Generalized hydromechanical model for stomatal responses to hydraulic perturbations

Generalized hydromechanical model for stomatal responsesto hydraulic perturbations

H.W. Kwon a,b, M.Y. Choi a,nQ1

a Department of Physics and Astronomy and Center for Theoretical Physics, Seoul National University, Seoul 151-747, Republic of Koreab School of Physics, Korea Institute for Advanced Study, Seoul 130-722, Republic of Korea

A U T H O R - H I G H L I G H T S

� We model the ‘soil–plant–air’ system describing stomatal responses to water.� Our model is described by the dynamics of water and osmolytes transport in plant.� The turgor pressure of a cell is taken to be a power function of its volume.� Both stationary and nonstationary results agree qualitatively with the experimental data.� Our model can be extended further to describe stomatal responses to several environmental factors.

a r t i c l e i n f o

Article history:Received 29 October 2012Received in revised form10 September 2013Accepted 13 September 2013

Keywords:Wrong-way responseQ3TranspirationCavitationEmbolismConductanceLeaf excisionPhotosynthesis

a b s t r a c t

Stomata respond in a common pattern to various hydraulic perturbations on any part of the ‘soil–plant–air’ system: initial transient ‘wrong-way’ responses and final stationary ‘right-way’ responses. In order todescribe this pattern on the basis of statistical physics, we propose a simple model where turgor pressureof a cell is taken to be a power function of its volume, and obtain results in qualitative agreement withexperimental data for responses to a variety of hydraulic perturbations: Firstly, stationary stomatalconductance as a function of the vapor pressure deficit divides into three regimes characterized bysensitivities of the stomatal conductance and the transpiration rate with respect to vapor pressuredeficit; secondly, for every hydraulic perturbation, the initial transient ‘wrong-way’ responses alwaysappear; thirdly, on condition that water is supplied insufficiently, stomatal oscillations are oftenobserved; finally, stomatal responses following leaf excision exhibit, after the initial transient wrong-way responses, slow relaxation to stomatal closing. In particular, comparison of areoles having differentnumbers of stomata demonstrates that areoles with small numbers of stomata tend to provoke lack ofwater in the soil as well as in the plant. In addition, our model also describes well dependence of thestomatal conductance on temperature. It may be extended further to describe stomatal responses toother environmental factors such as carbon dioxide, light, and temperature.

& 2013 Published by Elsevier Ltd.

1. Introduction

Stomata respond to various environmental factors, e.g., water,temperature, carbon dioxide, light, and so on. Above all, water is themost important factor to plants due to material transport and cooling.In comparison with studies on stomatal responses to temperature,carbon dioxide, or light, stomatal responses to water have beenstudied for a longer time. Although the latter studies have producedexperimentally and theoretically good results, the key mechanism ofthe stomatal responses still remains unknown and no model describeswell both stationary and non-stationary responses.

It has been found that stomatal conductance at stationarity dividesinto three regimes of high, medium and low humidity, characterizedby the sensitivities of the stomatal conductance and the transpirationrate with respect to vapor pressure deficit (Monteith, 1995). In thehigh- and medium-humidity regimes, the transpiration rate increasesas the humidity decreases. Entering the low-humidity regime, how-ever, the transpiration rate gets reduced. Explanations as to why thetranspiration rate reduces in the low-humidity regime include xylemcavitation (Oren et al., 1999; Buckley and Mott, 2002; Dewar, 2002)[for general discussion on xylem cavitation, see Jones and Sutherland,1991; Sperry et al., 1993, and Sperry, 2000], cuticular transpiration(Farquhar, 1978), patchy stomatal closure (Mott and Parkhurst, 1991),and abscisic acid (Mott, 1995; Zhang and Outlaw, 2001a, 2001b).Unfortunately, which is the key factor still remains to be a matter ofdebate.

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Contents lists available at ScienceDirect

journal homepage: www.elsevier.com/locate/yjtbi

Journal of Theoretical Biology

0022-5193/$ - see front matter & 2013 Published by Elsevier Ltd.http://dx.doi.org/10.1016/j.jtbi.2013.09.016

n Corresponding author. Tel.: þ82 2 880 6615; fax: þ82 2 884 3002.Q2E-mail address: [email protected] (M.Y. Choi).

Please cite this article as: Kwon, H.W., Choi, M.Y., Generalized hydromechanical model for stomatal responses tohydraulic perturbations. J. Theor. Biol. (2013), http://dx.doi.org/10.1016/j.jtbi.2013.09.016i

Journal of Theoretical Biology ∎ (∎∎∎∎) ∎∎∎–∎∎∎

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http://dx.doi.org/10.1016/j.jtbi.2013.09.016



mailto:[email protected]





To an abrupt change in the vapor pressure deficit (equivalently,change in humidity), initial stomatal responses are opposite to thefinal responses; these phenomena are called the initial transientwrong-way responses. It is in consensus that these responses arecaused by mechanical advantages of epidermis (DeMichele andSharpe, 1973; Wu et al., 1985; Spence et al., 1986; Klein et al.,1996). After initial transient wrong-way response, guard cells,however, must overcome the mechanical advantages of epidermisin order to produce final stationary right-way response. The draw-down model (Raschke, 1970; Lange et al., 1971) and the osmoticregulation model (Meidner, 1986; Grantz, 1990) were establishedas the overcoming mechanism, which was theoretically analyzedby Buckley and Mott (2002). Still, if the key factor – whatever it is– in producing the wrong-way response overwhelms the over-coming mechanism, then the wrong-way response appears againand then the right-way response does and so on, giving rise tooscillations of stomata. Indeed, under poor conditions of water,stomata often oscillate (Cowan, 1972; Farquhar and Cowan, 1974;Rand et al., 1981; Haefner et al., 1997; Jarvis et al., 1999; Wanget al., 2001). There are two suggestions for the mechanism toinduce oscillations: one is the osmotic regulation of guard cellsand the other is the stomatal patchiness, albeit the origin ofstomatal patchiness is not clear.

In response to excise a leaf from the plant, stomata open andthen close. Observation of this behavior, called the ‘Iwanoff effect’,dates back to Darwin in 1898. In spite of its long history, there arestill few models describing plausibly such stomatal responses tothe leaf excision.

Even though various models have already been suggested toexplain the above phenomena, none of them could explain all thephenomena. For instance, the model of Haefner et al. (1997), inwhich an areole on a leaf is viewed as a square lattice andhydraulic interactions between neighboring sites on the latticeare included, exhibits only one of the three regimes at stationarityand fails to reproduce the stomatal responses following leafexcision.

The purpose of this paper is to establish a model for stomatalstationary and non-stationary responses to various hydraulicperturbations. Specifically, we propose a model which describesaccurately the above phenomena and is as simple as possible inthe spirit of Occam's razor.

2. Model and method

2.1. Model description

This study is devoted to modeling responses of stomatalconductance (or equivalently, stomatal aperture) only to hydraulicperturbations. Accordingly, various environmental factors affect-ing the stomatal conductance are disregarded except water. Forexample, carbon dioxide, light intensity, or temperature areassumed to remain constant during stomatal responses to hydraulicperturbations, so that their effects do not come into action.We thus focus on how much the stomatal conductance varies,rather than how high it is.

In describing stomatal responses to various hydraulic perturba-tions, it is essential to include the water transport system in theplant, consisting of the root, xylem in the trunk, xylem in the leafveins, and several kinds of cells in leaves. However, the root andthe xylem in leaf veins are not specified explicitly in the modelbecause we are not concerned with their water potentials or watercontents; rather, they are merged into the xylem in the trunk andthus the word ‘xylem’ in this paper stands for the integratedsystem of the root, xylem in the trunk, and xylem in leaf veins. Inconsequence, the xylem plays the role of a passage through which

water is transported from soil to epidermal cells neighboring withthe leaf veins.

