To begin we first need to consider HOW SUBSTANCES ARE TRANSPORTED IN FLUIDS

Transcription

1 MEMBRANES Backgrund All animal tiue uch a mucle i cmped f a grup f cell and the lutin bathing them, the intertitial fluid. Within the cell i the intracellular fluid and the cell i bunded by a thin membrane (abut 10 nm thick) which baically eparate the intertitial fluid frm the intracellular fluid. Virtually all the phyical ubtructure within a cell are al bunded by membrane, in fact mt vlume f bilgical interet are bunded by membrane. Bth the intertitial fluid and the intracellular fluid cnit largely f water with variu ubtance dilved in them. Membrane allw me ubtance t pa thrugh them but nt ther and fr thi rean are referred t a emipermeable membrane. A membrane i aid t be permeable t a ubtance that can pa thrugh it and the ubtance that can pa thrugh i aid t be permeant. If a pecific mlecule can nt pa thrugh the membrane then we ay the membrane i impermeable t that pecific mlecule and the mlecule i aid t be impermeant. A imple mdel f thi i a membrane with hle paing thrugh it f a given diameter which allw maller mlecule thrugh but nt larger ne. Bld in the bdy mve thrugh bld veel which are mainly pipe and behave like rubber pipe. The preure in the bdy fluid utide bld veel i lightly negative (with repect t atmpheric preure) wherea the preure in the bld veel i pitive. Hence the wall need t cpe with the preure difference. One can think f the heart a a balln and the exce preure P = 2 /r, where i the urface tenin f the heart wall and 2r the diameter f the heart. In the bld veel (pipe) the exce preure P = /r. Bld tranprt nutrient and xygen which flw int the capillarie, the uter wall f which are membrane. The nutrient and xygen can pa thrugh the capillary wall membrane int the urrunding tiue (ie cell) and wate prduct leave the cell and enter the bld. Thi exchange f mlecule thrugh membrane i crucial in phyilgy and ther example are in the peratin f the kidney and the cntrl f the intertitial fluid. In rder t undertand thi mvement f mlecule acr membrane we need t cnider the prce in me detail. T begin we firt need t cnider HOW SUBSTANCES ARE TRANSPORTED IN FLUIDS Bulk Flw and Slvent Drag Cnider water flwing in a lng tube, the water mlecule are in cntact with ne anther. If ne water mlecule mve, it n lnger tuche it neighbur, and they ene it mtin. Thu, the water mlecule mve cllectively. Thi i bulk flw. In a tube uch a a bld veel, the bulk flw i caued by a preure difference between the end f the tube. If gluce i dilved in the water it will be carried alng with the water. The gluce i a lute and the water a lvent. There will thu be a flw f lute particle pat a pint in the tube. Thi prce i called lvent drag; the lute particle are dragged alng by the lvent. Thi i the prce by which red and white bld cell and dilved chemical like gluce, xygen and carbn dixide are carried thrugh bld veel. Diffuin Water will nly flw in a tube when there i a preure difference between the end, if thi i nt the cae water will nt flw in the tube. If, fr example, the gluce cncentratin in the water i unifrm alng the tube, it will nt flw either. If, hwever, there i gluce at ne end f the tube but nt the ther, it will gradually fill the tube t a unifrm cncentratin, even thugh the water i nt flwing. The rean i that the water and

