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Transport Processes in Responding Lipid Membranes: A Possible Mechanism for the pH Gradient in the Stratum Corneum Christoffer Åberg,* Håkan Wennerstro¨m, and Emma Sparr DiVision of Physical Chemistry 1, Chemical Center, Lund UniVersity, P.O. Box 124, SE-221 00 Lund, Sweden ReceiVed February 20, 2008. ReVised Manuscript ReceiVed April 24, 2008 The “acidic mantle” of the skin surface has been related to several essential functions of the skin, although the origin of the acidity is still obscure. In this paper, we investigate how different transport processes can influence the local proton concentration inside a membrane consisting of oriented lipid bilayers. This system is chosen as a simple model of the extracellular lipids in the upper layer of the skin, the stratum corneum. We present a theoretical model for diffusional transport over the membrane in the presence of an osmotic gradient and a gradient in CO2, taking into account the influence of these gradients on the lipid structure and the local electrostatics. We are also discussing the complications in applying the concept of pH to the stratum corneum. From this, we make the following conclusions: (i) The definition of pH in the stratum corneum is ambiguous, and thus, all statements regarding pH should always be related to a clear definition. (ii) A natural definition of pH in the stratum corneum can be proposed which takes into account local heterogeneity, local charges, and the fact that the stratum corneum is not in thermodynamical equilibrium. (iii) Diffusive transport across an oriented bilayer stack in the presence of an osmotic gradient and/or a gradient in CO2 can give rise to a substantial gradient in pH. (iv) The results from the simplified model can be correlated to experimental observations of pH in the stratum corneum.
1. Introduction The skin surface has an acidic character with pH varying between 4 and 6.1 This “acidic mantle” has been related to several essential functions of the skin, including the cutaneous antimicrobial defense,2,3 the permeability barrier formation,4,5 and the desquamation process.6,7 It has also been observed that several skin diseases are associated with a disturbance of the “normal” skin pH.8–10 In contrast to the acidic surface of the skin, the extracellular fluid inside the body is near neutral with pH around 7.4. The resulting gradient in pH between the skin surface and the tissue is considered to be located in the uppermost layer of the epidermis: the stratum corneum (SC).11 From both molecular and biological perspectives, it is clear that a change in pH of two units across the 10-20 µm that constitutes the SC is a dramatic event. Still, the origin of this gradient remains obscure. Several different suggestions, ranging from sweat secretion12 to active regulation * To whom correspondence should be addressed. Telephone: +46 46 222 8148. Fax: +46 46 222 4413. E-mail:
[email protected]. (1) Parra, J. L.; Paye, M.; The EEMCO Group. Skin Pharmacol. Appl. Skin Physiol. 2003, 16, 188. (2) Aly, R.; Maibach, H. I.; Rahman, R.; Shinefield, H. R.; Mandel, A. D. J. Infect. Dis. 1975, 131, 579–583. (3) Puhvel, S. M.; Reisner, R. M.; Amirian, D. A. J. InVest. Dermatol. 1975, 65, 525–531. (4) Behne, M. J.; Barry, N. P.; Hanson, K. M.; Aronchik, I.; Clegg, R. W.; Gratton, E.; Feingold, K.; Holleran, W. M.; Elias, P. M.; Mauro, T. M. J. InVest. Dermatol. 2003, 120, 998–1006. (5) Mauro, T.; Grayson, S.; Gao, W. N.; Man, M.-Q.; Kriehuber, E.; Behne, M.; Feingold, K. R.; Elias, P. M. Arch. Dermatol. Res. 1998, 290, 215–222. ¨ hman, H.; Vahlquist, A. J. InVest. Dermatol. 1998, 111, 674–677. (6) O (7) Ekholm, I. E.; Brattsand, M.; Egelrud, T. J. InVest. Dermatol. 2000, 114, 56–63. (8) Rothman, S. In Physiology and Biochemistry of the Skin; Rothman, S., Ed.; University of Chicago Press: Chicago, 2003; pp 221-232. (9) Farwanah, H.; Raith, K.; Neubert, R. H. H.; Wohlrab, J. Arch. Dermatol. Res. 2005, 296, 514–521. (10) Seidenari, S.; Giusti, G. Acta Derm.-Venereol. 1995, 75, 429–433. ¨ hman, H.; Vahlquist, A. Acta Derm.-Venereol. 1994, 74, 375–379. (11) O (12) Patterson, M. J.; Galloway, S. D. R.; Nimmo, M. A. Exp. Physiol. 2000, 85, 869–875.
by a sodium-hydrogen antiporter protein,13 have been given. The transport processes of other species across the SC could also influence the variation in the local H+ concentration. This forms the objective of the present study. The most important function of the SC is to serve as a barrier, protecting the body from uncontrolled water loss and uptake of hazardous chemicals from the environment. It is, however, important to bear in mind that even though the SC has a very low permeability, it is not totally tight. As an example, there is a non-negligible transepidermal water loss (TEWL) of about 100-150 mL per day and square meter of skin surface through the intact healthy skin.14 Moreover, respiratory gases such as O2 and CO2 pass through the skin to some extent.15,16 This illustrates that there are several simultaneous transport processes occurring across the SC. The gradients in, for example, water, temperature, and metabolic gases and in other species may further affect the molecular organization in the barrier membrane. The SC consists of 10-15 layers of dead flattened keratinfilled cells (corneocytes) embedded in a matrix of stacked lipid lamellae in an array similar to “bricks and mortar”.17,18 The main components of the SC extracellular lipids are ceramides, free fatty acids, and cholesterol.19 The extracellular lipids constitute the sole continuous regions of the SC, and molecules passing the skin barrier must be transported through them.20–23 (13) Behne, M.; Oda, Y.; Murata, S.; Holleran, W. M.; Mauro, T. M. J. InVest. Dermatol. 2000, 114, 797. (14) Nilsson, G. E. Medical Dissertation, Linko¨ping University, Linko¨ping, Sweden, 1977. (15) Hatcher, M. E.; Plachy, W. Z. Biochim. Biophys. Acta 1993, 1149, 73–78. (16) Stu¨cker, M.; Struk, A.; Altmeyer, P.; Herde, M.; Baumga¨rtl, H.; Lu¨bbers, D. W. J. Physiol. 2002, 538, 985–994. (17) Elias, P. M. J. InVest. Dermatol. 1983, 80, S44-S49. (18) Michaels, A. S.; Chandrasekaran, S. K.; Shaw, J. E. AIChE J. 1975, 21, 985–996. (19) Wertz, P.; Norle´n, L. In Skin, Hair, and Nails. Structure and Function; Forslind, B., Lindberg, M., Eds.; Marcel Dekker, Inc: New York, Basel, 2004; pp 85-106. (20) Simonetti, O.; Kempenaar, J. A.; Ponec, M.; Hoogstraate, A. J.; Bialik, W.; Schrijvers, A. H. G. J.; Bodde´, H. E. Arch. Dermatol. Res. 1995, 287, 465– 473.
