Planta

Planta (1992)187:14-25

9 Springer-Verlag1992

Compartmental distribution and redistribution of abscisic acid in intact leaves I. Mathematical formulation Stefan Slovik*, Mathias Baier, and Wolfram Hartung Julius von Sachs-lnstitut f/Jr Biowissenschaften, Lehrstuhl Botanik I, Universit~it W/irzburg, Mittlerer Dallenbergweg 64, W-8700 Wfirzburg, Federal Republic of Germany Received 19 October 1990; accepted 9 October 1991

Abstract. Using experimental information obtained in earlier studies on the permeabilities of mesophyll and guard-cell membranes to abscisic acid (ABA), and on stress-induced p H shifts in the apoplasm and in symplasmic compartments (Hartung et al., 1988, Plant Physiol. 86, 908-913; Hartung et al. 1990, B P G R G Monogr. 215--235), a mathematical model is presented which will permit computer analysis of the stress-induced redistribution o f ABA amongst different leaf cell types (mesophyll, epidermis, guard cells, phloem cells) and their compartments (cell wall, cytosol, chloroplast stroma, vacuole). Metabolism and conjugation of ABA and its transport in the xylem and the phloem are also taken into consideration. We ask whether the stressinduced redistribution o f ABA is fast and intensive enough to induce stomatal closure within a few minutes. The model can be adapted to any other weak acid or base, e.g. to other phytohormones (auxins, gibberellins), which differ from ABA, e.g. by their membrane conductances, anion permeabilities and p K a values. Our wholeleaf model can predict the time course and the compartmentation of, for example, phytohormone concentrations as a function o f changing source-sink patterns (e.g. by compartmental pH shifts in the leaf lamina). An analysis of the present knoWledge of the ABA physiology of leaves and studies on stress effects are presented in subsequent publications. In this communication we describe the whole-leaf model and present and discuss all necessary morphological (volumes, surfaces etc.) and physiological (pH, membrane conductances etc.) parameters o f an unstressed leaf of Valerianella locusta L. We draw fundamental conclusions by comparing determined and calculated A B A concentrations in the leaf-cell compartments. We found that the model predictions are close to measured data, and we conclude that in unstressed leaves ABA is close to flux equilibrium amongst the different tissues and compartments. Abbreviations: ABA=abscisic acid; CON=ABA conjugates;

HABA = neutral ABA species * To whom correspondence should be addressed

Key words: Abscisic acid (compartmentation in leaves) - Computer model (ABA compartmentation) - Drought stress (quantification) - Leaf (ABA compartmentation) - pH shift - Stomatal regulation Valerianella

Introduction The conductance of the stomata for water vapor and CO2 is regulated by the intercellular CO2 partial pressure, light, vapor-pressure gradients between leaf interior and exterior, and abscisic acid (ABA), a phytohormone which is involved in stomatal regulation under drought or salt stress (Zeevaart and Creelman 1988). Net synthesis of ABA in leaves under stress starts after closure of the stomata. Synthesis is slow. It cannot be important in eliciting rapid stoma closure (Pierce and Raschke 1980; D6rffiing 1983). Rather, the redistribution of bulk ABA amongst different leaf tissues and compartments is involved in stomatal regulation (Hartung et al. 1988; Hartung and Radin 1989). Because ABA is a weak acid, which can penetrate biomembranes in the protonated form, changes of pH values in different leaf tissues and compartments (e.g. by light, darkness, stress) must cause a redistribution, which affects the ABA concentration in the apoplasm. As fully differentiated guard cells possess no plasmodesmata, apoplasmic ABA concentrations are important for stromatal regulation. The only mathematical analysis of ABA distribution and transport that is available has been published by Cowan et al. (1982). Their work is restricted to a redistribution amongst different mesophyll-cell compartments and the apoplasm as a consequence of pH shifts in the illuminated and darkened chloroplast stroma. The aim of our work is to analyse quantitatively and in more detail the complex physiology of ABA in unstressed and stressed entire leaves in regard to stomatal regulation. In this communication we formulate the mathematical model and demonstrate some preliminary conclusions by comparing determined and calculated ABA concentrations in leaf cell

S. Slovik et al.: A B A distribution in leaves: I. Mathematical formulation

compartments. A detailed analysis of the present knowledge of the ABA physiology of leaves and studies on stress effects are presented in subsequent publications. Preliminary results of our model were published by Hartung et al. (1990).

15

chyma. Concerning water loss, we distinguish between stomatal and cuticular transpiration. The vascular bundles are completely embedded within the mesophytl. All symplastic spaces are embedded in the apoplasmic space, which is divided into three distinct matrices (mesophyll apoplast, epidermal cell wall and guard-cell wall). The xylem (water-filled dead cells, e.g. tracheids, sclerenchyma cells) is attributed to the mesophyll apoplasm.

D e s c r i p t i o n o f the m o d e l

Tissues and compartments. Our model (Fig. 1) considers entire leaves consisting of chlorenchyma cells (with cytosol, chloroplasts and vacuoles), epidermal cells (with cytoplasm and vacuoles), guard cells (with cytosol, chloroplasts and vacuoles) and sieve elements of the phloem (only cytosolic sap). All tissue and compartment volumes and their surface areas (see Tables 1, 2 and Appendix VII) are determined for greenhouse-grown leaves of Valerianella locusta, which are amphistomatic. We distinguish neither between the upper and the lower epidermis, nor between spongy and palisade paren-

Abscisie-acid fluxes at membranes. The phytohormone ABA is a weak acid. The concentrations of the charged anion species, A B A - , and the uncharged, undissociated species, HABA, are related by the reaction

(1)

HABA+HzO ~ ABA-+H30 +

Only HABA is membrane-permeant, while ABA- (in contrast to, e.g. I A A - ) is practically impermeant (Hartung 1983; Baier and Hartung 1988; Hartung and Slovik 1991). Therefore flux-inducing gradients at membranes

ai r

HzO,~

cuticle ==============5=====5======================= guard cell wall

========================5==============5============

II

epidermis cell wall I

II

I II,

EPIDERMis CELLS

:.:l

Ii::i

GUARD CELLS

III

:'-(. :.-.-~ -." . ":::.'.2.~L:.'.".'-'.".'..'-'.-'.:~'.: . . .

'"

Ill III III III Ill III III III

I:-:

vacuole

!1.2:.. :.i ii::

ill II I I Iii mll

Ii l l

vacuole

chloroplasts

-" " " ' A B A ' ( ~ ) ' H I~A'.','-','. '" : ' - . ", " . " " ' ' , ' . " "."-" ABA < ~

Ill

HABA

'. "- " "--Z--'----'----'--'----L__'--__L__L. . . . . ".'.' " ..... "'." "'-'11

b

CH,OJo

~

H20~

I

............ '-

ABA receptor

H20

.ell..ll

Y ,4-.--HA A

.)A:

(~>

_~-

I Hz0

I Hz0

substomatal cavity i,u,i H H20 ,4

..o.

ABA 4i

HABA

"PA ~

(~>-'ABA ~

;i )'. ] u g a t e ; acuole

1:[:: "I

roots

X.'-'-'.'-'-'.

