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Membranes, material transfer and enzymes
1. INTRODUCTfON Two-compartment, multleomponent, material tranefer aysteme - Comprised of homogeneous pbaeea llepwated by mllllpore flItera - bwe beer. ehldkd and their bebavlor mla1yzcd tbeoretlc~lly (Nlms, 1961;Ni~~~~and'Phurbe~. 1961; Salmlnen, 1963; Ekman, Raatas and Salmtnea. 1863). Thene model eystema are of interest becaaee they mlmlc in come degree the bdbavior of UvlLne material transfer eyateme. They exhlblt Ion ielectlvlly, are capabie of beitg brdught lnta statloeary at&es - the model equ*valent of basal c~tatea ln Mobglcal systems - ard during their evoIutlon toward a atatlonarv etxte. tone ma” mow agalnet their concentr;tlo” g&ente. In-deed %rl!nlnen (1863) wae able to demonstrate etntfonm’y states in WbIchthe concentratton ratlo of potassium IonP between tbe two pbaeea v/as g%+ter than one, while at the came time the CowentratIon rauo of eodIum ions was 1rse than ON, a altaatlon character1rrt1c lot’ the concentr;ttlan ratloa of theee Ions between the In&de aad cwtalde Of many 1lVIngcella.
me energy t&OUrCerequired to bring about
and to malntaln the nonequilibrium stationary states In these model systems was largely external. Biological systems achieve the requlslte rate of energy production to bring about basal states from eneymattca11y oata1yze.d chem1ca1 reactions, called collectively the metabolic pathways, whfch occur somewhere in the system. A better mcdel of b’olotical material transfer BVSterns might tbereZorewbea linear three-compktment system. ia ahloh noneauillbrIum ion distrltutlonpatterne Of (he Lltat1omwystates ar* the remWant of an enzyme catalyzed ohemlcal renetloll takhlg place Ill the central eompwtment, The chemical reaction weeted for the model systems was the ewymatlc hydrolysis of urea (NH&CO + Iis “%!.% 2Nff3 + COP, (1) since It seemed probirble that fluxea of sufftclent magnitude of the small reactant molecules throuuh celloohane membranes could be estabIlehG to pr&luce ion dl8trlbutlon patterns between the termlnal phahases,elgniflcantly different from equlllbrlum values, and thus verify one of the consequencee of a physical-chemical tkeory Of matei+ tmnefer. 2. MATERIAL9 AND METHODS Plaetic contrlnera were faehloned ‘n erwb a
L.F.NIMS
354
way that an apparatus mntaining compartmentsof about SO, 90, 30 cm3 respectively, separated by cellophane SO0dlalyzlng discs, couM be readily assembled. The nominal area of tbe membranes was some h cm~. MagnetIcally driven atirrir+z llare were pIa+ tn each corn rtment, and tli SolUtlonU contained therein I $,‘e etirred vigorously at a ccmstant rate during the couwe of ‘&I exwrlment. Tbta we done to tnfwe hornwaneity of the phases and to keep the effective thick1~8s of the dtffuslon barrlere - the membrane8 and
theirassociated
boundary
layers
- as con-
stant and as reproducible as psqble tn a particular experimental series. A echematic diagram of the apparatus as developed Is glveo in fia. 1. -Flat dialyztng membranes, designated as cellophane 300 by the Fischer Scienttfic Company, were selected for these ezmerlments. Thev were Impermeable to urease, q&e uniform in p&pertil?s, and sufficiently free of adventitious ptn hole8 that comparable experiments could be carrled out on different samples of tbe stock mate-W. Recent atudtes (Riley et al., 1984) make it seem ’ -‘-I that membranes of tkts type are compxxed cd a ‘-” dense surface layer supported by a spongy porn”.. . ?” Tbe major partton of the realstance to materia, u..r_- ” “n,bably ltea in the dense layer, about 1%of the tcwa, thickness of the membvane. The permeants seem to be molecularly dispersed In the denice layer, and material transfer tlwou~ this Layer Is probably accomplished by a dlffuslon process. The rest of the membrane appears tc ta quite porous and 16 ~r+x”mably readlty penetwble by water, small Leutrai mOleCuii9 and ealta. Mafeic Acic.
As the material transfer Bystem approaches a stationary state. a wnslderable hvdrostattc pressure dliferenc; develops between ti,e central compartment and phase (1. The maximum pressure dlfferences exceeded the breaktog strength of unsummrted cell~~bane membranes of some 8 cm2 ark; - about 7 ibpeersquare Inch. For &is reason, membrane one was suppvrt& either by a linen cloth disc or a stainless steel grid. Thi quantitative results obtained with the hvo supports were not the same. These differences are undbubtedly due to the geometrical parameters Of the complex Lurriers so formed. Tbe #UppOrt material screena one surface of the cellophane membrane from full pgrticipatton in the transfer process and thus gives an effective area less than the nominal area. The thickness of the barrter is the sum Of the thicknesseJ Of the two mlstirred tcwndarv kwers. tile ee,lwbnna membrane, and the ;oupdort &&&al. %&I&&& the effective area nor tblCkne88 Of the two composlte barriers ,c1knmvn or readily determlnable. Phase (t of UK apppratus v/a* perfu6ed at rates from 0.5 to 2 em3 ner minute with anuews
KC!. H-0
tine.
eantly with the system-was de&d reacbed a Practleal stat!OMrY &ate. The
,
tohive awldual
partment bliartered ocai.tonallY wttb tbe attainment Of P true stat*onvy .tPte. In mo,t exwriments. tke rate of loss of ena~mrttc acttvtt~ so slowthat L god iQpro;imation to thi atauonaryaate MI felt to have bell rcbteved. When th aytim was daemed to be In a stattonary atate. umpler of tke fluldr In the termtnal compartmenta were taken for chemical anatyds.
wae
Experiments were carried out at room temperahires, 22 l 2oc. The Lithium and wtasslum concentrations of the samples were determined by flame pbotometry. The chloride ion concentrations were determined in an Aminco-Cotlove chloride ion titration apparatus. The pfi was determined with a glass electrode and &‘&roan pH meter. The ammonia concentration of pbaee o was determined by aeratfon of a eamyl~ made alkaline with potassium carbonate. The ammo@. hberated was trapped in a solution of standard acid, and the excese acid was then titrated with standard alkali. The ammonia productIon rate of the system was calculated from the determined ammonia concentrations of phase d and the measured rate of perfusion. Two different commercial sources Of urea59 were avatlable. and the urease was not further purified. They bad nominal enzymatic acUvIUes Of 2% I) and 1000 units per g respectively. Kn the fndivldwal experiments. weighed aluounte of the impure ureaae preparations were added to the 60 cm2 volume fluid I” the central compartment. Beczwee complexity of the system, *t was difficult to obtain as high a degree of reproducibllity from experiment to ezqwfment as might be destrable. A sufficient number of experiments we, e done that the general behavior of such Bystems could be descrtkd. Focus to theee cxperlmente wde directed toward the stationary state ton dIetrlbotion patterns achieved in the terminal phases of the system and their relation to tile ,ate Of tbe cherntcal react*on occurring in the central compartment. Phase S ‘was in equillbrlum wltb the cent& compartment, and of oecessitv at a hkber wessure than abase a. The only 60~sin lik In the &tlonary state were the flows of the the chemfcal reaction through membrane oae. ff both phase (I and phase I3 are mnlntained at atmospheric preeewe, very amaLi flowa of all through membrane two become pobsfble, and the system czn than be ntd to be In the srwetton mode. Quant1tPuve rrtud1ea Of t11e preewe In tblr twstem and the comwsltlon Vt3riouS fluid &w?es when the sy&m, LB In the secretion mode to the rate of chemical resetlon will be reported at 1 b&r tinw.
of
of the
syitem
compunenta of
perme;mts
therelattonOf
3. EXPERIMENTAL
of the
RESULTS
llw qwntIfatIve data obtained on the campoMtons of phases II and S wfth respect to Ifthfum, eodlum. and potassfum tons In ewerat of the ifuasl-a!.utfowy &ate experfments are 5umma-
rized In table 1. Stnce in-these exaerimente the initial concentrations of the dlstrtkant ions in elch phase were Identical, it Is evident that during the approach to the stationary state, these ions must have at came time moved against their respecttve concentration gradients. Furthermore. the larzer the rate uf ammonla croductlon, ‘or the l&per the pH difference bet&en the termloaf ptxlses, the greater was the departure of the ion concentration ratios from unity. The system dlaplays a small degree of ion selectivity. In each experiment the pot8eeium ton C*nCClltrauOn is larger than the llthiuzn ion rat,% The chlorfde tons are distrlbuted fn the oppos!te seuee. Th&on selecl,vity Is much emaller than that seen In the maiority llvfr?g cell& Howevar .t is net due to cf&c~, and there is no reaeon to belleve that the phystCal-chemw.4 principles proposed for the bahavlor of the model system would not be equally appltcable to biologIcal 8ydem.e fn generni.
