To interpret the observed stomatal response delays, we couple the gas exchange model with a mathematical description of guard cell energetics. This is necessary because more responsive guard cells (i.e. faster stomatal movements) come at a higher energetic cost of operation, which, in turn, has an impact on the net carbon gain.
Stomatal movement costs Stomatal movements in response to variable environmental conditions are triggered by changes in guard cell osmotic potential, which, in turn, drive changes in water content and hence guard cell volume. The transport of osmoticum (chiefly K+ and sucrose) through the ion channels of the guard cell membrane is responsible for changes in osmotic potential (Zeiger, 1983; Vavasseur & Raghavendra, 2005; Shimazaki et al., 2007). During stomatal opening, the extrusion of H+ by a proton pump and malate2− synthesis inside the guard cell (Assmann et al., 1985; Assmann, 1999; Hanstein & Felle, 2002; Shimazaki et al., 2007) cause the entrance of osmoticum through the activated inward channels (Roelfsema & Hedrich, 2005; Vavasseur & Raghavendra, 2005). When stomata close, outward ion channels are activated and, presumably, the activity of the proton pump is reduced (Hanstein & Felle, 2002). Thus, the proton pump operation and malate synthesis require energy (as ATP) to proceed, whereas, in a first-order analysis, stomatal closing may be regarded as a comparatively passive mechanism (Assmann & Zeiger, 1987; Roelfsema & Hedrich, 2005). The current limited understanding of the precise mechanism driving the signaling and stomatal movements prevents a mechanistic description of all energetic costs of stomatal response to fluctuating light. Hence, in the following, a simplified representation of stomatal opening mechanics and energetics is employed to estimate the stomatal movement cost C(t). This modeling approach combines the limited experimental evidence available on stomatal movement energetics with existing theories and experimental results relative to gas diffusion through stomata. This approach is the first dynamic model of stomatal movement energetics, as previous attempts were restricted to the total energetic cost of a single opening (Assmann & Zeiger, 1987) or only considered steady-state conditions (Dewar, 2002; Buckley et al., 2003).
The opening cost per unit time C(t) (in units consistent with gas exchange measurements, i.e. μmol CO2 m−2 s−1) equals the marginal cost dc for a stomatal conductance change dg multiplied by the corresponding change in g per unit time, that is, C = dc/dg × dg/dt. The first term in the product accounts for the costs involved at different levels of stomatal aperture (i.e. c is expressed as μmol m−2), and the second term represents the speed of stomatal changes (see Eqn 1). To link these theoretical developments with measurable quantities and existing theories, the term dc/dg is further decomposed as dc/dμ × dμ/dg, where μ is the mean stomatal aperture and c (μ) is the cost to achieve a given μ.
Because of the 1 : 1 stoichiometry of ion uptake and proton extrusion (Raschke, 1975), the shape of the relationship c (μ) can, in principle, be inferred from experimental data linking stomatal aperture and guard cell cation concentration, and hence total ion uptake (Hsiao, 1976). Such dependence suggests that c (μ) can be described as:
- (Eqn 9)
where γ is the cost per unit leaf area needed to fully open the stomata and ν is a shape factor (Fig. 2a). To estimate γ, we consider the energetic costs of proton extrusion and malate synthesis for an individual stomata (following the rationale of Assmann & Zeiger, 1987), and scale the cost up to the leaf level, consistently with the other flux calculations. To proceed, we start from proton extrusion rates per unit stomatal aperture, which range between 0.2 and 5.14 μmol H+ μm−1 per stomata (data for Vicia faba and Commellina communis after Raschke & Humble, 1973; Gepstein et al., 1982; Inoue & Katoh, 1987). If 1 mol ATP is assumed to be necessary to extrude 2 mol H+ (Assmann & Zeiger, 1987) and the efficiency of ATP production (either through respiration or photophosphorylation) is c. 5 mol ATP per mol CO2 (Taiz & Zeiger, 2006), a range of opening costs of c. (0.64–5.58) × 10−7 μmol CO2 μm−1 per stomata are obtained. In the absence of more precise data, this range is assumed to account for most of the variability in guard cell metabolism when no water stress is present. To scale up the costs to the leaf level, these costs need to be multiplied by the species-specific maximum stomatal aperture and stomatal density. Using the values reported by Larcher (2003), a range of γ ≈ 13–3350 μmol CO2 m−2 of leaf area for the full aperture of all stomata is obtained. When considering a particular species with specific values for the maximum stomatal aperture and stomatal density, a narrower range may be found.
