Empirical models for aperture and volume versus turgor pressures
Observations of stomata of Tradescantia virginiana by Franks et al. (1998) allow stomatal aperture (a, µm) to be estimated from empirical functions of epidermal and guard cell turgor pressure (P_{e} and P_{g}, respectively, MPa):
(A1)
The functions f_{1}(P_{g})and f_{2}(P_{g}), which relate stomatal aperture to guard cell turgor at zero and maximum epidermal turgor, respectively (maximum epidermal turgor is considered equal to epidermal osmotic pressure, π_{e}, MPa) are given by Eqns A2 and A3:
(A2)
(A3)
The parameters ζ_{1} and ζ_{2} (both MPa) control the curvature of a versus P_{g}; a_{m} is the maximum aperture (µm); ξ is the unitless proportion by which maximum stomatal aperture is reduced at full epidermal turgor (π_{e}, MPa) relative to zero epidermal turgor; P̂_{g} pis the value of guard cell turgor (MPa) at which the curvature of a versus P_{g} becomes negative at high epidermal turgor (when P_{e} = π_{e}); and the second term in parentheses in Eqn A3 ensures the pore is closed when both guard cell and epidermal turgor are zero.
The data of Franks et al. (2001) allow guard cell volume (V_{g}, µm^{3}) to be inferred from P_{g}, using an empirical model of the form:
(A4)
where the terms c_{1}, c_{2} and c_{3} are constants for a given guard cell pair. Values for all of the parameters in Eqns A1–A4 are given in Table 1, and our parameter estimation procedures are discussed below under the heading Parameter estimation. We assume the relationship between P_{g} and V_{g} is not affected by P_{e}; no data exist to test this hypothesis, which is applied here for simplicity. Note this is equivalent to specifying a value of unity for Cowan’s (1977) variable σ.
Table 1. Parameter values Parameter name  Symbol  Value 


Maximum stomatal aperture  a_{m}  18·11 µm^{a} 
a versus P_{g} curvature parameter, low P_{e}  ζ_{1}  1·598 MPa^{a} 
a versus P_{g} curvature parameter, high P_{e}  ζ_{2}  0·7694 MPa^{a} 
Relative reduction in a_{m} by high P_{e}  ξ  0·509 (unitless)^{a} 
Inflection point in a versus P_{g} at high P_{e}  _{g}  2·015 MPa^{a} 
Polynomial constants for V_{g} versus P_{g}  c_{1}  –88·469 µm^{3} MPa^{−2}^{ b} 
Polynomial constants for V_{g} versus P_{g}  c_{2}  726·846 µm^{3} MPa^{−1}^{b} 
Polynomial constants for V_{g} versus P_{g}  c_{3}  4813·476 µm^{3}^{b} 
Source water potential  ψ_{s}  0 MPa^{a} 
[sourcetoepidermis resistance]·[aperture to conductance scaling factor]  r_{se}·γ  0·001309 mol air mmol^{−1} H_{2}O µm^{−1}^{c} 
Epidermal osmotic pressure  π_{e}  0·5252 MPa^{c} 
Gas constant  R  8·31441 MPa µm^{3} pmol^{−1} K^{−1} 
Leaf temperature  T  298 K 
Mathematical development
We assumed that the water potential of the epidermal cells (ψ_{e}, MPa) is in equilibrium with the water potential of other parallel evaporating sites (Nonami et al. 1990), and that this common water potential is determined by a balance between xylem supply (liquidphase flow from a source at potential ψ_{s}[MPa] through a resistance r_{se}[MPa (mmol H_{2}O mleaf2 s1)1]) and transpiration (vapourphase flow at a rate E[mmol H_{2}O mleaf2 s1]) from inner surfaces of cells in the leaf. E is defined as the product of conductance (g, mol air mleaf2 s1) and the evaporative gradient (D, mmol H_{2}O mol^{−1}air; formally, the mole fraction gradient of water vapour from leaf to atmosphere). Stomatal conductance (g_{s}, mol air mleaf2 s1) is assumed to be proportional to aperture by a constant factor γ (mol air mleaf2 s^{−1} µm^{−1}), and the boundary layer resistance is considered zero for simplicity (g = g_{s}). Collectively, these assumptions (Eqn A5) imply that ψ_{e} is a decreasing linear function of the rate of water loss (γaD), with slope given by r_{se} (Eqn A6):
(A5)
(A6)
Equation A6 can also be expressed in terms of epidermal turgor pressure (P_{e}) and epidermal osmotic pressure (π_{e}, MPa, assumed constant in the simulations presented here), for the purposes of predicting water flow across isothermal phase boundaries:
(A7)
This equation emerges from standard water relations, given the assumptions listed above. An independent expression for P_{e} can be obtained by solving the empirical/mechanical equation (Eqn A1) of Franks et al. (1998) that relates aperture to P_{e} and P_{g}:
(A8)
The existence of two independent expressions for P_{e} allows its elimination, removing one degree of freedom. We equate the righthand sides of Eqns A7 and A8 and solve for aperture:
(A9)