ratio mntewhat
if
4. PHYSICAL-CHEMICAL
THEORY
4.1. ThermodyMmics Of sttmiy The fmmal descrtptlon of
state processes
material transfer systems has been grealy facilitated by the development of a thermodynamics of steady atate processes (De Groot arzdMazur, 1962; Katcbalsky and Curran, 1965). Thermcdynamlc theory Is important since it gives precise relalions among the measurable variables or properties of a system, but it offera “0 clues ae to mechanism, and mechanical theory must be Invoked I” order to Luxlersta”d tile processes or tra”S!fCfr. When a psrtlcular barrier is permeable to j different material subs!a”ces, lhe basic relation descriptive 01 material lranefer through a” element of volume of a membrane g&en by thermodynamics IS I” which @fis lhe cbsmical po:e”tiPJ 61 B “butt%1 molecule, or the eleetrochemlcal potent@1 of an lo”: J; le the rate of flow o: j LWPunit area of the me&ane. and is called the fim; .fj”l. known as a phenomenologlcal cross coefficient. The quantity +jj obeys Onsager’s (1931b. b) reciprocal relatlOn* epuirl t0 Yji* “enron 118501.winted out that ea. 12) nxav be re3.arded as a generallzatlon af Fl&‘.‘l&v w&h takes account of the InteractIona between the rarlous chemical soecies flcwin~ t!xoueh a” element of volume oia barrier. B&.tly &eakl”g, Rck’s law 1s only appllcabls lo naultlcompoonent material transfer ny~tems wher. a11 permeable aubatancea traverse the barrier with the same speed, or each permeant cpll move through the tarrier by B seuarate aad 1”deDe”de”t“athway. It becomes in&&ate when two or mar; permesmts are flowing through the same element of volume with different velocltles. The flows of neceaslty must mutually Interfere by some pbyalca? proces8 or another. The flows are then said to be coupled, and the mag”lhld.3 Of the crosll coefflclentm la a meaare of the devxee of co”“ll~~1. The first step in the descrl$on of a s&e& is the decision ae to the namber of dlstl”ct wrmeants preeent. We will presums that seve&en are present in the model system aml will symbollze them as H 0. li30+, OH-. urea, Ii Ma (maleic acid). E&a-, Maul, CO3-Aq, &Oj, CO;, NH3.Aq, NH OH, NH& K+, M+, nd t 1”a steady state aeventern Cl‘. Thtemeans tha fhwes appear on the right hand elde of eq. (2). 81 addition &n)(n - 1) cross eaefflclpnts, wlw, (PI- 1) permevlts are present in L*l 1acItherma1 system, are required 1” the descrlptlon (Onsp+ij
IS
ger,
1945).
stncewe
kmw
very little
abxdmag-
nltudes of the cross coefficients *n the cellophane medium we will not attempt the complete descrlptlon of the model system, but will be content with the development of practical relations useful in the tmderstanding of the kbavlor of erperlmental systems. The first step in this development is the introduction of mechanical theory in order to give phy&al significance b the phenomenologlcal erws coefficients which appear I” eq. (2). Material transfer is a problem of motion. Whenever material substances are in motion. a descrlptlon of the motion is possible In terms of forces (Page, 1952). TO obtain an equalan Of motion in conde”sed phases, we first vleuualize an element of volume. The molecules In this element of volume are In thermal motion, lmt the average Ye&r velocity Of lbe molecules IS zero. If a” external force acts on the particles in the element Of vobmle, tire pwt1c1es undergo an i\cceleratio” in the directlon of the line of actfen of the force. The particles aqulre an average veloelty wblch is DOWa funcllan of the the. Wbllc the farce Is dletrllmted over the vcdume. it mav be assumeti to sotOR the center of m&.sofaif particles of a &en kind c~~“lalnedin the element of vo1ume. Tile weraE* ve1oeitv relltlve lo some : often iled lhe drlff YCframe of referew lOCIty - la alsO the velocuy 0, the center Of iwa!3e. The external forces of Interest to us here are conservative, that IS, the fields 0r force are characterized by a potential whose magnlbade Is a functlo” of the spcc coordInatea of Uw center of mpss, ad may also be * fu”ctIon of the time. The mngnltude of the force per unit volume IS related to the potential of the field PI +/v - -q tvd Cf (3) in which Fff Is the can~ervatlve force, Vls the magnitude of a” element of volume of the barrier, Cf IS the local coneentrat1on of f in fhls element of volume, a”d pad /q Is the ~adlent of the chemical potential 01 f II 1 Is D “m&al molecule, or the gradlent of the electrocbemlcal potentla1 of I if i Is an 10”. If the cenler of “~8s of f moves relatIvelv with rewect t” the center of mass of a”y o&e; chemtc~l apeeler which IS present In the bprrier, an lnter”al resistance to” mot,on 14 pnerated. The inter”a1 raisla”ce to motlon 1s presumably due to the composite action of Vpn der Waale, Coulomb and other Intermolecular
and the local Fij , Y = Ci Cj vfj “jj
to
zero.
Unforhinately o”r tierstandlng of condensed phases is so meager that rarely 5s itpossible to calculate the magnitudes of Local resistance coefficients from first principles. They are best regarded as emp(rlcal properties of a barrier. They can be related lo molecular frictional coeffleien?” as is often done (Onsager and Fuoas, 1932; Sple@er, 1958; &den. and Kabzhalsky, 1961; Mm@, 1961). Thus, +ij is a function of the elze amI shape of motecu!es as well as of the properties of the medtum in which the molecules are suep+r.ded. Enpertence Iodlcates that vij dwe Mt dewad UP” ofj when the relative YOIOCIties of motion are small. This Is the usual situation In biological barriers. However, vi’ does dewxl somewhat upon cf as well as upon I+Ie IOCal concentrations of all other substances present In the element of volume, I” the description that fo11ows. the system Is take” as Ilnear, so that ail varl”DLes become f”“CW.ns of the spafe coordinate x. Eq. (4) can be rnodlfled If the relative veloolty, “f’, is reof placed by the relatwe ve10c1ties “jnr a nd$,,, i andj with respect to the membrane. The vector relatton Uij = V*m - Ujm (5) allows
-api/ax+Jt
- Cf~jCf~~~* .
(6)
me
Jy
Cfcrt /c. -ZP.J jti I I 2 j+i
Eq. (10) can be written by Intwxluclng a straight
= Ci vi,,,
(71
In eq. (6) for their equiIn wh(ch frietlonal SS functiow of the local rcsts’anee nxafftetents.
the
in a more coefllcient
c dT../C. J D 1.
*ii = -j,i
Ofeq. (4)
We “ow introduce the concept of a LLux, dellned as the rate of now (moles per SW) per un*t a~~i( of membrane. The flux, Jf, Is given by the vector r4Lption
8IIbstltUtlen of the fluxes Valents gives B relation forces are represented concen’~atlo”s, the Ioeal
.
(8)
exactly.
(5)
the rewrltlng
= cj’ijCirJi,
is
- CirijJ,p
Steady statr flow is produced in the following way. The eo”>.ervative forces act to accelerate the particles, anJ tend to produce ever-lncreasi”g drlir veioews 01 the va~Lou8 species present in an elem#!nt of volume. As the dHit velocities increase, the fT1ctiolliil forces also increase in m?gnitude. Eventually the vector sum of a,, forces acting upon a given chemical species becomes zero, the accelerations vanish, and the drift velocities are thereafter conslant. The vector relation among the forces can be written Ffi+~j,Fij=O. (9) ji I When eq. (9) is obeyed the situation is called a stalionary state, for then all descriptive variables of a system are independent of the Lime. Such a srate in the model system is equivalent to basal siates Ln living systems. In Lhe majority of systems of interest to the biologirt. the descriptive varlnbles of the transfer systems are f”netlo”s of the time. However, the accelePationS in tne system are usually so small that cq. (9) can be used without introducing appredable errors in interpretation. Thus even when the deswlptlve variable.? are functions of the time, the material transfer systems are often called steady state systems by the bioioglst. The forces in eq. (9) CB” be replaced by their eq(livBlentB, eee eqs. (3) and (8). The resutt obtalned is
The definllive Ffj/Y
The equation
= CjrijJi”
(4)
in which Cf and Cj z!re the local concentrations of the chemical spewas i and j, a”d “jj is the relative velocity of motion of the centers of mSss of i with respect 1o j. The phenomenological coeffielentrji can be waa!ti 2 !zcz: ;is:siawe coefficienf,~and represents the interference in the motion of i caused by the presence of j in lhe same element Of volume. Eq. (4) is written in this way to take aecou~d of the fact that Fij @es to zero whenever C,. Cj, or oij goes
;; E;;&.!Zq.
fluxes.
Fe/V
equation
?1.I =o. compact deltned
(LC) form “s
(11)
becomes
w Thla equation Is of the Same form SS that given by the thermodynamics of steady state processes, but now Le phenomenalogical coefflelents have been @en physical Significance. The cross coefk4ent, ~fj, Ls a local resistance coef~lclent. and the stralatlt coeffleient..r,;. ts a function oi all the p&m cross c&rii&ents and the appropriSte local concentration rza~ios. It ~wbt be remembered Lt&; a Straight coefficfent
L. L=.MMs
358
lacludes a Lerm of the ty~::‘ C,~V<~/C~. These are the or.& terms in whicit explicit reference 1s made to the membrane a8 euch.
One difficulty in the direct appllcatio” of eq. (12) to the deacriotlon of the behavior of actual sy&ems lies 1” t6e fact that both the v’e and the J’S contained therein are wihnown functions of the soace coordinate. Some of the dtffteultv is “Ye&ne a new set Of fluxes, called for convenience component fluxes, arc introduced into the descriptive relations. The eIectric flux, JE, ~“ore commonly known 1” other units as the electric current, Is always avalIabIP for the deecriptio” of any material transfer *ystem contal”l”g Ionic perme^“ts. All mobile lone present in a” element oi volume “re posslbIe constltw e”ta of the electric flux, and since, I” the present eystem, the elsctrtc fIux 1s zero, the followbx reiation holds
if
tn which ei is the charge “umber of a” ion, poslWe or negative aa the ease may be. Two impllcattons of eq. (13) a~+ tmportant to the ondersta”d1ng Of matec1a1 transfer. First, it 18 evident tit tontc fluxes are not alI lndependent. Second, the only time a” Ionic flux does not depend upon the space coordinate 18 when the barrier is a homogeneous phase or when a single IO” carries the totallly of the electric current Uuough the barrier. I” the model eyetem, ten different ionic species are possIbIe co”stttue”t3 of the electric current. In the statlonarv atate characteristic of the boutiary condltlone lmposed upon the system, phase p 1s not ln direct eo”unu”lcatto” wlth the exter~l environment. The flux of three of the constituents of the electric flux, the Uthium, potasstum, and chtorlde tons, 1s zero in the etatkmar‘y atate. These che&lcal ape&s can be called ;ll.trlbuta”ts to dlstmgoish them from the remnlnlng permeante. WlIlCh can be called reactants. Urea does not partielpate in chemical tea=tlons in lbe membrane. Ita flux may be taken as a component flux composed of only one eonstltue”i I” the stationary et&e J orea = CO”&.
