Solute permeation through lipid membrane: Statistical Rate Theory approach

Transcription

1 /v x ANNALES UNIVERSITATIS MARIAE CURIE-SKŁODOWSKA LUBLIN POLONIA VOL. LXVI, 1, 2 SECTIO AA 2011 Slute permeatin thrugh lipid membrane: Statistical Rate Thery apprach W. Piasecki 1,* and R. Charmas 2 1 Department f Bichemistry, Faculty f Physical Educatin and Sprt in Biala Pdlaska The sef Pilsudski University f Physical Educatin in Warsa, Pland 2 The State Cllege f Cmputer Science and Business Administratin in Łma, Pland A ne mdel describing transprt acrss lipid membrane as develped based n the Statistical Rate Thery (SRT) f interfacial transfer. In ur calculatins e replaced lipid membrane ith ell-defined ctanl membrane. Accrding t SRT the rate f slute transfer acrss ater/ctanl interface depends n the rati f slute cncentratin in the bth phases and the ater/ctanl partitin cefficient. In the case f thick membranes and thick diffuse layers ur mdel predicts almst linear dependence f flux n cncentratin gradient. Hever, fr thin membranes and thin diffuse layers flux becmes nnlinear functin f cncentratin gradient, especially fr mre plar slutes and higher gradients. 1. INTRODUCTION It has been knn frm ver 100 years that the lipphilicity f cmpund is a majr factr gverning its transfer acrss lipid membrane [1, 2]. Because bilgical membranes are nt cnvenient t physicchemical studies, substitute substances hich prperties resemble lipid membrane ere frequently used in * Crrespnding authr: Wjciech Piasecki, Department f Bichemistry, Faculty f Physical Educatin and Sprt in Biala Pdlaska, The sef Pilsudski University f Physical Educatin in Warsa, ul. Akademicka 2, Biala Pdlaska, Pland, Tel , Fax , jciech.piasecki@af-bp.edu.pl

2 148 W. Piasecki and R. Charmas many experiments. The mst ppular substance f this type is ctanl. The cmmn measure f cmpund lipphilicity is ctanl/ater partitin cefficient P. Its values ere determined fr thusands f cmpunds used in pharmaclgical, txiclgical and csmetic research. The cmmn practice in these studies as the assumptin that the transprt f slutes thrugh lipid membrane as gverned by passive diffusin [3]. Hever there are evidences that interfacial transfer can be rate-limiting step during membrane permeatin [4, 5]. There are several experimental studies dealing ith the kinetics f the partitining beteen ater and ctanl. Their critical discussin can be fund in the revie ritten by Fisk et al. [6]. The rate cnstants f interfacial transprt can be determined by using different experimental techniques such as falling drps, hrizntal side-by-side diffusin cells, r rtating diffusin cell methd [7-9]. The interpretatin f the results f thse experiments as based n the linear nnequlibrium thermdynamics f membrane transprt [10]. In this apprach ne assumes that slute flux thrugh membrane [ml m -2 s -1 ]] linearly depends n the cncentratin difference n the bth sides f membrane. In this paper e uld like t apply the Statistical Rate Thery f interfacial transprt (SRT) t describe the kinetics f the partitining f slute beteen ctanl and ater. The SRT apprach as develped by Ward and it is based n the lcal thermdynamic equilibrium assumptin and relates the transfer kinetics ith the quantum mechanical prbability f crssing the interface by a single particle [11, 12]. The driving frce fr mlecule transfer in the SRT is the change in the micrscpic states distributins, hich is assciated ith the transfer entrpy (r equivalently ith the gradient f chemical ptential in bth phases). The SRT as successfully applied t describe the transprt rates f varius interfacial prcesses like liquid evapratin [13], gas adsrptin n slids [14], the permeatin f inic channels in bilgical membranes [15], the surfactant adsrptin at liquid interfaces [16], metal in adsrptin n bisrbents [17], and the relaxatin time f prtn adsrptin frm slutin n xides [18]. In the next sectin e ill develp the equatins based n the SRT apprach fr the rate f slute transprt acrss ctanl membrane. Next, e ill cmpare the SRT predictins ith the results given by the classical linear mdel fr t cmpunds sluble in ater but having different plarity, caffeine and butanl. 2. THEORY Accrding t the Statistical Rate Thery (SRT) the rate f interfacial transprt depends n the difference f slute chemical ptential in t phases (e.g. ctanl and ater phase) as flls,

