Plant Physiol. (1986) 80, 1006-10110032-0889/86/80/1006/06/$01.00/0

Compartmental Efflux Analysis: An Evaluation of the Techniqueand Its Limitations'

Received for publication August 29, 1985 and in revised form December 24, 1985

JOHN M. CHEESEMANDepartment ofPlant Biology, University ofIllinois, Urbana, Illinois 61801

ABSTRACT

Efflux analysis is an established tool for characterizing the exchangeproperties of multicomponent systems. In this report, we have simulatedseveral three- and four-compartment systems with error-free and imper-fect data, the errors being designed to mimic actuaL nonbiological varia-bility in isotope efflux studies. The data sets were analyzed using com-puterized nonlinear regression techniques to identify the important as-pects of actual experimental design (uptake times, efflux collectionschedules, and total efflux times), and to consider the possibility that aproperly designed and executed experiment might fail to resolve com-partmentation correctly. The results showed that for any of the systemssimulated, including those with error-free four-component data, a reason-able three-component fit was obtainable. Resolution of the additionalcompartment was not always possible. In correctly resolved systems,failure to estimate the correct decay constants was common, especiallywhen the half-times were separated by less than an order of magnitude.We conclude that efflux analysis, by itself, lacks the power to providereliable information about multicompartment systems.

Compartmental efflux analysis is a well-established analyticaltool for studies ofion transport in plants. Though it was originallyused for the testing of compartmentation models in both algae(5) and higher plants (7), in recent years the technique has beenmore widely used to characterize internal ion pool sizes andmembrane kinetic parameters or exchange constants. The usualanalysis of efflux data requires the assumption that the relevantcompartments of the experimental system (i.e. a root, a leaf, astorage tissue slice, or an algal cell) are functionally in series witheach other. If one accepts this hypothesis, the theory of com-partmental analysis is straightforward (14, 15); the amount ofsome component, say an isotope labeled ion, remaining insideof the system after a given time of efflux is appropriately de-scribed using the equation:

QT(t) = Ql e kl + Q20e-k2 + ... + Q ° e-k' (1)where Q,° is the content of compartment j at t = 0, kj is theassociated kinetic constant for transmembrane flux (kj = l/T =0.693/th), and n is the number of compartments.The mathematical description of a curve and its statistical

resolution from a data set are, however, considerably differentproblems. In preparing to use this technique in our laboratory,we became aware that the practice of efflux analysis is less firmlyestablished in virtually every regard, from selection of the initiallabeling conditions, to determination of the efflux collection

' Supported by grant PCM 83-04417 from the National Science Foun-dation.

schedule and to analysis of the results. We noted with particularconcern that in spite of the widespread availability of computersoftware for nonlinear regression, separation of components hascontinued to depend largely on the visual peelback or straight-edge regression technique (e.g. Ref. 6). To our knowledge, therehas been only one study in the ion transport literature which hasused a nonlinear regression program instead (11, 12).

In this paper, we will discuss the results of computer assisteddata analysis of computer simulated efflux experiments. The useof simulation to refine compartmental efflux methodology wassuggested by Walker and Pitman (15) who also provided a limitedexample, considering the effect of th separations on apparentcomponent efflux rates in a three-compartment model. Rygie-wicz et al. (12) also performed a limited simulation study toconsider the suitability of three computer programs to resolve athree-component system, with and without random errors.The simulations which we will present were performed to

determine the possible effects of sample collection schedules anduncontrollable errors on the evaluation ofresults under a numberof hypothetical situations with three and four compartments inseries. Though the series assumption is not above question (2-4,15), that aspect of the problem will not be questioned here.Particular attention will be given to factors which may preventsuccessful data analysis, either as identification of the incorrectnumber of compartments or as incorrect resolution of kineticparameters.