10.1021/la800543r CCC: $40.75 2008 American Chemical Society Published on Web 07/12/2008
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In the present work, we describe a theoretical model to couple the diffusional transport of water and CO2 to the variation in pH in a model membrane system composed of stacked oriented lipid bilayers. The bilayer stack represents the SC extracellular lipid matrix in the absence of the corneocytes. The lipid composition was chosen to include also titrating fatty acids, as these are important components of the SC extracellular lipids. One important part of the model is the osmotic gradient between the dry ambient atmosphere and the water-rich environment inside the body. In fact, this gradient is located across the SC.24 The osmotic gradient drives diffusive transport of water, and it can influence the lipid structure along the membrane by heterogeneous swelling or induced phase changes.25–27 Heterogeneous swelling implies that the local electrostatic potential between the charged bilayers varies along the stack, which affects the local pH. Another transport process that is treated in the present study is that of CO2, as the gradient in CO2 can affect the concentration of HCO3and CO32- along the membrane and thus influence the local pH. From this model, we are able to draw quantitative conclusions based on a set of parameters that are estimated to mimic the conditions of the SC. The paper is organized as follows: First, we describe the simplified model of a bilayer stack representing the SC extracellular lipids. We then discuss complications when it comes to generalizing the concept of pH, that is defined for a bulk system in thermodynamical equilibrium, to a system such as the SC, which is locally heterogeneous and charged and not at equilibrium. A natural definition is given. This is followed by a description of the model where we treat (i) the effect of an osmotic gradient, (ii) the effect of a gradient in CO2, and (iii) the effect of coupled osmotic and CO2 gradients, on pH. Finally, we discuss different ways of measuring pH in the SC in relation to the definition given and relate some experimental observations to the model.
2. The Bilayer Stack: A Simple Model for the SC Lipids Consider N parallel ordered bilayers, representing the extracellular lamellar SC lipids (Figure 1). The bilayers are assumed to contain titrating fatty acids, and the lipid composition is considered to be the same in all bilayers. On the upper side, the bilayer stack is exposed to the ambient air with a specified relative humidity, RH. On the lower side, the stack is in contact with the water-rich environment corresponding to physiological saline (0.16 M NaCl). These boundary conditions can be expressed in terms of the osmotic pressure, Π, or, equivalently, the water chemical potential, µw
kBT ln RH ) µw - µw0 ) -VwΠ
(1)
where Vw is the molecular volume of water, kB is the Boltzmann constant, T is the absolute temperature, and µw0 is the water chemical potential of bulk water. The first equality assumes ideal gas behavior of humid air, while the second is exact only for incompressible fluids. When discussing reaction equilibria, also (21) Potts, R. O.; Francouer, M. L. Proc. Natl. Acad. Sci. U.S.A. 1990, 87, 3871–3873. (22) Bodde´, H. E.; van den Brink, I.; Koerten, H. K.; de Hann, F. H. N. J. Controlled Release 1991, 15, 227–236. (23) Barry, B. W. Eur. J. Pharm. Sci. 2001, 14, 101–114. (24) Warner, R. R.; Myers, M. C.; Taylor, D. A. J. InVest. Dermatol. 1988, 90, 218–224. (25) Sparr, E.; Wennerstro¨m, H. Colloids Surf., B 2000, 19, 103–116. (26) Sparr, E.; Wennerstro¨m, H. Biophys. J. 2001, 81, 1014–1028. (27) Costa-Balogh, F. O.; Åberg, C.; Sousa, J. J. S.; Sparr, E. Langmuir 2005, 21, 10307–10310.
Figure 1. (a) An oriented stack of bilayers, representing the extracellular lipid matrix of the SC, exposed to a gradient in osmotic pressure and CO2. The osmotic gradient is directed outward, corresponding to that the water-rich environment inside the body has a lower osmotic pressure than the (generally) dry ambient atmosphere. The gradient in CO2 has the same direction, since the partial pressure of CO2 in typical ambient air, pCO2 ) 38 Pa, corresponds to a concentration of 12.9 µM, significantly lower than the physiological value of [CO2] ) 1.79 mM. (b) Schematic illustration of two bilayers separated by a distance h. In the modeling, the distribution of ions is found taking into account the chemical equilibrium of eq 2 and the electrostatic interaction with the surface. Only H+ ions are shown for clarity.
the concept of water activity, aw, is useful, and this is simply equal to the relative humidity RH. We can envision rather extreme osmotic gradients over this bilayer stack representing the SC extracellular lipids. The physiological saline solution corresponds to 99.4% RH, whereas the ambient atmosphere typically has a RH of 40-90%. The local interbilayer separation is strongly dependent on the local osmotic pressure, so a gradient in osmotic pressure implies a heterogeneous swelling of the bilayer stack.25 We also consider a gradient in CO2 over the bilayer stack. The partial pressure of CO2 in ambient air is typically 38 Pa, whereas physiological conditions correspond to around 5 kPa. Assuming the same solubility as in water, this corresponds to solution concentrations of 12.9 µM and 1.79 mM, giving the boundary conditions on the upper and lower side of the stack, respectively. We assume a high thermal conduction so that the temperature T ) 304 K (generally considered as normal skin temperature) is uniform throughout the system. This assumption is relevant for comparison with in vitro experiments on the SC. However, for a complete description of skin in vivo, a gradient in temperature should also be included in the model. In the numerical calculations, we have chosen the parameters to be relevant to the SC. In the extracellular SC lipids, the surface charge is mainly due to free fatty acids, which can undergo dissociation
R-COOH h R-COO- + H+
(2)
The (area) density of ionizable groups, S, was set to 1 per 100 Å2, corresponding to that approximately 25% of the alkyl chains in the bilayer are free fatty acids, the (intrinsic)28 pKa of the free fatty acids was set to pKa ) 4.8,29,30 and the bilayer thickness was set to to l ) 56 Å, which is reasonable for the long-chain (28) Engblom, J.; Engstro¨m, S.; Jo¨nsson, B. J. Controlled Release 1998, 52, 271–280.
