The Excitability of Plant Cells: with a Special Emphasis on Characean Internodal Cells Author(S): Randy Wayne Source: Botanical Review, Vol

The Excitability of Plant Cells: With a Special Emphasis on Characean Internodal Cells Author(s): Randy Wayne Source: Botanical Review, Vol. 60, No. 3, The Excitability of Plant Cells: With a Special Emphasis on Characean Internodal Cells (Jul. - Sep., 1994), pp. 265-367 Published by: Springer on behalf of New York Botanical Garden Press Stable URL: http://www.jstor.org/stable/4354233 Accessed: 12-11-2015 14:20 UTC

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This content downloaded from 128.84.125.94 on Thu, 12 Nov 2015 14:20:46 UTC All use subject to JSTOR Terms and Conditions THE BOTANICAL REVIEW VOL. 60 JULY-SEPTEMBER1994 NO. 3

The Excitability of Plant Cells: With a Special Emphasis on CharaceanInternodal Cells

RANDY WAYNE

Sectionof PlantBiology CornellUniversity Ithaca,New York14853, USA

I. Abstract, Ubersicht, Japanese Abstract, Resumen ...... 266 II. Introduction:Plants Sense Environmental Cues and Act Appropriately ...... 268 III. Recording the Voltage Changes During an Action Potential ...... 277 IV. The Ionic Basis of the Resting Potential ...... 281 V. The Ionic Basis of the Action Potential ...... 287 VI. Chloride Efflux Occurs in Response to Stimulation ...... 289 VII. Application of a Voltage Clamp to Relate Ions to Specific Currents ...... 292 VIII. The Action Potential at the Vacuolar Membrane ...... 299 IX. The Plasma Membrane Action Potential ...... 301 X. Identification of the Ca2,-Activated Cl- Channel on the Plasma Membrane Using Patch Clamp Techniques ...... 303 XI. Summary ...... 307 XII. Coda ...... 308 XIII. Acknowledgments ...... 314 XIV. LiteratureCited ...... 314 XV. Appendix A: List of Abbreviations ...... 335 XVI. Appendix B: Information Theory, Thermodynamics, and Cell Communication .. . 337 XVII. Appendix C: Membrane Capacitance ...... 340 XVIII. Appendix D: Derivations of the Nernst Equation ...... 344 XIX. Appendix E: Derivation of the Goldman-Hodgkin-Katz Equation ...... 356 XX. Appendix F: Derivation of the Hodgkin Equation That Relates Conductance to Flux ...... 362

'Dedicated to Professor Noburo Kamiya on the occasion of his eightieth birthday.

Copies of this issue [60(3)] may be purchased from the Scientific Publications Department, The New York Botanical Garden, Bronx, NY 10458-5125, USA. Please inquire as to prices.

TheBotanical Review 60(3): 265-367, July-September, 1994 C 1994 The New YorkBotanical Garden `65

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Now in the furtherdevelopment of science,we wantmore than just a formula.First we have an observation,then we have numbersthat we measure,then we have a law whichsummarizes all the numbers.But the realglory of scienceis thatwe canfind a way of thinkingsuch that the law is evident. -Richard Feynman(in Feynmanet al., 1963)

Mathematiciansmay flatterthemselves that they possess new ideaswhich mere humanlanguage is as yet unableto express.Let themmake the effortto express these ideasin appropriatewords without the aid of symbols,and if they succeedthey will not only lay us laymenunder a lastingobligation, but, we ventureto say, they will find themselvesvery much enlightened during the process,and will even be doubtfulwhether the ideasexpressed in symbolshad ever quite found their way out of the equationsinto theirminds. -James ClerkMaxwell (cited in Agin, 1972)

I. Abstract This review describesthe basic principlesof electrophysiologyusing the generation of an actionpotential in characeaninternodal cells as a pedagogicaltool. Electrophys- iology has proven to be a powerful tool in understandinganimal physiology and development,yet it has been virtuallyneglected in the study of plant physiology and development. This review is, in essence, a written account of my personaljourney over the past five years to understandthe basic principlesof electrophysiologyso that I can apply them to the study of plant physiology and development. My formal backgroundis in classical botany and cell biology. I have learned electrophysiology by readingmany books on physics writtenfor the lay person and by talking informallywith many patientbiophysicists. I have writtenthis review for the botanistwho is unfamiliarwith the basics of membranebiology but would like to know that she or he can become familiar with the latest informationwithout much effort. I also wrote it for the neurophysiologistwho is proficientin membranebiology but knows little about plant biology (but may want to teach one lecture on "plant action potentials"). And lastly, I wrote this for people interested in the history of science and how the studies of electrical and chemical communicationin physiology and developmentprogressed in the botanicaland zoological disciplines.

Ubersicht Dieser Uberblickbeschreibt die Grundprinzipender Electrophysiologieunter Ver- wendung eines Aktionspotentialsin internodalenZellen der characean Algen als padagogeshes Mittel. Die Elektrophysiologie hat sich beim Verstandnis von Tierphysiologie und entwicklung als wirksames Mittel erwiesen; dennoch ist sie bisher beim Studium der Pflanzenphysiologie und entwicklung praktish vernachldssigtworden. Dieser Uberblickist im Wesentlichenein schriftlichenBericht meiner personlichenBemuhungen in den letzten funf Jahrendie Grungprinzipender Electrophysiologiezu verstehen,um sie auf das Studiumder Pflanzenphysiologie und entwicklunganzuwenden. Meine Spezialitatist klassische Botanikund Zellbiologie. Elektrophysiologiehabe ich durch das Lesen von vielen Physikbuchernfur den Laien erlerut;daneben hatte ich auch die Gelengheit, informell mit vielen geduldigen Biophysikern daruberzu sprechen. Ich habe diesen Uberblick fur den Botaniker geschreiben, der mit den Grundlagender Membranbiologienicht vertrautist, der aber wissen mochte, dass

This content downloaded from 128.84.125.94 on Thu, 12 Nov 2015 14:20:46 UTC All use subject to JSTOR Terms and Conditions EXCITABILITYOF PLANT CELLS 267 er/sie ohne allerzu grossen Aufwand mit dem neuesten Stand der Wissenschaft vertrautwerden kann. Ich habe ihn weiterhinfuir den Neurophysiologengeschrieben, der mit der Membranphysiologie wohl vertraut ist, der aber wenig uiber Pflanzenbiologie weiss (aber velleicht eine Vorlesung fiber "Pflanzenaktionspotentials"halten mochte). Schlieslich habe ich ihn auch fuirden Lesergeschrieben, der sich furdie Wissenschaftsgeschichteinteressiert und dafuir, wie das Studiumder Elektrischen und chemischen Kommunikation in derPhysiologie und Entwicklungin den botanischenund zoologischen Disziplinen fortgeschrittenist.

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Iz-t>u Z L, -C 0 t I ArL Att -Et\ L rCI& I 4 $b DtH ,- t untestimoniode mi e n personalkduate l tzltimo c lios 4los que he etudiode fisiogI detsarl de,C,M plana s. n, tA cD3WTf -C -7 ti -E tE 6Dg iz A e 4t; tzb), ,8cM 6 b 2b Z X ,t I 1 -C Lttf Iti;d 6 -C 67D50 el estudioZItde lafisioogia de"dsaro de pantas.D Este estudioe, esenciamete Resumen

Este estudio analiza los principios basicos deeyectrofisiologia a utilizando la generacio'n de un potencial de accio'n en ce'lulas internodiales de algas characean como herramientapedagobgica. Electrofisiologia ha demostradoser un excelente mestodo parael entendimientode la fisiologdiay deldesanrollo animalpero ha sido ignoradaen el estudio de la fisiologisay deldesacollo de plantas.Este estudio es, esencialmente, un testimonio de mi experiencia personal durante los ulItimoscinco anios en los que he estado estudiando los principios ba'sicos de electrofisiologi'a para poder aplicarlos al estudio de fisiologlia y desaffollo de plantas. Mis estudios se concentraron en bota'nica cla'sica y biologi'a celular. He aprendido electrofisiologi'a a base de leer muchos libros de ffsica escritos para gente sin gran conocimiento de la materia, adema's he tenido el placer de hablar informalmente con muchos bioffsicos de gran paciencia. He escrito este estudio para el bota'nico que

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desconoce la base de biologia membranalpero al que le gustariasaber que el o ella puede familiarizarsecon la informacionma's actual sin mucho esfuerzo. Tambienlo escribiparael neurofisiologo,experto en biologiamembranal,pero que no sabe mucho sobre biologia de plantas (y que desee enseiiar una clase sobre los "potenciales de accion de las plantas"). Y finalmente,escribi esto paragente interesadaen la historia de la ciencia y en como los estudios de comunicacion electrica y quimica en la fisiologia y el desarrolloprogresaron en las disciplinas zoologicas y botanicas.

II. Introduction: Plants Sense EnvironmentalCues and Act Appropriately Every day we can observe the fact that plants respond to the environment.For example, plants respondto gravity. "Rememberthe little seed in the styrofoamcup: The roots go down andthe plantsgo up" (Fulghum,1989). Plantsalso respondto light and temperature.In a perennialgarden, each variety flowers at a certain time of the year because the plantsare capableof identifyingthe season by detectingboth the day length and the temperature.Plants can also perceive the time of day, and many plants have "sleep movements" where they change the orientationof their leaves so that they maximize the absorptionof light duringthe day and minimize the loss of heat at night. Plantsadjust their height andleaf shapedepending on whetherthey aregrowing in the sun or shade. In this case, they detect the differencein the qualityof light (i.e., color distribution)that occurs in full sun comparedto shade.Plants can determinethe directionof light, and they bend towardthe light in a way thatpresents the maximum surface area to the sun to maximize photosynthesis.After sensing the various envi- ronmentalstimuli, the sensor region must transmita signal to the respondingregion. In theory, this signal can be chemical (e.g., hormonal),physical (e.g., electrical or hydraulic),or both (Kudoyarovaet al., 1990; Malone, 1993; AppendixB). It has been known since Charles Darwin's time that many plants, like animals, respondto mechanicalstimulation (Bernard, 1974; Darwin, 1893, 1966;Davies, 1987, 1993a, 1993b;Galston, 1994;Jaffe, 1973, 1976;Jaffe &Galston, 1968;Pickard, 1973; Sibaoka, 1966, 1969; Simons, 1981, 1992; Smith, 1788; Thomson, 1932; Wayne, 1993). Insectivorousplants thatlive in nitrogen-and mineral-depletedareas get their required nitrogen and minerals by capturing and digesting insects. The sundew immobilizes an insect in its mucilage. Subsequently, the mechanical stimulation inducedby the insect tryingto escape is sensed by the tip of the tentacle.The sensitive tip thenproduces a series of actionpotentials, that is, pulse-like,propagating electrical disturbancesthat can communicateinformation (Fig. 1). In the case of the sundew, the action potentialsinduce the marginaltentacles to bend, pushingthe insect toward

Fig. 1. Action potentialsof variousorganisms. A. Action potentialsof Mimosa (the sensitive plant) and Dionaea (Venus fly trap)(redrawn from Sibaoka, 1966). B. Action potentialof a squid giant axon (redrawn from Hodgkin & Katz, 1949). Note the differencesin the ordinatesand abscissae in A and B. C. Receptor potentials of the sensory (trigger)hair of Dionaea inducedby a mechanicalforce (redrawnfrom Benolken & Jacobson, 1970). The "force" in these experiments is given in kilograms. This is incorrect:Since kilograms are a unit of mass, force should be given in newtons. Force (F, in N) can be obtainedfrom the mass (m, in kg), accordingto Newton's Second Law (F = ma), by multiplyingthe mass by its acceleration (a, in m s-2). However, Benolkin and Jacobsonneither estimatednor measuredthe accelerationin these experiments.Thus, we cannottransform the "force" given in kilogramsinto properunits. In any case, the first stimulatingforce (F) was greaterthan threshold and induced an actionpotential; the second stimulating

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T'ime, a force was below the thresholdand the resultingmembrane depolarization was smaller than the first. The rate of change in the force as well as the magnitudewas furtherreduced in the thirdstimulating force, and the change in membranepotential was even smaller than the first or second. Thus, a graded response occurredin responseto subthresholdstimuli. When the stimulusis subthreshold,a receptorpotential appears as an electrical replica of the stimulus.

This content downloaded from 128.84.125.94 on Thu, 12 Nov 2015 14:20:46 UTC All use subject to JSTOR Terms and Conditions 270 THE BOTANICAL REVIEW leaf's center (Williams & Spanswick, 1976). Although action potentials are not transmittedto neighboringtentacles, a slower hormonalsignal induces them to wrap around the insect and provide a secure trap (Williams & Pickard, 1972a, 1972b). Nearby secretorycells subsequentlyexude digestive enzymes "to the little stomach thus formed, and the plant makes a meal of its prey" (Pickard,1973). The lobes of a Venus fly trapcapture its prey by closing when an insect walking across it strikestwo hairs or one hair twice. This "primitive memory" ensures that the trap closes only aroundlive prey. The trapcloses only afterthe stimulatedsensory hairs communicate to the motorcells by means of an actionpotential (Benolken & Jacobson,1970; Brown & Sharp, 1910; Burdon-Sanderson,1873; Burdon-Sanderson & Page, 1876; Di Palma etal., 1961;Hodick& Sievers, 1989; Jacobson, 1965; Stuhlman&Darden, 1950; Williams & Bennett,1982). John Burdon-Sanderson, an uncle of J. B. S. Haldane,first studied the movementsand electrical properties of the Venusfly trapfor CharlesDarwin (Gillespie, 1970, 1972). Upon closure,the gland cells secreteenzymes thateffect digestionof the insect and uptakeof nutrients(Rea & Whatley, 1983; Rea et al., 1983; Robins, 1976; Robins& Juniper,1980; Scala et al., 1968;Schwab et al., 1969). The leaves of many sensitive plants, including Mimosa, fall and look dead and unappealingto a would-be herbivoreonce the grazinganimal touches it (Aimi, 1963; Allen, 1969; Setty & Jaffe, 1972; Tinz-Fuichtmeier& Gradmann,1991; Toriyama, 1954, 1955, 1957, 1967; Toriyama& Jaffe, 1972; Toriyama& Sato, 1968; Turnquist et al., 1993). Leafletcollapse may act as a defense mechanismsince the collapse results in an increasedexposure of sharpthorns (Eisner, 1981). Touching induces an action potential thatpropagates from the stimulatedregion to the rest of the plant (Sibaoka, 1962). This propagated"information" causes the rest of the plantto look unappealing also and to "deploy their thorns." It is also possible that mechanicalstimulation by the wind induces an action potentialthat causes the leaflets to collapse. This reduces the exposed surfacefrom which moisturemay be lost (D. F. Grether,pers. comm.). Plantsthat do not show obvious movementsalso elicit actionpotentials (Paszewski & Zawadzki, 1973; Scott, 1962). In Luffa,action potentials cause a transientinhibition of growth (Shiina & Tazawa, 1986a). Action potentialsmay be involved in phloem unloading and, consequently, the distributionof nutrientsin plants (Eschrichet al., 1988; Fromm, 1991). Action potentialspropagated through the phloem may also be involved in pathogenresistance (Wildon et al., 1992). In variousflowers, pollen falling on the stigma generates an action potential that may be involved in subsequent pollination, incompatibility, or maturationprocesses (Goldsmith & Hafenrichter, 1932; Sinyukhin& Britikov, 1967). At the level of stimulus-responsecoupling, plants and animal cells undergo many similar processes, including the generation and propagationof actionpotentials and perhaps stimulation by neurotransmitters,including acetylcholine (Dettbarn,1962; Fluck & Jaffe, 1974; Tretyn & Kendrick, 1991). Includingstudies on plantexcitability with studies of animalexcitability may help us understandthe evolution of our own nervoussystem (Hille, 1984, 1992). In orderto understandthe physiological basis of the complex behaviorsof animals, neurobiologiststook a reductionistapproach and began studyingthe long nerve cells of squids which are so large they were originally thoughtto be blood vessels (Cole, 1968; Eckert et al., 1988; Hodgkin, 1964; Junge, 1981; Katz, 1966; Lakshmi- narayanaiah,1969; Matthews, 1986; Stevens, 1966; Tasaki, 1968; Young, 1936, 1947). Studying simple systems allows the development of techniques and the formulationof theoriesnecessary for understandingmore complex systems. Likewise,

This content downloaded from 128.84.125.94 on Thu, 12 Nov 2015 14:20:46 UTC All use subject to JSTOR Terms and Conditions EXCITABILITYOF PLANT CELLS 271 in orderto understandthe complex behaviorsof whole plants,some plantphysiologists have turnedto the giant algal cells, such as Chara and Nitella, where vegetative and reproductivegrowth and development,adaptive physiological mechanisms, and nu- trientuptake and translocation can be readilystudied (Amino & Tazawa, 1989; Andjus & Vucelic, 1990; Barber,191 la, 191lb; Beilby et al., 1993; Bisson & Bartholomew, 1984; Bisson et al., 1991; Blatt & Kuo, 1976; M. Brooks, 1939; S. Brooks, 1939; Brooks & Brooks, 1941; Brown, 1938; Buchen et al., 1991, 1993; Coleman, 1986; Collander, 1949, 1954; Cote et al., 1987; Dainty, 1962, 1963a, 1963b; Dainty & Ginzburg, 1964a, 1964b, 1964c; Dainty & Hope, 1959; Diamond & Solomon, 1959; Ding et al., 1991, 1992; Dorn & Weisenseel, 1984; Forsberg, 1965; Gertel & Green, 1977; Green, 1954, 1958a, 1958b, 1959, 1960, 1962, 1963, 1964, 1968; Green & Chapman, 1955; Green & Chen, 1960; Gillet et al., 1989, 1992, 1994; Green et al., 1971; Hansen, 1990; Hejnowicz et al., 1985; Hoagland& Broyer, 1942; Hoagland& Davis, 1923a, 1923b; Hoaglandet al., 1926; Hodge et al., 1956; Hogg et al., 1968a, 1968b; Homble & Foissner, 1993; Homble et al., 1989; Hotchkiss & Brown, 1987; Kamiya& Kuroda,1956a; Kamiya & Tazawa,1956; Katsuharaetal.,1990; Katsuhara & Tazawa, 1992; Kersey et al., 1976; Kishimoto, 1957; Kiss & Staehelin, 1993; Kitasato, 1968; Kiyosawa, 1993; Kwiatkowska, 1988, 1991; Kwiatkowska& Mas- zewski, 1985, 1986; Kwiatkowskaetal., 1990;Levy, 1991;Lucas, 1975,1976, 1977, 1979, 1982; Lucas & Alexander, 1981; Lucas & Dainty, 1977a, 1977b; Lucas & Shimmen,1981; Lucas & Smith, 1973;Lucas et al., 1978, 1989;MacRobbie & Dainty, 1958; MacRobbie & Fensom, 1969; Maszewski & Kolodziejczyk, 1991; McCon- naughey, 1991; McConnaughey& Falk, 1991; McCurdy& Harmon,1992; Metraux, 1982; Metraux & Taiz, 1977, 1978, 1979; Metrauxet al., 1980; Miller & Sanders, 1987; Mimura& Kirino, 1984; Moestrup,1970; Morrisonet al., 1993; Mullins, 1939; Nagai & Hayama, 1979; Nagai & Kishimoto, 1964; Nagai & Rebhun, 1966; Nagai & Tazawa, 1962; Nothnagel & Webb, 1979; Nothnagelet al., 1982; Ogata, 1983; Ogata & Kishimoto, 1976; Ohkawa& Kishimoto, 1977; Ohkawa& Tsutsui, 1988; Okazaki & Tazawa, 1990; Okazakiet al., 1987; Osterhaut,1927, 1931, 1958; Osterhaut& Hill, 1930a, 1930b; Palevitz & Hepler, 1974, 1975; Pickard, 1969, 1972; Pickett-Heaps, 1967a, 1967b, 1968; Ping et al., 1990; Pottosinet al., 1993; Probine& Barber, 1966; Probine & Preston, 1961, 1962; Reid & Overall, 1992; Reid & Smith, 1988, 1993; Reid et al., 1989; Rethy, 1968; Richmondet al., 1980; Rudingeret al., 1992; Sakano & Tazawa, 1986; Shen, 1966, 1967; Shepherd& Goodwin, 1992a, 1992b; Shimmen & Mimura,1993; Shimmen& Yoshida, 1993;Sievers & Volkmann,1979; Spanswick, 1970; Staves et al., 1992; Steudle & Tyerman,1983; Steudle & Zimmermann,1974; Studener, 1947; Sun et al., 1988; Takamatsuet al., 1993; Takatori& Imahori, 1971; Taylor & Whitaker,1927; Tazawa, 1957, 1964; Tazawa& Kishimoto, 1964; Tazawa et al., 1994; Thiel et al., 1990; Tolbert& Zill, 1954; Tronteljet al., 1994; Tsutsui & Ohkawa, 1993; Turner,1968, 1970; Volkmannet al., 1991; Vorobiev, 1967; Walker & Sanders, 1991; Wasteneys & Williamson, 1992; Wasteneyset al., 1993; Wayne & Tazawa, 1988, 1990; Wayne et al., 1990, 1992, 1994; Weidmann, 1949a, 1949b; Weisenseel & Ruppert, 1977; Yao et al., 1992; Zanello & Barrantes, 1992, 1994; Zhang et al., 1989; Zhu & Boyer, 1992). The characean algae, which include Chara and Nitella, are thought to be the ancestorsof all higherplants (Chapman& Buchheim, 1991; Graham,1993; Graham & Kaneko, 1991; Grambast,1974; Groves & Bullock-Webster,1920, 1924; Imahori, 1954;Manhart & Palmer,1990; Pal et al., 1962;Pickett-Heaps & Marchant,1972; Wilcox

This content downloaded from 128.84.125.94 on Thu, 12 Nov 2015 14:20:46 UTC All use subject to JSTOR Terms and Conditions 272 THE BOTANICAL REVIEW et al., 1993; Wood & Imahori,1964,1965). Indeed,Pliny the Youngerconsidered the charophytesto be membersof the genus Equisetum(Plinius Secundus, 1469). The study of excitability in plant cells forms the foundation of the study of excitabilityin isolatedcells. Althoughexcitability in tissues andorgans of whole plants and animals were well known since the early work of Luigi Galvani, the electrical phenomena were too complicated to allow much progress (Green, 1953). A giant breakthroughcame when electricalmeasurements were made on single cells. Indeed, action potentials were observed in the isolated internodalcells of Nitella in 1898 by Georg Hormann,using extracellularelectrodes, approximately30 years before they were observed in isolated nerve cells by Adrian and Bronk (1928). In fact, action potentials(also known as negativevariations, action currents, and death currents) were so well studiedin characeancells by Blinks (1936), Osterhaut(1931), and Osterhaut's colleagues at the Rockefeller Institute,that Cole and his colleagues at the National Institutesof Healthbegan studyingexcitability in characeancells. It may be surprising to know that Cole and Curtis wrote the following in their 1939 paper "Electrical impedanceof the squid giant axon duringactivity":

Recently the transverse alternating current impedance of Nitella has been measuredduring the passage of an impulsewhich originated several centimeters away (Cole and Curtis,1938b). Thesemeasurements showed that the membrane capacity decreased 15 per cent or less while the membraneconductance increased to about 200 times its restingvalue. Also this conductanceincrease and the membraneelectromotive force decrease occurred at nearly the same time, which was late in the rising phase of the monophasicaction potential. Similar measurementshave now been made on Young'sgiant nervefiber preparation from the squid (Young,1936). Thesewere undertakenfirst,to determinewhether or not afunctional nervepropagates an impulse in a mannersimilar to Nitella, and second, because the microscopic structureof the squid axon corresponds considerably better than that of Nitella to the postulates upon which the measurementsare interpreted.

Characeanaction potentialscan be inducedby various stimuli, including suddenly lowered or raisedtemperature or pressure,UV irradiation,and odorants,as well as by a mechanicalstimulation or a depolarizingcurrent (Harvey, 1942a, 1942b;Hill, 1935; Kishimoto, 1968; Osterhaut& Hill, 1935; Staves & Wayne, 1993; Ueda et al., 1975a, 1975b; Zimmermann& Beckers, 1978). As in animalcells, the action potentialis an all-or-noneresponse and does not usually depend on the strengthof the stimulus. In characeancells thereis a refractoryperiod following an actionpotential, during which a stimulus cannot cause a second action potential. Details of the characeanaction potentialhave been reviewed by Kishimoto(1972), Hope and Walker(1975), Beilby (1984), Tazawa et al. (1987), and Tazawa and Shimmen(1987). The action potential is conductedin both directionsaway from the place at which the plasma membraneis first depolarizedpast the threshold.The conductionrate is 0.01-0.4 m s-l (Auger, 1933; Sibaoka, 1958; Umrath, 1933), depending on the conductivity of the medium (Sibaoka, 1958). This is faster than the conduction velocity in sponges (0.003 m g-1;Mackie et al., 1983) and slower thanthe conduction velocity of action potentialsin nerves, which is between 0.4 and 42 m s-l (Eckertet al., 1988; Matthews, 1986).