MATERIALS AND METHODS

Simulated efflux data were generated using Eq. 1 for severalthree- or four-compartment systems. Though more compart-ments are not necessarily impossible, the four-compartmentsystem offers sufficient complexity for our present purposes.Systems with more than four compartments also become expen-sive to resolve in terms of computer time. The parameters usedwill be identified according to their group in Table I as PG-A toPG-E.Three 'sampling' or 'efflux collection' protocols were used

(Table II). Protocol 1 was that used by Mills et al. (Ref. 6; theirFig. 2). Protocol 2 was based on the schedule of Rygiewicz andBledsoe (11; CS Bledsoe, personal communication) and wassomewhat more rigorous than the protocol illustrated in thefigures of Rygiewicz et al. (12). Protocol 3 is the schedule whichwe have used in our laboratory for the analysis of 24Na' effluxfrom the roots of intact Spergularia marina plants (results to bepublished separately). In all cases, sampling continued to 540min at 60-min intervals unless otherwise stated.Sampling of error-free data used to construct the curves in

Figures 1 and 2, and to test the sufficiency of the samplingprotocols in the absence of noice, was done simply by solvingEq. 1 for the desired time set using a simple program in BASIC.All samples using the unmodified parameters groups in Table I

1006

COMPARTMENTAL EFFLUX ANALYSIS

Table I. Parameter Groups (PG) Used with Equation I to GenerateSimulated Compartmental Efflux Data

PG Q ° th, Q20 th, Q30 th3 Q40 th4min min min min

A 300 1 390 20 7500 2400B 300 1 390 20 3500 1500 4000 4000C 300 1 200 20 190 60 7500 2400D 300 1 190 5 200 50 7500 2400E 5000 0.2 100 1 150 10 7500 2400QO arbitrary units.

Table II. Efflux 'Sampling'Protocols Used in Conjunction withEquations 2 and 5 to Generate Error-Free and Imperfect Data SetsAll protocols were continued to 540 min at 60 min intervals.

Protocol Sampling Times in MinutesI 1 2 5 8 10 15 20 30 40

50 60 75 90 120...

2 2 4 6 8 10 15 20 25 3035 40 45 60 75 90 105 120...

3 0.5 1 1.5 2 2.5 3 3.5 4 56 7 8 9 10 12 14 16 18

20 25 30 35 40 50 60 75 90105 120...

had at least five significant digits. Resolution of the Q° valuesfor Eq. 1, therefore, presented no computational difficulties andwill not be considered as a major question in this paper.

For imperfect data designed to simulate the errors and char-acteristics of actual experiments, imperfections were introducedas follows. First, the timing of samples was adjusted to includean uncertainty, e, of ±0.2 min. This was to simulate the timerequired to change the collection solutions based upon ourexperience. Thus, for each collection time ti,

ti = ti + e (2)where the subscript i signifies the position of the sample in thesequence. After solving Eq. 1 for the 4i, the resulting data setconsisted of the data pairs (ti, QT(ti)). (The tilde accent [-]denotes that inaccuracy is involved. The inaccuracy is unknow-able, however, so ti is used in the analysis ofthe data, as in actualexperiments.)

Second, the amount of 'efflux' between ti and ti + 1 wasdetermined as

Di = QT(ti) -QT(ti+1) (3)and the random 'counting' error was introduced,

Di=Di * si' (4)

where a' was a random error factor of mean 1.0 ± 0.05. (Inpractice, when using BASIC programs to generate errors, theactual error limits were set such that the apparent SD were asgiven above, though the random numbers themselves were uni-formly, not normally, distributed.)The differences generated by Eq. 4 are equivalent to the

amount of label collected in the efflux solution in a givensampling interval, AQ/At. These values may also be analyzed asa function of time (e.g. Ref. 6). Walker and Pitman (15) notedthat at very long times, because of the properties of derivativesof exponentials, the curves plotted as log(QT) versus t or aslog(dQT/dt) versus t should be parallel. They, and Mills et al.after them, incorrectly asserted, however, that this characteristiccould be used to judge the suitability of the three componentmodel. In fact, parallel lines will result with any system domi-

nated at long times by a single compartment, regardless of thenumber or arrangement of much more rapidly exchanging com-ponents.

