FREE DIFFUSION INTRODUCTION

Transcription

1 2 FREE DIFFUSION INTRODUCTION In th chapter, we conder the mplet of tranport procee: the pave dffuon of a olute that occur when t electrochemcal potental on the two de of a permeable barrer are dfferent. Indeed, th proce o mple that t fal to repreent many apect of tranport n lvng ytem. Nonethele, t doe decrbe ome apect of bologcal tranport qute well, and t alo provde a bae cae whoe behavor can be compared agant that of more complex tranport mechanm. Th chapter dvded nto two ecton; the frt deal wth free dffuon of nonelectrolyte, and the econd wth that of onc pece. The prncpal property varable determnng the flux of a nonelectrolyte t permeablty, a quantty that can n prncple be related to the dffuon coeffcent of the olute. Electrolyte dffuon n free oluton mot rgorouly decrbed by clacal electrodffuon theory. The flux equaton provded by th theory are very complex, and they have not een nearly a much ue a have approxmaton to them. Accordngly, empha wll be placed here on the prncple of electrodffuon, and on the approxmate oluton and pecal cae that are mot commonly ued. The equaton of free dffuon can decrbe a wde varety of tranport phenomena, ncludng teady and unteady tranport procee; procee that can be decrbed n one, two or three dmenon and n a varety of geometre; and procee n whch chemcal reacton and flud flow take place multaneouly wth dffuon. In th chapter, we wll dcu a mall ubet of thee, focung on the tool that are appled to lvng ytem. Comprehenve dcuon of dffuonal procee can be found n other text, uch a Crank' (1975) clac text, publhed thrty year ago and tll beng reprnted! A more lmted et of oluton, but wth more content bologcal applcablty, can be found n Trukey et al. (2004). M.H. Fredman, Prncple and Model of Bologcal Tranport, DOI: / _2, Sprnger Scence+Bune Meda, LLC

2 30 CH.2:FREE DIFFUSION 2.1. FREE DIFFUSION OF NONELECTROLYTES The frt tranport proce we wll conder the dffuon of a dolved nonelectrolyte acro a membrane or a mlar barrer. The dffuve proce drven by the olute' concentraton gradent. For now, the nature of the olvent not partcularly mportant, and t wll uually be undertood to be water, whch the mot common bologcal olvent. A wll be een n Chapter 7, much of the materal developed below equally applcable to dffuon through a lpd flm, uch a the hydrophobc regon of a cell membrane. In free dffuon through a membrane, the olute partcle move about by random Brownan moton, lke that n free oluton. The olute flux, whch a meaurable and reproducble quantty, eentally the reultant of thee eparate moton. Even though the path of a ngle olute partcle cannot be predcted, the conequence of an enormou number of thee path qute reproducble The Teorell Equaton The flux n free dffuon can be wrtten very mply, n a form propoed by Teorell (1953): Flux = Moblty Concentraton Drvng force. (2.1) In the mot commonly ued unt, the flux the number of mol of olute crong one quare centmeter of membrane per econd; t proportonal to the product of the olute moblty, whch meaure t eae of tranport and depend jontly on the barrer/olvent and the olute, a well a the temperature; the olute concentraton, whch meaure the amount of materal avalable to partcpate n the proce; and the drvng force for the dffuon of the olute. The choce of a proper drvng force dctated by thermodynamc conderaton that we wll not examne untl Chapter 6; for now, we wll ratonalze that choce by analogy wth electrcal phenomena. Frt, we recall that, when the chemcal potental of the olute the ame n the two phae boundng the membrane, the olute n equlbrum, and t flux acro the membrane zero. An analogou tuaton occur n electrcal crcut; when there no electrcal potental dfference, there no current flow. When the electrcal potental at two pont are dfferent, the potental gradent defne a feld, and charged partcle move n repone to t. The force actng on the charge the negatve of the electrcal potental gradent. The analogou drvng force for olute flux the negatve of the chemcal potental gradent: Drvng force =. (2.2) Almot every tranport proce wth whch we wll be concerned can be decrbed n term of a ngle patal coordnate perpendcular to the plane of the barrer. Callng that the x-drecton, the drvng force become:

3 PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT 31 d Drvng force =. (2.3) dx The mplct aumpton n th one-dmenonal treatment of gradent and fluxe that thee vector are orented perpendcular to the membrane plane and have neglgble component parallel to that plane. Th reaonable f the extent of the membrane much larger than t thckne, a uually the cae. The Teorell equaton can now be wrtten: J d dx, (2.4) Uc where U and c are the olute moblty and concentraton, repectvely. The flux J potve n the drecton of ncreang x. An ntegral drvng force can alo be defned, by ntegratng Eq. (2.3) acro the membrane: Integral drvng force a 0 d dx dx. (2.5) I In Eq. (2.5), a the thckne of the membrane. Phae I bathe the face of the membrane at x = 0, and Phae the face at x = a. From Chapter 1, the ntegral drvng force zero at equlbrum. The ntegral drvng force would appear to be far more convenent than the dfferental drvng force [gven by Eq. (2.3)] for decrbng tranport, becaue t baed on the chemcal potental n the two phae external to the membrane. Chemcal potental nde the membrane, whch mut be known to fnd the local dfferental drvng force, are generally unmeaurable. Fortunately, wth a few reaonable aumpton, Eq. (2.4) can be ntegrated to gve an expreon that relate the tranmembrane flux to the condton n the ambent oluton. Th we now do Integraton of the Teorell Equaton; Fck' Frt Law; Solute Permeablty In ntegratng Eq. (2.4), the temperature aumed to be unform and the effect of reure on the chemcal potental of the olute neglected; thee are qute reaonable aumpton for the ytem wth whch we wll be dealng. If, n addton, the oluton are aumed to be deal, then the chemcal potental can be wrtten very mply a μ = contant + RT ln c. (2.6) Dfferentatng wth repect to x, d dlnc RT 1 dc RT dx dx c dx. (2.7)

4 32 CH.2:FREE DIFFUSION Subttutng Eq. (2.7) nto (2.4), J dc U RT dx. (2.8) The olute dffuon coeffcent D related to the olute moblty through the Nernt Enten relaton, D = U RT. D often referred to a the bnary dffuon coeffcent (denoted D j ), a a remnder that t value depend on the dentte of both olute and olvent. Introducng the dffuon coeffcent nto Eq. (2.8), we obtan: J dc D dx. (2.9) Equaton (2.9) known a Fck' frt law of dffuon. Note that the flux potve f the concentraton gradent negatve. In the precedng dervaton, t wa aumed that Eq. (2.6) hold wthn the membrane, a though tranport proceeded through aqueou pore n whch the dependence of chemcal potental on oluton properte wa dentcal to that n the aqueou oluton at the membrane face. Th the frt of everal dervaton n whch the expreon for chemcal or electrochemcal potental n free oluton wll be ued to decrbe the thermodynamc tate of olute or olvent nde a tranport barrer. The tate of olute and olvent nde a complex, heterogeneou bologcal barrer not o neatly defned. Accordngly, t convenent to thnk of c (x) [and (x) when decrbng electrolyte tranport] a the concentraton (and potental) of a free aqueou oluton n equlbrum wth a thn membrane lce at x. The concentraton and potental of th equlbrum oluton can be qute dfferent from that of the true oluton phae at that pont n the membrane; however, nce the two phae are defned to be n equlbrum, the chemcal potental of the olute and olvent are the ame n each. A notable dfference between the concentraton of uch an equlbrum aqueou oluton and the true ntramembrane olute concentraton are when the olublty of the olute n the membrane dfferent from that n the ambent aqueou phae. Such the cae for dffuon through the lpd blayer of the cell membrane. The relatonhp between the olute concentraton n the lpd and n an equlbrum aqueou oluton expreed n term of the partton coeffcent of the olute between the two phae. Dffuon through lpd layer wll be decrbed n Chapter 7. Fck' frt law aume a omewhat more complcated form when the oluton are nondeal. In that cae, the olute chemcal potental mut be wrtten n term of actvty. The actvty, n turn, the product of the concentraton and the actvty coeffcent. Thu, Eq. (2.7) replaced by: d dln a d RT ln c ln d RT dx dx dx dx. (2.10) For nonelectrolyte, the actvty coeffcent of the olute can be aumed to depend on only c, whch n turn a functon of x. Thu, the followng ubttuton can be made: dln dln dlnc. (2.11) dx d lnc dx

