Dynamically optimizing stomatal conductance for maximum turgor-driven growth over diel and seasonal cycles

Abstract Stomata have recently been theorized to have evolved strategies that maximize turgor-driven growth over plants’ lifetimes, finding support through steady-state solutions in which gas exchange, carbohydrate storage and growth have all reached equilibrium. However, plants do not operate near steady state as plant responses and environmental forcings vary diurnally and seasonally. It remains unclear how gas exchange, carbohydrate storage and growth should be dynamically coordinated for stomata to maximize growth. We simulated the gas exchange, carbohydrate storage and growth that dynamically maximize growth diurnally and annually. Additionally, we test whether the growth-optimization hypothesis explains nocturnal stomatal opening, particularly through diel changes in temperature, carbohydrate storage and demand. Year-long dynamic simulations captured realistic diurnal and seasonal patterns in gas exchange as well as realistic seasonal patterns in carbohydrate storage and growth, improving upon unrealistic carbohydrate responses in steady-state simulations. Diurnal patterns of carbohydrate storage and growth in day-long simulations were hindered by faulty modelling assumptions of cyclic carbohydrate storage over an individual day and synchronization of the expansive and hardening phases of growth, respectively. The growth-optimization hypothesis cannot currently explain nocturnal stomatal opening unless employing corrective ‘fitness factors’ or reframing the theory in a probabilistic manner, in which stomata adopt an inaccurate statistical ‘memory’ of night-time temperature. The growth-optimization hypothesis suggests that diurnal and seasonal patterns of stomatal conductance are driven by a dynamic carbon-use strategy that seeks to maintain homeostasis of carbohydrate reserves.


Introduction
It has long been theorized that stomata optimally use resources to maximize photosynthetic carbon assimilation (A n ; see Table 1 for terminology ;Cowan 1977;Cowan and Farquhar 1977).We refer to these and similar hypotheses that optimize A n in relation to some metric of water stress (e.g.Wolf et al. 2016;Wang et al. 2020) as assimilation optimization hypotheses (AOHs).Assimilation optimization hypotheses predict the optimal stomatal conductance (g w ) as the conductance at which the marginal carbon profit of water (λ = dA n /dE, where E is transpiration) and marginal carbon cost of water (χ w ) are equal, unless the maximum χ w exceeds the maximum λ, in which case g w = 0 (Cowan and Farquhar 1977;Wang et al. 2020; Fig. 1), and stomata close as λ increases.λ is property of gas exchange and photosynthesis and is independent of optimization (Buckley et al. 2017; see Supporting Information-Equation S1.3.10), while χ w emerges from finding the mathematical solution to the optimization problem given the assumed costs or constraints on the AOH (Wang et al. 2020;Potkay and Feng 2023;Equation 3).Estimates of λ are sensitive to how A n and leaf dark respiration (R d ) are modelled (Fig. 1; see Supporting Information-Fig.S1).The inclusion of non-stomatal limitations (NSLs) to photosynthesis (decreased mesophyll conductance and/or photosynthetic capacities; here modelled as reduced apparent photosynthetic capacities under negative leaf water potentials; Zhou et al. 2013;Novick et al. 2016;Dewar et al. 2018) causes daytime λ to decline faster with g w than without NSLs (Fig. 1).For example, formulating R d as a function of leaf relative water content (RWC L ; Flexas et al. 2005;Galmés et al. 2007), here called hydraulically regulated respiration (HRR), nearly doubles night-time λ (when A n = −R d ; Fig. 1), since excessive transpiration dehydrates leaves, decreasing R d (increasing A n ; see Supporting Information-Fig.S1).When resources are scarce and needed, their costs (χ w ) are high, leading to more conservative use (stomatal closure; Fig. 1).For example, stomata should not transpire a finite store of soil-water (e.g.sourced by episodic rain events) rapidly; otherwise, it depletes the soil-water available for future transpiration, reducing the total carbon gain through legacy effects (Buckley et al. 2017;Feng et al. 2022).The optimal strategy would be to transpire all the available water, depleting soil-water just as the next rain event either will imminently occur (Manzoni et al. 2013) or is probabilistically 'anticipated' (Cowan 1986;Mäkelä et al. 1996).This soil-water-saving strategy is often referred to as a water-use strategy (Manzoni et al. 2013;Mrad et al. 2019).
Recently, Potkay and Feng (2023) theorized that stomata have evolved to maximize turgor-driven growth over entire lifetimes, which we refer to as the growth-optimization hypothesis (GOH).The GOH expands on AOHs by not only considering the optimal coordination between g w and gas exchange but also non-structural carbohydrate (NSC) reserves (C) and turgor-driven growth (G).Though some AOHs are conceptually founded upon growth maximization, they mathematically couple growth to photosynthesis (treating G ∝ A n ; Givnish and Vermeij 1976;Cowan 1982;Friend 1995), ignoring NSC storage and limitations.The GOH emphasizes  Assimilation optimization hypothesis; a hypothesis that states that stomata optimize a trade-off between A n and some additional cost or constraint eCO 2 Elevated atmospheric CO 2 concentration GOH Growth-optimization hypothesis; the hypothesis that stomata optimize turgor-driven growth GOSM Growth optimizing stomatal model; a model that predicts the g w that maximizes turgor-driven growth; a GOSM makes other modelling assumptions in additional to the GOH to reach a tractable solution HRR Hydraulically regulated respiration; referring to the dependence of R d on RWC L NSC Non-structural carbohydrate; large organic macromolecules that provide both the material and energy required for biological chemical reactions, the synthesis of other organic compounds and the growth of new biomass; NSCs buffer the asynchrony of supply and demand over diel, seasonal and decadal timescales and across plant organs NSL Non-stomatal limitation; referring to decreased mesophyll conductance and/or photosynthetic capacities; typically, relevant to leaf desiccation Potkay and Feng -Optimizing stomatal conductance for turgor-driven growth carbon as a fundamental resource and NSCs as storage.
The finite nature of NSCs imposes a cost to growth (represented by η; Equations 3 and 4).Spending NSCs to grow at one instance depletes the NSCs required for growth at later times, potentially hampering future growth as a legacy effect.Hence, stomata should maximize growth by storing NSCs when the potential for C accumulation is high and η is low (daytime, spring, summer) to be used when the potential for C accumulation is low and η is high (night-time, fall, winter, drought).Like the water-use strategy, the GOH suggests a carbon-use strategy, by which stomata open to slow the depletion of NSCs, especially when NSCs are valuable (small C, large η).NSC depletion is slowed in two ways.First, stomatal opening increases A n , accumulating NSCs.Second, stomatal opening reduces plant water potentials (ψ) (Sperry et al. 1998) and thus also cambial turgor (Hölttä et al. 2010), slows cell expansion and division (Lockhart 1965;Kirkham et al. 1972) and reduces growth's carbon demand.For systems with multiple limiting resources, when one resource becomes more limiting, the other resources become less costly (Farquhar et al. 2002).All else being equal, the cost of water (χ w ) should decline and stomata should open under low NSCs and/or high demand (low C and/or high η in Fig. 1).This coordination explains observations of more aggressive water use and accelerated declines in ψ and water content in NSCdepleted plants (O'Brien et al. 2014;Sapes et al. 2019) and in defoliated trees (Salmon et al. 2015, which Lauriks et al. 2021bLauriks et al. , 2022)).Alternatively, stomata should close, deprioritizing carbon gain, even when water is available, if growth is sink limited (Blonder et al. 2023).Potkay and Feng (2023) showed that the GOH captures realistic stomatal, growth and carbohydrate responses to environmental cues in steady state (when stomata, NSCs and their costs have equilibrated).However, steady state reflects perfect acclimation of gas exchange, carbon use and growth to constant environmental conditions, and ignores the temporal dynamics of the underlying feedbacks between g w , C and G that define the carbon-use strategy, instead describing their final equilibrium.Real environmental conditions vary over diel, seasonal and longer timescales, and similarly, g w , NSCs and growth follow distinct temporal patterns over these timescales (Smith and Stitt 2007;Maseyk et al. 2008;Rossi et al. 2013;Martínez-Vilalta et al. 2016;Mencuccini et al. 2017;Tixier et al. 2018Tixier et al. , 2020)).Indeed, NSCs lag steadystate NSC predictions, and the lag depends on the amplitude and mean of seasonal photosynthetic activity, the phenology of NSC demand and tree size (Oswald and Aubrey 2023).Here, we present transient optimal simulations explicitly considering the carbon-use strategy under varying environmental conditions.
Additionally, we interpret the existence and environmental responses of nocturnal stomatal movements through the GOH.Stomata typically close during the first few hours of night and then open, reaching a maximum nocturnal conductance just before dawn (Hennessey et al. 1993;Resco de Dios et al. 2013a, 2015) with nocturnal stomatal conductances (g w,N ) that are typically 5-40 % of daytime conductances (Caird et al. 2007).AOH models have been unable to , where A n and E are the leaf area-specific net carbon assimilation and transpiration rates, respectively [see Supporting Information-Fig.S1a and b].Solid thick lines in A and B show the traditional daytime and night-time calculations of λ, in which gross photosynthetic assimilation depends on leaf internal CO 2 concentration (c i ) and leaf temperature (T L ; see Supporting Information-Fig.S1c), and dark respiration similarly depends on T L .Dashed thick line in A and B show daytime λ with non-stomatal limitations (NSLs) and night-time λ with hydraulically regulated respiration (HRR), in which cases, gross photosynthetic assimilation and dark respiration decline with decreasing leaf water potential (ψ L ) and leaf relative water content (RWC L ; see Supporting Information-Fig.S1d), respectively.Markers denote the optimal g w .The range of the x-axis corresponds to values of RWC L that exceed the turgor loss point [see Supporting Information-Fig.S1d].In A, the daytime λ curve (without NSLs) and the Low χ w curve intersect at a large g w beyond turgor loss (g w = 0.886 mol•m −2 •s −1 ), corresponding to where χ w = λ = 0. Simulations were performed at atmospheric CO 2 and O 2 partial pressures of 410 and 207 mmol•mol −1 , respectively, T a = 25 °C, RH = 0.4, a solar elevation below the zenith of 0 radians, and incoming irradiance of 600 and 0 W•m −2 in daytime and night-time simulations, respectively, using parameters from Potkay and Feng (2023) and Table 1.NSLs and HRR are turned on by default, and we turned them off for traditional estimates of λ by setting φ c = φ j = 1 [see Supporting Information-Equations S1.3.3 and S1.simulate this nocturnal pattern (Zeppel et al. 2014;Yu et al. 2019), because their mathematical formulations for χ w exceed night-time λ (Wang et al. 2021; Fig. 1), predicting complete closure at night.Parenthetically, representing HRR lessens this disparity between χ w and λ (Fig. 1B).Wang et al. (2021) introduced an empirical 'fitness factor' (f f = χ w,N /χ w,D , where χ w,N and χ w,D are night-time and daytime χ w ) to predict nocturnal stomatal opening from χ w,D (χ w,N = f f •χ w,D ), where χ w,D was predicted by Wang et al.'s (2020) AOH, and f f ≈ 0.15 at 25 °C when ignoring NSLs and HRR.Their pragmatic approach cannot explain why f f < 1 or how f f should respond to environmental conditions.Justifying nocturnal stomata opening as optimal requires a theory for why night-time carbon costs of water are less than that of daytime (χ w,N < χ w,D , f f < 1).
The GOH suggests three hypotheses for why χ w declines from day to night.First, χ w scales with extensibility ( φ , which relates turgor pressure to relative volumetric growth; Lockhart 1965;Potkay et al. 2022), which is temperature dependent (Cabon et al. 2020;Peters et al. 2021), resulting from temperature dependent enzymatically regulated cell expansion (Cosgrove 2000) and following a modified-Eyring equation with a temperature optimum (Johnson et al. 1942;Parent et al. 2010; see Supporting Information-Fig.S1 in Potkay and Feng 2023).Since nights are cooler than days, φ and χ w should be smaller at night than day, potentially opening stomata (Fig. 1B).Second, χ w increases asymptotically with C (Potkay and Feng 2023), and thus χ w should decline from day to night as NSCs wane (Smith and Stitt 2007;Tixier et al. 2018; Fig. 1B).Third, χ w declines under high sink demand (η) and thus should decline at night when most volumetric growth typically occurs (Steppe et al. 2015;Zweifel et al. 2021; but see Mencuccini et al. 2017).Here, we test these three hypotheses for nocturnal stomatal opening.