Practically, the differences in the analyses occur at short timeswhen dQT/dt is rapidly changing, and thus assignment of themeasured AQ/IAt to a particular time is most uncertain. For thepurposes of this paper, therefore, we chose to proceed by themore usual method, summing the samples from last to first togive QT(tM), the apparent amount remaining at each time:

QT(ti) = QT(ti+1) + Di. (5)The data pairs (ti, QT()i)) were then analyzed by nonlinearregression.The observations generated in this way were not mutually

independent. Errors associated with any sample would, in theabsence of canceling errors, be propagated back to the firstsample. Despite the errors, however, the function Q7(t) decreasesmonotonically with time, the 'noise' causing the rate of decreaseto be faster or slower than expected. These are important char-acteristics of actual data which may affect data analysis, butwhich cannot be simulated by simply altering the independentvalues of QT(t).Although as Rygiewicz et al. (12) have shown, there are a

number of generally available nonlinear regression programssuitable for analysis of the data, for the purposes of this paper,we have used the DISCRETE regression program of Provencher(9, 10), Version la, and the Cyber 175 mainframe computersystem at the University of Illinois. This program has severaladvantages over others such as BMDP Program AR (1) or SASNLIN (13). It is completely automatic, neither requiring norallowing any initial guesses at the values ofthe regression param-eters, or of their number. The best solution, determined by theprogram based on the approximate probability test that it isbetter than the alternatives (a modification of Fisher's F-testcorrecting for moderate nonlinearity) is reported as are thesecond best, third best, and so on. The suitability of each fit isalso indicated by the signal-to-noise ratio, defined as

S/N = QT(0)/ayy (6)where ao, is the standard deviation of the fit (9). In preliminarytests, it was established that the DISCRETE program and BMDPProgram AR, when it converged, agreed in the results.The reporting of several possible solutions at once greatly

simplifies our comparisons, here, of fits with different numbersof parameters. In actual experimental situations, of course, thiswould allow the investigator to accept or reject the 'best' solutionbased on evaluation of the results in their entirety. As in BMDP,SAS, and other statistical program packages, a number of otherstatistics are included which may also help in that evaluation.We have found two of these particularly useful. First, the ap-proximate standard deviations are reported by DISCRETE foreach of the regression parameters. These are also reported, forexample, by BMDP, in which case they are referred to as 'asymp-totic standard deviations.' While they are not statistically iden-tical to errors in linear models, they are close approximationsand are useful indicators of probable gross inaccuracies, as whenthe best solution contains too many components. Second, theplot of residuals versus time can indicate systematic data errorsor, again, the possibility that the incorrect number of parametershas been identified. We have found these plots particularly usefulto suggest when data errors or noise have resulted in identifica-tion of too few compartments.

RESULTS AND DISCUSSIONThroughout the literature on efflux analysis in plant systems,

analyses have usually resulted in the identification of threecomponents in series with decay half-times of (a) less than 1 min

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Plant Physiol. Vol. 80, 1986

to, perhaps, 10 min, (b) tens of minutes, and (c) hours, rangingfrom perhaps 10 to 1000. These components have usually beenreferred to as the 'wall' or the free-space, the 'cytoplasm' (thoughearly reports carefully noted that this term was inaccuratelygeneral, e.g. Ref. 6), and the 'vacuole,' respectively. Figure 1shows a three component decay curve generated using PG-A(Table I). The points shown with the curve incorporate thesimulated sampling and counting errors for three simulations.These examples were chosen to illustrate the range of variabilitywhich resulted from the random errors.

Using all three sampling protocols in Table II, error-free datacould be correctly resolved. Similarly, all three protocols yieldedsatisfactory resolution of half-times under imperfect conditions:in nine simulations th for component 1 ranged from 0.85 to 1.15min, the worst error being 15%. For component 2, th rangedfrom 19.0 to 22.4 min, the worst error being 12%. For compo-nent 3, th varied from 2218 to 2578 min, the worst error being11%. The worst case approximated standard deviations of theestimated decay parameters were 9.8, 8.7, and 1.5% for thecomponents 1 to 3, respectively. Coefficients of variation be-tween the nine separate estimates of the decay parameters were14.2, 6.0, and 6.2%.These analyses also indicated the possibility of identifying too

many components. In two of the nine analyses, a fourth com-ponent was identified by the DISCRETE program in the 'best'solution, in one case effectively splitting the intermediate com-ponent (th2 = 14 min, th3 = 46 min), and in the other splittingthe long component (th3 = 530 min, t4 = 3800 min). In the lattercase, the error estimates associated with the half-times were morethan 250% for the two longer components, strongly suggestingthe error. In the former case ('+' in Fig. 1), the errors for thetwo intermediate components were 30 and 40%. These were notlarge enough to arouse suspicion when compared with the resultsof actual four-component experiments (see below). Q° for thesecond compartment was also split in a way which was notunreasonable (Q2' = 260, Q30 = 177). As seen in Figure 1, theappearance of this data set, when graphed, was similar to that ofthe other sets.

