Some possible sources of error in determining bulk

Some possible sources of error in determining bulk elastic moduli and other parameters from pressure-volume curves of shoots and leaves Y. N. S. CHEUNG, M. T. TYREE,A N D J. DAINTY

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Dopptrt~tt~ictit c!fBotntzy, U t ~ i ~ ~ e r of s i tTorotrto, y Toronto, Ontorio M5S IAI Received August 6. 1975

. J. DAINTY.1976. Some possible sources of error in C H E U N C Y. . N. S., M. T . T Y R E E and determining bulk elastic moduli and other parameters from pressure-volume curves of shoots and leaves. Can. J . Bot. 54: 75%765. Pressure-volume curves were constructed from well defined models and hypothetical shoots in which reasonable values of osmotic pressure and cell wall elastic moduli were specified for cell types of different relative volumes. The pressure-volume curves s o obtained closely resembled those of real shoots and leaves. Comparing the bulk parameters obtained from analysis of the constructed pressure-volume curves with the values defined in the models allows us to examine the sources oferror in their evaluation. The graphical valuesof the original bulk osmotic pressure and of the total volume of osmotic (symplasmic) water agreed very well with those defined: however, the osmotic pressure at incipient plasmolysis and the bulk elastic moduli estimated from the graph were generally lower than their actual values originally used in the models. We show that the apparent linear dependence of the bulk elastic modulus of sitka spruce reported by Hellkvist et (11. (1974) may not reflect a similar linearity for the elastic moduli of individual cells. C H E U N C Y. . N . S.. M. T . TYREEet J. DAINTY.1976. Some possible sources of error in determining bulk elastic moduli and other parameters from pressure-volume curves of shoots and leaves. Can. J. Bot. 54: 75b765. Les auteurs ont construit des courbes representant les rapports entre la pression et le volume a partir de modeles bien definis de tiges hypothetiques dans lesquelles des valeurs raisonnables de pression osmotique et des modules d'elasticite des parois cellulaires ont etC specifies pour les types de cellules de differents volumes relatifs. Les courbes pression-volume ainsi obtenues ressemblent etroitement a celles de tiges et de feuilles vkritables. Si on compare les parametres globaux obtenus e n analysant les courbes pression-volume fabriquees. avec les valeurs definies dans les modeles, on peut examiner les sources d'erreur contenues dans leur evaluation. Les valeurs graphiques de la pression osmotique globale originale et du volume total de I'eau osmotique (symplasmique) concordent tres bien avec celles qui ont Cte dgfinies au depart; cependant la pression osmotique h la plasmolyse incidente et les modules d'elasticite globaux estimes a partir du graphique sont generalement plus bas que leurs valeurs originalement utilisees dans les modeles. Les auteurs montrent que la dependance linkaire apparente du module d'elasticite global de I'epinette de Sitka rapportee par Hellkvist pt (11. (1974) pourrait ne pas montrer une telle linearite pour les modules d'elasticite des cellules individuelles. [Traduit par le journal]

al. 1964, 1965) has been very useful in studying Introduction In principle it would appear that a suitable the water relations of higher plants and could analysis of the quantitative relationship between potentially be applied in ecological studies water potential (Y) and volume of plant tissue (Cheung et al. 1975). The so-called 'pressureshould produce information about the elastic volume' curve for plant inaterial obtained by this moduli and osmotic pressure of the cells of that technique is normally curvilinear in shape and tissue. Such an analysis has been attempted by a can be described by either of the following few people and has been summarized and dis- equations (Tyree and Hammel 1972): cussed by Dainty (1975). Probably the best method for certain tissues of obtaining a Yvolume relationship is the pressure bomb. This paper tries to show that we have been laboring under some illusion in assuming that elastic [2] 1/P = VIRTN, = lln, when F ( V ) = 0. moduli, in particular, can be derived in such a way. P is the balance pressure in the bomb, Vo is the The pressure-bomb technique (Scholander et original volume of osmotic (symplasmic) water,

759

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CHEUNG ET AL.