Equation A9 shows that the theoretical hydraulic relationship between a, P_{e} and D (Eqn A7) reduces the empirical mechanical relationship between a, P_{e} and P_{g} (Eqn A1) to a semiempirical hydromechanical function allowing a to be inferred from only P_{g} and D (Eqn A9); (P_{g} and D are considered the only dependent variables because all other terms in Eqn A9 are assumed constant). The hydraulics and mechanics of guard cells also represent an independent set of theoretical and empirical constraints on D, P_{g} and a, and link these variables to ψ_{e}, guard cell water potential (ψ_{g}, MPa), the hydraulic resistance between guard and epidermal cells (r_{eg}, MPa (mmol H_{2}O m^{−2} s^{−1})^{−1}), guard cell volume (V_{g}), and guard cell osmotic content (n_{g}, pmol). One fundamental theoretical relationship among these variables is the equality of ψ_{g} with P_{g} − π_{g}. A gradient/resistance model of water flow provides another constraint: at steadystate; any flow from epidermal to guard cells (F, mmol H_{2}O mleaf2 s^{−1}) is balanced by evaporation from guard cells, and is determined by r_{eg} and the difference between ψ_{g} and ψ_{e}. Defining the rate of evaporation from guard cells (E_{g}, mmol H_{2}O mleaf2 s^{−1}) as a fraction (K, unitless) of the total rate of transpiration, E, the steadystate flow through the guard cell evaporating site is given by Eqn A10:
(A10)
The magnitude of K is a key factor distinguishing the two hypotheses under discussion in this study: the n_{g}regulation hypothesis does not require any drawdown between ψ_{e} and ψ_{g}, so K can be assumed to equal zero, but the r_{eg}regulation hypothesis does require a drawdown, and therefore positive F and K. However, for the purposes of this study, neither K nor the aperture/conductance scaling factor, γ, need to be considered in isolation from r_{eg}, because these three terms only appear as a product (K·r_{eg}·γ) in the critical equations. In solving Eqn A10 for guard cell water potential or turgor pressure, the product of K, r_{eg} and γ can thus be replaced by a single variable, r_{eg}*:
(A11)
(A12)
It is more illuminating in the present context to express π_{g} in terms of guard cell osmotic content, n_{g}, because, whereas π_{g} is influenced by a suite of hydraulic factors, n_{g} is directly controlled by metabolic processes. Recognizing that V_{g} can be inferred from P_{g} using the data of Franks et al. (2001) (Eqn A4), and that π_{g} is a simple function of V_{g} and n_{g} in ideal dilute solutions, π_{g} can be expressed as a function of n_{g} and P_{g} (Eqn A13):
(A13)
(A12b)
We now have seven variables (a, ψ_{s}, P_{g}, D, r_{se}, r_{eg}*, and n_{g}) and two constraints among these variables (Eqns A9 and A12b), leaving five degrees of freedom. To perform the semiempirical simulations presented in this study, we closed the system by using adding four empirical constraints (measurements of a, D, and ψ_{s}, and estimates of r_{se} · γ, in intact plants; details are provided in the Materials and Methods section in the main text) and one hypothetical constraint, the latter representing either the n_{g} or r_{eg}regulation mechanism for the humidity response. To represent the guard cell osmotic regulation hypothesis, r_{eg}* was set equal to zero, and to represent the drawdown hypothesis, π_{g} was held constant at each of three values (2·5, 3·0 and 3·5 MPa). (These choices of hypothetical constraints are explained in the next section of the Appendix, where the mathematical criteria for each hypothesis are developed.) The remaining variable (n_{g} or r_{eg}*, respectively) was then determined by iterative solution of Eqns A9 and A12b, as described below, and plotted against time and other system variables to show the inferred behaviour of the hypothetical control variable during the stomatal response to humidity.
Care is needed in implementing Eqns A9 and A12b in a numerical simulation, because, although P_{e} is not explicitly constrained to be nonnegative in the preceding derivation, P_{e} remains implicit in Eqns A9 and A12b. In conditions where plasmolysis occurs, P_{e} will implicitly become negative unless it is explicitly calculated by the computer code and constrained to be nonnegative; note that in this case, epidermal osmotic pressure must also implicitly be allowed to increase with further decreases in epidermal water potential. However, when P_{e} is zero, Eqn A9 may be replaced by Eqn A2, so the code never needs to change π_{e} explicitly.