Growth optimizing stomata model
Here, we review Potkay and Feng's (2023) growth optimizing stomatal model (GOSM).The GOH states stomata maximize an individual tree's whole-stem growth over its lifetime: where g w is the stomatal conductance to vapour, t is time and G is the whole-tree growth, which depends on the leaf areaspecific transpiration (E) and the NSC storage (C) as control and state variables, respectively.This maximization is subject to the constraint that C must be balanced by its supply and demand: where a L is the leaf area, A n is the leaf area-specific net assimilation, R M is the stem and root maintenance respiration, R G is the construction respiration and f c is the constant fraction of G + R G diverted to R G .Environmental conditions include the soil-water potential (ψ soil ), air temperature (T a ), the air's relative humidity (RH), irradiance on a surface normal to the solar beam (I s ) and solar elevation below the zenith (Φ).The difference in the temporal bounds over which growth is maximized (t 2 − t 1 ) is the timescale of optimization.The GOH suggests that the timescale of optimization spans an individual's lifetime.Pragmatically, we approximate an infinitely long timescale by reducing it to its smallest repeated units, solving Equation (1) over a cyclic annual period (Oswald and Aubrey 2023).Since how G responds to C and E is sensitive to tree size (Potkay et al. 2022;Potkay and Feng 2023), this cyclic approximation holds for mature trees, whose relative growth is slow and whose relative change in size is negligible (Valentine and Mäkelä 2012;Forrester 2021), but not young trees, whose fast growth dynamically alters the relationship between G, C and E. Our boundary condition's inability to describe small trees is an important limitation, considering that the GOH maximizes growth over a tree's entire lifetime, including trees' youngest stages.We propose alternative simulations for young trees in Supporting Information-Notes S1 and Section S2.
Solving Equations ( 1) and ( 2) through the calculus of variations (Witelski and Bowen 2015) with g w as the control variable and C as the state variable, Potkay and Feng (2023) derived a solution through the marginal carbon cost of water (χ w ; Fig. 1A), where η is the NSC-use efficiency, representing sink demand and the cost to growth of depleting NSCs (0 < η ≤ 1 − f c ), and ∂G/∂E < 0, since transpiration causes ψ and turgor to decline, slowing growth.Their solution also describes how η changes in time, For the full derivation of Equations ( 3) and ( 4), please see Potkay and Feng (2023; notably see Section S2 of their Supporting Information).Though Potkay and Feng (2023) recognized that η is dynamic, they presented solutions in which η was either a known, instantaneous constant or in steady state ( η = 0) for simplicity.Here, we describe the full dynamics of η as described by Equation ( 4).
The carbon-use strategy that opens stomata to conserve NSC is apparent from Equations ( 3) and (4).By Equation (4), the η•∂R M /∂C term is responsible for elevating η over time when C is small, since η•∂R M /∂C is always positive and greatest at low is responsible for diminishing η over time when either C is small or conditions are favourable for growth, either of which causes large ∂G/∂C (∂G/∂C ≥ 0, ∂ 2 G/∂C 2 ≤ 0, ∂G/∂C ∝ G).Their net effect of C on η is unclear from Equation (4) without a priori knowledge of the magnitudes of ∂R M /∂C and ∂G/∂C.Nonetheless, we later show η > 0 when C is small and η < 0 when C is large (Fig. 2).Hence, C and η feedback dynamically to stabilize each other.Large η opens stomata (smaller χ w by Equation 3; Fig. 1A), assimilating more carbon while reducing sink demand, accumulating NSCs and reducing η over time by Equation ( 4).This negative feedback suggests that the carbon-use strategy seeks to maintain homeostasis of NSCs, which is indeed supported by past studies (Smith and Stitt 2007;Schönbeck et al. 2018;Dickman et al. 2019).
In Potkay and Feng's (2023) GOSM, A n is modelled by a big-leaf version of the Farquhar et al. (1980) model with Potkay and Feng -Optimizing stomatal conductance for turgor-driven growth temperature-dependent photosynthetic capacities, leaf temperatures are predicted from the foliar radiation budget, ψ is predicted from steady-state hydraulics, G is formulated after Potkay et al.'s (2022) turgor-driven growth model (in which an NSC-unlimited potential turgor is estimated empirically from simulated ψ, which determines an NSCunlimited potential G 0 that sets the maximum for realized G), and G and R M are temperature dependent and NSC limited following Michaelis-Menten kinetics (Thornley 1970(Thornley , 1971)).We modified the GOSM [see Supporting Information-Notes S1 and Section S1] by simplifying the mathematics for stem xylem hydraulics and G, quickening the computation of ∂G/∂E with negligible effects on predictions [see Supporting Information-Fig.S4].We added three new processes: (1) temperature-dependent hydraulic conductances, which limit stomatal opening under cold weather (Lintunen et al. 2020), (2) NSLs, here modelled by reducing apparent photosynthetic capacities under excessively negative ψ (Zhou et al. 2013;Novick et al. 2016;Dewar et al. 2018), since they may be key to explaining stomatal behaviour when conditions are unfavourable for growth (Potkay and Feng 2023), and (3) HRR, which may be partially responsible for explaining nocturnal stomatal opening (Fig. 1B).

