INTRODUCTION

Top Abstract Introduction Results & Discussion References

The passive transport rates of small molecules across biological membranes are often explained, at least qualitatively, by means of a bulk-phase solubility diffusion model (Fettiplace and Haydon, 1980; Finkelstein, 1976; Hanai and Haydon, 1966; Paula et al., 1996). This model, which may be traced to the formulation of Overton's rules nearly a century ago (Overton, 1899), describes the permeation process in terms of the partitioning of the permeant into the membrane, followed by its diffusion through the membrane, where the properties of the membrane are assumed to be adequately represented by a bulk lipid (e.g., hydrocarbon) solvent. It is well known, however, that permeabilities across biological membranes and model lipid bilayers depend strongly on both the degree of lipid chain packing in the membranes (Lande et al., 1995; Worman et al., 1986; Xiang and Anderson, 1995b, 1997) and the size of the permeating solute (Stein, 1986; Walter and Gutknecht, 1986; Xiang and Anderson, 1994c). The effects of lipid chain packing on permeability are most clearly demonstrated by the dramatic increase in transmembrane transport rates that occurs because of a gel-to-liquid crystalline phase transition (Carruthers and Melchior, 1983; Jansen and Blume, 1995; Papahadjopoulos et al., 1973; Xiang and Anderson, 1997). Membranes that are more "ordered" as a result of polar headgroup composition, or because of increased cholesterol concentrations or lower temperatures, or monolayers under high lateral pressures have greater resistances to permeation than predicted from a bulk solubility diffusion model, often by orders of magnitude (Bar-On and Degani, 1985; Brahm, 1983; Finkelstein, 1976; Magin and Niesman, 1984; Peters and Beck, 1983; Sada et al., 1990; Todd et al., 1989a,b). A steep size selectivity is exhibited by both biological membranes and lipid bilayers, as noted by several groups, including Lieb and Stein (1969, 1971, 1986), Walter and Gutknecht (1986), and ourselves (Anderson and Raykar, 1989; Xiang and Anderson, 1994b). These phenomena also cannot be accounted for by bulk solubility diffusion theory.

Although substantial progress has been made recently, the molecular mechanisms responsible for the chain-ordering effects on permeability and the role of permeant size have not been well understood historically. It has been common to interpret both the effects of chain ordering on permeability (Lande et al., 1995) and the steep dependence of permeation rates on permeant size (Stein and Nir, 1971; Stein, 1986) exclusively in terms of changes in solute diffusivity within membranes. Walter and Gutknecht (1986) argued, for example, that the effects of solute size on partitioning into a bilayer membrane are unimportant on the basis of the similarity in solubility of short-chain n-alkanes in lipid bilayers (Miller et al., 1977). On the contrary, large size effects on solute partitioning into interphases are evident from the high resolution attainable in chromatographic separations on the basis of subtle differences in size and shape (Wise et al., 1981). More recently, others have shown both theoretically (Marqusee and Dill, 1986) and experimentally (DeYoung and Dill, 1988, 1990; Xiang and Anderson, 1995b) that increasing chain ordering within lipid bilayers substantially reduces solute partitioning into bilayers. These effects are particularly evident in the more ordered regions of the bilayer, as confirmed in neutron diffraction experiments (White et al., 1981) and molecular dynamics simulations (Marrink and Berendsen, 1994, 1996; Xiang and Anderson, 1998). Further complicating the situation, statistical mechanical theory recently developed by the authors (Xiang and Anderson, 1994a) and molecular dynamics simulations conducted in these laboratories (data not shown) suggest that the size selectivity in partitioning is amplified with increases in bilayer chain ordering.

These recent results suggest that structure-transport relationships developed solely on the basis of lipophilicity or hydrogen bonding potential without consideration of molecular size effects and the influence of bilayer composition (i.e., chain ordering) on these size effects may be unreliable. Walter and Gutknecht (1984) noted, for example, that literature data for the incremental free energy changes accompanying the addition of a methylene group to various homologous series of permeants derived from transport studies across a variety of model bilayer and biological membranes were highly variable, ranging from nearly zero to 900 cal/mol. In addition to the inappropriate treatment of unstirred layer effects in some of these studies pointed out by Walter and Gutknecht, the differences in membrane composition and complexity in terms of lipid chain packing (e.g., gel and liquid-crystalline phases) may also have contributed to the variability in the effects of permeant chain length on permeability.