3 Slute permeatin thrugh lipid membrane: Statistical Rate Thery apprach 149 = R ex μ μ μ μ exp exp (1) RT RT here is the net flux thrugh interface, and μ and ptentials f the slute in ctanl and ater phase, respectively. μ are the chemical R ex is the exchange rate beteen t phases at equilibrium. The SRT is based n the assumptin f lcal thermdynamic equilibrium, s e can use chemical ptentials in eq (1), hever in the SRT e d nt assume that investigated system is near equilibrium. Equatin (1) describes t transprt prcesses: flux frm ctanl t ater and frm ater t ctanl. In the equilibrium ttal flux is equal t zer. Starting frm the flling expressins fr slute chemical ptential in bth phases, μ = μ 0 + RT ln a (2a) μ = μ 0 + RT ln a (2b) 0 here μ and a dente standard chemical ptential and the activity f the 0 slute in ctanl phase, and μ and a dente the same quantities in ater phase, e can transfrm the equatin (1) int, = R ex 0 0 μ μ a exp RT a 0 0 μ μ a exp RT a (3) Taking int accunt the thermdynamic definitin f the partitin cefficient P, 0 0 μ μ P = exp RT and replacing activities by cncentratins e finally btain, (4) c Pc = Rex (5) Pc c

4 150 W. Piasecki and R. Charmas The equatin (5) describes transfer kinetics thrugh ctanl/ater interface accrding t the SRT. This equatin predicts that flux thrugh interface depends n slute cncentratin rati in bth phases, partitin cefficient P and the exchange rate at equilibrium beteen rganic and aqueus phase R ex. The Statistical Rate Thery predicts nnlinear dependence f flux n the cncentratins in bth phases. Accrding t the classical descriptin f slute transprt kinetics thrugh liquid interface the flux depends linearly n the cncentratins in the bth phases [6, 7]. = k c k c (6) here k is the rate cnstant fr crssing the interface frm ctanl t ater and k is the reverse. The rate cnstants k and k are related ith the partitin cefficient by the relatin P = k / k. The analysis presented abve is the starting pint fr kinetic descriptin f the system depicted in Figure 1. There are t cmpartments cntaining aqueus slutins separated by membrane hich pres are cmpletely filled ith ctanl. Such system is ften used t investigate skin permeability (here a slute is disslved in aqueus vehicle applied n the skin and penetrate thrugh stratum crneum t viable epidermis) [19, 20]. A slute hich crss the membrane has t vercme three diffusin barriers (t in aqueus phase and ne in ctanl) and t interfaces: ater-ctanl and ctanl-ater. When bulk phases are ell stirred the drp f cncentratin prfile can nly be bserved in thin layers adjacent t ater/ctanl bundary. The slute transprt in this layers is due t passive diffusin. In steady-state cnditin cncentratin gradients in diffuse layers are linear and the flling equatins based n the Fick s first la f diffusin can be ritten: D = ( c1b c 1 h ) (7a) 1 D0 = α ( c1 c2 h ) (7b) D = ( c2 c2 ) (7c) h 2 b

5 Slute permeatin thrugh lipid membrane: Statistical Rate Thery apprach 151 here α dentes prsity f membrane (ther symbls are defined in the descriptin f Figure 1). Membrane c 1 Diffuse Layer c 1b c 2 Water c 1 Octanl c 2 Water Cncentratin Diffuse Layer h h h c 2b Distance Fig. 1. T aqueus slutins separated by ctanl membrane. Bulk cncentratins f substance in t ater cmpartments are dented by c 1 b and c 2 b, the cncentratins at uter and inner membrane interfaces are dented by c 1 and c 2, and c 1 and c 2. h is the thickness f diffuse layer in ater and h the thickness f ctanl membrane, hich prsity (i.e. fractin f free vlume in membrane) is α. Accrding t the Statistical Rate Thery fluxes thrugh ater-ctanl and ctanl-ater interface can be expressed as, P c1 c1 = α Rex (8a) c1 Pc1