HAIA

'i [ L ~ A - c ~

roots

A

•

+ + + + + =

tracheid lumina sclerenchyma lumina xylem cell walls phloem cell walls chlorenchyma cell walls mesophyll apoplasm

0

~

chloroplasts

Mesophy|l ceils

Fig. 1. Overview of compartments and metabolic pathways considered by model analysis. We distinguish between chlorenehyma cells, epidermal cells, guard cells and phloem cells, embedded in three distinct apoplasmic spaces and partially connected by plasmodesmata. Transpiration is divided into stomatal and cuticular water loss. Abscisic-acid fluxes are divided into mass flow, diffusion, membrane permeation and carrier transport. Metabolic turnover rates occur in the cytosol, in the vacuoles and in the apoplasm. We distinguish between free A B A and its conjugates. Most morphologi-

cal and physiological parameters are determined experimentally for Valerianella locusta L. Cn~o,~,~ = concuctance of the teaf cuticle to water (the index indicates the interface between epidermal cell walt and air); Cn~o.gwa = conductance of the leaf cuticle to water (interface between guard cell wall and air); CH2o,~, = conductance of stomata to water (interface between substomatal cavity and air); HzO ~ ---- water concentration in air; H 2 0 i = water concentration in intercetlulars; PA = phaseic acid; P, X, Y = degradation products o f ABA-anions or o f the undissociated molecular species H A B A

16

S. Slovik et al, : ABA distribution in leaves: I. Mathematical formulation

Table 1. Volumes (V), concentrations (SumABA) and pH values of unstressed leaf compartments in the light. Additionally, indices denoting compartments are defined. Most morphological and physiological parameters are determined experimentally for Valerianella tocusta (Appendices II, III and VII) Compartment

Index V

SumABA pH

(t0 -6" m - 3 - m -2)

(nM)

Mesophytl cells Cytosol Chloroplast stroma Vacuole

mp mc

7.07 11.31

9.0~ 35.7~

7.30 7.90

mv

122.94

2.8

6.33

Epidermal cells Cytoplasm Vacuole

ep ev

4.76 42.84

55.1 b 5.6b

7.30 ~ 6.30 ~

Guard cells Cytosol Chloroplast stroma Vacuole

gp gc

0.0647 0.0301

426.4 ~ 467.6 a

8.16 8.20

gv

0.8531

195.7

6.80

Phloem Cytosolic sap

fp

1.09

12.0a

7.60

Apoplasm Mesophyll Epidermis Guard cells

mw ew gw

17.73 7.47 0.18

4.1 4.1 4.1

6.50 6.00 6.30

a Determinations for the cytoplasm (cytosol and chloroplasts are close to equilibrium); Appendix III b Determinations for protoplasts (cytoplasm and vacuole may be in equilibrium); Appendix III Estimations from chlorenchyma cells; Appendix III d Assumption; there are no data for Valerianelta; Appendix Ill

Table 2. Surface areas A and conductances

C of membranes and other interfaces between adjacent model leaf compartments (Appendices IV and VIII). The indices denoting adjacent compartments are defined in Table 1. In this table, indices denoting areas between defined compartments are summarised as the 'sum' of the indices of the adjacent compartments. Most morphological and physiological parameters are determined experimentally for Valerianella locusta L.

Compartment I

a r e d e t e r m i n e d b y the a b s o l u t e p H at e a c h side o f the m e m b r a n e , b y the pKa o f A B A a n d b y the p e r m e a b i l i t y o f H A B A . T h e p e r m e a b i l i t y coefficients for H A B A differ b e t w e e n different m e m b r a n e s ( p l a s m a l e m m a , t o n o p l a s t , c h l o r o p l a s t envelope) o f different l e a f tissues ( G i m m l e r et al. 1981; Baier a n d H a r t u n g 1988; D a e t e r a n d H a r t u n g 1990). I n the t o n o p l a s t o f g u a r d cells a n A B A c a r r i e r has been identified. It a c c u m u l a t e s A B A w i t h i n the v a c u o l e (Baier a n d H a r t u n g 1988). Compartmentation o f A B A metabolism. Synthesis a n d d e g r a d a t i o n o f A B A is l o c a t e d in the c y t o s o l ; c h l o r o plasts d o n o t p l a y a d o m i n a n t role in A B A m e t a b o l i s m ( H a r t u n g et al. 1980; H a r t u n g et al. 1981; B r a y a n d Z e e v a a r t 1985; C o w a n a n d R a i l t o n 1986; L e h m a n n a n d G t u n d 1986). T h e f o r m a t i o n a n d m o b i t i s a t i o n o f A B A c o n j u g a t e s is l o c a t e d in the v a c u o l e ( K a i s e r et al. 1985; L e h m a n n a n d G l u n d 1986). In o u r m o d e l we d o n o t distinguish between h y d r o l y s a b l e a n d n o n - h y d r o l y s a b l e conjugates. T h e p o t e n t i a l o f m e t a b o l i s i n g A B A a n d its c o n j u g a t e s is n o t d i s t r i b u t e d e q u a l l y a m o n g s t different leaf tissues. In g u a r d cells neither A B A synthesis n o r d e g r a d a t i o n o r t u r n o v e r o f c o n j u g a t e s c a n be o b s e r v e d ( D t r f f i i n g 1983; Behl a n d H a r t u n g 1986; L a h r a n d R a s c h k e 1988). In the c y t o s o l o f c h t o r e n c h y m a cells, h o w e v e r , A B A synthesis as well as A B A d e g r a d a t i o n occurs. A d d i t i o n a l l y , c h l o r e n c h y m a cells are c a p a b l e o f c o n j u g a t i n g o r m o b i l i s i n g A B A in the v a c u o l e ( L e h m a n n a n d G l u n d 1986). N o t h i n g is k n o w n a b o u t A B A m e t a b o lism in the c y t o s o l o f sieve tubes. T h e r e f o r e , in o u r m o d e l , A B A m e t a b o l i s m in the p h l o e m is a s s u m e d to be absent. U n f o r t u n a t e l y , there is o n l y i n c o m p l e t e i n f o r m a tion o n A B A m e t a b o l i s m in e p i d e r m a l cells. I n the m o d e l the e p i d e r m a l cells are c o n s i d e r e d to possess the s a m e

Compartment tI

Index

A (m2-m -2)

C (t0 9 - m . s - ~ )

Mesophyll Apoplasm Cytosot Cytosol Cytosol

Cytosol Vacuole Chloroplast Cytosol (Epid.)

mwmp mpmv mpmc mpep

32.30 29.42 36.31 0.0024

25 5 5000 1950a

Epidermis Apoplasm Cytosol

Cytosol Vacuole

ewep epev

8.48 7.90

250 ~ 13b

Guard cells Apoplasm Cytosol Cytosol

Cytosol Vacuole Chloroplast

gwgp gpgv gpgc

0.405 0.377 0.062

246 12.6 5000 c

Phloem Apoplasm

Cytosolic sap

mwfp

0.532

50

Cell walls Mesophyll Epidermis

Epidermis Guard cells

mwew ewgw

0.076 0.025

519.5" 1935~

Calculated as DABA/Ax;Appendix IV b Estimations from guard cells; Appendix IV c Estimations from chlorenchyma-cell chloroplasts; Appendix IV

S. Slovik et al. : ABA distribution in leaves: I. Mathematical formulation metabolic capabilities as the chlorenchyma cells. Finally, we take into account that ABA degradation also occurs in the mesophyll (Loveys and Robinson 1987) or epidermal apoplasm (own observation).