To make the model hydromechanically concrete, we regardwater as an incompressible liquid in view of the extremely smallcompressibility (of the order 10�4 MPa), and adopt the continuityequation describing water flow.

2.2. Water flow pathway

While it is apparent that water is absorbed into a plant via itsroot and water vapor is transpired from its leaves to the ambientair, the exact pathway of water transport from xylem to theevaporating sites still remains unknown, thus making it necessaryto choose one of the several candidates for the pathway. Amongthem, we consider the water flow pathway as illustrated in Fig. 1[similar to design 2 in Zwieniecki et al., 2007]. This pathway hastwo characteristics: first, direct water flow from xylem to meso-phylls is blockaded and second, direct water flow between guardcells and mesophylls is also inhibited. In fact, allowing them turnsout not to give significantly different results. On the other hand,the water flow from the epidermal cell to the mesophyll is allowed(Zwieniecki et al., 2007) in accord with the good evidence that amesophyll cell interacts hydraulically with adjacent epidermalcells (Nonami and Schulze, 1989). In addition, the guard cell is inhydraulic contact only with the epidermal cell.

In the model, while it is considered that evaporation into asubstomatal cavity occurs mainly on the surface of the mesophylland the guard cell wall also contributes to it, the evaporation from anepidermal cell to a substomatal cavity is not taken into account andonly its cuticular transpiration to the ambient air is considered.

2.3. Stomatal region and lattice model

When one draws imaginary middle lines between every pair ofneighboring stomata as the Voronoi partition in mathematics orthe Wigner–Seitz cell in solid state physics, the leaf is divided intoas many regions as there are stomata: each region contains onestoma, one substomatal cavity, a pair of guard cells, a pair ofsubsidiary cells, a few epidermal cells and mesophylls, and so on.We call it a ‘stomatal region’ (corresponding to a unit cell on alattice). Fig. 2 illustrates the L� L square lattice of an areolesurrounded by leaf veins; each site on the lattice represents onestomatal region. At a site (i.e., in a stomatal region), epidermal cellsinteract hydraulically with guard cells and mesophylls, whereasthe second characteristic of the water flow pathway does not allowthe guard cells to interact hydraulically with the mesophylls.In addition, the interactions between neighbouring sites are achievedonly through the epidermal cells: While an epidermal cell at a siteinteracts hydraulically with the epidermal cells at the four nearestneighbouring sites (above, below, left, and right in Fig. 2), themesophyll does not interact with one in the neighbouring site.

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Fig. 1. Water flow pathway diagram. Solid and dashed arrows indicate water flowsin the forms of liquid and vapor, respectively. The root between soil and xylem andleaf veins between xylem and epidermal cells are not specified. Water flows fromxylem to mesophylls as well as between guard cells and mesophylls are notallowed.

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Water flows between cells are indicated by bi-directed arrows inFig. 2. On the other hand, the border lines (marked with thick solidlines) in Fig. 2 represent the leaf veins, which are merged intoxylem in the model. The epidermal cells at the sites adjacent to aleaf vein also take water from the vein (or xylem in the model) butthey do not directly interact with the epidermal cells at the sitesacross the vein.

Even though there are a few epidermal cells and a pair ofsubsidiary cells in a stomatal region, it is assumed in the model thattheir hydraulic states are always identical (or the relaxation time isvery short). Namely, all of them have identical turgor pressures,osmotic potentials, and water potentials; this makes it sufficient tospecify only one epidermal cell in the region. In consequence, theepidermal cell gives the mechanical back pressure to the guard cellsin the region and interacts hydraulically with the epidermal cells inneighboring regions or with xylem if the stomatal region is adjacentto the xylem. Similarly, all mesophylls and a pair of guard cells in astomatal region are also represented by a mesophyll and one guardcell, respectively.

2.4. Mathematical model – dynamics

Similar to Ohm's law in electromagnetics, the water current isgiven by the product of the hydraulic conductance and thenegative gradient of the water potential between two compart-ments. Combined with the continuity equation (in the spatiallyintegrated form), the water current inward a compartment con-tributes positively to the change in the number of water moleculesinside the compartment and the outward water current contri-butes negatively to that. As a result, the rate of change in thenumber of water molecules in the xylem reads

dNx

dt¼ αsxðψ s�ψxÞ�nl∑

½j�αxeðψx�ψ j

eÞ; ð1Þ

where N is the number of water molecules (in units of mmol), ψthe water potential, and α the hydraulic conductance. The sub-scripts ‘x’, ‘s’, and ‘e’ denote the xylem, soil, and epidermal cell,respectively, whereas ½j� indicates all the neighboring sites of theleaf veins. Here nl designates the number of leaves on the plant.The variables and parameters are listed in Table 1.

In general the permeability of water into a cell (called endos-mosis) is larger than that out of the cell (called exosmosis) (Kamiyaand Tazawa, 1956; Dainty, 1963). However, since transport of waterfrom one cell to a neighboring cell always involves one exosmosisand one endosmosis, the hydraulic conductance between thosecells does not depend on the direction of water flow (at leastbetween cells of the same kind), and remains symmetric, e.g., the

hydraulic conductance from xylem to an epidermal cell is thesame as that from an epidermal cell to xylem, αxe ¼ αex.

The number of water molecules in an epidermal cell in astomatal region changes due to cuticular transpiration, the waterflow into or out of guard cells and mesophylls in the region, andthe flow into or out of epidermal cells in the neighboring regions.The epidermal cells adjacent to leaf veins are affected additionallyby the water flow into or out of the xylem. In consequence, thewater flow equation governing the number of water molecules inthe epidermal cell at site i takes the form

dNie

dt¼ �eic�αegðψ i

e�ψ igÞ�αemðψ i

e�ψ imÞ

þ∑⟨j⟩i

αeeðψ je�ψ i

eÞþχ iαxeðψx�ψ ieÞ; ð2Þ

where ec denotes the cuticular transpiration rate from the epider-mal cell to the ambient air across the cuticle layer on the leaf.The superscripts i and j refer to sites on the lattice, running from1 to L2, and ⟨j⟩i indicates all the sites neighboring with the ith siteexcept those across the vein. Further, the character function χ itakes the value unity or zero depending on whether or not the ithsite neighbors on the leaf vein; the subscripts ‘g’ and ‘m’ stand forthe guard cell and the mesophyll, respectively.

Similarly, the water flow equations for the guard cell and forthe mesophyll read:

dNig

dt¼ �eigþαegðψ i

e�ψ igÞ ð3Þ

and

dNim

dt¼ �eimþαemðψ i

e�ψ imÞ; ð4Þ

where eg and em denote the evaporation rates into the substomatalcavity from the guard cell and from the mesophyll, respectively.

There is a time lag between evaporation into the substomatalcavity and transpiration from the cavity to the ambient air throughthe stomatal pore: the former lags behind the latter. The presenceof this lag is supported by the gap between the equilibration timein the stomatal pore after a variation in the ambient humidity andthe equilibration time between the two neighboring cells (Cowan,1972). Therefore we model in such a way that the total rate ofevaporation from the guard and mesophyll cells approachesexponentially to the stomatal transpiration rate with the corre-sponding relaxation time:

ddtðeigþeimÞ ¼

1τE½Ei�ðeigþeimÞ�; ð5Þ

where τE is the relaxation time in the evaporation rate and E thestomatal transpiration rate.

Even though the light intensity and the ambient concentrationof CO2 are kept constant during stomatal responses to hydraulicperturbations, variations in the stomatal aperture result inchanges of the diffusion rate of CO2; consequently, the internalconcentration of CO2 may change. On the other hand, during thewhole time that the stomata respond to a hydraulic perturbation,the change in the amount of osmolytes caused by photosynthesisis neglected because the former time scale (�minutes) is evenshorter than the latter time scale (�hours). For that reason, it isassumed that the change in the amount of osmolytes in the guardcell is caused only by the active uptake and passive emission of theosmolytes. The number of osmolytes in a guard cell tends toincrease at the rate that the guard cell takes up the osmolytes fromthe neighboring epidermal cells actively, i.e., consuming ATP. Onthe other hand, it tends to decrease at the rate that the osmolytesflow into the cells adjacent to the guard cell passively, i.e.,by means of diffusion (Dewar, 2002; Buckley et al., 2003).