A more systematic characterization of solute permeability with respect to the states of lipid packing in lipid bilayers is essential to a thorough understanding of molecular mechanisms for solute transport across biological membranes. The difficulties in investigating the effects of permeant size on permeability arise from the fact that changes in permeant size are usually accompanied by changes in lipophilicity, with the latter effects often overshadowing the effects of permeant size alone.

In this study we will investigate the effects of permeant size on membrane permeability in a way markedly different from previous studies. Namely, we will examine by means of an NMR line-broadening method the size and shape selectivity of dipalmitoylphosphatidylcholine (DPPC) bilayers for the transport of a series of seven monocarboxylic acids differing in chain length and degree of chain branching (Fig. 1) as a function of lipid bilayer packing, characterized by the bilayer free surface area. The use of a homologous series of monocarboxylic acids minimizes the effects of hydrogen bond potential on the analysis of solute shape and size effects. Our aim, in part, is to examine the hypothesis that increasing lipid chain order will be accompanied by increased membrane selectivity to permeant size and shape. To accomplish this we will apply a chain ordering correction factor to the bulk solubility-diffusion model predicted permeability coefficients to account for the discrepancies between the observed values and those calculated neglecting chain-ordering effects (Xiang and Anderson, 1997). We will then develop an empirical relationship between the permeability decrement due to chain ordering and the bilayer packing properties described by the membrane free surface area. The slope of the linear relationship between the natural logarithm of the chain-ordering correction factor (i.e., the permeability decrement) and the inverse of free surface area will be shown to depend linearly on permeant size when expressed in terms of permeant cross-sectional area.

View larger version (48K): [in this window] [in a new window]   FIGURE 1   The side and head-on views of space-filling models of seven monocarboxylic acids used in this study as permeants. The area of the head-on view gives the minimum cross-sectional area of the molecule.

	EXPERIMENTAL

Materials

DPPC was purchased from Avanti Polar Lipids (Pelham, AL). Formic acid (99%) and [¹⁴C]formic acid (>98%) were purchased from Sigma Chemical Co. (St. Louis, MO). [³H]Acetic acid, [¹⁴C]propionic acid, and [¹⁴C]butyric acid were obtained from American Radiolabeled Chemical (St. Louis, MO). Acetic acid (99.8%), propionic acid (99+%), butyric acid (99+%), valeric acid (99+%), isovaleric acid (99%), trimethylacetic acid (99%), deuterium oxide (99%), and hexadecane (99%) were purchased from Aldrich Chemical Co. (Milwaukee, WI). All other reagents were obtained commercially and were of analytical reagent grade. Polycarbonate membranes and membrane holders were obtained from Nuclepore (Pleasanton, CA).

Large unilamellar vesicle liposome preparation

A detailed description of the experimental procedure has been published elsewhere (Xiang and Anderson, 1995a). In brief, large unilamellar vesicles (LUVs) were prepared by a modified combined technique of Bangham et al. (1965) and Olson et al. (1979). The DPPC lipids were accurately weighed, dissolved in chloroform, evaporated to a dry thin film under nitrogen gas, and left under vacuum for 2 h at ~50°C. A deuterated aqueous solution containing 5-30 mM permeant was then added to a final lipid concentration of 10 mg/ml. The lipids were then hydrated by repeated vortexing and shaking above the main transition temperature (41°C). The multilamellar vesicles formed were then forced through a 0.1-0.2-µm polycarbonate membrane filter 18 times to form LUVs before the NMR transport experiments.

Permeability coefficient determinations

The permeability coefficients for the seven monocarboxylic acids across DPPC bilayers were determined at various temperatures by the ¹H-NMR line-broadening method developed by Alger and Prestegard (1979) and further validated in a recent study by the authors (Xiang and Anderson, 1995a). The experiments were performed on a Bruker-200 NMR spectrometer (Bruker Instruments, Billerica, MA) operated in the Fourier transform mode at 200 MHz. Samples were equilibrated for 20 min at a given temperature controlled by a standard variable-temperature accessory (BVT1000; Bruker). Each spectrum was the average of 32-1000 acquisitions separated by 2-5-s pulse delays. The spectra were Fourier transformed and phased with an Aspect 3000 computer. The resonance frequencies of selected protons in the permeants located inside and outside the vesicles, _i and _o, were separated by adding an impermeable shift reagent, Pr(NO₃)₃ (final concentration, 0.5-3 mM), to the sample before the spectral acquisitions. The binding capacity of the chemical shift reagent (Pr³⁺) and its effects on solute permeability have been studied previously (Xiang and Anderson, 1995a). The DPPC main transition temperature was found to remain constant (41 ± 0.5°C) upon the addition of 5 mM Pr³⁺, and the permeability coefficient for acetic acid in DMPC/CHOL was shown to be independent of [Pr³⁺] up to 40 mM. The concentration of free Pr³⁺ available for binding to the outer vesicle surface is lower than the total Pr³⁺ concentration because of complex formation between carboxylic acids and Pr³⁺. For example, at [Pr³⁺] = 5 mM and an acetic acid concentration of 0.05 M (pD 6.32), only ~25% of the total Pr³⁺ would exist in the uncomplexed form.

