Phloem Mobility of Xenobiotics - Plant · PDF filetrapping in the basic phloem is not the only...

Plant Physiol. (1988) 86, 803-8100032-0889/88/86/0803/08/$01 .00/0

Phloem Mobility of XenobioticsI. MATHEMATICAL MODEL UNIFYING THE WEAK ACID AND INTERMEDIATE PERMEABILITY

THEORIES

Received for publication June 16, 1987 and in revised form October 30, 1987

DANIEL A. KLEIERE. I. Dupont de Nemours and Company, Experimental Station, Agricultural Products Department,Wilmington, Delaware 19898

ABSTRACT

A passive diffusion model has been developed which simultaneouslyaccounts for the dependence of phloem mobility on permeability and aciddissociation. The model is consistent with the observation that the additionof an acid moiety to an otherwise phloem immobile compound may en-hance that compound's ability to move in the phloem. However, acidtrapping in the basic phloem is not the only enhancement factor. Acidfunctionalization also lowers the effective permeability usually towards itsoptimum value. The unified theory predicts that for a given acid disso-ciation constant there is an optimum permeability and conversely for agiven permeability there is an optimum dissociation constant.

According to one theory, the phloem mobility of a compounddepends upon the presence of a weak acid functionality withinthat compound (2). Another theory deals with compounds whichare nonelectrolytes at physiological pH and ascribes efficientphloem mobility to compounds whose membrane permeabilitiesfall within some optimum range (6, 8). The optimum range isdetermined by attributes such as plant length, leaf size, phloemsap velocity, etc. The latter theory has been developed into amathematical model (8). Neither of these theories invokes acarrier mechanism and the two theories are not necessarily inopposition. This paper describes an extension of the mathemat-ical model (8) that unifies the two theories. The extended modelexplains in a straightforward fashion the enhanced phloem mo-bility of weakly acidic compounds without invoking a carriermechanism. While the model provides for acid trapping withinthe sieve tubes, the mobility of weak acids is in large part dueto their intermediate permeability.

THEORY

The model used is an extension of that of Tyree et al. (8). Itconsists of a linear plant of length LI (Fig. 1). A xenobiotic isassumed to have been applied over a length l* of the leaf whoselength is 1. The length of the leaf, petiole, and stem is 0.5L,while the root system accounts for the other half of the plant'slength. Phloem sap is assumed to flow through a sieve tube ofradius r. The velocity of the phloem sap is assumed to rise linearlyfrom near zero at the leaf tip until it reaches a maximum valueof v -which is maintained throughout the petiole and stem. Xe-nobiotic is assumed to enter the sieve tubes in the leaf zone bypassive diffusion driven by a concentration gradient. For an acidic

I A complete list of abbreviations and symbols appears in Table I.

Table I. Abbreviations and Symbols Used in the Text

L: Length of plant in m1: Length of leaf in m1*: Length of leaf in m over which compound is ap-

pliedr: Radius of sieve tube in mv: Maximum velocity of phloem sap in m/ss: Distance from leaf tip in mHA: An undissociated acidA -: Conjugate base of HA[HA]L: Concentration of HA (M) in apoplast[HA],: Concentration of HA (M) within sieve tube at a

distance s from leaf tip[A - l Concentration of A - (M) in apoplast[A -],: Concentration of A - (M) within sieve tube at a

distance s from leaf tipC,.O: Total concentration of xenobiotic (M) in leaf apo-

plast; total of [HA]L and [A -

C,(s): Concentration of xenobiotic (M) within the sievetube at a distance s from the leaf tip; total of[HA], and [A -]

Cf: Concentration factor (unitless); the ratio of C,(0.9 L) to C,,0

[H +- i: Hydrogen ion concentration (M) within sieve tube[H + ]o: Hydrogen ion concentration (M) within apoplastPHA: Permeability of HA in m/sPA: Permeability of A- in m/sKa: Acid dissociation constant of HA (M)pKa: Log (1/Ka)KO,,: Octanol-water partition coefficientnHA. Number of moles of HAnA : Number of moles of A-Q: Area of cylindrical sieve tube element (m2)v': Flow velocity (m/s) of phloem sap at a distance s

from leaf tip

substance, HA, the total concentration of xenobiotic in the leafapoplast, C,O, is defined as the sum of the concentrations of HAand A- where A- is the conjugate base of HA. All concentra-tions may be assumed to be in mol/L. Analogously, C,(s) isdefined as the total concentration of xenobiotic at point s in theleaf sieve tube. Generally, C,,O will exceed C,(s) for s < 1, thatis within the leaf. Hence, within the leaf portion of the plantthere is a net flow of xenobiotic into the sieve tube.