5 PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT 33 Subttutng Eq. (2.11) nto (2.10): The Teorell equaton then become d 1 dc dln RT 1 dx c dx d lnc. dln dc J URT 1 dlnc dx. (2.13) Defne an augmented dffuon coeffcent D * by * d ln D D 1 dlnc. (2.14) For an deal oluton, D * = D. By ubttutng Eq. (2.14) nto (2.13), a flux equaton obtaned that look almot dentcal to Eq. (2.9), and can be regarded a a generalzaton of Fck' frt law to nondeal oluton: * J D dx dc. (2.15) Fck' frt law, a generalzed above, now ntegrated acro the membrane to yeld an expreon for flux n term of the tranmembrane concentraton dfference. To et up the ntegraton, Eq. (2.15) rewrtten a J dx D dc. (2.16) * In the teady tate, the olute flux ndependent of x. Aume that the ame true of D *; Eq. (2.16) can then be ntegrated acro the membrane and olved for J : * ( I ) D c c J. (2.17) a The flux n Eq. (2.17) baed on a unt area of membrane, o t can be contnuou at the nterface x = 0 and x = a only f the entre cro-ecton of the barrer avalable for tranport. Furthermore, the aumpton that the expreon for chemcal potental a a functon of concentraton are the ame n both barrer and bath mple that the olute dffue through the ame olvent a that n the ambent phae. The only barrer for whch thee aumpton hold would be a thn tagnant water flm omehow mantaned between two well-trred aqueou bath. The olute permeablty of uch a thn flm defned a the olute flux per unt concentraton dfference: k 0 J J D, (2.18) c c c a I 0 0 *

6 34 CH.2:FREE DIFFUSION where we have ued the upercrpt 0 to ndcate that dffuon take place through a thn aqueou flm. The form of Eq. (2.18) ha been adopted to decrbe olute tranport n bologcal ytem. For uch ytem, the barrer not a thn aqueou flm, and the permeablty not gven by D */a. The olute permeablty of a bologcal barrer n general an expermental property, obtaned by dvdng the meaured flux of a olute by t tranmembrane concentraton dfference: k J, (2.19) c where J the meaured flux. Radolabeled tracer are often ued to meaure permeablty; the numerator and denomnator of the rght-hand de of Eq. (2.19) are replaced by the tracer flux and the tranmembrane dfference n tracer actvty. Even when the olute doe not cro the membrane by free dffuon, the expermental permeablty decrptve of the tranport behavor of the ytem. Such emprcal permeablte, though not alway eay to nterpret n phycal term, are nonethele ueful for comparng olute tranport rate and for predctng fluxe under mlar condton. There are ome cae n whch permeablty can be etmated from a dffuon coeffcent and membrane thckne. If the barrer a tablzed thn flm of a olvent mmcble wth water, the permeablty of the olute determned by the partton coeffcent, the flm thckne and the bnary dffuon coeffcent of the olute n the olvent that compre the membrane [ee the econd paragraph of the note followng Eq. (2.9), and Chap. 7]. If the membrane poee large ntertce or pore uch that dffuon through them the ame a that n free oluton, the permeablty gven by D */a, where the vod fracton n the membrane. Our nablty to predct membrane permeablty a pror reflect our gnorance of many factor that nfluence the tranport of a gven olute through a gven membrane. Some of thee factor, partcularly applcable to tranport through water-flled paage, are temzed below: The vod fracton mentoned above, or the fracton of the preented area of a membrane that occuped by pore, are often unknown. If the pore are not hghly connected, ther retance to dffuon wll depend on ther tortuoty; f the olute mut dffue down a tortuou path, t wll cro more lowly. Tranport depend crtcally on the dameter of the paage along the length of the dffuon path. The wall of pore gve re to a vcou drag that retard the dffuonal proce, and the degree of retardaton reman gnfcant for pore a large a ten tme the olute dameter. Th effect, whch wll be dcued n detal n Chapter 7, become greater when the dameter of the pore cloer to that of the olute. When the pore only lghtly larger than the olute, the latter mut cro the membrane by ngle-fle dffuon, and the augmented dffuon coeffcent no longer the

7 PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT 35 approprate meaure of olute moblty. Of coure, f the pore ze le than the olute ze, then the olute doe not cro the membrane at all, rrepectve t aqueou dffuon coeffcent. In mall pore, there conderable opportunty for chemcal and phycal nteracton between the olute and the wall. Such nteracton nclude adorpton and bndng. Furthermore, f the pore wall are charged, Donnan effect can caue the concentraton of charged olute nde the pore to be markedly dfferent from the concentraton n the adjacent oluton. There can alo be nteracton between the olvent and the wall of the pore, whch can caue the effectve olvent vcoty to dffer from that n free oluton, thereby (ee 2.1.4), affectng the olute dffuon coeffcent. Once the permeablty pecfed, the tranmembrane flux predcted by J = k c. (2.20) Equaton (2.20) the expreon mot commonly ued to decrbe the pave free dffuon of a nonelectrolyte acro a barrer. It can alo decrbe the tranport of an on n the abence of an electrcal potental gradent. Some llutratve value of olute permeablty are gven n Table 2.1. Table 2.1. Nonelectrolyte Permeablte of Three Cell Membrane Ox erythrocyte a cm/ Urea Glycol Dethylene glycol Glycerol Ehrlch acte tumor cell (moue) b cm/ Galactoe methyl glucoe Sorboe Glucoamne Human erythrocyte c cm/ Ethanol Glycerol Thourea Urea a Davon and Danell (1952). b Crane et al. (1957). c From ummary n Leb and Sten (1986).

8 36 CH.2:FREE DIFFUSION Untrred Layer Aume that c I > c ; by our conventon, c > 0, J > 0, and olute move from Phae I to Phae. Before a olute molecule can cro the membrane, t mut frt fnd t way from the bulk of Phae I to the membrane nterface at x = 0. Two mechanm are avalable to accomplh th: dffuon, whch decrbed by equaton lke thoe gven above, and convecton, n whch flud moton carre olute from the man body of the phae to the proxmty of the membrane. If one of the phae the nde of a cell, convecton lmted, and exchange between the bulk of the cytoplam and the membrane nterface largely by dffuon. Smlarly, dffuon uually the domnant tranport mechanm n the extracellular pace on the other de of the cell membrane. In many expermental tuaton, however, convecton can be ntroduced by trrng. The nfluence of trrng doe not extend unattenuated to the membrane oluton nterface; a thn, effectvely untrred layer adjacent to the membrane reman. Solute croe th layer only by dffuon, and t flux properly defned by Eq. (2.18); at x = 0, I D I I J ( c ) I b c, (2.21) where D the olute dffuon coeffcent n the th phae, the thckne of the correpondng untrred layer, and the ubcrpt b denote the bulk phae concentraton of the olute; c the olute concentraton at the nterface between the membrane and the th phae. For nondeal oluton, the augmented dffuon coeffcent would be ued n place of D. Smlar conderaton apply to the tranport of olute from x = a to the bulk of Phae. The olute concentraton profle hown n Fgure 2.1. Three retance n ere eparate the two bulk phae. The olute flux equal to the overall concentraton dfference dvded by the um of thee retance, each of whch nverely proportonal to a permeablty: J c c I b b I I / D 1/ k / D. (2.22) Here, k the true permeablty of the membrane. The apparent permeablty of the membrane, J /c b, the recprocal of the denomnator n the precedng equaton. The characterzaton of the untrred layer (or dffuon layer, a t alo known) a a harply defned boundary layer contanng all of the dffuonal retance outde the membrane clearly an approxmaton. When the boundng phae untrred, there are concentraton gradent throughout. In the preence of trrng, convectve effect are abent at the membrane oluton nterface and ncreae wth dtance from the membrane urface. Notwthtandng the lmtaton of the untrred layer concept, t doe provde a convenent mean for ncludng dffuonal retance outde the membrane n the equaton for olute flux, and for characterzng the magntude of uch retance. The quantty can be regarded a the thckne of a layer of the external phae whoe retance to dffuon the ame a that actually preent outde the membrane.