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The propagationvelocity of an actionpotential along a cell dependson how rapidly the depolarization(i.e., the decreasein membranepotential) diminishes with distance. The membranewill become depolarizednext to the initial site of depolarization,and the potential will increase(i.e., hyperpolarize)with distanceaway from the initial site of depolarizationuntil the membranepotential is equal to the resting potential. This is known as electrotonictransmission. An action potentialcan be propagatedonly if sufficient currenttravels down the cell to depolarizethe adjacentmembrane below the thresholdin membranepotential. A new action potential will begin only if the membranepotential of the adjacentmembrane is depolarizedpast the threshold. The ionic currentresponsible for depolarizingthe membraneand propagatingthe actionpotential can travelthrough the cytoplasmor acrossthe plasmamembrane. The currentmoves along a potentialdifference in a mannerdescribed by Ohm's Law:

I = -EIR' = -EG (1) where I is currentdensity (in A m-2) E is potential (in V) R' is the specific resistance(in Q m2) G is the specific conductance(in siemens per m2 (S m-2) or Q-1 m-2).

The magnitudeof depolarizationof the adjacentmembrane depends on the amount of currenttraveling through the cytoplasm, which in turn depends on the relative resistances of the cytoplasm and the membrane.In orderto select for the movement of the ionic currentthrough the cell from one end to the other,instead of throughthe membraneand outside the cell (which would negatethe depolarizingeffect of the ionic current),the resistanceof the cytoplasmmust be low and the resistanceof the plasma membranemust be high (like an insulatedwire). Thus,fast actionpotentials propagate along nerve cells whose cytoplasmicresistances have been lowered by making their diameterslarge and whose plasmamembrane resistance has been increasedby surround- ing the axon with a high-resistancemyelin sheath.Thus, the differencein propagation velocity betweenvarious excitable cells dependson theirstructure. As in excitablecells in animals,the actionpotential in characeancells is transmittedfrom cell to cell (Sibaoka & Tabata,1981; Spanswick1974b; Spanswick & Costerton,1967; Tabata, 1990). Characeaninternodal cells exhibit a responseto electricalstimulation that is similar to the contractionresponse skeletal muscles displayfollowing an electricalstimulation by nervecells. In both cases, the responseis nicknamedE-C coupling.In animalcells, it refers to excitation-contractioncoupling, where electricalexcitation causes muscle contraction(Ebashi, 1976). In characeancells, E-C coupling refers to the fact that electrical stimulationcauses the cessation of protoplasmicstreaming (Findlay, 1959; Hill, 1941; Hbrmann,1898; Kishimoto& Akabori, 1959). The protoplasmof characeancells constantlymoves aroundthe peripheryof the cell like a rotatingbelt at a rateof 104 m s-1, in a processknown as protoplasmicstreaming (Allen & Allen, 1978; Corti,1774; Ewart, 1903; Goppert & Cohn, 1849;Kamiya, 1959, 1981;Kamiya & Kuroda,1956b; Shimmen, 1988; Shimmen & Yokota,1994). Protoplas- mic streaming,which is drivenby the same interactionsbetween actin and myosin that cause contractionin muscles, allows the mixing and transportof essential molecules throughthe protoplasmof large cells where diffusion would be limiting (Fig. 2).

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Vm ..:: -. En Ac , _

Fig. 2. Diagramof characeaninternodal cell. The cell is essentiallya rightcircular cylinder with a volume equal to (nr2l)] and a surface areaequal to 2itrl + 2irr2.The length (1,in m) of a "typical" cell dependson the species but may vary from 10-2 m in Nitella to 3 x 10-1 m in Nitellopsis. The diameter(2r, in m) varies from 0.3 x I0-3 m in Nitella to l0-3 m in Chara.C, chloroplasts;V, vacuole; Ac, actin bundles;Pm, plasma membrane; En, flowing endoplasm; Ex, stationary ectoplasm; W, extracellular matrix (= cell wall). (Redrawnfrom Tazawaet al., 1987.)

The time it takes for the averageparticle to diffuse a given distance througha cell can be calculatedusing Einstein's randomwalk equation(Berg, 1993; Einstein, 1926; Perrin, 1923; Pickard,1974):

t = x21(2D) (2) where t is the time (in s) x is the average distancethe particlesmove (in m) D is the diffusion coefficient (in m2 s-l).

Note that equation2 describes the simple case where diffusion is restrictedto one dimension. Diffusion can also be described in two dimensions (x and y) where t = r2l(4D) and r2 = x2 + y2. In threedimensions (x, y, and z), t = r21(6D)and r2 = x2 + y2 + z2 (Berg, 1993). According to the above equation,the time it takes for a particleto diffuse a given distance is governed only by its diffusion coefficient. The diffusion coefficient of a sphericalparticle in solution depends on the thermalmotion of the particle;consequently, the diffusioncoefficient is proportionalto temperature.Boltzmann's Constant (k, in J K-1) is the coefficientof proportionalitythat relates the energyof the particleto the temperature.The kineticenergy [(1/2)mv2]of a particlemoving in threedimensions is a consequenceof the thermalmotion of the particleand is equalto (312)kTwhere Tis the absolutetemperature (in K). The velocity of a particleof mass mj (in kg) is equalto (3kTlmj)112.If we considerthat the particle is largecompared with water, then we can use the equationsof fluid mechanicsto describethe movementof the particlethrough an aqueousenvironment. Thus, the movement of the particle depends on the hydrody-

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namic radius of the particle (rH,in m) and the viscosity of the solution (f1, in Pa s). Formally, the diffusion coefficient D is given by the Stokes-Einsteinequation:

D = kT/(6JtrHrl) (3)

where D is the diffusion coefficient (in m2 s-1) k is Boltzmann's Constant(1.38 x 10-23J K-1) T is the absolutetemperature (in K) rHis the hydrodynamicradius (in m) ri is the viscosity of the solution (in Pa s).

If we assume that the viscosity of the fluid phase of the cytoplasm is 0.004 Pa s (Luby-Phelpset al., 1986) andthat most small metaboliteshave ahydrodynamicradius between 10-10and 10-9 m, then most diffusion coefficients will be between 5 x 10-11 and 5 x 10-10m2 s-I at 298 K. Thus, a "typical" metabolitewill take between 25 and 250 ms to diffuse all the way across a 5 x 10-6 m cell, but it will take between 103 and 104s (2.8 h) to diffuseacross a 10-3m diam.characean internodal cell and 107and 108 s (4 months)to diffusealong the long axis of a 10 x 10-2m long characeaninternodal cell. Diffusionis thuslimiting in largecells (i.e., slowerthan the rates of enzymaticreactions), and mixing by protoplasmicstreaming (i.e., convection)becomes necessary.In general, the largerthe cell, the moreorganized the protoplasmicstreaming. In the naturalworld, cessation of streamingresults not from electricalexcitation but from the mechanicalstimulation of internodalcells by a predatoror otherobjects such as falling sticks. This mechanicalstimulation generates a depolarizationof the plasma membrane that is known as a receptorpotential (Kishimoto, 1968). The receptor potential is an amplificationmechanism whereby the minuscule mechanicalenergy of the tactile stimulus is transducedby the receptor of the stimulus (in this case a mechanoreceptor)into electrical energy that is amplifiedin proportionto the magnitude of the initial stimulus.(Energy is not createdfrom this amplificationprocess but is releasedin the formof ionic currentsfrom the electricalenergy already stored across the cell membranein the formof a membranepotential.) The receptorpotential usually lasts as long as the stimulus is present and can be consideredan electrical replica of the stimulus. However, the stimulus must be given rapidly enough or the receptor shows a slow decrease in its sensitivity to the stimulus, known as accommodation (Kishimoto, 1968; Staves & Wayne, 1993). The actin- or tubulin-basedcytoskeleton appearsnot to be involved in the functionof this mechanoreceptor(Staves & Wayne, 1993); however, the involvement of a membraneskeleton (Faraday& Spanswick, 1993) has not been ruled out. The receptorpotential is not self-perpetuating,and thus the depolarizationdue to the receptorpotential decreases with distance away from the activatedreceptor. [The theoreticaldescription of how potentialsand currentsspread through cells, known as "cable theory,"was originally workedout by Lord Kelvin (1855; Wheatstone, 1855) to describe the electrical propertiesof the transatlantictelegraph cable. He showed thatthe potentialdrops linearly along an insulatedcable placed between a batteryand groundjust like the temperaturedrops across a substanceplaced between a heat source and a heat sink.] If the stimulusis greatenough andfast enoughto cause the membrane potential to depolarize below a certain threshold level, an action potential will be

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Id A AL~~~~

Id ~~~~~~B

PM ?

Id.C

VM D);4G () =

Fig. 3. Diagramsof the cable propertiesof a nerve cell (or othernonvacuolated cell), a myelinatednerve cell, and a characeaninternodal cell (or other vacuolatedcell). A. In a nerve cell, the depolarizingcurrent (Id)crosses the plasma membrane(PM) andtravels through the cytoplasmand back out throughthe plasma membrane.The amountof currentthat moves throughthe cytoplasm and thus is capable of depolarizing adjacentmembrane areas depends on the relative resistance of the cytoplasm and the plasma membrane. The greater the ratio of membrane resistance to cytoplasmic resistance, the greater the amount of depolarizingcurrent that travels through the cytoplasm.The cytoplasmicresistance is inverselyproportional to the radiusof the axon. If the cytoplasmicdepolarizing current is large,then the electrotonictransmission will depolarize an adjacentmembrane area enough to surpassa thresholdof a voltage-gatedchannel and initiate an action potential in the adjacent membranepatch. B. In a myelinated nerve, the membrane resistanceis increasedby tightly wrappingSchwann cells aroundthe axon. Schwanncells have membranes that are almost purely lipid, so the axon becomes analogous to a wire wrappedwith an insulator.The Schwann cells are discontinuousat points known as the nodes of Ranvier,where the plasma membraneof the axon is in communicationwith the ions in the extemal medium and thus, at these points, a sufficient depolarizingcurrent traveling through the cytoplasmwill initiate an action potential.C. In a Chara cell, a depolarizing current can follow many pathways, including the plasma membrane,the cytoplasm, the vacuolar membrane(VM), and the vacuole. A characeaninternodal cell is thus analogous to an insulated wire within an insulated wire. This makes the pathway of currentmore difficult to calculate than for (vacuole-less) nerve cells. Note: In all the cases mentionedabove, the depolarizingcurrent will travel in all directionsso thatthe currentin any directionwill be inverselyproportional to the resistancein thatdirection. elicited (Fig. 3). An actionpotential is a largetransient depolarization of the membrane potentialthat is self-perpetuating(i.e., regenerative) and thus allows the rapid transmission of informationover long distanceswithout any loss of signalstrength (information). Thus, receptorpotentialsare useful for signal transduction within a single small cell, andaction

This content downloaded from 128.84.125.94 on Thu, 12 Nov 2015 14:20:46 UTC All use subject to JSTOR Terms and Conditions EXCITABILITY OF PLANT CELLS 277 potentials are necessary for signal transmissionin a giant cell or between cells. The action potential generatedin one area of a characeancell is responsible for causing the cessation of protoplasmicstreaming throughout the cell (Sibaoka& Oda, 1956). When streaming stops, the outer layer of the flowing protoplasmgels (i.e., forms cross-bridgeswith the actincables) andprevents any leakageof protoplasmthat may occur as a resultof a small-animalattack (Kamitsubo et al., 1989; Kamitsubo& Kikuyama,1992). Moreover, the stimulatedcell may becomeisolated from the neighbor- ing cells since the intercellularpassage of substancesthrough small tubes known as plasmodesmatais also inhibited.The cessationof streaminginhibits transport through the plasmodesmataindirectly since, in the absenceof streaming,unstirred layers form around the openingsof the plasmodesmataand arrival of substancesat thepore becomes diffusion limited(Ding & Tazawa,1989). The cessationof cytoplasmicstreaming has no effect on plasmodesmataltransport in smallcells wherediffusion is notlimiting (Tucker, 1987).

III. Recording the Voltage ChangesDuring an Action Potential

The action potential in characeaninternodal cells can be observed using simple electrophysiologicaltechniques (Fig. 4; Brooks & Gelfan, 1928; Umrath,1930, 1932, 1934, 1940). According to Alan Hodgkin (1951),

The most satisfactory way of recording the electrical activity of a living cell membraneis to inserta small electrodeinto the interiorof the cell and to measure the potential (i.e voltage) of this electrode with referenceto a second electrode in the external medium. This method was first employed in plant cells (see Osterhaut,1931) and has now been widely used in animal tissues.

When a glass microcapillaryelectrode is placed into the vacuole and the reference electrode is placed in the externalmedium, an action potentialis observed following stimulation.The action potential appearsto be composed of two components:a fast and a slow one (Fig. 5; Findlay, 1970; Findlay & Hope, 1964a; Kishimoto, 1959; Shimmen & Nishikawa, 1988). If we inserttwo microcapillaryelectrodes in the cell, we can resolve thatthese two componentsare the resultof two actionpotentials; a fast one that occurs across the plasma membraneand a slow one that occurs across the vacuolarmembrane (Findlay & Hope, 1964a). The averageresting potentials across the plasma membraneand vacuolarmembrane are -0.180 V and -0.010 V, respectively. [Note: In both cases the potential on the exoplasmic side of the membrane (E-space) is considered to be zero by convention and the protoplasmicside of the membrane(P-space), i.e., the side that is facing the cytosol, is negative.] During an actionpotential, the plasmamembrane depolarizes to approximately0 V and the vacuolarmembrane hyperpolarizes to approximately-0.050 V. The changes in the membranepotential, accordingto the Goldman-Hodgkin-KatzEquation, are due to the ionic currentsthat flow as a consequence of a change in the membrane permeabilityof each ion. The change in membranepermeability that occurs duringan actionpotential can be visualized by measuring the specific conductance of the membrane.The specific conductanceis an electricalmeasure of the overallmembrane permeability to all ions. The specific conductancesof the plasma membraneand vacuolarmembrane at rest are 0.83 siemens m2 (S m-2) and 9.1 S m-2, respectively.The specific conductances

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B Am oEv Em

Evc

Fig. 4. Diagramof setup used to measuremembrane potential. A. A characeaninternodal cell is placed in an acrylic block in which three holes (0.4-1 cm diam.) have been drilled. The cell parts situatedin a groove in the region between the threewells is coated with silicone grease or petrolatum(Vaseline) so that the threewells areelectrically isolated andcurrent can only travelacross the plasmamembrane and through the cells). A referenceelectrode (containingca. 100 mol m-3 KCl and 1-2% agar) is placed in the middle chamberand a microcapillaryelectrode (filled with 3000 mol m-3 KCl) is placed in the cytoplasm of the cell part in the middle chamber.The electricalpotential across the plasma membraneis measuredwith an electrometer.The cell is electrically stimulatedby applyinga depolarizingcurrent to the middle chamber. The currenttravels throughthe cell in all directions and is returnedto the other terminal of the current generatorby way of the wires placed in the two end chambers.Similarly, in orderto measure membrane resistance,current pulses (squarewaves of approximately1 07 A, 1 Hz) are appliedto the middle chamber. The currenttravels from the currentgenerator through the cell in all directionsand is returnedto the other terminalof the currentgenerator by way of the wires placedin the two end chambers.B. In orderto measure both the plasma membranepotential and the vacuolarmembrane potential, two microcapillaryelectrodes are placed in the cell part in the middle chamber.One electrode is placed in the vacuole and the other is placed in the cytoplasm. Thus, the electricalpotential across the plasma membrane(Eco) can be measured by comparingthe voltage measuredwith the electrodein the cytoplasmwith the referenceelectrode in the bath(0 V by convention).The electricalpotential of the vacuolarmembrane (EVC) is measuredby comparing the voltage measuredby the electrodes placed in the vacuole and cytoplasm. Using the "patchclamping convention,"the voltage of the electrodein the vacuole (or E-space)is consideredto be zero andthe potential of the cytoplasmicside of the vacuolarmembrane is negative. Note thatin earlierwork, the conventionwas to consider the cytoplasmic side (P-space) of the vacuolar membraneto be zero, and consequently the vacuolar membranepotential on the vacuolarside was positive. The sum of the potentialsof the plasma membraneand vacuolar membranes(Evo) can be determinedsimultaneously by comparingthe voltage between the electrode in the vacuole with the reference electrode. Remember that the polarity of the membranepotentials across the vacuolarand plasma membrane are oppositelydirected, the sum of the two potentials is less hyperpolarizedthan the potential across the plasma membranealone. (Redrawnfrom Shimmen & Nishikawa, 1988.) change with time during an action potential (Fig. 6). The peak conductancesduring the action potential of the plasma membraneand vacuolarmembrane are 30 S m-2 and 15 S m-2, respectively(Cole & Curtis,1938; Findlay& Hope, 1964a;Oda, 1962). In the squid giant axon, the specific conductanceof the plasma membraneincreases from 10 S m-2 to 400 S m-2 duringan actionpotential (Cole & Curtis, 1939). In both Nitella and squid,the specific conductanceof the plasmamembrane increases approximately 40 times duringan action potential. While the specific conductanceis an importantmeasure of membranepermeability, it does not tell us which ions are permeatingthe membraneand carryingthe current

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0.05 A > 0.00-' n

0 ai.

0.15- e 0.05 \

-0.20- I li - l l l l l I E E~~~~~~~~~~~~~0.000 1 2 3 4 5 6 7 8 9 10 Time,a .0

0.05- 0 -0.105 (

-0.15-: C~~~~~~~~~~~~ -0.201 f 0 1 2 3 4 S 6 7 8 9 10 a.

0.00-

> -0.022-

e -0.04- / Time, a -0.06--

:-0.08-.

-0.t0- 0 1 2 3 4 5 6 7 8 9 1 0 Tima, s

Fig. 5. Voltage-time graphsduring an actionpotential in characeanintemodal cells. An action potential was initiatedelectrically at 1 second. A. The time course of membranepotential changes across both the plasma membraneand vacuolarmembrane (Evo). B. The time courseof membranepotential changes across the plasma membraneonly (Ec,). C. The time course of membranepotential changes across the vacuolar membraneonly (EVC).(Redrawn from Findlay & Hope, 1964a.)

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0.05 50 AI >0.00 - ; 4 E

-0.05 30 0 a . ? 20 -Q.10 l l - 0 .0 0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~. E -0.15 10

-0.20 0 0.0 0.5 1.0 1.5 2.0

6 0.056 a I potentialwa ntitdeecrcly t0tme.Teiecureopeii emrnonutn E (0

..~ 0.004 0~~~~~~~~~~~~~~ 0~ Time, a C C0 20 -0.05 2 c. Eo

Ecos-0.10th0lsammrn G.Tetm oreo h lsammrn oeta hne r oie o 0 ~~~5 10 15 20 25 Time. a

Fig. 6. Graphsof conductancechanges duringan actionpotential in characeaninternodal cells. An action potentialwas initiatedelectrically at 0 time. A. The time courseof specific membraneconductance changes across the plasmamembrane (G). The time course of the plasmamembrane potential changes are given for comparison (E). B. The time course of specific membraneconductance changes across the vacuolar membrane(G). The time course of the vacuolarmembrane potential changes are given for comparison(E). In these cases, the specific membraneconductances were determinedfrom voltage clampingexperiments. Note the differenttime scales in the two graphs.(Redrawn from Findlay & Hope, 1964a.) that leads to the action potential.Next, we will determinewhich ions are responsible for carryingthe currentsthat cause the membranevoltage changes that characterize the action potential.

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IV. The Ionic Basis of the Resting Potential Before we can understandthe ionic basis of the action potential, we must first understandthe ionic basis of the restingpotential and ask, How do the ion concentrations in the cell and extracellularmilieu affect membranepotential? The electrical potential across a membraneis determinedto a large extent by the difference in ion concentrationsacross the membrane.Therefore, it is importantto know the concentrationsof the majorions on each side of a membrane.The concentrationsof C1-,Na+, K+, and Ca2+in the externalmedium, protoplasm, and cell sap of characeancells as reportedby Lunevsky et al. (1983) are given in Table I. As a comparison,the concentrations of theseions in theprotoplasm and the extracellular milieuof a squidaxon are also given (Eckertet al., 1988).When we comparethe characean internodalcell to the squidaxon, we see firstthat the plantcell has threecompartments thatbathe two excitablemembranes, whereas the animalcell has only two compartments thatbathe one excitablemembrane. This very fact makes the axona muchsimpler system to studysince it is geometricallysimilar to the transatlantictelegraph cable, thus making cable theoryeasily appliedto the analysisof the results.This is the reasonthat Kenneth Cole andHoward Curtis (1939) beganstudying the giantsquid axon rather than continue with characeaninternodal cells. Secondly, we see that the Na+ concentrationin the externalmedium is drasticallydifferent in the two cell types. The difference in the concentrationof each ion across a differentiallypermeable membrane creates the driving force for diffusion across the membrane. As each permeantcation diffuses acrossthe membrane,it leaves behinda negativecharge (e.g., protein with ionized carboxyl groups) that will tend to attractthe cation back to the original side. Thus, thereis a chemicalforce thatcauses the cationsto move from high concentrationsto low concentrationsand at the same time sets up an electricalforce thattends to move the cations back.Because the membraneis permeableto some ions (e.g., K+) and impermeableto others (e.g., proteins), at equilibriumthere is no net movement of charge and the chemical force equals the electricalforce (see Appendix D). There is thus a stable electrical potential that results from the initial uneven distributionof ions, as was shown by LeonorMichaelis (1925) using driedcollodion and apple skins as "the prototypeof what we will call a semipermeablemembrane." The unequal distributionof ions thus gives rise to a transmembranepotential difference that is electrically equivalentto a battery. The unequaldistribution of chargeis only possible becausethe nonconductinglipid bilayer that separatestwo conductingaqueous solutions acts as a capacitor, that is, a component that has the capacity to separatecharges (Appendix C). Otherwise,the membrane potential difference (i.e., the battery) would eventually run down. A capacitoris a componentthat resists changes in voltage when currentflows through it (much like a pH bufferresists changes in pH). The specific membranecapacitance is a measure of how many charges (ions) must be transferredfrom one side of the membraneto the otherside eitherto set up or to dissipatea given membranepotential. The specific membranecapacitance of all membranesis approximately0.01 F m-2 (Cole, 1970; Cole & Cole, 1936a, 1936b; Cole & Curtis, 1938; Curtis& Cole, 1937; Dean et al., 1940; Findlay, 1970; Fricke, 1925;Fricke & Morse, 1925; Williams et al., 1964). Using this value, it is possible to calculate the minimumnumber of ions that must diffuse across the membranein order to establish or dissipate a membrane potential using the following relation:

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Table I The ionic compositionof characeaninternodal cells and squid nerves (concentrationin mol m-3)

Characeaninternodal cell Squidnerve

External Protoplasm Vacuole External Protoplasm

Cl- 0.4 22 162 560 40-100 Na+ 0.1 5 34 440 49 K+ 0.1 110 103 22 410 Ca2+ 0.1 <0.001 12 10 <0.001

Note:For intact cells the extracellular milieu is easyto controlthe protoplasmic ion concentration is probablyunder strict regulation, but the vacuolar ion concentration probably varies tremendously with the age and growing conditions of the cell, and this will affect the shape of the action potential.

qj = -(Cm Em)l(zje) (4) where q is the numberof ions thatdiffuse across a membraneper unitarea (in ions mr-2) m is the specific membranecapacitance (0.01 F m2 = 0.01 C V-1 m-2) Em is the membranepotential set up by the diffusion of ions (in V) z; is the valence of the ion e is the elementarycharge (1.6 x 1019 C ion-1) qjzjeis equal to the charge density (in C m-2). Using the equationfor membranecapacitance (4) andthe equilibriumpotentials that we will calculate below, we will see that, at equilibrium,the initial concentrations change by less than 0.001% after the establishment of an equilibrium potential (Plonsey & Barr, 1991). This is importantto ensure that there is no violation of the principle of electroneutrality,which states that in a volume greaterthan a given size (e.g., macroscopicdimensions) the numberof positive chargesmust equal the number of negative charges. Electroneutralityis an outcome of the fact that any net charge sets up an electric field thattends to attractoppositely chargedparticles (as described by Coulomb's Law) that consequentlyrestore the net chargeof the volume to zero. As a consequence of a membranecapacitance, and the diffusion of ions across a differentiallypermeable membrane, an equilibriumpotential (or Nernst potential)is established (Nernst, 1888, 1889). The resultantequilibrium potential for each ion species is describedby the Nernst Equation:

= Ej (RTlzjF)ln (ne/np) (5) where E is the equilibriumpotential for ion j (in V) on the protoplasmicside of the membrane,assuming the potentialof the exoplasmic side is 0 V R is the gas constant(8.31 J mol-1 K-1) Tis the absolutetemperature (in K) z; is the valence of ion j

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Table II The equilibriumpotential (in V) of each ion across the plasma membraneand vacuolar membraneof a characeaninternodal cell and the plasma membraneof a squid nerve.