In the stem and petiole, the concentration of xenobiotic in theapoplast surrounding the sieve tube (including the xylem) is as-sumed to be zero. Hence, some of the xenobiotic leaks into theapoplast as the phloem sap carries the remaining portion towardsthe root. The xenobiotic which leaks into the apoplast is assumed

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Plant Physiol. Vol. 86, 1988

B. VELOCITY PROFILE:

0.0*.I

--------------- ; ------ 0.5 L

/---- 1.0 L

Vebocity

FIG. 1. Definition of plant parameters. A, plant dimensions. L is thetotal length of the plant. The length of the leaf, petiole and stem takentogether is 0.5 L. The length of the root is 0.5 L; I is the length of theleaf. The radius of the sieve tube is r. B, Velocity profile assumed forthis model. The velocity rises linearly from near zero at the leaf tip toa value of v at the base of the leaf. The phloem sap velocity is maintainedat the constant value of v throughout the petiole and stem. At the baseof the stem the sap velocity begins to fall linearly towards a value of nearzero at the bottom of the root.

to return quickly to the leaf through the transpiration stream.As the mathematical elaboration of the model will confirm (Ap-pendix A), what is needed to concentrate the xenobiotic in theroot zone is a compound that is permeable enough to penetrateinto the sieve tubes within the leaf zone but is not so permeableas to leak back into the apoplast during its passage to the rootzone. For an acidic substance, both the neutral and anionic spe-cies are assumed capable of permeating the sieve tube mem-brane, but the neutral species is assumed to have a permeabilitythat is approximately 4 orders of magnitude greater than that ofthe anionic species. This assumption follows from the observationthat the ionized form of a carboxylic acid lowers the octanol-water partition coefficient by 3.7 orders of magnitude relative tothe undissociated form (3). Acid-base equilibrium is assumed tobe maintained at all times and at all points in both the apoplastand in the phloem sap within the sieve tubes. Furthermore, thefluids in the apoplast and phloem are assumed to be buffered.Once in the root zone, the phloem velocity is assumed to decreaselinearly with distance until reaching a value of nearly zero at theroot tip.Using the above assumptions, differential equations governing

the rate of flow of xenobiotic through sieve tube elements of theleaf, stem, and root are derived. A steady state is then assumedso that the amount of xenobiotic entering a sieve tube elementin a given period of time is just balanced by the amount leaving.The resultant steady state equations for both the neutral andanionic species are similar in foirm to those derived by Tyree etal. (8). They are thus solved in an analogous fashion. Details ofthe derivation are given in Appendix A. If the concentrationfactor Cf is defined as the ratio of total concentrations, C,(0.9L)ICr., where C,(0.9L) is the total concentration of xenobiotic withinthe phloem sap at a point deep in the root zone and C,,O is thetotal concentration at the point of application in the leaf apoplast,then the following equation may be derived:

Cf = C,(0.9L)/C,O= {([H+], + Ka)/([HsI]o + Ka)} (1)

[ (a)([H IOPHA + PAKa) 1[H+]l(PHA + b) + Ka(PA + b)J

x exp{-c([H+]iPHA + PAKa)I([H+]i + Ka)}

In this formula, Ka is the acid dissociation constant, [H+]i and[H+]o are the hydrogen ion concentrations inside and outsidethe sieve tube respectively, PHA and PA are the permeabilitiesof the undissociated acid and conjugate base respectively, in m/s, and a, b, and c are parameters which characterize the linearplant:

a = 5.0 1*/ib = rvl2lc = 2.609L - 2 1(1 - ln(lll*))/rv

Here L is the length of the plant, I is the length of the leaf, l*is the length of the leaf over which xenobiotic has been applied,r is the sieve tube radius (all in m), and v is phloem sap velocity(m/s) in the petiole and stem.We note that Eq. 1 passes smoothly into equation 14 of Tyree

et al. (8) in the limits when Ka = 0 and when Ka is very large.In the former case the permeability appearing in the limitingequation is that of the undissociated species HA, and in the lattercase the permeability appearing in the limiting equation is thatof the anion, A-. Also observe that Eq. 1 passes smoothly intoEq. 14 of Tyree et al. when [H+] = [H+]i = [H+]0. In thisinstance the permeability appearing in the limiting equation is

Pave = ([H+IPHA + KaPA)I([H+I + Ka)which is the average permeability of HA and A - at the prevailingpH.

