Plant Physiol. (1977) 60, 259-264

A Model Relating Root Permeability to Flux and PotentialsAPPLICATION TO EXISTING DATA FROM SOYBEAN AND OTHER PLANTS1

Received for publication October 4, 1976 and in revised form April 11, 1977

BURLYN E. MICHELDepartment of Botany, University of Georgia, Athens, Georgia 30602

ABSTRACT

A model that relates hydranulic permeability to water flux and togradients in pressure potential and solute potential was tested usingsoybean (Glycine max) plants. Water flux was varied by additions ofpolyethylene glycol 6,000 around one portion of a divided root systemand by changing the light intensity and CO2 concentntion around theplants. The data are compatible with the model only if the hydraulicpermeability varies with flux; however, the data were insufficient forrigorous testing. Three sets of published data fit the model only ifhydraulic permeability varies. Evidence originally presented as involvingconstant hydraulic permeability is shown, rather, to require variablehydranulic permeability.

A considerable body of data seemed explainable only if thepermeability of plants to water varies with changes in rates ofwater flux (19, 20, 22). Fiscus (4) proposed a model couplingwater and solute fluxes to explain at least some of the data cited.Of that available to him, only the data of Lopushinsky (11) andMees and Weatherley (12) were adequate for testing; and Fiscusbelieved his model to be compatible with their data. Dalton et al.(3) independently proposed the same model and also indicated itto be consistent with the data of Mees and Weatherley. Newman(20) used various methods to reject the conclusions of Fiscus andof Dalton et al. Newman also suggested a two-membrane, three-compartment model as more realistic than the single membrane,two-compartment model of Fiscus; nevertheless he concludedthat these models predicted essentially the same water flux-pressure gradient relationship.

This manuscript reports the testing of the model of Fiscus (4)and Dalton et al. (3) for applicability to data obtained in mylaboratory from soybean divided root systems (10), reexaminesthe model's applicability to amenable published data, and con-siders other evidence regarding root permeability to water.

MATERIALS AND METHODS

Experimental. Soybean (Glycine max [L.] Merr. cv Bragg)plants were grown with divided root systems in solution cultureas previously (16) except for using half-strength nutrient solutionand improved containers. The latter consisted of side by sidePlexiglas compartments, each 3 x 3 x 29.5 cm internal dimen-sions, into which 240 ml of solution was placed (Fig. 1). Im-provements were greater sensitivity for measuring liquid ab-sorbed by roots, automation in recording the amounts absorbed,and prevention of mechanical injury to roots. The value of

I This research was supported by National Science Foundation GrantGB-21016.

uninjured roots is particularly important in preventing entry ofPEG2 (9, 16).To begin treatment, the nutrient solution was drained from

one compartment and replaced by nutrient solution plus PEG.To end treatment, the compartment was drained, flushed withdistilled H2O three or four times, and refilled with nutrientsolution. Before, during, and after treatment, water absorptionrates of both sides of divided root systems were monitored.The /,, of the nutrient solution was -0.37 bar. Concentrations

required for. desired 8,, values of PEG were obtained frompublished values (17). The PEG content of each slightly dilutedsolution recovered from treatment of roots was measured gravi-metrically (17). Some measurements of the water potentials ofleaf discs and of decapitated root exudates were obtained using aWescor C-51 sample chamber.

Full light from fluorescent and incandescent lamps was 0.3 calcm-2 min-' (Eppley pyrheliometer) or 43,200 lux (Weston illu-mination meter). Half-light (no incandescents) was 0.12 cal cm-2min-' or 21,600 lux. Temperature mean was 25 C and vaporpressure was maintained between 10.9 and 12.8 mm Hg. TheCO2 level outside the growth chamber usually was well above400 ul/l. The chamber was not gas tight; to provide a consistentCO2 level, supplemental CO2 was added automatically to main-tain 500 j,u/l during nonexperimental and some experimentalperiods. To achieve high transpiration rates, 300 ,ul/l was usedduring most of the experimental periods being reported. Thiswas obtainable during light hours partly because of photosyn-thesis in test and nontest plants and partly by absorption of CO2into KOH solution in trays on the floor of the chamber.At harvest, the top was severed at the hypocotyl and weighed.