Time (min)

FIG. 1. Semilogarithmic plot of error-free (-) and imperfect (C,O, A, +) 3-component data sets for simulated compartmental effluxanalysis experiments. Each symbol type denotes a separate simulation.The examples represent the range of variability between trials in 9repetitions. The data set designated (+) was incorrectly analyzed as

having 4 components (tb2 = 14 min; tj!3 = 46 min).

PG-A was also found to be undemanding in other simulatedsituations. Correct half-times could almost always be resolvedwith total sampling periods of 120 min. As noted in "Materialsand Methods," large differences between Q°values presented nocomputational problem. Thus, reduction of Q30 from 7500 to750 made no difference in resolution of the half-times. This alsoindicates that unless one can be assured that isotopic equilibriumwill be attained by the end of the loading period, the length ofthat period in an actual experiment is not critical. Similarly,increasing Q,' to 7500 had no effect, indicating that the 'quickrinse' usually used to remove adhering solution would not alterthe apparent th values as long as timing were begun at thebeginning of the rinse period.The simplicity of the analysis of PG-A results from the widely

separated decay half-times. This separation is also essential tothe use of the visual peel-back or straight-edge regression meth-ods. Unfortunately, the differences between the results in Figure1 and data generated using the other parameter groups are notparticularly obvious. Figure 2a compares the error-free curvesfor the five parameter groups, and Figure 2b plots the values forgroups B to E relative to those of PG-A. In the most extremecase (disregarding the large fast component in PG-E), the maxi-mum difference was only 1.5%.The significance of Figure 2 lies in the now common, if

unstated, assumption that three is the 'correct' number of com-partments. It is easily seen that this assumption will result inidentification of that number so long as the curves are analyzedvisually. Subjective evaluation is inherent to some degree in mostnonlinear computer regressions as well, however, since both thenumber of parameters sought and their initial value estimatesare usually required as inputs. Table III shows the half-times andapproximate standard errors for three-component fits to theerror-free and imperfect data simulated with the four-componentparameter groups B to E. In no case were the uncertainties ofthe estimates suspiciously large.

In several cases for each parameter group, three-componentfits were either 'best' or were flagged by the program as 'signifi-cant possibilities.' This exemplifies an alternative potential error,the failure to identify the correct number of decay componentsin spite of having searched for them. Unfortunately, nonlinearregression methods, unlike linear methods, do not always con-

verge upon stable parameter estimates in the time or number ofiterations allowed for a single program run (1, 13). Thus, it maybe impossible to decide when the appropriate solution has beenachieved, particularly when using programs accepting only fixednumbers of parameters.Three other possible origins of this component oversight error

are noisy data, insufficient separation ofthe decay constants, andan inadequate sampling protocol. These may occur alone or incombination, and the problems may not always be resolvable.For example, parameter group B, with two slowly decayingcomponents, could not be resolved with any of the samplingprotocols, largely because of the lengths of the two slow compo-nents. Experimental and biological constraints may preclude thatresolution regardless of sampling regime (e.g. Refs. 3 and 7).PG-C has a similar relative component separation problem, in

this case involving the intermediate time scale. Error-free datacould be resolved using any of the sampling protocols. However,imperfect data resulted in resolution of four components in onlysix of eight trials. The correct half-times were found only once,and then the intermediate component decay parameters were

associated with large approximate standard errors (Table IV).These errors are comparable to those noted above when fourcomponents were found for the three-component simulation.The effects of separation of decay parameters on resolution of

Q° values was considered briefly by Walker and Pitman (15)who found that, when th, was lengthened with respect to ti2, QI0spilled over into Q2'. In separate analyses (not shown), we found

1008 CHEESEMAN

COMPARTMENTAL EFFLUX ANALYSIS

160.5 _...

105.0 bQ.

102.5 qj

100.0 iz

'-N07 9;Q

FIG. 2. a, Semilog plots (left axis) of error-free 3- and4-component decay curves generated using the parametergroups shown in Table I. b, Linear plots (right axis) of the4-component data sets as percentages of the values in the3-component set (PG-A).