9 PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT 37 Fgure 2.1. Concentraton profle n the preence of untrred layer, and n the abence of olvent flow. The actual tranmembrane concentraton dfference, c I c, le than the overall concentraton dfference, c b I c b. It can be een from Eq. (2.21) that c approache c b a the thckne of the untrred layer approache zero; otherwe, the olute flux would become nfnte. When thee two concentraton are aumed to be dentcal (an aumpton that often made n practce, and wll be made lberally n the chapter to follow), the phae ad to be well-trred. Although vgorou trrng can reduce the effectve thckne of the untrred layer, t cannot be reduced to zero; the well-trred aumpton alway an approxmaton. The effect of a varety of trrng moton on olute flux are analyzed n Pedley (1983). In the preence of untrred layer, the concentraton dfference drvng the tranmembrane flux le than the dfference between the bulk phae concentraton (ee Fg. 2.1). Solute permeablte calculated ung the latter drvng force can be erouly underetmated f the retance of the untrred layer an mportant fracton of the total nterphae retance. Th more lkely to be the cae f the membrane permeablty hgh. The neglect of untrred layer effect can alo lead to error n the calculated parameter of carrer-baed tranport ytem (Chap. 4). The dffuon coeffcent of mall olute n the cytoplam are not known very well, o t dffcult to make good etmate of the error n cell membrane permeablty caued by ntracellular dffuonal retance. In uch cae, t common to

10 38 CH.2:FREE DIFFUSION aume that there no dffuonal retance on the cytoplamc de. The true cell membrane permeablty underetmated when th approach ued; however, permeablte that are derved n th way can be compared wth the permeablte of other olute derved mlarly, or ued to predct flux, a long a the bulk cytoplamc concentraton of the olute ued n the flux equaton. We wll ee n Chapter 6 that, for many olute/membrane combnaton, a tranmembrane concentraton dfference nduce a olvent flow, termed omo, n the drecton of the more concentrated oluton. The olute concentraton profle n the untrred layer curved when omo (or any tranmembrane olvent flow) preent. The effect of untrred layer on tranport n the preence of omo dcued n Chapter 10. Equaton (2.22) decrbe the teady-tate tranport of olute acro a ere of retance, for the cae n whch two of the retance are untrred layer and the thrd the membrane telf. The /D rato n the equaton are mply the recprocal of the permeablte of the ndvdual untrred layer. Equaton (2.22) can be regarded a a tranport equvalent of Ohm' Law for the voltage-drven current through a number of retor n ere; here, the voltage replaced by the bulk concentraton dfference, the current by olute flux, and the ohmc retance by the recprocal of the permeablte of each barrer. Th decrpton of the flux through ere barrer can be appled to many bologcal tranport procee, uch a tranport though a ngle layer of cell, where olute enter acro one face of the cell, croe the cytool, and then ext acro the other face; tranport through a cell upported by a permeable layer of extracellular matrx; or tranport through a ere of cell layer, a n epthela (Chap. 10). In uch cae, the general equaton for olute flux J c m 1 k 1, (2.23) where c the overall concentraton dfference and k the olute permeablty of the th of m barrer. A above, the recprocal of the denomnator of Eq. (2.23) the apparent permeablty of the compote barrer Applcaton of Soluton Theory A conderable body of theory ha been developed to decrbe free dffuon n oluton. Mot of th theory cannot be drectly appled to bologcal ytem, for reaon that have already been preented. One applcable product of oluton theory the Stoke Enten equaton, whch dentfe the varable that have the greatet nfluence on the dffuon coeffcent. In general, the dffuon coeffcent depend on the olute (naturally), the olvent, the concentraton of the olute (or compoton, for a multcomponent oluton), and temperature. Enten (1908) ued Stoke' Law to derve the followng approxmate expreon for the dffuon coeffcent of a phercal olute:

11 PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT 39 RT D, (2.24) 6N where olvent vcoty, olute radu, and N Avogadro' number. Stoke' Law decrbe the drag on a phere movng through a homogeneou flud of nfnte extent. Implct n th applcaton of Stoke' Law are the aumpton that olute molecule are much larger than thoe of the olvent, and that the nfluence of the oluton boundare (e.g., the wall of a pore) neglgble. Equaton (2.24) how that the mot mportant olute property affectng the dffuon coeffcent t ze (and hape; the equaton more complex for nonphercal olute), and the mot mportant olvent property t vcoty. Equaton (2.24) predct that the dffuon coeffcent nverely proportonal to the olute radu; that, the D product contant. Th condton met by the data n Table 2.2, even though the olute molecule are not much larger than thoe of the olvent. In bologcal ytem, th mple nvere relaton apple only to dffuon through large paage. When the ze of the pore not much greater than that of the olute, the permeablty depend on pore radu a well a olute radu. The effect of pore ze on olute permeablty wll be dcued n Chapter 7. Table 2.2. Tet of the Stoke Enten Equaton Dffuon coeffcent n Solute aqueou oluton radu, a at 25ºC, D a D 10 5 Solute nm cm 2 / nm-cm 2 / Methanol Urea Glucoe Glycerol Sucroe Raffnoe a Data from Schafer and Barfu (1980). The predcted effect of olvent vcoty on the dffuon coeffcent ha often been ued to nterpret and extrapolate expermental permeablty data. From the Stoke Enten equaton, the dffuon coeffcent expected to vary nverely wth olvent vcoty. If the temperature dependence of permeablty parallel that of the recprocal of the vcoty of water, th taken a evdence that the olute croe the membrane va water-flled pore. The permeablte of other dffuonal tranport route (e.g., acro the lpd phae of the cell membrane) are conderably more entve to temperature than the permeablty of an aqueou pore. Smlarly, f t known that a olute ue an aqueou pore to cro a membrane, then the temperature

12 40 CH.2:FREE DIFFUSION dependence of the vcoty of water can be ued to predct the olute permeablty at one temperature from the meaured permeablty at a dfferent temperature Fck' Second Law and Convectve Dffuon Fck' frt law one of the equaton mot commonly ued to decrbe bologcal tranport by free dffuon. It can readly be generalzed to any coordnate ytem: J = D c. (2.25) where J the flux vector n three-pace. In the teady tate, the law of ma conervaton appled to the pece J = 0. (2.26) Subttutng Eq. (2.25) nto (2.26), (D c ) = 0. (2.27) Equaton (2.27) the teady-tate form of Fck' econd law of dffuon, alo known a the dffuon equaton. When the dffuon coeffcent unform, the equaton mplfe further to 2 c = 0. (2.28) The dffuon equaton ha been olved n numerou geometre, for a wde varety of boundary condton. Table 2.3 ummarze ome ueful form of the teady-tate dffuon equaton. Table 2.3. Some Form of the Steady-State Dffuon Equaton 1. Cartean coordnate (x,y,z) 2 dc a) 1-dmenonal: D 0 dx 2 b) 3-dmenonal: c c c D x y z Cylndrcal coordnate (r =radal coordnate, z =longtudnal coordnate, no azmuthal varaton) D d dc a) r-varaton only: r 0 r dr dr b) r- and z-varaton, dfferent dffuon coeffcent n r- and z-drecton: D 2 r c c r Dz 0 2 r r r z 3. Sphercal coordnate, r-varaton only: D d dc r r dr dr 2 0 2