6 152 W. Piasecki and R. Charmas c 2 Pc2 = α Rex (8b) Pc2 c2 Applying the classical apprach e can rite [6], = α ( k c k c 1 ) (9a) 1 = α ( k c k c 2 ) (9b) 2 In steady-state all fluxes defined abve are equal t and (8ab) can be cmbined in ne expressin, st. The equatins (7abc) 2 αd P st 2 h + 4 1b 2b 2h αrex D αr st ( ) st c c ( c + c ) st = 1b 2b ex (10) The equatin (10) hich is based n the Statistical Rate Thery f interfacial transprt is nnlinear equatin ith reference t the flux and cncentratins and can be slved numerically. The equatins (7abc) and (9ab) can als be jined in ne expressin, hich is the standard frmula used t calculate flux acrss lipid membrane, P ( c1b c2b) st = (11) 2P h h D αd αk In equatin (11) driving frce fr slute transprt is the cncentratin difference in the numeratr f eq (11). The flux resistance expressed by the denminatr f eq (11) is divided int individual transprt resistances due t the diffusin thrugh unstirred diffuse layers in aqueus and rganic phases and the interfacial transfer. 3. RESULTS AND DISCUSSION S far the interfacial transfer cnstants have been determined by using the standard interfacial transprt mdel assuming linear dependence beteen slute cncentratin and its flux acrss membrane and applying the transprt resistance summatin apprach [4, 5, 7]. Because in literature e culd nly find kinetic parameters determined by using the abve apprach (ithut publicatin f ra

7 Slute permeatin thrugh lipid membrane: Statistical Rate Thery apprach 153 experimental data), e have decided t check the Statistical Rate Thery predictins in the flling ay. We decided t cmpare fluxes thrugh ctanl membrane (depicted in Figure 1) predicted by t appraches presented in the Thery sectin. We used the equatins (10) and (11) t calculate the flux as a functin f slute cncentratin in ne f aqueus phases. It as assumed that the vlumes f t aqueus phase are the same and equal t 100 cm 3. The ttal amunt f substance disslved in the system as equal t 10-2 mle. Additinally, it as assumed that in the starting pint f ur calculatins 90 % f slute as presented in ne cmpartment and the rest (10%) as disslved in the secnd. This initial separatin f slute beteen t aqueus cmpartments as necessary because the ur derivatin f transprt equatins as dne assuming steady-state cnditin. Calculatins ere perfrmed fr t cmpunds ith different lipphilicity: butanl and caffeine ith partitin cefficients P = and P = 0.68, respectively. These cmpunds ere chsen because accrding t Fisk et al. fr substances ith lg P bel 1.2 the kinetics f interfacial transfer shuld cntrl the verall rate f transprt [6]. The ther parameters used in calculatins ere taken frm Msher s paper [7] and cllected in Table 1. We chse Msher s data because they ere btained using side-by-side diffusin cells setup, in hich ne can cntrl interfacial area and hydrdynamics f the system. Additinally, similar setups are cmmercially available and used t determine the permeability f drugs and csmetic ingredients acrss the skin. Figure 2 depicts butanl and caffeine fluxes thrugh ctanl membrane ith thickness h = 100 μm and prsity α = as a functin f bulk cncentratin f slute in ne f aqueus phases. The vlume f ctanl in membrane is small enugh, s it can be neglected in the mass balance f slute (fr slutes ith lg P < 1.2, hich are cnsidered in this paper). As it flls frm Figure 2 the bth mdels predict identical butanl flux in the hle cncentratin range. Fr caffeine, hich is mre plar cmpund than butanl, e bserve deviatin f SRT flux frm linearity fr higher cncentratin gradient. We assumed that the thickness f unstirred diffusin layer in aqueus slutin as equal t h = 200 μm. This value is cnsistent ith Msher s estimatins. He fund that unstirred diffusin layer in aqueus slutin had thickness h = 176 ± 56 μm, and the thickness f stagnant diffusin layer in ctanl as equal t h = 97.6 ± 9.6 μm [7]. On the ther hand Miller, using falling drps technique, estimated that h = 9.1μm and h = 1.2 μm [21].