Flux balance of different tissues. The flux situation between leaf and shoot as well as between different tissues of a leaf is complex (Fig. 1). The water conductances of the stomata and of the epidermal cuticles define the transpiration rate and thereby transport o f dissolved ABA from the roots to the mesophyll apoplast. Concomitantly, an apoplasmic mass flow o f dissolved ABA from the mesophylt apoplasm to the epidermal cell wall and from there to the guard-cell wall occurs. The total stomatal transpiration can be attributed to different cell types (guard cells, epidermal cells, chlorenchyma cells) contributing to evaporation from the cell wall-intercellular interface areas in the substomatal cavity. Thus, the water flow rate to the bulk epidermis and from there to guard cells can be estimated depending on the actual stomatal transpiration rate. The transport rate o f ABA is calculated on the basis o f the water flow rate within the cell walls connecting mesophyll, epidermal and guard cells. Simultaneously, ABA diffuses amongst the three apoplasmic spaces defined above according to the actual concentration gradients between them. The same is true for plasmodesmata directly connecting the cytosol o f mesophyll and epidermal cells. They balance the mass transport and the diffusion o f ABA. Fully differentiated guard cells possess no plasmodesmata (Weyers and Hillman 1979). A final complication is the flux of H A B A from the mesophyll apoplasm into the phloem cell 'cytosol', which is exported permanently by the lamina of the leaf. No attempt has been made to model stomatal closure in response to ABA-receptor interactions in or at guard cells. The following section demonstrates how to translate the physiological network o f ABA in leaves as described here and reviewed in Fig. 1 into a mathematical network. It should be noted that we present a discrete model using a standard description of re-equilibration across membranes between two well-stirred solutions of finite volume.

17

where J is the flux density per unit membrane area between compartments j and k; c is the conductance o f the barrier separating the compartments for a particular molecular species, a n d X j and Xk are the activities o f the permeable species, in compartments j and k. As the concentrations o f ABA species are small in vivo, the H A B A activities can be replaced by.their concentrations. The concentration changes of the species in conapartmerits j and k during the integration time step dt are dXj = - J - A - V [ 1 - d t

(Eq. 1.2)

dXk = + J" A - V~- ~ 9 dt,

(Eq. 1:3)

where A is the membrane area separating compartments j and k. Vj and Vk are the volumes o f the compartments j and k considered. Consequently, the values o f the concentrations at the next integration step X(t + dt) are related to the previous values X(t) by X(t + dt) = X(t) + dX

for each compartment. The integration step dt is made very small to minimise the iteration error. These are simplified explanations which just stress the principles o f integrating ordinary differential equations numerically 1.

Calculation of the concentrations of all molecular species. In the equations given below we will use denotations which are variables in our computer program. They are defined in Table 1 and 2; see also Appendix L The standard dissociation equation for reaction 1 is K~ABA =

[ A B A - ] . [H +1 , [HABA]

General description. Unlike the mathematical approach of Cowan et al. (1982), the problems analysed in this work cannot be solved analytically with reasonable investment of time and work. Rather the calculation of the time course o f fluxes o f ABA amongst different leaf tissues and compartments,-and the adjustment o f the concentration o f the A B A - derived from the permeating neutral molecule is a typical task o f integrating a system of ordinary differential equations. At cacti integration step, changes in the concentration o f ABA species in the compartments can be calculated if individual conductances of 'the membranes separating the compartments are known. Fluxes are J = C - (Xj--Xk),

(Eq. 1,1)

(Eq. 1.5)

where KaABA = 1.778 9 10 -s m M is the equilibrium constant o f ABA (pKa = 4.75), [ABA-] is the concentration of dissociated ABA, [HABA] o f undissociated ABA and [H +] is the concentration o f protons in each compartment. SumABA is the total amount: SumABA = [HABA] + [ABA-]

Mathematical formulation

(Eq. 1.4)

(Eq. 1.6)

Employing Eqs. 1.5 and 1.6 the pH-dependent concentrations o f both molecular species o f ABA can be calculated: [HABA] =

SumABA 1 + KaABA/[H +1

(Eq. 1.7)

1 The program module of the model on ABA physiology is incorporated into an universal model environment written in Turbo Pascal 4.0 (MS-DOS 3.3). For calculations we employed a PC (180386processor, 33 MHz, and a 80387 mathematical co-processor). All results can be visualised graphically on screen (EGA or VGA). The compiled program 'ABA-LEAF.EXE' and all PASCALsource codes of the model are available on request from the senior author on sending him a formated DS HD 5Vd' diskette (I.2 MB). For model modifications TURBOPASCAL(version 4.0 or higher, Borland) is necessary

18 [ABA-] =

S. Slovik et al.: ABA distribution in leaves: I. Mathematical formulation SumABA 1 + [H+I/KaABA

= S u m A B A - [HABA] (Eq. 1.8)

In the computer program this calculation has to be made for all 12 compartments separately at each iteration loop.

Flux densities of ABA [it lipid membranes. F r o m the results o f Eqs. 1.7 and 1.8 thefollowing.flux equation can be calculated (cf. Eq. 1.1): JnAaA = CnA,A " ( [ H A B A l j - [HABA]0

(Eq. 1.9)

In the computer program, Eq. 1.9 is performed separately at each iteration loop for nine lipid membranes (cf. Table 2). In addition to the passive diffusion o f H A B A through the tonoplast of guard cells there is a saturatable influx component (carrier). We define its influx rate EABA employing a simple hyperbolic formulation: VABA" SumABA EABA -- KmAaA+ SumABA '

(Eq. 1.10) H 2 0 , t ( T ) = 18- 10- 6. exp[ - 5047.8/(T + 273.15) + 17.167]

where K m A B A is the affinity parameter, an apparent K m, and VA,A is the Vmaxo f the saturatable membrane flux component.

Diffusion of ABA amongst different tissues within the apoplasm and in the plasmodesmata. According to Fick's law the diffusion process o f ABA (i) within plasmodesmata connecting theepidermal cytosol and the cytosol of mesophyll cells, (ii) between mesophyll cell wall and epidermal cell wall, and (iii) between epidermal cell wall and guard-cell wall can be formulated as

Dkag = D ABA

ASumABA Ax

and plasmodesmata, but also by a small mass flow of water containing dissolved ABA within the cell walls and plasmodesmata. While the flux density of water is dependent on stomatal aperture and - to a minor extent - also on the cuticular water permeability and on the waterexport rate via the phloem sap, the water distribution within the lamina and therefore the SumABA redistribution, until the water evaporates either into the substomatal cavity or more directly into the atmosphere via the cuticular layer, is sensitive to leaf morphology. We do not try modelling ~water relations of entire leaves in detail. A s the stomata usually close before a significant decrease of the water potential o f leaf laminae can be measured (e.g. Blackman and Davies 1985), we assume a constant water content in all leaf compartments. The listing of the saturation concentration o f water vapor (i.e. at 100% relative humidity) as a function of temperature is summarised, for example, in Appendix III o f Nobel (1983). Employin~ a fitting routine, this listing can be substituted by a mathematical function, which is valid for physiologically relevant temperatures:

(Eq. 1.13) H2Osat(T) is the saturation concentration o f water vapor, given in m o l . mol-~, and T is the temperature in degrees centigrade (~ The factor 18 9 10-6 is the volume of one mole of pure water given in m 3 9mol-1. If the relative humidity RHa of the,atmospheric air and its temperature T a is known, then employing Eq. 1.14 the water concentration [H20]. in the air surrounding the leaf can be calculated to be [HEO]a = RHa" H20~t(T~).