2 gluce mlecule are nt at ret, even thugh the water i till. The mlecule are mving with an average peed which i prprtinal their ablute temperature. Becaue a liquid i quite dene they dn't mve very far befre they cllide with anther mlecule and change directin. Their mtin i cmpletely randm, we refer t thi prce by the term randm walk. When the water i flwing, a drift f all the mlecule in ne directin i uperimped n thi randm mtin in all directin. Cnider an imaginary bundary at me pint alng the tube, dividing the tube int tw regin A and B. In a hrt time interval abut half f the gluce mlecule (repreented by circle in the diagram) immediately t the left f thi pint will travel briefly t the right and cr the bundary (the ther half will be travelling t the left, ince randm mtin t the left r t the right i equally prbable). At the ame time, abut half f the gluce mlecule immediately t the right f thi pint will travel left and cr the bundary. If the cncentratin f gluce mlecule n the left i the ame a n the right, the net reult i n change f gluce cncentratin ince a many mlecule mved frm left t right a mved frm right t left. Suppe, hwever, (a in the diagram) that the tw adjining regin have different gluce cncentratin. Any mlecule i equally likely t mve t the right r left. Half f the n the left (regin A) will mve right, and half f the n the right (regin B) will mve left. There will be a net mvement f gluce frm left t right, nt becaue gluce mlecule prefer t travel t the right, but becaue there were mre n the left t wander t the right than there were n the right t wander t the left. Thi prce i called diffuin. Diffuin require a change in cncentratin with ditance, r the exitence f a cncentratin gradient. The Tranprt f ubtance thrugh membrane The preence f a cell membrane greatly implifie the decriptin f the mvement f a ubtance thrugh the bdy becaue all dilved ubtance mve thrugh membrane much mre lwly than they d in either the intertitial fluid r the intracellular fluid.thu a negligibly mall cncentratin r preure gradient in either the intertitial fluid r the intracellular fluid uffice t bring the ubtance up t the membrane a rapidly a the ubtance mve thrugh the membrane. The mvement thrugh the membrane i the rate cntrlling mechanim. Nw cell membrane are ~10nm thick and they are emi-permeable. The implet mdel i t cnider a number f pre piercing the membrane, many bilgical membrane cntain pre. A membrane pierced by pre will be permeable t mlecule which are mall enugh t pa thrugh the pre and impermeable t the that are t large. Water flw thrugh the pre (bulk flw) if there i a preure difference acr the membrane. Uually all membrane are permeable t water, it i the permeability f ther ubtance that are f bilgical interet. Slute mlecule which are mall enugh t enter the pre will be carried by lvent drag if there i bulk flw. They will al diffue if there i a cncentratin gradient alng the pre. In mt cae the flw f a lute i a cmbinatin f bth effect. In the diagram the pen circle are water mlecule, the lid circle are mall lute mlecule and the large black blb i a prtein mlecule. (a) and (b)

3 are eentially the ame,the nly difference i that in (b) the pre fllw a mre trtuu path than in (a). In (a) & (b) water flw ( like bulk flw thrugh a pipe) if there i a preure difference between the tw ide f the membrane. If there i n preure difference acr the membrane the lute mlecule can diffue if there i a lute cncentratin difference between the tw ide f the membrane. If there i a preure difference bulk water flw will ccur and the lute diffuin wuld be uperimped (if there i a cncentratin difference a well) n the flw f the lute with the water. If there are n pre a in (c) mall mlecule can till dilve in the membrane and then diffue. Omi If a membrane i permeable t water and impermeable t lute, then water will flw int the mre cncentrated lutin t try t equalie the cncentratin thi i called mi. Omtic Preure A very intereting effect ccur if we have tw cmpartment f pure water eparated by a emipermeable membrane. Cnider, fr example, a U-tube with a membrane dividing it int tw regin a hwn belw. If the pure water can pa thrugh the membrane, there i n bulk flw when the preure n bth ide i the ame. If we nw intrduce n the right me ugar (lute) mlecule which cannt pa thrugh the membrane (remving me f the water mlecule t keep the preure initially the ame n either ide f the membrane) we dicver that, even thugh the preure i the ame n bth ide, water flw frm left t right. The intrductin f the lute mlecule reduce the number f water mlecule per unit vlume n the right. Even thugh the water mlecule are mving cllectively, and the phenmenn i nt diffuin, there are mre water mlecule n the left than n the right and flw ccur. Water will cntinue t pa thrugh the membrane in an attempt t equalie the number f water mlecule per unit vlume n each ide and a a cnequence etablihe a preure difference between the tw ide, ince the level n the right hand ide increae and that n the left decreae. Eventually thi preure difference i ufficient t tp further flw f water frm left t right. Thi preure difference required t tp the flw i called the OSMOTIC PRESSURE, f the lute. Anther way f aying thi i that we can make the flw f water zer if we increae the preure n the right. The amunt by which it mut be increaed i the mtic preure f the lute. Thermdynamic f Tw Cmpnent Sytem (Slvent and Slute) In a tw cmpnent ytem, uch a water (lvent) and a lute (gluce), in which a lute cncentratin gradient exit, there i an exchange f particle between the regin f different lute cncentratin. In thermdynamic we intrduce the cncept f a difference f chemical ptential, between the regin exchanging particle t accunt fr the mvement f particle. Analgu t bject mving dwn an incline frm a higher ptential energy t a lwer ptential energy. The chemical ptential,, i a functin f preure, P, temperature, T, and the mle fractin, X, f particle r i.e. (P,T,X ). r r r