(14)
““d Jurea does not dewnd upon the space coor~dlnate. While the flux of the malelc compawnt. Jl& Id zero In the stationary etnte, its co”ntltuents do partlclIMe In chem;oal rea.ctto”a. The “Mete component CM be regarded ~8 composul of three oonetttoente. whose fkxes we depend-
ent upon the epaee coordinate, J&
= JIf*f&
+ JIfMa-
or + .Jm=
= cl.
(15)
The fluxes of the carbon dioxide component, JC03, the ammonia component, .&3, ti the water component, Jw, are each composed of several constituents. The relation can be writte” as follows:
Jcoz =Jcoz~ct
+ JQCO~ + JHCOj+ Jco5 = const.
JNH3 = JNffpns Jw = JH~
+ JNwo,,
+ “NH; = eon&
+ JH~c,++ Jon- + Jszm
+ Jco2Aq + 4x2c03+Jncoj
+ JNH,&
+ Jm40ff
= eonst.
(16)
It Is obvious from the discussion dven atwe that the fluxes of the various ehemikl speffes barrier are not Independent. They may affect one a”ouwr because Of the mutual electrostatic interacllons of the conatituente. or because the eo”stltue”te paructpate I” to”lzatla” or “eutrallxatlo” reactions.
present tn a
Whenever the component flux Is flnlte, a convenient functlo” to introduce Into a descr i;tlon of materID transfer Is the ratto of the co”stitue”t Such a ratto LBcalled flax to the component flu a material transfer number, and is defined formally *B
Jcs = %Jcp
(17)
I” tbls equation,the subscrtptcs dealg”mtes the constituent, and the symbol cp deolg”ates the component. The materlal transfer number 1~ posltlve if the dIrectlo” vex?tors of the co”stttuents and component nuxes are sb”iIer1y oriented, but “egatwe If they are opposed. The absolute “lag”ltode Of materla1 trM#fer numbers ranges from zem to i”fi”lty, and It 1s a fonctton of the spwe coordinate. A” lmporm”t property ehared by materlal transfer numbers *s that
As is apparent from eqs. U3), (15) and (16). a given substaanee can be a constituent of mote than u,,e of the comr,rnent Iluxes. This Eaactresults in interesting relitions among the transfer numbers of a particular chemical species with respect to different components. Furthermore, it also means that alter&e hut equivalexd sets of material transfer numbers can be selected for the description of any system. We will select the following set for the description of the present model system: +OpAq
CO2 = ‘CO3.Aq+02
NH3 JNH3.Aq = ‘mi3.~q
3
J~~3 3
in whick a new set of C~IISS coefficients appear. These cross eoefitcients can be d&gnated loul species-campollen cross coefficients. Tkey zre the algebraic sutns of the pertinent local reflistawe coefixients weighted b:r the appropriate material transfer numbers. They may be poaitive or negative in .sig~~and also are functions of the space coordinate 4.6.
Chemicnl
reoclions
b the Dawie*
In this system numerous chemical reutlons occur m the tarrier, of which membrane ODCie but a part. These continue even when the system is in a stationary state. For example, the thenical species Hz0 carp be said :o be a partietpwt in the following set of chemicti reactions:
Hz0 + H2G = H30+ + OH-
JH30+ = f;,O+ J,., ,
~~13+ H2Ma = H~O++ HMaHz0 + HMa = H30+ + Ma= Hz0 + HzC03 = H30+ + HCOj Hz0 + HCOi E A30’ A co; 2H30 + NH: = H@ r NH*~H .YHz0 + NQOH z= NH3*Aq ,,‘H2G + H9C03 = CO9’Aq
.
the stationnry state, deepfte the fact that chemical reacttans are occurring, the lae+i concentratlons of all chemical species we not functions of the time. If we apply the principles of material balance to each element of volume of the barrier, with thle condltlon In mind, we obtabl
IR
A&+/Jr = (rate of formatton - rate of destruction) of i Jp&z
-
‘(t;2Ma + t;p”j&l.
in whichA ia the
(19)
is obtained by substltutlon of theothermembers Intoeq. (15). If we now replace the fluxee ln eq. (12) with thatr equivalents PB &en by eq. (19) and c6llect the terms, we 0btaIn z+local equluon, The last member
aI eqs. (19:
,
(22)
effec:iYe are;i af the Lwrfer. The r L&Bof formatton or destructlon of 1 LBthe slgeblatc sum of all ckemlcal reactions in wkick i is a prodocl, or tn which i is destroyed. To estimate the space dependence of the flux uf a partleular CkemiCal species, we WOULd need to know the local Coheentratlons of all reactants 3nd the magnitude oi the particular reaction ve-• loclty coefficients. We do not have this ltiormatlon for the media III which these chemtfnl rewtlons OCCUP- the media provided by cellophane or by the membranes of btolo&aZ mateta
transfer systems. This lack of Information 15 one of the main reasone why. from the physIcalchemical point of view, the complete desertp?ton of material transfer cannot be made at the present tbne. If any attempt is made at a complete descriptlon, one must keep in mind that from a macroscopic potnt of vtew, each element of volume in a barrier 1s electrlcally neutral. This impses a restrtctton on the local Concentration of ions, for they most be related as
cj %Ci
= il.
The addition of eq. (XV to our system of equnti3ns makes it possible in principle to gtve format relations for the eonCentratton prof1les of all chemical species within the membrane when the syetem LB I” a stationary state. The formal equations for the concentration profiles or any other variable whose maenitude ts a function of the space coordinate w&Id be functions of the cross coesficlents of eq. (12), the reactIon veklcity coefficients of the chemical reactions occurring tn the barrier, the compoettton of phase o. the geometric variables such as the effecuve areas and thickness of the barrier, and the rate of the enayme reactto” taken place in the central compartmenL From what has bee” dlsedseed so far it should be apparent that this would be a formidable mathematical undertaking even for a SyStSm as 5Uperfictally slmple as the mode1 system may appear to b.?. we WI11not attempt this task here but wtll proceed to the derlvatlon of certain Dractlcal eqoattons descrtptive of actual experiments. 4.7. Chenricd veoclion in lhe cenlral compartment In the stationary state the concentrattons of the components of the chemical reaction taking place in the central compartment are ttme fdependent. This implies that the rates of ftow of the oompunents thin@ barrier one in moles per set are numeriealIy equal to their ratce of formation or destruction. The rats of formarlon or destruction of any one of the cumponents can be selected as the index of the chemical reaction rate, stnce they are all rel.&d by the atotci:,ometric coefficients of th!, reactton. If the rate of fo=“IattOn Of amUOnia, QN,+,, tS SeleCted as the index of the chemical reactton rate, the fluxes can be wrx1en A&es
= @NH2 9 A&
= ‘%NHs (2414
in which A ie the effective area of the barrier. quanti
Substituting these
Thus we discover that the stationary state gradtent of chemical or electrochemical potential of any permeant in barrier one Is determined by the rate of the enzymatic catalyzed reaction taktng place in the central compartment and soms of the vartous local resistance mzoefficients multirlled by the aDDroDrtate material numbers. Iit is u&i InPq.(25) to deeignate a reactant, then the term in the brackets on the right hand side includes the atratght coefftctent 01 the reactant. acd LB more likelv to be dependent u*‘* the l&al coneentratlo~-eee eq. (11) - than when i refers to a distrihutant. if we define an integral spectes-Componee. reststance coeffielent of a barrier to the ehemicat epeetes i aa equal (0
transfer
etc., in which x2 - x1 is the effective thickness of the barrier, it is possible to integrate eq. (26) over the space coordtnate (see also Kirkwwd, 1954). The result obtained is b A tii =
I b -MRturea+& -R;cO2) -R&&Qd
(21)
It Is convenient, In order
save space, to delp fine a net drag coefftctent $infis as
(20) In which case eq. (2s) can be wrttter, ae
-Abtij=“tNHs6NHs/A .
(26)
The net drag Coefficient Can be poaltlve or negative, large or small. Although it is made up of the linear sum of maw terms. its essenttai nature should be clear. fis magnitude and stgn are saslly determined when t destgnates an ton. The stationary state electrochemlc~l potential of the ion Is dstermtned by placing els&odes reverslble to the Ion in the two homogsneow, pbaaes in eontaet with the barrier (Ntma, 1963;
Nims and Kuo, 1954). A” Emff will appear between the terminals of the probe electrodes which 1s equal to Emfi = Awi/ziF
= -“1N~3i+,H3/ziFA
(30)
in which df Le the Charge “umber of the ion, poeitlve or negative as the case “ray be, and F is Faraday’s eanstant. The recent development of many “ion specific” reversible membrane electrodes promises to make this a general method for the determbmtion of $,,eaet./A for many ions
aild membranes.
The quantity
oNH3,
In eq.