Since experimental values for permeabilities are not readilyavailable, an empirical relationship was sought between perme-ability P and the more easily measured or estimated octanol-water partition coefficient K,,. The octanol-water partition coef-ficient of a substance is the concentration of the substance in theoctanol phase of a two phase octanol-water system divided bythe concentration in the water. The following linear relationshipbetween permeability and octanol-water partition coefficient wasderived from data found in the literature (1) for the permeationof 70 compounds through Nitella cell membranes:

log P = 1.20 log(K0w) - 5.85 (2)r = 0.85

In this equation P has the units m/s and r is the correlationcoefficient for the least squares fit. Using Eq. 2 the expressionfor the concentration factor Cfwas recast in terms of the log(K0w)of the xenobiotic.The coefficients on the right hand side of Eq. 2 may be con-

sidered as plant parameters. These coefficients will depend uponthe thickness and viscosity of the sieve tube membranes, andultimately upon the chemical composition of the membranes.Plant cell membranes may differ from Nitella membranes in asignificant way. In fact, a relationship consistent with tightermembranes has been determined for potato tuber discs (5). Thus,the permeability values predicted by Eq. 2 may in fact be toohigh. This point is worthy of further investigation.

RESULTS

Phloem sap is maintained at a pH more basic than that of thesurrounding apoplast. Thus, in order to model the phloem mo-bility of acids, we have taken the pH of the phloem sap to be8.0 and that of the apoplast to be 6.0 (9). For our standard longplant we have taken a length of 1.0 m, a leaf length of 0.05 m,

A. PLANT DIMENSIONS:

Keaf

I10.5 L

0.5 L

Petlboe &Stem

Root-tApoplaisl~ 1Sieve Tube

(Radkje r)

804 KLEIER

I 1.



PHLOEM MOBILITY OF XENOBIOTICS: MATHEMATICAL MODEL

a sieve tube radius of 5.0,um, and a phloem transport velocity0.1 mm/s. These parameters are characteristic of potato plants(8). The xenobiotic was assumed to be spread over the entireleaf (1 = l* = 0.05 m).The calculated log Cf values as a function of pKa and log(K0w)

are provided in Table II and plotted in Figures 2 and 3. It is tobe noted that we usually take log(K0w) for the undissociated acid(HA) to be 3.7 greater than that of the conjugate base (A -).For a given pKa, Figure 2 demonstrates that these is an optimumlog(K0w) which represents a balance between the high permea-bility required for the xenobiotic to permeate the sieve tubemembranes within the leaf and the low permeability required forthe compound to remain largely entrapped on its path to theroot zone. The improvement in phloem mobility to be expectedfrom acid functionalization is evident when the curves labeledpKa = 14 and pKa = 4 are compared.

Just as there is an optimum log(K0w) for a given pKa, there isan optimum pKa for a given log(K0w). An oily acid with a highpKa favors rapid entry into the phloem within the source leafbut makes for a leaky passage through the stem and petiole. Alow pKa enhances the ability of the xenobiotic to remain in thephloem en route to the root zone but slows entry into the phloemat the source. Again, the optimum pKa represents a balancebetween these opposing considerations.From this model it is clear (Fig. 3) that a xenobiotic which is

both nonacidic and highly permeable (high log[K0w]) representsa very poor candidate for phloem mobility. Such a substancewould fall into the deep well of the depicted surface. It wouldreadily enter the sieve tubes of the treated leaf, but would justas readily be lost from the symplast as the compound moveddown through the petiole and stem. Such compounds have beentermed pseudoapoplastic (7). However, one way of drawing acompound out of the well is to functionalize it with an acidicsubstituent. This will have the effect of simultaneously loweringthe permeability of the parent and decreasing the pKa, bothchanges tending to move the compound up and perhaps over thelip of the abyss. Thus, the weak acid theory and the intermediatepermeability theory of phloem mobility are seen to be incor-porated into a single model. It should be noted, however, thatthe model, while incorporating an acid trapping factor in thepresence of a pH differential, predicts improved mobility for anacid functionalized compound even when the pH differential iszero (Table III). Thus, acid trapping usually improves the mo-bility of weakly acidic compounds, but much of the improvementis simply due to reduction of the effective permeability upon acidfunctionalization.