(Eq. 1.14)

(Eq. 1.11)

w h e r e DAa A is the diffusive flux along the effective path-

length Ax, driven by the concentration difference ASumABA, and characterised by the diffusion coefficient D ABA (m 2 9 s- 1) of bulk ABA in the particular medium. Apparent diffusion conductances DABA/Ax are calculated in Appendix IV. Employing Eq. 1.11, we define the diffusive fluxes DABA o f A B A - or H A B A within inhomogeneous ~compartments on a leaf-area basis as DABA =" D ABA9 ( S u m A B A j - S u m A B A 0 9 A/Ax.

(Eq. 1.12) A is the area passed by ABA, i.e. it is the interface area within plasmodesmata or within the apoplasm per unit leaf area. The area A and all other variables are defined in Tables 1 and 2 for all three diffusive fluxes considered in the leaf model.

Stomatal transpiration. The actual ABA concentration in different compartments is not only determined by the permeability of membranes to H A B A in different tissues and by diffusion processes o f SumABA within cell walls

T h e same can be done for the water concentration [H20]i within the intercellular spaces of the leaf, employing the leaf temperature Tle~f: [H20]i = RHi" H20~.t(TIo~r).

(Eq. 1.15)

In Eq. 1.15 the relative humidity RHI of the intercellular air is close to saturation. We employ RHi = 1.0. Using Eqs. 1.13 to 1.15, the stomatal transpiration mass flow MH~o,i. can be calculated on the basis o f the actual mean stomatal apperture CH20,ia(m 9 S-1 or m 3 9m -2 9 s-x) o f both faces of the Valerianella leaf measured on a leafarea basis" M,~o.~.

=

C H 2 0 , ia "

([H20]i--[H2O]a).

(Eq. 1.16)

It is not really known where water evaporation is focussed in the substomatal cavity. Usually it is assumed that it may be focussed near guard cells, but it is unknown how substomatal cuticles confuse the situation. Thus, we attribute the total stomatal transpiration rate Mn20, ia to the mesophyll, epidermis and guard calls on the basis of their fractions FracI (see Appendix VII) of

S. Slovik et al. : A B A distribution in leaves: I. M a t h e m a t i c a l formulation

interfaces to the substomatal cavity. It is sufficient to define M n 2 o , ewi = M n 2 o , ia " FracIr MH20,gwi = MH20, ia"

and

Fraclgwi

(Eq. 1.17)

19

Mass flow of ABA in cell walls and plasmodesmata. The transport rate M A B A . . . . from the roots into the mesophyll apoplasm is calculated readily by multiplying the water flow rate MH2O. . . . by the ABA concentration SumABAx in the sylem sap of the petiole:

(Eq. 1.18) MABA . . . .

for the stomatal water evaporation of epidermal cells and guard cells respectively. Thus, Fraclm~ for the mesophyll is readily defined as

The export flow lamina is

FracIm~i = 1 - (FracI,wi + FracIgw~).

MABA, fp ---- MH20,fp " S u m A B A f p ,

(Eq. 1.19)

Cuticular transpiration. Like stomatal transpiration, the cuticular transpiration rate Mmo .... of epidermal cells and MH,o,gw, of guard cells is based on Eq. 1.13 to 1.15. The flux equations are similar to Eq. 1.16. For epidermal cells we obtain Mu~o.... = CH20. . . . " ([H2Oli-[H2013 9(2-Agw~) (Eq. 1.20) where A~,. is the absolute surface of guard-cell cuticles on a leaf-area basis. The factor 2 defines two lamina faces. Cn2o.... is the mean cuticular water conductance for both epidermal layers. Accordingly, for guard cells we obtain Mayo,w,. = Cu~o.... 9([H20]i- [H20]~) 9Agw.. (Eq. 1.21) At this point is should be stressed that the boundarylayer resistance at the leaf surfaces which depends on wind velocity is not considered separately. The effective conductances CH2o,ia and Cn2o.... can be adapted in order to take into account this complication.

(Eq. 1.25)

= MH20 . . . . " SumABAx MABA,fp

of the phloem sap leaving the (Eq. 1.26)

where Mn2o,fp is the mass flow of the phloem sap and SumABArp is its actual ABA concentration. Concerning the mass flow of ABA within cell walls and plasmodesmata, difficulties occur because water is transported in a mixed symplasmic-apoplasmic manner. As biological membranes have high permeabilities for water, but are relative to water - practically impermeable to molecules such as sucrose (Nobel 1983, p. 40) or ABA (see conductances in Table 2), a mixed transport of ABA can be disregarded. For this reason, ABA is transported only by flowing water in the plasmodesmata and cell walls. For mesophyll-cell walls we write MABA . . . . .

=

Mn2o. . . . .

" SumABAmw"

fmwew"

(Eq. 1.27)

MABA ..... is the transport rate into the epidermal cell wall, Mn2o. . . . . is defined in Eq. 1.23, SumABAmw is the actual ABA concentration in the mesophyll apoplasm, and fmwewis the fraction of cell-wall area relative to the total cell area passed by the water flow along its way to the epidermis (see Appendix VII). Similarily, we define

MABA,0w~ = MH20,ewgw" SumABA0w 9f~w~

(Eq. 1.28)

for ABA flowing into the guard-cell wall, and

Mass flow of water into the epidermal cell wall and from there into guard-cell walls. From Eqs. 1.21 and 1.18 the total water flow r a t e M H : o , ewgw into the guard-cell wall can be added up: MH20, ewgw ~ MH20,gwa-~- MH20,gwi.

= MH20, ewgw-'k MH20 . . . . -t- M n 2 o , ewi.

(Eq. 1.23)

The total water flow r a t e MH2 o . . . . from the roots into the mesophyll apoplasm is the sum of stomatal and cuticular transpiration rates and the water-export rate via the mass flow of the phloem sap (MH~o,fp, cf. Appendix V): Mn2o. . . . = MH20,ia"[- MH20 . . . . -+- MH20, gwa-'~-0 . 8

"

SumABAmp " fmpop

(Eq. 1.29)

for ABA flowing within the plasmodesmata connecting mesophyll cells and epidermal cells.

(Eq. 1.22)

The water flow rate from the mesophyll into the epidermis is the sum of the results of Eqs. 1.17, 1.20 and 1.22: MH20 . . . . .

MABA,~pop = MH2O . . . . .

"

MH20,fp.

(Eq. 1.24) The factor 0.8 represents a typical relative water content of the phloem sap.