The proton peak(s) for the methylene group adjacent to the carboxylic acid group in acetic acid, propionic acid, butyric acid, and valeric acid exhibit the greatest chemical shift in the presence of paramagnetic ions and thus were used for the permeability determinations. However, proton coupling in propionic acid, butyric acid, valeric acid, and isovaleric acid splits the proton resonances into two to four peaks, precluding an accurate determination of the line broadening due to solute transport. This was minimized by irradiating these protons through a separate decoupling channel. The collapsed singlet peak had a linewidth of 1.0-2.2 Hz in LUVs in the absence of chemical shift reagent. For isovaleric acid and trimethylacetic acid, the proton peak(s) for the methyl groups are the strongest and most symmetrical and thus were used for the permeability determinations.

The lifetime of the permeant inside the vesicle, _i, was obtained using the following linewidth expression in the slow exchange limit, |_i  o|T_2,i 1 (Piette and Anderson, 1959):

(1)

where is the full linewidth at one-half the maximum peak height, and T_2,i is the spin-spin relaxation time, which includes heterogeneous line broadening in the absence of exchange. The linewidth in the absence of exchange (1/T_2,i = 2.6-5.0 Hz) was obtained at a low temperature and/or high pD, where the permeation rate is negligible.

Previous studies in these laboratories have shown that the permeabilities of ionized carboxylic acid permeants are negligible in the pD range of interest (Xiang and Anderson, 1995a). Thus the permeability coefficient for the neutral species, P_m, can be expressed as

(2)

where V is the entrapped volume, A_t is the vesicle surface area, and K_a is the dissociation constant in D₂O. The V/A_t ratio was determined from the vesicle hydrodynamic diameter (d), as obtained from dynamic light scattering (DLS) measurements according to the formula V/A_t = d/6. DLS experiments were conducted with a goniometer/autocorrelator (model BI-2030AT; Brookhaven, Holtsville, NY) and an Ar⁺ ion laser (M95; Cooper Laser Sonics, Palo Alto, CA) operated at 514.5-nm wavelength. One drop of LUV suspension was placed in a 13 × 75 mm cleaned glass test tube and brought to a volume of 2 ml with the same filtered buffer solution. The sample was placed in a temperature-controlled cuvette holder with a toluene index-matching bath. Autocorrelation functions were determined for a period of 100 s with a 10-80-µs duration at 90° and analyzed by the method of cumulants.

Partition coefficient determinations

Hexadecane/water partition coefficients for the series of monocarboxylic acids were measured using the shake flask method at a pH 2 units below the pK_a of the corresponding acid to ensure that >99% was in its un-ionized form. The organic solvent was first washed three times with an equal amount of de-ionized water. The organic solvent (2-3 ml) and 1 ml of an aqueous solution containing 3 × 10² to 1 × 10⁴ M of "cold" permeant or 1-20 µCi of radiolabeled permeant were placed in a test tube and mixed with magnetic stirring in an incubator at a preset temperature (25-50°C) for 24 h. The sample was then centrifuged to remove any emulsified water from the organic phase. Aliquots of both phases were carefully taken for high-performance liquid chromatography (HPLC) analyses of "cold" compounds (acetic acid, propionic acid, butyric acid, valeric acid, isovaleric acid, and trimethylacetic acid) or for liquid scintillation counting (LS1801; Beckman Instrument Co., Fullerton, CA) of radiolabeled compounds (formic acid, acetic acid, propionic acid, and butyric acid). Duplicate measurements were performed with two different permeant concentrations to evaluate the effects of permeant self-association in hexadecane on the measured partition coefficients. To minimize the potential effects of more lipophilic impurities in the radiolabeled compounds (not a problem if HPLC is used for concentration analyses), the organic phase was replaced with fresh solvent, and the above experimental procedure was repeated until the radioactivity in the organic phase reached a plateau value. The partition coefficient was calculated as the ratio between the molar concentrations in the hydrocarbon solvent and water.