In order to provide a feeling for the effect of plant parameterson phloem translocation, we have also calculated the phloemmobility of a series of compounds as a function of pKa andlog(K0w) for a shorter plant (Table IV). Our standard short plantis identical to the standard long plant except that its length is 15cm and the sap velocity is taken as 0.3 mm/s. As expected, thecalculated Cf values are generally much larger for the short plantthan for the long plant. This is due to the much shorter transitdistance through regions of the plant where the xenobiotic canleak back into the surrounding apoplast.As specific examples, we note that benzoic acid with a pKa of

4.2 and a log(K0w) of 1.87 should fall near the maximum on theright forward edge of the surface depicted in Figure 3. On theother hand, a nonacidic compound such as N-methylpyridiniumchloride with a pKa nominally of 14 and a log(K0w) of -3.3should fall near the maximum on the back edge of the surfacedepicted in Figure 3. Modest structural modifications of theseparent compounds may have significant effects on phloem mo-bility. In order to test the model, we have performed a systematicstudy of the effect of structural modifications of benzoic acidsand N-alkylpyridiniums on phloem mobility. In the case of the

benzoic acids we have attempted to vary the pKa while main-taining a roughly constant log(K0w). In the case of the N-alkyl-pyridiniums we have varied the length of the alkyl chain in orderto study the effect of changing the log(K0w) while maintaining aconstantpKa. The results of these systematic studies are reportedin a companion paper (4).

APPENDIX

Mathematical Elaboration of the Model. Consider a cylindricalsieve tube element of length ds and radius r located at a distances from the leaf tip (Fig. 4). Let nHA and nA- represent the numberof moles of undissociated acid HA and conjugate base A-, re-

spectively. The steady state condition requires that the numberof moles of a given species entering the sieve tube element equalthe number of moles leaving so that

dnHAldt = dnA, dt = 0 (Al)for every element along the length of the plant. For each suchelement there are three distinguishable surfaces through whichxenobiotic may enter or exit. If phloem sap flow is from top tobottom (Fig. 4), xenobiotic may enter through the top circularface and exit through the bottom circular face. Additionally, itmay permeate through the cylindrically shaped plasma mem-brane separating the apoplast from the living tissue of the sievetubes.

Consider first the permeation of the membrane. We assumethat once a given amount of either HA or A - penetrates thelipid membrane, dissociation or association occurs instanta-neously to the degree required to maintain the prevailing equi-librium in its environment (apoplast or sieve tube).Permeation of neutral HA takes place under the influence of

a concentration gradient, ([HA]0 - [HA],), across the plasmamembrane. Similarly, permeation ofA - is assumed to take placeunder the influence of a concentration gradient ([A - ], - [A - ]U.[HA]L and [A - ] are concentrations at point s in the sieve tube,while [HA], and [A -], are concentrations in the surroundingapoplast. The concentrations [HA], and [A -] are assumed tobe uniform within the application zone (i.e. within a distance l*from the tip of the leaf). [HA], and [A -], are approximated aszero below the application zone. The rate of permeation of HAthrough the membrane is assumed to be proportional to theproduct of the concentration difference and the surface area Qof the cylinder surrounding the element:

(dflHAldt)perm = PHAQ([HA]o - [HA]s) (A2)The constant of proportionality PHA is by definition the perme-ability of the membrane for the undissociated species HA andhas the dimension of velocity. A similar equation can be writtenfor the anion A- with a permeability constant PA. We usuallytake the permeability constant of the anion to be 3 to 4 ordersof magnitude less than that of the undissociated conjugate acid.We do not explicitly take into account the effect of electric fieldson the permeation of the charged species.

In order to maintain acid-base equilibrium within that portionof the sieve tube located in the leaf, only a certain fraction ofthe xenobiotic transported across the membrane into the sievetube as HA can remain associated there and only a certain frac-tion of the xenobiotic transported as A- can remain dissociatedupon arrival. Similarly, in those regions of the plant below thepoint of application where the sieve tube is expected to leakxenobiotic, only a certain fraction of the xenobiotic transportedacross the membrane as HA can originate from undissociatedHA, the remainder must arise from A- which associates im-mediately before entering the membrane, and only a certainfraction of the xenobiotic transported across the membrane asA- can originate from A-, the remainder must arise from un-dissociated HA which dissociates immediately before entering.

805



806 KLEIER Plant Physiol. Vol. 86, 1988

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Log Cf -1.