Derivation of the mathematical criteria for the mechanism of the feedback humidity response
Stomatal aperture declines as the evaporative demand increases. Under most conditions, this also causes transpiration rate (E) to increase with D, consistent with evidence that stomata respond directly to the rate of water loss from leaves (Mott & Parkhurst 1991). This is often described as a feedback response of g to E, or of E to D, which contrasts with occasional reports of an apparent ‘feedforward’ response in which E declines with increasing D at high values of D (Farquhar 1978; Monteith 1995; Franks et al. 1997). In the following analysis, we consider only the ‘feedback’ domain of the response, because the mathematical criteria for producing a decline of aperture with increasing E are much simpler.
To produce the observed steadystate and transient ‘wrongway’ responses of stomatal aperture to humidity, we must express these responses mathematically, in the context of the epidermal mechanical advantage. Formally, changes in epidermal turgor pressure must be overcompensated by changes in guard cell turgor to cause a decline in aperture with transpiration rate. The magnitude of this overcompensation is given by the ratio of the sensitivities of aperture to epidermal and guard cell turgors:
(A14)
(A15)
Note that the mechanical advantage, formally defined by Eqn A15, is highly sensitive to both P_{g} and P_{e} (Fig. 1b). From Eqns A14 and A15, the total dependence of aperture on transpiration rate, E, can be expressed in differential form as:
(A16)
Noting that E=γaD, with γ assumed constant in the present study, we can rewrite Eqn A16 as:
(A16b)
This transformation will clarify later steps in the derivation, by preserving the linkage of K, r_{eg} and γ in the variable r_{eg}*, and of r_{se} and γ as a product (which was estimated as a single parameter; see Materials and Methods). Therefore, since ∂a/∂P_{g} is always positive (Sharpe et al. 1987; Franks et al. 1998) in Fig. 1a, the parenthetical term in Eqn. A16b must be negative if aperture is to decline with increasing transpiration rate:
(A17)
(A18)
(A19)
By combining Eqns A17, A18 and A19, we identify the following general criterion:
(A20)
Equation A20 states that, in order for guard cell turgor to overcome the residual mechanical advantage (m − 1), at least one of the following two quantities must always be sufficiently large: (i) a metabolically controlled decline in guard cell osmotic pressure (–dπ_{g}/d[aD]), or (ii) an epidermalguard cell drawdown factor (r_{eg}*) and its sensitivity to relative changes in transpiration rate (dr_{eg}*/d[lnaD]). These requirements are weighted inversely by the hydraulic supply resistance to the epidermis and its sensitivity to normalized transpiration (the denominator on the left side of Eqn A20).
Equation A20 can be greatly simplified under conditions where transpiration does not substantially alter hydraulic supply, either by causing source water potential (ψ_{s}) to decrease or by drawing down leaf water potential enough to cause cavitation in the hydraulic supply pathway to the epidermis, decreasing r_{se}. If ψ_{s} and r_{se} are constant, then from Eqn A5,
(A21)
The parenthetical quantity in Eqn A21 can be multiplied by the first term in Eqn A20 to give
(A22)
Invariance of r_{se} also eliminates the second term in the denominator of Eqn A20, which, together with Eqn A22, simplifies Eqn A20 to
(A20b)
The hypothesis that the humidity response is due solely to regulation of guard cell osmotic pressure, π_{g} (with no need for a water potential gradient between epidermal and guard cell) is best represented by the assumption that r_{eg}* is zero, which can be interpreted in two ways: no evaporation occurs from guard cells (K = 0) or the resistance between epidermal and guard cells is negligible (r_{eg} = 0). This hypothesis reduces the criterion to Eqn. A20c, which states that changes in guard cell osmotic pressure must outpace changes in epidermal turgor by a factor equal to the residual mechanical advantage (m − 1):
(A20c)
The alternative hypothesis, that the humidity response is due solely to nonzero r_{eg} (with no assistance from active regulation of guard cell osmotic pressure), is best represented by the assumption that π_{g} is constant (dπ_{g} = 0). In this case, the criterion reduces to Eqn. A20d,
(A20d)
which states that the sum of the epidermalguard cell hydraulic drawdown factor (r_{eg}*) and its sensitivity to relative changes in transpiration rate, both expressed relative to the sourcetoepidermis hydraulic resistance factor (r_{se} · γ), must exceed the residual mechanical advantage.