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Fig. 2. Lattice model for a leaf [modified from Haefner et al., 1997]. An areole on aleaf is considered as a L� L square lattice on which each site represents onestomatal region having a pair of guard cells, a few of epidermal cells, and a few ofmesophylls. (Left) Thick solid lines indicate leaf veins. (Right) Bi-directed arrowsrepresent water flows between two cells.

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The destination of the passive flow of the osmolytes is just theepidermal cell in the model. In particular, we adopt the inter-pretation by Buckley et al. (2003) that the resistance againstdiffusion is proportional to the turgor pressure of the epidermalcell. In consequence, the number Sg of guard cell osmolytes

(in units of mmol) evolves according to

dSigdt

¼ 1τS

βτ�πie�πi

g

Pie

!; ð6Þ

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Table 1Variables and parameters.

Symbol Name Value Units

VariablesNx;e;g;m

a Number of water molecules – mmolαsx Hydraulic conductance between soil and xylem – mmol MPa�1 s�1

Ve;g;ma Volume – μm3

Se;ga Number of osmolytes – mmolψx;e;g;m

a Water potential – MPaPx;e;g;m

a Turgor pressure – MPaπe;g;m

a Osmotic potential – MPaeg;ma Evaporation rate – mmol s�1

ec Cuticular transpiration rate – mmol s�1

E Transpiration rate – mmol s�1

A Stomatal aperture – μmgs Stomatal conductance in the absence of boundary layer effects – mmol kPa�1 s�1

g Stomatal conductance in the presence of boundary layer effects – mmol kPa�1 s�1

ɛe;g;ma Cell wall rigidity – MPa

Parameters Plant or internal parametersαxe Hydraulic conductance between xylem and the epidermal cell 2.307�10�8b

mmol MPa�1 s�1

αee Hydraulic conductance between neighboring epidermal cells 1.154 �10�8bmmol MPa�1 s�1

αeg Hydraulic conductance between the epidermal cell and guard cell 2.307�10�9bmmol MPa�1 s�1

αem Hydraulic conductance between the epidermal cell and mesophyll 6.921�10�9bmmol MPa�1 s�1

α0sx Hydraulic conductance αsx in the absence of xylem cavitation 8.375�10�6bmmol MPa�1 s�1

T0 Reference temperature in xylem cavitation 300b K~ψ 0x

Inflection point on the xylem cavitation curve at T ¼ T0 �1.36b MPaγ0 Sensitivity of αsx to ψx at ψx ¼ ~ψ x and T ¼ T0 2.00b MPa�1

τE Relaxation time in evaporation rate 1.445�1012b s mmol�1

τS Relaxation time in the number of osmolytes 4.336�1012b s mmol�1

τ ATP concentration in guard cell 4.27�10�12b mmol ATP μm�3

β Sensitivity for osmolytes uptake 1.17�1012c ðmmol ATP μm�3Þ�1

bx Slope of ψx to Nx 1.83�10�4bMPa mmol�1

ρ Density of water at T¼300 K and T¼280 K 5.536, 5.555�10�11d mmol μm�3

κg Gradient of guard cell wall rigidity to turgor pressure 6.73 73.65e,f Unitlessκe Gradient of epidermal cell wall rigidity to turgor pressure 4.64 70.93e,f Unitlessκm Gradient of mesophyll cell wall rigidity to turgor pressure 1.09 70.22e,f Unitlessɛ0g Guard cell wall rigidity at plasmolysis 3.24e MPa

ɛ0e Epidermal cell wall rigidity at plasmolysis 0.487e MPa

ɛ0m Mesophyll cell wall rigidity at plasmolysis 0.287e MPa

V0g

Guard cell volume at plasmolysis 3180e μm3

V0e

Epidermal cell volume at plasmolysis 8000b μm3

V0m

Mesophyll cell volume at plasmolysis 6000b μm3

μ Ratio of guard cell to mesophyll evaporation rate 1b Unitlesss Scaled stomatal conductivity 2.768 �10�10b

mmol kPa�1 μm�1 s�1

AM Maximum stomatal aperture 18.11g μmη1 ; η2 A vs. Pg curvature parameter at low Pe and at high Pe 1.598, 0.7694g MPaξ Relative reduction in AM by high Pe 0.509g Unitless~P g Inflection point in A versus Pg at high Pe 2.015g MPa

gb Boundary layer conductance 1.384 �10�9bmmol kPa�1 s�1

ν Ratio of cuticular to stomatal transpiration rate 0.05b Unitlessnl Number of leaves in plant 121b Unitless

Parameters Environmental or external parametersψ s Soil water potential – MPaT Temperature – KD Vapor pressure deficit – kPa

OtherskB Boltzmann constant 8.31�109 MPa μm3 mmol�1 K�1

a Subscripts ‘s’, ‘x’, ‘e’, ‘g’, and ‘m’ stand for ‘soil’, ‘xylem’, ‘epidermal cell’, ‘guard cell’, and ‘mesophyll cell’, respectively.b Values chosen to fit the data.c Values measured by Buckley et al. (2003).d Values adapted from Lide (1990).e Values adapted from Franks et al. (2001).f Values distributed according to log-normal distributions, each described by the mean 7 standard deviation.g Values measured by Buckley and Mott (2002).






where τS is the relaxation time, β the sensitivity for osmolyteuptake, τ the concentration of ATP in the guard cell, πe and πg theosmotic potentials of the epidermal and guard cells, respectively,and Pe the turgor pressure of the epidermal cell.

Since guard cells and epidermal cells in a stomatal region areassumed to be represented by a single guard cell and a singleepidermal cell, respectively, in the model, the rate of change in thenumber of epidermal cell osmolytes is equal in magnitude andopposite in sign to that of guard cell osmolytes:

dSiedt

¼ 1τS

πie�πi

g

Pie

�βτ

!: ð7Þ

With the assumption that the number of mesophyll osmolytesremains constant in time, Eqs. (6) and (7) are expressed in terms ofthe total number St of osmolytes in the region:

dStdt

¼ ddtðSigþSieþSimÞ ¼ 0; ð8Þ

which establishes the conservation of the total number of osmo-lytes in the region.

In consequence, our model is described by a system of ð5L2þ1Þcoupled first-order differential equations, which consists of onedifferential equation (1) for xylem and five kinds of differentialequations (2)–(6) for each stomatal region, with L2 correspondingto the total number of sites or stomatal regions on the latticesurrounded by leaf veins.

2.5. Mathematical model – relations between variables

The above (5L2þ1) differential equations involve variablesmore than 5L2þ1 and we thus need a few relations between thevariables in order to handle these differential equations. First, thewater potential in a cell is determined by the turgor pressure andthe osmotic potential in the cell:

ψλ ¼ Pλþπλ ðλ¼ g; e;mÞ: ð9ÞThe contributions of the matric potential and gravity to the waterpotential have been neglected because only 0.02 MPa/m isrequired to drag up water via the xylem (Taiz and Zeiger, 2002)and every cell in a leaf lies essentially at the same height. Thesuperscripts labeling sites have been dropped here and henceforthwill be omitted whenever no ambiguity is expected. Namely,Eq. (9) holds for all kinds of cell at each site. In addition, theosmotic potential in a cell is given by the van't Hoff equation:

πλ ¼ �SλkBTVλ

ðλ¼ g; e;mÞ: ð10Þ

where kB is the Boltzmann constant, T the temperature, and V thevolume of the cell. The xylem osmotic potential is practically zerosince there are a few osmolytes in xylem.