Transport Processes in Responding Lipid Membranes Table 1. Common Parameters [I0] pH0 1/S T r pKa Π0 λ H l N Di kCO2
0.16 M 7.4 100 Å2 304 K 80 4.8 29,30 109 Pa 33 2 Å 33 6.0 × 10-21J 34 56 Å 31,32 1000 2 × 10-9 m2/s 35 1.7436
ion concentration in bulk pH in bulk density of ionizable groups temperature dielectric constant of water pKa of free fatty acids magnitude of repulsive short-ranged force decay length of short-ranged force Hamaker constant bilayer thickness number of bilayers diffusion coefficient CO2 partition coefficient
SC lipids.31,32 The number of bilayers was set to N ) 1000. All parameters are summarized in Table 1. It should be noted that the choice of the parameters is not so crucial in the sense that variations in these only cause minor quantitative differences of the calculated results and does not change the overall general qualitative trends.
3. Definition of pH in the SC The pH of a system is a direct measure of the chemical potential of protons. However, there are two basic problems of applying the fundamental definition
µi ≡
( ) ∂G ∂ni
T,p,nj
of the chemical potential to protons. One difficulty arises from the fact that H+ never occurs as an isolated species in solution. In water, one rather has clusters such as H3O+ and H5O2+ and one has to be careful in the definition of nH+. A more serious problem arises from the fact that the protons are charged. There exists a discussion of the relevance of single ion activities that goes back to the early days of solution thermodynamics.37,38 The challenge is to account for the electrostatic interactions that can be of macroscopic range. For bulk solutions, one has arrived at practical solutions to these problems and pH is routinely measured in such systems. In the SC, protons experience an environment that is locally heterogeneous with respect to both charge distribution and shortrange interactions. Let us illustrate the complications arising from these circumstances by considering an equilibrium lyotropic lamellar liquid crystal formed by charged amphiphiles. At equilibrium, the chemical potential and thus the pH are constant throughout the system. There are on the other hand strong variations in the local proton concentration. This is caused both by interactions with the charged amphiphile headgroups and by the fact that protons avoid the apolar regions. It is not trivial to measure pH of such a system. A conceivable approach would be to equilibrate the system with a small aqueous volume. Ions and protons can then exchange between the system to establish (29) Jukes, T. H.; Schmidt, C. L. A. J. Biol. Chem. 1935, 110, 9–16. (30) White, J. R. J. Am. Chem. Soc. 1950, 72, 1859–1860. (31) Ohta, N.; Ban, S.; Tanaka, H.; Nakata, S.; Hatta, I. Chem. Phys. Lipids 2003, 123, 1–8. (32) Bouwstra, J. A.; Gooris, G. S.; van der Spek, J. A.; Bras, W. J. InVest. Dermatol. 1991, 97, 1005–1012. (33) Rand, R. P.; Parsegian, V. A. Biochim. Biophys. Acta 1989, 988(3), 351– 376. (34) Israelachvili, J. Intermolecular and Surface Forces, 2nd ed; Academic Press Limited: London, 1992. (35) Cussler, E. L. Diffusion: Mass Transfer in Fluid Systems; Cambridge University Press: Cambridge, 1997. (36) Battino, R.; Evans, F.; Danforth, W. J. Am. Oil Chem. Soc. 1968, 45, 830–833. (37) Pethica, B. A. Phys. Chem. Chem. Phys. 2007, 9, 6253–6262. (38) Guggenheim, E. A. J. Phys. Chem. 1929, 33, 842–849.
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an equlibrium with respect to these species. The resulting pH in the bulk water phase is then interpreted as the pH of the lamellar system. It is important to realize that the water is not at equlibrium, and in practice this means that a measurement has to be made before extensive water exchange has taken place. For a theoretical calculation of the pH, one can make use of the Poisson-Boltzmann equation to account for the electrostatic effects. The chemical potential of ions can then be calculated from the concentration in the midplane of the aqueous layer in the lamella39
µi ) µi0 + kBT ln ci(midplane) + zieΦ(midplane)
(3)
where ci(midplane) and Φ(midplane) are the concentration and electrostatic potential at the midplane, respectively, and zie is the charge of the ion. The SC is normally not an equilibrium system with a gradient in the thermodynamic variables in the perpendicular direction. We will use the assumption that ions do not penetrate across lipid bilayers, while we expect that the equilibration is fast within the respective aqueous layers. For such a case, we evaluate the local pH using eq 3. Thus, we see the local pH as that of a bulk aqueous phase that provides an equilibrium with the lamellar aqueous layer with respect to protons and other ions.
4. Osmotic Gradient When a bilayer stack is exposed to an osmotic gradient, it responds by heterogeneous swelling.25,26 This will also affect the local H+ concentration between the bilayers, since the local electrostatic potential changes. It is therefore crucial to specify what forces contribute to the local osmotic pressure in the bilayer system. 4.1. Interbilayer Forces. We take into account three different contributions to the local osmotic pressure between two bilayers
Π(h) ) Πedl(h) + Πrep(h) + Πdisp(h)
(4)
Here, Πedl is a repulsive force due to the overlapping electrical double layers of the two charged interfaces.40 Πrep is the repulsive short-ranged exponentially decaying force (sometimes called “hydration force”) present in all charged and uncharged bilayer systems33,41
Πrep(h) ) Π0e-h⁄λ
(5)
where Π0 is the force at zero separation, and λ is a decay length. The attractive dispersion interaction, Πdisp, is given by
Πdisp(h) ) -
(
H 1 2 1 + 6π h3 (h + l)3 (h + 2l)3
)
(6)
where H is the Hamaker constant and l is the bilayer thickness.42 In a mean-field approach, the electrostatic interaction, Πedl, is conventionally modeled by solving the Poisson-Boltzmann equation for the electrostatic potential. We will assume only monovalent ions, for which the mean-field approximation is a good one.40 For the case of two bilayers, one considers two planar parallel uniformly charged surfaces, representing the “lipid headgroup plane” on each side of the aqueous layer, with a surface charge σ separated a distance h. A solution is known for the case where the surface charge is not known a priori but is determined (39) Jo¨nsson, B.; Wennerstro¨m, H. J. Colloid Interface Sci. 1981, 80, 482– 496. (40) Evans, D. F.; Wennerstro¨m, H. The Colloidal Domain s where Physics, Chemistry, Biology and Technology Meet; Wiley-VCH: New York, 1999. (41) Israelachvili, J. N.; Wennerstro¨m, H. J. Phys. Chem. 1992, 96, 520–531. (42) Parsegian, V. A.; Fuller, N.; Rand, R. P. Proc. Natl. Acad. Sci. U.S.A. 1979, 76(6), 2750–2754.
