Physics 178/278 - David Kleinfeld - Fall checked Winter 2014

Transcription

1 Physics 178/278 - David Klinfld - Fall chckd Wintr Elctrodiffusion W prviously discussd how th motion of frly dissolvd ions and macromolculs is govrnd by diffusion, th random motion of molculs that rsults from fluctuating forcs. This pictur largly holds in th prsnc of wak forcs that mrly add a slight bias to th motion, such as th cas of lctric filds across mmbran pors. In particular, in th prsnc of an lctric fild th motion is limitd by th collisions so that th vlocity, as opposd to acclration, is proportional to th forc. W hav v D ( r, t) = µ E( r, t) (1.1) = µ V ( r, t) whr v D ( r, t) is known as th drift vlocity and µ is th mobility. Th jok is that lif is diffusion with a small drift vlocity. How big ar ths trms? Without proof, w not that v D 10 1 cm = 10 2 µm in s ms mmbrans and v D 10 5 cm 4 µm = 10 in cytoplasm. s ms W can now calculat th flux du to th lctric fild as J D ( r, t) = C( r, t) v D ( r, t) (1.2) = µc( r, t) E( r, t) = µc( r, t) V ( r, t) Thus th total flux, thrmal as wll as forc drivn, is At quilibrium, J( r, t) = 0 and thus or J( r, t) = D C( r, t) µc( r, t) V ( r, t) (1.3) C( r) = C( r ) µ D (V ( r) V ( r )) V = V ( r) V ( r ) = D ( ) C( r) µ ln C( r ) (1.4) (1.5) but w prviously showd that this quilibrium potntial is just givn by th Nrnst formula, i.., V = k BT ln ( ) C( r) C( r ) (1.6) 1

2 Thus µ = D (1.7) W can stimat th siz of v D for mmbrans and compar it with th RMS thrmal vlocity, v th, sinc th concpt of th drift vlocity holds only if v D << v th. For a mmbran, th lctric fild is of ordr E = V k BT 1 so that v L L D = µ E D 1 D k B. On th othr hand, w discussd arlir that th scal of T L L th thrmal vlocity is v th λ D. Thus τ λ v D λ L v th 0.1 v th (1.8) so that th thrmal vlocity is rlativly small for distancs largr than th collision lngth, a rathr intuitiv rsult. On th othr hand, for larg lctric filds may may anticipat a dviation from a simpl linar rlationship btwn E and v D, a topic w will rturn to. Sinc w prviously argud that th thrmal vlocity can b stimatd from quipartition, i.., v th kb T m, w hav v D c kb T mc 2 λ L (1.9) from which w can stimat th absolut siz of th drift vlocity for mmbrans to b v th 10cm/s. W can now put all of th formalism togthr to gt a modifid flux for th diffusion quation, i., ( J( r, t) = D C( r, t) + ) C( r, t) V ( r, t) (1.10) At this point, lt s cool it with th gnral form for 3-dimnsions and focus on th cas of currnt through a por of cross sctional ara A that spans a mmbran of thicknss L. W furthr assum that th lctric fild is uniform (probably not tru, but it allows us to mak som usful progrss), so that V (x) = V (0) x, w hav L an quation for th lctrical currnt, I, I = J(x)A = DA ( dc(x) dx + ) C(x) V L (1.11) whr A is th ara of th mmbran. This quation is in th form of dc(x) dx + const C(x) = const, which w can solv dirctly to obtain I = D V C(0) AC(L) (1.12) L 1 V This is known as th Nrnst-Plank rlation. Th intrsting, and ssntial fatur, is that th I V curv is nonlinar for voltag changs on th ordr of 25mV away from th rvrsal potntial. V 2

3 FIGURE - Nrnst-Plank.ps If w includ th possibility of a valnc, z = ±1, ±2, ±3, tc, th gnral form of th Nrnst-Plank rlation bcoms I(V ) = z 2 D L V zv k C(0) B T AC(L) 1 zv (1.13) and is particularly strong for divalnts, such as Ca +2. Th ssntial physics is that it is asir for currnts to flow from high concntrations to low concntrations, so that th conductanc (slop of th I V ) is largr whn ions mov from high to low, rathr than from low to high, concntration. This proprty in known as rctification, and is a normal proprty of any cll mmbran. Th limiting currnts ar Ohmic, with z 2 D L I(V ) z 2 D L ( ) V AC(L) if V >> ( ) V AC(0) if V << Ths asymptotic rlations ar oftn good approximations, and for th bttr or wors most physiologists assum that th I-V rlation is Ohmic. But do this at your pril. Hagiwara mad this assumption about th Inward Rctifir for K + and cam to som rronous conclusions about his data! 1.1 Th rsting potntial of a nuron In stady-stat, th total currnt across th cll mmbran is zro. Whn only a singl ionic spcis can pass across th mmbran, i.., whn th currnt is carrid by only a singl ionic spcis, th stady-stat potntial is also th quilibrium potntial and w rcovr th Nrnst potntial. In gnral, howvr, multipl ions contribut to th currnt flow across th mmbran. W focus on a cll with just 3 of ths, Na +, K + and Cl. Thr is a Nrnst-Plank currnt associatd with ach ion, i.., ( ) D I Na +(V ) = A L Na + ( ) D I K +(V ) = A L K + ( ) D I Cl (V ) = A L Cl In stady stat, w must hav V V V [Na + ] in [Na + ] out V 1 V [K + ] in [K + ] out V 1 V [Cl ] in [Cl + ] out V 1 V (1.14) (1.15) (1.16) I Na +(V ) + I K +(V ) + I Cl (V ) = 0 (1.17) 3