Abscisic-acid metabolism. As detailed information on kinetic parameters are rare, we simplified the mathematical formulation of ABA metabolism to its most essential features. Total synthesis rates in the cytosol of chlorenchyma (RnABA,mp)and epidermal cells (RnABA,op)are defined as model constants (see Appendix V1). The degradation rates of ABA in the cytosol of chlorenchyma (RABA,mp) and epidermal cells (RABA,op) are defined as pseudo-first-order reactions with the corresponding velocity constants being kABA,mpand kABA,ep: RABA,mp = kABA,mp " SumABAmp

(Eq. 1.30)

RABA, ep = kABA, ep " SumABAop

(Eq. 1.31)

In chlorenchyma and epidermal cell vacuoles, both ABA conjugation and its mobilisation from the conjugates are

20

S. Slovik et al.: A B A distribution in leaves: I. Mathematical formulation

defined as pseudo-first-order reactions with the corresponding velocity constants being kABA,mvand kAaA,,~for the conjugate formation and kcoN,.v and kcoN,evfor conjugate hydrolysis 9 The net reaction rates of ABA conjugation in or at the chlorenchyma vacuoles (RABA,m~) and epidermal vacuoles (RmA,ov) are

The epidermal cell wall balances two mass flows, two diffusion flows, one membrane permeation and one catabolic rate:

dSumABAew =

[(MAB A. . . . .

DABA,ewgw -- JHABA. . . . p"

-- MABA, ewgw+ DABA . . . . .

--

Aewep)/Vew-- RABA, ew] " d t (Eq. 1.37)

RABA, m v =

kABA,mv " SumABAmv-kcoN.~v " C O N m v

(Eq. 1.32) RABA,ev = kABA, ev " SumABAe,,-kcoN,ev

" CON0v. (Eq. 1.33)

CON~v and CONe, are the concentrations of the vacuolar conjugates in the mesophyll cells or epidermal cells, respectively. There are indications that degradation of ABA takes place in the apoplasmic space of the mesophyll wall (RABA,mw) and-or in the epidermal wall (RABA,ew), which we define as simple Michaelis-Mententype reactions with the corresponding affinity parameters b e i n g KmABA'mw and KmARA' ~w- VARA,mw and VAnA,eware the maximal reaction rates (Vma0: VABA,~w " SumABA~w RABA'mw = KmABA,mw+ SumABAmw VABA,ew" SumABAew RABA, ew = KmABA,ew+ SumABAow

dSumABAgw = [(MABA, ewgw+DABA, ewgw--JrlABA,gwgp " A~,,gp)/V~]- dt (Eq. 1.38) Now we calculate the ordinary differential equations of the four cytosolic compartments of the symplasm. The chlorenchyma cytosol balances one mass flow and one diffusive flow within plasmodesmata, three membrane permeations and two metabolic rates: dSumABAmp = [(JHABA . . . . p" Amwmp-- JHABA,mpme "Ampmc -- JHABA,mpmv " Ampmv -- MABA,mpep -- DABA, mpep)/Vmp +

(Eq. 1.39)

+ RHABA,mp -- RABA,mp] " d t

(Eq. 1.34) (Eq. 1.35)

Differential budoet equations. This part is the core of the model interrelating all equations formulated above to a mathematical network 9 Differential budget equations are written for a short time interval dt (integration step)9 Their general structure was demonstrated under General description in this section. The concentration increments which result during the iteration step dt in all 12 compartments for ABA and its conjugates are calculated and added to the values already extisting in these compartments. Before summation, a common unit has to be defined9 It is m o l . s - 1 . m -3. Therefore, membrane fluxes JHABA and EABA, which are calculated on a membrane-area basis (mol 9m -2 9s-1), must be multipled by the membrane area per leaf area. These area values are summarised in Table 2. We thus arrive at flux rates per m 2 of leaf. Now JHABA and EABA are compatible to the mass flow MABA and the diffusion flow DABA, both already defined per unit leaf area (mol 9m - 2 . s- 1). Finally all fluxes mentioned must be divided by the volume per unit leaf area (m 3 9m - 2) of the compartment considered, in order to find the concentration increments. These volumes are summarised in Table 1. After this, the reaction rates RHABAand RABA, which are already given in m o l . m - 3 . s - l , can be added. First we calculated the ordinary differential equations for the three apoplasmic spaces. The mesophyll apoplasm balances two mass flows, one diffusion flow, two membrane permeations and one catabolic rate: dSumABAmw = [(MABA, maw -- MABA . . . . . -- DABA . . . . . -- JHABA . . . . p" Amwmp - JHABA.mwfp " Amwfp)/Vmw - RABA,mw]

9 dt

The guard-cell wall balances one mass flow, one diffusion flow and one membrane permeation:

(Eq. 1.36)

The epidermal cell cytosol balances one mass flow and one diffusi;ee flow within plasmodesmata, two membrane permeations and two metabolic rates: dSumABA~p = [(JHABA . . . . p " A~,,,r

epev" Aepev +

MABA, mpep q'- DABA, mpep)/Vep "+ RHABA, ep -- RABA, ep] " d t

(Eq. 1.40) The guard-cell cytosol balances three membrane permeations and one carrier-driven membrane transport: dSumABAsp = {[JHABA,gwgp " A~-JHABA,gpge -- (JHABA,gpgv + EABA, gpgv) " Agpgv]/Vgp} 9 dt

"

Agpg~ (Eq. 1.41)

The phloem cell sap balances one membrane permeation and its own mass flow: dSumABAfp = [(JHABA,mwfp"Amwfp--MABA,fp)/Vfp] 9 (Eq. 1.42) Now we calculate the ordinary differential equations of the two different chloroplast stromata. The chlorenchyma stroma balances only the chloroplast envelope permeation: dSumABAmc = (JHABA,mpme 9

"dt

(Eq. 1.43)

Also the stroma of the guard-cell chloroplasts balances only the chloroplast-envelope permeation: dSumABAgc = (JHABA,gpgr 9 Agpgo/Vgc) 9dt

(Eq. 1.44)

Finally, we calculate the ordinary differential equations of the three different vacuolar compartments. For two vacuoles two separate equations must be calculatedTone

S. Slovik et al. : ABA distribution in leaves: I. Mathematical formulation

Table 3. Measured cytoplasmic ABA concentrations under selected experimental invitro conditions, in-situ concentrations calculated after Eqs. 1.6, 1.7, 1.8, 3.3 and 3.4 (Appendix II1), and ABA concentrations in the cytosol and in the chloroplast stroma after equilibration of the model leaf

Cell type

Chlorenchyma Epidermis Guard cells

21

In vitro Measured data Medium Cytoplasm pH [ABA] [ABA] (~M) (laM) 6.0 5.8

9.1 4.0

In situ Corrected data Apoplasm Cytoplasm pH [ABA] [ABA] (riM) (nM)

172 6.5 0.0551~ 6.0 1280 6.3

4.1 4.1 4.1

25.4 55.1 439.5

Model data Cytosol Stroma. [ A B A ] [ABA] (nM) (nM) 43.6 52.5 222.7

173.2 244.1

Calculated on the basis of the total ABA content of Vaterianelta epidermal strips by subtracting the ABA content of guard cells Compartment volumes and pH values are taken from Table 1 (see also Appendix 111) for ABA and one for its conjugates. Concerning free ABA, the chlorenchyma-cell vacuole balances one membrane permeation and one metabolic rate: dSumABAmw = (JnABA,rapmv" mmpmv/Vrav-RABA,mv)"dt (Eq. 1.45) Concerning the conjugates o f ABA, the chlorenchymacell vacuole balances only one metabolic (net) rate: dCON~v = RABA~v" dt

(Eq. 1.46)

Eq. 1.47 balances one membrane permeation and one metabolic rate o f epidermal vacuoles: dSumABAo~

=

(JHABA,epev

"