After exudation measurements, using a pipette attached withrubber tubing to the decorticated stump, divided root systemswere blotted between paper towels and weighed. Based ongrowth rate measurements, weights at the start of the experi-mental period were estimated; and, using straight line interpola-tion, intermediate values were calculated to permit computationof best estimates of rates. Treated roots were assumed not togrow during exposure to PEG but to enlarge normally at allother times.To permit expression of J, as water flux, estimates of root

surface areas were needed. By assuming a density of 1 and anaverage diameter of 0.5 mm, multiplication by 80 cm2 g-' con-

2 Abbreviations: PEG: polyethylene glycol 6,000; Jr: volume flow,equivalent to water flux; L: hydraulic permeability; ip: water potential;*,: pressure potential; t/,: solute or osmotic potential; a: reflectioncoefficient; J,: solute flux; n: effective number of ions per mol of solute(2.3 for Hoagland No. 2 nutrient solution based on -nRTC = *, =-0.74 bar); R: gas constant; T: Kelvin temperature; C: mol cm-3 ofsolute; k: coefficient of active solute absorption; w: coefficient of solutepermeability at Jl = 0; S: selectivity; subscript 1: ambient; subscript 2:within xylem; superscript u: untreated; superscript t: treated; r: correla-tion coefficient.

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Plant Physiol. Vol. 60, 1977

42 = p,, equation 2 simplifies to J, = -knRT/aoift1. With , =-nRTC, nRT = 5.7 x 104 bar cm3 mol-' for the nutrientsolution used; and J1. becomes (15.4 x 104)klo-. That k could begreater than 1 x 10-11 mol cm-2 sec-1 seems unlikely becauseany value above this requires q2 to be equal to 4is1 at rates ofwater flux that are much too high (8, 18). This value of k iswithin the ranges used by Fiscus (4), Dalton et al. (3), andNewman (20).Maximum developable root pressure is q,,2 at Jl. = 0. On the

untreated side of the divided root system, pu was always 0;therefore qP42 = At,,u (equation 3) and, at J,. = 0, AP,)U = o*/w(reduced equations 1 and 2). Setting maximum root pressure torange between 0.2 and 5 bars and k, between 0.1 x 10-11 and1.0 x 10-11 mol cm-2 sec-', the above equation set correspond-ing values for w. All of these were used, in equation 4 and withthe relationship of S to Atq, to relate o-ip, to J,. for four limitingcombinations of k and maximum root pressure, within which anyactual combinations should fall (Fig. 2).Because PEG does not penetrate cell walls (14), its t/i should

be equivalent to the ql, of roots immersed in unpressurizedsolutions. Rewriting equation 1 as

J, = -L(IP2 - q/pl + O-Aql,)

FIG. 1. Schematic diagram (not to scale) of an apparatus for testingplants with divided root systems. Only parts for one compartment of onecontainer are shown. The large circle in the root compartment representsa cavity in the base protected from root entry by a 100 mesh stainlesssteel screen thimble. The small circle is a cavity accepting a tube tocomplete an air-lift water pump for aeration and circulation of compart-ment liquid. The dashed lines represent the growth chamber walls. Anaquarium pump pushes growth chamber air for 24 compartmentsthrough a Y-tube into two, wide mouth, pint jars for distribution througha set of 12 capillary tubes in each jar. Numbers 1, 2, and 3 are solenoidvalues. The drain line is clamped during filling. Solenoid 3 usually isactuated by the controller for a preset time period when the liquid leveldrops below the thermistor, permitting the thermistor to self-heat. Eachevent is recorded. The thermistor is located in a small column attachednear the top front of each compartment. The column provides an accessi-ble location free of roots and surface disturbance.

verted fresh weights to approximate areas. An error in averagediameter, though modifying absolute values, would not changegeneral relationships or introduce errors in determining modelapplicability.

Analysis. Although Fiscus (4) and Dalton et al. (3) usedosmotic pressure notation, and Newman (20) continued thisusage, q., notation is preferred and will be used here. Basicequations are

J, = -L(Atp + o-tp) (1)

41 = -4182J,./nRT = k + wAtt,8 - (1 - )(t81 + Ps2)Jr/2nRT (2)

Atpl = P2 - pl and AtO,, = 'PS2 - Ol (3)

To permit equation 1 to be used without knowledge of 4N2 and,therefore, Aqi,, equation 2 was solved for 4/I2 and this valuesubstituted in the right hand equation 3. Simplification showedA+s = -S+ip,, where

S = (o- + nRTk/tlt,J,.)/('12 + o-/2 + nRTco/J.) (4)

Dalton et al. (3), by using ip,1 rather than (qp,, + tP,2)/2, obtainedslightly different forms for equations 2 and 4.