The partition coefficients obtained using both "hot" and "cold" compounds and two different solute concentrations were generally within the experimental error, suggesting that self-association was not a significant factor in the partition coefficient determinations.

An HPLC system consisting of a syringe-loaded sample injector (Rheodyne model 7125; Rainin Instrument Co., Woburn, MA), a solvent delivery system (110B; Beckman Instrument Co., San Ramon, CA) operated at a flow rate of 1.0-1.2 ml/min, a dual-wavelength absorbance detector (model 441, Water Associates, Milford, MA) operated at 214 nm, an integrator (model 3392A; Hewlett-Packard Co., Avondale, PA), and a reversed-phase column packed with 5-µm C18 300 Å (Jupiter, 4.6 mm i.d. × 25 cm; Phenomenex Co., Torrance, CA) was used at ambient temperature for the analyses of the "cold" monocarboxylic acids taken during the partition experiments. Mobile phases containing 2.5%-40% acetonitrile, depending on the analyte lipophilicity, and 0.01 M phosphate buffer (pH 3.0) were employed.

pK_a determinations in D₂O

The ionization constants for the series of monocarboxylic acids in deuterated water were measured by a pD titration. An accurately weighed amount of each acid was dissolved in 2.0 ml D₂O to yield a final concentration of 0.03 M. pK_a determinations were carried out at 24°C under nitrogen by slow addition of 0.03 M NaOD while monitoring the apparent pH with a standard pH meter (PHM82; Radiometer, Copenhagen, Denmark) calibrated with standard buffer solutions. The pD values were obtained by adding 0.40 units to the corresponding pH readings (Glasoe and Long, 1960). Plots of the pD values versus the titrant volumes added were fitted numerically, including a correction for changes of the ionic strength using Davies' equation (Perrin and Dempsey, 1974). The dissociation constants obtained are presented in Table 1.

	RESULTS AND DISCUSSION

Top Abstract Introduction Results & Discussion References

Transport experiments were conducted in large unilamellar vesicles composed of DPPC. The chain packing properties within these DPPC bilayers, as characterized by the bilayer free surface area (see Eq. 8), were varied by changes in temperature, which also brought about changes in bilayer phase structure (gel and liquid crystalline phases). Seven monocarboxylic acids differing in chain length and the degree of chain branching as shown in Fig. 1 were used as permeants. This permeant series offered several advantages for the purposes of this study. One was the technical advantage of being able to determine the permeability coefficients for this series of relatively lipophilic permeants by the NMR line-broadening method, in contrast to other, slower transport methods that rely on separations of intravesicular permeants from the external solution (e.g., size exclusion chromatography, ultrafiltration, or dialysis). Moreover, the size and shape of permeants within this series could be varied in a more systematic manner. Furthermore, the structural changes involved only variations in the number of methylene groups, which have relatively small and well-documented effects on solute lipophilicity in comparison to most substituents. This rendered less problematic the selection of an appropriate bulk reference solvent to correct for the effects of changes in permeant lipophilicity on permeability, as the incremental free energy for the transfer of a single methylene group from water to various bulk organic solvents is essentially independent of the nature of the solvent (849 cal/mol in heptane compared with 712 cal/mol in octanol; Davis et al., 1972, 1974).

Permeability coefficients: expectations from bulk solubility-diffusion theory

As noted earlier, bulk solubility-diffusion theory assumes that solute partitioning from water into and diffusion through a lipid bilayer or biomembrane resembles that within a homogeneous slab of bulk solvent such as olive oil, octanol, or hydrocarbon. The permeability coefficient predicted from this model (P_o) can be expressed as

(3)

where K_hc/w and D_o are the organic solvent (i.e., hexadecane)/water partition coefficient and the diffusion coefficient in the organic solvent, respectively, and  is the effective thickness of the homogeneous layer of organic solvent.

