Blackwell Science, LtdOxford, UKPCEPlant, Cell and Environment0016-8025Blackwell Science Ltd 2003? 2003 261017671785 Original Article Stomatal model T. N. Buckley et al. Plant, Cell and Environment (2003) 26, 1767–1785 A hydromechanical and biochemical model of stomatal conductance T. N. BUCKLEY1, K. A. MOTT2 & G. D. FARQUHAR1 1Environmental Biology Group and Cooperative Research Centre for Greenhouse Accounting, Research School of Biological Sciences, The Australian National University, GPO Box 475, Canberra City, ACT 2601, Australia and 2Department of Biology, Utah State University, Logan, UT 84322–5305, USA ABSTRACT mimic. This limits their usefulness as tools for probing sto- matal and leaf functioning and constrains the confidence A mathematical model of stomatal conductance is pre- with which their predictions can be extended to future cli- sented. It is based on whole-plant and epidermal hydrome- mates. To address these limitations, several authors have chanics, and on two hypotheses: (1) the osmotic gradient attempted recently to model g in a more mechanistically across guard cell membranes is proportional to the concen- explicit fashion (e.g. Dewar 2002; Gao et al. 2002). However, tration of ATP in the guard cells; and (2) the osmotic gra- those models were based on assumptions about epidermal dient that can be sustained per unit of ATP is proportional water relations and stomatal hydromechanics that are incon- to the turgor pressure of adjacent epidermal cells. In the sistent with recent experiments and they calculated guard present study, guard cell [ATP] is calculated using a previ- cell osmotic pressure (pg) from irradiance or photosynthetic ously published model that is based on a widely used variables in a phenomenological fashion, much like the mesophyll photosynthesis. The biochemical model of C3 Jarvis and Ball–Berry models (Jarvis 1976; Ball et al. 1987) conductance model for Vicia faba L. is parameterized and discussed above. Our goal was to develop and present a tested As with most other stomatal models, the present model for g that overcomes some of these limitations. model correctly predicts the stomatal responses to varia- Many stomatal responses are driven by changes in pg, tions in transpiration rate, irradiance and intercellular CO2. which is determined partly by solute influx in response to Unlike most other models, however, this model can predict a proton-motive force created by plasma membrane H+- the transient stomatal opening often observed before con- ATPases (e.g. Tominaga, Kinoshita & Shimazaki 2001). ductance declines in response to decreases in humidity, soil Although this is a well-established paradigm in stomatal water potential, or xylem conductance. The model also physiology, it is rarely incorporated explicitly into models explicitly accommodates the mechanical advantage of the of stomatal conductance. One of the few attempts to do so epidermis and correctly predicts that stomata are relatively (Farquhar & Wong 1984) assumed that g itself, rather than insensitive to the ambient partial pressure of oxygen, as a pg, is proportional to the concentration of ATP in photo- result of the assumed dependence on ATP concentration. synthetic cells, which could be calculated from the bio- chemical photosynthesis model developed by Farquhar, Key-words: gas exchange; guard cell; photosynthesis; sto- Caemmerer & Berry (1980). That conductance model pre- mata; transpiration. dicted observed responses to irradiance, temperature, CO2 partial pressure, O2 partial pressure and leaf chlorophyll INTRODUCTION content. However, it could not predict any response to hydraulic factors such as humidity or water supply to the A model of stomatal conductance (gsw or simply g; see leaf, because it did not explicitly include the hydromechan- Table 1 for a list of symbols) is required to predict plant gas ical context that links guard cell osmotic pressure to sto- exchange accurately. Most models of leaf and canopy gas matal conductance. A single value of pg can produce a wide exchange use a phenomenological model for g (e.g. Jarvis range of stomatal apertures and conductances, depending 1976; Ball, Woodrow & Berry 1987; the latter modified by on the relationships between guard cell turgor pressure and Leuning 1995; and more recently by Tuzet, Perrier & Leun- volume, between guard and epidermal cell water potentials ing 2003). These models have been successful because they and between stomatal aperture and guard and epidermal are mathematically simple, and because they agree with cell turgor pressures. direct measurements of g under many conditions. However, Intensive study of these hydromechanical factors reveals it is difficult to interpret their mathematical structures in a paradox. When the rate of water loss from the leaf is terms of the regulatory mechanisms