Because o, k, and w were not measured, reasonable estimateswere selected for evaluation. The value of a- for nutrient solutionions is likely to be within 0.7 to 1.0 (5, 13); and, because theeffect on the model within this range is small (Fig. 2), themidpoint value of 0.85 was utilized in most places.

Relationships expressed by the model for certain special casesproved useful in selecting appropriate values for k and w. When

(5)and adding increasing concentrations of PEG to a treated sideuntil J,.' approached 0, should have brought (qP2 - 'I4p + 419t)to near 0 also, regardless of the magnitude of L'. Using q,s of thePEG to predict q'l, and Figure 2 to predict oAtp,j, close approxi-mations of q142 could be reached. Considering q42 = OU2, remem-bering that pl = 0, and obtaining o-Aqi,u from Figure 2, valuesfor Lu at that particular J,u were calculated. These values werethen used to test experimental data against those predicted bythe model.

After finding that the experimental data were incompatiblewith the model when L was constant, several procedures wereused to predict possible values for variable L. Because untreatedplants exposed to constant conditions exhibited a predictablerhythm of J,. throughout a day and because root L has beenreported to fluctuate diurnally (21, 22), L was considered aspossibly varying in direct proportion to the Jl. of a typical,untreated plant. It might also be possible for L to vary in acircadian rhythm not related on a one to one basis to the Jl, soexaggerated circadian patterns of variation also were considered.Finally, the possibility that L, without regard to a rhythmicpattern, might be a function of any temporary J., was investi-gated. For the latter, all data were grouped in simultaneous

Jv (cm sec-' x le)FIG. 2. Relationships of o-Atp,8 to J, for four limit combinations of k

and maximum root pressure. Lines are drawn only for a- = 0.85, but keypoints for o- = 0.7 and 1.0 are indicated. See equations 1, 2, and 4 in thetext.

260 MICHEL

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Plant Physiol. Vol. 60, 1977 APPLICATION OF A ROOT PERMEABILITY MODEL

pairs, treated side during one measurement period versus un-treated side during the same period. A value of LU, when J,t wasnear 0, was obtained as outlined above. For other values of J,.and JrU, arbitrarily selected values of Lu were substituted insimultaneous equations (equation 5) to find matching values forL'. These paired values for Lu and LI were plotted versus Jr.Iteration permitted best values to be picked.By having set limits for oa, k, co, and Aq,U at J, = 0, so that

o'Aqi', became some function of J, (Fig. 2), extremes of exudationrates of decapitated roots became predictable (with Aqi,. = 0, Jr= -LoTAtk, where L may be constant or a function of Jr). Inaddition, extremes of 41/2 for exudates became predictable.

RESULTS

The lag time from a change in conditions, even from darknessto bright light and the reverse, was no more than 15 min (Fig. 3,upper). Summing the number of additions by 15-min periodsproduced totals not varying by more than two, except over longperiods where circadian rhythms were responsible (Fig. 3,lower). Such results permitted treatment periods to be kept shortwithout destroying confidence in the values found and madepossible several successive treatments during a portion of the daywhen absorption rates were normally high. Untreated plantstypically showed a steady increase in absorption from a relative

7 '