To use bulk solubility-diffusion theory to predict permeability coefficients as temperature is varied, the temperature dependence for permeant partitioning into (and diffusion through) an organic solvent that most closely mimics the physicochemical properties of the barrier domain in the bilayer membrane under investigation must be known. A large body of evidence has shown that the barrier domain in lipid bilayers behaves like a hydrocarbon solvent with respect to intermolecular electrostatic interactions (Stein, 1986; Xiang and Anderson, 1994c). Moreover, certain dynamic properties in the bilayer interior, including chain reorientation rates and microviscosity resemble those in bulk hexadecane (Bell, 1981; Brown et al., 1986; Venable et al., 1993). Thus hexadecane, which resembles the lipid chains in DPPC in terms of chain length and degree of saturation, was used as a model solvent in this study. Fig. 2 shows the Arrhenius plots of the hexadecane/water partition coefficients for the series of monocarboxylic acids used in this study. The molar free energies, enthalpies (obtained from the slopes of linear fits of the data in Fig. 2), and entropies of transfer from water to hexadecane are presented in Table 1. The transfer enthalpies (H°) are in the range of 3.2-5.4 kcal/mol and generally increase with the chain length, suggesting that permeant dehydration contributes only a small fraction of the large activation energies observed for permeability of the monocarboxylic acids across DPPC bilayers (vide infra). Increases in H° with chain length were also found for the transfer of short-chain alkanols (C₁-C₅) from water to hydrocarbons (Nemethy et al., 1963). These values are approximations, as they are assumed to be constant over the entire temperature range explored (25-50°C). Large transfer heat capacities are typically observed (DeYoung and Dill, 1990), however, which may change the slopes of Arrhenius plots from which the molar enthalpies of transfer were obtained. The partition coefficient for valeric acid is slightly larger than that for isovaleric acid, which is consistent with the view that chain branching decreases the molecular surface area and thereby reduces the partition coefficient due to the reduced hydrophobic interaction in water (Grant and Higuchi, 1990). However, the partition coefficient for trimethylacetic acid is about twice as large as that for valeric acid, whereas H°_hc/w is smaller for trimethylacetic acid. These results may be attributed to the fact that the three methyl groups in trimethylacetic acid may interfere with solvation of the carboxylic acid group by water, effectively making it less hydrophilic. Steric hindrance of solvation of the ion produced on ionization of highly hindered aliphatic acids also decreases acid strength (Hine, 1975), as demonstrated by a 0.2-unit higher pK_a value for trimethylacetic acid than that for valeric acid.

View larger version (20K): [in this window] [in a new window]   FIGURE 2   Arrhenius plots of the hexadecane/water partition coefficients for formic acid (), acetic acid (), propionic acid (), butyric acid (), valeric acid (), isovaleric acid (+), and trimethylacetic acid ().

The diffusion coefficients of monocarboxylic acids in bulk solvents are determined primarily by their molecular sizes (Albery et al., 1967), whereas the polar carboxylic acid residue is expected to play a minor role, at least in nonpolar hydrocarbons (Chan, 1983). As a result, the diffusion coefficients for the monocarboxylic acids used in this study were estimated from a relation developed from the experimental diffusivity data for a series of alkane homologs in hexadecane at 25°C (Hayduk and Ioakimidis, 1976) and the assumption that the diffusion coefficient varies inversely with solvent viscosity:

(4)

where  is the viscosity (cp) of hexadecane and V_s is the solute volume (ų). The molecular volumes for the series of diffusants were calculated from an atomic additivity method (Edward, 1970). The size dependence in Eq. 4 (n = 0.83) is stronger than that predicted by the Stokes-Einstein relation (n = 0.33), though less than that typically observed for solute diffusion in polymers (n = 1.1-3.8) (Lieb and Stein, 1969).

Because of the small changes of bilayer density with temperature, the thickness of the acyl chain region in the bilayer, , is related to the bilayer reduced surface density  (vide infra) by  = l_o, where l_o (= 38.4 Å) is the end-to-end length of a fully extended DPPC molecule.

The permeability coefficients (P_o) from bulk solubility-diffusion theory are calculated at different temperatures by combining the sets of partition coefficients, diffusion coefficients, and bilayer thickness data described above. These results are presented in Fig. 3 for later comparison with the experimental permeability coefficients. Bulk solubility diffusion theory predicts a weak dependence of permeability on temperature, as indicated by the relatively small activation energies E_a derived from the slopes of the ln P_o versus 1/T data in Fig. 3, which are listed in Table 1 (10.3-12.7 kcal/mol).

View larger version (22K): [in this window] [in a new window]   FIGURE 3   Arrhenius plots of the permeability coefficients for formic acid, acetic acid, propionic acid, butyric acid, valeric acid, isovaleric acid, and trimethylacetic acid across DPPC LUVs. , Experimental permeability coefficients (Pm); +, permeability coefficients predicted from bulk solubility-diffusion theory (Po).