13 PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT 41 An addtonal contrbuton to the olute flux are f the oluton telf movng. Let u return to the one-dmenonal cae decrbed by Eq. (2.9). If the oluton flowng n the x-drecton at a velocty v, the olute flux augmented by a convectve term equal to the product of the oluton velocty and the local concentraton: The three-dmenonal equvalent of Eq. (2.29) dc J D vc. (2.29) dx J = D c + c v, (2.30a) where v now a vector. Subttutng Eq. (2.30a) nto Eq. (2.26), the equaton for teady-tate convectve dffuon wth a unform dffuon coeffcent become D c v c c v0. 2 Bologcal oluton can be regarded a ncompreble, and t can be hown that ncompreblty mple v = 0, o the teady-tate convectve dffuon equaton become D c v c 0. (2.30b) 2 When v unform (a t would be, n the one-dmenonal cae), Eq. (2.29) can be ntegrated to gve an expreon relatng flux, the concentraton boundary condton, and velocty. It eay to add a olute convecton term to the more general form of the dffuon equaton gven n Table 2.3, but t not eay to olve the equaton that reult. Numercal mulaton uually requred. The ue of Eq. (2.26) to decrbe ma conervaton n the teady tate mple that the dffung pece nether produced nor conumed n the regon of nteret. Th aumpton wll generally apply throughout th text. In the Appendx to Chapter 10, the one-dmenonal convectve dffuon equaton [Eq. (2.29)] wll be extended to nclude change n pece concentraton reultng from fluxe acro the regon boundare. In Chapter 11, Fck' econd law, wth [Eq. (2.30b)] and wthout [Eq. (2.27)] convecton, wll be generalzed to nclude chemcal reacton wthn the regon and tme-dependent behavor Jutfcaton of the Steady-State Aumpton: Tme Scale n Bologcal Tranport Vrtually all of the tranport procee decrbed n th text are teady-tate procee; that, the concentraton n the ytem both the external boundary condton and the condton nde the barrer are aumed to be ndependent of tme. When there nether producton nor conumpton of the pece of nteret nde the membrane, ma conervaton mple that the teady tate flux atfe Eq. (2.26). In the common one-dmenonal decrpton of membrane tranport, Eq. (2.26) become

14 42 CH.2:FREE DIFFUSION mply dj /dx = 0; that, the flux baed on a unt membrane urface area the ame at every cro-ecton n the membrane. We ued th fact to ntegrate Fck' frt law, and we wll ue t agan. Another mplcaton of the teady-tate aumpton that the flux contant n tme. But th create an apparent contradcton: how can the boundary condton reman contant n the face of a perpetual flux? Clearly, they cannot, and the eay fx to potulate, at leat for the purpoe of analy, that the boundng oluton are nfnte n extent, o ther compoton do not change even when olute lot or ganed. Infnte ytem are convenent to potulate but rare n the real world. Happly, mot bologcal membrane do experence a relatvely table mleu becaue of homeota the tendency of lvng ytem to mantan a contant nternal envronment, whch nclude, for ntance, the compoton of the extracellular flud that defne the boundary condton for olute tranport nto and out of cell. The mantenance of reaonably contant boundary condton for bologcal tranport thu accomplhed by other agence (uch a the kdney) outde the ytem under tudy. Of coure, lvng ytem do experence change n ther envronment that caue change n tranport rate; thee may reflect a falure of homeotatc mechanm or a udden challenge to the ytem occaoned externally or by the behavor of the organm. Some bologcal procee, uch a regulatory event, are nherently dynamc. In uch cae, the boundary condton for tranport cannot be regarded a contant, and teady-tate oluton would no longer eem to apply. The proper approach to decrbng tranport under thee crcumtance depend on the rate at whch the boundary condton change. If they change lowly compared to the rate at whch the tranport proce can adapt to that change, tranport can be regarded a quateady; that, the tranport rate at any tme equal to the teadytate flux correpondng to the boundary condton at that tme. If the boundary condton change more rapdly, a full unteady-tate oluton of the dffuon equaton wthn the membrane neceary. The tme for the tranport rate to adapt to changng boundary condton the tme needed for the concentraton profle n the membrane to change to the profle approprate to the new boundary condton. Ung a Fourer ere oluton n lab geometry, We (1996a) obtaned a tme contant, t d = a 2 /( 2 D ), for the approach to the teady tate nde a homogeneou membrane wth an arbtrary ntal concentraton profle, expoed at t = 0 to new concentraton boundary condton at each face. A mght be expected, t d horter for thnner membrane and for more rapdly dffung olute. One tranport proce for whch the boundary condton depend on tme the dffuon of a olute nto or out of a cloed compartment, uch a a cell. The rate at whch the concentraton n the cell change related to the olute flux and the urface area and volume of the cell. Aume the nteror of the cell well-mxed and Phae, o flux nto the cell potve. The rate of change of the number of mole of olute n the cell, n, gven by dn dt J S, (2.31)

15 PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT 43 where S the urface area of the cell. The concentraton of olute n the cell equal to the number of mol of olute per unt volume: c = n /V, where V the cell volume (we aume for mplcty that all porton of the cell are acceble to the olute). Combnng thee two equaton and Eq. (2.17), wth D ntead of D *, we obtan I dc D( c c ) S V. (2.32) dt a Eq. (2.32) readly ntegrated for a contant ambent concentraton, c I. The ntracellular concentraton follow a decayng exponental n tme, wth a tme contant t c = av/(d S). The tme contant horter when the olute pae through the membrane more readly (low thckne, hgh dffuon coeffcent) and when the urface-tovolume rato of the cell large. The quateady approxmaton approprate f t d << t c, or a << 2 V/S. Interetngly, the dffuon coeffcent doe not appear n the crteron, becaue t affect the tme contant of both procee mlarly: even a a hgh membrane dffuon coeffcent allow the concentraton profle nde the membrane to adapt more quckly, t alo caue the ntracellular concentraton to change more rapdly. Generally, the quateady approach ha proven adequate for decrbng bologcal tranport n the preence of changng boundary condton. It an mportant aumpton underlyng tme-dependent applcaton of compartmental analy, a modelng technque for complex ytem that wll be dcued n Chapter 8. Of coure, change n the boundary condton are not the only tool by whch lvng ytem elct change n flux. A we hall ee, uch change partcularly, rapd change are n mot cae obtaned by alterng the tranport properte of the membrane themelve FREE DIFFUSION OF ELECTROLYTES The free dffuon of electrolyte conderably more complex than that of nonelectrolyte. The bac flux equaton for electrolyte the electrodffuon equaton. Th nonlnear equaton oluble, but the general oluton are o complex that they have rarely been appled to bologcal ytem. A general oluton of the electrodffuon equaton, and a number of pecal cae, are gven below Dfference between Electrolyte and Nonelectrolyte Dffuon There are two prncpal dfference between the dffuon of electrolyte and nonelectrolyte: 1. Charged olute are ubject to electrcal force when electrotatc potental gradent are preent. Accordngly, the drvng force for electrolyte tranport the gradent of the electrochemcal potental rather than that of the chemcal potental. 2. Snce any electrolyte oluton mut contan at leat one anon and one caton, there are alway at leat two olute pece. The extence of mul-