rL

10 I| &*&bk9

~~~~~~~~~~~~~~~~~~~~ ~~~~~~k 12 |sk

14 sf i a i I I5

I~~~~~~~~~~~~~~~~~~~149119 20;hss|&Zk||ssssse6o 2l|ssssssss|esss 1721 I L L 22 kks|k22 23| *fi23 24 .... . .a. ....I

HR OF DAY

FIG. 3. Upper: record of replacement of absorbed liquid by bothsides of a soybean divided root system. Photographs of the event re-

corder paper were cut and assembled in hourly segments, appropriate hrof day being indicated at each end. Upper record in each pair is fromtreated side (each event = 0.44 ml); lower, from untreated side (eachevent = 0.49 ml). Treated side was drained and refilled starting at 1030,1200, 1345, and 1530 to produce tp5, lowering from PEG of 1.8, 3.8,5.6, and 7.4 bars, respectively. After draining at 1715, treated side was

washed four times before refilling with nutrient solution. Light came on a

few minutes after 0800 and went off at 2200. Lower: Fifteen-min totalsof water increment additions to both sides of the soybean divided root

system. Increment counts are from record in upper portion of figure.

value of about 0.77 at 1 hr after lights on to 1.00 at about 5 to 6hr after lights on followed by a gradually steepening rate ofdecrease to about 0.64 at 13 hr after lights on (10).

Progressively increased concentrations of PEG slowed andeventually nearly stopped water absorption from the treated side(Fig. 4). In the example shown, the treated side had originallysupplied more than half of the water. Marked reduction on thetreated side resulted in slight reduction of the total absorbed(Table I), but no symptoms of plant stress ever were evident.Values for J,U and J1 (Table I) were calculated from absorp-

tion rates and the data of Table II as described under "Materialsand Methods."When J4 = 1.3 x 10-7 cm sec-', ql+2 - I+ (equation

5) should have been close enough to 0 to permit accurateassessment of Lu ("Materials and Methods"). The first two andfourth limiting combinations (Fig. 2) gave L :o10 X 10-7 cm

sec-1 bar-'. Because these three combinations also gave insignif-icantly different calculated values for 4'p2, only data using thefirst are presented. The third combination gave LU 13.8 x 10-7cm sec-' bar-'. To make 142 = C42 in the absence of PEG,corresponding values for LI were 12.9 x 10-7 and 17.8 x 10-7cm sec-' bar-'. These values for L were then used for each set ofconditions (Table I) to calculate paired values for 41p2 and 'p,2(Fig. 4). Because of the manner of determining LI in the absenceof PEG, ip2 closely matched 41p2, so most such matched pairs(Table I) were omitted from Figure 4.Two types of results cast serious doubt that L could be con-

stant within the context of the model. First, although L valueswere obtained by methods that caused tp2 to match q/P2 bothbefore treatment and at the highest PEG concentration, largediscrepancies exist between predicted values for Op2 and P12 atintermediate PEG concentrations. Second, predicted Op12 valuesat moderate to low JrU appear to be too high. Leaf ip measure-ments from comparable, untreated plants ranged from -7.2 to-9.3 bars (unpublished). If 75% of the resistance in soybeans isacross the roots (1), /P'2 values would be expected in the range of-5 to -6.5 bars. Direct measurement of untreated soybeanhypocotyl (unpublished) showed values within this range. Thecalculated values (Fig. 4, L constant) were much closer to 0.

Table I. Exp,erinentaZ Donditions, Absorption Rates, and FDiuxes vor a SoybeanDivided Foot Suster- Drirnc .curZu Periods for Co>o Success--Di -Das

Timel Light Co PEG Total JU it2 s Absorp v v

3 -hr UI1 1 ars c3 hr- c sec- 1071000 43,200 300 0.0 23.0 29.8 38.51130 43,200 300 -1.8 25.9 52.0 25.61315 43,200 300 -3.8 25.1 66.6 9.01500 43,200 300 -5.6 24.4 70.7 2.61645 43,200 300 -7.4 24.0 70.5 1.31900 43,200 300 0.0 23.1 33.6 34.5

0030 0 500 0.0 2.8 4.3 3.8

1000 43,200 300 0.0 24.4 31.1 38.41100 21,600 300 0.0 16.5 19.8 27.21245 21,600 300 -3.5 17.0 46.4 2.51408 21,600 300 -5.6 17.6 49.1 1.21515 43,200 300 -5.6 29.2 78.4 4.91700 43,200 500 -5.6 21.0 57.2 2.51900 43,200 500 0.0 19.5 27.8 27.2

2330 0 500 0.0 3.7 5.5 4.91 Midpoint of the one hr measurement period.

Table II. Pertinent -ata Collected at Harvest of" So:ear Divided FootSusterm and L-a,cuZated Values for 70 hr Prior to H-arVest

Measurement Description At Harvest 72 hr Prior