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log KowFIG. 2. Plot of the log Cf versus log Ko for two values of the pK,,. Cf is the concentration factor (see text for definition). High values of C,

imply good mobility. K,,, is the octanol-water partition coefficient of the undissociated acid and K& is the acid dissociation constant. These curveshave been determined for a standard long plant with a length of 1.0 m, leaf length of 0.05 m, and phloem sap velocity of 0.0001 m/s. The pH ofthe phloem sap was taken as 8.0 and that of the surrounding tissue as 6.0.

Since the phloem is assumed to be buffered, these fractions aredetermined by the requirement that [HA]J[A -] = [H + Is/Karemain unchanged after transport of an aliquot of xenobiotic.Thus, the effective rate of transport of neutral HA due to per-meation of both HA and A- is given by

(dnHAldt)perm-

PHAQ([HAI, - [HA],) x {[HA],/([HA], + [A- ])}+ PAQ([A -] - [A ]-) x {[HA],/([HA]. + [A -D]}

The first term accounts for the rate change of nHA due to per-meation of HA while the second term accounts for the ratechange of nHA due to permeation of A -. Within the applicationzone the factor in braces in the first term represents the fractionof HA which remains associated after permeation into the sievetube and the factor in braces in the second term represents thefraction of A- which associates after permeation into the sievetube. Below the application zone the factor in braces in the firstterm represents the fraction of outward permeating HA whichoriginates from associated HA and the factor in braces in thesecond term represents the fraction of outward permeating Awhich originates from associated HA.Use of the equilibrium relationship, Ka = [H+],[A ]-1[HA]S

- [H+][A-]j[HA],, division by the volume of the sieve tube

element, V = lrr2ds, and use of the expression Q = 2irrds, forthe surface area of the cylinder surrounding the element thenyields the following equation for the permeation rate:

(d[HA]S/dt)Perm = {(PHA + (PAKaI[H+IO))[HAIO-(PHA + (PAKI[H + ]))[HA]S} (A3)

x (2/r)Ifl + (KJ[H+],)}The rate change in concentration due to entry of xenobiotic

from the top face of the element and exit from the bottom faceis given by:

d[HA]0Idt= -v5(d[HAI)ds) - [HA],(dv5/ds) (A4)Here V, is the flow velocity of the phloem sap at point s in

the sieve tube. Combining Eqs. A3 and A4 and using the steadystate assumption d[HA]sldt = 0 yields the following differentialequation:

{(PHA + (PAKI[H+]O))[HA]O-(PHA + (PAKa/[H+Is))[HA]S}x (21r)/{1 + (Ka/[H+]s)}-vs(d[HAl0Ids) -[HA]s(dvslds) = 0

(A5)Multiplying Eq. A5 by minus one and using the following symbols

Ki = 1/{1 + KaI[H +Is}Po= PHA + (PAK./[H ]0)Ps PHA + (PAKaI[H ]s)

807



Plant Physiol. Vol. 86, 1988

Eq. A6 is the working differential equation of this model. So-lution of this equation following Tyree yields the followingexpression for the concentration of HA at point s = 0.9L whichis 90% of the length of the plant from the leaf tip:

[HA]0 9L1[HA]I - {aKjP,PJ(K,P, + b)} x exp(-cKjPj) (A7)where a, b, and c are defined in the Theory section above.The equilibrium condition requires [A- 109L = Ka[HA](.9L/

[H+10.9L. Thus,

[A- 10 9L1[HA]I - {Ka/[H + 1(09L} X {aK,P0/(K,P, + b)}x exp( - cKiP) (A8)

If we define the concentration factor Cf as the ratio of xenobioticin the sieve tube at s = 0.9L to that of the xenobiotic in theapoplast of the leaf, we obtain

Ct = ([HA](.9L + [A ](0.9L)/([HA], + [A -],)= {[H+].J([H+], + K,)j} (A9)

x {([HA]J0 L, + [A-- 10.9L),[HA],,lwhere we have used [A -] = Ka[HA]o/[H+1o. Substituting Eqs.A7 and A8 into Eq. A9 and replacing Ki, P, and P, with theirdefinitions gives Eq. 1.

Acknowledgments-The author is grateful to Drs. Francis Hsu. James Sanborn.James Steffens, Paul Porter. and John Carr for their encouragement and supportfor this project. The author is also indebted to Mark Lipton and James Hegedusfor programming and graphics support. I would also like to thank Drs. R. H.Bromilow and B. T. Grayson for useful comments on this manuscript.