Dynamic simulations
We applied the GOSM and parameters of Potkay and Feng (2023) for Scots pine to perform both year-long and day-long simulations of the g w , A n , E, λ, η, C, ψ L and G that maximize growth under cyclic environmental conditions.Parameters for new processes are shown in Table 2. Four original parameters related to sink demand from Potkay and Feng (2023) were adjusted to produce realistic NSC ranges and NPP:GPP ratios in year-long simulations (Table 2; see Supporting Information-Fig.S5).The η and C dynamics (Equations 3 and 4) are simulated by explicit finite difference with 0.5-h time step with cyclic boundary conditions (η(t 1 ) = η(t 2 ), S6).Atmospheric conditions are based on NOAA reanalysis data (NCEP/NCAR Reanalysis 1; Kalnay et al. 1996) for forest stands in Northern Spain (Tillar Valley; Poblet Forest Natural Reserve; Prades Mountains), where several site-specific hydraulic and growth parameters were estimated (Potkay et al. 2021(Potkay et al. , 2022)).Details concerning environmental forcings and numerical solutions are presented in Supporting Information-Notes S1, Section S2, Figs.S2, S3 and S5.
We present cyclic year-long simulations over an average year (t 1 = 0, t 2 = 365 days), reflecting environmental conditions that vary within and among days and seasons, but not across years [see Supporting Information-Figs.S2 and S3].Additionally, we compare year-long simulations to day-long simulations for individual days of the year (DOY; t 1 = 0, t 2 = 24 h) and to Potkay and Feng's (2023) steady-state solution to the GOSM.Two steady-state simulations in which Ċ = 0 and η = 0 were performed.First, stomata were optimized for instantaneous midday environmental conditions, and second, stomata were optimized for diel-averaged constant conditions [see Supporting Information-Notes S1 and Section S3].The steady-state solution to diel-averaged conditions is similar to the cyclic day-long simulation, except C and η are constant within a day instead of being solved by Equations ( 2) and (4), and boundary conditions are not defined in steady-state simulations.

Nocturnal stomatal opening
To test our hypotheses that diel NSCs and temperature fluctuations explain nocturnal stomatal opening, we numerically solved C and φ(T a ) combinations that simulate realistic nocturnal stomatal conductance (g w,N ).Values of 0.005 and 0.04 mol•m −2 s −1 were chosen as end members considering that g w,N is typically 5-40 % of daytime g w (Caird et al. 2007), assuming daytime g w = 0.1 mol•m −2 •s −1 , based on mid-growing season values from our year-long simulation (Fig. 2A).Simulations were performed at Potkay and Feng's (2023) default environmental conditions, except RH and I s , which were set to 0.5 and 0 W•m −2 , respectively, and η = 0.44•(1 − f c ) based on our year-long simulation (Fig. 2E).We did not directly test our hypothesis that diel η fluctuations explain nocturnal stomatal opening, because simulated η is nearly constant (Figs.2-5) and thus cannot explain nocturnal stomatal opening.Although η rises nocturnally for DOY 1-146 and DOY 305-365 (Fig. 2E), thereby lowering χ w from day to night, the increase in η is far too slight to achieve the small nocturnal χ w required for nocturnal stomatal opening.All else being equal, η would have to increase ~7 % nocturnally relative to its daytime value (for f f = 0.15 and a daytime η of 0.44•[1 − f c ]).However, the simulated η change between day and night never exceeds 0.2 % and is negative for approximately half of the year (Fig. 2E).
With a minimal NSC storage of C = 60 mol, when g w,N is most sensitive to NSC changes, and the corresponding φ that produced g w,N = 0.04 mol•m −2 •s −1 at same reference conditions, we simulated the instantaneous response of g w,N to leaf-to-air vapour pressure deficit (VPD; D L ; by varying RH), soil water potential (ψ soil ) and atmospheric CO 2 concentration (c a ), keeping C constant.These environmental responses were repeated at elevated (+25 %) and depleted (−25 %) NSC storages to represent delayed responses caused by slower changes in NSC status associated with VPD, drought and eCO 2 (Muller et al. 2011;Dietze et al. 2014;Du et al. 2020).According to a meta-analysis of NSCs under various experiment treatments by Du et al. (2020), the average effects of warming, drought and eCO 2 on total NSC concentrations are 9 %, −4 % and 24 %, respectively.Hence, our ±25 % was chosen as a maximal value, coinciding with the greatest change in g w,N .