8 154 W. Piasecki and R. Charmas Tab. 1. The parameters values used in the calculatins. The data ere taken frm Msher s paper [7] r assumed by us. In calculatins it as assumed that t aqueus cmpartments had the same vlumes (100 cm 3 ) and ttal amunt f butanl r caffeine in the system as equal t 10-2 mles. Parameter Butanl Caffeine Octanl-ater partitin cefficient P Diffusin cnstant in ater D [ 10 cm s ] Diffusin cnstant in ctanl D [ 10 cm s ] Thickness f unstirred diffusin layer in ater h [μm] (assumed) 200 r r 20 Membrane thickness h [μm] (assumed) 100 r r 10 Membrane prsity (assumed) Water-t-ctanl rate cnstant k [ μm s ] Octanl-t-ater rate cnstant k [ μm s ] The exchange rates in equilibrium R ex [ ml m s ] determined in this paper by using the SRT fr ctanl membrane separating t aqueus slutins Revie f the literature reveals that the thickness f stagnant diffusin layers cnstitutes significant uncertainty in the determinatin f kinetic parameters fr ctanl/ater interface. Fr a lng time it had been believed that the slute transfer rate acrss lipid membrane is determined by the diffusin near the interface and ithin the membrane, hever the cases in hich the interfacial transfer kinetics determined the verall passage rate ere reprted. Fr example, Guy and Hnda fund significant energy barrier fr the interfacial transfer f methyl nictinate acrss ctanl membrane [5]. Next, Miller shed the experimental evidences that the interfacial transfer is a rate-limiting step during passive membrane permeatin [8], hever his interpretatin as immediately criticized by Hladky [22]. After a fe years, Miller respnded in a mre detailed experimental study, hich cnfirmed his initial suggestins, that the interfacial transfer can be the bttle-neck fr slute flux acrss membrane [4]. 5

9 Slute permeatin thrugh lipid membrane: Statistical Rate Thery apprach Flux [μml/cm 2 /s] Butanl C 1b [ml/dm 3 ] Flux [μml/cm 2 /s] Caffeine C 1b [ml/dm 3 ] Fig. 2. The fluxes predicted by the SRT using eq (10) (slid red line) and the classical apprach using eq (11) (dashed blue line) fr butanl and caffeine transprt acrss ctanl membrane as a functin f bulk cncentratin f slute in ne f aqueus phases. The parameters used in the calculatins ere cllected in Table 1. It as assumed that the thickness f unstirred diffusin layer in aqueus slutin is equal t h W =200μm, and membrane thickness is equal t h =100μm. The determined thickness f unstirred diffusin layer can change depending n applied experimental technique. In rtating diffusin cell experiments the

10 156 W. Piasecki and R. Charmas thickness f the diffusin layer depends n the square rt f the rtatin speed and can be calculated using Levich equatin [6]. Recently the applicatin f scanning in-sensitive micrelectrdes has enabled the determinatin f cncentratin prfiles f permeating in species [23]. Fr example fr sdium and calcium ins unstirred layer thickness as fund t be higher than 100 μ m. These discrepancies in the estimatins f unstirred diffusin layer thickness (abut ne rder f magnitude) may be f great imprtance fr slute flux acrss interface. When e assumed that ctanl membrane as ten times thinner than befre and diffuse layer thickness in ater as equal t h = 20 μm, e btained the fluxes depicted in Figure 3. Bth fr caffeine and butanl e bserve that SRT flux becmes nnlinear and greater than the flux predicted by eq (11) fr slute cncentratins abve 0.07 M. The analysis f all results fr caffeine and butanl suggests that interfacial transfer fr mre plar cmpunds (like caffeine) shuld exhibits mre nnlinear behaviur in cmparisn ith mre lipphilic cmpunds (like butanl). This is prbably the result f higher energetic barrier fr the transfer f plar cmpunds acrss ater/ctanl interface. 4. CONCLUSIONS The Statistical Rate Thery predicts that the slute flux acrss ctanl membrane is nnlinear functin f cncentratin gradient. The calculated rates f permeatin fr caffeine and butanl suggest that nnlinear behaviur shuld be mre visible fr mre plar cmpunds. Thick unstirred diffuse layers adjacent t the interface smth ut nnlinear dependence f slute flux predicted by SRT and therefre the results btained are similar t thse given by the linear mdel f membrane permeatin. The SRT frmalism, hich as develped here, is quite general and can be used t interpret slute flux acrss any lipid membrane (ctanl, silicn, isprpyl myristate etc.).