16 44 CH.2:FREE DIFFUSION tple pece and, correpondngly, multple fluxe lead to two concept that are only when electrolyte tranport condered. The frt of thee concept electroneutralty: the concentraton of potve charge n a mall ample volume equal the concentraton of negatve charge. Th condton can be wrtten a follow: zc 0. (2.33) The econd concept onc current. Ion movng n oluton carry current jut a electron do n metal conductor. The contrbuton of each pece to the current denty equal to the product of the pece' flux and t charge. The current denty obtaned by ummng thee contrbuton: I I z J. (2.34) where I = z J the contrbuton of the th on to the current. Note that the unt of I a gven above are mol of charge per quare centmeter of tranport area per econd. If the rght-hand de of Eq. (2.34) multpled by the Faraday, the unt become coulomb per quare centmeter per econd; that, ampere per quare centmeter The Electrodffuon Equaton The flux of the th on n free oluton, lke that of the nonelectrolyte n the precedng ecton, equal to the product of the moblty of the on, t concentraton, and the approprate drvng force, whch n th cae d / dx. The drvng force can be wrtten n term of the chemcal and electrotatc potental gradent: d d d z. (2.35) dx dx dx The chemcal potental gradent treated a n the prevou ecton, and the flux equaton become RT dc d J Uc z dc c dx dx URT Ucz d. (2.36) dx dx Equaton (2.36) the electrodffuon equaton, whch the mot common tartng pont for decrbng free dffuon n electrolyte oluton. It alo known a the Nernt Planck equaton. A n the analy of the Donnan equlbrum, concentraton are ued rather than actvte, to facltate the ue of the electroneutralty condton n olvng the equaton. A a conequence, the oluton of th equaton neglect drect on on nteracton durng the tranport proce, and the analy to follow trctly hold only for oluton more dlute than thoe found n lvng ytem. The electrodffuon equaton defne the dependence of onc flux on the gradent n concentraton and electrotatc potental n the membrane or barrer acro whch tranport take place. Thee gradent are not generally meaurable. It therefore derable to ntegrate the equaton, o that the fluxe can be related to the condton at the membrane urface.

17 PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT 45 The reult of th ntegraton wll reveal a much more complex dependence of flux on boundary condton than wa the cae when nonelectrolyte were condered. In dlute oluton, the flux of an uncharged pece depend on only the concentraton of that olute n the bathng oluton; th o even when other olute are preent. In electrolyte oluton, however, the flux of each on depend on the concentraton of all on n the boundng oluton, and not n a mple fahon. In addton, for any partcular par of boundng compoton, the fluxe, and hence the tranmembrane current denty, depend on the tranmembrane potental dfference. Th llutrated n Fgure 2.2a. Shown n Fgure 2.2b are two common expermental tuaton. 1. Short Crcut. Electrode n the two bathng oluton are connected by an external crcut, hortng out the membrane and brngng the potental dfference acro t to zero. The current denty meaured under hortcrcut condton called the hort-crcut current. Pave on fluxe at hort crcut are drven by only concentraton gradent and can therefore be decrbed by the ame equaton a are ued to decrbe the flux of nonelectrolyte. Accordngly, onc fluxe at hort-crcut can be expreed n term of the membrane permeablte of the on, followng Eq. (2.20). It eay to calculate the hort-crcut current from the membrane properte and boundng compoton, nce at hort-crcut the fluxe of the on are ndependent of one another and the flux of each can be calculated from Eq. (2.20). 2. Open Crcut. At open crcut, there no net tranport of charge acro the membrane; that, I = 0. The zero-current condton more typcal of unmanpulated bologcal ytem. The external path between the two de of the barrer n Fgure 2.2b doe not ordnarly ext, and electroneutralty demand that equal amount of potve and negatve charge cro the barrer. At open crcut, the quantty of expermental nteret the potental dfference acro the membrane. The term dffuon potental alo ued to decrbe the potental dfference that develop acro a membrane when the current zero and all flux pave. Under certan aumpton, the electrodffuon equaton can be ntegrated to compute the open-crcut potental and fluxe from the compoton of the oluton on the two de of the membrane and the onc moblte wthn t; the more general oluton are preented frt. Before proceedng, t hould be remarked that many cellular and ntracellular membrane contan actve tranport ytem that generate a net onc flux and correpondng actve onc current, I a. The zero-current condton n th cae a I I 0, where I the pave current decrbed by the electrodffuon equaton.

18 46 CH.2:FREE DIFFUSION Fgure 2.2. (a) A oluton of the electrodffuon equaton, for an uncharged membrane at 25ºC. The compoton of the oluton n Phae I 68 mm NaCl, 15 mm KHCO 3, and 68 mm RCl, where R a large caton whoe moblty one-tenth that of Na; n Phae, the RCl ha been replaced by NaCl. The moblte of the on are taken from Table 2.4, and the membrane modeled a an aqueou flm 1 cm thck. Dahed lne denote the hort-crcuted ( = 0) and open crcuted (I = 0) condton. After Fredman (1970). (b) Short-crcut and open-crcut condton. In the former, the ammeter meaure the hort-crcut current; n the latter, the voltmeter meaure the opencrcut potental dfference.

19 PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT Integraton of the Electrodffuon Equaton The ntegraton of the electrodffuon equaton complcated by the nonlnearty of the equaton. The nonlnearty are from the econd term on the rght-hand de, becaue the onc concentraton and potental gradent are functon of locaton n the membrane. The econd term wa abent when nonelectrolyte were condered. In the late nneteenth century, a number of nvetgator, ncludng Planck (1890) and Behn (1897), reported oluton of the electrodffuon equaton. We wll not reproduce thee dervaton here, lmtng ourelve ntead to the aumpton and fnal reult. Both nvetgator made the ame aumpton: 1. There are n on n the ytem, and Eq. (2.36) hold for each of them; the moblty of each on ndependent of poton (x) or local compoton. The phae boundng the two face of the membrane, whoe thckne a, are denoted Phae I and Phae. The concentraton of the th on n the jth phae denoted c j. 2. All on are unvalent. Th retrcton can be omtted, but the oluton more complcated when the valence of the on are not all the ame. 3. At every pont n the membrane, the local compoton electroneutral. Th aumpton trctly fale whenever the electrc feld E = d/dx nonunform, but the devaton from electroneutralty almot alway trval. The relaton between the nonunformty of the electrc feld and the departure of the oluton from electroneutralty are a follow: a volume of oluton that not electrcally neutral contan a net charge called the pace charge,, whoe local concentraton equal to z c. Thu, when the electroneutralty condton [Eq. (2.33)] atfed, = 0. The pace charge concentraton related to the gradent of the electrc feld through the Poon equaton: de/dx = ( / ), where the permttvty of the barrer. Thu, the electroneutralty aumpton trctly correct (.e., = 0) only when the feld unform (.e., de/dx = 0). When electroneutralty aumed and the electrodffuon equaton olved accordngly, the calculated electrotatc potental not generally a lnear functon of x; hence the feld not unform and the electrolyte oluton cannot be electrcally neutral. For bologcally relevant boundary condton, th ncontency unmportant. Ung the electrc feld gradent obtaned by olvng the electrodffuon equaton wth the electroneutralty aumpton, the pace charge denty can be computed from the Poon equaton. Th value of nevtably order of magntude le than the concentraton of the electrolyte oluton telf. 4. The ytem n the teady tate, o all onc fluxe are ndependent of x. The Planck oluton gve a trancendental expreon for the membrane potental = I a a functon of the boundng compoton. The open-crcut potental obtaned by olvng