Fresh Weight, Top (g) 60.3 52.0Leaf Area, One Side 7(d2) 19.1 16.5Fresh Weight, Untreated Roots (g) 13.2 11.4Fresh Weight, Treated Roots (g) 13.3 11.9Exudation Rate, Decap. Roots (crr3 hr-1) 0.22Exudation Jv, Decap. Roots (cm sec"1 x 107) 0.29Root Area/Leaf Area 1.11

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Plant Physiol. Vol. 60, 1977

2.6 70-1.3 4.3-3.8 44-2.JU_jt (cm sec-l x107)

FIG. 4. Paired values of predicted upper root xylem pressure poten-tials for untreated and treated sides of divided soybean root systemsunder different combinations of experimental conditions. JrU - Jrt valuesand conditions are in Table I. Vertical lines: limit 1, constant L; horizon-tal lines: limit 3, constant L; clear: limit 1, variable L; diagonal lines:limit 3, variable L. Limits 1 and 3 are in Figure 2.

Moreover, predicted *p2 night values were about -0.5 and 0.5bar (Fig. 4). Corresponding values would be about -0.7 and-0.3 bar. Highest soybean stem and leaf t, measured at night(15 and unpublished) have been near -2 bars.Because constant L has the model predicting widely different

simultaneous values for *4p2 and 442 and values for *4Ip2 that are

too high at moderate to low J.u, curves for variable L were

obtained by procedures outlined under "Methods."Considering L to vary in direct proportion to the circadian

rhythm of J, for a typical, untreated plant reduced slightly someof the discrepancies cited above but increased others. Exaggerat-ing the rhythmic variations of L had a similar effect. Neitherconstant L nor any circadian pattern of variation of L seemed topermit the model to account for the results obtained.

Following procedures outlined in "Methods" for combination1 (Fig. 2), the 13 pairs of J, during light hours (Table I) yieldedthis straight line relationship between L and J., L = 0.118Jr +(1.70 x 10-7) (Fig. 5). Regression analysis modified this to L =

(0.121 + 0.004)J, + (1.60 ± 0.16 x 10-7) r2 = 0.973, which isinsignificantly different from the original. In all instances buttwo, predicted values for 4'I2 and 442 are within 0.7 bar of eachother (Fig. 4). In one of the exceptions, 1st day and -7.4 barsPEG, a reduction in JVt by 0.6 x 10-7 cm sec-1 would havebrought 442 into line. With the rate of absorption so near 0, a

measurement error of this magnitude could have occurred easily.The second exception, 2nd day and -3.5 bars PEG, would haverequired a measuring error much greater than was likely.

Results for combinations 2 and 4 were sufficiently comparablewith those for combination 1 to warrant omission here.

Combination 3 yielded this relationship, L = 0.138J, + (4.10X 10-7), and predicted values for 41p2 and 442 shown in Figure 4.Fit was nearly as good as for combination 1, but predicted valuesfor 1442 at low Jr appear to be too high.

In general, a linear relationship between L and Jr comes muchcloser to matching predicted values of *p42 and 4p2 than eitherconstant L or L varying in circadian rhythm.The exudation rate of the example used (Table II) was among

the lowest for 28 plants measured, for which the range was 0.25X 10-7 to 1.27 x 10-7 and the mean was 0.64 x 10-7 cm sec-I.Combinations 1 and 2, with constant L, and combinations 1, 2,and 4, with variable L, predicted values within this range (TableIII). A value of k closer to 0.1 x 10-11 than to 1.0 x 10-11 molcm-2 sec-I appears to be favored by such results.

Ten comparisons of the 4102 values of exudates from decapi-tated roots in full strength nutrient solution ranged from 0.5 bargreater to 0.5 bar less than p,, with a mean of nearly 0.2 barmore negative. Only combination 3 with variable L (Table III) isexcluded from this range. The data are not adequate to help witha decision; however more precise data of this type would bequite useful.

DISCUSSION

The results indicate that if the model demands L to be a

constant, the model fails to predict reasonable values of internalwater potential. A linear relationship between J, and L pro-

vided, with one major exception, a fit between measured andmodel-predicted values considered to be within experimentalerror. The poor data available regarding exudation rates and ip8