Membrane permeability coefficients: deviations from bulk solubility-diffusion theory

Fig. 4 shows the proton magnetic resonance peak(s) for the methylene group adjacent to the carboxylic group in butyric acid in DPPC LUVs in the absence and presence of chemical shift reagent (Pr³⁺) and with and without coupling to the neighboring methylene group. In the absence of Pr³⁺, decoupling of the methylene group by irradiating at a radio frequency corresponding to the central position of the adjacent methylene group leads to a narrow singlet peak (~1.5 Hz) for the methylene group (Fig. 4 B). Addition of Pr³⁺ shifts the proton peaks of the extravesicular permeant downfield (~140 Hz), whereas that of the permeant entrapped in the internal aqueous space is broadened (Fig. 4, C and D). The degree of line broadening, which is attributed to the exchange of permeant across the LUVs, was found to depend on pD and temperature. The permeability coefficient can be calculated according to Eqs. 1 and 2 from the observed linewidth and solution pD.

View larger version (9K): [in this window] [in a new window]   FIGURE 4   Proton magnetic resonances for the methylene group adjacent to the carboxylic group in butyric acid in DPPC LUVs at pD 6.09. (A and B) In the absence of chemical shift reagent and at 21°C. (C) In the presence of 2 mM Pr3+ at 32°C. (D) In the presence of 2 mM Pr3+ at 38°C. The decoupling power was off for the spectrum in A and on for the spectra in B-D. The sweep width was 2.8 kHz and the pulse width was 4.0 µs. The decouple width was 10 Hz.

Temperature dependence

Methylene group contributions

(5)

View larger version (14K): [in this window] [in a new window]   FIGURE 5   Dependence of the observed permeability coefficients (Pm) and those predicted from bulk solubility-diffusion theory (Po) on the number of carbon segments in the permeants. , DPPC bilayers at 30°C (gel phase); , DPPC bilayers at 45°C (liquid-crystalline phase); , predicted Po values at 30°C. The × and + symbols are for isovaleric acid and trimethylacetic acid, which have the same number of carbon segments as valeric acid.

The permeability decrement due to chain ordering depends on free surface area and permeant cross-sectional area

A large body of evidence from this and other laboratories (DeYoung and Dill, 1988, 1990; Xiang and Anderson, 1995b, 1997) indicates that the failure of homogeneous solubility diffusion theory to predict permeabilities in lipid bilayer membranes stems from the neglect of the effects of chain ordering in lipid membranes as characterized by order parameters and other related properties. We have previously shown that the permeability coefficient predicted from bulk solubility-diffusion theory must be adjusted downward by a factor f to correct for chain-ordering effects (Xiang and Anderson, 1997). Thus the permeability decrement due to chain ordering is defined as the ratio between the experimental and bulk phase model-predicted permeability coefficients:

(6)

We have recently proposed that the effects of bilayer packing on transbilayer permeabilities as quantified by f can be rationalized in terms of a dependence of permeability on the two-dimensional packing structure, as characterized by the free surface area per lipid molecule, a_f, as depicted below:

(7)

where a_s is a characteristic cross-sectional area of solute and f_o and  are constants independent of permeant size and bilayer packing structure. Similar to the definition of free volume (Bondi, 1954), the mean free surface area per lipid molecule is related to the area occupied by one phospholipid molecule, A, by

(8)

where A_o (= 40.8 Å²) is the area of a phospholipid molecule in the crystal (Tardieu et al., 1973) and  (= A_o/A) is the reduced surface density. The surface density data, as reported in our previous work (Xiang and Anderson, 1997), were obtained from acyl chain order parameters (DeYoung and Dill, 1988; Morrow et al., 1992) based on the existence of one-to-one correspondence between these two bilayer properties, regardless of the chemical structure of the lipid (Boden et al., 1991; Nagle, 1993).

The model described by Eq. 7 predicts an inverse linear correlation between ln f and the free surface area. This prediction was verified in a study of the permeability of acetic acid in distearoylphosphatidylcholine, dipalmitoylphosphatidylcholine, dimyristoylphosphatidylcholine, and dilauroylphosphatidylcholine bilayers and their mixtures with cholesterol, at various temperatures both above and below the gel-to-liquid crystalline phase transition temperatures (Xiang and Anderson, 1997). The present study, employing permeants varying in molecular size, allows us to confirm the inverse dependence of ln f on a_f and to test another novel feature of this model, the dependence of the permeability decrement on the cross-sectional area rather than the molecular volume of the permeant.