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through a surface equilibrium like in eq 2.43 Though this specific case is useful, we also need the solution of related, but different, cases. Assume that the intervening aqueous solution is in equilibrium with a bulk containing (for simplicity) only monovalent ions. We let Φ ) 0 in the bulk and denote the total concentration of cations (anions) in the bulk by [I]. In particular, some of the cations are H+ with a bulk concentration [H+]. For this case, the Poisson-Boltzmann equation simplifies to
r0
d2Φ ) e[I](exp(eΦ(x) ⁄ kBT) - exp(-eΦ(x) ⁄ kBT)) dx2
(7)
where r is the dielectric constant of water and 0 is the dielectric permittivity of a vacuum. Placing the origin of the coordinate system at the midplane between the surfaces, the boundary conditions are, by symmetry
|
dΦ )0 dx x)0
(8)
and by Gauss’s theorem and electroneutrality
r0
|
dΦ eR )σ)dx x)h⁄2 S
Figure 2. Variation of H+ concentration with osmotic pressure between two bilayers. The higher osmotic pressure, the smaller the separation between the bilayers. The H+ concentration at the midplane between bilayers is heavily affected (dashed curve), showing a large increase as the bilayers are pushed together with increased osmotic pressure. In contrast, the H+ concentration at the bilayer interface is largely unaffacted (dotted curve). The system is assumed to be in equilibrium with a bulk at physiological conditions, so the pH of the system is constant regardless of osmotic pressure (solid line).
(9)
In the latter equation, R is the (unknown) degree of ionization of the surface charge, which must fulfill the chemical equilibrium
Ka )
R c (h ⁄ 2) 1 - R H+
(10)
where cH+(h/2) ) [H+] exp(-eΦ(h/2)/kBT) is the H+ concentration at the surface. Here, Ka is the acidity constant of the free fatty acids in the bilayer, which is generally different from that in solution.28,44 We have neglected this effect and put pKa ) 4.8.29,30 Equations 7–10 define the problem of finding Φ(x) uniquely. A solution can be found in terms of elliptic functions.43 Once the electrostatic potential is known, we can find other quantities of interest, such as, for example, the ion concentration profile between the surfaces. In particular, the contribution to the osmotic pressure is given by
Πedl(h) ) [I]kBT(exp(eΦ(0) ⁄ kBT) + exp(-eΦ(0) ⁄ kBT)) (11) With the three different contributions, eqs 5, 6, and 11, specified, the dependence of the total osmotic pressure, eq 4, on the bilayer separation, h, is known. With the Poisson-Boltzmann theory, also the ion distribution is known as a function of osmotic pressure. In the numerical evalutation of this model, we have chosen the parameters to be relevant to the SC (see Table 1): r was put to 80; for the short-range repulsive force, we used Π0 ) 109 Pa and λ ) 2 Å as typical values;33 and the Hamaker constant was set to H ) 6.0 × 10-21 J, typical for hydrocarbon-water systems.34 (The subscript 0 has been added on [I] and pH in Table 1 for later convenience.) Figure 2 illustrates the relation between the H+ concentration distribution and the osmotic pressure (RH) for the two extreme cases: the midplane and the surface. At positions between the midplane and surface, the concentration profile lies between these two curves. The figure demonstrates clear differences between H+ concentrations in bulk, at the midplane and at the surface. We can understand the origin of these differences in terms of the local electrostatic potential, as illustrated in Figure 3. (43) Ninham, B. W.; Parsegian, V. A. J. Theor. Biol. 1971, 31(3), 405–428. (44) Chan, D.; Perram, J. W.; White, L. R.; Healy, T. W. J. Chem. Soc., Faraday Trans. 1 1975, 71, 1046–1057.
Figure 3. Schematic illustration of the electrostatic potential (solid curve) between bilayers, compared to bulk (dashed line). (a) For large separations, the electrostatic potential at the midplane is almost the same as that in bulk. (b) For smaller separations, significant changes can be seen at the midplane, corresponding to a large change in ion concentration. Note that the potential as well as its derivative at the wall are largely unaffected by separation, implying that the concentration at the wall is constant.
4.2. Negligible Ion Transport. When discussing the swelling of the bilayer stack due to the osmotic gradient, the problem solved above is not applicable in general. This is because it was assumed that the aqueous solution between the two bilayers was in equilibrium with a certain bulk solution. In the bilayer stack, it is only the layer on the lower side of the stack that is in equilibrium with the physiological bulk. To simplify matters, we will consider the limit of negligible ion transport. This is justified by the very low permeability of a lipid bilayer to ions.45 In this limit, the number of ions within each aqueous layer is constant in time but unknown. However, the SC is a part of a living system, and there are other transport process going on. In particular, one process that could be considered important is the desquamation of the skin. In the present model, one can imagine that the bottom interbilayer solution is in equilibrium with a physiological bulk. This layer then moves toward the skin surface and loses its contact with the bulk. In the limit of negligible ion transport, this implies that the number of ions within each aqueous layer is the same along the stack, except for titration events within the layers as in eq 2. Furthermore, the actual number of ions is obtained from an assumed equilibrium with a physiological bulk, representing the initial contact at the lower side of the stack. (45) Hauser, H.; Phillips, M. C.; Stubbs, M. Nature 1972, 239, 342–344.
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Figure 4. Variation of H+ concentration with osmotic pressure between two bilayers, under the assumption that the number of ions in between the layers is constant. With this assumption, the pH of the system is allowed to respond to an increased osmotic pressure (solid curve) and shows a similar trend as the H+ concentration at the midplane (dashed curve). The H+ concentration at the bilayer is essentially unaffected (dotted curve). The results can be interpreted as the variation in pH and H+ concentration along the bilayer stack, for the case that ion transport through the bilayers is negligible.