Characeaninternodal cell Squidnerve

Plasmamembrane Vacuolarmembrane Plasmamembrane

C1- 0.103 -0.051 -0.026 Nat -0.100 0.049 0.075 K+ -0.180 -0.002 -0.075 Ca2+ 0.059 0.121 0.118

F is the FaradayConstant (9.65 x 104 coulombs mol-1) ne and npare the concentrationsof a given ion on the exoplasmic andprotoplasmic sides of the membrane,respectively (in mol m-3). We are (simply but wrongly) assuming that the activity coefficients of the ions are identical and equal to 1 on both sides of the membrane.See Appendix D for the derivationof the NernstEquation.

The equilibrium potentials for each ion across the plasma membrane and the vacuolarmembrane calculated from the NernstEquation (at 298 K) are given in Table II. The equilibriumpotentials across the plasma membraneof a squid axon are given for comparison. The equilibriumpotentials of all the ions diffusing across a membranecontribute to the resting membranepotential. David Goldman (1943) and Hodgkin and Katz (1949) showed that at equilibriumthe net flow of all ions throughthe membraneis zero and the equilibriumpotential is determinedby thepermeability of the membrane to a given ion (Pj) accordingto the following equationknown as the Goldman-Hodg- kin-KatzEquation:

Ed = (RT/F) In PK [Ke]+ PNa [Nae]+ PCl [Clv] (6) PK [KP] + PNa[Nap] + PCI[Cle] where Ed is the diffusion potential (in V) on the protoplasmicside of the membrane, assumingthe potentialof the exoplasmic side is 0 V R is the gas constant(8.31 J mol-1 K-1) T is the absolutetemperature (in K) F is the FaradayConstant (9.65 x 104 coulombs mol-1) PK, PNa, and PCCare the permeabilitycoefficients of K+, Na+, and Cl- ions, respectively (in m s-1) Ke Nae, and Cle are the concentrationsof K+, Na+, and Cl- on the exoplasmic side of the membrane(in mol m-3) Kp,Nap, and Clpare the concentrationsof K+,Na+, and Cl- on the protoplasmic sIde of the membrane(in mol r-3). See AppendixE for the derivationof the Goldman-Hodgkin-KatzEquation.

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The resting potential of the plasma membraneof characeaninternodal cells and squid axons is approximately-0.18 V and -0.075 V, respectively, indicatingthat for both cell types PK >> PNa> Pcl and the membranepotential is determinedto a large extent by the passive diffusion of K+ (MacRobbie,1962; Spanswicket al., 1967). The greaterthe permeabilityto K+,the more hyperpolarizedthe membranewill tend to be. As we will see later, a hyperpolarizedmembrane is a stable membrane,compared to a depolarizedmembrane. Thus increasingthe K+permeability has a stabilizingeffect on the membrane.Notice that when PK >> PNa > PCI,the Goldman-Hodgkin-Katz Equationreduces to the NernstEquation for K+ (equation2). If the plasma membranepotential was only a consequence of the diffusion of K+ from the protoplasm(=100 mol m-3) to the extracellularsolution (0.1 mol m-3), the cell would have a membranepotential of -0.18 V. According to equation4, 1.125 x 1016K+ mn2would have to diffuse across the plasma membranein orderto "charge the capacitance"of the membraneto -0.18 V. If we assumethat the typical characean internodalcell is a cylinder0.5 x 10-3 m in diameter(d) and 3 x 10-2 m long (1),it has a surface area(idl) of 4.7 x 10-5 m2. Thus, 5.3 x 1011 K+ ions must diffuse across the plasma membraneof one cell to make a membranepotential of -0. 18 V. A cylindrical internodalcell with the above dimensions will have a volume (ntr2l)of 5.89 x 109 m3. Thus, there were initially 5.89 x 10-7 mol K+ in the cell. Using Avogadro's Number, we see that there were initially 3.5 x 1017K+ in the cell. Thus, only [(5.3 x 1011)/(3.5x 1017)]x 100%= 0.00015% of the initial K+ ions must diffuse out of the cell in order to charge the capacitance of the membrane and create a membrane potential of -0.018 V. A. closer look at the Goldman-Hodgkin-KatzEquation shows that when PNa and Pcl are nonzero, the resting potential will be less than that predictedby the Nernst potentialfor K+.However, the plasmamembranes of bothcell types have electrogenic pumps thatpump out cations at the expense of ATP to help generatea more negative membranepotential. Characeancells have a H+-ATPase(Spanswick, 1974a, 1980, 1981) and squid nerves have a Na+/K+ATPase (Eckertet al., 1988). The membrane potential of the characeanplasma membraneis usually between -0.18 and -0.30 V. The characean plasma membrane has three stable states. The K-state, where the membranepotential (more positive than-0.18 V) is determinedpredominantly by the diffusion of K+;the pump state, where the electrogenicH+-pumping ATPase contrib- utes to the restingpotential of approximatelybetween -0.18 and -0.30 V; and a third stable state which occurs at high pH where the permeabilityto H+ or OH-becomes so large thatthe diffusionpotential is determinedby the permeabilitycoefficients for H+ or OH- (PH or POH; Beilby & Bisson, 1992; Bisson & Walker, 1980, 1981, 1982). While the plasma membranesof characeancells in the pump state and high pH state are excitable, and cells in the K+-stateare inexcitable, we will only talk about the actionpotential in cells thatare just at the interfacebetween the K+-stateand the pump state (-0.18 V), since it has been shown that the action potentialis mostly due to the changes in the conductanceof the passive diffusion channelsand not the electrogenic H+-pumpingATPase (Kishimotoet al., 1985). The vacuolarmembrane of characeancells also has an electrogenicH+-pumping ATPase as well as an electrogenic H+-pumpingpyrophosphatase (Shimmen & MacRobbie, 1987; Takeshige et al., 1988; Takeshige & Tazawa, 1989) capable of hyperpolarizingthe membrane potential to values greater than -0.25 V, yet the vacuolar membranepotential (= -0.01 V) is similar to the equilibriumpotential for

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Table III The electrochemicalpotential (in V) of each ion across the plasma membraneand vacuolar membraneof a characeanintemodal cell and the plasma membraneof a squid nerve

Characeaninternodal cell Squidnerve

Plasmamembrane Vacuolarmembrane Plasmamembrane

Cl- -0.283 0.041 -0.049 Na+ -0.080 -0.059 -0.150 K+2+ 0 -0.008 0 Ca2+ -0.239 -0.131 -0.193

If we assumethat an ion is movingfrom a P-space(initial state) to anE-space (final state), we cantell whetherthe movement is spontaneous(passive, or exergonic) in thisdirection by multiplyingthe electrochemicalpotential with the product of thevalence of theion times the Faraday (zF). Whenthe product of theelectrochemical potential and zF is negative,cations and anions are accumulatedpassively (spontaneously) and efflux is active. Whenthe product of theelectrochemical potential and zF is positive,cations and anions are accumulatedactively and efflux is passive(spontaneous). Inreality, active transport usually results from the energy available from the electrochemical gradient of H+established by theprimary H+-pumping ATPase. Secondary transporters then move the actively accumulatedions across the membrane.

K+ and Cl-, indicatingthat thereis a large permeabilityto K+ and Cl- that effectively short-circuitsthe pumps (Rea & Sanders, 1987; Tyerman,1992). The difference between the resting membranepotential (Em) and the equilibrium potentialfor an ion (E ) acrossthat membrane, known as the electrochemicalpotential, is an indicationof whetheror not thation is in passive equilibrium,actively takenup, or actively extruded.Assuming that the membranepotentials across the characean plasma membraneand vacuolarmembrane are -0.18 V and-0.01 V, respectively,the electrochemicalpotentials (Em - E1)across each membranecan be calculatedand are given in Table III. The electrochemicalpotentials across the plasma membraneof a squid axon are given for comparison(assuming Em = -0.075). At the resting potentials of the characean and squid plasma membrane, K+ is passively (spontaneously)distributed across the plasmamembrane [ZF(Em - EK)= 0]; Cl- is actively (not spontaneously)accumulated against its electrochemicalpotential gradient [zF(Em- EC, )< 0] and moves passively out of the cell; Na+ is actively extrudedagainst its electrochemicalpotential gradientand moves passively into the cell [zF(Em- ENa)< 0]; and Ca2+is also actively extrudedagainst its electrochemical potentialgradient [ZF(Em -ECa) < 01and moves passively into the cell. On theother hand, at the restingpotential of the characeanvacuolar membrane, C1-, K+, Na+, andCa2+ are activelyextruded from the protoplasmto the vacuoleacross the vacuolarmembrane. The activemovement of ions requiresenergy and is thusan endergonicreaction. The passive movementof ionsis anexergonic process which releases energy. This energy can be coupled to endergonicprocesses to do work,including the cotransport of substances. In orderto betterunderstand the electrophysiologyof excitablecells, it helps to draw an equivalentcircuit that explains the electricalanalogs in the membrane.First, let us assume that at the resting potential,the sum of all ionic currents(IIj, in A m-2) is 0

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A B

c 4~4j r E r) g

Fig. 7. Equivalentcircuit and structuralinterpretation of a Nitella plasma membrane.A. The equivalent circuit as interpretedby Cole and Curtis(1938). An equivalentcircuit is an electrical circuit that is made up of the minimum number of electrical components (resistors, capacitors, batteries, etc.) that, when connected, can mimic complicatedelectrical (or other) behaviors.E representsthe electromotiveforce of a cell of a battery;Cm represents the membranecapacitance; and r4represents the resistanceof the membrane that is in parallelwith the membranecapacitance. B. Structuralinterpretation of the equivalentcircuit. The electromotive force or membranepotential (E) is determinedby the differencein the K+ concentrationon both sides of the membrane.The lipid bilayer, by virtue of its thickness (2.7 x 1079 m) and its relative permittivity (3), separates two conducting layers and thus acts as a capacitor. The conductance (the reciprocalof resistance)is providedto a large extent by the proteinaceouspores throughthe membrane.

A. The model put forth by Cole and Curtis(1938) includes the observationsthat the membranepotential is due to an electromotiveforce (the equilibriumpotential due to the unequaldistribution of ions) and is equivalentto a batterythat is in series with a conductance(proteinaceous ion channels)and in parallelwith a capacitor (as a result of the lipid bilayer). Their model is shown in Fig. 7A and a structuralinterpretation of the equivalentcircuit is given in Fig. 7B. In orderto describe the resting and action potentialin giant squid axons, Hodgkin (1964) partitionedthe series conductanceand electromotive force of the equivalent circuit into a numberof conductances(one for Na+, one for K+, and one for the ions thatdo not contributeto the actionpotential-the leak) and a numberof electromotive forces due to the equilibriumpotentials of each ion (Fig. 8). The contributionof the electromotiveforce for each ion to the overall membrane potential depends on the ability of that ion to carry current. The fact that the electromotive force and the conductancefor an ion are in series means that if the conductanceis zero for a given ion, then the electromotiveforce for that ion will not contributeanything to the membranepotential, even if there is a large electromotive force. Since the conductancefor all ions except K+ is so small in the resting cell, the membranepotential is essentially equal to EK (the equilibriumpotential for K+). The currentdensity (Ij, in A m-2) carriedby each ion dependson both the specific conductance of the membranefor that ion (gj, in S m-2) and the electrochemical potential for that ion (Em- Ej). This relationshipis just anotherform of Ohm's Law where the specific membraneconductance for an ion (gj) is equal to the reciprocalof the specific resistanceof the membraneto thation (rj, in Q m2).

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Cl RKVNa R L ++ VNa -VK DL1

Fig. 8. Equivalent circuit of a squid axon according to Hodgkin (1964). V representsthe membrane potential.VNa, VK, and VLrepresentthe electromotiveforce acrossthe plasmamembrane(i.e,. the difference between the membranepotential and the Nernstpotential) for sodium ions, potassium ions, and ions that do not contributeto the actionpotential (i.e., the leak), respectively.RNa, RK, and RL represent the (variable) resistance of the plasma to sodium, potassium, and leak ions, respectively. C representsthe membrane capacitancethat is in parallel to the resistances. The electrical resistance of the cytoplasm, in which the microcapillaryelectrode resides, is denotedby r.

I = -gj(Em - E) = -(Em - Ej)Irj (7) where Ii is the currentdensity (in A m-2) carriedby ion j g* is the partialspecific conductance(in S m-2) of the membranefor ion j Jm is the membranepotential (in V) E is the Nernst potentialor equilibriumpotential (in V) for ion j rj'isthe partialspecific resistanceof the membrane(in Q m2) for ionj.

We use BenjaminFranklin's convention that positive currentis carriedby positive charges (Cohen, 1941). So a currentcarried by Cl- is negative and a currentcarried by Na+ or K+ is positive. (Consequently,an inwardcurrent is carriedby the influx of a cation and the efflux of an anion). At rest the sum of all currentsequals zero. We can now use the equivalentcircuit of the resting membraneto help us understandthe action potential.We will see that there is a change in the membranepotential during an action potentialbecause thereis a change in the conductance(or the permeability) of the membraneto certainions.

V. The Ionic Basis of the Action Potential The basic principlesinvolved in the generationof an action potential are the same in animal, plant, and fungal cells. The specifics of the action potentials are different

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0.05 1

> ~~~2

-0.00 C ~~~3 0 0-

.0 -0.05

-0.10 II 0 1 2 3 4 Time,8s%lO35 Fig. 9. The effect of sodium on the action potentialof squid neurons.Trace 1 shows an action potential that was initiatedin axons bathed in artificialsea water (ASW) containingNaCl. Traces 2-5 show action potentialsinitiated after various periods of time afterthe Na+in the ASW was thenreplaced with Choline-Cl. It takes a certainamount of time for the Na+concentration near the membraneto decrease.(Redrawn from Hodgkin & Katz, 1949.) and result from the fact that,in general,animal cells live in solutions that are isotonic and are full of ions, while plants and fungi live in dilute solutions that are very close to distilled water. The capabilityof cells to live in dilute solutions is a consequence of the presence of a rigid extracellularmatrix (cell wall) thatcan supportand prevent the lysis of the plasma membranethat otherwise would occur when the osmotic pressure on the two sides of the membranediffer by more than 100 Pa. (Plant and fungal cells can build up pressure differences 1000-10,000 times greaterthan this value.) Therefore,the "animaltype of action potential"is not possible in plant cells (and vice versa), because the ion concentrationssurrounding a plant cell are different from those found in the extracellularmilieu of animalcells. Hodgkinand Katz (1949) foundthat external Na+ is requiredfor the actionpotential of squid axons. Upon stimulationin normalseawater, the squidaxon depolarizesfrom approximately-0.075 V to +0.04 V at the peakof the actionpotential. However, when the Nat is replaced by choline, both the rate of depolarizationand the amplitudeof the action potential decreases (Fig. 9). Hodgkin and Katz (1949) then proposed the sodium hypothesis,which states thatthe massive depolarizationand overshoot of the membranepotential to positive potentials results from an influx of Na+. Using the radioactivetracer 24Na+, Keynes (1951) then showed thatelectrical stimulation causes an 18-fold increase in Na+ influx. Later it was found that tetrodotoxin(TTX), the deadly poison found in and (mostly) removed from the Japanesepuffer fish eaten as sashimi, prevents the action potentialby specifically blocking Nat currentsin giant axons (Narahashiet al., 1964). Not all animal cells have Na+-based,TTX-inhibited

This content downloaded from 128.84.125.94 on Thu, 12 Nov 2015 14:20:46 UTC All use subject to JSTOR Terms and Conditions EXCITABILITYOF PLANT CELLS 289 action potentials. The Na+-based action potentials occur only in higher animals. Evolutionarily,they begin to appearwith the Coelenterates.The more ancient and ubiquitousaction potentialsresult from "Ca2+spikes" (Hille, 1984, 1992). The equilibriumpotential for Na+ (ENa) is not large enough in characeancells to account for the large depolarizationthat occurs duringan action potential (Williams & Bradley, 1968). Upon stimulation,the characeanplasma membrane depolarizes to approximately0 V. While the electrochemicalpotential of Na+ will tend to drive Na+ into the cell at the resting potential,it will cease to move passively into the cell once the potential depolarizes to -0.100 V and the electrochemical potential for Na+ approaches0 V. Thus Cl- and Ca2+ions have positive equilibriumpotentials and are the only ions that are capable of depolarizingthe membraneto 0 V. The cation Ca2+ can depolarize the membraneby enteringthe cell, and the anion Cl- can depolarize the membraneby leaving the cell. Thus, while the action potentialsof the squid and characeanplasma membraneare similarin termsof the voltage changes, the ions that carrythe depolarizingcurrents are different. Characeancells have a vacuolarmembrane in additionto the plasma membrane. The membranepotential of the vacuolar membranehyperpolarizes from -0.01 V (protoplasmicside negative) to -0.05 V (protoplasmicside negative)during an action potential. Using similar logic, we see that Cl- is the only currentcarrier capable of bringing the vacuolarmembrane membrane potential to the -0.05 V it attainsat the peak of the action potential. In this case, Cl- will move from the vacuole to the protoplasm.

VI. ChlorideEfflux Occurs in Response to Stimulation

Using a thermodynamicapproach, we have determinedthat Ca2+and Cl- ions are capable of carrying the currentthat leads to the change in the plasma membrane potentialthat occurs duringan actionpotential in characeancells. There are two ions that are candidatesfor the charge carrierduring the action potential, and there were two schools of thoughton which was the charge carrier:One thoughtthat Ca2+was importantand the other thought that Cl- was important.Hope (1961a, 1961b) and Findlay (1961, 1962), working in Australia,noticed that the magnitudeof the depolarization that occurred during an action potential depended on the external Ca2+ concentration,where the depolarizationincreased as the externalCa2+ concentration increased, and, further, that external Ca2+ was required for the action potential (Findlay & Hope, 1964b). They suggested that the action potential was due to an increasein the permeabilityto Ca2+and Ca2+was the current-carryingion. However, using 45Ca2+as a tracer,they were unable to observe any change in 45Ca2+influx (probably due to the difficulty in separatingwall-bound 45Ca2+from cytoplasmic 45Ca2+).Thus the role of Ca2+in the action potential remainedenigmatic (Hope & Findlay, 1964). Fifteen years later,Hayama et al. (1979) detecteda Ca2+influx, again bringingthe action of a Ca2+influx into the picture. On the other hand,Mullins (1962), workingin the United States, proposedthat the action of Ca2+was to activate the mechanismresponsible for Cl- efflux. Gaffey and Mullins (1958), Mullins (1962), and Hope and Findlay(1964) loaded internodalcells with 36Ch-and measuredits efflux in the restingcell and following stimulation.They found that there is an increase in Cl- efflux duringan action potential.Mailman and Mullins (1966) measuredthe Cl- efflux using a Ag/AgCl electrode, confirming the

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Table IV Cl- efflux in a characeaninternodal cell at rest and duringan action potential.

Efflux(mol m-2 s-1)

Restingcell 1 x 10-8 Excitedcell 100x 10-8

data obtained using radioactivetracers and improving on the time resolution from hours using the tracertechnique (Hope et al., 1966) to seconds using the electrode technique (Coster, 1966). Typical Cl- efflux values are given in Table IV. Given a membranedepolarization of 0.18 V and a specific membranecapacitance of 10-2 F m2, an increase in the efflux to 1.125 x 1016Cl- m-2 (= 1.87 x 10-8 mol m2, using Avogadro's Numberas a conversionfactor) would be needed to discharge the membranecapacitance and depolarizethe membraneto 0 V (equation4). Assum- ing that the action potentiallasted 1 s, this would be equivalentto an efflux of 1.86 x 108 mol m-2 s-1 Thus, the increase in the efflux in response to stimulationis more than enough to overcome the membranecapacitance and accountfor the depolarization observed duringthe action potential. We can estimatethe changein membranepermeability to Cl- (PCl)during excitation using the following formula (Dainty, 1962) based on the Goldman flux equation (Goldman, 1943): - J PCl[(ZClF/RI)Em]{ncl nClexp[(zClF/RI)Em] (8) ( exp[(zclF/RT)Em] 1 where Jcl is the flux of Cl- in (mol m-2 s-l) Em is the diffusion potential(in V) on the protoplasmicside of the membrane, assumingthe potentialof the exoplasmic side is 0 V. R is the gas constant(8.31 J mol-1 K-1) T is the absolutetemperature (in K) F is the FaradayConstant (9.65 x 104 coulombs mol-1) Zcl is the valence of Cl- (-1) PC,is the permeabilitycoefficient of the membraneto Cl- (in m s-l) nCl?is the concentrationof Cl- on the exoplasmic side of the membrane(in mol m-3) nCllis the concentrationof Cl- on the protoplasmicside of the membrane(in mol m-3). See AppendixE for a derivationof the Goldman-Hodgkin-Katz Equation.

Given the observed fluxes of C1- and assumingthat the cytosolic and extracellular concentrationsof Cl- remain constant at 22 and 0.4 mol m-3, respectively, and the membranepotential is -0.18 V, we find that the average permeabilitycoefficient of the membranefor Cl- rises approximately100-fold from 4.5 x 10-10 m s-1 to 4.5 x 10-8 m s-l duringan action potential. We can also relate the Cl- flux (Jcl) measuredchemically to the conductanceof the

This content downloaded from 128.84.125.94 on Thu, 12 Nov 2015 14:20:46 UTC All use subject to JSTOR Terms and Conditions EXCITABILITYOF PLANT CELLS 291 membrane(gcl) for Cl- measuredelectrically using the following equation(Hodgkin, 1951; Hodgkin & Keynes, 1955; Hope & Walker, 1961; Keynes, 1951; Linderholm, 1952; Smith, 1987; Teorell, 1953; Ussing, 1949a, 1949b; Ussing & Zerahn, 1951):

9Cl = (ZCL2F2/RT)JC1 (9) where gcl is the partialspecific membraneconductance for Cl- (in S mi-2) ZcI is the valence of Cl- (-1) F is the FaradayConstant (9.65 x 104coulombs mol-1) R is the gas constant(8.31 J mol-1 K-1) T is the absolutetemperature (in K) Jcl is the unidirectionalflux of Cl- (in mol m-2 s-1). The resultis only approximatesince the equationonly holds when the Cl- ions move across the membraneindependently of each other and the membranepotential is at the equilibriumpotential for Cl- (i.e., +0.103 V). Many ions probablymove through long narrowpores and interactwith each other, invalidatingthe use of this equation (MacRobbie, 1962). Moreover,the membranepotential never reaches the equilibrium potentialfor Cl-. However, using equation9, we can approximatethe partialspecific conductanceof the restingmembrane (gcl = 0.037 S m-2) andof the excited membrane (gcl = 3.7 S m-2). Thus gcl contributesapproximately 4% to the overall conductance of the resting membraneand about 12% to the overall conductance of the excited membrane. Following the massive depolarization,the membranerepolarizes again due to the efflux of K+. This is truefor axons as well as for characeancells (Hodgkin& Huxley, 1953; Keynes, 1951). K+ efflux has been determinedby measuring32K+ fluxes or by using flame photometry (Kikuyama et al., 1984; Oda, 1976; Spear et al., 1969; Spyropouloset al., 1961). The K+ efflux is equal to the Cl- efflux. They are both 1.8 x 10-5 mol m-2 impulse-1 (Oda, 1976). Therefore,during excitation, PK is approximately 2.3 x 10-5 m s-5 (calculatedusing equation8). This is 1000 times greaterthan the resting PK (Gaffey & Mullins, 1958; Smith, 1987; Smith & Kerr, 1987; Smith et al., 1987a, 1987b). The necessity of a K+ efflux for the subsequentrepolarization is supportedby the observationthat a K+-channelblocker, tetraethylammonium (TEA) causes a prolongationof the action potentialin some but not all cells (Belton & Van Netten, 1971; Shimmen & Tazawa, 1983a; Staves and Wayne, 1993; Tester, 1988). The massive efflux of Cl- and K+ from the cell is accompaniedby waterloss and a transientdecrease in turgor(Barry, 1970a, 1970b).The waterloss resultsin a transient change in the volume of the cell that can be detected with a position transducer (Kishimoto & Ohkawa, 1966) and by laser interferometry(Sandlin et al., 1968). Now we can begin to accountfor the membranepotential changes thatoccur during a characean action potential by the movement of Cl- and K+. We can make an equivalentcircuit to help us visualize the changes in membranepotential that results from the changes in conductance. Figure 10 shows an equivalentcircuit of a characeancell. The resistorssymbolize the channels that provide the conducting pathway for a given ion. The batteries representthe electomotive force for each ion due to the unequaldistribution of ions (i.e., Nernstpotential or equilibriumpotential). The capacitorrepresents the abilityof

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withKresistancesandcapacitances,a . Th p mpa AP

rr c

th c too - Vt h h sf TP;

V ,-

Fig. 10. Equivalentcircuit of a characeancell. PM and VM representthe plasmamembrane appd vacuolar membrane,respectively. The external, cytosolic, and vacuolar solutions are representedby e, c, and v, respectively. Both the plasma membrane and the vacuolar membrane have electromotive forces and conductances(1/resistance) for potassium,sodium, chloride, and calcium ions. The two membranes,both with resistancesand capacitances,are in parallel.The plasmamembrane has a H0-pumpingATPase, which has a conductance.The vacuolarmembrane Hn-pumpinghas a ATPaseHi-pumpingpyrophosphatase,and a both of which have conductances.The electrical resistanceof the cytoplasm, in which the microcapillary electrode resides, is denotedby r. the lipid bilayer to separatecharge. The plasma membranehas an electrogenic H+- pumpingATPase with an electromotiveforce of f.46V (due to the hydrolysisof ATP; Blatt et al., 1990). The potentialdifference across the plasmamembrane (approximately .180 V, inside negative)vue is definedas membranepotential of theprotoplasmic side minus the membranepotential of the exoplasmic side (defined as 0 V). The vacuolarmembrane has a negative potentialdifference that is also defined as the membranepotential of the protoplasmicside (-0.010 V) minus the membrane potential of the exoplasmic side(detined as 0 V). The membranepotential of the vacuole is determinedto a largeextent by the K+-dependentelectromotive force since the K+ conductanceis so high. This conductanceeffectively shunts the H+-pumping ATPase and pyrophosphataseon the vacuolarmembrane so they do not contribute significantly to the vacuolarmembrane potential. Note that the potential differences across the plasma membraneand the vacuolarmembrane are oppositely directed.