(30) is easily estimated fror~ experimeotaUy determined concentrations of ammonia in the efnuent from phase a and the Known rate of perfusion of phase d in the mcdel system. For a purely chemical description of materiel transfer *y*teroe. the electrochemical potential differences can be replaced by chemical potential differences of ion pairs. The formal relation is AbUi Abpj AbUti?) = 7 - 7 sod eq. (29) for an IO” pair
5. PRACTICAL
v
becomes
RELATfONS
Tin chemlca~potential difference of a neutral chemical f!f?ecie~ (1” the two sides of a barrier in an Iwthefmal material transfer syetem IS IgIve” w Abut = UfAbP + RTAb
In Cpf
(33)
in whlchflf
is the ‘average’ partial molar volume of 1, md vf is the aethlty coefficient. A slmllar expr*IuI*o” for iD” pars of OPpoSlte charge is Abptfj, E VcfjjAbP+
ab+ = -[$-lb1 mupea+ R”,‘, -R%o,) - R&i31
RTAb In CiCiY(Q) . (34)
U We WlVe eq. (33) far AbP, ad replace Abfli by itll equlvale”t 28 given I” eq. (ZV, we obtatn for “,e neutral material of which the membranes in the model system are formed
An abbreviated
form of eq. (35) can be written
Ablp=_tob’ mNn3~Nn3/~mA) + AblDnz (36) Eq. (35) is particularly Lnstructive, for it tells us that when the model system is 1” a statlonary state, the pressure difference across a barrier is the resultant of two forces - the “et drag force of the reactants upon the membrane as they flow through and the osmotic force which is proportional to the concentration ratio of the membrane material at its two surfaces. The auantitv abp mieht as we,, have been witteo imp, ior the pr&ore difference appeare only acroee the membrane subetzmce. and not in the unstirred boundary layers. The barrier must have a portion with come of the properties of a solkd, that is, it must have sufficient mechanical srrength to withstand such a pressure difference. Within tl;? cellopha”e membranes, as well a6 within many biobgical membranes, the actoal local concentrations of the pero.aa”ts are likely to be small, therefore the contributton of Abn, to the oresewe difference is also likelv to be small, or thr pressure difference be&es a functlon almost entirely of the type and the rate of the cnemlcal rea&ns occur&g in the ayetern. In plant materials, the natures of the ?imiting membranes and of the react+zta flo;ving throw& these membranea rre such that the Iwee differences !” mechanical pressure dnvelops ilc?oes the barriers. The dtur”al variation in the pressure difference observed in trees, for exLmple (Scholander et al., 1965), LB undoubtedly due to a change in the typos of flowing throueh membranes as the svstetn alternates between photosynthetic and resilratory activity. The mechanical difference in pre~e”~‘e between the Inside and outslde of ~many animal cells, the mammallan erylhrocyte in Lte physiologlcal environment, for example, is very small. Thla means, In the terms of this desrriptlon, that the natures of the red cell membrane and tbe particular reactants flowing through it i” the basal state are such that the net drag force on the membrane ia also new zero. If we let i 1” eq. (33) deelg”ate the chemical apecles H30, a” ewation Is obtaIned In which the mecbanloal pressure diff~irnot ac!‘oee the barrter !e no*:: e&pressed in tot’,“8 of the net drrg
reactants
L. F. NlM8
362
force of the reactants upon Hz0 and the osmotic prewure difference of H20. The equ’.on Le
pair between phases I and p 16 primarily a function of the chemical reaction rate in the central
Abll, = _(J”
eq. (40) it is possible to obtain the following relations for the concentration ratios of the distributant ione, CL-, Li+, and K+ between phases a and p,
lf20,Nfi3QNH3'ulf20A)+ 3%f20~
(37) The nuperscript bl LBused to indicate that this preseur~: difference appears accoee barrier one. Since phas~a 0 of the model system is neither a eowee nor sink of any reactant, the only etaiio,,ary state posetbie at barrier two ts one of equilibrium, and the pressure difference here is Gwen bv _ . Ab2, = Ab21$f,0 . (38)
CLm,=(-
This ccwzlition is !-mownas a Dorwn equlltbrlum since the central compirtment of the model eyetern contains the charged tmpermeant ions, the enzyme and Its protein impurities. This presare difference 1s aulte small. much less than thatat barrier one; where thi netdrag force LB the major factor producing the pressure dlffer5.2. Ion concenlralion ratios By methods similar to those described above, CM obmn expressims *or the f”nctton A In C&? y(jj) for each barrier of the three-oompartmen i system or for the compoaltd barrier made up of the two cellophane membranes, the supportIng materials, the four unotlrred bauxkry layers, and the central compartment. Fur example, the following relauon e?a?te between the terminal ohaees 01and 0.
v/e
%Cl $ bj bl UK’HZO fi2O,Nfi3 - *HCl,NH3 )
Ccl-
bj fW,NfQ 0~~3 ART
- &+,K+),Nfi3j)
d
uKC1- %Cl
1n-E
c”~ =(
%20
bl %‘~NKs
- ‘$+,li+),NH3)
q,,
+ cP$ Ii+ BCI PKC1 ln
%+&&CL ’ (39)
The pressure difference between phases (I and l3 an eqwkion almilar to eq. (33). EltmPa from es. (39) yields
(41)
Prom eqs. (41) It Is &dent that the ratlos of the dtstributant Ions are functions of the rata of ehemlcal reactton as well 98 of the pll dlfferencee between the terminal phase& That euch 18 the fact ts evident in the data presented Ia table 1. 5.3. ion sslecttm ratios It the second member of eq. (41) 1s subtracted third member, we obtain an eapresclo~~ for an ion selection ratto. The eqoatton is from the
The last termin e&(40) is small and can be neg. k&d in the derivation of I>ractieol refattone. Note that the ,#Ifunctions pertain only to krrlar one of the three-CompPrtment model syatem, the barrier ln which idt drag force6 are fldte. Eq. (40) *tatee that the actktty rrtto of an tar
- ++,+,,,,,
4,,
(42)
- In*. KC1 LtCI
The last term Ln eq. (42) is near zero for the condlttons obtained In the model system, and eq. (421 states that the ion selection ratlo denends p&aruy upon the ret &an resistance’cbeificlente, the partial motar volumes of apprapriate ion pairs, and the rate of chemtcal reactlo”. It is nearlv btdaoendent of the oH differences exfor tile fact that net overall reelstance coeffIcients depend somewhat upon the local bydrogal ton CoocentPatlOn8. Eq. (42) suggests that a plot of. the logarithm of the ion eelectlon ratlo agalnet QNH3 would be lnformatlve. The elope of the resultlng curve at any point would have the magnitude
cept
In animal material tranqfer systems, like that of the human crythrocyte to blood, the pressure difference between the inside and outside of the cells ie near zero. and simpler eqw,ttons will then be descriptive of the ton selection ratios. In such caeee ea. (42) reduces to
c’$c$+
ln+‘c’;;. + =-%+,m+uact,Qtact.
/hRT.
(44)
ExperImental veriftcatlon of eq. (42) has been obtained by Thurber and Thompson (1967). Their studlee of !he human red cell disclosed the fact the logexithm of the ton,setectton ratlo was a “ear linear function of Qbct./AT over the tmperature range of 4% to body temperatures. fix their studlos, the rate of glycolysis of glucose bv the red cell wa8 altered bv olacine the ineub&on tubes at various t&p&at&s. From their Pesulte It ap,,eare that those pro,,ert,es of the red cell membrane controlling Us resistance to material tranefer do not depend to any great degree upao the temperature.
that
t
Ftge. 2 and 3 are plots of the data obtained tn the three-compartment apparatoe. Wlthln the experimental errors of Ihe observations. It ap,xars that the logarithm of the Ion selectlo” ratto *s a “early llncar function of the ehemlcal reaction rate. nowever, the elopes Of the 1tnes tn the tvlo n&e of experlmente differ. We presume that this dlffermee is doe to differences In the unknown and not eaolly determlnable effective thicknesses ti arDll of the two typos of barrier used. For example. if the dlfference was entirely due to the area, the cloth supported cellophane membrnoe would have an effecUve area of about one hdf ttit Of the Bt‘%l &Fed membrane.
SUf,tX.rted CettOFti,E
6. DISCUSSION Despite the rather involved mathematical con8tderations nece%?a~y to obtain the final worklna eauatione for the deseriotlon of selectton ratios tn stationary state eystemo, the principles under whleh model material transfer eysterns operate are rahtlvely simple. The movement of substances through membranes can be
_ .
L. F.N,Ms~
361 accountrd
far as the resultant of the actions of
conservattve and fricttonal forces. A dlstrlbutant speciea can mwe against Its concentratton gradtent If the net drag force, due to all other permeant species, i8 greater than the conservative farce acting on the species. The flow of dietrtbutant species relative to the barrier or the membrane ceases when the conservative force becomes equal to ihe net drag force. In such a case, the system 18 in a ~tatlonary state, which Is maintained as long 88 a steady supply ergy-yielding chemlcal molecules ts avstkable io the system, and the products of the chemleal reaction ape removed from the system at a steady rate. The asymmetry of the present system artses less from epecial properttes of the two barrierr; than from the fact that the system is open ta reactants from only one side or the other, just as in the general btological eysbm. However. If the sy’ltern has membranes of ereatly differ&t pr&tles, such ae high den& of fixed posttlve or negative charges, the functional asymmetry of the system with respect to the dfstritutants could be enhanced. It should be evident that the behavtor of the system does not depend upon the volume of the intermediate phase. Thus, if the tarrier in the system was tien to inpltie both membranes and the intesmedlate phases, then the system could be called a two-coinpartment system wtth chemical reacttons taktng place in a, ‘membrane’. The ayatem, as it stands. IS 111ustrat1veof the s1tuation, luck as edst 3 between the blood (phaas a), cells ourroundlna a kidnev tubule (the intermedhte phaee and the two m&nbranesj, and ths lumea at the ktdney tubule (phase ~3), etc. The model system la tltustrative of one Important point. Tha relatioh between the ion dlstrlhution pattern and the pr’oductlon rate of a reactant requires the slmulbmeoua determination of five varlablos. Aarelv in exoertments on ltving systems haa such a*task~ &en attempted, 11 the system IS not In a stattonaly atate. irnny mm’e measured variables are reaulred. even tf’a steady state approxtmatton la made,’ &xl the mzthemattcal compledty is much Increased. In the physical-chemical descrtptton, the concepts of drag forces, nr reststance coefftctents and chemical reaction rates, replace the more generally referred to ideas of ‘acttve ‘membrane pumps’. The major dlfflculty of a complete physical-chemical descrsptton of matvrlnl transfer through membranes in nmount of tnformarton which is-required to determine in an erpltctt faehiru, the magnitudes of all of the qnanttttse
transfer’or
Dment Bvstems Is the
multleom-
that cullectively constttute a ‘$ function. liowever, the magnitudes of the 0 functions can be determmed by sultable experiments. and these magnttudea for a variety of substances and membranes would greatly clarify our knowledge of what actually DOCUPB withln the membranes of biological matertal trawler systems.