Since xylem is made up of dead cells and has lignin, it may beregarded as a vacant ‘rigid’ cylinder with a given volume. There-fore we may take approximately the linear relationship betweenthe number Nx of water molecules in the xylem and its waterpotential ψx: ψx¼bx Nx with appropriate constant bx. The evolu-tion equation for the xylem water potential thus reads

dψx

dt¼ bx

dNx

dt¼ bx αsxðψ s�ψxÞ�∑

½j�αxeðψx�ψ j

eÞ" #

: ð1′Þ

In general, αsx, the hydraulic conductance between soil andxylem, depends on the xylem water potential. The xylem cavitation(or the xylem embolism) reduces the hydraulic conductance of thexylem by generating air bubbles. To accommodate such reduction,we take specifically the soil-to-xylem hydraulic conductance in thesigmoidal form as observed in, e.g., Thuja occidentalis (Jones and

Sutherland, 1991):

αsxðψxÞ ¼α0sx

21þ tanh½γðψx� ~ψ xÞ��

; ð11Þ

where the hysteresis in the xylem cavitation (Dewar, 2002) isneglected for simplicity. Here, α0

sx measures the maximum soil-to-xylem hydraulic conductance, i.e., in the limit ψx-1 and γ denotesthe sensitivity of the conductance to the xylem water potential atψx ¼ ~ψ x, with ~ψ x corresponding to the inflection point on thexylem cavitation curve in Eq. (11). Both γ and ~ψ x are assumed todepend on the temperature T via

γðTÞ ¼ γ0T0

Tð12aÞ

and

~ψ xðTÞ ¼ ~ψ 0xTT0

; ð12bÞ

where T0 is the temperature scaling factor, ~ψ 0x the inflection point of

the xylem cavitation curve at T¼T0, and γ0 the sensitivity of thehydraulic conductance to the xylem water potential at T¼T0 andψ x ¼ ~ψ 0

x. (Note that the product γðTÞ ~ψ xðTÞ does not depend on thetemperature T.) Although the parameter T0 is in fact redundant,

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-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-1.2 -1 -0.8 -0.6 -0.4 -0.2 0

π (M

Pa)

ψ (MPa)

EpidermisMesophyll

0

1

2

3

4

5

3000 4000 5000 6000 7000

P g (M

Pa)

Vg (μm3)

Fig. 3. (a) Guard cell turgor pressure Pg versus guard cell volume Vg and (b) osmoticpotential π versus water potential ψ for epidermal cells and mesophylls. In (a) threesets of data points, corresponding to three guard cells, are redrawn from Frankset al. (2001). Each set of data is fitted to Eq. (14) with values of (κg , ɛg0, Vg

0) given byð10:7;0:037;3250Þ for filled circles ð�Þ, ð5:97;0:668;2800Þ for filled triangles ðmÞ,and ð3:51;4:66;3490Þ for open squares ð□Þ. In (b) two sets of data points,corresponding to epidermal cells and mesophylls, are redrawn from Nonami andSchulze (1989). Each set of data is fitted to Eq. (16) with values of (κ, ɛ0, π0) given byð4:64;0:487; �0:953Þ for epidermal cells ð�Þ and ð1:086;0:287; �2:248Þ for meso-phylls ð○Þ.






i.e., removable via the transformation γ0T0-γ0 and ~ψ 0x=T0- ~ψ 0

x, itis convenient to maintain the units of γ0 and ~ψ 0

x.In contrast to the case of xylem, a given number of water

molecules in a cell does not determine uniquely its turgor pressurebecause its volume is not constant. Therefore, another relationbetween the turgor, volume, and number of water molecules isneeded. For guard cells, the experimental data in Franks et al.(2001) indicate that the rigidity ɛg of the guard cell wall is relatedlinearly to the turgor:

ɛg � VgdPg

dVg

� �¼ κgPgþɛ0g ; ð13Þ

where κg denotes the gradient of the cell wall rigidity with respectto the turgor pressure and ɛ0g the guard cell wall rigidity at zeroturgor (called incipient plasmolysis). Solving Eq. (13), we obtain theguard cell turgor pressure, up to an additive constant ɛ0g=κg, as apower function of the volume with the exponent κg:

PgðVgÞ ¼ɛ0gκg

Vg

V0g

!κg

�1

" #; ð14Þ

where V0g is the plasmolysis guard cell volume. Fig. 3(a) illustrates

the relation between the guard cell turgor pressure and the guardcell volume, given by Eq. (14), with appropriate values of κg, ɛ0g,and V0

g.Also plotted are the experimental data of Franks et al. (2001).

Estimating the mean and standard deviation based on the threefitted values of κg as a sample, we obtain κg¼6.7373.65.

Eq. (9), together with Eqs. (10) and (14), gives the guard cellwater potential as a function of the guard cell volume and thenumber of guard cell osmolytes:

ψ gðVg; SgÞ ¼ɛogκg

Vg

Vog

!κg

�1

" #�SgkBT

Vg: ð15Þ

Using the density of water N=V � ρ, we obtain the time derivativeof the guard cell volume, which takes the place of Eq. (3):

dVg

dt¼ 1ρ½αegðψ e�ψ gÞ�eg�: ð3′Þ

On the other hand, in the case of epidermal cells or mesophylls,there are no experimental data available for the relations amongthe turgor, volume, and the number of water molecules. For thatreason, it is assumed that the turgor pressures of epidermal cellsand mesophylls are also given by power functions of theirvolumes, respectively, in a similar manner to Eq. (14). The result-ing equations together with Eqs. (9) and (10) lead to the waterpotential as a function of the osmotic potential for both epidermalcells and mesophylls:

ψλðπλÞ ¼ɛ0λκλ

π0λ

πλ

!κλ

�1

" #þπλ ðλ¼ e;mÞ; ð16Þ

where π0e and π0

m denote the epidermal and mesophyll osmoticpotentials at zero turgor, respectively. We obtain ðκe; ɛ0e ;π0

eÞ andðκm; ɛ0m;π0

mÞ from experimental data in Nonami and Schulze(1989), as shown in Fig. 3(b); the good agreement of the resultsof Eq. (16) with the experimental data is encouraging andsupportive of the model proposed here. The fitted values of κeand κm are 4.6470.93 and 1.0970.22, respectively. Unlike thecase of guard cells, however, V0

e and V0m are not inferred from the

data. Similar to Eq. ð3′Þ for the guard cell, we thus derive the timederivatives of the epidermal cell volume and of the mesophyllvolume:

dVe

dt¼ 1ρ

��ec�αegðψ e�ψ gÞ�αemðψ e�ψmÞ:

þ∑⟨j⟩αeeðψ j

e�ψ eÞþχ iαxeðψx�ψ eÞ�

ð2′Þ

and

dVm

dt¼ 1ρ½αemðψ e�ψmÞ�em�: ð4′Þ

In addition, the epidermal cell and mesophyll water potentialsobtain the same form as Eq. (15) with the subscript ‘g’ replaced by‘e’ and ‘m’, respectively.

The ratio μ of the guard cell evaporation rate eg to themesophyll evaporation rate em into the substomatal cavity isassumed to be constant in time, i.e.

eg ¼ μem: ð17Þ

This allows one to write Eq. (5) in terms of only the mesophyllevaporation rate:

demdt

¼ 1τE

Eμþ1

�em

� �: ð5′Þ

We next note that the stomatal conductance grows with thearea of the stomatal pore. Since the length of the pore along thelongitudinal axis of the guard cell hardly alters, the pore area isdetermined by the perpendicular width of the pore, called thestomatal aperture. Accordingly, the stomatal conductance gs isproportional to the stomatal aperture A:

gs ¼ sA; ð18Þwhere s is the scaled stomatal conductivity. The stomatal apertureis determined by the guard cell turgor pressure and the ratio of theepidermal cell turgor pressure to its osmotic potential. Theaperture function used in our model is taken from Buckley andMott (2002) and takes the form

AðPg; Pe=πeÞ ¼ f 1ðPgÞþPe

πe½f 1ðPgÞ� f 2ðPgÞ� ð19Þ

with

f 1ðPgÞ ¼ AMexpðPg=η1Þ�1expðPg=η1Þþ1

¼ AM tanhPg

2η1

� �

and

f 2ðPgÞ ¼ ð1�ξÞAM 1þexp~Pg�Pg

η2

!" #�18<:

� 1þexp~Pg

η2

!" #�19=;;

where AM denotes the maximum stomatal aperture, ξ measuresthe relative reduction in the stomatal aperture by high epidermalturgor pressure, ~Pg corresponds to the inflection point on thecurve of the stomatal aperture versus guard cell turgor pressure athigh epidermal turgor pressure, and η1 and η2 are related to thecurvatures of the curve at low epidermal turgor and at highepidermal turgor, respectively.