measurements of exudate were compatible with but not reallydefinitively supportive of variable L within the model. Not onlybetter exudation data but also measurements in the lower stemand 4112 measurements other than of exudates are needed if thefeasibility of the model is to be proved.Even though J, and L values obtained can only be considered

relative, because average root diameter was only estimated, therange obtained for exudation Jr values and L values at exudationare within the range found or cited by others (8, 18).Newman (20), considering the model to require constant L,

showed that the data from decapitated tomato root systems ofMees and Weatherley (12) were incompatible with the model.Calculated L values from three experiments reported by Mees

ho, lo

0 8.0

T.0

:6E

-i4

40Jv (cm *ec-I x leO)

FIG. 5. Hydraulic permeability coefficient versus water flux throughsoybean roots for limit combinations 1 and 3 of the coefficient of activesolute absorption and maximum root pressure (Fig. 2). Paired valueshave the same symbol.

Table III. Exudation Liquid Fluxes and Solute Potentials Predicted byCertain Limit Combinations of Constants Inserted in the Model

Limit Combination11 2 3 4

Constant L Jv (crm sec-1 x 107) 1.28 0.88 6.71 1.91

*s2 (bars) -0.50 -0.46 -0.87 -0.57

Variable L Jv (cm sec-1 x 107) 0.71 0.27 3.87 0.34

Ps2 (bars) -0.83 -0.55 -1.36 -0.60

All a = 0.85; k for 1 and 2 = 0.1 x 10-11, for 3 and 4 = 1.0 x 10-11mol cm-2 sec-1; A*p at Jv = 0 for 1 and 3 = -5.0, for 2 and 4 = -0.2 bar.These values for a, k, and Aip determined the values of u (methods); so

that u for 1 = 0.017 x 10-11, for 2 = 0.425 x 10 -11, for 3 = 0.17 x 10-11,and for 4 = 4.25 x 10-11 mol cm-2 sec-1 bar-1.

262 MICHEL

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Plant Physiol. Vol. 60, 1977 APPLICATION OF A ROOT PERMEABILITY MODEL

and Weatherley, referred to but not given by Newman (20)(though values for one had been reported by him earlier [19])are related to J, expressed in arbitrary units in Figure 6. BecauseMees and Weatherley reported Jr as rates of rise in differentsized capillary tubes and without relation to root system size,direct comparison among experiments was impossible; there-fore, the J4 values were normalized in relation to A&5 in the 1.39to 1.60 bars range. Although all but one of the Aq,, values are insequence, there are obvious discrepancies in relative distancing.No vertical normalization of L was attempted, so values arerelative only within each curve and not among curves. Regard-less, Figure 6 does indicate that L was approximately linearlyrelated to J, at Ap, below about 1.5 bars, but L became nearlyconstant at higher values of Aqi,. Such results contrast sharplywith those for soybean (Fig. 5) in which the linear relationshipextended to Atp, > 7 bars. Nevertheless, within the context ofvariable L, the model appears applicable to both the tomato dataof Mees and Weatherley (12) and the soybean data of this paper.

Lopushinsky's data for decapitated tomato roots (Fig. 6 in ref.11) supply values for all terms of equation 1 but L and a-. Theycan be compared directly with the similar data of Mees andWeatherley (12). Lopushinsky used full strength Hoagland solu-tion without citing a reference or assigning a ,s. Mees andWeatherley used Hoagland No. 2 nutrient solution and indicated

= -0.88 atm. I used the latter solution but measured s, to be-0.74 bar. Newman (20) used -0.4 bar as qi,, of Lopushinsky'sexternal medium when showing his "tangent test" to rejectFiscus' model for Lopushinsky's results. The model would not berejected if uMl were more negative than -0.7 bar, which it musthave been. To make the best possible comparison with theresults of Mees and Weatherley, these values were assigned: 1P,1= -0.89 bar and a- = 0.76 (19). The results (Fig. 6), normalizedwith the data of Mees and Weatherley for Jv and AuJ,, essentiallycorroborate those of the latter in showing L to vary nearlylinearly with J, in a lower range before becoming almost constantin a higher range. Lopushinsky's data are compatible with Fis-cus' model only with L variable.Brouwer (2) reported L to vary both with distance from the

