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1 Suorting Information Wolf et al 73/nas65443 SI endix From Eq 4, i = a 6 Substituting this into [3] and rearranging gives = 6 2 ½6V + ð a + kþš + ½V a V ΓŠ, [S] where V is V R and Γ is the O 2 comensation oint: VΓ* + kr V R We calculate / from [S], using the imlicit function theorem: = ð a + kþ ðv a V ΓÞ 32 6V ð a + kþ [S2] Both otimization criteria [9] and [] can be exressed in the form =, which using [S2] roduces the generic otimization criterion, ð a + kþ ðv a 2V ΓÞ 32 6V ð a + kþ = ð a + kþ ðv a V ΓÞ = ½32 6V ð a + KÞŠ ð a + kþ ðv a V ΓÞ = 32 6V ð a + kþ, and using the fact from [S] that 62 6V = ð a + kþ ½V a V ΓŠ, we have 6 2 6V = 32 6V ð a + kþ = gs ð a + kþ ð32 6V Þ 6ðV Þ We now use a relationshi obtained from [S] that gs½v a V ΓŠ ð a + kþ ð6 6V Þ=, to convert the above equation to = g gs ½V a V ΓŠ s 6ðV Þ V " 6 2 V a Γ # = 6 2 ½V a V ΓŠ Tyical values for the arameters in mmol m 2 s and m are V 5, Γ* 3, R, and k 3 g,, and a 3, and so Γ 25 and V 4 With tyical values of such as (mmol m 2 s ) and of (mol m 2 s ) such as 5, we see that the second term on the RHS will be small unless stomates are nearly shut lso, we can aroximate /V as zero This yields sffiffiffiffiffi 6 ffiffiffiffi = [S3] a Γ For the WUEH, = λ E Θ ðψ L Þ dψ L, and for the MH: = Θ ðψ L Þ ψ L Using E = ½w w a Š from Eq 6, de/dg in the WUEH otimization criterion [9] is simly the saturation deficit: ½w w a ŠEqs6and 7 imly R ψ s ψ L KðψÞdψ ½w w a Š = fter differentiating both sides of this exression with resect to ψ L = D using the imlicit function theorem, Kðψ L Þ Thus, = λ½w w a Š ½w wašθ ðψ LÞ Kðψ L Þ for the WUEH otimization and = ½w wašθ ðψ LÞ Kðψ L qþ for the MH We now define a beta function ½w w as βðψ L Þ = aš Thus, the otimal stomatal conductance [S4] is sffiffiffiffiffi 6 gs ot βðψ L Þ ffiffiffi, [S4] a Γ w* w a qffiffiffiffiffiffiffiffiffiffiffi Kðψ where β WUEH ðψ L Þ = L Þ qffiffiffiffiffiffiffiffiffiffiffiffi λkðψ L Þ Θ ðψ L Þ for the criterion [9], and β OH ðψ L Þ = for the criterion [] To derive the full Kðψ L Þ Θ ðψ L Þ suite of formulas for the beta function in Table simly substitute K max for Kðψ L Þ and/or Θ min for Θðψ L Þ so that Θ ðψ L Þ is zero SI endix 2 Model Derivation Measurements show that desiccating leaves show a broad decline in transiration and stomatal conductance in resonse to lowered water otential as in Fig In a series of elegant theoretical and emirical aers, Peak, Mott, and coworkers develoed a new hysicochemical model of stomatal conductance (74 76), which contains several of the mechanisms resonsible for the effect of leaf water otential on stomatal aerture Fig S shows the elements of leaf structure and function included in this model In Peak and Mott s model, the water otential in a leaf at the terminus of a xylem element, ψ L, asses into the liquid vaor interface on the surface of the adjacent substomatal ore (think of a meniscus at the bottom of the ore, Fig S) It crosses a resistance between the xylem and the interface and so the water otential in the interface ψ i is less than ψ L (Fig S) The average water otential of the air next to the guard cells, ψ g, is lower than ψ i for two reasons: (i) a dro in temerature during evaoration into the substomatal ore and (ii) diffusion of a small amount of outside air into the ore We modified Peak and Mott s model to account for the effect of stomatal aerture on the diffusion of outside air into the ore (see below) ir in a substomatal ore is always almost comletely saturated This is why transiration can be modeled accurately as a constant times vaor ressure deficit, as in Eq 5 onetheless, the water otential in a guard cell is aroximately the water otential in the substomatal ore adjacent to the guard cell: ψ g ψ g, and the ressure P g, which determines guard cell turgor, is equal to the sum of ψ g and the osmotic otential of guard cell cytolasm π g (turgor ressure = P g = ψ g + π g ψ g + π g, Fig S) How