4 It i related t the Gibb free energy, G, by: G = r ( r n r) = 1 N 1 2 N 2 where N r i the number f mle f 23 particle r, 1 i water and 2 lute. (1 mle cntain 6.022x10 mlecule - Avgadr Number, N ). Fr a pure cmpund (ie pure water) G= N, and thu in thi cae the chemical ptential i the Gibb free energy per mle f pure cmpund (i.e G/N ). The chemical ptential f the lvent (r=1) in a lutin (lvent lute) i given by 0 1(P,T,X 1) = 1 (P,T) RT ln(x 1) 0 where 1 (P,T) i the chemical ptential f the pure ubtance and R i the ga cntant (R=NAk where k i the Bltzmann cntant). A Fr ur ytem f water and lutin eparated by an impermeable membrane, the lutin i at a preure P and ince it cntain lute we mut write the chemical ptential a (T, P, X ), thu 0 1 (T, P, X 1) = 1 (P,T) RT ln(x 1) 1 1 T be in equilibrium, the chemical ptential f the lvent mut be the ame n bth ide f the membrane. (Preure P meaured in water at membrane, P meaured in lutin at membrane). Thu (T, P, X ) = (P,T) r (P,T) RT ln(x ) = (P,T) hence (P,T) - (P,T) = - RT ln(x ) On the left hand ide f the lat equatin the nly change i due t a change in preure. Recalling that the chemical ptential f a pure cmpund i equal t the Gibb free energy per mle we can ue dg 1 = V 1 dp - S 1 dt where dg 1 i the change in Gibb free energy per mle (i.e change in ), V 1 the mlar vlume f cmpnent 1 (which we will aume i independent f P, water incmpreible t a gd apprximatin) and S 1 i the entrpy per mle f cmpnent 1. Thu at cntant T we have dg 1 = V 1 dp = V 1. Hence fr equilibrium we mut have V 1 = - RT ln(x 1). It i uually mre cnvenient t give the cncentratin f the lutin in term f the lute. Thu ince X 1 X 2 = 1 (Sum f ml fractin), we can write = - (RT/ V 1 ) ln(1- X 2) If X i mall (i.e. a dilute lutin), ln(1- X ) = -X - (X ) /2 - ~ -X, thu = (RT/ V 1 ) X2 3 Let the cncentratin f lute, in mle per m, be C 2, then C 2 = N 2/V ttal (N 2 = n f mle f cmpnent 2)

5 Again, if the lutin i dilute X 2 = N 2 /(N 1 N 2) ~ N 2/N 1, thu = (RT/ V 1 ) N 2/N1 Nw fr a dilute lutin V ttal ~ V 1 N 1, thu we finally have fr the mtic preure 3 = RT C 2 r alternatively = NAk TC 2 = ktc 2 where c 2 i the cncentratin f mlecule per m. Omtic preure i the exce preure that wuld have t be applied t prevent the flw f water thrugh the membrane. It i the ttal partial preure f all the pecie that cannt pa thrugh the membrane. The effect f a change in ttal preure acr the membrane ariing frm ther caue i identical. Tranprt acr a membrane Permeable membrane Let P i = preure inide cell r bld veel and P = preure utide Then fr equilibrium P i = P. If P i > P utward flw If P < P inward flw i Semi-permeable membrane Can define the mtic preure inide and utide with repect t pure lvent i.e. = RT C ( C mle per m 3 ) Then if the hydrtatic preure are equal (i.e. P i = P, n preure difference), we mut have fr equilibrium = i If hwever i > we wuld get inward flw (i.e. imagine a U tube which i t hrt t etablih the preure difference, then water will cntinue t flw thrugh the membrane) A preure difference and a cncentratin difference Since i the preure neceary t prevent flw, the net driving preure P d = P - (minu ign ince ppite directin). Hence fr equilibrium P = P, r P - = P d di i i If P - > P i i inward flw P - < P utward flw i i Same a flw thrugh a pipe with a preure difference acr the end. The flw fr a given difference in driving preure i exactly the ame, whether the difference i caued by a difference in mtic preure r in ttal preure, r me cmbinatin.