With the boundary layer effects taken into account, thetranspiration rate from the substomatal cavity to the ambient airreads

E¼ gD¼ gs1þgs=gb

D¼ sA1þsA=gb

D; ð20Þ

where g is the stomatal conductance including the boundary layereffects, gb the boundary layer conductance and D the vaporpressure deficit (Buckley, 2005). In addition, it is assumed thatthe ratio ν of the cuticular transpiration rate ec (from theepidermal cells) to the stomatal transpiration rate E (from the

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substomatal cavity) is constant in time:

ec ¼ νE: ð21ÞEqs. (1′)–(5′) and (6) with Eqs. (9)–(11), (14), and (19)–(21) give

a complete description of the model, and are to be solved. It is ofinterest that the differential equations (1′)–(5′) and (6) areinvariant under the change in the time scale: t-ct, accompaniedby αsx-αsx=c, αxe-αxe=c, αee-αee=c, αeg-αeg=c, αem-αem=c,τE-cτE, τS-cτS, s-s=c, and gb-gb=c. (Note that these para-meters are plant specific and related to the time scale.)

2.6. Numerical simulation

The relations among the dynamical variables supplement theevolution equations for the variables, and make the modeldescription complete. Unfortunately, however, the resulting sys-tem of coupled nonlinear differential equations resists analyticsolutions and we thus resort to numerical methods. Specifically,we employ the fourth-order Runge–Kutta method programmed inC language to integrate the differential equations. Given the initialconditions for the dynamic variables, the system is in the out-of-equilibrium state for given external parameters, inducing thevariables to undergo rapid changes initially. In order to observerelevant responses only to given perturbations, we allow thesystem to relax in advance, so that it is in the stationary statebefore turning on the perturbation.

After the system is prepared, the following three kinds ofperturbations are introduced: the system is first perturbed by asmall increase in the vapor pressure deficit. Responding to it, thesystem attains to a new stationary state. Iteration of this proceduregives the relationship between the vapor pressure deficit and thestomatal conductance at stationarity. Similarly, the relationshipbetween the soil water potential and (stationary-state) stomatalconductance is obtained. Subsequently, the prepared system isperturbed by large changes in the vapor pressure deficit or soilwater potential. After these perturbations, dynamic responses ofstomata are probed. Finally, we simulate leaf excision by settingthe soil-to-xylem hydraulic conductance equal to zero, whichcorresponds to no more water supply from the soil.

According to the lattice model illustrated in Fig. 2, the leaf ismodeled as a few areoles, which are essentially decoupled witheach other due to leaf veins. An areole is then modeled as a systemof L2 stomatal regions arranged into an L� L square array. Weconsider various sizes, up to L¼19, and present mostly results forthe system of L¼9, i.e., the system consisting of 81 stomata.Simulations of the system of larger size in general do not yieldqualitative difference in the overall results.

The parameter values used in simulations are listed in Table 1.Note that in our simple model the turgor pressure of a cell is takento be a power function of its volume, based on the direct Pg–Vg

data and the indirect πe�ψ e and πm�ψm data. Since the powersof the functions κ's, describing gradients of cell wall rigidity to theturgor pressure, are the essential parameters of the model, we takeinto account variations over stomatal regions, obtained fromfitting of data in Franks et al. (2001) and Nonami and Schulze(1989). Namely, those parameters are taken to be randomlydistributed over the system, according to the log-normal distribu-tions (ensuring the positiveness of κ). The mean values andstandard deviations of such distributed parameters are also listedin Table 1 in the form of ‘mean 7standard deviation’.

2.7. Nullcline analysis

In case that there includes only one stoma in an areole, thenullcline analysis can help to understand the system, describedby Eqs. (1)–(6) [for general discussion on the nullcline analysis,

see, e.g., Haefner, 2005]. Setting Eqs. (1)–(5) equal to zero andusing Eq. (21), we obtain the nullcline equation for the transpira-tion rate En as a function of ψ n

x:

Enðψ n

xÞ ¼αsxðψ n

xÞnlð1þνÞðψ s�ψ n

xÞ; ð22Þ

where asterisks denote stationary values. Similarly, the nullclineequation for the epidermal water potential ψ n

e is derived fromEq. (1) in the form

ψ n

eðψ n

xÞ ¼ψ n

x�αsxðψ n

xÞ4nlαxe

ðψ s�ψ n

xÞ: ð23Þ

In principle, eliminating the variable ψ nx in Eqs. (22) and (23) leads

to the relation between En and ψ ne; in practice, however, we may

not obtain the relation analytically due to the nonlinearity ofEq. (11). Nonetheless, the parametric plot of Eqs. (22) and (23)with ψ n

x as a parameter gives the relation numerically. Here we call

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0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1

E/Em (unitless)

1

2

3

0

1

2

3

4

5

6

0 1 2

D (kPa)

1

2

3

1×19×913×13

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2

g/g m

(uni

tless

)g/

g m (u

nitle

ss)

D (kPa)

1

2

3

1

2

3

MonteithT=27°CT=7°C

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2

E/E m

(uni

tless

)E/

E m (u

nitle

ss)

D (kPa)

1

2

3

1

2

3

1

2

3

0

20

40

0 40 80

1

2

3

1

2

3

1

2

3

Fig. 4. Relations between the stationary stomatal conductance g, vapor pressuredeficit D, and the transpiration rate E. gm and Em denote the maximum values of thestomatal conductance and of the transpiration rate, respectively, at temperatureT¼27 1C. In (a)–(c), experimental data represented by filled circles are adaptedfrom Monteith (1995) whereas solid and dotted lines plot results of numericalsimulations of the 9�9 system for the soil water potential ψ s ¼ �0:3 MPa, attemperatures T¼27 1C and 7 1C, respectively. Depending on the slopes of g–D, E–D,and g–E curves, the stationary stomatal conductance divides into three regimes.Shown in (d) are three E versus D curves for three different numbers of stomata inan areole: only a single stoma (solid line), 81 stomata (dotted line), and 169stomata (dot-dashed line) in an areole. Simulations are performed at temperatureT¼27 1C. Em indicates the maximum transpiration rate of the 9�9 system atT¼27 1C and ψ s ¼ �0:3 MPa. Inset shows the case of one stoma over extendedranges of D. Other parameter values are given in Table 1.






the relation the ‘water nullcline’ since it is characterized byconstant numbers of water molecules.