20 48 CH.2:FREE DIFFUSION I N ln ln U U N N N V V N N N ln ln N I I I I I (2.37) for, where = exp(/rt), U j U c, caton j V j Uc, anon j and N j the total concentraton of the jth phae, defned a N j n c. The Behn oluton provde the fluxe a well a the membrane potental, and cont of a et of n + 1 equaton that are olved multaneouly: 1 j gln N I RT N, (2.38a) J URT N N c c ( = 1,,n). (2.38b) a N e N I - z / RT ( )( e I ) ( gz 1) - z / RT I A before, flux from Phae I to Phae potve. If the compoton {c I } and {c } are pecfed, and the moblte are known, then Eq. (2.38) conttute n + 1 equaton n n + 2 unknown: an unpecfed contant, g; the membrane potental, ; and n fluxe, {J }. Snce there one more unknown than there are equaton, one of the unknown, or a functon of them, mut be pecfed. Generally, th ether the membrane potental or the tranmembrane current denty. If the potental pecfed, Eq. (2.38a) can be olved for g, and Eq. (2.38b) gve the fluxe drectly; a more dffcult teratve procedure requred f the current gven and the membrane potental and fluxe are ought. Strctly peakng, the upercrpt I and denote the potental and concentraton jut nde the membrane face. However, n applyng the precedng oluton, the concentraton that are generally ued are thoe n the boundng phae, and the membrane potental meaured n the external oluton a well. Often, no harm done when th approxmaton made, but error can are f a boundng phae contan charged pece that cannot enter the membrane, or f the membrane contan fxed charge that cannot leave (Fg. 2.3). In uch cae, the correct (.e., ntramembrane) boundary condton for Eq. (2.37) and (2.38) are related to the compoton and potental n the bathng oluton by the Donnan equlbrum expreon of the prevou chapter. The oluton gven above mut be further modfed when the membrane tructure charged, becaue the concentraton of the charge on the membrane mut be ncluded n the electroneutralty condton (ee 2.2.6).

21 PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT 49 Fgure 2.3. Boundary condton for oluton of the electrodffuon equaton. In the example hown here, there one moble caton (C) and one moble anon (A) n the ytem. Both boundng phae are well trred. Phae I contan a negatvely charged pece that cannot enter the membrane, o there a Donnan equlbrum at x = 0. Inde the uncharged membrane and mpermeantfree Phae, c C = c A, by electroneutralty. The correct boundary condton for the electrodffuon equaton are thoe at the flled crcle. The potental dfference between Phae I and equal the um of the calculated membrane potental and the Donnan potental at the Phae I nterface. Before proceedng to the condton under whch mpler oluton of the electrodffuon equaton can be obtaned, we hould oberve that the equaton lnear n c ; therefore a partal ntegraton can be carred out ung an ntegratng factor. Th procedure demontrate that the on flux the product of three term: U RT, I c exp( z / RT) c, and the recprocal of the ntegral of exp[z (x)] acro the membrane. Snce (x) generally not known a pror, th equaton cannot gve flux drectly; however, the frt term how that the flux proportonal to the moblty of the on, and the econd demontrate that the flux of an on at equlbrum zero (ee alo 2.2.4) Some Specal Cae Equlbrum. By ettng J = 0 n Eq. (2.38b), we obtan the condton under whch the th on n equlbrum acro the membrane. The flux zero when the lat factor n the numerator zero: c - / I e z RT c 0. (2.39)

22 50 CH.2:FREE DIFFUSION Fgure 2.4. Potental-drven current, unform compoton. The electron flow n the external crcut and the current acro the membrane are both drected from Phae I to Phae becaue the tranmembrane current defned a the flow of potve charge [Eq. (2.34)]. It eay to how that Eq. (2.39) precrbe that the th on n equlbrum when t Nernt potental equal to the membrane potental. Th concluon wa alo reached n Chapter 1. Unform Compoton. The oluton of the electrodffuon equaton proceed more drectly when the compoton of the oluton on the two de of the membrane are the ame, and the compoton nde the membrane conequently unform. When a potental appled acro the membrane, the on mgrate acro, drven by the electrc feld, and generate an onc current; th tuaton llutrated n Fgure 2.4. The concentraton gradent of each on zero, o Eq. (2.36) become: d J Ucz dx. (2.40) Snce c ndependent of x, Eq. (2.40) can ealy be ntegrated to gve the onc flux a a functon of the mpoed potental dfference: Ucz J. (2.41) a Each flux proportonal to the potental dfference acro the membrane. It follow trvally from Eq. (2.34) that the membrane current denty mlarly proportonal to the membrane potental; hence, when the compoton unform, the membrane obey Ohm' Law. If the fracton of membrane area avalable for tranport, and the tranport path are uffcently large that the effect of the pore wall on the tranport rate can be neglected, the conductance of the membrane g (mol/cm 2 -ec-v) I a Ucz. 2

23 PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT 51 In electrcal unt, conductance meaured n emen; 1 S = 1 ohm 1 = 1 ampere/volt. The membrane conductance n S/cm 2 obtaned by multplyng the prevou value of g by. If, n Eq. (2.40), the electrotatc potental gradent regarded a the drvng force for tranport, then, accordng to the Teorell equaton, the product U z aume the role of a moblty. Indeed, the abolute value of th product termed the electrcal (or electrophoretc) moblty of the on, u = U z. The electrcal moblte of everal bologcally mportant on are preented n Table 2.4. Table 2.4. Electrcal Moblte of Several Bologcally Important Ion at 25ºC (Robnon and Stoke, 1965; Dave, 1968) Electrcal moblty, cm 2 /ec-v, Ion u x 10 4 H L 4.01 Na 5.19 K 7.62 NH Mg 5.50 Ca 6.17 Cl 7.92 NO HCO Dffuon Potental of a B-Ionc Sytem. An explct oluton for the dffuon potental can be obtaned from Eq. (2.38) f the ytem contan only one anon and one caton, of equal charge. In th cae, the anon and caton fluxe are equal, nce the current zero, and the membrane potental RT U U C C U U A A c ln c I, (2.42) where c electrolyte concentraton and the ubcrpt on the moblte denote the caton and anon. The dffuon potental ndependent of membrane thckne, and depend on only the moblty rato U C /U A (dvde numerator and denomnator by U A to ee th) and the concentraton rato c /c I. The orgn of the dffuon potental eaet to explan for th cae n whch only two on are preent. If the membrane permeable to both the anon and caton of a alt whoe concentraton dfferent on each de of the membrane, both on wll cro. The onc fluxe mut be equal when the current zero, even though the moblte of the two on are not generally the ame. The dffuon potental develop to compenate for th dfference n moblty by ncreang the electrochemcal potental drvng force for the on havng the lower moblty, and decreang that for the more moble on. The potental pull the le moble on acro the membrane, whle re-