Annual trends
Annual trends in midday g w , A n , E, λ, η, C, ψ L and G from our year-long simulation are shown in Fig. 2.During the beginning and end of the growing season, λ is small, suggesting that stomata are nearly maximizing A n instantaneously (dA n /dE approaching 0) at these times.Counterintuitively, our GOSM predicts that g w is greatest during the cold, dormant periods at the beginning and end of the year, resulting from high RH, low T a [see Supporting Information-Fig.S3] and small VPD (Fig. 2A).Large g w under low VPD is unsurprising, considering AOHs and other empirical formulations for g w often predict similar results (Oren et al. 1999;Katul et al. 2009).In fact, low VPD conditions are often ignored in studies for this reason (Ewers and Oren 2000), and predictions are often corrected by simply constraining g w below a parameterized maximum g w (i.e.g w = min(g w opt , g w max ), where g w opt is the optimal solution, and g w max is the maximum conductance; Mäkelä et al. 1996) or by constraining VDP above a threshold (D L ≥ 0.05 kPa in CLM5; Franks et al. 2018).These large g w had little impact on A n and G (Fig. 2B and H), which control C and η dynamics (Equations 3 and 4), since A n and G were both temperature limited (Bernacchi et al. 2001;Parent et al. 2010), and A n was light limited [see Supporting Information-Fig.S3].
Midday values of A n and E begin to rise from their winter values on the ~45th and 66th DOY.Rising E at the beginning of the growing season weakly affects the early seasonal trend of ψ L , which is also modulated by temperature seasonality due to the temperature dependence of hydraulic conductance.In fact, by midseason, ψ L becomes less negative over time, despite rising E, because of the positive effects of warmth on hydraulic conductance, reducing the potential difference required to drive flow.Soon after A n and E elevate, T a rises enough for growth to begin.Throughout the

S6
) only on DOY 174-210 and 309-334 when midday C was nearly constant across days in the year-long simulation (i.e. at the seasonal maximum and minimum C; see Supporting Information-Fig.S7).Day-long, year-long and both steadystate simulations of A n compared well on these days, and day-long simulations and steady-state simulations to dielaveraged conditions agreed for all variables.In addition to A n , day-long and both steady-state simulations agreed only in terms of η.Meanwhile, steady-state simulations to instantaneous midday conditions agreed with year-long simulations for E and G.However, for g w , λ, C and ψ L , year-long predictions and steady-state predictions to instantaneous conditions differed from each other as well as from day-long simulations and diel-averaged steady-state simulations.Notably, day-long and steady-state simulations over-predicted C more often than not.The steady-state solution to instantaneous midday conditions predicted infinite C over the growing season, because midday whole-canopy photosynthate input (a L × A n ) exceeds the maximum possible midday sink demand (G 0 /(1 − f c ) and R m,0 in Potkay and Feng 2023), meaning that there is no optimal stomatal behaviour that prevents NSCs from accumulating between days at these times according to Potkay and Feng's (2023) instantaneous steady-state strategy; instead, a dynamic perspective is necessary.For day-long simulations and diel-averaged steady-state solutions, these C disparities declined while approaching DOY 172 and 325 when yearlong-simulated midday C was nearly constant across days as source-and sink demand were near equal.

Diel trends
We present diel trends from our year-long simulation for individual days throughout the growing season, including an example of the early growing season (DOY 133;

Nocturnal stomatal opening
Our GOSM requires either very cold temperatures (T a ≈ 5 °C) or unrealistically small NSCs to simulate realistic g w,N [see Supporting Information-Fig.S10], meaning that nocturnal stomatal opening would be simulated only during the winter and early spring.Since growth is both temperature-and NSC limited, these required small T a and C would negate virtually all growth at night.Simulated diel C and η changes are too small to significantly alter χ w between day and night (Figs. 3-5; see Supporting Information-Figs.S8 and S9).Though χ w varies slightly between day and night, these variations arise predominantly from diel T a changes.These results do not support our hypotheses that diel T a , C and η variations explain nocturnal stomatal opening.Upon changing φ to a relatively small value that produced g w,N = 0.04 mol•m −2 •s −1 at our reference conditions and C = 60 mol ( φ equal to ~5 % of the value in Table 2), we simulated the environmental responses of g w,N at constant C (Fig. 6).By reducing φ to lessen χ w,N , we effectively apply Wang et al.'s (2021) approach with a constant 'fitness factor'.At a given C, hydraulic stress (VPD, soil drought) closes stomata, while CO 2 concentration has no immediate effect on g w,N (Fig. 6).Depleting C opened stomata (Fig. 6), while C accumulation closed stomata nocturnally (Fig. 6).Conversely, long droughts deplete NSCs and open stomata at night (Fig. 6).