11 Slute permeatin thrugh lipid membrane: Statistical Rate Thery apprach Flux [μml/cm 2 /s] Butanl C 1b [ml/dm 3 ] Flux [μml/cm 2 /s] Caffeine C 1b [ml/dm 3 ] Fig. 3. The fluxes predicted by eq (10) (slid red line) and eq (11) (dashed blue line) fr butanl and caffeine transprt acrss ctanl membrane. The parameters used in the calculatins ere cllected in Table 1. It as assumed that the thickness f diffuse layer and membrane as ten times smaller than befre ( h = 20 μm and h = 10 μm ).

12 158 W. Piasecki and R. Charmas 5. REFERENCES [1] Q. Al-Aqati, Nature Cell Bilgy, 1999, [2] A. Missner and P. Phl, Chem. Phys. Chem. 2009, 10, [3].F. Nagle,.C. Mathai, M.L. Zeidel and S. Tristram-Nagle,. Gen. Physil. 2008, [4] D.M. Miller, Bichimica et Biphysica Acta 1991, 1065, [5] R.H. Guy and D.H. Hnda, Internatinal urnal f Pharmaceutics 1984, 19, [6] P.R. Fisk, M.G. Frd and P. Watsn. in (Cmptn, R.G. and Hancck, G., eds.) Applicatins f kinetic mdelling, Elsevier, Amsterdam 1999, pp [7] G.L. Msher, Pharmaceutical Research 1994, 11, [8] D.M. Miller, Bichimica et Biphysica Acta 1986, 856, [9] W.. Albery,.F. Burke, E.B. Leffler and. Hadgraft,. Chem. Sc. Faraday I 1976, 72, [10] S.T. Hang, AIChE urnal 2004, 50, [11] C.A. Ward, R.D. Findlay and M. Rizk, urnal f Chemical Physics 1982, 76, [12].A.W. Ellit and C.A. Ward. in (Rudzinski, W., Steele, W.A. and Zgrablich, G., eds.) Equilibria and dynamics f gas adsrptin n hetergeneus slid surfaces, Elsevier, Amsterdam 1997, pp [13] C.A. Ward and G. Fang, Physical Revie E 1999, 59, [14] W. Rudzinski and T. Panczyk, urnal f Physical Chemistry B 2000, 104, [15] F.K. Skinner, C.A. Ward and B.L. Bardakjian, Biphysical urnal 1993, 65, [16] P. Chen and A.W. Neumann, Cllids and Surfaces A 1998, 143, [17] W. Plazinski and W. Rudzinski, Langmuir 2009, 25, [18] W. Piasecki, urnal f Physical Chemistry B 2006, 110, [19]. Hadgraft. in (Cmptn, R.G. and Hancck, G., eds.) Applicatins f kinetic mdelling, Elsevier, Amsterdam 1999, pp [20] K.D. McCarley and A.L. Bunge, urnal f Pharmaceutical Sciences 2001, 90,) [21] D.M. Miller, Bichimica et Biphysica Acta 1991, 1065, [22] S.B. Hladky, Eurpean Biphysical urnal 1987, 15, [23] P. Phl, S.M. Saparv and Y.N. Antnenk, Biphysical urnal 1998, 75,