VII. Application of a Voltage Clamp to Relate Ions to Specific Currents 'Upuntil now we have been talking aboutexperimental setups where the currentis applied as a stimulus and we recordthe changes in membranepotential. This experi-

This content downloaded from 128.84.125.94 on Thu, 12 Nov 2015 14:20:46 UTC All use subject to JSTOR Terms and Conditions EXCITABILITYOF PLANT CELLS 293 ment can be done in reverse by using a voltage clamp, which was designed by K. S. Cole in 1949 to measure the current that flows when the membranepotential is "clamped"to a given voltage (Sherman-Gold,1993; Standen et al., 1987). No net current flows when the membrane potential is clamped to the resting potential, whereas thereis a currentflow when the membranepotential is clampedto potentials less negative than the "thresholdpotential for excitation." When the membrane potentialremains constant, the changein the amountof currentthat flows is a measure of the conductance.The voltage clamp is superiorto the traditionalcurrent clamp in that it can rapidly clamp the membraneand allow one to distinguish the "instan- taneous" currentsthat flow from the dischargeof the membranecapacity (capacity culTents)from the time-dependentionic currents(Hodgkin et al., 1952; Kishimoto, 1964). The pioneeringwork on voltage clampingthe giant squid axons by Alan Hodgkin, Andrew Huxley, and BernardKatz establishedthe classical principlesof membrane biophysics (Hille, 1984, 1992). When Hodgkin and his colleagues clamped the membranepotential at values less negative than the thresholdpotential (-0.056 V), they saw a time-dependentcurrent. First there was an inwardcurrent and thenthis was replacedby an outwardcurrent (Fig. 1 A). They noticed that when they removed the Na+ from the external medium, the early inward currentdisappeared but the late outwardcurrent remained (Fig. 1IB). By subtractingthe currenttrace obtainedin the absence of Na+ from the one obtained in the presence of Na+, they obtained a difference tracethat represented the Na+ current.They assumedthat the currenttrace obtained in the absence of Na+ representedthe K+ current.This was confirmed by removing the naturalprotoplasm by perfusionand substitutingit with eitherK+-con- tainingor K+-freefluid andobserving that the late currentdisappeared when they used K+-free fluid. By dividing the magnitudeof the Nat currentor the K+ currentat a given time by the membranepotential which was held constant by the clamp, they could calculatethe changein Na+and K+ conductances with time thatoccurred during an action potential (Hodgkin, 1964). Hodgkin and his colleagues found thatthe restingmembrane potential results from the fact that gK > gNa > gc. The action potential occurs after a stimulus causes an increasein the conductanceto Na+ (gNa).Thus, upon stimulation,a substantialamount of Na+ moves down its electrochemicalpotential gradient from the externalmedium into the axoplasm.This inwardcurrent causes a shift in the membranepotential toward the Na+ equilibriumpotential (i.e., depolarization).The membranepotential consequently shifts away from the K+ equilibriumpotential so that the electrochemical potential gradientnow drives an increasedefflux of K+. This is accompaniedby an increase in the conductance to K+ (g90. The increased conductance for K+ will repolarizethe membraneback to its restinglevel. The repolarizationof the membrane to its resting level due to K+ efflux is facilitated by the time-dependentdecrease in gNa (i.e., inactivation). Since the K+ efflux overlaps the Na+ influx, the membrane never depolarizesall the way to ENa.The peak depolarizationis somewherebetween EK and ENaand can be predictedfrom the Goldman-Hodgkin-KatzEquation using the time-dependentrelative permeabilitiesand concentrationsof K+ and Na+ (see Eckertet al., 1988; Junge, 1981; Matthews, 1986). Lunevsky et al. (1983), working in the formerSoviet Union, used a voltage clamp to study the action potential across the plasma membraneof the characean alga Nitellopsis. They insertedone electrode in the protoplasmand one in the medium to

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210 K K,Na

S N~~~a E: -10 C)

-20 I 0 1 2 3 4 5 Time, s x 103

04 3 E B

Na i, X2

C) C)

a0.

0 2 4 6 8 10

Time, 8 x 103

Fig. 11. A. Currentrecordings from a squid axon. Separationof currentscarried by Nat and K+ using a voltage clamp. K,Na representscurrents measured in the presenceof extracellularNae and intracellularK+. K representsthe currentsmeasured in the absence of extracellularNa+. (In this case the Na' was replaced by choline). Na representsthe Na+ current,which was obtainedby subtractingthe K+current from the total current.Note: By convention,inward currents are always downward.B. Changein conductanceover time. The conductancefor Na+ and K+were determinedby dividingthe currentdensity (in A m-2) carriedby Na+ or K+ by the potential at which the membranewas held by the voltage clamp (0.056 V). (Redrawnfrom Hodgkin, 1964.)

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7 5 4

Fig. 12. Setupof voltage clampto studycharacean cells. An internodalcell is placedin athree-chambered acrylic holder (1). A microcapillaryelectrode (2), used to recordthe membranepotential, is placed in the cytoplasm. The voltage is followed by a preamplifier(4) and passed to a voltage clamp amplifier(5). The voltage clamp amplifiercompares the membranepotential to the desiredpotential and outputscurrent to the Ag/AgCl wires (3) so thatthe membranepotential is held at a desiredvalue. The currentis then passed to a currentmonitor (6) andan oscilloscope or oscillograph(7) so thatthe currentand voltage can be recorded simultaneously(7). (Redrawnfrom Lunevskyet al., 1983.) measurethe membranepotential. The membraneis clampedto the desiredvoltage by passing just enough currentthrough the cell using Ag/AgCl wires (Fig. 12). Using a step-voltage clamp, where the membranepotential can be varied in a stepwise manner, Lunevsky et al. (1983) obtained a series of currenttraces that represented current movement through the plasma membrane at each membrane potential (Fig. 13). They resolved their data into three currents:the quick transient, the slow transient,and the steady state current(in additionto the leakage current). The first, quick transientcomponent lasts for several hundredmilliseconds. When the membraneis depolarizedto -0.05 V it appearsas an inwardcurrent. It changes from an inward currentto an outwardcurrent as the membranepotential is clamped to more and more depolarizedvalues. The reversalpotential for this current-that is, the voltage wherethe currentis neitherinward nor outward (Ij = 0 A m-2)-is between -0.060 and-0.020 V in artificialpond water (APW). This rapid, transient current could be studied betterif we could block the second current,so we'll talk about the second currentand then come back to the first. The second, slow transientcurrent appears when the membraneis depolarizedbelow a thresholdbetween -0.09 and-0.12 V andthe maximumpeak inwardcurrent is seen at potentials0.01 to 0.02 V morepositive thanthe threshold.The slow transientcurrent exhibits activation-inactivationkinetics that are similarto, althoughslower than, the Na+ currentin squidaxons. The reversalpotential of the slow transientcurrent depends on the external Cl- concentration.When the externalCl- concentrationis increased

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\ -0.02 V "0,4 <.035V035

Fig. 13. Currentrecordings from a characeaninternodal cell. The membranewas clamped at various potentials and the currentthat flowed throughthe cell was recorded.Vp, and Ip, represent the potentialof the plasma membrane and the current density through the plasma membrane,respectively. The reversal potential for the first quick transientcurrent is -0.045 V in this cell. (Redrawnfrom Lunevskyet al., 1983.)

3.25-fold, from 32 mM to 104 mM, the reversalpotential changes from -0.010 V to -0.040 V in a mannercompletely accountedfor by the Nernst Equation,as Cl- is the only ion carryingthe current.Further evidence that the second slow transientcurrent is carriedby Cl- comes from the observationthat this currentis inhibitedby the well- known Cl-channel blockers, ethacrynicacid and anthracene-9-carboxylicacid (Fig. 14). The peak currentcarried by Cl- ions is between 4 and 7 A m-2. The net flux can be linked to the observed currentusing the following equation:

Jcl = IClI(ZClF) ( 10) where JCl is the flux of C1- (in mol m-2 s1l). 'Cl is the electric currentcarried by Cl- (in A m-2) ZClis the valence of Cl- (-1) F is the FaradayConstant (9.65 x i04~coulombs mol-1).

This means that at the peak of the action potential the instantaneousrate of Cl- efflux must be 4-7 x 10-5 mol m-2 s-1 When the C1- currentis inhibitedby ethacrynicacid, it is possible to study only the first quick transientcurrent. The first currentrapidly activates in 100-250 ms and slowly inactivates over the next 0.5-1 s (Fig. 15). The channel that passes the first currentpassescations, including Na+, K+,and Ca2+, although under normal conditions it will pass Ca2nions selectively.

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-0.05 5 IP I 4L -0.153

10.5 Am

ils

Fig. 14. Inhibitionof the Cl- currentby the Cl- channelblocker ethacrynicacid. Trace 1 representsthe initial current and traces 2-5 represent currentsmeasured at successive 10-minute intervals after the applicationof ethacrynicacid. (Redrawnfrom Lunevskyet al., 1983.)

The voltage-clamp experimentsshow that membranedepolarization can activate two transient(inward) currents: The first one is carriedby Ca2+ions moving into the cell and the second one is carriedby Cl- ions moving out of the cell. Thus, there is an initial quick transient increase in the conductance for Ca2+, followed by a slow transientincrease in the conductancefor Cl-. Following these two transientcurrents there is a steady state currentthat probably is carriedby K+, as determinedby its reversal potential (Fig. 16). The changes in conductances over time have been estimatedfrom the work of Lunevskyet al. (1983). Beilby and Coster(1979a, 1979b) carriedout a Hodgkin-Huxleytype analysison theirvoltage-clamp data which cannot be reconciledat the presenttime with thatof Lunevskyet al. (1983). Hironoand Mitsui (1983) also have done a Hodgkin-Huxleytype analysison the activation-inactivation of the combinedCa2+ and Cl- currents.A complete analysis and reconciliationof the data will have to await new methods for specifically identifying and completely blocking each individualionic current. Lunevskyet al. (1983) postulatedthat the Cl- channelis activatedby an increasein the protoplasmicCa2+ concentration that occurs as a consequenceof the activationof the Ca2+conductance. The activationof the Cl- channel is inhibited when cells are treated with the Ca2+ channel blocker, La3+ (Tsutsui et al., 1987a, 1987b, 1987c). Moreover, Ca2+is requiredfor the repolarizationof the membranepotential which results from K+ efflux across the plasmamembrane (Shimmen & Tazawa, 1983a) as well as the activationof a K+channel responsible for the repolarizationof the vacuolar membrane(Katsuhara et al., 1989; Laver, 1992; Laver & Walker, 1991). Now let's study the excitabilityof each membraneseparately instead of observing two membranesin series. First we'll look at the vacuolar membrane.The vacuolar membraneis studied by gently permeabilizingthe plasma membrane(with 5 mM EGTA and no Ca2+ at 4?C) so that the plasma membrane becomes completely permeableto ions and small molecules yet keeps macromoleculesin the protoplasm

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0.15~~OPNi 8

05.Am~2O

1 s

Fig. 15. Activation-inactivationof the first current.The C1-current has been inhibited in this cell by ethacrynicacid. This transientcurrent represents the passage of monovalentions (60 mol m-3 K+ or Na') througha cationchannel. This channelprobably functions as a Ca2' channelunderphysiological conditions, where the Na+,K+, and Ca2+concentrations are approximately 0.1 mol m-3 each. (Redrawnfrom Lunevsky etal., 1983.)

C0J 60 E c" 50

40 C

C 30 0 ~Ca2 2

20 C)

0 1 ~~~~~~~2 3 45 Time. s Fig. 16. Change in conductanceover time. Estimateof partialspecific conductancesderived from data presentedin Figures 13-15.

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p . v

Fig. 17. Diagram of a permeabilized cell. I and P represent intact and permeabilized cells, respectively. In the intact cell, both the plasma membraneand the vacuolar membraneare semipermeable. In the permeabilized cell, only the vacuolar membrane is semipermeable. (Redrawn from Tazawa & Shimmen, 1987.)

(Shepherd& Goodwin, 1989; Fig. 17). Since ATP leaks out (and streamingstops), the cells must be bathedin Mg2+-ATP,which resultsin a reactivationof protoplasmic streaming(Shimmen & Tazawa, 1983b). The action potential of the vacuolarmem- branecan also be studiedon protoplasmicdroplets (Kobatake et al., 1975; Takenaka et al., 1975).

VIII. The Action Potential at the VacuolarMembrane In plasma membrane-permeabilizedcells of Chara,the vacuolarmembrane action potentialcan be activatedby increasingthe Ca2+concentration of the externalmedium from 0 to 1 gM (Fig. 18; Kikuyama,1989). The increasedCa2+ concentration on the protoplasmic side of the vacuolar membrane causes the vacuolar membrane to depolarize for as long as the Ca2+is present;however, a short pulse will activate a normal-lookingdepolarization and repolarization.When the Cl- concentrationon the protoplasmicside is increased,the degree of Ca2+-induceddepolarization decreases as predicted by the Nernst Equationif Cl- is the only current-carryingion. The Cl- currentflows througha specific Cl- channel,since the Cl- channelblocker, anthracene- 9-carboxylicacid, completelyeliminates the Ca2+-induceddepolarization (Kikuyama, 1989). In orderto confirmthat a Cl- channelwas actuallyopening, Kikuyama(1988) measuredthe flux of Cl- from the vacuole using a Ag/AgCl electrode.He found that an increase of the Ca2+concentration on the protoplasmicside of the membranefrom 0 to 5 gM caused an increase in the flux of Cl- from the vacuole by approximately 106 mol m-2 s-1. If the naturalcell sap is replacedby perfusionwith a Cl--free cell sap, no Cl- flux can be measured,indicating that the observedCl- flux originatesfrom the vacuole (Kikuyama,1988). How is the vacuolarmembrane action potential activatedin intact cells? In intact cells, it is generatedwhen the plasmamembrane has been excited, indicatingthat there must be a couplingmechanism between the two membranes(Findlay, 1970; Kikuyama & Tazawa, 1976). Ca2+is assumedto be the intracellularcoupling agent since:

1. The [Ca2+],increases upon excitation from approximately0.1 to 10 FtMas measured by aequorinluminescence (Kikuyamaet al., 1993; Kikuyama& Tazawa,

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I l 0. 0_ 300s

E Time

o ~~~Time

A9C c

0 CL

E ~~~300s 0

Time

Fig. 18. The effect of Ca2+ on the vacuolarmembrane action potential in plasmamembrane permeabilized cells. A. The Ca2+ concentrationon the protoplasmicside of the vacuolarmembrane of permeabilizedcells was increased at various times (black bars) and the membranepotential was recorded. A downward deflectionrepresents a hyperpolarization.B. The Ca2+concentration on the protoplasmicside of thevacuolar membrane of permeabilized cells was increased (down arrow) or decreased (up arrow), and the C1L concentrationof the mediumwas measured.The Cl- concentrationwas measuredwith a Ag/AgCl wire and given in volts, where, in accordancewith the Nemst Equation,a 10-fold change in Cl- causes a 0.059 V change in the potential of the wire. C. The Ca2+ concentrationon the protoplasmicside of the vacuolar membraneof permeabilizedcells was increasedat varioustimes (black bars) and the membranepotential was recorded. This experimentwas done in the presence or absence of A-9-C. A downwarddeflection representsa hyperpolarization.(Redrawn from Kikuyamna,1988, 1989.)

1983; Williamson & Ashley, 1982). The influx of Ca2~icessfo Om o m-2 s51 to 3 x 10O7Mol M-2 5-1 upon excitation(llayama et al., 1979). 2. Replacementof externalCa2+ with Ba2+,Mg2+, orMn2+ does not inhibitthe plasma

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A ____

B ______

Fig. 19. Diagramof a vacuolarmembrane-free cell. A representsthe intactcell, B representsa cell whose naturalcell sap has been replacedwith artificialcell sap, and C representsa cell in which the cell sap was replaced with a solution that causes the disintegrationof the vacuolarmembrane and the dilution of the cytosol. (Redrawnfrom Tazawa & Shimmen, 1987.)

membraneaction potential (Barry, 1968), butinhibits the generationof the vacuolar membraneaction potentialand the cessation of streaming(Kikuyama, 1986). 3. Microinjection of Ca2+ into the protoplasm induces an action potential at the vacuolarmembrane (Kikuyama, 1986).

It seems clear that the vacuolarmembrane action potentialresults from the stimulation by Ca2+,the "messenger"activated by the plasmamembrane action potential.

IX. The Plasma MembraneAction Potential The actionpotential at the plasmamembrane can be studiedby endoplasm-enriched fragments using centrifugation(Beilby & Shepherd, 1989) or by making vacuolar membrane-freecells (Tazawa& Shimmen, 1987; Williamson, 1975). This is done by cutting off the ends and perfusing the cell with a Ca2+-freesolution (Fig. 19). The vacuolar membranesubsequently becomes unstable and breaks. The plasma mem- branethen becomes the only surfacemembrane. The compositionof the two solutions on each side of the plasmamembrane can be readilyadjusted to study the excitability of the plasma membrane(Tazawa & Shimmen, 1987). In rapidlyperfused, vacuolar membrane-freecells, containing the Ca2+-chelator EGTA, the Cl- channel in the plasma membraneis inactivated(Mimura & Tazawa, 1983) and the Ca2+current can be studied in isolation (Fig. 20; Shiina & Tazawa, 1987a). There are at least three classes of Ca2+channels in the plasma membraneof characean internodal cells that are differentiatedbased on their localization and pharmacologicalproperties (Staves et al., 1993; Wayne et al., 1993). In a resting Nitellopsis cell, the Ca2+(Sr2+) flux is approximately130 x 10-9 mol m-2 s-1, and it increases to 1640 x 109 mol m-2 s-1 during an action potential.These ionic fluxes can be related to the ionic currentsby multiplying the fluxes by zF (19.3 x 104 coulombs mol-1). Thus, the current resulting from the flux of Ca2+ transiently increasesfrom 0.025 A m-2 to 0.316 A m-2 duringan actionpotential. [Note thatthese

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0.06 A

E 0.03 E

i0.00.

S~~~-0.25 -0.20 -0.1 5 -0.1 0 -0.05 0.bo Membrane Potential,, V

-0.05-

0 >~~~~~~~~~~~~ L -0.10 C

E 0.15

-0.20 0 12 3 45 Tl-me,01.2 X Fig. 20. The Ca+ currentin rapidlyperfused, vacuolar membrane-free cells. A. The currentdensity of vacuolarmembrane-free cells was measuredwith a voltage clamp.B. The membranepotential of a vacuolar membrane-free cell was measuredwith a cuffent clamnp.The membranewas excited by a depolarizing cuffent. (Redrawnfrom Shiina & Tazawa, 1987a.)

fluxes observed in tonoplast-freeNitellopsis cells are higher than those observed in normalChara cells (Staves, unpublishedresults)] . The cuffent thatpasses throughthe Ca2+channel involved in the generationof the action potential is blocked by La3+

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(Shiina& Tazawa,1987a). The shapeof the actionpotential of the plasmamembrane is differentin vacuolarmembrane-free cells comparedwith intactcells. In vacuolarmembrane-freecells the actionpotential is typicallyrectangular in shape,indicating that the Ca2+current does not showtime-dependent inactivation. This activation may be regulated throughthe phosphorylationof a Ca2+channel, since the Ca2+channel can be activated only when it is dephosphorylated(Shiina & Tazawa,1986b; Shiina et al., 1988). In slowly perfused vacuolar membrane-free cells, the plasma membrane Cl- channel is still functional (Shiina & Tazawa, 1988). In these cells, membranedepo- larizationcauses a large inward current(Fig. 21). Cl- ions carry the inward current since the large depolarizationand the Cl- efflux are inhibited by the Cl- channel blocker A-9-C. The Cl- channel is activatedin response to extracellularCa2+ since the Cl- currentand C1-efflux requireextracellular Ca2+ and are inhibitedby the Ca2+ channel blocker,La3+ (Shiina & Tazawa, 1987b).

X. Identificationof the Ca2+-ActivatedCl- Channel on the Plasma MembraneUsing Patch Clamp Techniques

Up to now we have talked about the specific conductanceof the membraneto a given ion, and we have assumed that this conductance is due to the presence of channels. Channelsare single proteinmolecules embeddedin the membranethat act like little regulated(i.e., gated)pores throughwhich ions can pass. Recently, thepatch clamp techniquehas made possible the visualizationof single channels(Hamill et al., 1981; Sakmann & Neher, 1983). With this technique, a tiny piece of the plasma membrane (10-12 m2) can be pulled away from the cell, thus making certain the identity of the membrane in which the channel resides (Fig. 22). The isolated membranepatch no longer has a naturalmembrane potential, so a voltage clamp is used to set (and vary) the membranepotential to any given value. Typically, channels are activated at certainpotentials and a currentcan be measured.When the channels are closed, no currentpasses; and when the channels open, a small currentin the picoampere range can be observed. It is possible to observe the behavior of single channels opening and closing. Okiharaet al. (1991, 1993), using patch clamping techniques, have actually identified the Ca2+-activatedC1- channels in the plasma membranethat are responsiblefor the action potential(Fig. 23). The currentpassing throughthe channels is observed only when the Ca2+concentration on the protoplasmic side of the membraneis about 1 ,M. When the Ca2+ concentrationis either lower or higher, the currentdecreases, indicatingthat a small amount of Ca2+ is needed to activate the channel and greater amounts cause it to inactivate. This behaviormay account for the kinetics of the "slow transientsecond" current observed by Lunevsky et al. (1983) in their voltage clamp experimentson intact cells. The reversalpotential of the Ca2+-activatedcurrent is positiveand depends on the Cl- concentrationin a mannerpredicted by the NemstEquation if Cl- is the only ion carrying the current.The Ca2+-activatedCl- channelis also voltage-dependent.This channelis activatedupon depolarizationto voltagesless than-0.16 V, indicatingthat a membrane depolarization(i.e., receptorpotential alone) is necessary but not sufficient for the activation of the Cl- channel that is responsiblefor the realizationof the massive depolarizationand increase in conductancecharacteristic of the action potential. The magnitudeof the Cl- currentobserved (I.) in a patchdepends upon the number

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I~E Time E 120s

~ o

A AA A A 7 6 5.4 4.8 7 pCa

Control A9C Wash

joEomono4 n TimeE 120s

AV A V A v pCa

Fig. 21. The action potential in vacuolarmembrane-free cells. A. The effect of Ca2+ on the Cl- efflux and membranepotential. Cl- efflux was measuredwith Ag/AgCl wires in the externalmedium. Membrane potential was measuredwith a microcapillaryelectrode. pCa representsthe -log[Ca2+].B. The effect of A-9-C on Cl- efflux and membranepotential. (Redrawn from Shiina & Tazawa, 1987b.)

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Fig. 22. Diagramof a patchclamp setup.A patchof membraneis held by a high-resistanceseal to a glass microcapillarypipette. One side of the membraneis exposed to the bathand the otherside of the membrane is exposed to the pipettesolution. The electrodeis connectedto a patchclamp amplifier,which in principle is identicalto a voltage clamp amplifierexcept thatthe currentsthat flow aresmaller (picoampere compared to microampere).The patchclamp amplifiercan issue a commandto hold the membraneat a given voltage and measurethe currentthat flows at the commandvoltage [see Hamill et al. (1981) for details]. of Cl- channelsin the patch (m), the probabilityof a channelbeing open (PO),and the currentpassed by a single channel (Is) accordingto the following relation:

Io=bPIs (11) where IOis the currentpassed throughthe patch (in A) b is the numberof channelsin the patch (dimensionless),assuming that all the channels are identical POis the probabilityof a channelbeing open (dimensionless) Is is the currentpassed by a single channel (in A).