Dynamic optimization and plants' timescales of optimization
Plants are theorized to optimally coordinate their supply and demand for resources over relevant timescales.Natural selection acts over this timescale of optimization, since selection has generated plants' adaptive regulatory patterns, which remains a major uncertainty (Dewar et al. 2009).Ideally, the timescale should be the period over which the objective is integrated (t 2 − t 1 in Equation 1).Individual plant traits may operate over different timescales, and they certainly differ in the bounds of integration (t 2 − t 1 ) that modellers use to predict them through mathematical optimization.It is common to choose longer bounds for traits that change slower (e.g.assuming that quickly changing g w is instantaneously optimized; Wolf et al. 2016).However, rapid kinetics do not necessarily imply shorter mathematical bounds (Feng et al. 2022).For realistic predictions, the choice of bound length does not necessarily have to equal or even approximate the true timescale over which natural selection acts.The mathematical bounds of integration must only describe the shortest portion of the timescale in which the temporal dynamics of the chosen constraints (soil−water in water-saving AOHs, NSCs in the GOH) repeat (e.g. the timescale is treated as the sum of repeated periods that are identical or effectively the   same in a stochastic sense).Though the timescale for stomata has been theorized as plants' lifetime (Cowan 1986) and may be even longer, water-saving AOHs often approximate the timescale as many repeated soil-water dry-downs between rain events (Cowan 1982;Mäkelä et al. 1996;Manzoni et al. 2013;Mrad et al. 2019).Conversely, we have approximated the timescale as multiple repeated years, the shortest period over which NSC trends are cyclic (Oswald and Aubrey 2023).
Since stomatal conductance, NSCs, and growth rates follow near-cyclic trends over diel and seasonal timescales (Smith and Stitt 2007;Maseyk et al. 2008;Rossi et al. 2013;Martínez-Vilalta et al. 2016;Mencuccini et al. 2017;Tixier et al. 2018Tixier et al. , 2020)), we simulated plant responses that maximize growth over individual days (t 2 − t 1 = 24 h) and a full year (t 2 − t 1 = 365 days) and compared these dynamic simulations to Potkay and Feng's (2023) steady-state solution (Figs. 2-5; see Supporting Information-Figs.S8 and S9).Year-long simulations captured realistic seasonal and diel patterns, supporting the GOH.Cyclic day-long simulations, however, were possible only for the middle and end of the growing season when changes in C between days were small in year-long simulations and when maximum possible sink demands exceeded instantaneous carbon input (when steady-state simulations would predict finite C).On these days, year-long, day-long and dielaveraged steady-state simulations generally compared well in terms of A n , though not in terms of other predicted variables [see Supporting Information-Fig.S7], especially not in their C predictions.This suggests that cyclic boundary conditions (C(t 1 ) = C(t 2 )) and steady state are inappropriate to describe NSC use for an individual DOY, since real NSCs change over multiple days and are thus non-cyclic across days.The amplitude of simulated NSC cycles depends on the duration of the simulation and shrinks as t 2 − t 1 becomes smaller, approaching the steady-state solution as t 2 − t 1 approaches zero.In day-long simulations, t 2 − t 1 is small, and thus day-long and diel-averaged steady-state simulations approximate each other [see Supporting Information-Fig.S7].Hence, day-long simulations over-predict C for the same reason that steady-state simulations do: they ignore the fact that instantaneous A n during the peak growing season exceeds the average A n over a year, thereby overestimating C. Sink demands during the peak growing season similarly exceed their annual averages, but to a much lesser extent, since plants respire throughout the entire year, while peak photosynthesis occurs over a smaller window (at least in seasonal environments).More appropriate boundary conditions for C in dynamic day-long simulations would set C(t 1 ) to a realistic predetermined value.C(t 2 ) would be treated implicitly (explicit definition is non-essential).However, the appropriate value for C(t 1 ) is unclear.It could be chosen from measurements of whole-tree NSCs, although these estimates are rare, or it could be taken as the value solved by the full year-long simulation at the specified DOY.The latter choice, however, makes the day-long simulation unnecessary since the year-long simulation makes all of the same predictions over a longer period.The dynamic version of the GOSM presented here is an approvement upon Potkay and Feng's (2023) steady-state GOSM, which cannot capture how C and η evolve dynamically over hours to seasons.Hence, the dynamic GOSM better describes how stomata respond to environmental variation as modulated by the changes in NSC storage (C) and the cost of sink demand (η).
Both year-long and day-long simulations predicted that maximum diel G typically occurs in the late afternoon when air is warmest and C is slightly elevated (Figs.3-5; see Supporting Information-Figs.S8 and S9), clashing with expectations and observations that most volumetric growth and turgor-driven cell expansion occur at night (Steppe et al. 2015;Zweifel et al. 2021; but see Mencuccini et al. 2017).Accordingly, most volumetric growth is expected at night when ψ and turgor are greatest.This expectation, however, ignores temperature limitations on expansion (Cabon et al. 2020;Peters et al. 2021).In other simulations in which T a is constant throughout day and night (not shown), most growth indeed occurs at night when ψ and turgor are largest.Volumetric growth (e.g.cell expansion, division) is only one aspect of growth (Hilty et al. 2021), while our GOSM describes growth gravimetrically through an  and C), a soil water potential of 0 MPa (in A and C), a solar elevation below the zenith of 0 radians, and an incoming irradiance of 0 W•m −2 , using parameters from Potkay and Feng (2023) and Table 1.Air temperature (T a ) was set to 25 °C for most calculations (e.g.leaf temperature, R d , λ), except for exclusively the calculation of the extensibility ( φ ), for which T a = 8.07 °C.Potkay and Feng -Optimizing stomatal conductance for turgor-driven growth implicit assumption that expansion and cell wall hardening are synchronized (i.e.wood density is assumed constant in Potkay et al. 2022).Hence, it is not entirely appropriate to compare our simulations of gravimetric growth to observations of volumetric growth, which are known to follow distinct temporal patterns (Rathgeber et al. 2016).For example, cell wall hardening is expected to be NSC dependent (Hölttä et al. 2010;Cartenì et al. 2018;Friend et al. 2023) and temperature dependent (Cuny et al. 2015;Cuny and Rathgeber 2016), suggesting that, unlike expansion, hardening may be faster during the day, which is consistent with our simulations.Nonetheless, our representation of growth should be improved.Future versions of the GOH should model the phases of wood formation separately to distinguish between volumetric growth, gravimetric growth, and their distinct dynamics and sensitivities to environmental conditions by integrating ideas from xylogenesis models (Eckes-Shephard et al. 2022).The GOH's objective function (Equation 1) would be rewritten as maximizing volumetric instead of gravimetric growth, and we expect changes to our predictions of g w and C. Water-use efficiency (χ w ) and C would both be higher (resulting in lower g w ) than currently predicted (Fig. 2) in the early growing season when cambial and developing xylem cells are dividing and expanding (larger ∂G/∂E term in Equation (3) because of shorter, more-concentrated period of expansion than our current growth scheme) with little NSC deposition, since cell wall hardening would have not begun.Conversely, χ w and C would both be lower (with higher g w ) in the late growing season as increasingly more developing xylem cells end their expansion (∂G/∂E approaching zero) and transition into cell wall hardening, when most NSC deposition occurs.