Setting Eq. (6) to vanish and using Eqs. (10) and (14), we havethe nullcline equation for Vn

g:

Vn

gðSne ;Vn

eÞ ¼ðS�SneÞkBT

SnekBTþβτPeðVn

eÞVn

e; ð24Þ

where S� St�Sm is constant and Sng ¼ S�Sne. With the help ofEq. (24), we derive the two distinct nullcline equations for thetranspiration rate:

En

1ðSne ;Vn

eÞ ¼sAðSne;Vn

eÞ1þsAðSne;Vn

eÞ=gbD ð25Þ

from Eq. (20) and

En

2ðSne ;Vn

eÞ ¼ð1þμÞμ

αeg½ψ eðSne;Vn

eÞ�ψ gðSne;Vn

eÞ� ð26Þ

from Eq. (3) with dNg=dt ¼ 0. After obtaining the relation Vn

eðSneÞfrom the contour En

1ðSne ;Vn

eÞ ¼ En

2ðSne;Vn

eÞ, we have the other relationbetween En and ψ n

e (called the ‘osmolyte nullcline’) via insertingVn

eðSneÞ into Eq. (25) or (26). (Note that the osmolyte nullclinedepends on the water vapor deficit D but not on the soil waterpotential ψs, whereas the water nullcline depends on ψs but not on D.)Now, the qualitative properties of the system can be illustrated byplotting the water nullcline and the osmolyte nullcline in the phasespace ðEn, ψ n

eÞ.

3. Results

In this paper, we consider the responses of an areole, not thoseof each site. For that reason, the stomatal aperture Ai, the stomatalconductance gi, and the transpiration rate Ei (i¼1,…,L2) areaveraged over the L2 sites and we use the averaged quantities A,g, and E in this section. Note that the averaged quantities over L2

sites are different from the corresponding quantities in the 1�1system or in the mean-field model which cannot bear the effectsof variations of κ's and interactions between neighboring sites.

3.1. Stationary stomatal conductance

Simulating the first kind of perturbation, we obtain therelationship between vapor pressure deficit D and stationarystomatal conductance g and that between D and the transpirationrate E. In Fig. 4(a)–(c), the stationary stomatal conductance as afunction of the vapor pressure deficit is divided into three regimes:regime 1 [corresponding to regime C in Monteith, 1995] lies at lowvalues of D, in which g changes little but E increases dramaticallywith the increase of D; regime 2 [or regime A in Monteith, 1995] atintermediate values of D, in which g decreases rapidly but E stillincreases with D; regime 3 [regime B in Monteith (1995)] at highvalues of D, in which both g and E decrease as D is increased.

Characteristic features of the relationships in each regime aresummarized in Table 2 in terms of the signs of the first-orderderivatives. Fig. 4(a)–(c) also shows the dependencies of g–D, E–D,and g–E curves on temperature T. These temperature dependen-cies explain qualitatively the phenomena observed in real leaves:The lower the temperature, the lower the conductance (Buckley

et al., 1999). In addition, Fig. 4(d) illustrates howmuch water vaporwould be transpired if only one stoma were surrounded with leafveins. In this case, the transpiration rate is higher than that of ausual leaf; even the soil as well as the plant will dry up eventually.Moreover, this 1�1 system does not possess regime 3, as shownin the inset of Fig. 4(d). Fig. 5 presents the result of the nullclineanalysis, which agrees with the simulation result [see the data forthe 1�1 system in Fig. 4(d)]. In addition, Fig. 5 shows fourdynamic trajectories after changes of ðVg;VeÞ from the fixed pointðVn

g;Vn

eÞ at D¼2 kPa, ψ s ¼ �0:3 MPa, and T¼27 1C to ð0:95Vn

g;Vn

eÞ,ð1:05Vn

g;Vn

eÞ, ðVn

g;0:95Vn

eÞ, or ðVn

g;1:05Vn

eÞ.Similar to Fig. 4, Fig. 6 displays the stomatal conductance g

versus the soil water potential ψ s instead of the vapor pressuredeficit D at two temperatures.

3.2. Initial transient wrong-way responses

Changing rapidly the vapor pressure deficit or the soil waterpotential, we examine how stomata respond and show the results inFig. 7. It is pleasing that Fig. 7(a) reveals the initial transient ‘wrong-way’ responses and the final stationary ‘right-way’ responses toabrupt changes in the vapor pressure deficit, observed in experiment.

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Table 2Characteristic features of three regimes.Q4

dg/dD dE/dD dg/dE

Regime 1 � 0 40 � 0Regime 2 o0 40 o0Regime 3 o0 r0 40

1

6

-1 -0.5 0

E* /Em

(uni

tless

)

ψ∗e (MPa)

WN (ψs=-0.3 MPa)WN (ψs=-0.9 MPa)

ON (D=2 kPa)

ON (D=1 kPa)

1.05V*g

0.95V*g 1.05V*

e

0.95V*e

Fig. 5. Water nullcline (WN) with ψ s ¼ �0:3 MPa (solid line) and �0.9 MPa (dottedline) and osmolyte nullcline (ON) with D¼1 kPa (filled square) and 2 kPa (opencircle) at T¼27 1C. The water nullcline does not depend on the water vapor deficit Dwhereas the osmolyte nullcline is independent of the soil water potential ψ s. Fourdirected lines depict the dynamic trajectories starting from ðVg;VeÞ ¼ ð0:95Vn

g;Vn

eÞ,ð1:05Vn

g ;Vn

eÞ, ðVn

g;0:95Vn

eÞ, or ðVn

g;1:05Vn

eÞ near the fixed point at D¼2 kPa,ψ s ¼ �0:3 MPa, and T¼27 1C. Em indicates the maximum transpiration rate of the9�9 system at T¼27 1C and ψ s ¼ �0:3 MPa.

0

0.2

0.4

0.6

0.8

1

-1 -0.5 0

g/g 0

(uni

tless

)

ψs (MPa)

T=27°CT=7°C

Fig. 6. Stationary stomatal conductance g versus the soil water potential ψs,obtained from numerical simulations at two distinct temperatures, T¼27 1C (solidline) and 7 1C (dashed line). The vapor pressure deficit is taken to be D¼1 kPa andg0 denotes the stomatal conductance at ψs¼0. Values of other parameters are givenin Table 1.






In Fig. 7(a), experimental data are redrawn from Buckley and Mott(2002), which illustrates responses of the stomatal aperture. Forcomparison, we plot in Fig. 7 responses of the stomatal aperture,rather than the stomatal conductance. After a sudden increase in thevapor pressure deficit from 0.4 kPa to 0.6 kPa, the stomatal apertureis shown to increase for a while, decrease for a longer time thereafter,and reach a stationary value smaller than the initial value. On thecontrary, after a sudden decrease in the vapor pressure deficit from0.6 kPa to 0.4 kPa, the stomatal aperture reduces a little, growsrapidly, then attains to a larger value than the original one.

Fig. 7(b) shows the stomatal responses to sudden changes inthe soil water potential: the stomatal aperture responds to adecrease in the soil water potential from �0.3 MPa to �0.6 MPain a similar pattern to the case of the increasing vapor pressuredeficit. When the soil water potential is raised from �0.6 MPa to�0.3 MPa, the stomatal response is similar to that followingreduction of the vapor pressure deficit.

3.3. Stomatal oscillations

When the environmental and plant parameters are assigned inpoor conditions of water, e.g., high vapor pressure deficits, low soilwater potentials, low hydraulic conductances, etc., the stomatalconductance often responds in the form of oscillations to abruptchanges in the vapor pressure deficit (Cardon et al., 1994). Typical suchoscillations obtained from simulations, together with experimental

data, are exhibited in Fig. 8. Whether the stomatal oscillations appearor not depends on the assigned values of the distributed parameters,as known (Haefner et al., 1997), e.g., the standard deviations of κ's.Likewise, the frequencies and decay times depend on the realizationsof those distributed parameters, as sketched in Fig. 8.