A kinetic analysis of the results shows that the increase in the observed current induced by an increase in the Ca2+ concentrationon the protoplasmicside of the membrane from 0.1 to 1 gM results from. an increase in the probability of the Ca2+-activatedCl- channelbeing open. Likewise, the decreasein the currentobserved when the Ca2+concentration increases from 1 to 10 IM resultsfrom a decreasein the probabilitythat the Ca2+-activatedCl- channel will be open. This appearsto be part of the molecular mechanism to explain the activation-inactivationkinetics of the

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Vp,v pCa A 7 6 5 -0.02

. - -

01 -~-

-0.2

Ip,A VP, V -02 -0.1 B

0 7 qX10-12 06 *5 . x10-12

07 PO 0 6 0.3

C t ' / 0.2 t~~~~.1

-0.2 -0.1 Vp, V

Fig. 23. Patch clamping analysis of the Ca2+-activatedCl- channel. A. Patch clamp analysis of inside out patches treatedwith variousconcentrations of Ca2+in the bath and held at various commandvoltages. The observed currentsare discrete, which is due to the opening and closing of Cl- channels. There are probably3 or 4 channels in the patch. B. I-V curves of single channels. C. The effect of Ca2+ on the open probabilityof the Cl-channel. Vpand Ip representthe electricalpotential of the plasma membraneand the currentthrough the plasma membrane,respectively. P0 representsthe open probabilityof the Cl- channel and pCa = -log (Ca2+). (Redrawnfrom Okiharaet al., 1991.)

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.0 00*

Ca2+ . . 0a.0 *

0 0 @0 0 MR 2

1*P

CI K ~~~~~AT 5. Actomyosin

,PK'PP

6 V

Ci-

Fig. 24. Summary diagramof the generationof an action potential and the subsequentcessation of streaming.A mechanicalstimulus, perceived by a mechanoreceptor(MR), causes a depolarizationof the plasma membraneknown as the receptorpotential (1). The receptorpotential is probablydue to the activationof Ca2' channels(2). If the increasein the intracellularCa2' concentrationis sufficient,the membranepotential further depolarizeswhen Ca2+and the accompanyingdepolarization activates the C1-channel in the plasmamembrane (3). This resultsin an inwardcurrent carried by Cl- (i.e., C1-efflux). Subsequently,the membranepotential moves away from the equilibriumpotential for K+. Thus, K+ moves out of the cell down its electrochemical gradientand the membranepotential repolarizes (4). Thepermeability of themembrane to K+may also increase as a conseuence of the activationof a Ca2t-activatedK+ channel. The Ca + that enters the cell in response to mechanicalstimulation then diffuses across the 5-20 ,um protoplasm at a velocity of approximately 1 jm s-1 (5) and activates the Cl- channel of the vacuolar membrane.Cl- moves from the vacuole into the protoplasm,causing a slower vacuolarmembrane action potential,which is observed as a hyperpolarizationof the vacuolarmembrane (6). Ca2+also activates a K+ channel on the vacuolarmembrane which results in a repolarizationof the vacuolarmembrane potential. The transientincrease in protoplasmicCa2+ stops streamingas the wave of Ca2+ moves throughthe cell. Protoplasmicstreaming stops because the increasedCa2+ activates a proteinkinase thatphosphorylates the heavy chain of myosin and inhibits the interactionof actin and myosin, thus inhibiting the proteins that providethe motive force for streaming(7). Eventuallythe elevated Ca2' concentrationreturns to the resting level due to the pumping out of Ca2' by Ca2+-ATPaseson the plasma membraneand sequesteringinto internalcompartments (8).

inward Cl- current.Patch clamping has provided a visualizationof membranephe- nomenathat would have been beyond the wildest ideationof JuliusBernstein (1912), the creatorof the membranetheory of excitation.

XI. Summary A mechanicalstimulus causes a depolarizationof the plasma membraneknown as the receptorpotential (Fig. 24). The receptorpotential is probablydue to the activation

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of Ca2+ channels.Probably the value of the receptorpotential is directlyrelated to the numberof Ca2+channels activated.If a sufficient numberof Ca2+channels are not activated, the increased Ca2+ influx will not cause a large enough increase in the intracellularCa2+ concentrationto activate the processes necessary to cause the regenerativeaction potential, since Ca2+influx must be much greaterthan the Ca2+ efflux broughtabout by the Ca2+-pumpingATPases in the plasma membrane.If the increase in the intracellularCa2+ concentration is sufficient, the membranepotential furtherdepolarizes when Ca2+ and the accompanyingdepolarization activates the C1- channelin the plasma membrane.This resultsin an inwardcurrent carried by Cl- (i.e., Cl- efflux). Subsequently,the membranepotential moves away from the equilibrium potentialfor K+.Thus K+ moves out of the cell down its electrochemicalgradient and the membranepotential repolarizes.The permeabilityof the membraneto K+ may also increase as a consequenceof the activationof a Ca2+-activatedK+ channel. The Ca2+ that enters the cell in response to mechanical stimulationthen diffuses across the 5-20 gm protoplasmat a velocity of approximately1 Rm s-I and activates the Cl- channel of the vacuolar membrane.Cl- moves from the vacuole into the protoplasm,causing a slower vacuolarmembrane action potentialwhich is observed as a hyperpolarizationof the vacuolarmembrane. Ca2+ also activatesa K+channel on the vacuolar membranewhich results in a repolarizationof the vacuolarmembrane potential.The transientincrease in protoplasmicCa2+ stops streamingas the wave of Ca2+moves throughthe cell. Protoplasmicstreaming stops because the increasedCa2+ activates a proteinkinase thatphosphorylates the heavy chain of myosin and inhibits the interactionof actin andmyosin, thusinhibiting the proteins that provide the motive force for streaming(Tominaga et al., 1986). Eventuallythe elevated Ca2+concentration returnsto the resting level due to the pumpingout of Ca2+by Ca2+-ATPaseson the plasma membraneand sequestrationinto internalcompartments.

XII. Coda

In this paperI have triedto show thatwork on excitabilityof plants and particularly of characeaninternodal cells has advancedconsiderably over the past century.We are just beginningto know the extentto which actionpotentials in higherplants are similar to or different from those in characeancells (Hodick & Sievers, 1986; lijima & Hagiwara, 1987; Iijima & Sibaoka, 1981, 1982, 1983, 1985; Samejima& Shibaoka, 1982; Tazawa, 1984). Likewise, we arebeginning to realize the extent to which action potentials may be involved in long-range communicationin higher plants (Davies, 1987; Shiina & Tazawa, 1986a;Wildon et al., 1992). Could actionpotentials communicate some of the informationneeded for whole-plantdevelopment and responsesto environmentalstimulation? Living organismsuse both chemical (e.g., hormonal)and physical (e.g., electrical) messengers for long-range (i.e., cell to cell) information transfer.It is noteworthythat animalphysiologists are split approximately50%150% between those who study chemical communicationand those who study electrical communication,while plant physiologists are split approximately99%/I% between those who study chemical communicationand those who study electrical communication, with a few studyingboth (Bandurski,1991; Bates & Goldsmith,1983; Cleland et al., 1977; Felle et al., 1986; Liu et al., 1991; Lund, 1947; Pickard, 1984). This unequaldivision amongplant physiologists is especially puzzling afterwe realize that plant electrophysiologyplayed such a prominentrole in the early historyof electrical

This content downloaded from 128.84.125.94 on Thu, 12 Nov 2015 14:20:46 UTC All use subject to JSTOR Terms and Conditions EXCITABILITY OF PLANT CELLS 309 communication.I suggest that theremay be five possible explanationsthat explain the unequaldivision that has been so prominentduring the last half of the twentiethcentury:

1. the discovery of auxin, a ubiquitousand omnipotentchemical messenger 2. institutionalnationalism, racism, and sexism 3. the belief thatplant cells do not have a plasma membrane 4. the apprehensionof plantphysiologists in using physical laws to model biological systems 5. the use of plants in parapsychologicalwork.

Perhapsthe splitinto a chemicalthesis of communicationand an electricalantithesis of communicationin plantsresulted from the notion that one typeof messengermust be more importantthan the other (the "one God, one Messenger"theory). Thus, the great successes of theprominent, chemical-oriented plant physiologists Frits Went and Kenneth Thimann (documentedin Went & Thimann,1937) in purifyinga chemicalmessenger (the plant hormoneauxin) that caused so manyprofound physiological and developmental responses left thephysical-oriented plant physiologists powerless to continuetheir work on electrical communicationin plants.Indeed, since that time, plant growthand developmenthave almostbecome synonymouswith the studyof hormones. Auxin was discoveredin the 1930s, a time when the world was in political turmoil. Perhapsthe division of the worldinto the Allies and Axis exacerbatedthe birthof the thesis-antithesisconviction. The discoveryof auxincoincided with the rise of Adolph Hitler in Germany. While Germany had been the "center"of plant physiological researchprior to the discovery of auxin, America (particularlyCal Tech) became the "center"of plantphysiological researchafter the discovery of auxin (Boysen-Jensen, 1936; Brauner& Rau, 1966; Bunning, 1939, 1948, 1953; Goebel, 1920; Haberlandt, 1890, 1901; Kohl, 1894; Nemec, 1901; Pfeffer, 1875; Skoog, 1951; Stern, 1924; von Guttenberg,1971). Thusthe false dichotomywas set up:Choose the physical-oriented plant physiology dogma established by W. Pfeffer's school in Germany or the chemical-orientedplant physiology dogmaauthorized by F. Went's andK. Thimann's school in America. Interestingly, following World War II, the political alliance between Germanyand Japanalso led to a scientific alliance, and physical-oriented plant physiology also became prominentin Japan. I believe it possible that racism and sexism furthercompounded the problem in America since the results and conclusions of an Indian man (JagadischandraBose) and an Americanwoman (BarbaraPickard), the two majorproponents of the import- ance of electrical communicationin plants (Bose, 1906, 1913, 1926, 1985; Pickard, 1973, 1984), were somewhatoverlooked. Interestingly, the work of Sir J. C. Bose that showed that plants have almost the same life reactionsas humansinspired the Nobel LaureateRabindranath Tagore to dedicateto him a book of poems entitled Vanavani (Voice of the forest) (Kripalani,1962). Another reason why the study of electrical communicationin plants lacked popu- larityby plantphysiologists may have arisenfrom the belief held by some biophysical plant physiologiststhrough the late 1950s, that plant cells do not have a differentially permeableplasma membrane but only a cytoplasmicsurface layer or "plasmalemma"(see Briggs, 1957;Briggs & Hope, 1958;Briggs & Robertson,1948, 1957;Briggs et al., 1961; Dainty, 1962; Hope, 1951; Hope & Robertson,1956; Hope & Stevens, 1952; Hope & Walker,1975; MacRobbie & Dainty, 1958;Robertson, 1992; Walker, 1955, 1957).This

This content downloaded from 128.84.125.94 on Thu, 12 Nov 2015 14:20:46 UTC All use subject to JSTOR Terms and Conditions 310 THE BOTANICAL REVIEW view was championedby the prominentbiochemical (Briggs & Haldane, 1925) and biophysicalplant physiologist G. E. Briggs,who with R. Robertsonwrote in 1957:

It is generally accepted that there is, between the vacuole and the outside of a plant cell, a barrier or barriers which offer a high resistance to the passage of many solutes including salts of strong acids and bases. The tonoplast which separates the vacuolefrom the cytoplasmis accepted as one such barrier, but many assume that there is also another barrier [the plasma membrane]at the outside of the cytoplasmadjoining the cell wall. Althoughwe believe that no such barrier existsfor manysolutes we do not propose to consider all the evidence in detail, nor is it thepurpose of this article to surveythe historyof what we believe to be erroneous concepts.

This conclusion came from the applicationof Ockam's razor (Ockam, 1487; cited in Hope & Walker,1975). The logic was thatsince animalcells have only one limiting membrane,it stood to reason thatplant cells, too, have only one limiting membrane. It was obvious that plant cells have a visible vacuolarmembrane, surrounded by the protoplasmic "free space," so there seemed to be "no reason" to have a second membrane(Hope & Walker, 1975: 32). Even today many botanistsuse the term "plasmalemma,"which was first used by Mast in 1924 and defined by him in 1926 to signify the outer layer of the protoplasm of Amoebathat may or may not be a distinctdifferentially permeable barrier. Plasma- lemma was introducedby Mast (1926) as a noncommittalterm to replacecontroversial terms like Zellhaut, Plasmahaut, or Plasmamembran,used by Pfeffer to indicate a differentiallypermeable membrane; or Hdutchen(third layer, pellicle, or pellicula), used by Buitschli,Rhumbler, Schaeffer, Chambers, and Howlandto indicatea surface layer surroundinga colloidal or foam-like protoplasm,a protoplasmwhich itself is capable of causing differentialpermeability. Plasmalemmawas supposedto be a harmoniousword that fit somewherebetween Pfeffer's (1877) operationaldefinition of the plasma membraneand Buitschli's(1984) opinion:

The surface of the protoplasm has been referred to as the "Hautschicht", "Hautplasma"or best of all "hyaloplasma"..... I have decided to designatethis semipermeablelayer [the surface of theprotoplasm] as "plasmahaut"or "plasmamembran".... The "plasmamembran"can only be defined by its osmotic propertiesand not by its morphologicalcharacteristics. (Translatedfrom Pfeffer, 1877).

I am only doubtfulwhether the so-called protoplasmic membranepossesses the importancefor osmoticprocesses which Pfefferascribes to it; on the ground of which processes Pfeffer was reallyfirst led to assume its existence; they are now also supposed by him to prove its presence, even when direct observation shows nothing of itfor certain. (Biitschli, 1984)

"Plasmalemma"was then introducedinto the plant literatureby Plowe (193 la):

Mast (1924) has given us the term "plasmalemma"for the thin external layer

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which is the "membrane"if an osmoticfunction exists. Thisterm seems free from the difficultiespresented by both "ectoplast"and "plasmamembrane. " It has seemedpermissible,therefore, to extendits use to thebotanical world, and to employ it to denotea distinct,differentiated layeron the outersurface of theplantprotoplast.

Incidently, the two paperspublished in 1931 by Plowe presentdefinitive evidence showing that in the plant cells she studied, the "plasmalemma"was a differentially permeable "membrane"(Plowe, 1931a, 1931b). Given the preponderanceof the evidence that the "plasmalemma"of the plant cell is a differentially permeable membrane,it is both meaningfuland correctto refer to it as a plasma membrane. The study of electricalcommunication in plantsrequires the applicationof physical laws to biological systems, and the applicationof physical laws to organismsusually requiresthe use of the most simple organismsavailable or the isolation of single cells. The apprehensionof plant physiologists in using simple lower plant systems and physical laws to model biological systems is evident in the following quote by F. C. Steward(1935) in his assessmentof Osterhaut'swork:

It is in these operations that undue weight seems to have been given to certain features, not applicable to cells in general, of what is after all an obscure organism [Valonia]. Lastly, there seems grave danger that the admittedlyingenious and extensive workdevoted to models maydivert attention from thefundamentalfacts that, not only are they far removedfrom physiological reality, but also that the very principles upon which they are based are such thatjudgement must be reserved concerning their applicabilityto the general problem of salt absorptionin vivo.

This view is still prevalenttoday. As late as 1986, Stewardwrote in the Treatiseof Plant Physiology:

This chapter [Solutes in Cells] has developed its main theme with reference to cells of angiosperms, that is, the cells of the most highly organized land plants which encounterin nature the most complexenvironments. The earlier chapter (25) properly gave much attention . .. to the uses made of aquatic plants in investigations on cell physiology, especially in studies which capitalized on specialfeatures of their morphology.As, for example, the so-called giant cells of membersof the Siphonalesand the large internodalcells ofCharaand Nitella. Interestingas these organismswere, and are, the specialfeatures thatmake them attractive experimentalobjects also render them,from the standpoint of this chapter, atypical and not representativeof the plant kingdomas a whole.

It appearsthat the philosophy(Osterhaut, 1924) andexperiments of W. Osterhaut- the scientist so instrumentalin enticing the animalphysiologists Curtisand Cole and, later, Hodgkin and Huxley into studying simple systems that could be physically modeled-had the opposite effect on plantphysiologists. In physics, fundamentallaws were discoveredby choosing simple "ideal"systems like gases or dilute liquids, but not solids, to demonstratethe existence and behavior of atoms (Feynman,1965, 1985). Hydrogenatoms, not tin atoms,were used to explain the quantumnature of the atom. Likewise, animal behavioristsstudy the electrical

This content downloaded from 128.84.125.94 on Thu, 12 Nov 2015 14:20:46 UTC All use subject to JSTOR Terms and Conditions 312 THE BOTANICAL REVIEW behavior of isolated nerve cells in orderto understandcomplex behaviors from the knee-jerkresponse to the regulationof the heartbeat.However, within the scientific realmof plantphysiology, the presentationof the "botanicalequivalent of an ideal gas in a real world"(i.e., algal cells) has typically met with the reactionthat they are "far removed from physiological reality." While progressis being made in applyingthe laws of physics to the study of plants at the organ level (Cosgrove, 1993a, 1993b, 1993c; Green, 1981; Niklas, 1992; Silk, 1984; Thompson, 1963) and the suborganellarlevel (Jeanet al., 1989), the single cell has been under-utilizedin applying these laws to the study of plant physiology and development. The study of plant growth and development at the single-cell level complementsinvestigations done on the whole plant, isolated organelles,and protein complexes (Delbriick, 1970). Once a cellular approachis taken, it is very importantto identify the cell type that is respondingto the physiological or developmentalstimulus (Wayne, 1992). This is necessary since the signal transductionchains and responsecharacteristics of a given cell type in a multicellularorganism may vary(Hamill et al., 1991). Perhapsthe reason that we have no evidence for the involvement of cyclic AMP (cAMP) in plants is because the average cAMP concentrationof all cells in seedlings, and not just the cAMP concentrationin the respondingcells, was assayed (Amrhein, 1977; Brown & Newton, 1973; Elliot & Murray,1975). This experimentalapproach would be analogous to treating a whole calf with light or hormones and then grinding it all up to determinewhether or not the level of cAMP has changedor not. I believe that, even today, plant physiologists prefer the vitalistic approach to science ratherthan the unificationist approachthat is based on the hypothesis that there are certain fundamentallaws of naturethat can describe physics, chemistry, and biology in mathematicalform (Brillouin, 1949; Elasser, 1958; Loeb, 1912, 1916; Lotka, 1925). There is still a strong belief that there is a "gravedanger" in applying the fundamentallaws of physics to plant physiology. Stewardwas wary of Osterhaut applying electrical theory to membrane transportsince membrane transportde- pended on respiration,a component not included in the Nernst Equation.However, Kitasato and Spanswick later showed the relationship between the product of respiration (ATP) and membranetransport when they discovered the electrogenic pump and the equivalent electrical circuit that includes both the Nernst Equation and the ATP-utilizing electrogenic H+ pump. In my opinion, more progress can be made and has been made by building up a synthetic plant physiology from fundamentalprinciples (unificationist approach)than from describing phenomena and making interpretationswithout considering whether or not the interpretations obey the laws of physics. I hope this review inspiressome people to applybiophysical principles to the study of plant physiology and development.Using the common language of mathematics and principles of physics allows a ready transferenceof informationbetween related fields (Rothstein, 1951). Since all measurementsare based on physical principles,we have a chance of understandingour instrumentsas well as understandingthe plant if we include biophysical thinking in our approach.When we make theories in plant physiology and development, we are implicitly or explicitly assuming that certain mathematical or physical principles are either true or false. Implicit and wrong assumptionscan mislead the field for years. Interestingly,while sometimes the assumptionsof physical principles are met in

This content downloaded from 128.84.125.94 on Thu, 12 Nov 2015 14:20:46 UTC All use subject to JSTOR Terms and Conditions EXCITABILITYOF PLANT CELLS 313 biological systems, often they are not (Bohr, 1933; Donnan, 1927; Klieber & Gradmann,1993; Schrodinger, 1946; Szent-Gyorgi, 1960). This gives biologists a chance to broadenthe currentlaws of physics or, if you like, "naturallaws." According to Brillouin (1949),

We have been looking, up to now,for a physiochemicalinterpretation of life. It may well happen that the discovery of new laws and of some new principles in biology could result in a broad redefinitionof our present laws of physics and chemistry,and produce a complete change in point of view.

This is not so farfetchedas it seems when we realize that botanists have already played a leading role in the developmentof new physical principles.It was the work of the botanist Robert Brown (1828, 1866) on the movement of organelles in the cytoplasm of pollen that gave rise to the acceptanceof the reality of atoms and the determinationof Avogadro's Number(Einstein, 1926; Perrin, 1923). And it was the work of botanistsH. de Vries, C. von Nageli, W. Hofmeister,and W. Pfeffer that led J. H. van't Hoff to apply the gas laws to dilute solutions and develop the Boyle-van't Hoff Law, which relatesconcentration to osmotic pressure(Ling, 1984). The botanist Friedrich Reinitzer discoverd liquid crystals (Brown & Wolton, 1979). And the physicist George Stokes used his own studies on the fluorescence of chlorophyll extractedfrom bay leaves to develop the relationshipbetween absorptionand fluorescence known as Stokes' Law (Harvey, 1957; Lloyd, 1924). In fact, the botanist Andrew Caesalpinus,who was the Chairof Medicine at the University of Pisa from 1597 to 1592, was Galileo's teacher(Brewster, 1846). Some animal physiologists of the nineteenthcentury, including Adolf Fick, Her- mann von Helmholtz, and Jean Poiseuille also contributedgreatly to physics. They were membersof a "new school of physiologists"who rejectedvitalist explanations of life and attemptedto describe physiological phenomenain terms of physical and chemical principles. J. B. S. Haldane (1924) and R. A. Fisher (1930) contributed greatly to mathematicsin the twentieth century as a consequence of their need to describe genetics quantitatively.These nineteenth-and twentieth-centuryscientists tried to unite naturalhistory (e.g., botany,zoology, physical geography,geology, and mineralogy) with naturalphilosophy (e.g., mathematics,physics, astronomy, and chemistry). Natural philosophy and natural history were separated based on the assumption that the former but not the latter was susceptible to mathematicaland experimentaltreatments (Huxley, 1902). Lastly, althoughthe study of electrical communicationin plants had alreadybeen ignored by the plant physiology communityfor years, duringthe 1960s the study of electrical communicationin plants became taboo following the publicationof high- profile, irreproduciblework that reportedthat plants that were connected to a lie detector were able to perceive dangerand gave a perceptiblechange in the electrical resistance of a leaf while witnessing the murderof brine shrimp (Backster, 1968; Horowitz et al., 1975). As a result of the renewed interestin ionic messengers,particularly Ca2+, in plant development (Trewavas, 1986), coupled with the successful introduction of the vibratingelectrode (Nuccitelli, 1986) andpatch clamp techniques to plantphysiology (Hamill et al., 1981; Hedrich& Schroeder,1989), many more plantphysiologists are bridging the gap between the chemical-orientedthesis of communicationand the

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physical-orientedantithesis of communicationto forma synthesisthat recognizes the importanceof bothchemical and electrical communication in consideringthe mechanisms responsiblefor plantdevelopment and the responsesof plantsto the environment. I will end with a passage from a letterBenjamin Franklin wrote to Peter Collinson on 29 April 1749:

Chagrineda little that we have been hithertoable to produce nothingin this way of use to mankind;and the hot weather coming on, when electrical experiments are not so agreeable, it is proposed to put an end to them for this season, somewhathumorously, in a partyofpleasure, on the banksof theSkuylkil. Spirits, at the same time, are to be fired by a spark sentfrom side to side through the river,without any other conductorthan the water;an experimentwhich we some time since performed,to the amazementof many.A turkeyis to be killedfor our dinner by the electrical shock, and roasted by the electricaljack, before afire kindled by the electrified bottle: when the healths of all thefamous electricians in England, Holland, France, and Germany are to be drank in electrified bumpers [small thin glass tumblers,nearly filled with wine, and electrified as the bottle. Thiswhen brought to the lips gives a shock, if theparty be close shaved, and does not breathe on the liquor (Cohen, 1941)], underthe discharge of guns from the electrical battery.

XIII. Acknowledgments Our laboratorywork is supportedby NASA. Thanksto Drs. G. Bailey, A. Furuno, 0. Hamill, M. Lea, C. Leopold, S. Meissner, T. Mimura,K. Niklas, L. Palevitz, M. Rutzke, J. Saienz-Badillos,R. Spanswick,A. Stachowitz, M. Staves, R. Turgeon, S. Wayne, and S. Williams for their comments on the manuscript.I would also like to thankmy parents-for everything.