The mathematical formulation of this problem starts by finding the total number of ions within an aqueous layer for a certain bilayer system that is in contact with a bulk. The concentration profile in the layer can be found from the model described in section 4.1, and to find the total number of ions (per area) we integrate this profile over the spatial coordinate, x. When it comes to the cations, we also have to add the H+ that are chemically bound to the surface:
ctot - ) [I0]
∫0h ⁄2 exp(eΦ0 ⁄ kBT) dx 0
(12a)
R0 h0⁄2 exp(-eΦ0 ⁄ kBT) dx + ctot + ) [I0] 0 S R0 h0⁄2 exp(-eΦ0 ⁄ kBT) dx + cHtot+ ) [H+ 0] 0 S
∫
∫
(12b)
Dw(x) dµw cw(x) kBT dx
where Dw(x) is the local water diffusion coefficient, cw(x) is the water concentration, and µw(x) is the water chemical potential. The gradient in water chemical potential across the aqueous layers is negligible compared to the gradient over the lipid layers, and we further assume ideal solution behavior so that
cw(x) ∝ exp(∆µw ⁄ kBT) where ∆µw ≡ µw - µ0w is the difference in water chemical potential from the standard state. Under the condition of steady-state flux, one can then show that the position x as a function of water chemical potential is given by25
x(∆µw) )
N × ∆µwN ⁄ kBT ( kBT e - e∆µw0 ⁄ kBT)
∫∆µ∆µ
w
(12c)
∫0h⁄2 exp(eΦ ⁄ kBT) dx ) ctot-
(13a)
[I]
∫0h⁄2 exp(-eΦ ⁄ kBT) dx + RS ) ctot+
(13b)
[H+]
∫0h⁄2 exp(-eΦ ⁄ kBT) dx + RS ) cHtot
(13c)
+
Jw ) -
w0
where Φ0(x) and R0 refer to the solution we find when assuming equilibrium. As above, [I0] and [H+ 0 ] are the total concentration tot tot of ions and H+, respectively, in bulk (Table 1). Here, c, c+ , tot+ and cH denote the number (per area) of anions, cations (including H+), and H+, respectively. Equation 12 holds for the layer on the lower side of the stack that is in equilibrium with the bulk. For subsequent layers, the condition of equilibrium with a bulk is instead replaced by the conditions
[I]
that for this case there is a gradient in actual pH (solid curve) and not only in H+ concentration. This occurs since we have abandoned the assumption of an equilibrium with a bulk. In fact, the gradient is rather large and shows qualitatively similar behavior as the H+ concentration at the midplane (dashed curve). 4.3. Variation in H+ Concentration along the Bilayer Stack. Figure 4 can be used to predict the pH (or H+ concentration at the bilayer interface or midplane) at any position along the bilayer stack, once the osmotic pressure at that position is known. In particular, for a given ambient RH, the pH at the upper side of the bilayer stack, corresponding to the skin surface pH, can be found from the ordinate corresponding to the ambient RH abscissa. To find the spatial profile, we must find the variation of osmotic pressure with position. This is given by the water transport across the bilayer stack. To obtain this, we write the water flux, Jw, with a generalized form of Fick’s first law40
Here, [I] and [H+] are not known a priori but rather are found from the full solution of eqs 7–10, supplemented by the extra conditions in eq 13. For a given separation h, [I] and [H+] correspond to concentrations in a bulk that would be in equilibrium with the bilayer system. In other words, this enables us to find what we have interpreted as the pH of this system. An efficient algorithm for a numerical solution of this problem can be found in the Supporting Information. Figure 4 shows the variations in H+ concentration with osmotic pressure (RH) in the limit of negligible ion transport. We observe
∆µw′⁄ kBT
(l + h(∆µw ′))e
d∆µw ′ (14)
where ∆µwN and ∆µw0 are the water chemical potential at the upper and lower sides of the bilayer stack, respectively, and N is the total number of layers. Figure 5 shows the spatial variation of H+ concentration across the bilayer stack for three different osmotic gradients. The osmotic pressure on the lower side of the stack is kept constant at Π ) 0.81 MPa, corresponding to physiological saline (0.16 M NaCl); at the upper side, the osmotic pressure is varied, corresponding to different ambient RH. For a negligible gradient, the bilayer stack would be homogeneously swollen, and the local H+ concentration would be the same within each aqueous layer. For a small gradient (Figure 5a), corresponding to a humid atmosphere, the bilayer stack responds by heterogeneous swelling. This is due to a continuous difference in the local osmotic pressure along the bilayer stack. As illustrated in Figure 3 above, a change in separation also changes the local electrostatic potential, leading to a redistribution of the constant amount of ions. This results in a higher concentration at the midplane (dashed curve). This effect increases with increasing osmotic gradient (Figure 5b and c), corresponding to progressively lower RH, since the separation between bilayers on the upper side decreases. The actual pH is similarly, though less, affected along the bilayer stack (solid curve). The H+ concentration at the bilayer interface is basically unaffected (dotted curve). The position where the curves end corresponds to the total thickness of the bilayer stack, and a comparison of parts a-c in Figure 5 shows that the bilayer stack is less swollen as the osmotic gradient is increased.
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5. Carbon Dioxide Gradient The gradient in CO2 can also affect the H+ concentration profile along the bilayer stack. Since CO2 in water forms carbonate and bicarbonate ions, a gradient in CO2 necessarily gives a gradient in H+ concentration. To describe this, we present a model of diffusive transport of CO2 across a homogeneous bilayer stack, coupled with chemical reactions. At ambient, as well as physiological, conditions the dominant reaction of the CO2–water system is
CO2 + H2O h HCO3- + H+
(15)
From the stochiometry of this reaction, it follows that the production per unit volume, si, satisfies the relation
sCO2 ) -sHCO3- ) -sH+ If we include a production term in the continuity equation, we find at steady-state
si )
dJi dx
and thus, the following relation must hold between the fluxes
d (J + Ji) ) 0 dx CO2
i ) HCO3-, H+
(16)
If the reaction rate is sufficiently fast compared to diffusion, chemical equilibrium will be maintained at all times, so that
Keq )
cHCO3-cH+ cCO2aw
(17)
where Keq is the equilibrium constant for the reaction, aw ≡ RH is the water activity, and we have assumed that the activities of CO2 and the ions can be approximated by their concentrations. It is then convenient to work in terms of concentrations and write Fick’s first law in the aqueous layers in the form
Ji ) -Diw
dciw dx
(18)
where Dwi and cwi are the diffusion coefficient and concentration, respectively, in water. In the bilayers, we write it as
Ji ) -Dil
dcil dx
(19)
where instead Dli and cli are the diffusion coefficient and concentration, respectively, in the bilayer. Consider the limit of negligible ion transport through the bilayer. In this limit, Fick’s first law, eqs 18 and 19, inserted in eq 16 reads w DCO 2
w d2cCO 2
dx2
+ Diw l DCO 2
d2ciw dx2
l dcCO 2
dx
)0
i ) HCO3-, H+ (20)
)0
(21)
for the water and lipid layer, respectively. Furthermore, we assume that the number of other ions between the bilayers is constant and given by a bilayer system in equilibrium with a physiological bulk (just as we did for the osmotic gradient for the case of negligible ion transport, section 4.2). However, when considering the CO2 gradient, we have an additional source of ions from the reaction in eq 15. The crucial point that enables us to still make use of the assumption of constant
Figure 5. H+ concentration as a function of spatial coordinate, x, along a stack of bilayers exposed to an osmotic gradient. At the lower side of the stack (x ) 0), the osmotic pressure is kept constant, corresponding to physiological conditions; at the upper side, it corresponds to an ambient atmosphere with RH ) (a) 99%, (b) 90%, and (c) 60%. Transport of ions between bilayers was neglected. A large increase in H+ concentration at the midplane between each pair of bilayers (dashed curve) occurs, mainly in the lower side of the stack. This leads to a concomitant decrease in pH (solid curve). A lowering of the pH at the upper side of the stack, corresponding to the skin surface, is seen for an increasing osmotic gradient.