Carbon-use strategies
Much like how some AOHs suggest that stomata follow water-use strategies that moderate current transpiration to save soil-water for future uptake (Manzoni et al. 2013;Mrad et al. 2019), the GOH suggests that stomata also follow carbon-use strategies to maintain NSC homeostasis (Smith and Stitt 2007;Schönbeck et al. 2018;Dickman et al. 2019).When NSCs are high, η declines (Equation 4; Fig. 2), both of which close stomata (Fig. 1A), increasing ψ, turgor and G, while reducing A n and NSCs through decreased carbon supply and increased demand.When NSCs are low, η rises, and stomata open to improve A n and reduce turgor and G, accumulating NSCs through increased supply and decreased demand.This carbon-use strategy was evident in year-long optimizations (Fig. 2) and day-long optimizations, though diel C − η feedbacks were far smaller than seasonal feedbacks.These regulatory, negative feedbacks are driven by C changes predominately and η changes to a lesser extent, since seasonal and diel η changes were small (Figs.2-5; see Supporting Information-Figs.S8 and S9).These results suggest that carbon, in addition to water, is a resource that limits stomatal opening (Buckley 2023).
The carbon-use strategy can explain why maximal photosynthetic stimulation by eCO 2 and coinciding heightened stomatal sensitivity typically occur at the beginning of the growing season (low χ w during the early growing season in Fig. 2; Quentin et al. 2015;Urban et al. 2019;Sanches et al. 2020;Lauriks et al. 2021bLauriks et al. , 2022) ) when NSCs deplete to meet growth demands (Palacio et al. 2018;Tixier et al. 2020).Since larger seasonal NSC fluctuations occur in more seasonal climates (Fermaniuk et al. 2021), we expect seasonal variations in photosynthetic and stomatal sensitivity to eCO 2 to be pronounced in boreal ecosystems.Similarly, the carbonuse strategy can explain stomatal behaviour's apparent coordination with NSC depletion during droughts.According to the GOH, drought-induced NSC depletion has a negative effect on χ w that counters the rise of χ w due to hydraulic stress, slowing stomata closure, consistent with observations of aggressive water use and accelerated desiccation in NSCdepleted plants (O'Brien et al. 2014;Sapes et al. 2019) and defoliated trees with smaller stem NSC reserves (Poyatos et al. 2013;Salmon et al. 2015).Unlike the GOH, these two phenomena (seasonal CO 2 sensitivity, stomata-NSC-drought feedbacks) cannot be explained by most AOHs (e.g.Cowan and Farquhar 1977;Mäkelä et al. 1996;Manzoni et al. 2013;Wolf et al. 2016;Sperry et al. 2017), which do not account for NSCs and their role in stomatal behaviour and plant water use (Blonder et al. 2023).Only a few AOHs capture NSC-dependent stomatal behaviours, including AOHs that consider the downregulation of photosynthesis under elevated leaf sugar concentrations (predicted by phloem transport models; Nikinmaa et al. 2013;Hölttä et al. 2017;Dewar et al. 2022) and an AOH, which explicitly defines a cost of maintaining leaf osmotic potentials (Deans et al. 2020), which we interpret as implicitly incorporating a cost of the sugars that generate osmotic potential.In these AOHs, leaf sugar concentrations impose a cost to g w , while all NSCs (sugars, starches) impose a constraint to g w in our GOSM.Additionally, NSCs in our GOSM are not leaf specific; all NSCs are relevant regardless of organ.In addition to the direct control of NSCs on g w , NSC demand (represented by η) modifies g w in the GOSM, which no AOH directly considers (although phloem transport-based AOHs indirectly consider sinks, which impact sugar concentrations).
Our GOH emphasizes NSCs as a key limiting resource by explicitly incorporating the NSC balance as a constraint on growth.This NSC constraint does not mean growth may not also be water limited (Lempereur et al. 2015;Eckes-Shephard et al. 2021); water conservation over time merely does not currently factor into the current mathematical set-up of our optimization problem (the GOSM nonetheless represents the effects of water limitation on growth and hydraulics).Hence, our current model may offer an incomplete view of stomal behaviour, and the relative contributions of carbon and water limitations to optimal stomatal behaviour remain unknown.Future versions of the GOH should consider both carbon and water as constraints to describe both the carbon-use strategy and the water-use strategy and identify their relative contributions to stomatal opening and closure.Because we have not constrained the objective (Equation 1) by soil-water, we limited our simulations to scenarios of constant and high soilwater availability.Nonetheless, even without the water-use strategy, the GOH would predict lower g w and G and higher C if transpiration depleted soil-water, since the GOH captures these responses under soil-water stress in steady state (Potkay and Feng 2023), and dynamic simulations tend to follow steady-state trends as an attractor.Contrarily, in the absence of hydraulic limitations, water-saving AOHs would predict unrealistically large g w in our simulations with constant and high soil-water content, since plant-controlled soil-water depletion is prerequisite to their theory.These large g w can be shown by Manzoni et al.'s (2022) heuristic model for g w for a plant transpiring at a constant rate during a soil-water dry-down, in which g w is inversely proportional to the dry-down length (see their Equation 11).In our modelling scenarios with constant and plentiful soil-water, the dry-down length is effectively zero, leading to infinite g w according to Manzoni et al. (2022).Other water-saving AOHs would predict finite but still similarly large g w under constant and high soil-water content.The inclusion of hydraulic limitations also constrains g w from being excessively large in water-saving AOHs (Mrad et al. 2019;Lu et al. 2020); nonetheless, soil-water dry-downs, which are absent from our simulations, are fundamental to these modified water-saving AOHs.

Nocturnal stomatal opening
We hypothesized that the GOH explains nocturnal stomatal movements, particularly through diel fluctuations in T a , η, and/or C (carbon-use strategy), since these variables control χ w (Fig. 1).Nocturnal stomatal patterns have been thought to maximize growth (Resco de Dios et al. 2016) and involve NSCs (Lasceve et al. 1997;Easlon and Richards 2009).Support for our hypotheses was mixed.On one hand, either very low T a or unrealistically small C were prerequisite to realistic g w,N [see Supporting Information-Fig.S10], nullifying growth at night when most expansion typically occurs (Steppe et al. 2015;Zweifel et al. 2021), and simulated diel changes in C and η were too small for the diel variations in χ w required to open stomata nocturnally (Figs.3-5; see Supporting Information-Figs.S8 and S9).In fact, diurnal NSC changes are known to be small in trees (e.g.Tixier et al. 2018) and might be more relevant in smaller plants that deplete NSCs faster (Smith and Stitt 2007;Oswald and Aubrey 2023).On the other hand, after lowering φ , our GOSM captured several nocturnal stomatal behaviours, including how diminished daytime assimilation (e.g.shading) increases g w,N (Barbour et al. 2005;Easlon and Richards 2009) by storing fewer NSCs as well as decreased g w,N under eCO 2 (Wheeler et al. 1999;Zeppel et al. 2012;Resco de Dios et al. 2016) upon acclimation by storing more NSCs, and under high VPD (Barbour and Buckley 2007;Howard and Donovan 2010) and soil drought (Zeppel et al. 2012(Zeppel et al. , 2014;;Chowdhury et al. 2021;Wang et al. 2021) (Fig. 6).Both eCO 2 and short periods of hydraulic stress are known to cause NSCs to accumulate (Dietze et al. 2014;Mitchell et al. 2014;Du et al. 2020;Grossiord et al. 2020), which cause night-time stomatal closure according to the GOH (Fig. 6).Conversely, long periods of hydraulic stress may deplete NSCs, opening stomata nocturnally (Fig. 6).The GOH suggests that g w,N has only a delayed response to CO 2 concentrations as a long-term carbon-use strategy, while g w,N responds to hydraulic stress in both instantaneously through immediate growth declines (Lempereur et al. 2015;Tumajer et al. 2022) and in a delayed manner due to the slower acclimation of NSCs and the carbon-use strategy.This instantaneous response on g w,N is always negative, while the delayed response may be either positive or negative depending on whether NSCs are depleted or accumulated, respectively, and thus depending on the duration and nature of hydraulic stress.The GOH may also explain positive and neutral responses of g w,N to VPD (Barbour et al. 2005;Daley and Phillips 2006;Dawson et al. 2007;Howard and Donovan 2010;Zeppel et al. 2012;Resco de Dios et al. 2013b;Wang et al. 2021) if elevated VPD causes such NSC accumulation.However, we cannot explain reports of nocturnal opening under eCO 2 (Wheeler et al. 1999;Zeppel et al. 2012;Resco de Dios et al. 2016).We are unaware of any studies reporting a NSC decline under eCO 2 , and hence the literature does not support NSC acclimation as a reason for nocturnal stomatal opening under eCO 2 .Similarly, eCO 2 -induced changes in leaf area cannot explain nocturnal opening, since eCO 2 tends to increase leaf areas (Lauriks et al. 2021a).Instead of opening, however, such an increase in leaf area would close stomata further by increasing χ w (Manzoni et al. 2022;Potkay and Feng 2023).Nonetheless, our GOH offers a step forward to identify the evolutionary advantage of nocturnal stomatal opening, which AOHs are generally unable (Zeppel et al. 2014;Yu et al. 2019) unless employing an empirical 'fitness factor' to predict χ w,N from χ w,D (Wang et al. 2021).
Capturing nocturnal stomatal opening with realistic night-time expansion may require framing the problem in a probabilistic manner that accounts for plants' statistical 'memory' of environmental conditions (Buckley et al. 2017; see Supporting Information-Notes S1, Sections S4.1 and S4.2).Indeed, nocturnal stomatal behaviour is regulated by circadian rhythm (Hennessey et al. 1993;Resco de Dios et al. 2013a, 2015, 2020), which probabilistically 'anticipates' a range of environmental conditions.Following this idea, we derived a simple expression for Wang et al.'s (2021) 'fitness factor' accounting for plants' thermal 'memory' [see Supporting Information-Equation S4.2.6]: where D are the mean φ for the 'anticipated' night-time and daytime temperature distributions, respectively, and φ (T a,N ) and φ (T a,D ) are the φ evaluated at the actual night-time and daytime temperatures, respectively.While our deterministic GOSM (Equation 3) implicitly assumes plants' 'anticipation' of temperature is perfectly accurate and precise, Equation (5) does not.By accurate, we mean the 'anticipated' temperature distribution is centred on the actual temperature, and by precise, we mean the variation of the 'anticipated' temperature distribution is small.If 'anticipation' is accurate and relatively precise, Equation (5) predicts realistic f f only on cool days and nights [see Supporting Information-Fig.S11], confirming our previous results [see Supporting Information-Fig.S10].In fact, as long as 'anticipation' is accurate, Equation (5) cannot predict realistic f f (~0.05 < f f < ~0.25 at 25 °C) on warm nights regardless of the precision [see Supporting Information-Fig.S12].To capture realistic f f , 'anticipation' must be inaccurate, 'anticipating' night-time temperatures cooler than actuality [see Supporting Information-Fig.S13], enabling simultaneous nocturnal stomatal opening and expansion.This thermal inaccuracy could arise from a selective 'memory' and from the fact that leaf temperature (T L ) is not the same as temperatures in other growing tissues (approximated here by T a ).If stomata 'anticipated' that T a was always some function of actual T L based on a selective 'memory' of only daytime experiences (when on average, T a < T L for T a < ~28 °C; Michaletz et al. 2016), stomata would 'anticipate' temperatures cooler than actuality at night (when T a ≈ T L in actuality).Alternatively, the thermal inaccuracy might reflect a cautious strategy to safeguard against incurring penalties of cold nights that are unaccounted in the GOH framework.Potkay and Feng -Optimizing stomatal conductance for turgor-driven growth