ACKNOWLEDGEMENTS
paper is tpsed *aa
The work upon whtch this performed at the Brookhaven Nattonal Laboratory under the auspices of the US Atomic Energy C0~lUt0fdOn. The author wishes to thank Miss R.Butera and Mrs. tJ.Schnappauf far thetr technical asststanee in obtaining the quantitativedata reported herein. REFERENCES
MEMBRANES,
365
MATERIAL TRANSFER AND ENZYMES
Page, I., 1952, Introdvctian to theOletirJ3, physics tTl. vaa Nostrand co., Inc., New York] p. 63. Riley, R., J. 0. Gardner and U. Mertan. &964. Cellulose acetate membranes. Electron microscopy of struc?ure. Science 143. 801. Salminen, S., 1963, ?. model of traneport of Bodium and potaesIum ions, Nature 200, 1069.
Schokmder,
P. F., H. T. Rammcl,
E.A. Hemmlngsen.
E. D. Brad~treet
1966, Sap pressure
and
in vascular
plunts, SciEnee 148. 339. Splegler, KS., 1968, Transport processes in tonic membranes. Trans. Faraday SoC.54. I#%. Thurber, R. E. nnd A. MI.Thompeon. 1967. Station3qstale sodium nnd potassium ion distributions of human erythrucytes, Am.J.Physiol.212, 877.
me energy t&OUrCerequired to bring about
and to malntaln the nonequilibrium stationary states In these model systems was largely external. Biological systems achieve the requlslte rate of energy production to bring about basal states from eneymattca11y oata1yze.d chem1ca1 reactions, called collectively the metabolic pathways, whfch occur somewhere in the system. A better mcdel of b’olotical material transfer BVSterns might tbereZorewbea linear three-compktment system. ia ahloh noneauillbrIum ion distrltutlonpatterne Of (he Lltat1omwystates ar* the remWant of an enzyme catalyzed ohemlcal renetloll takhlg place Ill the central eompwtment, The chemical reaction weeted for the model systems was the ewymatlc hydrolysis of urea (NH&CO + Iis “%!.% 2Nff3 + COP, (1) since It seemed probirble that fluxea of sufftclent magnitude of the small reactant molecules throuuh celloohane membranes could be estabIlehG to pr&luce ion dl8trlbutlon patterns between the termlnal phahases,elgniflcantly different from equlllbrlum values, and thus verify one of the consequencee of a physical-chemical tkeory Of matei+ tmnefer. 2. MATERIAL9 AND METHODS Plaetic contrlnera were faehloned ‘n erwb a
L.F.NIMS
354
way that an apparatus mntaining compartmentsof about SO, 90, 30 cm3 respectively, separated by cellophane SO0dlalyzlng discs, couM be readily assembled. The nominal area of tbe membranes was some h cm~. MagnetIcally driven atirrir+z llare were pIa+ tn each corn rtment, and tli SolUtlonU contained therein I $,‘e etirred vigorously at a ccmstant rate during the couwe of ‘&I exwrlment. Tbta we done to tnfwe hornwaneity of the phases and to keep the effective thick1~8s of the dtffuslon barrlere - the membrane8 and
theirassociated
boundary
layers
- as con-
stant and as reproducible as psqble tn a particular experimental series. A echematic diagram of the apparatus as developed Is glveo in fia. 1. -Flat dialyztng membranes, designated as cellophane 300 by the Fischer Scienttfic Company, were selected for these ezmerlments. Thev were Impermeable to urease, q&e uniform in p&pertil?s, and sufficiently free of adventitious ptn hole8 that comparable experiments could be carrled out on different samples of tbe stock mate-W. Recent atudtes (Riley et al., 1984) make it seem ’ -‘-I that membranes of tkts type are compxxed cd a ‘-” dense surface layer supported by a spongy porn”.. . ?” Tbe major partton of the realstance to materia, u..r_- ” “n,bably ltea in the dense layer, about 1%of the tcwa, thickness of the membvane. The permeants seem to be molecularly dispersed In the denice layer, and material transfer tlwou~ this Layer Is probably accomplished by a dlffuslon process. The rest of the membrane appears tc ta quite porous and 16 ~r+x”mably readlty penetwble by water, small Leutrai mOleCuii9 and ealta. Mafeic Acic.
As the material transfer Bystem approaches a stationary state. a wnslderable hvdrostattc pressure dliferenc; develops between ti,e central compartment and phase (1. The maximum pressure dlfferences exceeded the breaktog strength of unsummrted cell~~bane membranes of some 8 cm2 ark; - about 7 ibpeersquare Inch. For &is reason, membrane one was suppvrt& either by a linen cloth disc or a stainless steel grid. Thi quantitative results obtained with the hvo supports were not the same. These differences are undbubtedly due to the geometrical parameters Of the complex Lurriers so formed. Tbe #UppOrt material screena one surface of the cellophane membrane from full pgrticipatton in the transfer process and thus gives an effective area less than the nominal area. The thickness of the barrter is the sum Of the thicknesseJ Of the two mlstirred tcwndarv kwers. tile ee,lwbnna membrane, and the ;oupdort &&&al. %&I&&& the effective area nor tblCkne88 Of the two composlte barriers ,c1knmvn or readily determlnable. Phase (t of UK apppratus v/a* perfu6ed at rates from 0.5 to 2 em3 ner minute with anuews
KC!. H-0
tine.
eantly with the system-was de&d reacbed a Practleal stat!OMrY &ate. The
,
tohive awldual
partment bliartered ocai.tonallY wttb tbe attainment Of P true stat*onvy .tPte. In mo,t exwriments. tke rate of loss of ena~mrttc acttvtt~ so slowthat L god iQpro;imation to thi atauonaryaate MI felt to have bell rcbteved. When th aytim was daemed to be In a stattonary atate. umpler of tke fluldr In the termtnal compartmenta were taken for chemical anatyds.
wae
Experiments were carried out at room temperahires, 22 l 2oc. The Lithium and wtasslum concentrations of the samples were determined by flame pbotometry. The chloride ion concentrations were determined in an Aminco-Cotlove chloride ion titration apparatus. The pfi was determined with a glass electrode and &‘&roan pH meter. The ammonia concentration of pbaee o was determined by aeratfon of a eamyl~ made alkaline with potassium carbonate. The ammo@. hberated was trapped in a solution of standard acid, and the excese acid was then titrated with standard alkali. The ammonia productIon rate of the system was calculated from the determined ammonia concentrations of phase d and the measured rate of perfusion. Two different commercial sources Of urea59 were avatlable. and the urease was not further purified. They bad nominal enzymatic acUvIUes Of 2% I) and 1000 units per g respectively. Kn the fndivldwal experiments. weighed aluounte of the impure ureaae preparations were added to the 60 cm2 volume fluid I” the central compartment. Beczwee complexity of the system, *t was difficult to obtain as high a degree of reproducibllity from experiment to ezqwfment as might be destrable. A sufficient number of experiments we, e done that the general behavior of such Bystems could be descrtkd. Focus to theee cxperlmente wde directed toward the stationary state ton dIetrlbotion patterns achieved in the terminal phases of the system and their relation to tile ,ate Of tbe cherntcal react*on occurring in the central compartment. Phase S ‘was in equillbrlum wltb the cent& compartment, and of oecessitv at a hkber wessure than abase a. The only 60~sin lik In the &tlonary state were the flows of the the chemfcal reaction through membrane oae. ff both phase (I and phase I3 are mnlntained at atmospheric preeewe, very amaLi flowa of all through membrane two become pobsfble, and the system czn than be ntd to be In the srwetton mode. Quant1tPuve rrtud1ea Of t11e preewe In tblr twstem and the comwsltlon Vt3riouS fluid &w?es when the sy&m, LB In the secretion mode to the rate of chemical resetlon will be reported at 1 b&r tinw.
of
of the
syitem
compunenta of
perme;mts
therelattonOf
3. EXPERIMENTAL
of the
RESULTS
llw qwntIfatIve data obtained on the campoMtons of phases II and S wfth respect to Ifthfum, eodlum. and potassfum tons In ewerat of the ifuasl-a!.utfowy &ate experfments are 5umma-
rized In table 1. Stnce in-these exaerimente the initial concentrations of the dlstrtkant ions in elch phase were Identical, it Is evident that during the approach to the stationary state, these ions must have at came time moved against their respecttve concentration gradients. Furthermore. the larzer the rate uf ammonla croductlon, ‘or the l&per the pH difference bet&en the termloaf ptxlses, the greater was the departure of the ion concentration ratios from unity. The system dlaplays a small degree of ion selectivity. In each experiment the pot8eeium ton C*nCClltrauOn is larger than the llthiuzn ion rat,% The chlorfde tons are distrlbuted fn the oppos!te seuee. Th&on selecl,vity Is much emaller than that seen In the maiority llvfr?g cell& Howevar .t is net due to cf&c~, and there is no reaeon to belleve that the phystCal-chemw.4 principles proposed for the bahavlor of the model system would not be equally appltcable to biologIcal 8ydem.e fn generni.