Values of ln f calculated from the experimental (P_m) and bulk solubility-diffusion model predicted (P_o) permeability data for the series of monocarboxylic acids (cf. Fig. 3) are presented in Fig. 6 as a function of the membrane free surface area, a_f. The free surface area was varied by changes of temperature, which also induced changes in bilayer phase structure. Values of f vary from nearly 1 for the smallest permeant (formic acid) in liquid crystalline bilayers to <1 × 10⁴ for the permeant with the greatest cross-sectional area, trimethylacetic acid, in a highly ordered gel-state bilayer. As noted in Fig. 6, approximately linear relationships are observed between ln f and the mean free surface area for all monocarboxylic acids studied. Least-squares fits of the permeability decrement data according to Eq. 7 are also shown in Fig. 6. The slopes and correlation coefficients obtained are listed in Table 2. The slope, which measures the sensitivity of the permeability coefficient of a given permeant to membrane chain ordering, increases significantly from formic acid to acetic acid, then remains approximately constant from acetic acid to valeric acid, and again increases significantly for isovaleric acid and trimethylacetic acid, two permeants that have the same number of carbons and molecular volume as valeric acid.

View larger version (18K): [in this window] [in a new window]   FIGURE 6   Natural logarithms of the permeability decrement, f, for formic acid, acetic acid, propionic acid, butyric acid, valeric acid, isovaleric acid, and trimethylacetic acid across DPPC bilayers versus the inverse of the reduced free surface area, Ao/af. The solid lines are derived from least-squares fits.

Fig. 1 displays space-filling models of the monocarboxylic acids used in this study, constructed with a molecular graphics program, Insight II (Biosym Technologies, San Diego, CA). Views from two different perspectives are shown. The cross-sectional areas along the long axes (i.e., the minimum cross-sectional areas) as represented by the "head-on view" in Fig. 1 are presented in Table 1. Values of the slopes obtained from the ln f  A_o/a_f profiles in Fig. 6 are plotted versus the minimum cross-sectional areas of the monocarboxylic acid permeants in Fig. 7. A linear least-squares fit of the data yielded a slope of 0.072 Å², with a 95% confidence range (S-plane) of 0.032-0.11 Ų, providing a value of  = 2.7. Thus the slope is significantly different from zero. The excellent correlation of r = 0.94 is substantially higher than the correlation of r = 0.59 obtained for a similar plot of the slopes versus molecular volume (not shown).

View larger version (11K): [in this window] [in a new window]   FIGURE 7   The correlation between the negative slopes for the ln f  Ao/af profiles obtained in Fig. 6 and the minimum cross-sectional areas for the monocarboxylic acid permeants.

The above results verify that Eq. 7 has the correct functional form. Thus the correction factor that accounts for the reduction in permeability in lipid bilayer membranes due to chain ordering is shown to depend exponentially on the ratio of permeant size to the free surface area of the membrane, both of which can be determined in independent experiments. We now consider the underlying molecular mechanisms that may account for the functional form of Eq. 7, with particular emphasis on the role of permeant size.

Previously, the effects of permeant size on permeability have been analyzed almost exclusively in terms of changes in permeant diffusivity (Stein, 1986; Walter and Gutknecht, 1986). In turn, our understanding of molecular diffusivity comes primarily from studies in simple liquids and polymers. Among many diffusion models, the free-volume concept has been most successful in describing various diffusion processes in simple liquids as well as in more complex polymers (Hildebrand, 1977; Vrentas, 1977). In the original diffusion theory of Cohen and Turnbull (Cohen and Turnbull, 1959; Turnbull and Cohen, 1970), translational diffusion of a molecule was assumed to occur when statistical redistribution of free volume opens up a cavity of a critical size in the immediate vicinity of the molecule. As a result, the diffusion coefficient D is proportional to the probability of finding a free volume with a size equal to or greater than that of the diffusant, V_s, which is usually assumed to be an exponential function:

(9)

where D_o is a preexponential factor,  is a constant to account for any overlap of free volumes, and V_f is the mean free volume. The Cohen-Turnbull theory has been used to predict the effects of permeant size on transport across both biological and model lipid bilayer membranes (Stein, 1986).