6 Significance in bld veel A bld flw thrugh the capillarie, which are typically 1000 m lng by 7 m in diameter, xygen and nutrient leave the bld and g t the cell. Wate prduct leave the cell and enter the bld. Diffuin i the main prce that accmplihe thi tranfer. The bld cell mve in plama, which cnit f water, electrlyte, mall mlecule uch a gluce and carbn dixide, and large prtein mlecule. All but the large prtein mlecule can pa thrugh the pre in the capillary wall. Outide the capillarie i the intertitial fluid, which bathe the cell. The cncentratin f prtein mlecule in the intertitial fluid i much le than in the capillarie. Omtic preure i an imprtant factr in determining the preure in the intertitial fluid and therefre it vlume. Typically the mtic preure inide a capillary, i = 28 trr (3700 Pa) and utide the capillary (intertitial fluid), 0 = 5 trr (660 Pa). Nte thee preure are with repect t atmpheric preure which i taken t be zer. [ The actual atmpheric preure i 760 trr r 100,000 Pa (1 trr = 133Pa)]. The ttal preure in the intertitial fluid i abut -6 trr (-800 Pa). It i maintained belw atmpheric preure by the rigidity f the tiue. The driving preure f water and mall mlecule utide the capillary i therefre: P d0 = P 0-0 = -6-5 =-11 trr (1460 Pa) The ttal preure within the capillary drp frm the arterial t the venu end, cauing bld t flw alng the capillary. A typical value at the arterial end i 25 trr (3300 Pa); at the venu end it i 10 trr (1300 Pa). If the preure drp alng the capillary i linear, the ttal preure v. pitin are a pltted n the left. Subtracting frm thi the mtic preure due t the large mlecule which can t pa thrugh the capillary wall give the curve fr the driving preure, P di. (b) hw the ttal and driving preure in the intertitial fluid and (c) cmpare the driving preure inide and utide the capillary. The preure i larger inide in the firt half f the capillary and i larger utide in the ecnd half. The reult i an utward flw f plama thrugh the capillary wall in the firt half and an inward flw in the ecnd half. There i a very light exce utward flw. Thi exce return t the circulatin thrugh the lymphatic ytem, a ytem f veel and lymph nde which parallel the vein and enter the venu circulatin near the heart. The balance hwn in (c) can be diturbed in variu way and uually reult in water cming ut f bld int intertitial fluid leading t welling called edema.(eg. welling f the ankle and leg) 1). High bld preure alng the capillary. All preure inide i raied therefre utward flw ccur ver a greater ditance 2). A reductin in the mtic preure f the plama becaue f a lw prtein cncentratin (Malnutritin). 3). An increaed permeability f the capillary wall t large mlecule, which effectively reduce the mtic preure. Thi i the edema aciated with an injury which lead t lcalized welling. EDEMA ue a diuretic t try t remve exce water; if thi den t wrk, then ultra-filtratin.