V, is the volulne of water expressed, V = Vo - V,, RT is the gas constant times the absolute temperature, N, is the total number of osmoles of solutes in all the living cells in the shoot, x is the osmotic pressure, and F(V)/V is the volumeaveraged turgor (VAT) pressure of the cells of the shoot (Tyree and Hammel 1972). The nonlinear region of the P-V curve is a resultant of both turgor and osmotic pressure changes in the cells, whereas the linear region can apparently be attributed to the osmotic component alone. Tyree and Hammel (1972) have shown that the VAT pressure of some species can be approximated by an empirical expression of the form

able values of osn~oticpressure and cefl wall elastic moduli, E, are specified for different cells of various relative volumes. The model allows us to compare the bulk values of n, V, and E from the analysis of the constructed P-V curve to the actual cell values in the hypothetical shoots, thus allowing us to explore the possible sources of error in interpreting P-V curves. Some new data on PvATvs. E (as defined by equation 14) of real shoots is presented together with a reanalysis of the data published by Tyree and Hammel (1972).

where P,,,, is the maximum VAT pressure (when the leaf water potential is zero). One deficiency of equation 4 is that PvATnever reaches zero no matter how large Vc is; nevertheless, equation 4 seems to fit the data for sitka spruce over most of the range of PvAT. In this paper, P-V curves will be discussed that have been constructed from well defined models of hypothetical shoots in which reason-

and upi is the water volume of the ith cell a t which P l i first equals zero; we call this the volume at incipient plasmolysis. Assuming E' remains constant, equation 7 can also be written as

Theory

Relatioti bet ween P (B~lnncePrc.ssi~ic) NIICIV, (tlie Volutue Expres.~rd) [3] PvAT= VAT pressure = E*((V - Vp)/Vp)" The relation between P and tlie two components of water potential for an ith cell of a where Vp is the volume of the symplasm when shoot in the pressure bomb has been shown by PvATfirst reached zero, and E* is an arbitrary Tyree and Haininel (1972) to be constant. We have been able to fit data for Acei P = R7'(t1'/vi) - P l i sacc17ari1tn Marsh., Populus b a l s a n ~ f ~ ~L., i a Fiaxi17ilspen~~sj~Iv~~tiica ~ a r s h .and , Git7kgo biloba L. or to this empirical equation, although Hellkvist et 01. (1974) failed to obtain a good fit with sitka spruce (Picea sitcllo~sis(Bong.) Carr.). We now where ni is the nurnber of osmoles of solute in suggest that this empirical expression no longer the cell, vi is the cell water volume, and rri and be used because two arbitrary constants (E* and P,' are the osmotic and turgor pressures of the 11) are needed to fit the data and the constants ith cell. But . . . bear no simple relation to the modulus of [6] ni=n,'vo'/~l=noi~~oi/(uoi-u,i), elasticity of the cell walls. Using a bulk modulus, defined by where noi and 11,' are the osmotic pressure and the volume of the cell water at maximum turgor and v,' is the volume expressed from the ith cell. Hellkvist et 01. (1974) obtained a good fit for P t i can be related to the elastic modulus of the sitka spruce, in which the bulk modulus ap- cell and cell volume by peared to be a linear function of VAT pressure. E' = dP,i/(dvi/u,') = dPti/df, If we let E' = upVAT, where cr is a constant, then [7] it follows that where . . . . [8] f = (v' - vP')/up' = (v,' - u,' - vP')/vp1.

[91

pti = E ~ ( ( u, ~ u , ~- u ~ ~ ) ~ u ~ ~ )

Substituting equations 6 and 9 into 5, rearranging, and solving the quadratic equation for v,' gives

760

C A N . J . BOT. VOL. 54, 1976

TABLE I. The cell types of the hypothetical shoots of Fig. 1 and Fig. 2

upf can be calculated from equation 9 when no water has been expressed (uei = 0):

[I\]

P , ~ ~ = ~ ~ ' = E ~ ( ( U ~ ~ - U ~ ~ ) / U ~ ~ )

where P,c,iis the maximum turgor pressure of the ith cell (when the water potential is zero); so

Cell type

noi,bar

v,'/ V,, %

E',

bar

The shoot in Fig. 1 10 20

90 10

100 100

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The shoot in Fig 2. Thus the uei from the ith cell a t any given P before incipienl plasmolysis can be calculated froin equations .10 and 12 provided we start . with values of no', u,', and E'. A1 or after the occurrence of incipient plasmolysis, P equals ni, and uci can be calculated by rearranging equation 6 into Adding together all the values of u,' for all the cells in a model shoot yields V,.

6.0 7.0 8.0 9.0 10.0 1 1 .O 12.0 13.0 14.0

5.5 8.5 11.5 15.5 18.0 15.5 11.5 8.5 5.5

175 175 175 175 175 175 175 175 175

To obtain a P-V curve comparable to those comrnolily observed will require clioosing a noi of about 20 bars for the set of ~ " used s or picking cell types with an average E' of about 45 bars.