number of ions between the bilayers is the fact that the dominant reaction, eq 15, always produces the ions in pairs. Thus, the total number of H+ within the bilayers, excluding the ones that were already present at the lower side of the bilayer stack, must be equal to the corresponding number for HCO3-. If we assume w w w similar diffusion coefficients in water,DHCO - ) DH+ ≡ D , this 3 condition results in w w/ w w/ cHCO -(x) - cHCO - ) cH+(x) - cH+ 3 3
(22)
w* w* where cHCO - and cH+ are the concentrations that come from 3 being in equilibrium with a bulk at the lower side of the bilayer stack.
The solution in the two regions must satisfy the continuity conditions
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w l lim kCO2cCO ) lim cCO 2 2
(23a)
lim (JCO2 + Ji) ) lim JCO2
(23b)
xfhxfh-
xfh+
xfh+
where x ) h marks the interface between the two regions. Here, kCO2 is the partition coefficient between a bilayer phase and water. The second condition follows in the limit of negligible ion transport through the bilayer. The physical interpretation of this is that although no ions can pass through the bilayer, there can still be a finite flux at the interface due to a production of CO2 at this point. The CO2 then readily passes the bilayer. The problem defined by eqs 17 and 20–23 is solved by providing two boundary conditions in cCO2 at each side of the bilayer stack; its numerical solution is discussed in the Supporting Information. Figure 6 presents the result of this model for the H+ concentration variation along the bilayer stack. The calculations were performed for cCO2 ) 1.79 mM and 12.9 µM at the lower and upper side, respectively, corresponding to physiological (pCO2 w* ) 5 kPa) and typical ambient conditions (pCO2 ) 38 Pa). cHCO 3 w* + and cH were set to physiological values. In the absence of an osmotic gradient, the bilayer stack is homogeneously swollen, and we used a constant interbilayer separation h ) 81 Å (corresponding to physiological osmotic pressure). As typical diffusion coefficients in the liquid phase, we used Di ) 2 × 10-9 m2/s for all species35 and set the CO2 partition coefficient to kCO2 ) 1.7436 (Table 1). The gradient in CO2 is rather large, amounting to a more than 2 orders of magnitude difference across the bilayer stack, with the upper side having the lower concentration. This leads to a concomitant decrease in both HCO3- and H+ at the upper side. However, due to the concentration condition, eq 22, and the high amount of HCO3- inside the body, the effect is most clearly seen for H+. Figure 6 shows this effect, where, indeed, the decrease in cH+ across the stack is larger than 2 orders of magnitude. We note that the equilibrium constant Keq ) 4.27 × 10-7 M of eq 17 is rather small. This implies that transport through the aqueous phase is completely dominated by CO2. Neglecting the ion transport also through the aqueous phase is, in fact, an excellent approximation, agreeing completely with the result shown in Figure 6. This will prove useful in section 6 below.
6. Coupled Osmotic and Carbon Dioxide Gradients We have shown that the gradients in osmotic pressure and CO2 affect the H+ concentration gradient across a bilayer stack. In the following, we will couple these two gradients and study their combined effects on the H+ concentration. 6.1. Variation of Bilayer Forces with Osmotic Pressure and Carbon Dioxide Concentration. Consider a bilayer stack in the presence of both an osmotic gradient and a gradient in CO2. Both gradients will influence the interaction between bilayers and lead to a heterogeneous swelling. We consider the case of negligible ion transport. With only an osmotic gradient present (section 4), this case implies that the number of ions within each aqueous layer is constant (eq 13). When the bilayer stack is exposed also to a CO2 gradient, only the number of “inert” ions (i.e., all ions except H+ and HCO3-) can be assumed constant. Denoting the tot tot number of inert anions and cations (per area) by cand c+ , respectively, we write the requirement that these stay constant as
∫0h⁄2 exp(eΦ ⁄ kBT) dx ) ctoth⁄2 [I+]∫0 exp(-eΦ ⁄ kBT) dx ) ctot + [I-]
(24a) (24b)
where [I-] and [I+] are the unknown concentrations of the bulk solution the system is in (hypothetical) equilibrium with. In general, [I-] * [I+]. Hence, charge neutrality
[I+] + [H+] ) [I-] + [HCO3-] ≡ [I]
(25)
is an independent condition. As already stated, we further assume chemical equilibrium
Keq )
[HCO3-][H+] awcCO2
(26)
to hold. The water activity aw ) RH is given in terms of the osmotic pressure by eq 1 and is directly related to the interlamellar forces, eq 4. The unknown quantities depend on each other in a rather intricate way, but the problem is well-posed and can be solved numerically. An efficient algorithm for this is presented in the Supporting Information. Figure 7 shows the calculated results for the variation in H+ concentration with osmotic pressure for a bilayer system in contact with typical ambient CO2 partial pressure. The qualitative trends are similar to those obtained for the situation when only the osmotic gradient is considered (Figure 4). The effect of the CO2 dissociation equilibrium is a less acidic system. The H+ bulk and midplane concentrations vary with the osmotic pressure, while the H+ concentration at the wall remains constant. 6.2. Variation of H+ Concentration along the Bilayer Stack. The previous section describes how the bilayer separation and the ion profiles between the walls depend on the osmotic pressure and CO2 concentration. To obtain the relation to the spatial coordinates along the bilayer stack, this must be coupled to the (steady-state) diffusive transport of both water and CO2. These transport processes are intimately coupled by the chemical equilibrium in eq 15, which, in a similar way as above (eq 16), can be shown to yield
d J - JCO2) ) 0 dx ( w
(27a)
d J -J ) 0 dx ( HCO3- H+)
(27b)
d J + JCO2 + JHCO3- + JH+) ) 0 dx ( w
(27c)
However, we will make the simplifying approximation that all transport is due to CO2 and water. As stated at the end of section 5, this is an excellent approximation for the case treated there. In this approximation, the fluxes decouple, as can be seen from eq 27 when JHCO3- ) JH+ ) 0
dJw )0 dx
dJCO2 dx
)0
so that the flux is independent of spatial coordinate. This has two implications: (i) that the coordinate along the bilayer stack, x, also for this case is given by eq 14, with the difference being that h depends on both µw and cCO2, and (ii) that the variation in cCO2 can be found from
- PCO2(W) (cCO2(W) - cCO2(0)) ) JCO2 ) -PCO2(x)
(cCO (x) - cCO (0)) 2
2
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Figure 6. H+ concentration as a function of spatial coordinate, x, along a homogeneously swollen stack of bilayers exposed to a gradient in CO2. At the lower side of the stack (x ) 0), cCO2 corresponds to physiological conditions; at the upper side, it corresponds to a water phase in equilibrium with typical ambient air. In each aqueous phase CO2, HCO3-, and H+ were assumed to be in chemical equilibrium, and the transport of ions through the bilayers was neglected. Since the amount of CO2 at the upper side of the stack is lower than that at the lower side, the total amount of CO2 (whether in the form of CO2 or HCO3-) within a bilayer is also lower, leading to the observered lowering of H+ concentration along the stack. Since the upper side of the stack corresponds to the skin surface, this corresponds to a basic skin surface.