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V. Appendix A: List of Abbreviations a is (-Kjujzje/dx) b is the numberof channels in a patch (dimensionless) c is the speed of light (3 x 108 m s-1; Spiridonov,1986) d representsan infinitesimaldifference exp is the base of the naturallogarithm (2.72) e is the elementarycharge (1.60 x 10-19coulombs per particle;Spiridonov, 1986) f representsa function f(x) is 1/[exp(zjFEm/RT)- 1] where x = Em and is a constantwhen Em= Ej g is the accelerationdue to gravity(9.80 m s2) on the surfaceof the earthon an object of mass m (in kg), andis derivedfrom the gravitational1constant(G,667x11 Nm2kgN ; Spiridonov, 1986), the mass ofgrvttoalcntn the earth(me, 5.98 x 10Pkg)6.67 kg) Xand theh radius of the earth (re 6.38 x 106 m) using the equivalence of the following formulafor force: Force = mjg = Gg(memj/re2) gj is the partialspecific conductanceof the membranefor substancej (in S m-2) h is Planck's Constant(6.65 x 1034 J s; Spiridonov, 1986) h' is a height (in m) i is a subscriptor superscriptthat represents "in" j is a subscriptthat representssubstance j j' = I(-1) k is Boltzmann's Constant(1.38 x 1W23J K-1; Spiridonov,1986) k' = (nJni) = exp(zjBEj) I is length (in m) m is molecularmass of a substance(in kg mol-1) mjis the mass of substancej (in kg) n1is the concentrationof substancej (in mol m-3) ne is the concentrationof a given ion on the exoplasmic side of the membrane(the E-space, in mol m-3) np is the concentrationof a given ion on the protoplasmicside of the membrane(the P-space, in mol m-3) o is a subscriptor superscriptthat represents "out" q is the numberof chargedparticles that move across an area (in m-2) q' is the numberof chargedparticles (dimensionless) qj is the numberof ions j that move across an area(in m-2) r is the radius (in m) rHis the hydrodynamicradius (in m) rj is the specific resistanceof the membranefor substancej (in Q m2) t is time (in s) u is the mobility (in m2 J-1 s-1) Urnis the molar mobility constant(in mol m2 J-1 s-1) u' is the mobility (in m2 V-1 s-1) v is velocity (in m s-1) x is a distance (in m)

This content downloaded from 128.84.125.94 on Thu, 12 Nov 2015 14:20:46 UTC All use subject to JSTOR Terms and Conditions 336 THE BOTANICAL REVIEW z is the valence of an ion or electron (dimensionless) Za iS the valence of an anion (-) Zcis the valence of a cation (+) zj is the valence of substancej (dimensionless)

A is the area (in m2) C is the specific capacitance(in F m-2) Cmis the specific membranecapacitance (=1O-2 F m-2) C' is capacitance(in F) D is the diffusion coefficient (in m2 s-4) Dj is the diffusion coefficient of substance] (in m2 s-1) E is an electric potential(in V) Ed is the diffusion potentialdetermined by the G-H-K Equation(in V) Et is the total driving force on an ion (in V) Emis a membranepotential (in V) Ej is the equilibriumpotential of ion j (in V) E't is the potential (in V) at time t Eo is the potential (in V) at time 0 Eo. is the potential (in V) at time infinity F is Faraday'sConstant (9.65 x 104 coulombs mol-1; Spiridonov,1986) F' is force (in N) F* is the electric force between two chargedbodies (in N) G is the specific conductanceof the membrane(in S m-2) G' is Gibbs's free energy (defined as an intrinsicquantity, in J mol-1) H is enthalpy (defined as an intrinsicquantity, in J mol-V) I is electrical currentdensity (in A m-2 = coulombs m-2 s-1) I' is electrical current(in A = coulombs s-l) IOis the currentpassed througha patch (in A) I, is the currentpassed by a single channel (in A). I is electrical currentdensity carriedby ion j (in A m-2) I is information(in J K-1) Jtot, Jn are the total net flux (in mol m-2 s-l) Jp is the net flux due to pressure(in mol m-2 sl) Jg is the net flux due to gravity (in mol m-2 s-1) Jd is the net flux due to diffusion (in mol m-2 s-1) Jel is the net flux of chargedparticles (in mol m-2 s-1) Jj is the net flux of substancej (in mol m-2 s-1) Jw is the net volume flow of water (in m3 m-2 s-) JK is the net flux of K+ (in mol m-2 s-1) JNa is the net flux of Nat (in mol m-2 s-1) Jcl is the net flux of Cl- (in mol m-2 s-1) Ji is the influx (in mol m-2 s-l) JOis the efflux (in mol m2 s-l) Kj is the partitioncoefficient of substancej (oil/water;dimensionless) KE is kinetic energy (in J) Lpis the hydraulicconductivity (in m s-I Pa-l) NA is Avogadro's Number(6.02 x 1023mol-1) P is pressure(in Pa)

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Pj is the permeabilitycoefficient of substancej (in m s-1) POis the probabilityof a channel being open (dimensionless) R is the gas constant (8.31 J mol- K-1; Spiridonov,1986) R' is the specific resistance (in El m2) Rmis the specific membraneresistance (in Qi m2) S is entropy (defined as an intrinsicquantity, in J moV17K-1) T is absolute temperature(in K); with absolute temperaturescales, the "value of a degree"is independentof temperatureand 273.16 K is defined as the triplepoint of water (AmericanSociety for Testing and Materials,1974) Tl* is the thermalinformation (in J mol-1) 7' is tension (in N m-1) V is volume (in m3) V j is the partialmolal volume of a substance(in m3 mol-1) Y is JjdxlujzjedE Z is the total specific impedancedue to resistorsand capacitors(in Q i2) ZRis the impedancedue to resistors(in Q) Z( is the reactance(in Q)

B = FIRT= elkT A representsan infinitesimaldifference between state variables nt=3.14 1 represents"the sum of' ? = relative permittivity(dimensionless) ao is the permittivityof a vacuum (8.85 x 10-12F m-1) 1lis the viscosity (in Pa s) t is the chemical potential(in J mol-1) Rj is the chemical potentialof substancej (in J mol-1) Rj is the chemical potentialof substancej in the standardstate (in J mol-1) t is the membranetime constant(in s)

Note: All units are given in SI units and all constantsare roundedoff to two decimal places or less.

XVI. Appendix B: InformationTheory, Thermodynamics,and Cell Communication

The development and adaptive behavior of multicellularorganisms depends on communicationbetween cells. Communicationis the faithfultransfer of information. Informationtransfer can occur at many levels: between the environmentand the cell, between cells in a tissue, between tissues, and between organs.In orderto get a better understandingof communicationand methods of communication,we are going to define information in thermodynamicterms. It appears that information theory (Bialek, 1987; Block, 1992; Mehra, 1994; Mercer, 1981; Orlozi, 1991; Raisbeck, 1963; von Neumann, 1963; Young, 1987) may provide a sturdy framework for understandingthe informationtransfer involved in plant growthand development. According to Brillouin (1949, 1950, 1956, 1959, 1964), Shannon and Weaver (1949), Szilard (1964), Rothstein (1951), Tribus and Mclrvine (1971), and von Neumann (1963), informationcan be consideredequivalent to negative entropy (or

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negentropy)where entropyis equivalentto the numberof possible states. The greater the number of possible states, the greater the entropy and the greater the lack of information. I* = -AS (B.1) where I* is information(in J K-1) AS is entropy(in J mol-VK71)

It is useful to thinkabout information in energy units.In orderto do this we multiply by the absolutetemperature (T, in K).

T1*= -TAS (B.2) where TI*represents thermal information (in J mol-1)

We can relate the thermalinformation to Gibbs's free energy using the following formula:

AG'-=AH - TAS (B.3) where AG' is the Gibbs free energy (in J mol-1) AH is the enthalpy(in J mol-1) Tis the absolutetemperature (in K; and is assumedto be constant) AS is the entropy (in J mol-1 K71;Bergethon & Simons, 1990; Klotz, 1967; Lewis & Randall, 1923; van Ness, 1983).

I have defined Gibbs's free energy as an intrinsicquantity. This is in keeping with the spiritof JosiahGibbs, who definedit in termsof free energy(V, in kcal) perimplied mass of 1 kg. Gibbsused kilogramsinstead of moles becausein the nineteenthcentury, when the atomic and formulamasses were constantlybeing revised, moles were not reliable units comparedto the mass of a substance(Lewis & Randall, 1923).

SinceTI* = -TAS,

TI*= AG' - AH. (B.4)

We can relate thermal informationto the chemical potential of a system since AG'(defined as an intrinsicquantity) is relatedto the chemical potentialaccording to the following definition:

AG' =AA (B.5)

where AG' is Gibbs's free energy (in J mol-1) Ag is the chemical potential(in J mol-1)

T'hechemical potentialas conceived by Gibbs (1928) describesthe contributionof

This content downloaded from 128.84.125.94 on Thu, 12 Nov 2015 14:20:46 UTC All use subject to JSTOR Terms and Conditions EXCITABILITYOF PLANT CELLS 339 every type of energy (includingconcentration effects, mechanicalenergy, etc.) to the system. Thus,

Ag = Aif + RTA(ln n) + zFAE + PAV + F 'Ax +AT'A + mgAh' +... (B.6) where

Ag is the difference in the chemical potential (in J mol-1) of the system in the final state minus the initial state g' is the chemical potentialof the system in the standardstate (in J mol-1) and Ag', and is assumedto be equal to 0 J mol-1 R is the gas constant(8.31 J mol-VK-1) T is the absolutetemperature (in K) A(ln n) is the differencein the naturallog of the concentrationof a substancein the final state minus the initial state (dimensionless) z is the valence of a substance(dimensionless) F is the FaradayConstant (9.65 x 104 coulombs mol-1) AE is the electrical potentialdifference of the system containingthe substance in the final state minus the initial state (in V or J coulomb-1) V is the partialmolar volume of a substance(in m3 mol-P) AP is the difference in the pressureon the system containingthe substancein the final state minus the initial state (in Pa) F ' is the translationalforce appliedto the system (in N) Ax is the difference in the dimensionof the system containingthe substancein the final state minus the initial state (in m mol-1) A is the surface areaof the system (in m2 mol-4) AT' is the difference in the tension of the surface of the system containingthe substancein the final state minus the initial state (in N m-1) m is the molecularmass of a substance(kg mol-1) g is the accelerationdue to gravity (9.80 m s-2) Ah' is the differencein heightof the system containingthe substancein the final state minus the initial state (in m).

Thus, by combiningequations B.4 andB.5, and since AR' = 0 J molPl, we get

Tl* = RT A (ln n) + zFAE + PAV + F 'Ax + AT'A+ mgAh'- AH. (B.7)

EquationB.7 tells us that electrical, mechanical,and gravitationalenergy, as well as chemical energy,can be used for information.This informationcan be used to direct plant growthand development(Sinnott, 1946). Thus, in thermodynamicterms, informationcan be releasedor storedfollowing a changein the concentrationof a substance [A (ln n)], a change in electricalpotential (AE), or a change in volume (AV),distance (Ax or Ah), and/or tension (AT'). AH (at constantpressure) represents energy con- vertedinto heat. The thermalinformation of the message is positive only if the energy in the stimulus(or stimuli)is greaterthan AH, the level of thermalnoise in the system. At room temperature(298 K), the minimumheat content of (or thermalnoise in) the system is equalto (312)RT,or 3714.6 J mol-1. The energyabsorbed by any one receptor must be greaterthan (312)(R/NA)Tor (312)kTwhich is 6.2 x 10-21J.

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Informationcan also be released and stored following a change in other types of energy, including light energy [hNAAvwhere h is Planck's Constant(6.65 x 10-34J s), NA is Avogadro's Number (6.02 x 1023 mol-1), and Av is the difference in frequencyof the light absorbedor radiated(in s-1), assumingthat the frequencyof the initial state is 0], or by changingthe sensitivityof a receptor(protein) to substance(n), an electrical potentialdifference (AE),mechanical deformation (P, F' and AT'). The concept of energy has changed over the past 200 years since Count Rumford (BenjaminThompson) observed that the heat producedduring the boringof a cannon was roughly equivalent to the mechanical energy expended. Later, James Joule determined the quantitativerelationship between mechanical energy and heat by measuringthe change in temperatureof water in which a paddle wheel was turning (Joule, 1852). Joule also determinedthe relationshipbetween electrical energy and heatby measuringthe changein temperatureof wateras electricitywas passed through it. Otherexperiments showed the equivalenceof chemical energy and heat by adding reactantsto a solution (Joule, 1852). These and otherexperiments led to the principle of the equivalence beween various types of energy and the powerful concept of the conservationof energy. This review treatselectrical energy and chemical energy as equivalent in providing informationneeded for plant growth and development, and focuses on the methodsand results gained from studieson how electricalenergy plays a partin informationtransfer in plant cells.

XVII. Appendix C: Membrane Capacitance

A capacitoris composedof an insulatinglayer sandwichedbetween two conducting layers (Cole, 1928; Horowitz & Hill, 1989). Capacitance(C' in F) is the propertyof a membranethat resists changes in voltage. It is a measureof the chargein coulombs that must be added to the membranein order to raise its potential by 1 V. The magnitudeof the capacitanceis dependenton the electricalproperties of the insulating layer or dielectric (E-, in F m-1), the area of one side of the insulatinglayer (A, in m2), and the thickness of the insulating layer (x, in m). The capacitance can be determinedfrom the following equation:

C' = (F-F-O)Alx (C.1) where C' = capacitance(in F) ? = relative permittivityof the insulatinglayer (3 for lipids and 80 for water, dimensionless) F- is the permittivityof a vacuum (8.85 x 10-12F m-1) A is the surface areaof one side of the insulatinglayer (in m2) x is the thickness of the insulatinglayer (in m).

Since the capacitancedepends on the surface areaof the capacitor,we usually talk about the specific capacitance(C, in F m-2) which is normalizedwith respect to area and is equal to C'/A. The specific capacitanceof many membranesis approximately 0.01 F m-2 (Cole, 1970; Cole & Cole, 1936a, 1936b; Cole & Curtis, 1938; Curtis & Cole, 1937; Dean et al., 1940; Findlay, 1970; Fricke, 1925; Fricke & Morse, 1925; Williams et al., 1964).

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The specific membranecapacitance (Cm, in F m-2) is thought to be a propertyof the nonconductinglipid bilayer since it can be accountedfor by the propertiesof the bilayer (Cole, 1970). Assuming the relative permittivityof lipids is approximately3 and the thickness of the lipid bilayer is approximately2.7 x 10-9 m, the specific membranecapacitance will be approximately10-2 F m-2, accordingto equationC.2.

Cm= (?so)/x. (C.2)

The capacitanceof a materialor membranecan be determinedalgebraically from the definitionof a capacitorgiven in equationC. 1 or C.2, respectively.These equations can also be used to determinethe identity of the dielectricmaterial or the areaof the membraneif the specific capacitanceis known. The capacitancecan be measuredempirically since the capacitanceis defined as the amount of charge per unit area (in coulombs m-2) that can be separatedwhen an electrical potential (E, in V) is placed across two conducting elements that are separatedby a nonconductingsubstance. When 1 V is placed across a 1 F m-2 capacitor,1 coulomb of positive chargebuilds up on the plate of the capacitorconnected to the positive terminalof the batteryand 1 coulomb of negative chargebuilds up on the plate of the capacitorconnected to the negative terminalof the battery(Horowitz & Hill, 1989). Conversely,capacitance can be defined as the numberof eitherpositive or negative chargesper unit areathat must be separatedin orderto establish an electricalpotential of 1 V. Since the capacitancerelates the chargeseparation to an electricalpotential, we get the following relation:

C = -(zeq)IE or E = -(zeq)IC (C.3) where C is the specific capacitance(in F m-2) z is the valence of the chargedparticle e is the elementarycharge (1.6 x 10-19coulombs) q is the numberof chargedparticles that move to one side of the capacitor(in m-2) E is the electrical potentialdifference across the capacitor(in V).

If we differentiateequation C.3 with respect to time, we get

dEldt = -d[(zeq)/C]/dt, (C.4)

Since C is a constant,we can take it out of the differentialand we get

dEldt= -(1/C) [d(zeq)/dt]. (C.5)

We can simplify. Since d(zeq)ldt is equal to the amount of charge moving per unit area per unit time (in coulombs m-2 s-1), it is equal to the currentdensity (I, in A m-2).

dEldt = -I/C (C.6)

This content downloaded from 128.84.125.94 on Thu, 12 Nov 2015 14:20:46 UTC All use subject to JSTOR Terms and Conditions 342 THE BOTANICAL REVIEW where C is the specific capacitance(in F m-2) (zeq)/dtis equal to currentdensity (I, in A m-2).

Thus a capacitordetermines how long it will take a given currentto cause a change in voltage. By combining equationC.6 with Ohm's Law,

I = -EIR' (C.7) where R' is the specific resistance(in Q m2), we see that

dEldt = -EI(R'C) or (dEIE)= -dtl(R'C) (C.8)

We can integrateequation C.8 to determinethe potentialdifference at any time t (in s). Rememberthat for the indefiniteintegral, JdEIE = In E +/- constantof integration, In x - In y = In (x/y), In and exp are inverse functions,and exp(ln x) = x; thus,

In E't - In Eo = -[tl(R'C)] or E't = Eoexp[-tI(R'C)] (C.9) where t is the time (in s) E't is the potential (in V) at time t Eo is the potential (in V) at time 0 R' is the specific resistance(in Q m2) C is the specific capacitance(in F m-2).

Thus, the membranepotential (Em) will not change instantaneouslyin response to a change in current,but will change logarithmicallyin a manner that depends on RmCm.RmCm determines the time it takes for the currentto charge the membrane capacity.For this reason,RmCm is called the membranetime constant (t, in s; Duncan, 1990). When we are consideringthe time dependence of how membranepotentials respond to a change in current,it is importantto take into considerationboth the capacitive currents and the ionic currents.Since in a typical characeancell Rm is approximately1 Q m2 and Cmis about 102 F m-2, the membranetime constant is approximately10 ms. Thus it takes 10 ms to reach 0.63 E,, where E.c is the potential (in V) at time infinity. It takes 20 ms to reach0.86 Ec,,,;30 ms to reach 0.95 Ec,,,;40 ms to reach 0.98 E,,,;and 50 ms (or 5RmCm),9to reach 0.99 E,,,. EquationC. 8 is essentially a restatementof Coulomb'sLaw, which relatesthe force (F*, in N) to the distance between two chargedspherical bodies. Coulomb's Law is given by the following equation:

F* = (zeq')(zeq')I(4nx2txO) (C. 10) where F* is the force of attraction(when F* is negative) or repulsion (when F* is positive) between two sphericalbodies (in N) q is the numberof charges (dimensionless) z is the valence (dimensionless) e is the elementarycharge (1.6 x 1019 coulombs)

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x is the distance between the two chargedbodies (in m) ? = relative permittivityof the insulatinglayer (dimensionless) cOis the permittivityof a vacuum (8.85 x 10-12F m-1).

Let's rearrangeequation C.10:

[(F*x)I(zeq')](4rrxo0?)= (zeq'). (C.11)

Now we can see that (F*x)I(zeq')is equivalentto the potential (in V) between the two spheres and can be designated -E (in V). (F*x)l(zeq') is defined as -E since positive currentmoves spontaneouslytoward negative potentials. If we assume that the capacitance(4xxso0, in F) is constant and we differentiate equation C.11 with respect to time, then after removing the capacitance from the differential(because we assume that it is constant)we get

-(dEldt) (4rrxs-0)= d(zeq')/dt (C. 12) where d(zeq)ldtis the positive currentthat flows between the two spheresand can be designatedI' (in A). Thus,

-(dEldt) (4itx??O)= I (C. 13) and

dEldt = -I'/(4trtvo). (C.14)

42txrepresents the "average"length (in m) of the lines of force between the two point spheres. The shortestline between the two spheres is a straightline x m long. The otherlines are curvedand get longer (andthe force gets weaker)as they approach infinity. If we line up many point spheresto form plates, the lines of force reinforce each other so that in essence the "average"length of the lines of force is equal to the distance between the plates (x, in m). Thus, equationC. 14 becomes

dEldt = -IT'(x?s0), (C. 15) and thus

dEldt = -I'/C' (C. 16) where C' is the capacitance(in F), defined in equationC. 1, and relates the change in voltage over time to the current. Up to now, we have made the assumptionthat the currentis constant.However, the magnitudeof the currentcan vary.The impedance(in Q) is a propertyof the membrane that relates the change in currentto the change in voltage when the membraneis clampedby a sinusoidallyvarying voltage (Horowitz& Hill, 1989;McClendon, 1927; Remington,1928). While the magnitudeof the currentis proportionalto the magnitude of the voltage (according to Ohm's Law), there will be a time delay if theycircuit containsa capacitor.A capacitorintroduces a time delay betweenthe changein voltage and the change in current.The time delay (also known as the phase change)introduced

This content downloaded from 128.84.125.94 on Thu, 12 Nov 2015 14:20:46 UTC All use subject to JSTOR Terms and Conditions 344 THE BOTANICAL REVIEW by the capacitordepends on the frequencyof the voltage. When the frequencyis 0 Hz (i.e., direct current,d.c.), the frequency-dependentresistance, or reactance, of the capacitoris infinite, and currentwill not pass throughthe capacitor;consequently, no time delay or phase change is observed. The time delay introducedby the capacitorcan be determinedby comparingthe sine wave of the outputcurrent with the sine wave of the inputvoltage. Althoughthis is conceptuallyeasy, it is mathematicallyrelatively difficult and so we use the algebra of complex numbers to representthe two sine waves and to determine the phase relationship. Complex numbershave a real partand an imaginarypart. In the case of impedance, the real partrelates to the impedancedue to the resistorsin the circuit [ZR(in Q) = R where R is the resistance of the resistors], and the imaginary part relates to the impedancedue to capacitorsin the circuit [i.e., reactanceZc (in Q) = l/j'wC' where j' =A(-1), to is the frequencyof the voltage change (in s-1), and C' is the capacitance of the capacitor (in F)]. Only the imaginary part of the impedance introduces a time delay or phase change. Thus, the imaginary impedance is the "frequency-dependent resistance" that introduces a phase delay in the currentrelative to the voltage change. Thus, the specific resistance(R', in Q m2) is the factorthat relates the currentto the voltage in a d.c. circuit, and the specific impedance (Z, in Q m2) is the factor that relates the currentto the voltage in an alternatingcurrent (a.c.) circuit.

XVIII. Appendix D: Derivationsof the Nernst Equation

T'heNernst Equationis a formulathat gives us informationon how two different forces (chemical and electrical, each in N mol-1) act on a concentrationof particles (mol m-3) to give us a certaindistribution of particles at equilibrium(Nernst, 1888, 1889; Nobel, 1991; Stein, 1986, 1990). It is assumed that the movement of each particle is independentof the movement of any other particle. The Nernst Equation describes the electrical potentialgradient that may arise when the initial distribution iS uneven. Like any robust equation,the Nernst Equationcan be derived from many starting points. One way is based exclusively on irreversiblethermodynamics (Katchalsky & Curran,1965). Irreversiblethermodynamics, in contrastto classical thermodynamics, considers what happens to a system as it approachesequilibrium instead of what happensat equilibrium. We derive the NernstEquation from irreversible thermodynamics by assumingthat all fluxes are a resultof an appliedforce (in N mol-1) on the particlesthat are moving:

Flux xForce. (D.1)

We also assume that the force must act on a given amountof substancein a given volume (i.e., concentration,in mol m-3). Thus,

Flux oxForce x Concentration. (D.2)

The units of the productof force x concentrationare N mi3 or force per unitvolume. According to the principles of irreversiblethermodynamics, a phenomenological

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Flux = Coefficient (Force per unit volume). (D.3)

The permeabilitycoefficient of a substancej (Pj, in m s-1) relates the flux of that substance to the difference in its concentration.The permeabilitycoefficient is a function of the diffusion coefficient (Dj, in m2 s-1) and the partitioncoefficient (Kj, dimensionless). The partitioncoefficient is the ratioof the solubility of the substance in water and oil or any hydrophobicsubstance that best representsthe membranein question (Stein, 1986). The partitioncoefficient relatesthe concentrationof a particle in the aqueoussolution surroundingthe membraneto the concentrationof the particle in the "oily"membrane (oil/water). The permeabilityalso dependson the distance(x, in m) over which the concentrationdrop occurs. The permeabilitycoefficient is given by the following equation:

Pj = (DjKjlx). (D.4) In biology, various phenomenologicalcoefficients exist that relate a given type of movement (flux) to a driving force. In additionto the permeabilitycoefficient (Pj, in m s-1), which relatesthe flux of the substance(Jj, in mol m-2 s-1) to the concentration gradient(dnj, in mol m-3),

Pj = -Jjl(dnj), (D.5) otherphenomenological coefficients exist. These include the specific conductanceof the membrane(G, in S m-2) which relatesthe flux of chargeor currentdensity (I, in A 2 or coulombsm2 s1) to the electricalpotential difference (dE, in V). [Thisis related to Ohm's Law wherethe coefficientis R (specificresistance in Q m2) andR' = 11G.]