Models In tlie following, we sliall use equation 10 or 13 to calculate V, as a function of P and conDiscussion struct tlie P-V curve from two kinds of hypothetical shoots. Pciral?ietrrs Shcij~itlgtile P-l.' Ciirue Constructing tlie P-V curves from known Type 1: Ce1l.r n,ith Diffc>ret~tnni's citid a Col?il~ion hypotlietical slioots gives some understanding Constcitlt E~ of the possible sources of error in evaluating I'lie P-Vcurve defined by a liypotlietical shoot bulk parameters sucli as no, n,, Vo, Vp, and E in i l l which each cell has tlie sanie E' but different a real slioot. The models indicate that tlie noi's is very similar to the one obtained by the smooth non-linear region of the P-V curve pressure-bomb technique on real slioots. Figures commonly observed can be accounted for by I and 2 show the P-V curves of tlie two such cells losing their turgor pressure at varied rates liypothetical shoots with cell types as defined in because of variations in either no1or E". Table I . The curve for the slioot consisting of Our P-V curve froni the liypotlietical shoot only two cell types (Fig. I) is very abrupt. with different noi's and the same and constant However, as .the osmotic pressures of the cell E"S gives an apparently good approximation to types become more diverse, a smoother curve is curves obtained fro111 real slioots. In a twig or a obtained. This is illustrated by the P-V curve of single leaf, noi is likely to vary aniong cells a hypothetical shoot consisting of nine cell types because of tlie differences in their physiological i l l Fig. 2 with a roughly normal frequency activities and (or) their location in relation lo a distribution of tlieir osniotic pressures. solute source. Variations in m a ~ n i t u d eof nni Tjpe 11: Cells wit11 Eqiicil noi but Diferent ~ " s have been reported in sugar beet b; Geiger et aj. Figure 3 shows the P-Vcurve of a hypothetical (1973) where cells in the phloeni tissue have x,' shoot consisting of nine cell types having of 30 bars compared to 13 bars in the ~nesophyll common noi's and different ~ " sas defined in cells. Our estimate froni another model not Table 2. The graph is also curvilinear and shown in tlie figures indicates that the existence resembles those P-V curves obtained in the of as few as six different nni's aniong cells in a pressure-bomb experiments. However, the drop shoot will give a P-V curve very comparable to in VAT pressure per unit amoulit of os~notic one experi~nentallyobtained. Diversity of noi in water expressed from this type of hypothetical plant samples is likely to be an important factor shoot is co~isiderablysteeper than that in Fig. 2. in shaping the non-linear region of a P-V curve.

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CHEUNG ET A L

VOLUME

E X P R E S S E D , % V,

FIGS.1 and 2. The pressure-volume curves of the hypothetical shoots with cell types having different x,"s and constant E ' . The cell types for the two shoots are defined in Table 1. Figure 1 is the curve of the hypothetical shoot consisting of two cell types. Figure 2 is the curve of the hypothetical shoot consisting of nine cell types. Refer to the Discussion for interpretation of the figures.

A n approximately normal distribution of no' vs. voi/Vo was assumed in Fig. 2, but a skewed or polymodal distribution would yield si~nilarP-V curves. The E ' anlong cells in a plant sample could also be different; the variation in E ' also contributes to make a P-Vcurve curvilinear (Fig. 3). The cells of real shoots are likely to have a range of values of nOi'sand ~ " s it; is clearly i~npossible to tell from a P-V curve what the real distribution in values of noi and E' is. I11 developing both models, it has been assumed that E~ of the individual cell remains constant during a change in cell turgor pressure. A hypothetical shoot with cells of uniform noi's but with E"S linearly dependent 011 the turgor pressure can produce an e~pressionsimilar t o equation 4, which also satisfactorily explains the non-linear region of the curve in sitka spruce (Hellkvist et al. 1974). It is uncertain whether s occurs; and if it does, this change in ~ " actually

to what extent it contributes to shape the P-V curve. Parameters f,.on? the P- V C~trve It is possible to obtain certain parameters, e.g., no, np, V,, Vp, and E from the P- V curve of the hypothetical shoot. Comparing these graphically obtained parameters with the values assumed in making the model allows us to ascertain some possible interpretive errors in the analysis of real shoots. The no and V, can be obtained by extrapolating the linear portion of the P-V curve to V, = 0 and I/P = 0, respectively. 'The no's obtained from Figs. 2 and 3 are 10.0 and 14.9 bars; the no's calculated from the assumed values by 12

C noi(;,'/ V,) are 10.0 and 15.0 bars, respectively.