Figure 2. (a) Relationship between aperture cost and stomatal aperture (Eqn 9). (b) Nonlinear effect of aperture on stomatal conductance (Eqn 11). (c) Carbon cost per unit time of stomatal opening, as a function of stomatal conductance (Eqn 12), for two stomatal opening response times (black lines, 4 min; gray lines, 8 min). For illustration, (b) also shows data for Aegopodium podagraria (circles; Kaiser & Kappen, 2000), Vicia faba (squares; Kaiser, 2009) and Zebrina pendula (dots; Bange, 1953), collected in well-ventilated chambers. Fitting Eqn 12 to the presented data leads to exponents β = 0.34 (dotted gray line), 0.48 (dash-dotted gray line) and 0.80 (solid gray line) for A. podagraria, V. faba and Z. pendula, respectively. The 1 : 1 line (solid black line) can be interpreted as Stefan’s diameter law (Brown & Escombe, 1900). In (c), gmax = gopt = 0.5 mol m−2 s−1 (i.e. saturating light conditions during sunflecks).
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The shape factor ν in Eqn 9 can be set to unity when the cost increases linearly with aperture, or can be greater than unity when the extrusion of protons becomes increasingly costly as the aperture reaches its maximum and a stronger proton gradient across the guard cell wall has to be overcome, as suggested by Assmann & Zeiger (1987). In the following analyses, it is conservatively assumed that a moderate nonlinearity prevails (i.e. ν = 2).
The relationship between stomatal conductance and aperture, g(μ), can be inferred from experimental data (Bange, 1953; Ting & Loomis, 1965; van Gardingen et al., 1989; Kaiser & Kappen, 2000, 2001; Kaiser, 2009) or a number of theories (Brown & Escombe, 1900; Patlak, 1959; Cooke, 1967; Parlange & Waggoner, 1970; Troyer, 1980; Lushnikov et al., 1994; Vesala et al., 1995). In general, these theories and experimental studies (Fig. 2b) predict that stomatal conductance and aperture scale as:
- (Eqn 10)
where the exponent β > 0 depends on the geometry of the stomata (its shape and depth) and the wind velocity (Bange, 1953; Lee & Gates, 1964; Ting & Loomis, 1965; Waggoner & Zelitch, 1965; Parlange & Waggoner, 1970; van Gardingen et al., 1989; Kaiser, 2009). This expression recovers the classical Brown & Escombe’s (1900) result when β = 1 (often referred to as Stefan’s diameter law), although interferences between adjacent stomata, elongation of the stomatal pore and depth of the diffusive pathway may yield β < 1 even for high boundary layer conductance (Ting & Loomis, 1965; Parlange & Waggoner, 1970; van Gardingen et al., 1989; Kaiser, 2009). For a generic β, the relative aperture is then computed as:
- (Eqn 11)
where gmax is the maximum stomatal conductance. A direct consequence of the nonlinearity in μ(g) is that a small change in aperture when stomata are closed results in a relatively large conductance gain. At high apertures, the gain in conductance decreases.
Finally, obtaining dg/dt from Eqn 1, an analytical expression for the total instantaneous costs of stomatal opening can be derived as:
- (Eqn 12)
where C(t) is set to zero when dg/dt ≤ 0. In Eqn 12, the parameter κ = νβ−1 is in the range 2–6 based on realistic values of β and ν. Here, we consider an intermediate value κ = 4 that accounts for mild nonlinearities in both c(μ) and μ(g) relationships (Eqns 9, 11; Fig. 2c), and gmax can be estimated from gas exchange data. It should be noted that, despite τop being the only time constant explicitly included in the above formulation of C(t) (because stomatal closure is assumed to be passive), because of the relevance of the previous history of stomatal conductance, the integrated cost over a certain time period depends on both time constants of the stomatal response. Indeed, rapidly closing stomata increases the total costs of stomatal movements, because low g at the end of the dark period causes higher opening costs at the beginning of the subsequent light period.