Supporting Information
The following additional information is available in the online version of this article -Figure S1.Simulated net assimilation, leaf temperature, and leaf water content corresponding to marginal carbon profits and costs of water shown in Fig. 1.
Figure S2.Probability distributions parameters for the interannual variation in air temperature at Northern Spain site.
Figure S3.Annual trends in diel minimum and maximum conditions at Northern Spain site.Figure S11.Fitness factor for combinations of 'anticipated' mean daytime and night-time temperatures, assuming that plants' 'anticipation' of air temperature is perfectly accurate.
Figure S12.Fitness factor for combinations of standard deviations and skewness of 'anticipated' daytime and nighttime temperature distributions, assuming that plants' 'anticipation' of air temperature is perfectly accurate.
Figure S13.Fitness factor for varying accuracy or ability to correctly 'anticipate' air temperatures.
Notes S1.Description of model changes, environmental forcing and numerical solution, and probabilistic growth maximization.
Notes S2.MATLAB code for model and plotting/analyzing data, including forcing data and simulation outputs.

Sources of Funding
A.P. and X.F.acknowledges the support from National Science Foundation CAREER award DEB-2045610.
Constant describing R G ; f c = R G /(R G + G) < 1, and when dC/dt = 0, f c = R G /(a L •A n − R M ) f f 'Fitness factor'; f f = χ w,N /χ wefficiency (NSCUE); cost of depleting NSCs associated with the profit of growth; 0 < η ≤ (1 − f c ) λ Marginal carbon profit of water; λ = dA n /dE φ 'Effective' extensibility, relating the average over the stem of the difference between turgor pressure and a threshold to the whole-stem relative volumetric growth rate

Figure 1 .
Figure1.Daytime (A) and night-time (B) values of marginal carbon profit of water (λ = ∂A n /∂E; thick lines) and marginal carbon cost of water given by the growth-optimization hypothesis (GOH) (χ w ; thin lines) for potential values of stomatal conductance (g w ), where A n and E are the leaf area-specific net carbon assimilation and transpiration rates, respectively [see Supporting Information-Fig.S1a and b].Solid thick lines in A and B show the traditional daytime and night-time calculations of λ, in which gross photosynthetic assimilation depends on leaf internal CO 2 concentration (c i ) and leaf temperature (T L ; see Supporting Information-Fig.S1c), and dark respiration similarly depends on T L .Dashed thick line in A and B show daytime λ with non-stomatal limitations (NSLs) and night-time λ with hydraulically regulated respiration (HRR), in which cases, gross photosynthetic assimilation and dark respiration decline with decreasing leaf water potential (ψ L ) and leaf relative water content (RWC L ; see Supporting Information-Fig.S1d), respectively.Markers denote the optimal g w .The range of the x-axis corresponds to values of RWC L that exceed the turgor loss point [see Supporting Information-Fig.S1d].In A, the daytime λ curve (without NSLs) and the Low χ w curve intersect at a large g w beyond turgor loss (g w = 0.886 mol•m −2 •s −1 ), corresponding to where χ w = λ = 0. Simulations were performed at atmospheric CO 2 and O 2 partial pressures of 410 and 207 mmol•mol −1 , respectively, T a = 25 °C, RH = 0.4, a solar elevation below the zenith of 0 radians, and incoming irradiance of 600 and 0 W•m −2 in daytime and night-time simulations, respectively, using parameters fromPotkay and Feng (2023) and Table1.NSLs and HRR are turned on by default, and we turned them off for traditional estimates of λ by setting φ c = φ j = 1 [see Supporting Information-Equations S1.3.3 and S1.3.4] and b r * = 0 [see Supporting Information-Equation S1.3.13].
Figure1.Daytime (A) and night-time (B) values of marginal carbon profit of water (λ = ∂A n /∂E; thick lines) and marginal carbon cost of water given by the growth-optimization hypothesis (GOH) (χ w ; thin lines) for potential values of stomatal conductance (g w ), where A n and E are the leaf area-specific net carbon assimilation and transpiration rates, respectively [see Supporting Information-Fig.S1a and b].Solid thick lines in A and B show the traditional daytime and night-time calculations of λ, in which gross photosynthetic assimilation depends on leaf internal CO 2 concentration (c i ) and leaf temperature (T L ; see Supporting Information-Fig.S1c), and dark respiration similarly depends on T L .Dashed thick line in A and B show daytime λ with non-stomatal limitations (NSLs) and night-time λ with hydraulically regulated respiration (HRR), in which cases, gross photosynthetic assimilation and dark respiration decline with decreasing leaf water potential (ψ L ) and leaf relative water content (RWC L ; see Supporting Information-Fig.S1d), respectively.Markers denote the optimal g w .The range of the x-axis corresponds to values of RWC L that exceed the turgor loss point [see Supporting Information-Fig.S1d].In A, the daytime λ curve (without NSLs) and the Low χ w curve intersect at a large g w beyond turgor loss (g w = 0.886 mol•m −2 •s −1 ), corresponding to where χ w = λ = 0. Simulations were performed at atmospheric CO 2 and O 2 partial pressures of 410 and 207 mmol•mol −1 , respectively, T a = 25 °C, RH = 0.4, a solar elevation below the zenith of 0 radians, and incoming irradiance of 600 and 0 W•m −2 in daytime and night-time simulations, respectively, using parameters fromPotkay and Feng (2023) and Table1.NSLs and HRR are turned on by default, and we turned them off for traditional estimates of λ by setting φ c = φ j = 1 [see Supporting Information-Equations S1.3.3 and S1.3.4] and b r * = 0 [see Supporting Information-Equation S1.3.13].