that they presumably experimentally increased (for example, by decreasing ambient humidity), leaf turgor and stomatal aperture both Correspondence: T. N. Buckley. E-mail: tom_buckley@alumni. decline in the steady state (Shackel & Brinkmann 1985; jmu.edu. Monteith 1995; Mott & Franks 2001). However, pressure © 2003 Blackwell Publishing Ltd 1767 1768 T. N. Buckley et al. Table 1. Mathematical terms used in this paper Name Symbol Value Units Terms in the model (Eqn 6) -2 -1 Stomatal conductance to water vapour gsw, g – mol air m s ATP concentration t – mmol ATP m-2 Hydromechanical/biochemical response parameter b 1.17 ± 0.27b [mmol ATP m-2]-1 Residual epidermal mechanical advantage M 0.98b unitless Guard cell resistive advantage r 0 a unitless b -2 -1 -1 Effective hydraulic resistance to the epidermis R 0.0456 MPa [mmol H2O m s ] b Epidermal osmotic pressure pe 0.525 MPa a Apoplastic osmotic pressure pa 0 MPa -1 Leaf-to boundary layer H2O mole fraction gradient Ds 10 [5–30] mmol H2O mol air a Source water potential ys 0 MPa Turgor-to-conductance scaling factor c 0.105b mol air m-2 s-1 MPa-1 Terms in simplified form of the model (Eqn 7) Guard cell advantage a – unitless -2 -1 ATP-saturated stomatal conductance gm – mol air m s ‘Michaelis constant’ for a Kg – unitless Hydroactive compensation point g – unitless Other terms in the model derivation Epidermal mechanical advantage [fitted value] m []mˆ 1.98b unitless Guard cell osmotic pressure pg – MPa Water potential of z yz – MPa Turgor pressure of zPz – MPa -2 -1 -1 Resistance from y to zryz – MPa [mmol H2O m s ] -2 -1 -1 Effective hydraulic resistance to the guard cells Rg MPa [mmol H2O m s ] Fraction of transpiration that occurs from zfz – unitless -2 -1 Leaf transpiration rate E– mmol H2O m s -2 -1 -1 Boundary layer resistance to water vapour rbw – [mol air m s ] Terms in the ATP submodel -2 -1 Leaf net CO2 assimilation rate A– mmol CO2 m s -2 -1 Rate of respiration that continues in the dark Rd – mmol CO2 m s Photorespiratory CO2 compensation point G* – Pa Intercellular CO2 partial pressure pi – Pa c Michaelis constant for RuBP carboxylation Kc 40.4 Pa 3c Michaelis constant for RuBP oxygenation Ko 2.48 ¥ 10 Pa Light-limited potential electron transport rate J– mmol e– m-2 s-1 b – -2 -1 Light-saturated potential electron transport rate Jm (2.02 ± 0.48)·|Vm| mmol e m s b Curvature parameter for J(I,Jm) qj 0.908 ± 0.030 unitless Incident photosynthetically active irradiance I 1100 [50–1600] mmol photons m-2 s-1 Product of absorbance and effective quantum yield F 0.195 ± 0.020b electrons photon-1 3 Ambient O2 partial pressure pO2 (2.10 [0.2–4]) ¥ 10 Pa Ambient CO2 concentration ca 365 [50–1000] p.p.m. 5 a Atmospheric pressure pt 10 Pa ATP concentration t – mmol ATP m-2 -2 ATP concentration when Wc > Wj tj – mmol ATP m -2 ATP concentration when Wj > Wc tc – mmol ATP m a -2 Basal ATP level provided by other processes to 1.6 mmol ATP m d -2 Total concentration of adenylates (t + [ADP]) at 12.6·|Vm| mmol AxP m d -2 Concentration of photophosphorylation sites p 2.5·|Vm| mmol sites m -2 Potential RuBP pool size Rp – mmol RuBP m -2 Total concentration of Rubisco active sites Et – mmol sites m -1 -1 Rubisco turnover number kc – CO2 site s Carboxylation rate: -2 -1 Limited by CO2 and Rubisco, but not by RuBP Wc – mmol CO2 m s -2 -1 Limited by RuBP and CO2, but not by Rubisco Wj – mmol CO2 m s 1b -2 -1 Limited by Rubisco only Vm (8.86 ± 0.215) ¥ 10 mmol CO2 m s d -2 -1 Limited by potential RuBP pool size only Vr 2.27·|Vm| mmol CO2 m s Values are given where appropriate; where ranges are given in brackets, standard values are given in italics, and for parameters estimated by gas exchange, standard deviations are given, preceded by the ± symbol. Sources: aassumption; bAppendix 4; cCaemmerer et al. (1994); d -2 -1 Farquhar & Wong (1984). The notation |Vm| means the numerical value of Vm, i.e. Vm/[mmol CO2 m s ]. The subscripts z and y are placeholders for e, g, x, or m, referring to epidermal cells, guard cells, xylem, and mesophyll cells, respectively. Where experimental precision was known, non-significant digits are subscripted but retained for accuracy. © 2003 Blackwell Publishing Ltd, Plant, Cell and Environment, 26, 1767–1785 Stomatal model 1769 probe experiments suggest that equal reductions in guard SYNOPSIS OF THE MODELLING APPROACH cell and epidermal turgor should cause stomatal aperture Our model, like several other recent efforts (Dewar 1995, to increase; this is because aperture responds negatively, 2002; Haefner et al. 1997; Gao et al. 2002), is based on five and more strongly, to the ‘backpressure’ of epidermal cells assertions that form a mathematical ‘closed loop.’ These than to the opening force provided by guard cell turgor are: (1) stomatal conductance is proportional to stomatal (Franks, Cowan & Farquhar 1998).