can an evaoration rate calculated using the aroximation of saturated ore air (Eq 5) remain accurate throughout a eriod in which ψ g changes enough to oen and close stomates? The answer is that water otential declines raidly as air loses moisture For examle, a decline in the relative humidity (h) of ore air from fully saturated to 99%, which has little effect on the evaoration rate redicted by Eq 5, means that the water otential of ore air declines from MPa to aroximately MPa at 2, which causes a substantial decrease in guard cell turgor It is useful to think of the effects xylem and atmosheric water otentials on guard cell turgor described above as the extensor muscles of a leaf, which work to oen stomates when ψ L increases or vaor ressure deficit decreases Leaf structure also contains Wolf et al wwwnasorg/cgi/content/short/65443 of5

2 an oosinystem that can be thought of as flexor muscles, which work to close stomates in resonse to an increase in ψ L and to oen them in resonse to a decrease This is called the wrong-way resonse because it would, if unoosed, cause stomates to oen in resonse to desiccation (3) Stomatal aerture is the result of the balance between the relative strengths of the right-way and wrong-way resonses The wrong-way resonse is caused by mechanical advantage of the eidermal cells surrounding the guard cells (3) s eidermal cells dessicate, they shrink, which has the effect of ulling oen the stomata if left unoosed In Peak and Mott s models, liquid hase water moves easily from the water air interface of the substomatal ore to the cells of the mesohyll and eidermis, so that the water otential of the cells is equal to ψ i (ψ i = ψ m = ψ e, Fig S) Because the resistance from the xylem to the water air interface of the substomatal ore is small, ψ i, ψ m, and ψ e are within a few tenths of a megaascal of ψ L Unlike mesohyll or eidermal cells, stomatal guard cells in Peak and Mott s models are hysically isolated from the water in the eidermal and mesohyll cells, which surround them, and exchange water only in the gas hase in the substomatal ore The internal ressure inside the eidermal cells (P e = ψ e + π e ) exerts mechanical force on the guard cells, which ush back because of their own internal ressure (P g ) Stomatal conductance is simly roortional to the difference between these forces (73, 74), = χ P g mp e if Pg > mp e, and = otherwise, [S5] where m is a mechanical advantage caused rimarily by the fact that eidermal cells outnumber guard cells (m 2) If mp e > P g, then the stomate is literally squeezed shut by the eidermal cells, and if P g > mp e, then the guard cells internal ressure increasingly forces the stomate oen as P g increases relative to P e Peak and Mott s work is very elegant, but has a major limitation in that the sensitivity to water otential in the leaf xylem is in the wrong direction: Stomata oen in resonse to dehydration, owing to the mechanical advantage of the eidermis over the guard cell Because both ψ g and ψ e are equal to ψ L minus a small term (caused by one or more dros in otential), achangeinψ L induces the same change in both ψ g and ψ e and thusthesamechangeinp g and P e (all else being equal: Δ ψ L = Δ P g = Δ P e ) But because the mechanical advantage (m) ineq is greater than one, the change in P e has a larger effect on stomatal aerture than the equal change in P g, causintomates to oen when xylem otential falls and close when xylem otential rises Here, stomates actively regulate the osmotic otential of guard cells (π g ) with an energy-consuminolute um Part of the urose of the active regulation is to overcome the wrong-way resonse that would occur if the assive forces were unoosed lthough the algorithm that controls this um is not fully understood, suerabundant data on the behavior of stomates constrain its functional form (7) Water demand In what follows, we assume that leaf boundary layer conductance [g b (s w ) from Eqs 3 and 5 in the main text] is infinite