Aepe~/Ve~- RABA,ev)"dt (Eq. 1.47)

Concerning the conjugates o f ABA, the epidermal vacuoles balance only one metabolic (net) rate: dCONo~ = RABA,ev " dt (Eq. 1.48) F o r the guard-cell vacuoles only ABA has to be considered as the balance for one membrane permeation and one carrier-driven membrane transport: d S u m A B A g = [(JnxaA,gpgv"~ EABA,gpgv) " A u , g v / V ~ ] "dt (Eq. 1.49)

Discussion Model results are close to the real situation and to mea~ sured data. We think that our leaf model respects all known morphological and physiological complications. We tried to approximate the real in-situ situation mathematically. Even factors, which exert only a small, but marked influence on ABA compartmentation (cuticular transpiration,, mass flow o f water and ABA amongst different tissues within the lamina) are included. Almost half o f all model parameters summarised in Table 1, Table 2 and in the Appendix are leaf-anatomy parame-

ters. The importance of morphology for physiological questions can also be stressed by regarding the standard formulation of the relaxation coefficient ~ (given in seconds), which characterises the compartmental efflux kinetic while approaching equilibrium: V z - A- C

(Eq. 2.1)

The compartment volume V (m 3- m - 2 ) and its surface area A (m 2- m-2), both given on a leaf-area basis (Table 1 and Table 2), largely determine the relaxation coefficient z [s]. Only the membrane conductance c (m" s-1) is a physiological parameter. We therefore determined microscopically all morphological parameters o f greenhouse-grown Valevianella locusta L. leaves thoroughly and in detail. All morphological parameters are determined on a leaf-area basis. Consequently, in heterobaric leaves (Daley et al. 1989) model calculations remain appropriate within coherent patches. The in-situ coherent apoplasm is divided artificially into three interrelated spaces, because model calculations need an approximately homogeneous water free-space. We assume that at least the apoplasms o f specified tissues and cell types have homogeneous ABA concentrations. In Table 3, measured cytoplasmic ABA concentrations are compared with calculated ABA concentrations in flux equilibrium within the model leaf. Model results are close to measured data. We compare and discuss experimental data, independent computer data and small deviations between them in detail for all leaf compartments in the communication o f Slovik and Hartung (1992a). At the moment we conclude (i) that our leaf model indeed approximates the real in-situ situation and (ii) that the observed compartmental ABA concentrations in unstressed leaves seem tO be close to the flux equilibrium of ABA.

Appendix L The system o f indices. F o r general considerations, the indices j and k denominate source (j) and sink (k) compartments. F o r detailed formulations, indices as summarised in Table 1 consisting o f two letters are used to identify a compartment in a defined leaf tissue or cell

22

S. Slovik et al.: ABA distribution in leaves: I. Mathematical formulation

type. The first index denominates the cell type or tissue, the second one identifies the cell compartment or cellwall part. For cell or tissue types, the first indices m, e, g, f, stand for chlorenchyma cells of the mesophyll (m), epidermal cells (e), guard cells (g) or leaf fibres (fascicles) (f). The second indices p, c, v, w stand for cytosol (p), chloroplast (c), vacuole (v), and cell wall water (w). The indices a, i and x can stand alone and denominate air surrounding the leaf (a), the intercellular air space (i), or the xylem sap before it enters the leaf lamina (x). Indices as summarised in Table 2 consisting of four letters are used to identify surfaces and border areas. This indices are the 'sum' of the indices denoting the adjacent compartments. 1I. p H values in different compartments and tissues. The pH values are summarised in Table 1. primp in the chlorenchyma cytosol, Prim, in the chlorenchyma vacuole, pHgp in guard-cell cytosol and pHgv of guard-cell vacuoles are determined for Valerianella by compartmental effiux analysis (Baier and Hartung 1988; Daeter and Hartung 1990). The pHfp of the phloem sap has been determined for isolated vascular strands of Plantago maior (Baier and Hartung 1991). The values pHmc= pHgc of the chloroplast stroma are estimates in the light after Heldt et al. (1973). The pHmw of the mesophyll cell wall has been measured, using a pressure chamber, in the exudate of the petioles of Valerianella (data not shown) and of Gossypium hirsutum (Hartung et al. 1988). pHmw= 6.5 is indeed a typical apoplasmic pH of many angiosperms (cf. Pfanz and Dietz 1987, who used a fluorescent technique). In the light, the pHcw=6.0 of epidermal cell walls and pHgw= 6.3 of guard-cell walls of Commelina communis is taken from Edwards and Bowling (1986). As there is no experimental evidence about pH values of the compartments in epidermal cells, we assume the same values pHCp= p r i m p and pHcv = pHmv as in chlorenchyma cells. It should be mentioned that our model environment allows us to change pH values in different compartments and tissues with different, independent time courses. This is an important prerequisite for modelling stress kinetics.

assume a local flux equilibrium between the cytosol and the chloroplast stroma of the specified cell types (chlorenchyma and guard cells), employing the cytoplasmic pH values and volumes as defined in Table 1. The total concentration of free ABA in epidermal cells of Valerianella has been determined as the difference of the ABA content of epidermal strips which are almost chlorenchyma-cell-free minus the recalculated (see Table 1 and below) in-situ ABA content of whole guard cells enzyme-linked immunosorbent assay (ELISA); data not shown. Using this epidermal-cell bulk concentration, we estimate the concentrations SumABAep in the cytoplasm and SumABAev in the vacuole of epidermal cells assuming partial flux equilibrium between cytoplasm and vacuole. Leaf compartments may have had time enough to come into local equilibrium before harvesting for experiments. Such partial flux equilibria as employed for epidermal protoplasts, guard-cell cytoplasm and chlorenchyma cytoplasm are calculated as follows. The mean concentration SumABA of two compartments j and k is defined as . . . .

SumABA . . . . =

SumABAj 9Vj + SumABAk 9Vk (Vj + V0 (Eq. 3.1)

We assume that SumABAj and SumABA k are in local flux equilibrium across a membrane which is permeable only to the uncharged HABA molecule. Thus we can write (cf. Eq. 1.7) SumABAj (1 + K.ABA/[H +]j)

SumABAk (1 + K.ABA/[H +]k)"

(Eq. 3.2)

Substituting SumABAj or SumABA~ in Eq. 3.1 by Eq. 3.2 we obtain the equilibrium concentrations SumABAj and SumABAk in both compartments j and k: SumABA~o~ 9(Vj + Vk) SumABAj = 1 + K,ABA/[H+]k

(Eq. 3.3)

1 + K,ABA/[H +]j " Vk + Vj IlL Concentrations in different compartments of different leaf tissues. All concentrations of ABA measured in unstressed leaves and used as initial conditions for model calculations are summarised in Table 1. [HABA] and [ABA-] are concentrations of the undissociated ([HABA]) and the dissociated ([ABA-]) species of ABA. The pKa = 4.75 of ABA employed is valid for physiologically relevant ionic strengths (Gimmler and Hartung 1988). SumABA is defined as the sum of [HABA] and [ABA-] for each model compartment. The concentration of ABA, SumABA, in the cytoplasm (cytosol+chloroplast in green cells) has been determined by effiux analysis for mesophyll cells (Daeter and Hartung 1990) and for guard cells of Valerianella locusta (Behl and Hartung 1986). We obtain only cytoplasmic bulk concentrations, but not the distribution of ABA between cytosol and chloroplast stroma. As the relaxation-time constant of chloroplasts for HABA is small (about 1.5 min), we