i= 1

V,'s obtained from the graphs also nearly equal I1

the expected Vb by

C voi. There seems to be good i= 1

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CAN. J. BOT. VOL. 54. 1976

0

I I0

1 20 VOLUME

I 30

I

I 50

40

EXPRESSED,

agreement between the estimated (graphical) a n d the actual values f o r no a n d V,. T h e bulk osmotic pressure of a plant a t incipient plasmolysis, rr,, is rather difficult to determine. According t o the definition of Tyree a n d Hanimel (1972) incipient plasmolysis of a shoot occurs when the V A T pressure becomes zero. T h e presence of a small n u m b e r of cells with rrOi's substantially higher ( o r E"S much lower) t h a n most will give the plant tissue a non-zero V A T pressure a t a certain P, even though most of the cells in the tissue have lost their turgidity. Consequently, the rc, of a shoot, by definition, will actually be the maximum incipient plasmolysis osmotic pressure of a particular cell type in the plant. T h u s , rc, SO defined, unlike rr, o r V A T pressure, is n o t a bulk o r averaged parameter but rntl~er nn extr'eine value. T h e volume of osmotic water a t incipient plasmolysis, V,, is linked t o the location of rc, o n a P-V curve, V, equals V, minus the V, a t the point rr,; thus Vp is also a n extreme value. T h e position of rr, ( o r V,) o n the P-V curve is difficult t o locate with certainty. T o illustrate the kind of errors t h a t c a n arise, let us examine

I 60

I

I

70

% V,

TABLE 2. The cell types of the hypothetical shoot of Fig. 3 Cell type

'

bar

u,'/

V,, %

E',

bar

Fig. 2. T h e rr, obtained f r o m this curve would a p p e a r t o be a b o u t 14 bars, whereas the correct extreme rr,' in the model should be a r o u n d 15 bars. T h e possible errors a r e even more clearly seen in Fig. I . T h e rr, is a t point A , not point F. But if the pressure-volume curve were truncated a t point A o r B, considerable errors would result in the evaluation of rc, a n d Vp as well a s no a n d V,. This suggests t h a t there would generally be a n underestimation of the rr, obtained from the P-V curves of real shoots a n d a concomitant overestimation of V,. Judging f r o m

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CHEUNG ET AL.

0

10

20

30

I 40

VOLUME EXPRESSED,

I

50

I 60

I 70

I

% V,

FIG.3. The pressure-volume curve of the hypothetical shoot consisting of nine cell types having the same no1and different ~ " s The . cell types of the shoot are defined in Table 2.

the model, it appears that the r, might not reflect the presence of cells with substantially higher roi's (for the whole shoot) that constitute less than 5% of Vo. The Evaluation of E A value for a bulk modulus of elasticity, E, can be obtained from the non-linear part of the P-V curve. If we define a bulk E analogous to equation 7, [I41 E = dPV,,/dF where Pv,, = the VAT pressure and F = (Vo - Vp - Vc)/Vp; E can be evaluated by measuring the slope of the curve obtained by plotting P,,, vs. F. However, E from equation 14 bears no simple relation to E' because V, cannot be related in a simple way to the v,"s of individual cells and because Vp cannot be accurately evaluated from the P-V curve for the reasons given in the previous section. Figures 4 and 5 show the relation between E'S and VAT pressure of the hypothetical shoots of Figs. 2 and 3 and of several real plant species. The E'S thus obtained normally show strong dependence on the

VAT pressure. In several instances, E varies roughly linearly with VAT pressure. However, this linear relation seems to be only incidental and does not hold for higher values of VAT pressure where E is more nearly constant. In this respect, Acer saccharurn, Pop~rl~rsbalsamifer, Nothofag~rsbetuloides, and Pilgerodendron uviferu differ from sitka spruce (Hellkvist et al. 1974). The values of E derived from the plot of VAT pressure vs. F are smaller than the E ~ ' S used in the hypothetical shoot models in Figs. 2 and 3. This appears to be due to the erroneous estimates of r p and Vp and because the, parameters in equation 14 can not be related in a simple way to the assumed values of E'. The strong dependence of E on VAT pressure makes it rather meaningless to quote an E without at the same time specifying the VAT pressure at which it is measured. Comparison of E'S between species should be made at comparable VAT pressure. The elastic modulus obtained near or at maximum VAT pressure, E,,,, may be a more reliable estimate of E' used in the hypothetical shoots. Table 3 shows the E,, for

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CAN. J. BOT. VOL. 54, 1976

VOLUME-AVERAGED

TURGOR PRESSURE, BAR

FIGS.4 and 5. T h e relation of the bulk elastic modulus (E) to the volume-averaged turgor (VAT) pressure for five plant material samples and two hypothetical shoots. In Fig. 4, the line AA is for the Abies cot~colorshoot, B B is for the hypothetical shoot shown in Fig. 2, CC is for the Pilgerodencl,.otz ~ruijern(twig I ) , D D is for the N o ~ h o f n g ~betlrloidrs is shoot. In Fig. 5 , EE is for the hypothetical shoot shown in Fig. 3, F F is for the Acer sncclrn~.lrttr(August) leaf, and GG is for the Poplrlus bnlsnrnifero shoot. Data for the shoots of Abies corrcolor, Notlrofog/rs betuloides, and Pilgerorletrdrotr ~ r u f i r owere obtained by reanalysis of results from Tyree and Hammel (1972).