G = -IldE. (D.6)

We also have the hydraulicconductivity (Lp, in m s-1 Pa-l) which relatesthe volume flow of water (Jw, in m3 m-2 s-1) to the difference in pressure(dP, in Pa).

Lp = -JwfdP. (D.7)

Since the concentrationof water,nw (in mol m-3), is approximately55,000 mol m-3 (55 M), we can multiplyJw by nw to get the flux (in mol m-2 s-1). This is related to the Hagen-PoiseuilleLaw, where the volume flow of water (Jw, in m3 m-2 s-1) is related to the difference in pressure(dP, in Pa) by the coefficient ntr2/81r,where r is the radiusof the tube (in m), I is the length of the tube (in m), and 1jis the viscosity of the fluid (in Pa s). Irreversiblethermodynamics does not tell us anything about the mechanism of movement.But it does specify very clearly the directionof movementand whetheror not the movement in a given directionis spontaneous.Figuring out the direction of movement with respect to the directionof the force can be somewhatconfusing, so I will begin by defining the importantconventions and concepts.

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1. Fluxes occur as a result of a force. 2. The magnitudeof the flux dependson the magnitudeof the force andthe magnitude of the phenomenologicalcoefficient. 3. The directionof the net flux is determinedexclusively by the chemical potentialin each compartment. 4. The net flux always occurs from higher chemical potential to lower chemical potential. 5. Since the difference in chemical potential always is defined as the chemical potential of the final state minus the chemical potentialof the initial state, the net flux always occurs in a directionexactly opposite to the differenceof the chemical potential.That is, the flux is spontaneous(i.e., passive) when the difference in the chemical potentialis negative.

The chemical potential (gj, in J mol-1) is defined as

jij=J,Uj + Rln nj + zjFE + VJP + mjgh'+... (D.8) where ,j is the chemicalpotential of substance in the standardstate (in J mol-1). The standardstate is definedas 298 K, 1 molar,zero electricalpotential, 0.1 MPa of pressureand zero height above sea level. Theother terms describe how the chemicalpotential of the substancechanges when it is in "nonstandard" conditions. (In standardconditions, all otherterms become zero.) n is the concentrationof the substance(in mol m-3) z is the valence of the substance(dimensionless) is the partial molar volume of the substance (im m3 mol-1). This is the differentialchange in volume of a solution when a differentialamount of solute is added.To visualize this, considerhow muchthe volume of a 10-3 m3 (one liter) solutionof waterchanges when you add one mole of solute. This value is typically close to 3 x 10-5 m3mol-1 (for NaCl and KCI,it is about 2.7 x 10-5 and 3.7 x 10-5 m3 mol-1, respectively). m is the mass of the substance(in kg) E is the electricalpotential of the compartment(in V) P is the hydrostaticpressure (above or below 0.1 MPa)of the compartment(in Pa) h' is the height of the substanceabove or below sea level (in m) F is the FaradayConstant (9.65 x 104coulombs mol-1) R is the gas constant(8.31 J mol-1 K-1) T is the absolutetemperature (in K) g is the accelerationdue to gravity (9.80 m s

Since, when we talk about cells, we will be talking about the distributionof ions between compartmentsthat are both at essentially the same pressureand height, we can eliminate the last two terms and simplify the equation:

=, + R7Tlnn + zFE1 (D.9a) ,2 =J + RIlnnn2 + zFE2 (D.9b) where g, and g2 referto the chemical potentialsof substancej in compartments1 and 2, respectively.

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It may seem thatthe hydrostaticpressure that develops across a plantcell can be so large that we should not eliminate the pressureterm. In fact, we can, as I will show you. If we have a solute with a partialmolar volume (V j) of 3 x i0- m3mol-1 and a hydrostaticpressure of 0.5 MPa acting on the solution on one side of the membrane (inside the cell) and a hydrostaticpressure of 0 MPa acting on the solution outside, the difference in the chemical potentialin the two compartmentswill be -15 J mol-1. By comparison,if we have a 10-fold concentrationdifference between two compartments, the differencein the chemical potentialRTln(n2/n1) of those compartmentsis -5702 J mol-1; or if we have a -0.05 V difference in voltage across the membrane, the difference in the chemical potential (zFA E) of the two compartmentsis -4825 J mol-1. Therefore,we can eliminatethe pressureterm since it contributesless than % to the chemical potential. Thus, it is valid to consider the electrochemicalpotential of each compartmentto be equal to equationsD.9a and D.9b. The difference in chemical potential (g2 - gl) is defined as the chemical potential of the final state (which, for a spontaneousprocess, is the statewith the lowest or least positive chemical potential)minus the initial state (which, for a spontaneousprocess, is the state with the highest or most positive chemical potential). Under conditions where a changein enthalpyis negligible, this meansthat we aredefining the difference in chemical potential to be the state with higher entropyminus the state with lower entropy.

( 92-g1) = (g + Rlln n2 + zFE2) - (gt + Rln nI + zFE1). (D. 10)

2 - ,1 is negative for a spontaneous,passive, or exergonic process; positive for a nonspontaneous,active, or endergonic process; and zero for a process which is at equilibrium. Let's consider the flux of an unchargedparticle where z = 0. The difference in the chemical potential (g2 - RI) is then given by

(R2- g1) = (g + Rln n2) - (g + Rln nl). (D. 1 1)

We can simplify the equationusing algebraand the rules of logarithms:

(g2 - Itl) = (Rnn n2) - (RTnnnl) (D.12) (g2 - g1) = RT(lnn2 - Innl) (D. 13) (g2 - g1) = RT[ln(n2/nl)] (D.14)

When n1 > n2, the flux is from ni to n2. The initial state is nl. The final state is n2 andthe differencein chemicalpotential is always written(by convention)as final state minus initial state. The net flux due to diffusion (Jd, in mol m-2 s-1) is proportionalto the difference in the force per mole (in N mol-1) in the final state minus the initial state.The force per mole is equalto the differencein chemicalpotential (dg, in J mol-1) over the distance(dx, in m) the concentrationgradient exists. The flux is relatedto the force by the following relation:

Jd ?x(-dt/dx). (D. 15)

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The gradient of the chemical potential (in J mol-1 m-1 or N mol-1, which is force per mole) is defined as the difference in chemical potentialdivided by the distance. The gradientin chemicalpotential is oppositein sign to the drop in chemicalpotential. The drop is defined as the chemical potential of the initial state minus the chemical potential of the final state, divided by a distance. The flux also depends on the concentrationof substance(n, in mol m-3) the force acts upon:

Jd u(n)(-dgldx). (D. 16)

Note: (n)(-dp1dx)is in units of force per unit volume (N m-3) and (u/NA) are the constantsof proportionalitythat relate the flux to the force per unit volume.

Jd = (u/NA)(n)(-dIdx) (D. 17)

where NA is Avogadro's Number(6.02 x 1023mol-1) u is the mobility (in m2 j-1 s-1). The mobility can also be given in units of velocity (in m s-1) per unit force (in N) and thus relates the velocity of a particleto the force exertedon it.

Since the derivative of ln(n2/n1)= dnln = (n2 - n1)ln where n1 and n2 are the concentrationsat points 1 and 2, respectively and n is the concentrationof the whole system at equilibriumwhich is equal to the concentrationat the midpoint between points 1 and 2. Since RT is constant,the derivativeof RT is RT;then

dg = dRTln(n21n1)= (RT)(lfn)(n2- n1), (D.18) and equationD. 17 becomes

Jd= (u/NA)(n)(-RT)(11n)(n2- n1)/dx. (D.19)

Rearrangeterms:

Jd= -(u/NA)(RT)(n)(1lIn)(n2 - nl)/dx. (D.20)

Simplify, since (n)(Iln) = 1:

Jd = -(u/NA)(R7)(n2- nl)/dx. (D.21)

We will use this form of the equationto derive the NernstEquation. This equation relatesthe flux of particles(Jd, in mol m-2 s-1) to the force per volume on the particles thatresults from a concentrationgradient [RT(n2 - n1)/dx,in N m-3] and the mobility of the particle (u/NA, in mol m2 J-1 s-1). Sometimes ulNAis referredto as the molar mobility constant (um, in mol m2 j-1 s-1). We can derive the usual form of Fick's Law from this equation since RINA= k whereR is the gas constant(8.31 J mol-1 K-1), NAis Avogadro'sNumber (6.02 x 1023 mol-1), and k is Boltzmann's Constant(1.38 x 10-23J K-1). Thus,

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Jd = -(ukT)[(n2 - n1)/dx]. (D.22)

Using Einstein's definition of the diffusion coefficient,

D = ukT, (D.23) we can put equationD.22 in the following form:

Jd = - (D)I[(n2- n )/dx], (D.24) which, by substitutingdn for (n2 - nl), can be written

Jd = -(D)(dn/dx), (D.25) which is the usual form of Fick's FirstLaw. It states thatthe drivingforce for the flux comes from the difference in chemical potential. In this case, the difference in chemical potential comes exclusively from the difference in concentration. The diffusion coefficient relates the magnitudeand directionof the flux to the magnitude and sign of the concentrationgradient. In the above derivationwe assumedthat n I and n2 are not equal and that eitherz or dE equals zero. However, if nI = n2, andif z and Vare not 0, the differencein chemical potential (g2 - R) is given by

(g2 - 91) = (9 + ZFE2) - (It + zFEI). (D.26) Since ,u -, =0,

(I2- L1) = (zFE2) - (zFE0). (D.27) Like the flux of unchargedparticles, the flux of chargedparticles (Jel, in mol m-2 s-1) is also proportionalto the negative difference in chemical potential in two compartments(-dgt):

Jel OC(dglidx). (D.28)

Rememberthat the gradientof the chemical potential(in units of J mol-I m-1 or N mol-1, which is force per mol) is defined as the difference in the chemical potential divided by the distance. The gradientin chemical potentialis opposite in sign to the drop in chemical potential. We call the flux Jel since the only drivingforce is electrical.The flux also depends on the concentrationof substance(n, in mol m-3) the force acts upon:

Jel (x (n)(-dR/dx). (D.29)

(u/NA) are the constantsof proportionalitythat relate the flux to the force per unit volume. NA is Avogadro's Number(6.02 x 1023mol-1) and u is the mobility in units of (M2 J-1 s-1). Thus,

Jel = (UINA)(n)(dpWdx). (D.30)

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According to equationD.27, dg = (zFE2)- (zFE1);thus,

Jel = (u/NA)(n) -[ (zFE2 - zFEl)Idx]. (D.3 1) We can simplify the equationusing algebra:

Jel = -(u/NA)(n)(zFE2 - zFE1)/dx (D.32) Jel= -(u/NA)(n)(F)(zE2 - zE1)/dx (D.33) Jel= -(u)(F/NA)(n)(zE2 - zE1)Idx. (D.34)

We will use this form of the equationto derive the Nernst Equation.This equation relates the flux of particles (Jel, in mol m-2 s-1) to the force per unit volume on the particles that result from an electrical gradient[Fn(zE2 - zED)/dx, in N m-3] and the mobility of the particle (u/NA, in mol m2 J-1 s-1). Sometimes ulNAis referredto as the molar mobility constant (um, in mol m2 j-1 s-1). When zEl > zE2, the flux is from E1 to E2. E2 is the final state. The difference in chemical potentialis always written(by convention)as final state- initial state. Thus, the flux is proportionalto the negative differencein chemical potential. (Note: Since both z and E can be eitherpositive or negative, we have an inherently confusing situation. For spontaneousprocesses, the initial state is always the most positive value of the productzE and the final state is always the least positive value of the productzE. With this convention,then, the difference (final - initial) is always negative for a spontaneous,passive, exergonic flux. When the difference is positive, the flux is nonspontaneous,active, or endergonic.) For example, for a monovalentcation (Zc= +) in an electric field that is between 0 V atA and 100 V at B, ZCEA= 0 V andZCEB = 100 V. The differencein electrochemical potentialis ZCEA- ZCEB= 0 V - 100 V = -100 V. The gradientis fromA to B. The flux will be spontaneousfrom B to A but will requireenergy for transportfrom A to B. The passiveflux will be proportionalto thenegative difference in theelectrochemical potential. For a monovalent cation (Zc= +) in an electric field that is between 0 V at A and -100 V atB, ZCEA= 0 V andZCEB = -100 V. The differencein electrochemicalpotential is ZCEB- ZCEA= -100 V - 0 V = -100 V. The gradientis from B to A. The flux will be spontaneousfromA to B butwill requireenergy for transportfrom B to A. The passive flux will be proportionalto the negativedifference in the electrochemicalpotential. For a monovalentanion (Za= -) in an electric field thatis between 0 V atA and 100 V at B, ZaEA=0 V and ZaEB= -100 V. The differencein electrochemicalpotential is ZaEB- ZaEA= -100 V - 0 V = -100 V. The gradientis from B to A. The flux will be spontaneousfromA to B but will requireenergy for transportfrom B to A. The passive flux will be proportionalto the negative differencein the electrochemicalpotential. For a monovalentanion (Za= -) in an electric field thatis between0 V atA and-100 V at B, ZaEA= 0 V and zaEB = 100 V. The difference in electrochemicalpotential is ZaEA- ZaEB= 0 V - 100 V = -100 V. The gradient will be from A to B. The flux will be spontaneousfrom B to A but will requireenergy for transportfrom A to B. The passive flux will be proportionalto the negative differencein the electrochemical potential. For electron transport(redox) reactionson the mitochondrialmembrane, thylakoid membrane,or plasma membrane,we use the reductionpotential for E and -1 for z to see which way electrons flow spontaneously.In this way, negative differences

This content downloaded from 128.84.125.94 on Thu, 12 Nov 2015 14:20:46 UTC All use subject to JSTOR Terms and Conditions EXCITABILITYOF PLANT CELLS 351 mean the flux is spontaneous.(Do not use the oxidation potential, which is just the reductionpotential x -1.) We can derive the usual form of Ohm's Law from equationD.34 since e = FINA, where e is the elementarycharge (1.6 x 10-19coulombs), F is the FaradayConstant (9.65 x 104 coulombs mol-1), and NA is Avogadro's Number(6.02 x 1023moh-1):

Jel= -(u)(n)(e)(zE2 - zEl)ldx. (D.35)

Factorout z:

Jel -(u)(n)(e)(z)(E2 - El)ldx. (D.36)

Since we will be talking aboutelectrons, let z = -1.

Jel = (u)(e)(n)(E2- E1)/dx. (D.37)

Multiply both sides by F where F is the FaradayConstant (9.65 x 104 coulombs mol-1).

(F)Jel = F(u)(n)(e)(E2 - El)/dx. (D.38)

Note that (F)Jelis in units of coulombsm2 s-1 or A m-2 and is thus currentdensity (1);(u)(c)(e)(lldx) is in units of coulombs2m2 J-1 S-, which is Q-1 m2 or S m-2, and is thus the specific conductanceor the reciprocalof the specific resistance(G = 1/R'). The potential difference (E2 - E1) is in units of volts = (E). Thus,

I=EIR' orE =IR', (D.39) which is Ohm's Law for the flow of electrons (wherez = -1) and 1 A = 1 coulomb/s; 1 V = 1 J/coulomband 1 Q = 1 Js/(coulomb)2in SI units. For the flow of positive current,Ohm's Law is written I = - EIR'. (D.40) We can now use equationsD.21 and D.34 to derive the NernstEquation. The flux of a particle due to more than one driving force (e.g., concentration,electrical, pressure,gravity, etc.) is

Jtot = Jd + Jel + Jp + Jg+ +(D.41) Note thateach flux may be in a differentdirection. Thus, while the individualfluxes may be large, the total net flux may be small or zero. In the case of a particle with only two driving forces (concentrationgradient and electrical gradient),we have the following situation:

Jtot= Jd + Jel (D.42) where

Jd= (uF/NA)(Rn7)((n2- nF)/dx (D.21) JelI -(u)(FINA)(n)(zE2 - zE,)Idx. (D.34)

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Thus,

Jtot = [-(u/NA)(RT)(n2 - nl)ldx] + [-(u)(F/NA)(n)(zE2 - zEl)ldx]. (D.43)

This equation,according to irreversiblethermodynamics, describes the initial flux of a particle given the initial conditions. As the flux continues, the concentration gradientand the electricalpotential gradient changes and we have to keep modifying the equationto determineeither the flux or the mobility coefficient. For this reason, this equation is usually used for two special cases: the starting condition or at equilibrium.The solution to this equation at equilibriumis known as the Nernst Equation.The Nernst Equationrelates the electrical forces to the chemical forces at equilibrium. At equilibrium,the net flux is zero. That is, Jtot= 0.

Jtot = [-(u/NA)(RT)(n2 - nl)ldx] + [-(u)(FINA)(n)(zE2 - zEl)ldx] = 0. (D.44)

After rearrangingterms, we get

-(u)(F/NA)(n)(zE2 - zE1)/dx = (uINA)(RT)(n2 - n1)ldx. (D.45)

Divide both sides by uzFINAto get

-(n)(E2- E1)fdx = (RTIzF)(n2 - n1)ldx. (D.46)

Divide both sides by n to get

-(E2 - El)ldx = (RTIzF)(l1n)(n2 - nl)ldx. (D.47)

Integratingequation D.47 will allow us to solve for the membranepotential that resultsfrom the diffusion of ionj if we know the concentrationof the ion on both sides of the membrane.In integratingthis equation we are assuming that the electrical potential gradientis linear across the membrane.Integrate the concentrationacross the membranethickness (dx) from side 1 to side 2. Rememberingthat, for the definite integral J2 (lIn)(dn) = ln n, and using the FundamentalTheorem of Calculus, we get

-(E2 - E1) = (RT/zF)ln(n2/nj). (D.48)

This equationtells us thatthe electricalgradient will always be the opposite, in sign, of the concentrationgradient for a cation. If n2 and E2 are always defined as the concentrationon the external side of the membrane(no, in mol m-3) and the electrical potential on the external side of the membrane (Eo, in V), respectively, and if n1 and E1 are always defined as the concentrationon theprotoplasmic side of themembrane (ni, in mol m-3) andthe electrical potentialon the protoplasmicside of the membrane(Ei, in V), respectively,then

-(Eo - Ei) = (RTIzF)ln(nJni), (D.49) and since by conventionEo = 0 V, then

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-(0 - Ei) = (RT/zF)ln(n0/ni). (D.50)

We can simplify to

Ei = (RTIzF)ln(n0/ni), (D.5 1)

which is the common form of the NernstEquation. We can also derive the NernstEquation from statisticalmechanics by startingwith Fick's Law (which describes the motion of particlesin a concentrationgradient) and Ohm's Law (which describes the motion of charged particles in an electric field). While thermodynamicsgives us informationabout the directionof the overallprocess, statisticalmechanics can be used to postulatea mechanismfor the process. Accordingto Fick's Law, therewill be a tendencyfor the particlesin any two groups of particles in communicationwith each other to mix. The net vector of motion will be from the dense group to the sparse group (or opposite in directionto the gradient in chemical potential). The magnitudeof the flux depends on the magnitudeof the concentrationgradient and the magnitudeof the diffusion coefficient (D, in m2 s-1).

Jd = -D (dnldx). (D.25)

The average flux is proportionalto temperature.This is reasonable,since diffusion is a consequence of the thermalmotion of particles.Therefore,

D ox kT (D.52)

wherek is Boltzmann's Constant(1.38 x 10-23J K-1) and Tis the absolutetemperature (in K). Boltzmann's Constantrelates the translationalenergy of the particle to the temperature. The average translationalkinetic energy (KE, in J) of a particleis equal to (312)kT. Hildebrand(1963) and Halliday and Resnick (1970; 382) show that (3/2)kTis equal to (1/2)mv2,where m is the mass of the particle(in kg) and v is the root mean square velocity of the particle (in m s-1). The higher the temperature,the more energy a particlehas and the fasterit moves. The coefficient thatrelates D to kTis the mobility u (in units of m2 J-1 s-1).

D = ukT; (D.23)

thus,

Jd = -ukT(dnldx). (D.53)

The mobility is a frictionalcomponent that relates the velocity of a particle(v, in m s-1) to the force (F', in N):

v =uF. (D.54)

This is the form of Stoke's Law which states thatthe frictionof a sphericalparticle depends on its hydrodynamicradius (rH, in m) and the viscosity (rnin Pa s) of the

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v = FV/(6ttrHrn). (D.55)

By substitutingD.55 into D.54, we see that

u = l/(6itrHip). (D.56)

That is, the mobility is inversely proportionalto the hydrodynamicradius of a sphericalparticle and the viscosity of the mediumthrough which it moves. Thus, after substitutingD.56 into D.23 we obtainthe following equation:

D = kT/(6rrHr), (D.57) and consequently,

Jd = -[(k)/(62trH1)](dn/dx). (D.58)

Thus, the flux (Jd) of particlesis proportionalto the concentrationgradient (dnldx) and its kinetic energy (relatedto kT), and is inversely proportionalto its size (related to rH)and the viscosity of the mediumthrough which it moves (rj). We can look at the flux of particles induced by an electrical gradientin a similar manner.The flux of particles(Jel, in mol m 2 s-l) is relatedto the currentdensity (I, in A m-2 or coulombs m-2 s-1) by the following relation:

Jel = II(ZF) (D.59) where F is the FaradayConstant (9.65 x 104coulombs mol-1) and z is the valence of the ion (dimensionless). Now use Ohm's Law for the flow of positive current:

I =-EIR'. (D.40)

Divide both sides by zF:

Il(zF) = -E/(R'zF). (D.60)

We will see that equationD.60 will be transformedinto the form of a typical flux equation since IlzF is equal to Jel and E/(R'zF) is equal to uzen(dEldx).In order to show this I will use dimensional analysis-that is, write out the units of each term and values of each constant: (coulombs m-2 s-1)/(9.65 x 104coulombs/mol) = - (V)/[(m2 Js/coulombs2)(9.65x 104 coulombs/mol)]. (D.61) Multiply both sides by 9.65 x 104: (coulombsm-2 s-)/(coulombs/mol) = -(V)/I(m2 Js/coulombs2)(coulombs/mol)]. (D.62)

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Cancel coulombs on both sides:

(mol m-2 s-1) = -(V)/[(m2 Js/coulombs)(mol)-1]. (D.63)

Multiply the right-handside by one (m2/M2):

(mol m-2 s-1) = -(m2/m2)(V)/[(Js/coulombs)(mol)(m2)]. (D.64)

Rearrangethe terms on the right:

(mol m-2 s-1) = -(V/m)(m2/Js)(coulombs)(mol/m3). (D.65)

Thus, the whole equation(in terms of units) is (mol m-2 s-1) =-(V/m)(m2/Js)(coulombs) (mol/m3); (D.66) or, in terms of symbols,

Jel = -(dE/dx)(u)(e)(n) (D.67) where Jel is the flux (in mol m-2 s-1) dEldx is the electricalpotential gradient or electric field (V m-1) u is the mobility (in m2 J-1 s-1) e is the elementarycharge (1.6 x 10-19coulombs) n is the concentration(in mol m-3).

Note: ue is also called the mobility (u', but is given in units of m2 V-1 s-1). EquationD.67 describes the flow of monovalentcations. In order to describe the flow of any ion we must multiply e by the valence (z).

Jel = - uzen(dEldx). (D.68) EquationD.68 is in the form of a flux equationthat relates the flux to the force of the electrical potentialgradient (zendEldx, in N mol m-3) and the mobility of the ion (u, in m2 J-1 s-1) Now we are ready to derive the NernstEquation from Fick's Law and Ohm's Law. The flux describedby Fick's Law of diffusion equals

Jd = -ukT(dnldx), (D.21) and the flux describedby Ohm's Law is equal to

Jel = -uzen(dEldx), (D.68) and the total flux (Jtot) is equal to Jtot = Jd + Jel *(D.42) After substitutingequations D.21 and D.68 into D.42, we get

Jtot = [-ukT(dnldx)] + [-uzen(dEldx)] (D.69)

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At equilibrium,Jtot = 0:

Jtot = [-ukT(dnldx)]+ [-uzen (dEldx)]= 0. (D.70) Thus, afterrearranging terms we get

-ukT(dnldx)= uzen(dEldx) (D.71)

Divide both sides by uzen:

-[(ukT)I(uze)](l1n)(dn/dx)= (dEldx) (D.72)

Cancel u:

- [(kT)I(ze)](1n)(dn/dx)= (dEldx). (D.73)

Integratingthe above equation will allow us to solve for the membranepotential thatresults from the diffusion of ionj if we know the concentrationof the ion on both sides of the membrane.In integratingthis equationwe are assumingthat the electrical potential gradientis linear across the membrane.Integrate the concentrationacross the membranethickness (dx) from outside to inside. Rememberingthat the definite integral Jo (lln)(dn) = ln n, and using the FundamentalTheorem of Calculus,we get

- [(kT)l(ze)]ln(ni/nO) = dE = Ei - Eo (D.74) Since - ln [xly] = ln [ylx],

[(kT)/(ze)]ln (noIni)= dE = Ei - Eo (D.75)

Define Eo = 0 (by convention):

[(kT)I(ze)]ln (n0/ni)= Ei. (D.76)

Multiply the left side by one (NAINA):

(NA/NA)[(kT)/(ze)]ln (no/ni)= Ei (D.77)

Since NAk= R and NAe= F, we get

Ei = (RT/zF)ln (nolni), (D.51) which is the familiarform of the NernstEquation.