ratio mntewhat
if
4. PHYSICAL-CHEMICAL
THEORY
4.1. ThermodyMmics Of sttmiy The fmmal descrtptlon of
state processes
material transfer systems has been grealy facilitated by the development of a thermodynamics of steady atate processes (De Groot arzdMazur, 1962; Katcbalsky and Curran, 1965). Thermcdynamlc theory Is important since it gives precise relalions among the measurable variables or properties of a system, but it offera “0 clues ae to mechanism, and mechanical theory must be Invoked I” order to Luxlersta”d tile processes or tra”S!fCfr. When a psrtlcular barrier is permeable to j different material subs!a”ces, lhe basic relation descriptive 01 material lranefer through a” element of volume of a membrane g&en by thermodynamics IS I” which @fis lhe cbsmical po:e”tiPJ 61 B “butt%1 molecule, or the eleetrochemlcal potent@1 of an lo”: J; le the rate of flow o: j LWPunit area of the me&ane. and is called the fim; .fj”l. known as a phenomenologlcal cross coefficient. The quantity +jj obeys Onsager’s (1931b. b) reciprocal relatlOn* epuirl t0 Yji* “enron 118501.winted out that ea. 12) nxav be re3.arded as a generallzatlon af Fl&‘.‘l&v w&h takes account of the InteractIona between the rarlous chemical soecies flcwin~ t!xoueh a” element of volume oia barrier. B&.tly &eakl”g, Rck’s law 1s only appllcabls lo naultlcompoonent material transfer ny~tems wher. a11 permeable aubatancea traverse the barrier with the same speed, or each permeant cpll move through the tarrier by B seuarate aad 1”deDe”de”t“athway. It becomes in&&ate when two or mar; permesmts are flowing through the same element of volume with different velocltles. The flows of neceaslty must mutually Interfere by some pbyalca? proces8 or another. The flows are then said to be coupled, and the mag”lhld.3 Of the crosll coefflclentm la a meaare of the devxee of co”“ll~~1. The first step in the descrl$on of a s&e& is the decision ae to the namber of dlstl”ct wrmeants preeent. We will presums that seve&en are present in the model system aml will symbollze them as H 0. li30+, OH-. urea, Ii Ma (maleic acid). E&a-, Maul, CO3-Aq, &Oj, CO;, NH3.Aq, NH OH, NH& K+, M+, nd t 1”a steady state aeventern Cl‘. Thtemeans tha fhwes appear on the right hand elde of eq. (2). 81 addition &n)(n - 1) cross eaefflclpnts, wlw, (PI- 1) permevlts are present in L*l 1acItherma1 system, are required 1” the descrlptlon (Onsp+ij
IS
ger,
1945).
stncewe
kmw
very little
abxdmag-
nltudes of the cross coefficients *n the cellophane medium we will not attempt the complete descrlptlon of the model system, but will be content with the development of practical relations useful in the tmderstanding of the kbavlor of erperlmental systems. The first step in this development is the introduction of mechanical theory in order to give phy&al significance b the phenomenologlcal erws coefficients which appear I” eq. (2). Material transfer is a problem of motion. Whenever material substances are in motion. a descrlptlon of the motion is possible In terms of forces (Page, 1952). TO obtain an equalan Of motion in conde”sed phases, we first vleuualize an element of volume. The molecules In this element of volume are In thermal motion, lmt the average Ye&r velocity Of lbe molecules IS zero. If a” external force acts on the particles in the element Of vobmle, tire pwt1c1es undergo an i\cceleratio” in the directlon of the line of actfen of the force. The particles aqulre an average veloelty wblch is DOWa funcllan of the the. Wbllc the farce Is dletrllmted over the vcdume. it mav be assumeti to sotOR the center of m&.sofaif particles of a &en kind c~~“lalnedin the element of vo1ume. Tile weraE* ve1oeitv relltlve lo some : often iled lhe drlff YCframe of referew lOCIty - la alsO the velocuy 0, the center Of iwa!3e. The external forces of Interest to us here are conservative, that IS, the fields 0r force are characterized by a potential whose magnlbade Is a functlo” of the spcc coordInatea of Uw center of mpss, ad may also be * fu”ctIon of the time. The mngnltude of the force per unit volume IS related to the potential of the field PI +/v - -q tvd Cf (3) in which Fff Is the can~ervatlve force, Vls the magnitude of a” element of volume of the barrier, Cf IS the local coneentrat1on of f in fhls element of volume, a”d pad /q Is the ~adlent of the chemical potential 01 f II 1 Is D “m&al molecule, or the gradlent of the electrocbemlcal potentla1 of I if i Is an 10”. If the cenler of “~8s of f moves relatIvelv with rewect t” the center of mass of a”y o&e; chemtc~l apeeler which IS present In the bprrier, an lnter”al resistance to” mot,on 14 pnerated. The inter”a1 raisla”ce to motlon 1s presumably due to the composite action of Vpn der Waale, Coulomb and other Intermolecular
and the local Fij , Y = Ci Cj vfj “jj
to
zero.
Unforhinately o”r tierstandlng of condensed phases is so meager that rarely 5s itpossible to calculate the magnitudes of Local resistance coefficients from first principles. They are best regarded as emp(rlcal properties of a barrier. They can be related lo molecular frictional coeffleien?” as is often done (Onsager and Fuoas, 1932; Sple@er, 1958; &den. and Kabzhalsky, 1961; Mm@, 1961). Thus, +ij is a function of the elze amI shape of motecu!es as well as of the properties of the medtum in which the molecules are suep+r.ded. Enpertence Iodlcates that vij dwe Mt dewad UP” ofj when the relative YOIOCIties of motion are small. This Is the usual situation In biological barriers. However, vi’ does dewxl somewhat upon cf as well as upon I+Ie IOCal concentrations of all other substances present In the element of volume, I” the description that fo11ows. the system Is take” as Ilnear, so that ail varl”DLes become f”“CW.ns of the spafe coordinate x. Eq. (4) can be rnodlfled If the relative veloolty, “f’, is reof placed by the relatwe ve10c1ties “jnr a nd$,,, i andj with respect to the membrane. The vector relatton Uij = V*m - Ujm (5) allows
-api/ax+Jt
- Cf~jCf~~~* .
(6)
me
Jy
Cfcrt /c. -ZP.J jti I I 2 j+i
Eq. (10) can be written by Intwxluclng a straight
= Ci vi,,,
(71
In eq. (6) for their equiIn wh(ch frietlonal SS functiow of the local rcsts’anee nxafftetents.
the
in a more coefllcient
c dT../C. J D 1.
*ii = -j,i
Ofeq. (4)
We “ow introduce the concept of a LLux, dellned as the rate of now (moles per SW) per un*t a~~i( of membrane. The flux, Jf, Is given by the vector r4Lption
8IIbstltUtlen of the fluxes Valents gives B relation forces are represented concen’~atlo”s, the Ioeal
.
(8)
exactly.
(5)
the rewrltlng
= cj’ijCirJi,
is
- CirijJ,p
Steady statr flow is produced in the following way. The eo”>.ervative forces act to accelerate the particles, anJ tend to produce ever-lncreasi”g drlir veioews 01 the va~Lou8 species present in an elem#!nt of volume. As the dHit velocities increase, the fT1ctiolliil forces also increase in m?gnitude. Eventually the vector sum of a,, forces acting upon a given chemical species becomes zero, the accelerations vanish, and the drift velocities are thereafter conslant. The vector relation among the forces can be written Ffi+~j,Fij=O. (9) ji I When eq. (9) is obeyed the situation is called a stalionary state, for then all descriptive variables of a system are independent of the Lime. Such a srate in the model system is equivalent to basal siates Ln living systems. In Lhe majority of systems of interest to the biologirt. the descriptive varlnbles of the transfer systems are f”netlo”s of the time. However, the accelePationS in tne system are usually so small that cq. (9) can be used without introducing appredable errors in interpretation. Thus even when the deswlptlve variable.? are functions of the time, the material transfer systems are often called steady state systems by the bioioglst. The forces in eq. (9) CB” be replaced by their eq(livBlentB, eee eqs. (3) and (8). The resutt obtalned is
The definllive Ffj/Y
The equation
= CjrijJi”
(4)
in which Cf and Cj z!re the local concentrations of the chemical spewas i and j, a”d “jj is the relative velocity of motion of the centers of mSss of i with respect 1o j. The phenomenological coeffielentrji can be waa!ti 2 !zcz: ;is:siawe coefficienf,~and represents the interference in the motion of i caused by the presence of j in lhe same element Of volume. Eq. (4) is written in this way to take aecou~d of the fact that Fij @es to zero whenever C,. Cj, or oij goes
;; E;;&.!Zq.
fluxes.
Fe/V
equation
?1.I =o. compact deltned
(LC) form “s
(11)
becomes
w Thla equation Is of the Same form SS that given by the thermodynamics of steady state processes, but now Le phenomenalogical coefflelents have been @en physical Significance. The cross coefk4ent, ~fj, Ls a local resistance coef~lclent. and the stralatlt coeffleient..r,;. ts a function oi all the p&m cross c&rii&ents and the appropriSte local concentration rza~ios. It ~wbt be remembered Lt&; a Straight coefficfent
L. L=.MMs
358
lacludes a Lerm of the ty~::‘ C,~V<~/C~. These are the or.& terms in whicit explicit reference 1s made to the membrane a8 euch.
One difficulty in the direct appllcatio” of eq. (12) to the deacriotlon of the behavior of actual sy&ems lies 1” t6e fact that both the v’e and the J’S contained therein are wihnown functions of the soace coordinate. Some of the dtffteultv is “Ye&ne a new set Of fluxes, called for convenience component fluxes, arc introduced into the descriptive relations. The eIectric flux, JE, ~“ore commonly known 1” other units as the electric current, Is always avalIabIP for the deecriptio” of any material transfer *ystem contal”l”g Ionic perme^“ts. All mobile lone present in a” element oi volume “re posslbIe constltw e”ta of the electric flux, and since, I” the present eystem, the elsctrtc fIux 1s zero, the followbx reiation holds
if
tn which ei is the charge “umber of a” ion, poslWe or negative aa the ease may be. Two impllcattons of eq. (13) a~+ tmportant to the ondersta”d1ng Of matec1a1 transfer. First, it 18 evident tit tontc fluxes are not alI lndependent. Second, the only time a” Ionic flux does not depend upon the space coordinate 18 when the barrier is a homogeneous phase or when a single IO” carries the totallly of the electric current Uuough the barrier. I” the model eyetem, ten different ionic species are possIbIe co”stttue”t3 of the electric current. In the statlonarv atate characteristic of the boutiary condltlone lmposed upon the system, phase p 1s not ln direct eo”unu”lcatto” wlth the exter~l environment. The flux of three of the constituents of the electric flux, the Uthium, potasstum, and chtorlde tons, 1s zero in the etatkmar‘y atate. These che&lcal ape&s can be called ;ll.trlbuta”ts to dlstmgoish them from the remnlnlng permeante. WlIlCh can be called reactants. Urea does not partielpate in chemical tea=tlons in lbe membrane. Ita flux may be taken as a component flux composed of only one eonstltue”i I” the stationary et&e J orea = CO”&.