Figure 7. Variation of H+ concentration with osmotic pressure between two bilayers in equilibrium with CO2 of typical ambient air. In the aqueous solution between the bilayers, CO2, HCO3-, and H+ were assumed to be in chemical equilibrium. The amount of other ions was assumed constant. The decrease in pH (solid curve) and -10log cH+ at the midplane (dashed curve) are smaller compared to the situation when the effect of CO2 is not considered (Figure 4). The H+ concentration at the bilayer interface is essentially constant (dotted curve) but lower compared to the case without CO2.
where the permeability of the part of the stack that lies between 0 and x is given by25
1 L(x) H(x) + ) l P(x) Dw kCO2DCO CO2 2 where H(x) is the total thickness of the water layers and L(x) is the total thickness of the bilayers up to the point x. Figure 8 shows how the H+ concentration varies with x in the bilayer stack for different osmotic gradients. The effect of the CO2 gradient is to make the decrease in pH less dramatic, though the upper side (corresponding to the skin surface) is still acidic. The qualitative behavior is similar to that in Figure 5 with a fast decrease over the lower layers and then an essentially constant value, both for the actual pH (solid curve) and -log cH+ at the midplane (dashed curve). The H+ concentration at the bilayer interface (dotted curve) is practically the same for all depths, although there is a minor decrease in the lower layers.
Figure 8. H+ concentration as a function of the spatial coordinate, x, along a stack of bilayers exposed to a gradient in both osmotic pressure and CO2. At the lower side (x ) 0), the osmotic pressure and cCO2 correspond to physiological conditions. At the upper side, the osmotic pressure corresponds to an ambient atmosphere with RH ) (a) 99%, (b) 90%, and (c) 60%, and cCO2 corresponds to a water phase in equilibrium with typical ambient air. Transport of ions through the bilayers was neglected, and in each aqueous solution between bilayers CO2, HCO3-, and H+ were assumed to be in chemical equilibrium. The amount of other ions between each pair of bilayers was set constant. The results are similar to the case of only an osmotic gradient (Figure 5): The pH (solid curve) and -10log cH+ value at the midplane (dashed curve) show a large decrease over the lower part of the stack. Comparing the respective values at the upper side of the stack, corresponding to the skin surface, we also find a decrease in pH with increasing RH, that is, with increasing osmotic gradient.
Note that the model used for the case of only a CO2 gradient does not take into account the effect of the local electrostatics between bilayers. Therefore, one cannot readily compare the results for the (almost) homogeneously swollen bilayer stack, Figure 8a, with the previous results in Figure 6.
7. pH in the SC We have presented a theoretical description for how osmotic and CO2 gradients can affect the H+ concentration along a bilayer stack. We have also discussed the definition of pH in heterogeneous nonequilibrium systems. The aim is now to couple the model and the fundamental views on pH in complex systems with existing experimental data on skin pH from the literature. 7.1. pH-Meter Method. A common method for measuring the skin surface pH is to apply a small volume (typically of the
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order of 10 µL) of water on the skin and then use a regular pH meter to measure the pH of this solution. Under controlled conditions (both when it comes to the actual measurement process and the environment in which it takes place, but also with the instructions given to volunteers), reproducible results can be found, at least at a one-digit precision level.1 By subsequent tape-stripping, this method can also be used in order to find the variation of pH with depth.11 The interpretation of the results from this kind of measurement is complex. What is measured is, in fact, an apparent pH due to an extraction of water-soluble components from the SC.46 It is then clear that a correct interpretation of these experiments demands a closer look at the transport of the different components of the SC. A detailed and quantitative description of this problem is hard to formulate, but it should be kept in mind that components that are not water-soluble are essentially absent in these measurements. Another effect that likely plays a role in these types of measurements is the fact that there is an equilibration of gaseous CO2 between the atmosphere and the water droplet. Indeed, pure water in equilibrium with ambient air has a pH of around 5.6, in close proximity to values reported for the skin surface pH. 7.2. Fluorescent Probe Method for the Bilayer Stack. Another method that has been used as a noninvasive tool for the determination of skin pH is the soaking of skin by a fluorescent probe with different lifetimes in its protonated, HP, and deprotonated, P-, states. Taking into account the effect of the local environment on the fluorescence of the probe enables a determination of the concentration ratio [P-]/[HP]. This analysis is nontrivial, but under the assumption that it can be done the combination of this kind of experiment with imaging techniques makes possible a three-dimensional view of how the [P-]/[HP] concentration ratio varies in the SC.47,48 Due to the inclusion of the electrostatic potential in the definition of pH given in section 3, it is difficult to extract the pH from the information obtained from these measurements. However, if we assume dilute solution behavior, one might hope to at least find the H+ concentration from
- 10log cH+ ) pKP + 10log
[P-] [HP]
(28)
where pKP is the acidity constant for the probe. Here, the question arises of what acidity constant to use. In general, for the complex environment of the SC, we have no clear answer to this question, but note that in the extracellular region the protonated form of the probe would typically be found in the bilayer and the deprotonated form would preferably be found in the aqueous layer. 7.3. Comparison between Experiment and Model. We conclude that it is difficult to make direct quantitative comparisons between experimental data and the calculated results due to problems described above. However, in a qualitative fashion, we note an agreement with a number of observations: Several investigations have shown an increase of skin surface pH after occlusion49,50 or rinsing with tap water.51 This is consistent with our model taking into account the effect of the osmotic gradient. The effect of occlusion is a decrease in Π at the skin surface, and it can be compared to the calculated effect (46) Rieger, M. Cosmet. Toiletries 1989, 104, 53–60. (47) Hanson, K. M.; Barry, N. P.; Gratton, E.; Clegg, R. M. Biophys. J. Annu. Meet. Abstr. 2002, B588. (48) Hanson, K. M.; Behne, M. J.; Barry, N. P.; Mauro, T. M.; Gratton, E.; Clegg, R. M. Biophys. J. 2002, 83, 1682–1690. (49) Aly, R.; Shirley, C.; Cunico, B.; Maibach, H. I. J. InVest. Dermatol. 1978, 71, 378–381. (50) Hartmann, A. A. Arch. Dermatol. Res. 1983, 275, 251–254. (51) Gfatter, R.; Hackl, P.; Braun, F. Dermatology 1997, 195, 258–262.