because this leads to a significantly simler formula without changing its qualitative behavior Thus, E demand is defined as the roduct of the bulk leaf conductance, gðunitsm air =rea Leaf =timeþ and the gradient in water vaor from inside a substomatal ore ðw, units M H₂O M air Þ to the air w a : E demand = w w a = gs Δw [S6] Leaf conductance is determined in large measure by the conductance of air across the ores of the stomata, which is actively regulated by higher lants in resonse to revailing environmental conditions (eg, light, O 2, ambient humidity, soil moisture) as well as hormonal signals (such as bscisic acid) Stomatal conductance is a bulk henomenon that can be thought of as a combination of the stomatal density er leaf, which may vary over growth and develoment timescales (89), and the aertures of the individual stomata themselves, which vary on timescales of minutes to hours (9) Physically based models of (eg, ref 3 roceeds from observations that aerture is determined by guard cell ressure P g and eidermal cell ressure P e (6) The stomatal aerture is ositively related to P g, oening as the guard cell becomes turgid, but negatively related to P e because the eidermal cells have a mechanical advantage m over the guard cells, = χ P g mp e if Pg > mp s and = otherwise, [S7] where χ is a roortionally constant maing guard cell ressure into stomatal conductance Because turgor ressure P is the sum of a (ositive) osmotic otential π and a (usually negative) water otential ψ, P = ψ + π (9), [S2] may be written = χ ψ g mψ e + χ π g mπ e [S8] We regard π e as a constant, reresenting the turgor loss oint of the leaf The water otentials (ψ g and ψ e ) are governed by a hysical hydrodynamic system that exhibits assive behavior in resonse to water otentials in the atmoshere or ustream from the leaf, whereas the guard cell osmotic otential (π g ) is determined by active biochemical rocesses s Buckley (3) analyzed in detail, the existence of the mechanical advantage of the eidermis over the guard cells requires an active mechanism to overcome a wrong-way resonse of stomatal conductance to leaf hydration or dehydration ctive rocesses Buckley and coworkers have, at various times, used different forms for the active resonse of the guard cell osmotic otential, all linking π g to π e by some factor B : π g = BP e π a ð73þ; π g = BP e ð3þ; π g π e = BP e ð77þ: [S9] [S] [S] To our knowledge, there is no a riori or hysical reason to choose one of these over another or to assume a linear form in the first lace Moreover, suerabundant observations show that increases linearly with hotosynthetic rate and is inversely roortional to *(T L ), where s is the O 2 concentration at the guard cell surface These behaviors must be mediated through changes π g and so must be built into the functional form for its regulation The biochemistry behind the regulation of the solute um that adjusts the osmotic otentials of guard cells is still highly uncertain, and so we roose to assume a functional form that yields the observed macroscoic behavior of stomates For now, we assume that π g = π e + f P e, ffiffiffiffiffiffiffiffiffi, where f is an increasing function of both P e and ffiffiffiffiffiffiffiffiffi Obviously, the regulation of π g requires energy exenditure and so one ossible way to interret f P e, ffiffiffiffiffiffiffiffiffi is as the adjustment of πg above π e, which otimally overcomes the wrong-way resonse caused by P e, but only if sufficient TPis available from hotosynthesis The hyerbolic deendence on might reflect cometition for TP between the dark reaction of hotosynthesis and solute um that regulates π g Thus, with π g = π e + f P e, ffiffiffiffiffiffiffiffiffi, Eq S3 becomes Wolf et al wwwnasorg/cgi/content/short/ of5