Figure 2 .
Figure 2. Annual trends from our year-long simulation in stomatal conductance (g w ), photosynthetic carbon assimilation (A n ), transpiration (E), marginal water-use efficiency (λ), NSC storage (C), growth rate (G), leaf water potentials (ψ L ) and marginal NSC-use efficiency (NSCUE; η) normalized by its maximum value (1 − f c ) that maximize growth under the cyclic environmental conditions shown in Supporting Information-Fig.S3.We present daily values at noon (1200 h).
water-use efficiency and stomatal closure.Additionally, η and C appear nearly constant throughout the day (Figs.3-5; see Supporting Information-Figs.S8 and S9), while dη/dt and C display opposing sinusoidal patterns, suggesting negative feedback between C and η like those seen in seasonal trends (Fig.2), although far smaller in magnitude, and displaying a reduced form of the carbon-use strategy at diel timescales.Though growth occurs during both day and night, maximum diel G is simulated in the late afternoon when air is warmest and C is slightly elevated (Figs 3-5; see Supporting Information-Figs.S8 and S9).

Figure 3 .
Figure 3. Diel trends from the year-long simulation during the beginning of the growing season (DOY = 133), including the stomatal conductance (g w ) (A), photosynthetic carbon assimilation (A n ) (B), transpiration (E) (C), marginal water-use efficiency (WUE; λ) (D), marginal NSC-use efficiency (NSCUE; η) normalized by its maximum value (1 − f c ) (E), NSC storage (C) (F), leaf water potential (ψ L ) (G) and growth rate (G) (H) as plant responses.Environmental conditions are shown in the bottom row, including trends in air temperature (T a ) (I), relative humidity (RH) (J), incoming irradiance on a surface normal to the beam (I s ) (K) and zenith angle (solar elevation below the zenith; Φ) (L).Day-long simulations for this day (DOY 133) did not converge to a result that satisfied the boundary conditions for C and η (i.e.Δη = 0 and ΔC = 0 isolines do not intersect; see Supporting Information-Fig.S6, e.g. of intersecting isolines).

Figure 4 .
Figure 4. Diel trends from year-long simulation during the peak growing season, particularly during peak annual NSC reserves (C) (DOY = 172), including the stomatal conductance (g w ) (A), photosynthetic carbon assimilation (A n ) (B), transpiration (E) (C), marginal water-use efficiency (WUE; λ) (D), marginal NSC-use efficiency (NSCUE; η) normalized by its maximum value (1 − f c ) (E), NSC storage (C) (F), leaf water potential (ψ L ) (G) and growth rate (G) (H) as plant responses.Environmental conditions are shown in the bottom row, including trends in air temperature (T a ) (I), relative humidity (RH) (J), incoming irradiance on a surface normal to the beam (I s ) (K) and zenith angle (solar elevation below the zenith; Φ) (L).Day-long simulations for this day (DOY 172) did not converge to a result that satisfied the boundary conditions for C and η (i.e.Δη = 0 and ΔC = 0 isolines do not intersect; see Supporting Information-Fig.S6, e.g. of intersecting isolines).

Figure 5 .
Figure 5. Diel trends from the year-long simulation during the end of the growing season (DOY = 307), including the stomatal conductance (g w ) (A), photosynthetic carbon assimilation (A n (B), transpiration (E) (C), marginal water-use efficiency (WUE; λ) (D), marginal NSC-use efficiency (NSCUE; η) normalized by its maximum value (1 − f c ) (E), NSC storage (C) (F), leaf water potential (ψ L ) (G) and growth rate (G) (H) as plant responses.Environmental conditions are shown in the bottom row, including trends in air temperature (T a ) (I), relative humidity (RH) (J), incoming irradiance on a surface normal to the beam (I s ) (K) and zenith angle (solar elevation below the zenith; Φ) (L).Day-long simulations for this day (DOY 307) did not converge to a result that satisfied the boundary conditions for C and η (i.e.Δη = 0 and ΔC = 0 isolines do not intersect; see Supporting Information-Fig.S6, e.g. of intersecting isolines).

Figure 6 .
Figure 6.Instantaneous response of nocturnal stomatal conductance (g w,N ) to vapour pressure deficit (D L ; A), soil water potential (ψ soil ; B) and atmospheric CO 2 partial pressure (c a ; C) at three NSC storages (75 %, 100 %, and 125 % of a reference storage size of C = 60 mol).Simulations were performed at atmospheric CO 2 and O 2 partial pressures of 410 μmol•mol −1 (in A and B) and 207 mmol•mol −1 , respectively, RH = 0.5 (in Band C), a soil water potential of 0 MPa (in A and C), a solar elevation below the zenith of 0 radians, and an incoming irradiance of 0 W•m −2 , using parameters fromPotkay and Feng (2023) and Table1.Air temperature (T a ) was set to 25 °C for most calculations (e.g.leaf temperature, R d , λ), except for exclusively the calculation of the extensibility ( φ ), for which T a = 8.07 °C.

Figure S4 .
Comparison of new stem hydraulics scheme toPotkay and Feng's (2023) original scheme.

Figure S5 .
Maximum and minimum NSC isolines and NPP:GPP ratio isoline for given maintenance respiration and extensibility parameter combinations.

Figure S6 .
Example of numerical search for cyclic solution.

Figure S7 .
Annual trends from year-long and day-long simulations.FigureS8.Diel trends from year-long simulations during seasonal peak in net assimilation.

Figure S9 .
Diel trends from year-long simulations during seasonal peak in woody growth.

Figure S10 .
Combinations of NSC storages and air temperatures that result in realistic nocturnal stomatal conductance.

Table 1 .
Glossary of key variables and acronyms.

Table 2 .
(Poyatos et al. 2013;Salmon et al. 2015)ximumcarboxylation rate; J max , maximum electron transport rate; ψ L ; leaf water potential; R d , leaf dark respiration; RWC L , leaf relative water content; T L , leaf temperature; T a ; air temperature; k L , leaf conductance per leaf area; k S , stem conductance; k R , belowground conductance; G, growth rate; R M , stem and root maintenance respiration rate; C, whole-tee non-structural carbohydrate (NSCs) pool.Potkay and Feng -Optimizing stomatal conductance for turgor-driven growth early growing season, g w declines.Meanwhile |ψ L |, η, A n , C, E, λ and G rise until peaking onDOY 119, 146, 163, 172,  195(approximately at peak I s ), 215 and 220 (at peak T a ), respectively, and then decline.During the growing season, midday g w ≈ 0.1 mol•m −2 •s −1 , comparing well to the upper bound of measurements from our site on healthy Scots pine(Poyatos et al. 2013;Salmon et al. 2015).The diel changes in η (i.e.dη/dt) inversely mirror C with an ~80-day lag (i.e.high and low C later cause η to decline and rise between days, respectively), displaying the carbon-use strategy.Nonetheless, η is approximately constant, suggesting that g w is controlled more by NSC dynamics (C) than by variations in sinkdemand (η).Day-long simulations satisfied both of our cyclic boundary conditions for η and C V