TABLF:3. T h e m a x i m ~ ~bulk m elastic modulus, E,,,,,, of plant species measured by the pressure bomb technique. All the values listed are from single measurements. T h e last digit in each of the E,,,,, values has been rounded off Plant species

E,nax, bar

Acer sncchnrlrtlr (June leaf) " (Aug. " ) " (Oct. " ) F~.ositrlrspet~t~sj~lcrrt~icn Ginkgo bilobn Poplrllrs bolsotlrife~~cr Abies corrcolor* Nothofagrrs brr~rloirles* Pi1gerodend1.011rruijero*, twig I " I1 Podocnrplrs nrrbiger~lrs* 'Data from Tyree and Haniniel

(1972).

several species; most of them seem to range from I00 t o 250 bars. One should clearly be cautious about the absolute value of E,,,:,, obtained by the pressurebomb technique. Conlparing the E' assumed in constructi~ig Fig. 2 (175 bars) and the E,,,:,, obtained as the slope of a VAT pressure vs. F plot (1 30 bars) indicates a discrepallcy of about 45 bars in E,,,:,,. A discrepancy is also observed between the grapllical E,,,:,, (63 bars) and the actual average E (70 bars) calculated from 11

C E ~ u , ~ / Vin, the hypothetical shoot in Fig. 3. i= I

The discrepancy undoubtedly can partly be attributed to the ambiguity in the meaning of V, compared with u p i of individual cells. Difficulties arise also in estimating the E,,, of a real shoot;

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CHEUNG ET AL

0

5

10

VOLUME- A V E R A G E D

besides the difficulty in estimating V,,, errors could arise from insufficient d a t a points on the P - V curve near ~ n a x i i n u ~turgor n pressure because of the very rapid d r o p of the V A T pressure per n nit a m o u n t of water expressed in that region. It appears that there is still n o unambiguous way of determining E by the pressureb o m b technique. Though E, a s presently defined, might be arbitrary a n d its accurate estimation might be difficult, we feel that the values obtained f o r a plant material can indicate the ability of a plant t o osmoregulate o r conserve water. A shoot o r leaf with a higher E,,,;,, will be able t o decrease its water potential inore rapidly . - with water loss than one with a smaller E,,,,: therefore cornparisons of Erll:,X between plants may be (see Cheung cJt a/. 1975).

Ack~~owledgment This work was made possible by N R C C grants A69 19 a n d A6459.

15

PRESSURE, BAR

C H F U N GY. . N . S . , M. T . T Y R ~ and . ~ . J. D A I N T Y1975. . Water relations parametel-s on single leaves obtained in a pressure bomb and some ecological interpretations. Can. J. Bot. 53: 1342-1346. DAINTY.J . 1975. Water relations of plant cells. 111 Encyclopedia of plant physiology. new series. Eclirc,tl by U. Liittge and M . G. Pitman. Springer-Verlag, Bcrlin. In press. G E I G E RD. . R.. R. T . G I A Q U I N T .S. A .A. S O V O N I C and K . R. J . FELLOWS. 1973. Solute distribution in sugar beet leaves in relation to phloem loading and translocation. Plant Physiol. 52: 585-589. H E L L K V I SJ.. ~ . .G . P. RICHARDS, and P. G. J A R V I S1974. . Vertical gradients of water potential and tissue water relations in sitka spruce ti-ees measured with the pressurechamher. J. Appl. Ecol. 11: 637-667. SCHOI.ANDER. P. F.. H. T . H A M M E LE. . D. BRADSTREET. and E. A. H E M M I N G S E1965. N . Sap pressure in vascularpl"nt~.Science.148: 339-346. S C H O L A N D EP.R .F.. H . T. HAM MI:^., E. A. H E M M I N G S E N . and E , D, BRADSTREET. 1964, Hydrostatic pressure a n d osmotic potential in leaves of mangroves and some other plants. P'roc. Natl. Acad. Sci. u.