Time (min)

Table III. Apparent Decay Half-Times (th) for 3-Component Fits to 4-Component Data Sets Generated UsingParameter Groups PG-B to PG-E (Table I)

Estimates approximate ± SD ofthe estimates are shown for selected, representative analyses. Italicized valueswere resolved for error-free data.

PG th th th Notesmin

B 1.00 ± 0.01 20.6 ± 0.13 2301 ± 2.5 (1)0.97 ± 0.05 20.3 ± 0.9 2353 ± 16 (1), (4)

C 1.06±0.03 31.8± 0.9 2335± 12 (1)1.71 ±0.12 37.8±2.9 2264±28 (1)0.90 ± 0.03 32.7 ± 1.3 2521 ± 19 (1), (5)0.98 ± 0.04 33.8 ± 1.2 2515 ± 17 (2)0.78 ± 0.04 27.2 ± 2.0 2256 ± 27 (3)0.75 ± 0.07 24.5 ± 2.5 2318 ± 38 (3), (4)0.89 ± 0.07 31.4 ± 3.1 2206 ± 43 (3), (5)

D 1.46 ± 0.08 29.4 ± 3.0 2332 ± 27 (1)1.53 ± 0.08 28.0 ± 2.7 2320 ± 24 (2)1.99 ± 0.11 42.9 ± 5.2 2252 ± 36 (1)1.10±0.09 16.0±2.1 2319±30 (3)1.42 ± 0.08 30.8 ± 4.0 2433 ± 37 (1)

E 0.205 ± 0.001 7.8 ± 0.3 2390 ± 8.6 (3)0.221 ± 0.004 3.0 ± 0.6 1761 ± 130 (3)0.265 ± 0.006 12.1 ± 7.3 2911 ± 1400 (3),(4)

a (1) Sampling protocol 1. (2) Sampling protocol 2. (3) Sampling protocol 3. (4) The 3 componentfit was the 'best' solution. (5) The 3 component fit was a 'significant possibility.' The approximateprobability that the 4 component solution was better was less than 0.95.

that error-free data could not be resolved if it were generatedwith PG-C modified such that th2 = 40 min. Even when thesampling protocol was modified for much more extensive datageneration, the analysis would not space th2 and th3 closer than afactor of two. The difficulty of resolution of similar decay con-stants may be largely associated with loss of computationalaccuracy during matrix inversions (9). The minimum resolvableseparation of decay constants can not, therefore, be defined.PG-D incorporated wider relative separation of decay times

but with the addition ofthe fourth component between the usual

'wall' and 'cytoplasm' time scales. Error-free data could again beresolved using any of the sampling protocols in spite of the factthat two wall half-times had elapsed by the point of first samplingwith protocol 2. Imperfect data were correctly resolved to fourcomponents in 10 to 14 runs, and half-times reasonably close tothe actual values were found five times. Table V summarizes theresults of the 10 simulations which converged on four compo-nents. This table also indicates the extent of the nonbiologicalvariability which might be expected in repeated analyses. It wasinteresting that between the failed imperfect and the perfect data

I-

1009

Table IV. Summary ofAnalyses ofSimulated 4-Component Data Sets Generated with PG-C (Table I), Indicating Nonbiological Variability whichMight Be Expected in Actual Experiments

Estimates ± approximate SD are shown for the 6 of 8 simulations in which 4 components were resolved. The asterisk (*) denotes the only analysiscorrectly estimating the half-times. The final line gives the mean, SD of the mean, and coefficient of variation for the combined analyses.

Sampling Protocols th, th2 th3 th4 Notesa

min1 1.6±0.13 22.5±9.5 109.4± 130 2433±28 (2)

0.87 ± 0.04 16.3 ± 20.0 40.0 ± 16 2546 ± 35 (2)2* 0.93 ± 0.03 20.9 ± 6.7 61.0 ± 27 2593 ± 50 (1)

0.91 ± 0.03 23.2 ± 3.6 97.3 ± 70 2320 ± 110 (1)3 0.42 ± 0.14 1.49 ± 0.49 31.1 ± 2.5 2280 ± 26 (3)

0.76 ± 0.08 11.4 ± 6.9 57.3 ± 26.0 2331 ± 110 (2)Summary 0.92 ± 0.39 (42%) 16.0 ± 8.4 (52%) 66.0 ± 31.2 (47%) 2417 ± 129 (5%)

a (1) 'Best' solution. (2) A 3-component solution was a 'significant possibility.' (3) A 5-component solution was a 'significant possibility.'

