs A10-A17. From Eq A10, dlnV/dN is simply 1/N. However, dlnJ/dN is different for the superleaf (Eqs A11-A13) and Badeck (Eqs A14-A17) models of potential electron transport rate. For the superleaf model, dlnJ/dN is calculated by differentiating the hyperbolic minimization function (Eq A3) with respect to N, solving for dJ/dN, and dividing by J: (The terms I* and α in Eq A52 are defi ned in Eqs A15 and A17.) As discussed in the text, invariance of dA/dN on short time scales is not generally possible, so the optimal criterion is actually invariance in the derivative of A averaged over a day. Because N is constant within a day, the derivative can enter the integral: The quantity 1/ν clearly increases in a saturating fashion as the water limitation diminishes (that is, as conductance increases). The partial derivative on the right side of Eq A53 thus represents an extreme value of 1/ν wherein water is not at all limiting (g tc → ∞) -i.e., it is the value of dA/dN that would obtain if g tc could increase arbitrarily to match any increase in photosynthetic demand, thereby keeping c i constant. We denote this limiting value by ν , and as before, replace g tc with g for clarity. ν is explained graphically by Fig. 7

A.5 Numerical Solution Techniques Used to Generate Figs. 2, 4 and 6
Optimal canopy nitrogen profi les were inferred using the model described above, in conjunction with a canopy light penetration model (de Pury and Farquhar 1997) and a simple model of intra-canopy aerodynamics (Leuning et al. 1995), as follows. The canopy was discretised in space, with layers 0.05 m 2 leaf m -2 ground thick (evaluated in the middle of each layer, i.e., at L = 0.025, 0.075 and so forth), and in time, with timesteps equal to one twelfth of the daylength. The daylength, which depends on geographic location and time of year, was calculated from equations presented by de Pury and Farquhar (1997). Linear summation was used to integrate all relevant variables (the values of each variable at each point were summed, multiplied by the ratio of the stepsize to the total domain).
The optimal values of g sc and N were calculated by a nested iterative solution procedure, as follows. (a) Approprite values of λ and η were arbitrarily chosen. (b) For one canopy layer (one value of L), a 'candidate' value of leaf N content (N) was posited. (c) For one timestep within this layer (one value of t), a candidate value of stomatal conductance (g sc ) was posited, and leaf temperature (T l ) was determined by iterative solution of energy balance (Eq A19) using values for incoming radiation, boundary layer conductance, and air temperature calculated from the radiation and aerodynamics submodels of de Pury and Farquhar (1997) and Leuning et al. (1995) (T l was adjusted until the ratio of the right-and left-hand sides of Eq A19 was within 10 -6 of unity). (d) (∂A/∂E) N was calculated for these values of t, L, N, and g sc , and T l , using Eq A40. (e) Steps (c) and (d) were repeated for different values of g sc until the ratio of (∂A/∂E) N and the chosen 'target' value of 1/λ was within 10 -5 of unity. (f) Steps (c) through (e) were repeated for each timestep within the layer, and the daily mean assimilation rate was calculated. (g) Step (f) was repeated for a slightly larger value of leaf N content (N (1+10 -10 )) to estimate ∂A/∂N numerically. (h) Steps (c) through (g) were repeated for different values of N until the ratio of ∂A/∂N and the chosen target value of 1/η was within 10 -2.5 of unity. (i) Steps (b) through (h) were repeated for all canopy layers, and the resulting canopy totals for assimilation, transpiration, and nitrogen content (A t , E t and N t , respectively) were calculated by linear integration.
Numerical values of λ, η and L t that yielded pairs of canopy profi les with nearly identical values of E t and N t from two different versions of the model (e.g., superleaf-or Badeckbased models, as for Fig. 4; see the legends for Figs. 2, 4 and 6 for details) were identifi ed as follows: (j) The entire procedure described in the preceding paragraph was repeated for many values of λ and η for each of the two model versions being compared. (k) For each 'canopy' thus generated, N t and E t were plotted against each other (with L t being the implicit variable). (l) Many pairs of N t vs E t curves (one for each of the two model versions being compared) were overlaid until an intersection point was found. At this intersection, the two model versions produced canopies with nearly identical values for both E t and N t (generally with different values of L t for each model version), allowing the two model versions to be be compared for the same resource constraints.