SumABAk =

SumABA . . . . 9(Vj + Vk) 1 + KaABA/[H+]j 1 + K~ABA/[H +]k " Vj + Vk

(Eq. 3.4)

All other concentrations are determined directly. The total concentration of ABA in the xylem sap and in the mesophyll apoplasm of Valerianella equals SumABAx ~ SumABAm, = 4.1 9 10- 6 m M (pressure chamber, ELISA assay; data not shown). We assume the same initial concentrations for all three apoplasmic spaces of the model leaf (SumABAmw=SumABAew -- SumABAgw). SumABAmw is determined in Valerianella mesophyll vacuoles (Daeter and Hartung 1990) and SumABAgv in Valerianella guard cell vacuoles (Behl and Hartung 1986). SumABAfp in the phloem sap of Valerianella is very difficult to determine experimentally

S. Slovik et al. : ABA distribution in leaves: I. Mathematical formulation and therefore still unknown. We will estimate SumABAfp by model calculation (cf. Slovik and Hartung 1992a). As we need any reasonable initial concentration, we chose arbitrarily SumABAfp= 12 p.M, which is about the mean SumABA concentration of free ABA in the leaf lamina of Valerianella leaves. It should be noted that the starting conditions o f the model calculations neither influence the principal behaviour of the model leaf, nor the final flux equilibrium achieved over a long time range. As an alternative, it would also be possible to start our model calculations with the total leaf content of free ABA. Finally, we must consider that all absolute compartmental equilibrium concentrations of SumABA in chlorenchyma cells and guard cells depend on the H A B A concentration in the medium used for in-vitro compartmental effiux analyses. The pH values and SumABA concentrations of the media used are summarised in Table 3. In situ the apoplasmic pH values and apoplasmic ABA concentrations are different (cf. Table 1, Table 3 and Appendix II). Using Eqs. 1.7 and 1.8, the expected cytoplasmic equilibrium concentrations of SumABA are calculated and summarised in Table 3. All expected compartmental ABA concentrations are calculated in more detail on the basis of Table 3 by employing Eqs. 3.3. and 3.4. The results are summarised in Table 1. C O N is the concentration of vacuolar conjugates of ABA. The total content of hydrolysable ABA conjugates in epidermal strips and in the rest of the leaf of Valerianella can be attributed to the vacuolar volumes of epidermal cells and mesophyll cells, (our own estimates are summarised in Table 1). We estimate CONey= 3.82 nM in epidermal cell vacuoles and CONmv=4.79 nM in mesophyll cell vacuoles (cf. Appendix V1). Thus, determined on a leaf-area basis, free ABA and its hydrolysable conjugates are almost equimolar in Valerianella leaves.

IV. Conductances and transport parameters for water and HABA among different compartments and tissues. Conductances are designated as C and given in m . s -1 (Table 2). CH20.ia = 0.0 l m ' s - 1 is the conductance for water vapor between intercellular space and air surrounding the leaf (stomatal aperture). CH20 . . . . = 2 - 1 0 - 4 m 9 s -1 is a typical conductance of thin epidermal cuticles (Larcher 1975). The apparent conductances within the apoplasm or plasmodesmata are calculated as quotients of DAaA/Ax and summarised in Table 2. As there are - to our knowledge - no direct measurements of the diffusion coefficent D ABA for ABA in water, we estimate it from the diffusion coefficients of other organic compounds by interpolation, employing the molecular weight of ABA. We obtain D ABA = 0.60 9 10 -9 m 2 9s -1 in free water (Weast 1975). D ABA,p 0.45" 10 -9 m z- s -1 is assumed to be the diffusion coefficient in the cytoplasm, which has about 75% water content. D ABA'w = 0.12 9 10 -9 m 2 9 s -1 is the diffusion coefficient within cell walls for ABA. The diffusion coefficient o f small molecules in the cell-wall matrix is about 0.2-fold that in free water (Nobel 1983). Touchard et al. (1989) determined a diffusion coefficient of D~ 0.31' 1 0 - 9 m 2 " s - 1 for C1- anions in flax cell walls. The diffusion pathway Axme = 0.23" 1 0 - 3 m be=

23

tween the centre of the mesophyll and the epidermis as determined morphometrically for Valerianella locusta is corrected by the factor n/2, which is half the circumference of a circle relative to its diameter. The mean half-distance amongst the stomata is Axog = 62 9 10-6 m. All conductances for H A B A of membranes are summarised in Table 2. The values CnABA. . . . p for the chlorenchyma plasmalemma and CnnBA,mpmv for the chlorenchyma tonoplast (Daeter and Hartung 1990), as well as CnA~A.~, o f the guard-cell plasmalemma and CnAnA,wgv o f the guard-cell vacuole (Baier and Hartung 1988) are determined for Valerianella by compartmental effiux analysis. CnABA,m,,fpo f the phloem-cell plasmalemma has been determined for isolated vascular strands o f Plantago major (Baier and Hartung 1991). Concerning the conductances o f epidermal cell membranes for ABA, only estimations exist so far. We assume that the H A B A conductances o f membranes in epidermal cells may have about the same value as in guard cells, which are related histologically and ontogenetically to epidermal cells. Measuring the deplasmolysis velocity, Url (1952) found six- to sevenfold higher membrane permeabilities for different osmotica (urea, glycerol, methyl urea, erythritol) in epidermal cells than in cortical chlorenchyma cells o f stems o f numerous plant species. The relaxation time o f deplasmolysis and the 'rate'-limiting membrane conductance (tonoplast or plasmalemma?) are inversely correlated (Eq. 2.1.). In guard cells and mesophyll cells the ABA conductance of the tonoplast is smaller than the ABA conductance o f the plasmalemma (Table 2). Url (1952) possibly compared tonoplast conductances of mesophyll and epidermal cells. The ABA conductance of the guard-cell tonoplast is only 2.5 times higher than o f the mesophyll tonoplast (Table 2). Thus, for model calculations we cautiously define CHABA, gpgv~---CHABA,epev for the tonoplasts of epidermal cells and CHABA,gwgp ''~-CHABA . . . . p for the plasmalemma (cf. Table 2), but detailed experimental data for ABA are not yet available. CHABA,mpme=CngBA,gpge o f chloroplast envelopes is defined after Gimmler et al. (1981), who studied the H A B A permealSility o f mesophyll chloroplast envelopes of Spinacia. F o r all membranes o f the model leaf a constant factor PAn for the permeability o f the anion A B A - relative to the corresponding membrane conductance to H A B A can be defined. As ABA is the only known p h y t o h o r m o n e which permeates membranes only in the undissociated molecule form, we define PAn = 0, that is, all membranes are completely impermeable to ABA. This is not true for other phytohormones (cf. Hartung and Slovik 1991). The parameters of the tonoplast carrier for ABA in guard cells are determined by Baier and Hartung (1988): VABA,gpgv "~- 17.1 9 1 0 - 1 2 m o l m - 2 " s -1 is the V~ax of the carrier (per m 2 o f tonoplast area) and KAaA.gpgv-~ 0.21 m M is the Km of the tonoplast carrier in guard cells.