3.4. Leaf excision

After a leaf is excised from the plant, water no longer flowsfrom the plant into the leaf. Nevertheless water within the veinsand cells of the leaf still transpires to the ambient air until thestomatal pores are closed. Since veins are merged into xylem inour model, we block water flow to the veins from soil (rather thanfrom the xylem) in order to simulate the leaf excision. Fig. 9exhibits the resulting stomatal responses to the leaf excision,consisting of initial transient ‘wrong-way’ responses, subsequentslow relaxation, and final stomatal closure. It is observed thatthese simulation results are in qualitative agreement with theexperimental data adapted from Powles et al. (2006) although theexperimental data display somewhat slower relaxation. However,

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4

5

6

7

0 20 40 60 80 100 120

A (μ

m)

t (min)

ψs : -0.3 to -0.6 (MPa)ψs : -0.6 to -0.3 (MPa)

4

5

6

7

0 20 40 60 80 100 120

A (μ

m)

t (min)

Buckley and MottD : 0.4 to 0.6 (kPa)D : 0.6 to 0.4 (kPa)

Fig. 7. Responses of stomatal aperture A to changes (a) in vapor pressure deficit Dand (b) in soil water potential ψ s. The system is perturbed suddenly at time t¼0(labeled by vertical dashed lines). (a) Filled circles ð�Þ correspond to the experi-mental data redrawn from Buckley and Mott (2002). The solid line plots stomatalresponses to an increase of D from 0.4 kPa to 0.6 kPa and the dotted line to adecrease from 0.6 kPa to 0.4 kPa at T¼27 1C and ψ s ¼ �0:3 MPa. (b) The solid lineplots stomatal responses to a reduction of ψ s from �0.3 MPa to �0.6 MPa and thedotted line to an increase from �0.6 MPa to �0.3 MPa at T¼27 1C and D¼0.3 kPa.Other parameters are given in Table 1.

0

0.1

0.2

0.3

0.4

0 20 40 60 80 1000

100

200

300

400

g/g m

(uni

tless

)

g (m

mol

m-2

s-1

)

t (min)

Fig. 8. Oscillations of the stomatal conductance g in time t. Filled circles ð�Þrepresent the experimental data in Cardon et al. (1994). In the simulations,performed at T¼27 1C and ψ s ¼ �1 MPa, the system is perturbed suddenly at timet¼0 by raising the vapor pressure deficit D from 1 kPa to 1.5 kPa. Values of otherparameters are given in Table 1, except for αem ¼ 6:922� 10�10 mmol MPa�1 s�1,αee ¼ 6:922� 10�9 mmol MPa�1 s�1, τS ¼ 216 s, and τE ¼ 204 s. Here gm in Fig. 4 isalso used. The frequencies and decay times depend on the realization of thedistributed parameters.

0

0.2

0.4

0.6

0.8

1

1.2

0 20 40

g/g 0

(uni

tless

)

t (min)

GlasshouseOutdoorT=27°CT=7°C

Fig. 9. Time evolution of the stomatal conductance following leaf excision at twodistinct temperatures. The leaf is excised at time t¼0 by setting α0sx ¼ 0 (verticaldashed line). Open circles ð○Þ and open triangles (▵), redrawn from Powles et al.(2006), represent the experimental data for glasshouse and outdoor plants,respectively. Solid and dotted lines show the stomatal conductance following leafexcision for D¼0.3 kPa and ψ s ¼ �0:3 MPa at temperatures T¼27 1C and 7 1C(corresponding to the glasshouse plant and the outdoor plant), respectively. g0 setsthe stomatal conductance at t¼0. Values of other parameters are given in Table 1,except for χ, gb, and α's, each multiplied by factor 2.4, and for τE and τS, each dividedby 2.4.






the temperature dependence appears not to agree: while theexperimental data show that the relative stomatal conductanceof the plant in a glasshouse, corresponding to the plant at the hightemperature, is a bit lower than that of the outdoor plant, thesimulation results hardly vary with the temperature and give evena slightly opposite trend. The origin of this discrepancy is not clearat this stage.

4. Discussion

Our model may appear partly similar to the existing models(Delwiche and Cooke, 1977; Haefner et al., 1997; Dewar, 2002;Haefner, 2005); however, we stress that our model is constructedfrom the precise definitions of relevant variables and based onstatistical physics. Most of all, the model produces results for thestomatal responses in good overall agreement with experimentaldata. Although there have also been proposed various models,where an areole in a leaf is viewed as a square lattice and hydraulicinteractions between neighboring sites on the lattice are included(Rand et al., 1982; Rand and Ellenson, 1986; Haefner et al., 1997;Ferraz et al., 2009), there are substantial distinctions: the modelsof Rand et al. (1982) and of Rand and Ellenson (1986) do notinclude dynamics for xylem and guard cells, and the results fromthe models of Haefner et al. (1997) and of Ferraz et al. (2009)exhibit only regime 2 in our model due to the inappropriateapproximation of the nonlinear differential equations into linearones. Further, it is not possible to observe the stomatal responsesfollowing leaf excision since xylem is not included in the models ofHaefner et al. (1997) and of Ferraz et al. (2009). In contrast, ourmodel takes into account explicitly the volume and the numbers ofwater molecules and of osmolytes as well as the water potential,turgor pressure, and osmotic potential in each cell. Moreover, the‘soil–plant–air’ system is organized by linking the system to thesoil. In consequence, it is possible to observe the stationarystomatal conductance in all the three regimes and stomatalresponses to changes in the vapor pressure deficit or the soilwater potential as well as stomatal responses following leafexcision.

There are several assumptions on which our model is built.For example, while osmolytes may indeed be transported fromepidermal cells to mesophylls and vice versa, the amount ofmesophyll osmolytes has been assumed not to change over thetime scale for stomata to respond to a given perturbation. Inparticular, it has been assumed without substantiation that onecell of each kind is representative of all the cells of that kind. Thisassumption is plausible if water transportation between two cellsof the same kind has a short relaxation time and if variabilityamong the cells is low. Otherwise, in order to obtain more preciseresults, one must include hundreds of variables associated with allthe cells in a stomatal region. It would, however, be formidable tohandle such a model, which requires an extremely long time tosimulate. This contrasts with our ‘cell-unitized’ model, the simula-tions of which give qualitatively good results, e.g., three regimes inthe stationary stomatal conductance, initial transient wrong-wayresponse, stomatal oscillations, and so on, in a reasonable time.

Instead of the assumption of incompressible water, we havealso considered the description of water in terms of the van derWaals equation, allowing variations in the density with pressure;this turns out not to change qualitatively the overall behavior,leading to essentially the same results. Accordingly, for simplicity,we have taken water as an incompressible fluid rather than tofollow the van der Waals equation.

Parameters also need mentioning. Although the values of a fewparameters are presented in existing literature (see Table 1), theseparameters are not consistent since the data in the papers were

not obtained from the same kind of plant. Furthermore, noexperimental data are available for some parameters in theexisting literature. For these reasons, it is not possible to compareour results quantitatively with experimental ones; this wouldrequire that all the parameters for a single plant should be given.Nevertheless, it has been demonstrated that our model withappropriately assigned parameters indeed yield results qualita-tively similar to experimental ones.

In the stationary state, the relationships between the vaporpressure deficit D, stomatal conductance g, and the transpirationrate E divide into three regimes as in Monteith (1995) or Dewar(2002), and the features of each regime agree with those inMonteith (1995). While regime 3 in which E reduces with theincrease of D still appears with no cuticular transpiration from theepidermal cell (ν¼0), it disappears in case that the xylem cavita-tion, Eq. (11), is not considered, i.e., if αsxðψx; TÞ ¼ α0

sx (not shown).In consequence, our results are supportive of the suggestion thatthe origin of regime 3 is the xylem cavitation (Oren et al., 1999;Buckley and Mott, 2002; Dewar, 2002), rather than the suggestionthat it is the cuticular transpiration (Farquhar, 1978) or the patchystomatal closure (Mott and Parkhurst, 1991). In addition, in regime 1,which is known to be difficult to observe experimentally(Monteith, 1995), a local maximum of the stomatal conductanceshows up on the g–D and g--E curves. Its appearance in our model,where abscisic acids (ABAs) are not considered, contrasts with thesuggestion of the dilution effects of abscisic acids (ABAs) as theorigin of the local maximum (Dewar, 2002). When the vaporpressure deficit vanishes, the guard cell water potential becomesequal to the epidermal cell water potential while the guard cellosmotic potential is lower than the epidermal cell osmoticpotential. Accordingly, the guard cell turgor is higher than theepidermal cell turgor. Both turgor pressures are absolutely high,which provide a large mechanical advantage of epidermis for thestomatal pore (Buckley and Mott, 2002). While the mechanicaladvantage of epidermis is large, a small vapor pressure deficit herereduces both turgor pressures evenly, thus enlarging the stomatalaperture. However, a large vapor pressure deficit tends to lowerthe mechanical advantage of epidermis indirectly and to reducethe turgor pressures unevenly. Namely, the guard cell turgorbecomes reduced much more than the epidermal cell turgor,which diminishes the stomatal aperture. This gives rise to thelocal maximum in our model.