Table V. Summary ofEstimated th Valuesfor the Simulated4-Component Data Sets Generated Using PG-D

Values are means ± SD of the means for the 10 of 14 trials in which4 components were identified. 'High' and 'low' denote the trials with thelongest and shortest estimates of th2 and t,3.Component ti, Coefficient of High Low

Variation

min % min min1 0.846 ± 0.376 45 1.17 0.352 5.19 ± 3.00 58 9.25 2.523 61.9 ± 41.7 67 107.7 30.94 2472 ± 202 8 2699 2393

150

100[t3

50

o

-50

5 10 15 20 25 30 35Somple Number

0 2 5 10 20 60 180 480Time (min)

FIG. 3. Residual plots generated by the nonlinear regression program,DISCRETE, for the 'best' solution in an unsuccessful resolution ofPG-E. Two components (th, = 0.17 min; th = 2165 min) were identified.Though incorrect resolution is indicated by the distinct pattern of theresiduals, the results are insufficient to improve that resolution. 'Sam-pling' was simulated with protocol 3.

sets, the maximum relative difference between the error-free andimperfect values at any time from 0 to 540 min was 1.1% (datanot shown).

Finally, PG-E was designed as the most strenuous of the tests.The most rapidly decaying component was very large, with ahalf-time of the same magnitude as the simulated timing inac-curacies. Estimates of contents at t = 0 reflected this; the coeffl-

cient of variation in 10 simulations was 25%. The two fastestcomponents were also separated only by a factor of five, andneither was longer than the time to first sampling in protocol 1

or 2. As might be expected, therefore, neither protocol 1 or 2could resolve the components even using error-free data. How-ever, the large wall component relative to the second and thirdcomponents was not the source of the resolution problems;reductions of the size ofthe wall component were without effect.Sampling protocol 3 resolved the components of error-free

data, but in 10 imperfect simulations, four components werecorrectly resolved only once. In the other nine trials, the degreeof failure ranged from identification ofone to three components.The presence of a distinct pattern in the residual plots (e.g. Fig.3) suggested the presence of more components in four cases, butcould not be used to resolve them. Again reduction of Q,° didnot alter the rate of successful convergence, nor did alteration oftotal sampling time.

CONCLUSIONS

In this paper, we have simulated results of compartmentalefflux experiments and analyzed the data statistically using com-puterized nonlinear regression techniques. Our objective was toevaluate the limitations of the experimental and analytical tech-niques; we have specifically not questioned the validity of thecompartmental model itself or the possible interpretations of theanalyses. Thus, if we accept the underlying mathematical as-sumption that the components are functionally in series, ourresults indicate several points of procedural interest with regardto the effects of loading times, sampling protocols, and totalefflux period duration on resolution of the efflux parameters.The results also indicate that simulation is a potentially usefultool in the planning of actual compartmental efflux analyses.

Perhaps the most important result here is the finding that awell-performed experiment may not give, and in some cases maybe unable to give, the right answer (i.e. to identify the numberand values of the decay parameters correctly), even if the as-sumptions themselves are valid. Thus the usual conclusion thatthe results are consistent with, or can be fit to, a three-compart-ment model may be coincidence; it may not actually be supportfor such a model. Though this does not indicate that compart-mental efflux analysis has no use, it does imply that it shouldnot be used alone. At the least, its integration with influx andpredictive simulation methods should be considered.

Finally, with potential difficulties ofthis type in a simple seriesmodel, it should be reemphasized that this assumption itself isby no means assuredly valid. Greater, physiologically significant,complexity would arise from the presence of parallel compart-ments such as plastids, a heterogeneous cell population, or a

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1010 CHEESEMAN Plant Physiol. Vol. 80, 1986

COMPARTMENTAL EFFLUX ANALYSIS

shoot. Elucidation of that complexity and its functional impor-tance in transport clearly warrants further study.

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11. RYGIEWICZ PT, CS BLEDSOE 1984 Mycorrhizal effects on potassium fluxes byNorthwest coniferous seedlings. Plant Physiol 76: 918-923

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