3 = χ ψ g + π e + f P e, mðψ e + π e Þ [S2] ψ g = RT L wi w ðt L Þ σðþð hþ, [S8] Later in SI endix 2, we describe constraints on the functional form of f(p e ) that come from emirical studies of stomatal conductance in desiccating leaves Passive rocesses In a grou of thoughtful exerimental and theoretical aers (74 76), Peak, Mott, and coworkers resented a new model for stomatal conductance, in which ψ g is in equilibrium with the water vaor in the stomatal ore adjacent to it, ψ g, and has no exchange of water with the eidermal cells adjacent to it In Peak and Mott s model, there is a relatively small resistance ðr Þ to water transort between the leaf vascular system and outside the vascular system () Outside the vascular system, the water otential of the eidermis (ψ e ), the mesohyll (ψ m ), and the liquid vaor interface inside the stomatal ore (ψ i ) are all considered equal in Peak and Mott s model, although there is considerable evidence of substantial resistance outside the xylem, which would make ψ e < ψ m (92 94) The extra resistance is easy to add without changing the qualitative outcome Recalling Fick s law, E = Δ ψ =r, where E demand is the evaorative flux out of the leaf from Eq S6: ψ i = ψ m = ψ e = ψ L r E demand = ψ L r Δw [S3] When the water evaorates from the liquid vaor interface into the ore, it exeriences a small dro in temerature and thus a second small dro in otential (74), ψ i = ψ L ðr + r 2 Þ Δw, [S4] where ψ i is the water otential of air in the substomatal ore immediately adjacent to the water vaor interface Because a small amount of outside air enters the ore and mixes with the nearly saturated air inside it, Peak and Mott reresent the water vaor artial ressure immediately adjacent to the guard cells inside the ore, w g, as a weighted average between the water vaor outside the leaf, w a, and the vaor immediately adjacent to the liquid vaor interface w i, w g = σw a + ð σþw i, [S5] where σ is the mixing ratio, which is thought to be small lthough the mixing ratio is constant in Peak and Mott s model, because outside air enters through the stomatal aerture, we assume that the mixing ratio is an increasing function of stomatal aerture σð Þ, such that σð = Þ = By definition, water otential ψ is ψ = RT w wðt LÞ, where R is the universal gas constant, is molar volume of liquid water, and w*(t) is the water content of saturated air at leaf temerature T L and so ψ g = RT L wg = RT L w ðt L Þ = RT L wi w ðt L Þ σðþ w ðt L Þ σðgs Þw a + ð σð ÞÞw i wi w a w ðt L Þ [S6] Because the air adjacent to the water vaor interface is nearly w saturated, i wðt LÞ = «, where «is small: ψ g = RT L wi ðw w ðt L Þ σðg ðtl Þ w a Þ sþ «[S7] w ðt L Þ ssuming the roduct of the two small numbers, σð Þ and «, is negligible, where h is relative humidity, h = w a /w* Because σð Þ is small, ψ g RT L wi w ðt L Þ ψ i σð Þ RT L ð hþ σð Þ RT L ð hþ «Finally, we aroximate the mixing ratio, σð Þ, by exanding to first order around =, where σ = dσðþ d : ψ g ψ i σ RT L ð hþ [S9] fter substituting [S3] into [S9], and recalling that ψ g is in equilibrium with ψ g, ψ g ψ L ðr + r 2 Þ Δw σ RT L ð hþ [S2] Substituting [S2] into [S], ψ L ðr + r 2 Þ Δw σ RT L ð hþ + π e = χb, + f P e,, mðψ e + π e Þ and using [S2], = χb + f [S2] ψ L ðr + r 2 Þ Δw σ RT L ð hþ + π e ψ L r Δw + π e, mðψ L r Δw + π e Þ [S22] Because the dro in otential from the terminus of the xylem to the water vaor interface is small [usually of order MPa or less (74)], we can aroximate f(ψ L r Δw + π e )as where f ψ L + π e, r Δwf ψ L + π e,,! f ψ L + π e, ffiffiffiffiffiffiffiffiffiffi s V Γ f ψ L + π e, =, P e and so [S22] may be aroximated as ψ L ðr + r 2 Þ Δw σ RT L = χb r Δwf ψ L + π e, ð hþ + π e + f fter solving Eq S23 for, we have ψ L + π e, mðψ L r Δw + π e Þ [S23] Wolf et al wwwnasorg/cgi/content/short/ of5