Sieve Tube

Leaf Tip

FIG. 3. Plot of log Cf versus log K0,o and pK,,. This surface has beencalculated for a standard long plant with a length of 1.0 m, leaf lengthof 0.05 m, and phloem sap velocity of 0.0001 m/s. The pH of the phloemsap was taken as 8.0 and that of the surrounding tissue as 6.0. For thisplot the permeability of the conjugate base A- was assumed to be zero.

K,,, is the partition coefficient for the undissociated acid. The surface isreminiscent of a waterfall viewed from above and to the side. Nonacidiccompounds (follow curve on back edge for pKa = 1 1) with log K0,, valuesclose to - 3 are near the lip of the falls. Increasing log K,,,, beyond - 3.0sends the predicted log Cf value plunging over the lip of the falls andinto the abyss of neoapoplastic compounds. The arrows point to the endsof a curve on the surface with a constant log K,,. value of - 2.0. Movingfrom the front edge (pK, = 1.0) to the back edge (pKa = 11) along thiscurve involves slowly climbing to higher log C1 values, reaching the lipat a pKa, value of approximately 6.0, then plunging into the abyss forpK& values greater than 6.0. Following this curve in the direction de-scribed is equivalent to following the row of Table II labeled log KO,,.- 2 from right to left.

yields an equation very similar to Eq. 1 of Tyree et al. (8):

(2K,/r)(PJHA], - Po[HA]o) + v,(d[HA1jds)+ [HA](dv,/ds) = 0 (A6)

1%

n dS

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FIG. 4. A sieve tube element within the leaf. The sieve tube elementhas a length ds and is located at a distance s from the leaf tip. Xenobioticwhich has been applied to the leaf is carried into the sieve tube elementby mass flow of the sap from above and by permeation through thecylindrical walls. Xenobiotic leaves the sieve tube element from thebottom by mass flow. In the steady state the amount entering the sievetube from above by mass flow and through the walls by permeation isequal to the amount leaving the sieve tube from below by mass flow.

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Plant Physiol. Vol. 86. 1988

LITERATURE CITED

1. COLLANDER R 1954 The Permeability of Nitella Cells to Non-electrolytes. Phvs-iol Planta 7: 420-445

2. CRISP CE 1972 Insecticides, In AH Tahori, ed, Proceedings of the 2nd Inter-national IUPAC Congress on Pesticide Chemistry, Vol 1. Tel Aviv, Israel,pp 211-264

3. HANSCH C, A LEO 1979 Substitution Constants for Correlation Analysis inChemistry and Biology. John Wiley and Sons, New York

4. Hsu FC, DA KLEIER, WR MELANDER 1987 Phloem mobility of xenobiotics 11.Bioassay testing of the unified mathematical model. Plant Physiol 86: (XX)-000

5. LICHTNER FT 1986 Phloem transport of agricultural chemicals. Itn J Cronshaw.

WJ Lucas, RT Giaquinta, eds. Phloem Transport, Alan R. Liss. Ness York.pp 601-608

6. PETERSON CA, PPQ DEWILDT, LV EDGINGTON 1978 A rationale for the am-bimobile translocation of the nematicide oxamyl in plants. Pestic BiochemPhysiol 8: 1-9

7. PETERSON CA, LV EDGINGTON 1976 Entry of pesticides into the plant symplastas measured by their loss from an ambient solution. Pestic Sci 7: 483-491

8. TYREE MT, CA PETERSON, LV EDGINGTON 1979 A simple theory regardingambimobility of xenobiotics with special reference to the nematicide, oxamyl.Plant Physiol 63: 367-374

9. ZIEGLER H 1975 Nature of transported substances. It MH Zimmerman, JAMilburn, eds, Encyclopedia of Plant Physiology, New Scries Vol I, PhloemTransport. Springer-Verlag, New York, p 92

810 KLEIER



Correction

Vol. 86: 803-810, 1988

Daniel A. Kleier. Phloem Mobility of Xenobiotics. L. Mathematical Model Unifying theWeak Acid and Intermediate Permeability Theories.

This is a corrected reprint of the above-referenced article, paginated as in the original, andwhich replaces it. The original printing contains errors in Tables III and IV that wereintroduced after the author approved the galleys.

Libraries in particular please note: This corrected version of Kleier's paper shouldbe bound in place of the original when volume 86 is sent for binding.

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