4 This condition is satisfid for only a singl voltag. Th algbra is asy to do it you rcall that on should solv for V rathr than dirctly for V (it is also asy if on ignors divalnts, lik Ca +2!). Th stady stat potntial is givn by (Goldman-Hodgkin-Katz quation) V SS = k BT ( ) D ln L K [K+ ] out + ( ) D + L Na [Na+ ] out + ( ) D + L Cl [Cl ] in ( ) D L K [K+ ] in + ( ) D + L Na [Na+ ] in + ( (1.18) D + L )Cl [Cl ] out Th fraction D is oftn calld th prmability, dnotd P. Actually, th prmability is dfind with th addition of a mystry fudg factor in front! W will s L blow that th prmability is dpndnt on concntration such that th channls ar significantly saturatd at physiological ion concntrations. As w mntiond on th first day, V SS typically has a valu of about -50 mv. Ion substitution xprimnts that maintain a constant total ion concntration hav confirmd this rlation. FIGURE - Goldman.doc 1.2 Rlations among Diffusiv procsss Sinc in th linar limit (i.., zv k B ± ) th I V rlation must rduc to Ohm s T law, i.., I = g πa2 V, w hav th rlation L g = (z)2 DC(L) (1.19) In gnral, w rcall that th diffusion constant (D), th mobility (µ), th viscosity (η), and th lctrical conductanc (g) ar all rlatd. W rcapitulat this blow, D = g = µ k BT (z) 2 C ions z = 1 η 6πa (1.20) whr th viscosity rlation for a particl of radius a is not rally usful for ions pr s. Lastly, as a practical issu, to convrt from physics units to chmical units, on rplacs k B by R = N A k B and by F = N A and masurs concntration in molar (mols/litr) rathr than ions pr cm Channl Saturation On xpcts that th currnt will not incras without bound as th ionic concntration incrass. This may occur as th kintics of channls limit th currnt. A simpl xampl of kintic limitations is to considr a channl with a singl ion binding sit. Th currnt will follow saturation charactristics for nzym binding 4

5 (Michalis-Mntn kintics), so that,.g., th abov limiting cas for th currnt (V >> k BT ) bcoms I(V ) z 2 D ( ) V πa 2 C(L) (1.21) L 1 + C(L) K q FIGURE - Channl-conductancs.ps Byond this ffct, stric hindranc and lctrostatic rpulsion will furthr limit th currnts. Hill has a nic discussion of this. Th bottom lin is that th saturation ffcts ar substantial nar physiological concntrations. Thus th prmability of channls will b implicitly concntration dpndnt, and formulas lik th Goldman-Hodgkin-Katz quation for th stady-stat mmbran potntial must b applid with prmabilitis chosn for, or corrctd for, th concntrations usd in th particular xprimnt. On furthr xpcts that th currnt will not incras without bound as th voltag incrass. This may occur as a dpltion rgion dvlops at on nd of th channl, and a spac charg rgion dvlops at th othr nd. In ffct, th voltag causs a concntration-basd limitation. An asy way to stimat this ffct is to considr th diffusion quation for a point sourc (por opning). At stady stat, C( r,t) t = 0 so that D 2 C( r) = I( )δ( r r ) (1.22) whr I( ) = I(V ). This is just Poisson s quation for a point charg, so that ( D 2 C( r) = I( ) 1 ( )) 1 4π 2 r r (1.23) Thus, intgrating ovr a hmisphr (so that th sourc doubls compard to th full sphrical solution) w hav C( r) = I( ) 1 + Constant (1.24) 2πD r This modifis th concntrations on opposit sids of th por to rad C(0) = C o (0) + I( ) 2πDa (1.25) C(L) = C o (L) I( ) (1.26) 2πDa Th currnt will saturat at a maximal valu whn th concntrations on opposit sids of th por ar qual, i.., C(0) = C(L). Thn th lctrical currnt is I( ) = πda [C o (L) C o (0)] (1.27) 5

6 An ordr of magnitud stimat of this spac charg ffct for a channl with a 10Å por diamtr is 30x10 12 Amprs. This is clos to th masurd saturation lvls for K + flow through th Ca 2+ -activatd K + channl. FIGURE - Spac-charg-or-not.ps Lastly, w can substitut th corrctd form for th concntration back into th Nrnst-Plant rlation. Thus, for xampl, th abov limiting cas for th currnt through th por bcoms (V >> k BT ) bcoms ( ) V πa 2 C(L) I(V ) z2 D L 1 + a 2L ( V ) (1.28) 6

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8 Squid Axon Extrnal Fluid [K + ] o in mm [Na + ] o in mm [Cl - ] o in mm Masurd RP (mv) Calculatd RP (mv) Diffrnc (mv) ASW A B C Sa Watr D E F G H I Adaptd from Hodgkin and Katz (1949) Chapt_4_goldman.doc

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