7 Vlume tranprt thrugh a membrane In me cae, e.g. kidney dialyi, it i al imprtant t knw hw lng it wuld take t remve a ubtance by tranprt thrugh a membrane. Define vlume fluence rate J v a the vlume flw per unit area per ec. Cnider pure water and a lute fr which the membrane i ttally impermeable, then J v i prprtinal t the difference in the driving preure. Write J v = L p ( P - ) where the cntant f prprtinality L p i called the hydraulic permeability. It depend n number and ize f pre and n the vicity. If the membrane i partially permeant t the lute, then the lute de nt reduce the flw much. We write = reflectin cefficient J v = L p ( P ) = 1 : membrane impermeable = 0 : membrane permeable We can lk at thi a different way if we write P=P d, where P d i the driving preure. Thi can be brken dwn int the driving preure fr water P and ne fr the permeant lute a fllw: d w P dw (1- ) i the driving preure fr permeant mlecule ( = P d) P = P dw (1- ) (1- ) i the mtic preure f all lute mlecule i the mtic preure f impermeant mlecule (1- ) i the fractin f lute which enter the membrane The reflectin cefficient,, mean that part f the lute mve freely with the water and part i reflected Subtituting fr = RT C. ( C i the difference in cncentratin f lute mle per m 3 ) Then J v = L p ( P RT C ) r J v = L p ( P kt c) ( c i the cncentratin difference in mlecule/m 3 ) Remember J v i the vlume flw f lvent (e.g. water). Vlume flw i ued in ultra filtratin which i ued t reduce the water frm bdy when diuretic d nt wrk. 2 2 Bld i paed thrugh filter which cntain a membrane typically f area 0.2 m with L p = 1 ml/ min/ m / trr. Cnnectin B in diagram i ued. Mlecule with mlecular weight 50,000 can be tranprted thrugh the pre f thi membrane. The filtratin rate i et by adjuting the external preure, via ultrafiltrate clamp, t cntrl preure drp acr membrane. Taking ut bld and remving water caue intertitial fluid change (decreae). Can be dangeru t remve t quickly. Old day bld letting! Thrwing away bld! Slute al might be able t pa thrugh the membrane, perhap if mlecule are mall, thu we need t lk at lute tranprt a well.

8 Slute tranprt Slute can either diffue thrugh the membrane r be dragged thrugh by the lvent flw (lvent drag). If lute cncentratin i the ame n bth ide f the membrane, there i n diffuin. Then the fluence rate i caued lely by lvent drag and i given by the prduct (lute cncentratin) x( lvent fluence) J = C Jv J = number f mle f lute flwing thrugh unit area f membrane per ec and J vi the vlume flw f lvent (due t ttal preure difference). Fr a partially permeant membrane we have t multiply thi by (1- ) (fractin f lute entering membrane) J = (1 - ) C J v If J v = 0, there will be n lvent drag (i.e. n ttal preure difference acr membrane). Hwever, if there i a cncentratin gradient then get diffuin, lute flux being prprtinal t the lute cncentratin difference acr the membrane. J C The cntant f prprtinality depend n the phyical prpertie f the membrane and al the diffuin cntant. The phyical prpertie f the membrane are encmpaed in a term, the membrane permeability r lute permeability. If the membrane ha pre, wuld depend n the pre ize, membrane thickne and number f pre per unit area. The diffuin cntant, D, i prprtinal t RT, thu J = RT C If bth prcee are preent then J = (1 - ) <C > J v RT C SOLVENT DRAG DIFFUSION <C > i average n the tw ide f membrane and J v = L p ( P RT C)

9 Artificial kidney Mdel Membrane i pru t the mall mlecule which it i neceary t remve. The dialyi fluid i mainly water but it i imprtant nt t remve me mall mlecule, Na, K and gluce at bdy cncentratin are added t the dialyi fluid. Define bdy fluid vlume, V (cntant). At time, t, it cntain N mle f lute and hence there i a cncentratin C f lute in the bdy (C = N /V if the lute i well mixed in bdy fluid). If we have a cled ytem the cncentratin C* f lute in the dialyi fluid wuld increae while C in the bdy decreae and thu after a time there wuld be n flw, ince J i prprtinal t (C-C*). Uually make V* V that C* remain effectively unchanged. Typically V ~ 40 litre, V*~ 100 litre. Mre recently a flw ytem i ued fr the dialyi fluid, thi i cntinuuly dicarded (hygiene f ytem very imprtant germ breed in the fluid). Fr a flw ytem C* remain cntant, V* =, P = 0. Then n lvent drag and lute i exchanged by diffuin J = RT C = RT [C C*] But J = number f mle f lute, N, flwing thrugh unit area/ec, thu Nw N = CV, therefre J = -(1/A) ( dn /dt) where A i the area f the membrane. (Minu ince dn /dt i negative). dn /dt = V dc/dt Thu J = -V/A dc/dt = RT [C C*] (C and C* are the cncentratin f lute in the bdy and dialyi fluid repectively). Therefre dc/dt = -A RT/V [C C*] Hence a C* i cntant thi equatin can eaily be lved t determine C a functin f t. t ~ hur fr dialyi.