(14)
““d Jurea does not dewnd upon the space coor~dlnate. While the flux of the malelc compawnt. Jl& Id zero In the stationary etnte, its co”ntltuents do partlclIMe In chem;oal rea.ctto”a. The “Mete component CM be regarded ~8 composul of three oonetttoente. whose fkxes we depend-
ent upon the epaee coordinate, J&
= JIf*f&
+ JIfMa-
or + .Jm=
= cl.
(15)
The fluxes of the carbon dioxide component, JC03, the ammonia component, .&3, ti the water component, Jw, are each composed of several constituents. The relation can be writte” as follows:
Jcoz =Jcoz~ct
+ JQCO~ + JHCOj+ Jco5 = const.
JNH3 = JNffpns Jw = JH~
+ JNwo,,
+ “NH; = eon&
+ JH~c,++ Jon- + Jszm
+ Jco2Aq + 4x2c03+Jncoj
+ JNH,&
+ Jm40ff
= eonst.
(16)
It Is obvious from the discussion dven atwe that the fluxes of the various ehemikl speffes barrier are not Independent. They may affect one a”ouwr because Of the mutual electrostatic interacllons of the conatituente. or because the eo”stltue”te paructpate I” to”lzatla” or “eutrallxatlo” reactions.
present tn a
Whenever the component flux Is flnlte, a convenient functlo” to introduce Into a descr i;tlon of materID transfer Is the ratto of the co”stitue”t Such a ratto LBcalled flax to the component flu a material transfer number, and is defined formally *B
Jcs = %Jcp
(17)
I” tbls equation,the subscrtptcs dealg”mtes the constituent, and the symbol cp deolg”ates the component. The materlal transfer number 1~ posltlve if the dIrectlo” vex?tors of the co”stttuents and component nuxes are sb”iIer1y oriented, but “egatwe If they are opposed. The absolute “lag”ltode Of materla1 trM#fer numbers ranges from zem to i”fi”lty, and It 1s a fonctton of the spwe coordinate. A” lmporm”t property ehared by materlal transfer numbers *s that
As is apparent from eqs. U3), (15) and (16). a given substaanee can be a constituent of mote than u,,e of the comr,rnent Iluxes. This Eaactresults in interesting relitions among the transfer numbers of a particular chemical species with respect to different components. Furthermore, it also means that alter&e hut equivalexd sets of material transfer numbers can be selected for the description of any system. We will select the following set for the description of the present model system: +OpAq
CO2 = ‘CO3.Aq+02
NH3 JNH3.Aq = ‘mi3.~q
3
J~~3 3
in whick a new set of C~IISS coefficients appear. These cross eoefitcients can be d&gnated loul species-campollen cross coefficients. Tkey zre the algebraic sutns of the pertinent local reflistawe coefixients weighted b:r the appropriate material transfer numbers. They may be poaitive or negative in .sig~~and also are functions of the space coordinate 4.6.
Chemicnl
reoclions
b the Dawie*
In this system numerous chemical reutlons occur m the tarrier, of which membrane ODCie but a part. These continue even when the system is in a stationary state. For example, the thenical species Hz0 carp be said :o be a partietpwt in the following set of chemicti reactions:
Hz0 + H2G = H30+ + OH-
JH30+ = f;,O+ J,., ,
~~13+ H2Ma = H~O++ HMaHz0 + HMa = H30+ + Ma= Hz0 + HzC03 = H30+ + HCOj Hz0 + HCOi E A30’ A co; 2H30 + NH: = H@ r NH*~H .YHz0 + NQOH z= NH3*Aq ,,‘H2G + H9C03 = CO9’Aq
.
the stationnry state, deepfte the fact that chemical reacttans are occurring, the lae+i concentratlons of all chemical species we not functions of the time. If we apply the principles of material balance to each element of volume of the barrier, with thle condltlon In mind, we obtabl
IR
A&+/Jr = (rate of formatton - rate of destruction) of i Jp&z
-
‘(t;2Ma + t;p”j&l.
in whichA ia the
(19)
is obtained by substltutlon of theothermembers Intoeq. (15). If we now replace the fluxee ln eq. (12) with thatr equivalents PB &en by eq. (19) and c6llect the terms, we 0btaIn z+local equluon, The last member
aI eqs. (19:
,
(22)
effec:iYe are;i af the Lwrfer. The r L&Bof formatton or destructlon of 1 LBthe slgeblatc sum of all ckemlcal reactions in wkick i is a prodocl, or tn which i is destroyed. To estimate the space dependence of the flux uf a partleular CkemiCal species, we WOULd need to know the local Coheentratlons of all reactants 3nd the magnitude oi the particular reaction ve-• loclty coefficients. We do not have this ltiormatlon for the media III which these chemtfnl rewtlons OCCUP- the media provided by cellophane or by the membranes of btolo&aZ mateta
transfer systems. This lack of Information 15 one of the main reasone why. from the physIcalchemical point of view, the complete desertp?ton of material transfer cannot be made at the present tbne. If any attempt is made at a complete descriptlon, one must keep in mind that from a macroscopic potnt of vtew, each element of volume in a barrier 1s electrlcally neutral. This impses a restrtctton on the local Concentration of ions, for they most be related as
cj %Ci
= il.
The addition of eq. (XV to our system of equnti3ns makes it possible in principle to gtve format relations for the eonCentratton prof1les of all chemical species within the membrane when the syetem LB I” a stationary state. The formal equations for the concentration profiles or any other variable whose maenitude ts a function of the space coordinate w&Id be functions of the cross coesficlents of eq. (12), the reactIon veklcity coefficients of the chemical reactions occurring tn the barrier, the compoettton of phase o. the geometric variables such as the effecuve areas and thickness of the barrier, and the rate of the enayme reactto” taken place in the central compartmenL From what has bee” dlsedseed so far it should be apparent that this would be a formidable mathematical undertaking even for a SyStSm as 5Uperfictally slmple as the mode1 system may appear to b.?. we WI11not attempt this task here but wtll proceed to the derlvatlon of certain Dractlcal eqoattons descrtptive of actual experiments. 4.7. Chenricd veoclion in lhe cenlral compartment In the stationary state the concentrattons of the components of the chemical reaction taking place in the central compartment are ttme fdependent. This implies that the rates of ftow of the oompunents thin@ barrier one in moles per set are numeriealIy equal to their ratce of formation or destruction. The rats of formarlon or destruction of any one of the cumponents can be selected as the index of the chemical reaction rate, stnce they are all rel.&d by the atotci:,ometric coefficients of th!, reactton. If the rate of fo=“IattOn Of amUOnia, QN,+,, tS SeleCted as the index of the chemical reactton rate, the fluxes can be wrx1en A&es
= @NH2 9 A&
= ‘%NHs (2414
in which A ie the effective area of the barrier. quanti
Substituting these
Thus we discover that the stationary state gradtent of chemical or electrochemical potential of any permeant in barrier one Is determined by the rate of the enzymatic catalyzed reaction taktng place in the central compartment and soms of the vartous local resistance mzoefficients multirlled by the aDDroDrtate material numbers. Iit is u&i InPq.(25) to deeignate a reactant, then the term in the brackets on the right hand side includes the atratght coefftctent 01 the reactant. acd LB more likelv to be dependent u*‘* the l&al coneentratlo~-eee eq. (11) - than when i refers to a distrihutant. if we define an integral spectes-Componee. reststance coeffielent of a barrier to the ehemicat epeetes i aa equal (0
transfer
etc., in which x2 - x1 is the effective thickness of the barrier, it is possible to integrate eq. (26) over the space coordtnate (see also Kirkwwd, 1954). The result obtained is b A tii =
I b -MRturea+& -R;cO2) -R&&Qd
(21)
It Is convenient, In order
save space, to delp fine a net drag coefftctent $infis as
(20) In which case eq. (2s) can be wrttter, ae
-Abtij=“tNHs6NHs/A .
(26)
The net drag Coefficient Can be poaltlve or negative, large or small. Although it is made up of the linear sum of maw terms. its essenttai nature should be clear. fis magnitude and stgn are saslly determined when t destgnates an ton. The stationary state electrochemlc~l potential of the ion Is dstermtned by placing els&odes reverslble to the Ion in the two homogsneow, pbaaes in eontaet with the barrier (Ntma, 1963;
Nims and Kuo, 1954). A” Emff will appear between the terminals of the probe electrodes which 1s equal to Emfi = Awi/ziF
= -“1N~3i+,H3/ziFA
(30)
in which df Le the Charge “umber of the ion, poeitlve or negative as the case “ray be, and F is Faraday’s eanstant. The recent development of many “ion specific” reversible membrane electrodes promises to make this a general method for the determbmtion of $,,eaet./A for many ions
aild membranes.
The quantity
oNH3,
In eq.