XIX. Appendix E: Derivationof the Goldman-Hodgkin-KatzEquation The flux equationsfor electrolytesand the Goldman-Hodgkin-KatzEquation are calculatedfrom the generalflux equation(based on Fick's andOhm's Laws). Remember, for a nonelectrolytethe permeability coefficient (Pj) is definedby the followingequation:

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JJ= P- (dnj) (E. 1) where J. is the net flux of a substance(in mol m-2 s-4) cn1is the difference in the concentrationof a substanceacross a membrane(in mol m-3), and Pj = KjujkTI(x) (E.2) where K1is the partitioncoefficient (oil/water)and is dimensionless u;is the mobility of the nonelectrolyte(in m2 J-1 s-1) k is Boltzmann's Constant(1.38 x 10-231 K1; = R/NA) R is the gas constant(8.31 J mol-1 K-1) NA is Avogadro's number(6.02 x 1023mol-1) T is the absolutetemperature (in K) x is the thickness of the membrane(in m), and, as defined by Einstein (1926),

D1= ukT (E.3) where Dj is the diffusion coefficient (in m2 s-1). However, in orderto determinethe permeabilityof an electrolyte,we must include the electrical force as well as the concentrationgradient in the equation(as we did in Appendix D).

Jj= [-ujkT(dnjldx)] - [ujnjzje(dEldx)]. (E.4)

We must also replace nj and dnj with Kjnj and Kjdnjwhere Kj is the partition coefficient for the electrolyte, since the concentrationin the membraneis not equal to the concentration we measure in the bulk aqueous solutions surroundingthe membrane but is related to that concentration by the partition coefficient (Kj, dimensionless). Water and the hydrophobicsolvent that best mimics the properties of the membranein question should be used to determine the partitioncoefficient (Stein, 1986).

Jj = [-ujkTKj(dnjldx)]- [ujKjnjzje(dEldx)]. (E.5) We will define the permeabilitycoefficient of an electrolyte by going througha series of algebraicmanipulations of this basic equation. Now we'll move the electricalterm [ujKjnjzje(dEldx)] to the left side of the equation:

Jj + ujKjnjzje(dEldx)= -ujkTKj(dnjldx). (E.6)

Factor out ujzje(dEldx)from the left side:

ujzje(dEldx){[(Jjdx)l(ujzjedE)] + njKj= ujkTKj(dnjldx). (E.7)

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Divide both sides by [(Jjdx)l(ujzjedE)]+ njKj: ujzje(dEldx)= -ujkTKj(dnjldx)l/[(Jjdx)l(ujzjedE)] + njKj}. (E.8) Multiply both sides by dxl(ujkT):

(dxluikT)[uzje(dE/dx)] = (dxlujkT){-ujkTKj/dnjldx)l[(Jjdx)l(ujzjedE)+ njK1] }. (E.9) Define B = elkT = FIRTand cancel like terms:

(zjB)(dE)= -{Kj(dnj)/[Jjdx/ujzjedE+ njKj] . (E.10)

If the term (JjdxlujzjedE) is defined as Yand all its partsare constant,we can easily integrateacross the membrane(from the outside to the inside) assumingthat dEldx is linear.This is known as the "constantfield assumption."We must use the substitution rule of integral calculus to get the definite integralJO [Kjdnjl(Kjnj+ Y)] = ln (Kjnj+ Y) and the FundamentalTheorem of Calculus.Then

(zjB)(dE)= -{ln[Kjnji+ (Jjdx/ujzjedE)]- ln[Kjnj0+ (Jjdx/ujzjedE)]}. (E.1 1) Rearrangeterms and rememberln x - ln y = ln (x/y):

(zjI3)(dE)= -ln{ [Kjnji+ (J dxlu zMedE)I/[Kjnfi + (JjdxlujzjedE)]}. (E. 12)

Remove ln by exponentiatingboth sides:

exp[(zjB)(dE)] -{ [Kjnj+ (JjdxlujzjedE)]I[Kjnio+ (JjdxlujzjedE)]}. (E. 13) Rearrangethe right side, rememberingthat - ln [x/y] = ln [y/x]:

exp [(zjf)(de)] = [KjnjO+ (JjdxlujzjedE)]/[Kjnji+ (JjdxlujzjedE)]. (E.13a)

Multiply both sides by Kjnjl+ (JjdxlujzjedE):

exp[(zjB)(dE)][Kjnji+ (JjdxIujzjedE)]= [Kjnj0+ (JjdxlujzjedE)]. (E. 14) Distributethe terms on the left side:

Kjnjlexp[(zj1B)(dE)]+ {(Jdx/u1z3edE)exp[(zj%3)(dE)]j = [Kjnj1+ (JjdxlujzjedE)]. (E. 15) Put all concentrationterms on the right side:

(JjdxlujzjedE)exp[(ziB)(dE)]- (J1dxlu1zjedE) = {Kjnjf- K1n,exp[(zj13)(dE)].(E.16) Factorout Jj from the left side:

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Jj[(dx/u zjedE)exp[(zjf3)(dE)] - (dxlujzjedE)] = {Kjnj? - Kjnjlexp[(zjB)(dE)]}. (E.17)

Divide both sides by (dx/ujzjedE)exp[(zjB)(dE)]- (dxlujzjedE):

J= {(Kjnj? - Kjnjiexp[(zjB)(dE)])I[(dxlujzjedE)exp[(zj13)(dE)] - (dxlujzjedE)}. (E. 18)

Factor out dxl(ujzjedE)on the right side:

Jj= {(Kjnf - Kjnjiexp[(z1B)(dE)])1[(dxlujzjedE)exp[(zj13)(dE)]- 1] }. (E. 19) Multiply the right side by one (B/B):

Jj = (B&I{(KjnjO - Kjnjiexp[(zj8)(dE)])/ [(dx/ujzjedE)exp[(zjB)(dE)]- 1] 1. (E.20)

Factorout Kj:

Jj = (KjB/B){ (nj? - njiexp[(zjf3)(dE)])/ [(dx/ujzjedE)exp[(zjB)(dE)]- 1] }. (E.2 1)

Since B = FIRT,

Jj = (BKjRTIF){(nj0- njiexp[(zjB)(dE)])I [(dxlujzjedE)exp[(zjB)(dE)]- 1 ] I . (E.22)

Define the permeabilitycoefficient of an electrolyteto be Pj, in m s-1: Pj = (KjujeR7)1(Fx). (E.2) Note thatthis is equal to KjujeI(fx),which is equal to KjujRTI(NAx),which is in turn equal to KjujkTlx.Thus, the permeabilitycoefficient of an electrolyte is identical to the permeabilitycoefficient of a nonelectrolyte.

Jj = Pj(zjf)(dE) { (njo - njiexp[(zjB)(dE)])I(exp[(zjB)(dE)] - 1)}. (E.23)

Since we integratedfrom the outside (initial state) to the inside (final state), Jj is defined as Ji - JOwhere the net flux equals the influx minus the efflux. Thus Jj is the net flux of substancej out of the cell. WhenJj is positive (andthe membranepotential is negative), the net passive (spontaneous)flux of both cations and anions is into the cell. When Jj is negative (and the membranepotential is negative), the net passive (spontaneous)flux is out of the cell for both anions and cations. The Goldman-Hodgkin-KatzEquation characterizes the membranepotential that develops across a membraneas a resultof the diffusionof ions down theirconcentration gradients(through a membranewith a specificpermeability coefficient for each ion). In orderto derive the Goldman-Hodgkin-KatzEquation, we assumethat the flux of cations is equal to the flux of anions. Thus at equilibriumthe principle of electroneutralityis maintained.That is,

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JK + JNa = JC1 (E.24) or JK+JNa-JCO=0 (E.25)

Define the fluxes of the majorions using equationE.23:

JK= PK (zB)(dE){ [K? - Kiexp[(zB)(dE)]]/[(exp[(zB)(dE)] - 1)] }, (E.26a)

JNa = PNa (zB)(dE){ [Na? - Naiexp[(zB)(dE)]]/[(exp[(zB)(dE)] - 1)] }, (E.26b)

Jci = PCi (zB)(dE){[Cl?- Cliexp[(zB)(dE)]]/[(exp[(zB)(dE)]- 1)]}. (E.26c)

Since ZK = 1, ZNa = 1, and ZC1= -1, then

JK = PK (B)(dE){ [K0 - Kiexp[(B)(dE)]]/[(exp[(J)(dE)] - 1)] }, (B.27a)

JNa = PNa (3)(dE){[Na? - Naiexp[(B)(dE)]]/[(exp[(B)(dE)] -1)]}, (B.27b)

Jcl = PCl (-B)(dE){ [ClO- Cliexp[(-B)(dE)]]/[(exp[(-B)(dE)] - 1)] }. (B.27c)

Simplify Jcl:

JCl =-PCi(B)(dE){[ClO-Cliexp[(-B)(dE)]]/[(exp[(-B)(dE)] -1)]}. (E.28)

Set JK + JNa - JCI = 0 accordingto equationE.25:

{PK(B)(dE)[(K - Kiexp[(B)(dE)])/[(exp[(B)(dE)] - 1)]]} + {PNa(B)(dE)[(Na0 - Na1exp[(B)(dE)])/[(exp[(B)(dE)] - 1)]] I + {PC1(B)(dE)[(ClO- Cliexp[(-B)(dE)])/[(exp[(-B)(dE)] - 1)]] I = 0. (E.29)

Divide all terms by B(dE):

{PK[(KO- Kiexp[(B)(dE)])/[(exp[(B)(dE)] - 1)]] I + {PNa[(Na0 - Naiexp[(B)(dE)])/[(exp[(B)(dE)] - 1)]] } + tPCI [(ClO- Cli exp[(-B)(dE)])/[(exp[(-fi)(dE)] - 1)]] 1 = 0. (E.30)

Simplify since l/{exp[(-B)(dE)] - 1} is equal to -exp[(B)(dE)]/exp[(B)(dE)]- 1; then

{PK[(K? - Kiexp[(B)(dE)])/[(exp[(1)(dE)] - 1)]] I + {PNa[(Na0 - Naiexp[(B)(dE)])/[(exp[(B)(dE)] - 1)]] I + {Pci(-exp[(B)(dE)])[CIO - ClIexp[(-B)(dE)]/[(exp[(B)(dE)] - 1)] } = 0. (E.3 1)

Cancel li/exp[(B)(dE)] - 1 from each term:

{PK [(KO- Kiexp[(B)(dE)])] } + {PNa [(Na0 - Naiexp[(B)(dE)])] I + {Pcl(-exp[(B)(dE)])[(C1I) - Cl1exp[(-B)(dE)])] I = 0. (E.32)

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Distributethe permeabilitycoefficients:

PKKO- PKKiexp[(B)(dE)] + PNaNaO- PNaNaiexp[(B)(dE)] + PCl(-exp[(13)(dE)])(C10)- PClCli(-exp[(B)(dE)])(exp[(-B)(dE)]) = 0. (E.33)

Simplify, rememberingthat -exp[(B)(dE)]exp[(-B)(dE)] = -1; thus,

PKKO - PKKiexp[(B)(dE)] + PNaNa0 - PN4Naiexp[(B)(dE)] + PCl(-exp[(J3)(dE)])(CI0)- PC1C1(-l) = 0. (E.34)

Simplify, since (-1)(-1) = 1:

PKKO- PKKiexp[(f3)(dE)] + PNaNa0 - PNaNaiexp[(B)(dE)] + Pcl{-exp[(B3)(dE)]}(CIO) + PC1Cli= 0. (E.35)

Move the negative in the fifth term outside the parentheses:

PKK0 - PKKiexp[(B)(dE)] + PNaNa0 - PNaNaiexp[(B)(dE)] - PCl{ exp[(1B)(dE)] } (C1?) + PCLC1I= 0. (E.36)

Rearrangeterms:

PKKO+ PNaNa0+ PC1CIi= PKKiexp[(B)(dE)]+ PNaNaiexp[(B)(dE)] + Pcl{exp[(B)(dE)]I(CO). (E.37)

Factorout exp[(B)(dE)]:

PKKO+ PNaNa0+ PC1Cli= exp[(f3)(dE)]{PKKi+ PNaNai + PC1Cl0I. (E.38)

Divide both sides by PKK1+ PNaNai+ PClCl0].

(PKKO+ PNaNa0+ PC1Cli)/(PKKi+ PNaNa1 + PCICl) = exp[(B)(dE)]. (E.39)

Remove the exponentby taking the naturallog:

ln [(PKKO+ PNaNa0+ PC1Cli)/(PKKi+ PNaNal + PC1Cl0)]= (B)(dE). (E.40)

Divide both sides by B:

dE = (1/B){ln[(PKKO + PNaNa9+ PC1Cli)/(PKKi+ PNaNai + PC1Cl0)]}. (E.41)

Replace 1/Bwith RTIF:

dE = (RT/F){ln[(PKKO + PNaNa0+ PC1Cli)/(PKKi+ PNaNa1 + PCICl0)]}. (E.42)

Since we integratedfrom outsideto inside and the FundamentalTheorem of Calculus

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states thatfl x dx = i - o, the membranepotential dE is definedas Ei - Eo. Since Eo = Oby convention, dE = Ei. Thus,

Ei = (RTIF)IIn [(PKKO + PNaNa0+ PC1Cli)/(PKKi+ PNaNa1 + PCICl0)]1, (E.43)

which is the common form of the Goldman-Hodgkin-KatzEquation and describesthe electricalpotential of the membraneas a consequenceof the diffusionof ions down their electrochemicalpotential gradient and the permeabilityof the membraneto each ion. The Goldman-Hodgkin-KatzEquation is extremelyuseful in calculatingpermeabil- ity coefficients; however, it assumes that the permeabilitycoefficient is independent of membrane potential and the concentrationof the permeating ion-that is, we assume the ions move throughthe membraneindependently. If these assumptionsare not met, the equationmust be modified.

XX. Appendix F: Derivationof the Hodgkin Equation That Relates Conductanceto Flux. The HodgkinEquation (text equation9) relates the unidirectionalflux of an ion (Ji or JO,in mol m-2 s-1) measuredchemically to the partialspecific conductanceof the membranefor an ion (gj, in S m-2) measuredelectrically (Walker& Hope, 1969). First we must define the partialspecific conductance.The partialspecific conductance of a membraneis defined from Ohm's Law:

gj = -IjlEt (F.1) where g is the partialspecific conductance(in S m-2) I is the net currentdensity carriedby the ion (in A m-2) htis the total electricaldriving force on the ion (in V).

The minus sign is included in equation F.1 so that the conductanceis defined as positive when the drivingforce is negativeand a cationis carryingthe currentflowing into the cell. The total electricaldriving force on the ion (Et, in V) is defined as the differencein the membranepotential (Em, in V) and the equilibriumpotential, or Nernstpotential, for that ion (Ej, in V), calculatedusing the Nernst Equation,where

Ej = (RTlzjF)ln (nolni) (F.2)

where E. is the Nernst potentialor equilibriumpotential for an ion (in V) R is the gas constant(8.31 J mol-I K-1) T is the absolutetemperature (in K) z. is the valence of the ion (dimensionless) I is the FaradayConstant (9.65 x 104 coulombs mol-1) no andni are the concentrationsof ionj on the exoplasmic side of the membrane (E-space) andprotoplasmic side of the membrane(P-space), respectively (in mol m-3).

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Thus, the total driving force (Et, in V) is defined by

Et = Em- Ej (F.3)

where Et is the driving force (in V) Emis the membranepotential (in V) Ej is the Nernstpotential or equilibriumpotential of the ion (in V).

Thus, the partialspecific conductancecan be expressedlike so:

gj = -Ij/(Em- Ej). (F.4)

However, experience tells us that most ionic currentsare not linear throughoutthe whole range of possible membranepotentials, and thus a partialspecific conductance must be defined for a small range in membranepotentials (dEt). Thus,

gj = -dIj/dEt, (F.5)

or, after expandingthe dE term,

gj = dIjld(Em- Ej). (F.6)

We can also define the drivingforce (Et, in V) as

Et = (1/zjB)ln {exp[(Em- Ej)zjB] (F.7)

since

Ej = (1/zjB)ln (nolni), (F.2a) In (noJni)= EjzjB, (F.2b) (nolni) = exp(EjzjB). (F.2c)

Substitutingequation F.2c into F.2 yields

= (1/z1B) ln [exp(EjzjB)]. (F.8) Thus,

Em- Ej = Em- (lI/zB) In [exp(Ejz1B)] (F.9) and d[Em- Ej] = d{Em- (1IzjB)ln [exp(Ejz1B)]. (F. 10)

Since ln [exp(x)] = x,

- = - d[Em Ej] d[Em (1/zjfB)(Ejzjf3)]. (F. 1) Afterrearranging the rightside of equationF.I 1, we get the differentialvoltage change:

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d(Em- Ej) = (lzjf3)d[(zjf3)Em- (zjB)Ej] (F.12) where (Zj13)Em- (zjB)Ej is a dimensionlessnumber that characterizes the magnitudeof the driving force for an ion. Next we must define the currentcarried by the ion. The currentcarried by ion j (Ij, in A m-2) is relatedto the net flux accordingto the following definition:

Ij = zjFJn (F. 13) where I. is the currentdensity carriedby the ion (in A m-2 or coulombs m-2 s-) z is the valence of the ion (dimensionless) } is the FaradayConstant (9.65 x 104 coulombs mol-1) Jn is the net flux of the ion (in mol m-2 s-1).

Now, to relatethe partialspecific conductanceto the net flux of an ion, we substitute equationF.13 into equationF.5 to get

gj = -d(zjFJn)/d(Em- Ej) . (F. 14) And substitutingequation F. 12 into F. 14, we get

gj = -d(ZjFJn)/(l1zjB)d[(ZjB)Em- (zjfB)Ej]. (F. 15) Since z; and F in the numerator are constants, let's take them out of the differential:

gj = -zjFdJn/(l1zjB)d[(zjB)Em- (zjB)Ej]. (F. 16) Combine constantsoutside the differential:

= -ZjFI(l/zjB) { dJnld[ (ZjB)Em - (zjB)Ej]1 (F. 17)

Simplify the constantsoutside the differential(since B = FIRT): g = -(zf2F2/RT){ dJn/d[(ZjB3)Em - (zjfB)Ej] } (F. 18) Simplify the constantsinside the differential:

gj = (zj2F21R) {dJn/d[(zj B)(Em - Ej)]} (F. 19) In orderto determinethe partialspecific conductancewe mustthus know the voltage dependenceof the flux {dJnId[(zjjB)(Em- Ej)]1. As this may not be an easily attainable or known quantity, we can simplify the equation by assuming that the membrane potential equals the Nernstpotential or equilibriumpotential for the ion. Thus, when we take the limit of equationF. 19 as d(Em- Ej) goes to zero, we find

gj = (Zf2F2IRT)Ji, (F.20a) gj = (z 2F21RT)J0 (F.20b)

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where g1is the specific membraneconductance for the ion (in S m-2) z is the valence of the ion (dimensionless) I is the FaradayConstant (9.65 x 104 coulombs mol-1) R is the gas constant(8.31 J mol-1 K-1) T is the absolutetemperature (in K) Ji andJO are the unidirectionalfluxes (influx andefflux, respectively)of the ion at equilibrium(in mol m-2 s-1).

Below is the derivationof equationF.20 from equationF.19:

First, let's replaceJn with Ji - Jo: g = -(z12F2/RT){d(Ji - Jo)Id[(zjI)(Em- Ej)]1. (F.21)

Now we must define the net flux Jn (in mol m-2 s-1) as Ji - JO(in mol m-2 s-1) where Ji and JOrepresent the influx and efflux of an ion, respectively. This equation was derived in Appendix E.

Jn= Ji - JO= -(KjujzjeEmldx){ 11[exp(zjf3Em) - 1] [no - niexp(zjBEm)].(F.22)

Ji is obtainedwhen ni =0; thus,

Ji = -(KjujzjeEJdx){ 1/[exp(zjBEm)- 1] }(no). (F.23)

JOis obtainedwhen n. = 0; thus,

JO= -(KjujzjeEmldx){ 1I[exp(zjBEm) - 1] I [niexp(zjfEm)]. (F.24)

In orderto simplify the algebrain the following expressions,let's define:

(-K1u1zeldx) = a 1/[exp zjBEm)- 1] = A(x)where x = Em and is a constantwhen Em= Ej. EquationsF.23 and F.24 become

Ji = (aEm)[fx)](no) (F.25) and

JO= (aEm)L(x)][niexp(zjBEm)]. (F.26)

Substitutingequations F.25 and F.26 into equationF.21 gives

gj = -(Zj2F2IRT){d(aEm)[tAx)] }[no -niexp(zjBEm)]/dzjB(Em - Ej). (F.27)

Take the constants z BJaEm, andf(x) out of the differential(remember that fx) is a constantwhen x =

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g=-(zf2F2/RTzjB){(aEm)fx)]Id [no-niexp(zjBEm)]/d(Em-Ej). (F.28)

Define k' = (no/ni)

gj = -(zf2F2/RTZiB){(aEm)[f(x)] }dni[k' - exp(zjBEm)]Id(Em- Ej). (F.29)

Remove the constantni from the differential:

gj = -(z 2F21RTzjB){(aEm)[Ax)] }nid[k' - exp(zjBEm)]Id(Em- Ej). (F.30) According to the Nernst Equation,

Ej = (RTlzjF)In (no/ni). (F.2) Thus,

EjzjB= In (nolni) (F.2b)

and

noni = exp(zjBEj). (F.2c) Thus, since k' was defined as nlni,

k' = exp(zj/3Ej). (F.3 1)

Substitutingequation F.31 into equationF.30 yields

g= -(zj2F2IRTZjB){(aEm)[f(x)]}nid(exp[zjBEj] - exp[zjB3Em])/d(Em- Ej)]. (F.32)

Multiply exp(zjBEj)- exp(zjBEm)by -1 to get exp(zjBEm)- exp(zjBEj),which will set up a correctderivative. Then multiply-(zj2F2/RTzjB) by -1 so the whole equation remainsequivalent.

gj = (zj2F2/RTzjf3){(aEm)[f(x)] }nid(exp[zjBEm] - exp[zjBEj)]/[d(Em - Ej)]. (F.33)

Now d[exp(z-BEm)- exp(zjBEj)]/[d(Em- Ej)] is a derivativethat fits the definition of a derivative:

g'(x) = lim [g(x) - g(a)]l(x - a) x -> a

Now we determinethe derivativeof d[exp(zjBEm)- e(zjBE-)]/[d(Em- Ej)] as Em > Ej. Remember,if g(x) = exp(zjBx)then g'(x) = zjBexp(zjBx).

lim d[exp(zjBEm) - e(zjBEj)]I[d(Em- Ej)] = zjBexp(zjBEm) Em *Ej

gj = (zj2F2/RTzjf{(aEm)[f(x)] }ni[zjBexp(zjf3Em)]. (F.34)

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Cancel zjB/z1B:

= (zF2/RI){ (aEm)[f(x)]}ni[exp(zjBEm)]. (F.35)

At the limit as Em approachesEj, Embecomes Ej, and so equationF.35 becomes

gj = (zj2F2IRT){(aEj)[f(x)] }nj[exp(zjBEm)]. (F.36)

Since, according to equation F.26, JO= (aEm)[f(x)][niexp(zjBEm)];and at the limit where Em approachesEj,

JO= (aEm)[f(x)][niexp(z}BEm)] = (aEj)[f(x)][niexp(zjBEj)]; (F.37) then after we substituteequation F.37 into equationF.36 we get

gj = (zi2F2/RT)(Jo). (F.20b)

And since at equilibriumJO = Ji, then

gj = (zi2F2IRT)(J,). (F.20a) Equations F.20a and F.20b are valid only in determining the partial specific conductance when the fluxes are measured at the equilibrium potential for that ion-that is, when the membranepotential equals the Nernstpotential for that ion. When both the fluxes and the conductances are measured at the equilibrium potentialbut are not equal when applyingequations F.20a or F.20b, then the assump- tions that underlie one of the equations are not true. Typically, this is because the independentmovement of ions assumedin the derivationof equationF.22 fromFick' s Law and Ohm's Law does not occur. If the ions bind to an asymmetricalchannel or a symmetricalchannel that has multiplebinding sites within the pore, thenthe independence assumptionwill be violated and the flux will not be proportionalto the ion concentrationat the equilibriumpotential. See Hodgkin and Keynes (1955) for an excellent explanationof the assumptionof independence.

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