10 Charged membrane and the mvement f in thrugh membrane There i uually an electrical ptential difference acr the wall f a capillary and acr a cell membrane. Thi i caued by layer f charge n each ide f the membrane which create an electric field in the membrane. If the lute cnit f in, the diffuin f in dwn their cncentratin gradient can generate a cunteracting vltage gradient. Fr example when a K in travere a pre in a membrane and leave a cell, - thi catin leave behind it a nn-permeating anin A. Thi eparatin f charge mean that there i an electric field in the membrane and wrk mut be dne by a pitive charge t carry it ut f the cell. (Ptential difference between 2 pint i the wrk dne againt electric frce in carrying unit pitive charge frm ne pint t the ther). The electric field in the membrane i unifrm becaue the eparated charge mut ditribute themelve unifrmly ver the it urface in rder t make the electric field zer everywhere in the inide and utide cnducting fluid. The reulting ptential nt nly influence the mvement f in but they are imprtant in undertanding the tranmiin f ignal in nerve. Define electr-chemical ptential, EC, a fllw - Imprtant in are Na, K and Cl EC = i (Z i e N A) VE where i i the chemical ptential All the exce charge i n the membrane, the intracellular fluid ( r axplam in nerve) and the intertitial fluid are neutral. If a ptential difference exit acr the membrane, then the wrk dne t mve an in acr membrane = Z i e V E, where (Z i e) i the charge n the in and V E the ptential difference. T mve ne mle f uch in, the wrk dne = N Z e V (N i Avagadr Number). A i E A Nw fr a flw f dn i mle f in dg = EC dn i VdP - SdT, which at cntant T and P becme dg = [( i (Z i e N A) V E] dni Fr a flw frm utide inide f dn i Fr equilibrium dg/dn i = 0, thu dg = {[ ] i in -[ ] i ut}dn i Z i e N A[V E in - V E ut] dni [ i] in -[ ] i ut = - Z i e N A[V E in - V E ut] But fr ideal lutin i = RT ln (X i). Taking V E ut = 0, we btain RT ln (X i in/ X i ut) = - Z i e N A V E in Rearranging V E in = -(RT)/ ( Z i e N A) ln (X i in/ X i ut) Nernt Equatin

11 Example Calculate the ptential difference acr a membrane due t the mvement f K in thrugh it, at 37C. X i in = mle/m and X i ut = 4 mle/m R= J/ml/K, N = x 10 and e = x 10 Culmb. A Since K i mnvalent Z i = 1, and T= 310K, hence RT/(Z i e N A) = If we cnvert t Lg 10 then we have (Lg = x Lg ) fr the Nernt Equatin e 10 V E in - V E ut = Lg 10 (4/155) = -95 mv A rule f thumb i that the ptential difference in mv i 60 Lg 10 (X i ut /X i in) -3 Cncentratin acr the nerve membrane (ml m ) Outide Inide Na 145 Na 15 K 5 K Other Cl 125 Cl Other Other 156 The charge eem t balance, but in practice there i a mall exce pitive charge -5 utide and a mall exce negative charge inide [exce 10 x ttal cncentratin]. Hwever nte mtly Na utide K inide ince V Ein in a nerve i pitive, the Nernt equatin implie that at equilibrium X X. True fr K, but nt Na. - The cncentratin f Na and Cl are much higher in the intertitial fluid than in the - intracellular fluid. The ituatin i revered fr K and the ther in, ay A. A urpriing fact i that a typical membrane with pre abut 0.3nm in diameter i rughly 100 time mre permeable t K than Na in. Nw an ilated Na in i maller than an ilated K in, hwever in water thee in becme hydrated and the number f water mlecule that can be held in the hydratin hell depend n the 2 charge denity. Nw the charge denity varie a q/(4 r ). Thu Na in have a higher charge denity becaue they are maller in and thu there are mre water mlecule in the hydratin hell f Na than there are in the hydratin hell f K. The hydrated diameter f K i 0.22 nm, wherea that fr Na i 0.34 nm, ince the pre are abut 0.3 nm in diameter hydrated K in can pa thrugh the pre mre eaily than hydrated Na in. i in i ut There are ther mechanim whereby in can travere membrane, in which the hydratin hell i remved and thu Na i mre permeant than K. Hwever thee prcee are mre cmplex and are beynd the cpe f thi cure.