4 h χ f ψ L + π e, ffiffiffiffiffiffiffiffiffi = hh + χ f ψ L + π e, ffiffiffiffiffiffiffiffiffi if > and = otherwise, where i ðm Þðψ L + π e Þ i i ðm Þ r + r 2 + Ω Δw RT L Ω = σ w ðt L Þ [S24] The functional form of Eq S24 imlies a simle functional form for f P e,, which controls the regulation of guard cell osmotic otential [recall that π g = π e + f P e, ffiffiffiffiffiffiffiffiffi ] If we assume f P e, = βðp e Þ + P e ðm Þ, [S25] where βðp e Þ is an increasing function of P e, then [S24] becomes = h + χ χγβðψ L + π e Þ i if > and = otherwise, β r + r 2 + Ω Δw ffiffiffiffiffiffiffiffiffi [S26] where β = dβðψ L + π eþ dðψ L + π eþ The assumed functional form for f P e, ffiffiffiffiffiffiffiffiffi makes gs roortional to and inversely roortional to Γ, as observed Because r is small, the following simler equation is accurate to within a few ercent: = χ ffiffiffiffiffiffiffiffiffi βðψ L + π e Þ + χ½r 2 + ΩŠΔw if > and = otherwise [S27] The final unresolved issue is the functional form of βðp e Þ large number of emirical relationshis have been reorted (7) ll show a more or less gradual decline of as ψ L decreases from zero Most start concave-down for ψ L close to zero and then switch to concave-u as in Fig (7) Linear βðp e Þ functions, like those in Buckley (3) or Buckley and coworkers (73, 77), largely lack both the initial concave-down shoulder at high ψ L and the concave-u ortion at low ψ L ot surrisingly, Weibull-like functions of jψ L j have concave-down and concave-u ortions in the right laces (Fig ) Summary The desiccation resonses in Fig are intriguing because the most direct way for a leaf to achieve otimal regulation of carbon lost to hydraulic costs would be to actively regulate stomatal aerture in resonse to changes in ψ L Our fusion model is derived from state-of-the-art models of the assive and active regulation of stomatal aerture described in a series of aers by Peak, Mott, and coworkers (74 76) and Buckley and coworkers (3, 73, 77) The assive resonses of stomates to surrounding hydraulic conditions are entirely mechanistic in these models, but active regulation is still constrained henomenologically (73, 77) The model roduced by our synthesis and extension of this literature should thus be viewed as a mechanistic and henomenological hybrid It includes erhas the simlest functional form for the active regulation of stomates that causes the Weibull-like relationshi of and jψ L j and the widely observed roortionality of and ffiffiffiffiffiffiffiffiffi, where s is the O 2 concentration at the guard cell surface (as in refs 2 and 5) The model is described in detail below and simlifies aroximately to χ ffiffiffiffiffiffiffiffiffi βðψ = s Γ L + π e Þ h i if > and = otherwise, + χ r 2 Δw + σ RTL ð hþ [S28] where r 2 is a water otential dro across the liquid vaor interface; Δw is the saturation deficit on the RHS of Eq 5; h is the relative humidity w a /w*(t L ); σ, R, and are constants; and βðψ L + π e Þ is a Weibull-like function of jψ L j that declines gradually from a maximum at ψ L = asψ L becomes increasingly negative and is then concave-u at very negative ψ L values, maintaininome nonzero level of conductance Like Leuning s emirical model (5), Eq S28 includes an aroximately hyerbolic deendence of on vaor ressure deficit (which traces to the dro in water otential from the xylem to the substomatal ore) Like the Ball and Berry model (2), the formula also deends on relative humidity, but with hyerbolic deendence on T L ( h), rather than being roortional to h However, unlike the Ball and Berry or Leuning models, Eq S28 includes a function, βðψ L + π e Þ, which controls the desiccation resonse of stomates to changes in water otential Wolf et al wwwnasorg/cgi/content/short/ of5

5 Fig S Schematic of the biohysical stomatal conductance model that shows resistances to water transort from leaf xylem to the atmoshere, as well as active and assive stomatal resonses P e and P g are internal ressure in the eidermal and guard cells, resectively Water otentials include leaf xylem (ψ L ), liquid vaor interface (ψ i ), the air next to the liquid vaor interface (ψ i ), the air next to the guard cells (ψ g ), guard cells (ψ g ), and eidermis (ψ e ) π e and π g are osmotic otentials of the eidermis and guard cells, resectively Wolf et al wwwnasorg/cgi/content/short/ of5