The stomata respond to various hydraulic perturbations in acommon pattern, composed of the initial transient wrong-wayresponses and the final stationary right-way responses. They are inqualitative agreement with a variety of experimental data.The initial transient wrong-way responses in our model are causedby the mechanical advantage of epidermis. Therefore, when theguard cell turgor pressure is low and thus the vapor pressuredeficit is high, the extent for the stomata to respond in the wrongway is small due to the low mechanical advantage of epidermis, asshown in Fig. 7.

The hydraulic conductance is defined to be the amount of waterto flow in or out per unit time per unit water potential difference.In general, the conductance between two cells depends on theircontact area, whereas the conductivity does not. While the contactarea between cells in our model has been considered to beconstant in time, the contact area in a real leaf may be dependenton the volumes of the cells: for example, the contact area maydecrease under poor conditions of water, which correspond in ourmodel to low hydraulic conductances (internal parameters) as wellas high vapor pressure deficits and low soil water potentials(external parameters). Indeed, the leaf resistance increases withreduction of the leaf water potential (DeMichele and Sharpe,1973). In our model, stomatal oscillations emerged only undersuch poor conditions of water, e.g., low-humidity, low hydraulic

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conductances and/or low soil water potentials. Accordingly, it issupported that hydraulic conductances between guard cells andepidermal cells are very low when stomatal oscillations occur(Cowan and Farquhar, 1977). In view of that the amount of guardcell osmolytes changes actively in our model, these results do notcontradict the suggestion of Losch and Schenk (1978) that activechanges in the amount of guard cell osmolytes induce stomata tooscillate.

Fig. 8 shows that there are discrepancies between the experi-mental data and simulation results: undamped oscillations withthe mean conductance decaying in experimental data versusdamped oscillations with the non-decaying mean in simulationdata. We believe that the non-decaying trend in simulationsresults from the conservation of Sg þ Se. Namely, if photosynthesisand/or exchanges of osmolytes with mesophylls were included,the decaying trend of the stomatal aperture would appear. Afterthe vapor pressure deficit increases abruptly, the stomatal aper-ture oscillates at a lower level than the original one and then thelower stomatal aperture decreases the CO2 concentration in theleaf and photosynthesis. In consequence, S � Sg þ Se becomesdecreased. Since the water nullcline is independent of S but theosmolyte nullcline depends on S, i.e., reduction of S results inreduction of E (and equivalently, A), the level of the stomatalaperture continues to decrease until the photosynthetic equili-brium is reached.

Since xylem in leaf veins and xylem in the trunk are unified inour model, leaf excision is achieved by means of cutting off thexylem from the soil, viz., by abolishing the soil-to-xylem hydraulicconductance. The veins are just water reservoirs, taking waterfrom the xylem and supplying it to neighboring epidermal cells;specification of the veins is thus expected not to alter significantlythe results. Both our model and the experiment of Powles et al.(2006) demonstrate that the stomatal conductance increases withthe temperature before the leaf is excised [see Fig. 4(a) forsimulation results]. In our model, this tendency persists after theleaf excision; on the other hand, the stomatal conductance of theoutdoor plant (at low temperatures) has been observed higherthan that of the glasshouse plant (at high temperatures) later,because the stomatal conductance of the outdoor plant increasesfor a longer time then decreases. This discrepancy comes from thefact that the transpiration rate indeed depends on the temperaturevia the diffusion constant Dw of water vapor, which satisfies theEinstein relation: Dw¼μw kB T with its mobility μw. According tothe Einstein relation, the lower the temperature, the smaller thediffusion constant. Therefore stomata at low temperatures closecompletely after a longer time than those at high temperatures. Onthe other hand, the relaxation times in simulation results turn outa little shorter than those of experimental data. This results fromthe assumption of constant cell-to-cell hydraulic conductances;in reality, the leaf water potential decreases with time and cell-to-cell hydraulic conductance in turn decreases, which results in thelong relaxation time.

The transpiration rate depends on the number of stomatasurrounded with the veins as shown in Fig. 4(d). In the caseL¼1, it is unreasonably high. In order to attain the stationary state,the plant absorbs water from the soil until the soil water potentialis lowered sufficiently, i.e., until the soil is very dry, and thus theplant will also dry up eventually. On the other hand, when L¼13,the transpiration rate is lower than that of L¼9 but still reason-able. Transient wrong-way responses of the system of L¼13appear as well, even though the extent is smaller than that ofL¼9 (not shown).

The size of areoles can be obtained from the data of thestomatal density ρs and vein density ρv defined to be the numberof stomata per unit area and the total vein length per leaf area,respectively. With the assumption that leaf veins form an L� L

square lattice for simplicity, the area of an areole takes the formð2=ρvÞ2. In consequence, we have the areole size in the form4ρs=ρ2

v (in units of the number of stomata). For instance, thestomatal and vein densities of Arabidopsis cotyledons are approxi-mately ρs ¼ 200 mm�2 (Ohki et al., 2011) and ρv ¼ 2:45 mm�2

(Haritatos et al., 2000), and the corresponding areole size is thusgiven by 133 in units of the number of stomata.

In addition to the results for the stomatal responses to varioushydraulic perturbations, the temperature dependence of stomatalconductances has also been obtained from our model: the higherthe temperature, the higher the stomatal conductance. It isremarkable that although the temperature dependence in ourmodel comes just from the van't Hoff equation for the osmoticpotentials, obtained results agree reasonably well with the experi-mental results of Buckley and Mott (2002) or Warrit et al. (1980).Such temperature dependence meets the necessity that plants athigher temperatures must transpire more in order to coolthemselves.

5. Conclusion and outlook

This study provides a mathematical model which shows boththe stationary and non-stationary state responses of stomata tovarious perturbations in water. The familiar continuity equation,the fluid version of Ohm's law, the incompressibility of water, andthe precise definitions of various potentials have been used in ourmodel, which has been simplified as much as possible but nomore, with undue distortion avoided. Together with a few plau-sible assumptions, our model is sufficient to explain diverseresponses of stomata, despite its relative simplicity. In conse-quence, our model yields qualitatively correct results for thestationary and non-stationary stomatal conductances, stomataloscillations, and responses to leaf excision. Further, plausibledependence of the stomatal conductance on the temperature hasbeen observed.

Finally, we point out that this is a first-stage model, bearingmuch room for extension. It may be extended to an integratedmodel, which describes stomatal responses to, e.g., water, tem-perature, carbon dioxide, and light intensity. One may also con-sider the possibility of long-range indirect interactions betweenepidermal cells across a vein, mediated by the change of the stateof xylem in the vein, and examine associated effects such asstomatal patchiness. Further extensions include building substo-matal cavities, which allows one to observe effects of transpirationdepending on the temperature, using the heat balance equation, toobtain the temperature of a plant from the temperature of theambient air, and introducing photosynthesis, to change the totalamount of osmolytes in a stomatal region.

Acknowledgments

This work was supported by the National Research Foundationof Korea through the Basic Science Research Program (Grant nos.2012R1A2A4A01004419 and 2011-0012331).

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Generalized hydromechanical model for stomatal responses to hydraulic perturbations