24 52 CH.2:FREE DIFFUSION tardng the flux of the more moble pece. Suppoe c > c I and U C >U A. Then the alt dffue from Phae to Phae I under t concentraton gradent, and the potental of Phae I become potve relatve to that of Phae (.e., > 0), o a to retard C and ncreae the drvng force for A. When U C = U A, there no moblty dfference to compenate for, and the dffuon potental zero. Dffuon potental can caue artfact n certan electrophyologcal experment, and t derable to avod them. Much ue made of concentrated KCl oluton (alt brdge) n uch etup, becaue the moblte of potaum and chlorde are almot dentcal. Conder the other extreme, n whch the moblty of one on much larger than that of the other, ay U C >> U A. In th cae, the membrane potental gven by Eq. (2.42) approache (RT/) ln c /c I. Snce there only a ngle electrolyte n the ytem, the argument of the logarthm alo c C /c C I ; thu the membrane potental approache the Nernt potental of the caton. We hall ee that the tendency of the membrane potental to approach the Nernt potental of the more (or mot) permeable on evdent for more complex electrolyte oluton a well. Th tendency ha been exploted expermentally to clamp the membrane potental at a elected value by bathng the two de of the membrane wth oluton contanng dfferent concentraton of an on (often potaum) to whch the membrane partcularly permeable. Actve and Pave Exchange wth a Cloed Compartment. In the teady tate, the net rate of entry of any pece nto a cloed compartment equal the rate at whch t conumed; otherwe, t concentraton n the compartment would change wth tme. Smlarly, when a compound ynthezed n a cloed compartment, the ynthe rate (le any conumpton of the materal nde the compartment) equal the rate at whch the ubtance leave the compartment. When the olute nether conumed nor produced wthn the compartment, t net entry rate mut be zero n the teady tate. Th the cae for mot on. A noted earler, cell membrane are capable of actvely tranportng ( pumpng ) on between the nteror of a cell and the extracellular flud. Suppoe that two catonc pece wth the ame valence, z, are exchanged acro the cell membrane, uch that, for each on of Spece 1 that pumped from Phae to Phae I, r on of Spece 2 are pumped from Phae I to Phae. In the teady tate, the net rate at whch each on croe the cell membrane the actve flux plu the pave flux mut be zero. Hence the pave flux of each on the negatve of t actve flux. Thu, for each on of Spece I movng pavely from Phae I to Phae, r on of Spece 2 move pavely from Phae to Phae 1: J 2 = rj 1, (2.43). where r the couplng rato or couplng coeffcent of the pump. Subttutng Eq. (2.43) nto Eq. (2.36), and rearrangng, Solvng for d/dx, dc2 d dc1 d U 2RT U2c2z ru1rt ru1c1z. (2.44) dx dx dx dx

25 PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT 53 dc1 dc 2 ru1 U d RT 2 dx dx dx z ru1c1 U2c. (2.45) 2 The functon n the parenthee on the rght-hand de equal to hence, druc ( 1 1Uc 2 2) dx dln( ru1c1 U2c2) ; ru c U c dx Integratng acro the membrane, and lettng z = 1, RT d dln( ru1c1u2c2). (2.46) z RT ln Uc ruc Uc ruc I I (2.47) Th equaton, whch obtaned drectly from the electrodffuon equaton, wa ued by Mulln and Noda (1963) to relate the membrane potental of keletal mucle to the tochometry of actve Na K exchange acro the mucle fber membrane. Equal Total Concentraton on the Two Sde of the Membrane: The Contant-Feld Equaton. Even though the concentraton of ndvdual on vary wdely n the body, the total onc concentraton, N, qute unform throughout (Table 2.5). The ntracellular onc content only 10% le than the extracellular value, o the oluton of the electrodffuon equaton for N I = N of ome nteret. Th oluton alo mpler and much more frequently ued than the more general oluton gven earler. Table 2.5. Typcal Ionc Content of Intracellular and Interttal Flud, and Blood Plama a Ion Plama Interttal flud Intracellular flud Na K Ca 1 1 <<1 Mg Cl HCO Phophate b SO 4 <1 <1 1 Lactate Total a Concentraton are n mm. Adapted from Guyton and Hall (2000). b Include larger molecule to whch phophate group are attached.

26 54 CH.2:FREE DIFFUSION The oluton for th pecal cae cannot be obtaned drectly from Eq. (2.38), whch become ndetermnate. We begn the dervaton by ung Eq. (2.36) to contruct two um: J dc d RT cz U dx dx, (2.48) n n n Jz dc d RT z c. (2.49) n n n 1 U 1 dx dx 1 In Eq. (2.49), ue ha been made of the aumpton, a wa made n the Planck and Behn oluton, that the on are monovalent, o z 2 = 1. The predomnance of monovalent on evdent from Table 2.5. Conder each of the four um on the rght-hand de of the two equaton jut wrtten. The frt um on the rght-hand de of Eq. (2.48) mplfed by nterchangng the order of ummaton and dfferentaton: dc d dn dx dx dx c. (2.50) The econd um on the rght-hand de of Eq. (2.48) zero, by the electroneutralty condton, Eq. (2.33). The frt um on the rght-hand de of Eq. (2.49) alo zero, becaue t the dervatve of a quantty that unformly zero: dc 0 d zc d 0 dx dx dx z. (2.51) The econd um on the rght-hand de of Eq. (2.49) N, by defnton. Thu, Eq. (2.48) and (2.49) can be rewrtten: J dn RT, (2.52) U dx Jz d N. (2.53) U dx Thee equaton are not eay to olve n the general cae, becaue the latter nonlnear. However, the oluton proceed ealy when N the ame on both de of the membrane. Frt we recall that the teady-tate flux ndependent of x; f we aume that the onc moblte are alo unform, then the left-hand de of Eq. (2.52) and (2.53) are contant. Thu the rght-hand de mut alo be contant, ndependent of poton n the membrane. From Eq. (2.52), dn/dx contant, o N a lnear functon of x. For the pecal cae of nteret here, N the ame at both de of the membrane; therefore, t mut be the ame throughout. Snce N unform and the rght-hand de of Eq. (2.53) contant, the electrc feld, d/dx, alo unform. If the potental gradent d/dx the ame everywhere n the membrane, t mut be equal to

27 PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT 55 I ( )/a /a. The electrodffuon equaton can then be wrtten n the followng form: dc J URT Uz c dx a. (2.54) The coeffcent of c n Eq. (2.54) ndependent of x, o the equaton can readly be olved for the flux a a functon of the condton on the two de of the membrane. The reult one form of the Goldman Hodgkn Katz contant-feld equaton: I -z / RT c c e J z U. (2.55) a -z / RT 1e The contant-feld equaton the equaton mot commonly ued to predct the on fluxe acro a membrane, and hence the membrane current, when the membrane potental and boundng compoton are pecfed. It clearly atfe the equlbrum condton: the flux of an on zero f t Nernt potental equal the membrane potental. The equaton can alo be ued to fnd the membrane potental when the current and boundng compoton are pecfed, but th generally requre a numercal oluton. An excepton when the current zero; n th cae, the contant-feld equaton can be manpulated to predct the dffuon potental: Uc Uc I RT caton anon ln. (2.56) I Uc Uc caton anon A ndcated, the um n Eq. (2.56) nclude ether all caton or all anon. Equaton (2.55) can be wrtten n nondmenonal form, n whch the nondmenonal flux a functon of two nondmenonal group: a nondmen- Ja I URTc onal potental z / RT whoe gn depend on the onc charge, and the tranmembrane concentraton rato C = c /c I : 1Ce 1e. (2.57) Plot of v., parameterzed by C, are hown n Fgure 2.5. The on flux zero when the membrane potental equal the Nernt potental (n nondmenonal unt, = ln C). A Eq. (2.55) ndcate and Fgure 2.5 demontrate, plot of flux v. potental are curved, o the membrane behave a a rectfer: equal and oppote devaton from the Nernt potental do not n general nduce equal and oppote on fluxe. Although the electrc feld gven by the electrodffuon equaton ndependent of x only when N I = N, the explct oluton for flux and potental gven above have een conderably more ue than the more unweldy Behn or Planck oluton. A noted earler, the ue of the contant-feld equaton n many bologcal applcaton