(30) is easily estimated fror~ experimeotaUy determined concentrations of ammonia in the efnuent from phase a and the Known rate of perfusion of phase d in the mcdel system. For a purely chemical description of materiel transfer *y*teroe. the electrochemical potential differences can be replaced by chemical potential differences of ion pairs. The formal relation is AbUi Abpj AbUti?) = 7 - 7 sod eq. (29) for an IO” pair
5. PRACTICAL
v
becomes
RELATfONS
Tin chemlca~potential difference of a neutral chemical f!f?ecie~ (1” the two sides of a barrier in an Iwthefmal material transfer syetem IS IgIve” w Abut = UfAbP + RTAb
In Cpf
(33)
in whlchflf
is the ‘average’ partial molar volume of 1, md vf is the aethlty coefficient. A slmllar expr*IuI*o” for iD” pars of OPpoSlte charge is Abptfj, E VcfjjAbP+
ab+ = -[$-lb1 mupea+ R”,‘, -R%o,) - R&i31
RTAb In CiCiY(Q) . (34)
U We WlVe eq. (33) far AbP, ad replace Abfli by itll equlvale”t 28 given I” eq. (ZV, we obtatn for “,e neutral material of which the membranes in the model system are formed
An abbreviated
form of eq. (35) can be written
Ablp=_tob’ mNn3~Nn3/~mA) + AblDnz (36) Eq. (35) is particularly Lnstructive, for it tells us that when the model system is 1” a statlonary state, the pressure difference across a barrier is the resultant of two forces - the “et drag force of the reactants upon the membrane as they flow through and the osmotic force which is proportional to the concentration ratio of the membrane material at its two surfaces. The auantitv abp mieht as we,, have been witteo imp, ior the pr&ore difference appeare only acroee the membrane subetzmce. and not in the unstirred boundary layers. The barrier must have a portion with come of the properties of a solkd, that is, it must have sufficient mechanical srrength to withstand such a pressure difference. Within tl;? cellopha”e membranes, as well a6 within many biobgical membranes, the actoal local concentrations of the pero.aa”ts are likely to be small, therefore the contributton of Abn, to the oresewe difference is also likelv to be small, or thr pressure difference be&es a functlon almost entirely of the type and the rate of the cnemlcal rea&ns occur&g in the ayetern. In plant materials, the natures of the ?imiting membranes and of the react+zta flo;ving throw& these membranea rre such that the Iwee differences !” mechanical pressure dnvelops ilc?oes the barriers. The dtur”al variation in the pressure difference observed in trees, for exLmple (Scholander et al., 1965), LB undoubtedly due to a change in the typos of flowing throueh membranes as the svstetn alternates between photosynthetic and resilratory activity. The mechanical difference in pre~e”~‘e between the Inside and outslde of ~many animal cells, the mammallan erylhrocyte in Lte physiologlcal environment, for example, is very small. Thla means, In the terms of this desrriptlon, that the natures of the red cell membrane and tbe particular reactants flowing through it i” the basal state are such that the net drag force on the membrane ia also new zero. If we let i 1” eq. (33) deelg”ate the chemical apecles H30, a” ewation Is obtaIned In which the mecbanloal pressure diff~irnot ac!‘oee the barrter !e no*:: e&pressed in tot’,“8 of the net drrg
reactants
L. F. NlM8
362
force of the reactants upon Hz0 and the osmotic prewure difference of H20. The equ’.on Le
pair between phases I and p 16 primarily a function of the chemical reaction rate in the central
Abll, = _(J”
eq. (40) it is possible to obtain the following relations for the concentration ratios of the distributant ione, CL-, Li+, and K+ between phases a and p,
lf20,Nfi3QNH3'ulf20A)+ 3%f20~
(37) The nuperscript bl LBused to indicate that this preseur~: difference appears accoee barrier one. Since phas~a 0 of the model system is neither a eowee nor sink of any reactant, the only etaiio,,ary state posetbie at barrier two ts one of equilibrium, and the pressure difference here is Gwen bv _ . Ab2, = Ab21$f,0 . (38)
CLm,=(-
This ccwzlition is !-mownas a Dorwn equlltbrlum since the central compirtment of the model eyetern contains the charged tmpermeant ions, the enzyme and Its protein impurities. This presare difference 1s aulte small. much less than thatat barrier one; where thi netdrag force LB the major factor producing the pressure dlffer5.2. Ion concenlralion ratios By methods similar to those described above, CM obmn expressims *or the f”nctton A In C&? y(jj) for each barrier of the three-oompartmen i system or for the compoaltd barrier made up of the two cellophane membranes, the supportIng materials, the four unotlrred bauxkry layers, and the central compartment. Fur example, the following relauon e?a?te between the terminal ohaees 01and 0.
v/e
%Cl $ bj bl UK’HZO fi2O,Nfi3 - *HCl,NH3 )
Ccl-
bj fW,NfQ 0~~3 ART
- &+,K+),Nfi3j)
d
uKC1- %Cl
1n-E
c”~ =(
%20
bl %‘~NKs
- ‘$+,li+),NH3)
q,,
+ cP$ Ii+ BCI PKC1 ln
%+&&CL ’ (39)
The pressure difference between phases (I and l3 an eqwkion almilar to eq. (33). EltmPa from es. (39) yields
(41)
Prom eqs. (41) It Is &dent that the ratlos of the dtstributant Ions are functions of the rata of ehemlcal reactton as well 98 of the pll dlfferencee between the terminal phase& That euch 18 the fact ts evident in the data presented Ia table 1. 5.3. ion sslecttm ratios It the second member of eq. (41) 1s subtracted third member, we obtain an eapresclo~~ for an ion selection ratto. The eqoatton is from the
The last termin e&(40) is small and can be neg. k&d in the derivation of I>ractieol refattone. Note that the ,#Ifunctions pertain only to krrlar one of the three-CompPrtment model syatem, the barrier ln which idt drag force6 are fldte. Eq. (40) *tatee that the actktty rrtto of an tar
- ++,+,,,,,
4,,
(42)
- In*. KC1 LtCI
The last term Ln eq. (42) is near zero for the condlttons obtained In the model system, and eq. (421 states that the ion selection ratlo denends p&aruy upon the ret &an resistance’cbeificlente, the partial motar volumes of apprapriate ion pairs, and the rate of chemtcal reactlo”. It is nearlv btdaoendent of the oH differences exfor tile fact that net overall reelstance coeffIcients depend somewhat upon the local bydrogal ton CoocentPatlOn8. Eq. (42) suggests that a plot of. the logarithm of the ion eelectlon ratlo agalnet QNH3 would be lnformatlve. The elope of the resultlng curve at any point would have the magnitude
cept
In animal material tranqfer systems, like that of the human crythrocyte to blood, the pressure difference between the inside and outside of the cells ie near zero. and simpler eqw,ttons will then be descriptive of the ton selection ratios. In such caeee ea. (42) reduces to
c’$c$+
ln+‘c’;;. + =-%+,m+uact,Qtact.
/hRT.
(44)
ExperImental veriftcatlon of eq. (42) has been obtained by Thurber and Thompson (1967). Their studlee of !he human red cell disclosed the fact the logexithm of the ton,setectton ratlo was a “ear linear function of Qbct./AT over the tmperature range of 4% to body temperatures. fix their studlos, the rate of glycolysis of glucose bv the red cell wa8 altered bv olacine the ineub&on tubes at various t&p&at&s. From their Pesulte It ap,,eare that those pro,,ert,es of the red cell membrane controlling Us resistance to material tranefer do not depend to any great degree upao the temperature.
that
t
Ftge. 2 and 3 are plots of the data obtained tn the three-compartment apparatoe. Wlthln the experimental errors of Ihe observations. It ap,xars that the logarithm of the Ion selectlo” ratto *s a “early llncar function of the ehemlcal reaction rate. nowever, the elopes Of the 1tnes tn the tvlo n&e of experlmente differ. We presume that this dlffermee is doe to differences In the unknown and not eaolly determlnable effective thicknesses ti arDll of the two typos of barrier used. For example. if the dlfference was entirely due to the area, the cloth supported cellophane membrnoe would have an effecUve area of about one hdf ttit Of the Bt‘%l &Fed membrane.
SUf,tX.rted CettOFti,E
6. DISCUSSION Despite the rather involved mathematical con8tderations nece%?a~y to obtain the final worklna eauatione for the deseriotlon of selectton ratios tn stationary state eystemo, the principles under whleh model material transfer eysterns operate are rahtlvely simple. The movement of substances through membranes can be
_ .
L. F.N,Ms~
361 accountrd
far as the resultant of the actions of
conservattve and fricttonal forces. A dlstrlbutant speciea can mwe against Its concentratton gradtent If the net drag force, due to all other permeant species, i8 greater than the conservative farce acting on the species. The flow of dietrtbutant species relative to the barrier or the membrane ceases when the conservative force becomes equal to ihe net drag force. In such a case, the system 18 in a ~tatlonary state, which Is maintained as long 88 a steady supply ergy-yielding chemlcal molecules ts avstkable io the system, and the products of the chemleal reaction ape removed from the system at a steady rate. The asymmetry of the present system artses less from epecial properttes of the two barrierr; than from the fact that the system is open ta reactants from only one side or the other, just as in the general btological eysbm. However. If the sy’ltern has membranes of ereatly differ&t pr&tles, such ae high den& of fixed posttlve or negative charges, the functional asymmetry of the system with respect to the dfstritutants could be enhanced. It should be evident that the behavtor of the system does not depend upon the volume of the intermediate phase. Thus, if the tarrier in the system was tien to inpltie both membranes and the intesmedlate phases, then the system could be called a two-coinpartment system wtth chemical reacttons taktng place in a, ‘membrane’. The ayatem, as it stands. IS 111ustrat1veof the s1tuation, luck as edst 3 between the blood (phaas a), cells ourroundlna a kidnev tubule (the intermedhte phaee and the two m&nbranesj, and ths lumea at the ktdney tubule (phase ~3), etc. The model system la tltustrative of one Important point. Tha relatioh between the ion dlstrlhution pattern and the pr’oductlon rate of a reactant requires the slmulbmeoua determination of five varlablos. Aarelv in exoertments on ltving systems haa such a*task~ &en attempted, 11 the system IS not In a stattonaly atate. irnny mm’e measured variables are reaulred. even tf’a steady state approxtmatton la made,’ &xl the mzthemattcal compledty is much Increased. In the physical-chemical descrtptton, the concepts of drag forces, nr reststance coefftctents and chemical reaction rates, replace the more generally referred to ideas of ‘acttve ‘membrane pumps’. The major dlfflculty of a complete physical-chemical descrsptton of matvrlnl transfer through membranes in nmount of tnformarton which is-required to determine in an erpltctt faehiru, the magnitudes of all of the qnanttttse
transfer’or
Dment Bvstems Is the
multleom-
that cullectively constttute a ‘$ function. liowever, the magnitudes of the 0 functions can be determmed by sultable experiments. and these magnttudea for a variety of substances and membranes would greatly clarify our knowledge of what actually DOCUPB withln the membranes of biological matertal trawler systems.
ACKNOWLEDGEMENTS
paper is tpsed *aa
The work upon whtch this performed at the Brookhaven Nattonal Laboratory under the auspices of the US Atomic Energy C0~lUt0fdOn. The author wishes to thank Miss R.Butera and Mrs. tJ.Schnappauf far thetr technical asststanee in obtaining the quantitativedata reported herein. REFERENCES
MEMBRANES,
365
MATERIAL TRANSFER AND ENZYMES
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