V. Membrane fluxes, diffusion, mass flow and reaction rates. Membrane permeation fluxes of H A B A are denoted as JnABA and given in mol 9 s - 1 . m - 2 membrane area. Diffusive fluxes o f H A B A or ABA are denoted

24

S. Slovik et al. : A B A d i s t r i b u t i o n in leaves: I. M a t h e m a t i c a l f o r m u l a t i o n

a s D A a A and given in m o l . s - 1 . m - 2 leaf lamina area. Mass flow o f water is denoted as Mn~o and given in m 3- s - 1 . m -2 leaf lamina area. Mass flows o f H A B A and ABA are denoted as MABA and given in m o l . s -1- m - : leaf lamina area. Carrier-catalysed membrane fluxes are denoted as EABA ('enzyme') and given in m o l . s -~- m -2 membrane area. Synthesis or other formation rates of ABA are denoted as RnAaA, degradation or conjugation rates o f ABA as RABA; both are given in mol. s -I- m -3. Mrl~O.fp = 0 . 6 3 3 . 1 0 -9 m a" s -1" m -2 is our estimation for the mass flow o f the phloem sap per time and per unit leaf surface area. This estimation is based on the flow velocity of the phloem sap typically being about 1 . 0 m - h -1 (Ziegler 1982), on the area o f phloem-cell lumina in a cross section o f petioles, on the information that normally about 20% o f the phloem area is sieve cells and sieve tubes, and on the corresponding leaf area. We obtain 11.4 9 10-6 m 2 sieve element area per m 2 leaf area for Valerianella. Very similar values can be determined for Plantago. Thus, we calculate:

MH:~

=

1.0-0.2" 11.4.10 -6 m 3- s -1 9m -2 (Eq. 3.5) 3600

VL Abscisie-acid metabolism: Synthesis, degradation and reversible conjugation. F o r standard calculations (unstressed leaves in the light), the synthesis rate o f ABA is set to zero in the chlorenchyma cells and epidermal cells: RIrlABA mp ----- RrlABA ep = 0 m o l - m - 3 - s-1. Velocity constants~ given in s "-'1, are denoted as kAaA (ABA is the substrate) or kcoN (conjugates o f ABA are the substrate). Concerning degradation rates in the apoplasm, we define VAag,~,w and VgBg,ewfor the mesophyll apoplasm or the epidermal apoplasm respectively; both a r e Vraax rates given in tool- m - 3 . s-1. The Km values are denoted as KmABA,m~ or K~AnA,o~, and given in raM. Unless mentioned elsewhere, in all compartments k g B A and gABg are set to zero in unstressed leaves. Epidermal and mesophyll vacuoles will be able to synthesise or degrade ABA conjugates automatically. The employed kinetical parameters a r e kABA,mv= 0.641 " S- 1 and kcoN.mv = 0.375 - S- 1 in mesophyll vacuoles, and kABA,o~ = 0.100" S-1 and kcos,0~=0.147 " S-1 in epidermal vacuoles. They are calculated from our own unpublished experimental data for Valerianella: (i) The ratio CON~, : ABAmv in mesophyll vacuoles is 1.71 M 9 M -1, (ii) CONe~: ABAo~ in epidermal vacuoles is 0.68 M - M - ~ . The ratios CON~: ABA~ in the vacuoles of both cell types (cf. Table 1 and Appendix III) may be close to the apparent equilibrium constant K, which will not be the same if both cell types synthesise a different mixture of hydrolysable ABA conjugates in their vacuoles: K

-

CONy ABAv

kAttA ' -

kcoN,v

(Eq.

3.6)

(iii) The absolute net conjugation rate of [3H]ABA in Valerianetla leaves, calculated on a volume basis, is 1.56 times higher in epidermal vacuoles compared to

mesophyll vacuoles (data not shown). The ratio of the conductances o f either the epidermal tonoplast: mesophyll tonoplast or the epidermal plasmalemma :mesophyll plasmalemma (or o f both) is about 10. The deplasmolysis velocity is governed by the smallest of both (tonoplast or plasmalemma) conductances (cf. Appendix IV). Therefore, the ratio o f the apparent velocity constants of the ABA conjugation in both tissues must be 10: 1.56: kABA.mv __ 6.41

(Eq. 3.7)

kABA, ev

(iv) The enzymatic net turnover of conjugates in the vacuoles o f both cell types will be governed by the permeability of the corresponding tonoplast membranes to HABA. To achieve fast equilibration of vacuolar ABA and CON, we selected high velocity constants on the arbitrary basis kAaA, ev = 0.10 9s -1. For guard cells it is known that there is no potential capability for synthesis, degradation or conjugation o f ABA (Behl and Hartung 1986). For standard calculations we study pure redistribution effects o f the ABA pool, which is only allowed to interfere with the vacuolar ABA conjugates.

VII. Leaf-anatomy parameters. A (m E - m-2) denotes the area separating the leaf compartments per unit leaf area and V (m 3 - m - z) the volume o f these compartments per unit leaf area. The unit leaf area is defined as the area o f the leaf projection, not the total surface. All anatomical parameters are determined for leaves o f Valerianella locusta. It should be stressed that most parameters are not estimations but determinations. Only the relative composition o f the cell lumina in different tissues consisting of vacuole: cytosol:chloroplast (if there are any) is difficult to determine. All volumes and most surface areas are summarised in Tables 1 and 2. N o t mentioned are A . . . . = 0.076 m 2 9 m-2, the area of the apoplasmic interface between mesophyll and epidermis, Aowgw = 0.025 m 2 . m - z , the area of the apoplasmic interface amongst epidermal cells and guard cells, A~pep = 2.43- l0 -3 m 2 - m - 2 , the area o f the symplasmic interface (plasmodesmata) between chlorenchyma cells and epidermal cells, and Agwa = 0.095 m 2. m -2, which is the cuticular surface attributable to guard cells. The fractions FracI are quotients of interface areas in the substomatal cavity. We employ FracIc~ = 0.2 for the relative fraction of epidermal cell wall - substomatal cavity - interfaces, and FracIgw=0.05 for the corresponding guard-cell wall interface. For correction o f the apoplasmic mass flow o f ABA, factors f are determined as areas of cell walls relative to the total cell area of cells in microscopic sections normal to the mass-flow direction within the regarded tissue: fmwew=0.045 for the apoplasmic mesophyll-epidermis interface, and f~,vgw=0.13 for the apoplasmic epidermal cell-guard cell interface. For symplasmic water fluxes in plasmodesmata from the mesophyll into the epidermal cell cytoplasm, fmp~p = 1.43- 10 .3 has been estimated using additional information of Nobel (1983).

S. Slovik et al. : ABA distribution in leaves: I. Mathematical formulation

VIII. Environmentalparameters. R H , = 1.0 is the relative humidity of the air (unstressed). Ta=20~ and T~oaf= 25~ are the temperatures of the air and of the leaf, given in degrees centigrade. Under these conditions, the transpiration rate is 3 . 2 m m o l . m - 2 . s - 1 , if CH2o, i~ = 0.01 m - s - I .

We are grateful to Professor U. Heber (Lehrstuhl Botanik I, University of Wiirzburg, FRG) for stimulating discussions. This work has been performed within the research program of the Sonderforschungsbereich 251 (TP 3 and 4) of the University of Wiirzburg. It has been supported also by the Fonds der Chemischen Industrie.

References

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