28 56 CH.2:FREE DIFFUSION can be jutfed by the fortunate fact that N doe not vary very much n lvng ytem. What make the contant-feld equaton more remarkable t utlty n pte of mportant dfference between on tranport n real bologcal membrane and the contnuum model mpled by the electrodffuon equaton. Thee dfference, whch wll become apparent n the chapter that follow, are more dramatc than a modet nonunformty n N. An llutraton of the ue of the contant-feld equaton to nterpret phyologc data preented n the next ubecton. Fgure 2.5. Nondmenonal repreentaton of tranmembrane flux gven by the Goldman Hodgkn Katz contant-feld equaton [Eq. (2.55)]. The varable are defned mmedately precedng Eq. (2.57). A wa the cae for nonelectrolyte tranport, the eae wth whch an on croe a bologcal barrer generally expreed n term of t permeablty, k. The flux equaton derved n and Eq. (2.38), (2.41), and (2.55) are baed on free oluton thermodynamc and are trctly applcable only to tranport acro a tagnant water flm. For uch tranport, and neglectng nondeal effect, the permeablty of the th on related to t dffuon coeffcent and moblty n free oluton by k 0 = D /a = U RT/a, analogou to the relatonhp for nonelectrolyte. And, a wa the cae for uncharged olute, on permeablte n bologcal ytem are expermental quantte, obtaned by meaurng the on flux under known condton and applyng the flux equaton preented above, wth U replaced by k a/rt. Wth th ubttuton, the frt two term n Eq. (2.55) become ( z / RT ) k. An mportant applcaton of the electrodffuon equaton predcton of the relatonhp between onc permeablte and the membrane potental. In all uch equaton derved above Eq. (2.37), (2.42), (2.47), and (2.56) the potental depend on the rato of lnear combnaton of moblte. In free oluton, the moblty and permeablty are proportonal, wth the proportonalty contant RT/a. If there a mlar proportonalty n bologcal membrane, t eay to how that the moblte

29 PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT 57 n thee equaton can be replaced by permeablte. Th wll be demontrated n the followng ecton Ionc Permeablty and the Retng Potental of the Cell An electrotatc potental dfference ext between the nteror of bologcal cell and the extracellular flud. Generally, the abolute value of th potental below 100 mv, wth the cell nteror negatve. Comparon of the Nernt potental of the prmary bologcal on K, Na, and Cl wth the membrane potental how that Cl generally cloe to equlbrum acro the cell membrane, but the caton are not. The nonequlbrum tate of the caton mantaned by actve tranport ytem n the cell membrane that pump potaum on nto the cell n exchange for odum on, whch are pumped out. The pump tochometry uch that the number of odum on pumped out exceed the number of potaum on pumped n; a a conequence, the pump generate an onc current acro the membrane. Equaton (2.56) ha been ued to etmate the relatve caton permeablte of the cell membrane. A noted earler, the total current acro the cell membrane equal the actve current due to the Na K pump plu the pave current, whch n th cae decrbed by the contant-feld equaton. The total current acro the membrane mut be zero, or ele charge accumulate n the cell. Thu, f there an actve current, the pave current cannot be zero. Equaton (2.56) wa derved under the aumpton that the pave current zero. Even though th not generally the cae for bologcal cell, the equaton ha been ued to decrbe the dependence of the cell potental on onc permeablte, under the mplct aumpton that the pave current cloe enough to zero that t effect on the cell potental can be neglected. Conderng only the three prmary on gven above, and lettng Phae I be the nde of the cell and Phae the outde, Equaton (2.56) become RT U c U c U c r ln U c U c U c I Na Na K K Cl Cl I I Na Na K K Cl Cl, (2.58) where r the cell potental. Snce Cl n equlbrum acro the cell membrane, r = E Cl : RT c r ln c Cl I Cl RT c ln c I Cl Cl. (2.59) Equatng the argument of the logarthm n the precedng two equaton, and rearrangng, c U c U c I Cl Na Na K K I I Cl Na Na K K c U c U c. (2.60) Subttutng Eq. (2.60) nto (2.59),

30 58 CH.2:FREE DIFFUSION RT U c U c r ln U c U c Na Na K K I I Na Na K K. (2.61) If the permeablty and moblty of each caton are related by U = k, the argument of the logarthm become k c k c Na Na K K I I Na Na K K k c k c. The proportonalty contant cancel out, effectvely replacng moblty by permeablty. Dvdng the numerator and denomnator of the argument of the logarthm by k K, k RT r ln k Na cna k K Na I cna k K c c K I K. (2.62) Equaton (2.62) ha been ued to etmate the Na/K permeablty rato from meaurement of the retng potental and ntracellular concentraton n oluton of known compoton. Value for nerve and mucle at ret range from 0.01 to 0.2; the permeablte of the two on n the red cell membrane are cloer to one another. The varaton of retng potental wth the permeablty rato llutrated n Fgure 2.6 for concentraton typcal of a nerve fber. Equaton (2.62) derved here a a pecal cae of Eq. (2.56), whch rele on the contant-feld aumpton. However, t can alo be derved by ettng r = 1 n Eq. (2.47), whch doe not aume a contant feld. When the couplng coeffcent unty, there no actve current, o the pave current at open crcut zero. Indeed, for th partcular cae, the contant-feld aumpton unneceary. In th applcaton of equaton baed on oluton theory to tranport n real bologcal ytem, an mportant caveat mut be tated. A we have already mpled, on and mot other olute do not cro bologcal membrane by dffung down fludflled path that can be regarded a mple extenon of the boundng oluton nto and through the membrane. Although the contant-feld equaton and other derved here from oluton thermodynamc can decrbe wth ome ucce the effect of boundary condton and membrane properte on fluxe and potental, the parameter (uch a permeablty) that we derve to ummarze the expermental reult may have a very dfferent phycal orgn than the ame parameter when ued to decrbe tranport n mple oluton. The permeablty rato of a cell membrane tell u omethng about how readly varou on cro, but t tell u very lttle about the phycal procee that accompany permeaton.

31 PRINCIPLES AND MODELS OF BIOLOGICAL TRANSPORT 59 Fgure 2.6. Effect of the odum/potaum permeablty rato on cell potental. The nternal compoton of the cell 14 mm Na and 140 mm K, and the ambent oluton 142 mm Na and 4 mm K. T = 37ºC Charged Membrane The membrane matrx can contan docated polar group and conequently poe a net charge. Other membrane, uch a the membrane of bologcal cell, exhbt a urface charge that due to expoed onzable group. We wll dcu the frt ntance here, and the nfluence of urface charge n the followng chapter. When extendng electrodffuon theory to membrane that contan free charge, the onzable group are aumed to be dtrbuted unformly wthn the barrer. Th aumpton a good one for ome ytem, uch a artfcal membrane made from on exchange ren, or gel-lke extracellular tructure, ncludng certan connectve tue. It le applcable to the charged cell membrane pore that we wll be dcung n Chapter 4. The tranport proce n unformly charged membrane decrbed by the model developed by Teorell (1935) and Meyer and Sever (1936), and whch llutrated n Fgure 2.7. When the membrane charge known, the Donnan equlbrum condton can be ued to compute the compoton of the oluton jut nde each membrane face; thee compoton are the boundary condton for the ntegraton of the electrodffuon equaton acro the membrane. The potental dfference between the oluton at the two de of the membrane equal to the algebrac um of the Donnan potental at each face and the tranmembrane potental dfference obtaned from the electrodffuon equaton.

32 60 CH.2:FREE DIFFUSION Fgure 2.7. The Teorell Meyer Sever (TMS) model. In the example hown here, a negatvely charged membrane bounded by two well-trred oluton of the ame electrolyte, CA. The concentraton c AI, c CI, c A, and c C the boundary condton for the electrodffuon equaton are n Donnan equlbrum wth the ambent phae. A typcal potental profle alo hown, aumng C more moble than A, and I = 0. The dfference I obtaned from the electrodffuon equaton. The oluton of the electrodffuon equaton more complcated than before becaue the electroneutralty condton nclude the fxed charge: zc z X0, (2.63) X where X charge concentraton and z X = ±1. The electrodffuon equaton wa ntegrated by Behn (1897) for a unformly charged membrane bathed by oluton all of whoe on have the ame valence [Eq. (2.38) were n fact obtaned from Behn' orgnal oluton by ettng X = 0]. The oluton gven n Harr (1972); t complexty, coupled wth uncertante regardng both the fxed charge concentraton n bo-