Galla (Galla et al., 1979) and later Vaz (Vaz et al., 1985a,b) and their co-workers modified the original free volume theory to describe the lateral diffusion of lipophilic probes in lipid bilayers in terms of free surface area rather than free volume. However, the dependence of permeant diffusive motion on molecular cross-sectional area rather than on molecular volume, as observed in studies of lateral diffusion in bilayers and in this study of transbilayer transport, is not unique to lipid bilayer membranes. Hildebrand and co-workers found that the diffusivity of dissolved gas molecules in bulk solvents depends on the cross-sectional area (V^2/3) of the diffusing molecules (Ross and Hildebrand, 1964). A study by Hayduk and Buckley (1972) also found that the bulk solvent diffusivities of linear or elongated molecules were in general higher than spherical or globular molecules with the same molar volume. Moreover, early studies of solute diffusion in polyisobutylene (Blyholder and Prager, 1960; Prager and Long, 1951) showed that the most elongated molecules have the highest diffusion rates (e.g., D = 2.64, 1.32, and 0.62 × 10⁹ cm²/s for n-pentane, isopentane, and neopentane, respectively), and straight-chain solutes have similar diffusion rates, regardless of their chain length (e.g., D = 4.81, 3.24, 2.64, 3.04, 3.16 × 10⁹ cm²/s for propane, n-butane, n-pentane, n-heptane, and n-octane, respectively). More recent studies by Vrentas (Vrentas and Vrentas, 1990) and others (Mauritz et al., 1990) further advanced the theory of Cohen and Turnbull by proposing that the elementary diffusive displacement may involve only a fraction of the total molecular length. As demonstrated in Fig. 8, because a solute molecule partitioned into the bilayer interior is preferentially oriented with its long axis along the bilayer normal (Mulders et al., 1986; Pope et al., 1984; Xiang and Anderson, 1994a), an effective transbilayer displacement may occur upon the opening of an adjacent free volume with a cross-sectional area equal to or greater than the minimum cross-sectional area of the diffusing permeant.

View larger version (25K): [in this window] [in a new window]   FIGURE 8   A schematic illustration of the effects of the free surface area of lipid bilayer membranes on the permeation of two permeants with the same molecular volume but different cross-sectional areas. (A) A lower free surface area. (B) A higher free surface area.

Whereas the effects of permeant size on permeability are typically treated in terms of changes in permeant diffusivity, large size effects on solute partitioning into interphases are evident by the high resolution attainable in chromatographic separations on the basis of subtle differences in the size and shape of the analyte (Wise et al., 1981). Schnitzer (1988) previously described the partitioning behavior of molecules into porous networks and membranes in terms of a free-volume model in which the partition coefficient of a bilayer membrane can be expressed by an exponential function:

(10)

A statistical mechanical theory developed recently by the authors also suggests that solute partitioning into the barrier region of lipid bilayers is disfavored by chain ordering and that the selectivity of the barrier domain for permeant size is amplified with increases in surface density (i.e., decreases in free surface area) (Xiang and Anderson, 1994a). These results have been confirmed in molecular dynamics simulations, which have demonstrated a strong size dependence for solute partition coefficients in the order chain region in lipid bilayers (Marrink and Berendsen, 1996; Xiang and Anderson, 1998) and that the size dependence is amplified at high surface densities (data not shown). Experimentally, we have also shown that the size dependence of solute permeation across egg lecithin bilayers can be better described by a hybrid model incorporating the effects of solute size on both diffusion and partition coefficients, even though more adjustable parameters are involved (Xiang and Anderson, 1994b).

Whereas studies of size effects on partitioning have not explored the issue of whether the dependence of the partition coefficient in bilayers correlates better with solute volume or cross-sectional area, previous studies have shown that because of the ordering of lipid molecules in bilayer membranes, solute molecules residing in the bilayer interior are oriented with their long axes preferentially along the bilayer normal (Mulders et al., 1986; Pope et al., 1984; Xiang and Anderson, 1994a). Theoretically, this alignment minimizes the work required to create a cavity to accommodate the solute molecule in the lipid bilayer (Xiang and Anderson, 1994a). Thus the resistance to the transbilayer movement of a solute in the bilayer interior due to the partitioning term may depend primarily on the cross-sectional area along the long axis of the solute.

In summary, this study has systematically explored the effects of permeant size and bilayer chain packing on permeability across gel and liquid-crystalline DPPC bilayers, using a series of seven monocarboxylic acid permeants differing in chain length and degree of chain branching. First, we defined the permeability decrement f as the ratio of the observed permeability coefficients to those predicted by bulk solubility-diffusion theory to account for the decreases in permeability coefficients due to chain ordering. This chain-ordering correction factor was empirically shown to depend exponentially on the ratio of permeant cross-sectional area to the mean free surface area of the membrane. This strategy of investigating simultaneously the effects of permeant size on permeability in bilayers varying in packing density enables us to establish, for the first time, a model (cf. Eq. 7) that combines the effects of bilayer chain packing and permeant size on permeability across lipid bilayer membranes.