of decreasing Π at the upper side of the bilayer stack. Indeed, the calculations in Figure 4 show an increase in pH with increasing hydration. Furthermore, a buildup of CO2, approaching physiological values, at the skin surface has been noted under occlusive dressing.49 Therefore, comparison with the calculations taking into account only the osmotic gradient, and not the coupled osmotic and CO2 gradients, is the most relevant. Studies showing an elevated skin surface pH following cleansing with alkaline soaps,51–53 or with a synthetic detergent formulated at the same pH as skin,54 are harder to interpret. Again, the major effect might be due to hydration, but the alkaline/ acidic components of the detergent will also play a role. When it comes to measurements of the pH gradient, fluorescent probe experiments on mouse skin have shown that -10log cH+ values of the extracellular regions remain essentially constant at 6.0 regardless of depth.48 Though the effect of the different partioning of the protonated and deprotonated form of the probe was not taken into account, the qualitative behavior can still be seen: The first term on the right-hand side of eq 28 is constant, and measurement of the ratio [P-]/[HP] gives the full trend in H+ activity, up to an additive constant. Assuming that the protonated form of the probe really is located in the bilayer, we can directly compare with our previous results for the H+ concentration at the bilayer interface. Indeed, Figure 8 shows an essentially constant concentration with position (dotted line), consistent with the experimental observation. Fluorescent probe experiments also show a clear difference between -10log cH+ values of the extracellular regions and corneocytes.48 Though the full analysis is complicated, we note that such an observation is not surprising considering the radically different environments. In other words, the observation is not necessarily an indication of lateral nonequilibrium effects.
8. Conclusions The definition of pH in a system such as the SC is ambiguous due to the local heterogeneity of the SC, the presence of charged species in the SC lipids, and the continuous transport processes occurring across the SC. Thus, any measurement of pH in the SC should always be directly related to a clear definition. We have presented a natural definition of pH that takes into account the problems encountered in the SC. The basic idea is to compare the local environment at a certain point with a bulk system, where the definition of pH is straightforward. The local pH is then defined as equal to the pH of that particular bulk system which corresponds to the local environment. This definition has been exemplified and applied in a simplified model of the extracellular lipid matrix. The effect of diffusive transport, specifically of water and/or CO2, on the pH gradient across the SC was investigated. The extracellular lipid matrix is the only continuous region of the SC, which motivates the use of a simplified model membrane system consisting of a stack of oriented lipid bilayers. The water gradient affects the interbilayer separation along the bilayer stack. Since the electrostatic potential between the charged bilayers is strongly dependent on the interbilayer separation, this gives a large effect on the interbilayer H+ concentration profile. The gradient in CO2 affects the local amount of HCO3-, CO32-, and, in particular, H+ between the bilayers. The main observation is that transport gives rise to a substantial gradient in pH across the bilayer stack. An osmotic gradient (52) Korting, H. C.; Braun-Falco, O. Clin. Dermatol. 1996, 14, 23–27. (53) Barel, A. O.; Lambrecht, R.; Clarys, P.; Morrison, B. M., Jr.; Paye, M. Skin Res. Technol. 2001, 7, 98–104. (54) Dikstein, S.; Zlotogorski, A. In Cutaneous InVestigation in Health and Disease; Le´veque, J.-L., Ed.; Marcel Dekker: New York, 1989; pp 59-77.
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causes a situation where the upper side of the bilayer stack, representing ambient conditions, is more acidic than the lower, representing physiological conditions inside the body. The gradient in CO2 gives rise to a pH gradient in the opposite direction. The effect of both an osmotic and a CO2 gradient is qualitatively similar to the effect of only an osmotic gradient. The calculations furthermore demonstrate a gradient in H+ concentration over the bilayer stack. The magnitude of this effect is strongly dependent on whether we consider the H+ concentration in the aqueous layer or at the bilayer interface. The general observation is an almost constant H+ concentration, -10log cH+ ) 6.0, at the bilayer interface in every layer along the stack. At the midplane between bilayers, on the other hand, the effect is much more pronounced. Here, the H+ concentration varies between the physiological value of -10log cH+ ) 7.4 and -10log cH+ ) 6.8 close to side of the bilayer stack that represents ambient conditions. The decrease in midplane H+ concentration is mainly occurring in the lower part of the bilayer stack, that is, in the part of the stack that is close to a physiological bulk.
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The model clearly simplifies the situation found in the SC in that it neglects, for example, transport through corneocytes and transport processes occurring in connection with hair follicles and sweat glands. However, the simplicity of the model allows us to draw quantitative conclusions based on a set of parameters chosen to represent the SC lipids. Within this model, it is possible to thoroughly explore the effects of this particular mechanism. The results show qualitative agreement with the effect of occlusion on skin surface pH. Also, the absence of a gradient in H+ concentration at the bilayer interface is consistent with fluorescence lifetime imaging studies of the pH gradient in the SC. Acknowledgment. E.S. and H.W. gratefully acknowledge financial support from the Swedish Research Council (Vetenskapsrådet) both through regular grants and a Linneaus programme. Supporting Information Available: Efficient algorithms for the numerical solution. This material is available free of charge via the Internet at http://pubs.acs.org. LA800543R