root tip and with Atp, The Vicia faba roots used were uniformlynear 1 mm diameter, so points plotted by Brouwer (Fig. 2 in ref.2) could be converted to units used here. Values for his fourzones covering a total root length of 10 cm were averaged toprovide better comparison with the soybean data for large por-tions of root systems. Although of a similar magnitude, Brou-wer's values for L were larger, changed over a 2-fold rather thanthe 3.4-fold range predicted for soybean, and formed a curvilin-ear relationship with J, (L increasing but little at higher Jr valuesreminiscent of tomato [11, 12]) (Table IV) rather than the linear

8 12Jt (ARBITRARY UNITS)

FIG. 6. Hydraulic permeability coefficient versus water flux for de-capitated tomato roots from Mees and Weatherley (12) and Lopushinsky(11). Abscissa was normalized according to A,,, in the range of 1.38 to1.60 bars. Ordinant was not normalized.

Table IV. Average Values of L, A*p, and Jv for the Apical 10 an of Viciafaba Roots from Brouwer (Fig. 2 in ref. 2) and Soybean L's at these Jv'sValues of A*p were obtained as follows: 1st row, in darkness; 2nd row, in

light; 3rd-6th rows, in light with decreasing *, around all but test root.

Vicia faba SoybeanL Ap v L

cm sec-1 bar-1 x 107 bars cm sec-1 x 107 an sec-1 bar-1 x 10710.9 -1.6 17.4 3.814.9 -2.0 29.8 5.218.8 -2.5 47.0 7.220.1 -2.8 56.3 8.320.7 -4.1 84.9 11.721.8 -4.4 95.9 13.0

one fitting soybean data (Fig. 5). Brouwer's results also showA*,, in darkness (Table IV, first row) to be -1.6 bars, which isclose to that suggested earlier in this paper for soybean.Some authors have assumed that a straight line relationship

between A*, or Atp,, and J, indicated constancy of L (6, 7) and/orthat the slope of the line is L (4, 6). The model of Fiscus (4) andDalton et al. (3) does predict a straight line with slope equal to Lat sufficiently negative values of At,5, however, not just anystraight line. The Atp,, intercept must be less negative than qi,,,(20), but also cannot be greater than 0 (equation 1).Consider the frequently cited results of Jensen et al. (7).

Although straight lines fit well their J, versus Ag,, data, extrapo-lated intercepts range from slightly below -0.1 to slightly above0.05 bar. Two things show that such intercepts do not permitusing the slopes of the lines, as they did, to be measures ofpermeabilities. First, with Atp values ranging only from -0.2 to-0.6 bar, the above intercepts are significant departures from 0,the value of +81. Second, because insignificantly different valueswere found with flux in the normal and reverse directions, anyeffect of crAu, must have been insignificant, with equation 1reducing to J, = -LAqp,; and the Ag, intercept should have been0 for J,/Atq, to equal -L. Calculation from individual values ofJ, and Atp,, for minimum and maximum differences of the lattershow L for stem, stem plus leaves, stem plus roots, and wholeplants of sunflower increasing by 21, 14, 22, and 15%, respec-tively, with a Aui,, change of only -0.4 bar. Of particular interestis the implication of departure from Darcy's law (constant L) notonly in roots but also in stem and leaf vascular tissue as well.Incidently, use of extrapolation to calculate L values at onevalue of J, greatly improved the comparison of whole plant Lwith that combined from separate parts made by the authors (7).

Consider, also, the data presented by Hailey et al. (6). Noneof the straight lines drawn in the range of large negative values ofAup, extrapolates to within Aui,, = 0 to up,1. Thus, the model ofFiscus (4) and Dalton et al. (3) must be rejected and the slope,Jr/At,p, cannot equal -L. Of interest and contrary to otherresults cited, experiments 2, 3, and 4 of Hailey et al. (6) showedL decreasing rather than increasing as Aup,, became more nega-tive.The overwhelming weight of evidence favors variable root

permeability to water. If L varies, so then may some of the other"constants" of the model; and the question of valid use of themodel when L is permitted to vary must be raised. Satisfactoryknowledge of these relationships for any species and possibleutility of the model as defined by equations 1 through 3 mustawait more nearly complete data.

Acknowledgment-I am grateful to H.-L. Lin for the soybean data.

LITERATURE CITED

1. BOYER JS 1971 Resistances to water transport,in soybean, bean, and sunflower. Crop Sci 1 1:403-407

2. BROUWER R. 1954 The regulating influence of transpiration and suction tension on thewater and salt uptake by the roots of intact Vicia faba plants. Acta Bot Neerl 3: 264-312

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