Accurate Potentiometric Determination of Lipid Membrane–Water Partition Coefficients and Apparent Dissociation Constants of Ionizable Drugs: Electrostatic Corrections

Abstract

Purpose

Potentiometric lipid membrane–water partition coefficient studies neglect electrostatic interactions to date; this leads to incorrect results. We herein show how to account properly for such interactions in potentiometric data analysis.

Materials and Methods

We conducted potentiometric titration experiments to determine lipid membrane–water partition coefficients of four illustrative drugs, bupivacaine, diclofenac, ketoprofen and terbinafine. We then analyzed the results conventionally and with an improved analytical approach that considers Coulombic electrostatic interactions.

Results

The new analytical approach delivers robust partition coefficient values. In contrast, the conventional data analysis yields apparent partition coefficients of the ionized drug forms that depend on experimental conditions (mainly the lipid-drug ratio and the bulk ionic strength). This is due to changing electrostatic effects originating either from bound drug and/or lipid charges. A membrane comprising 10 mol-% mono-charged molecules in a 150 mM (monovalent) electrolyte solution yields results that differ by a factor of 4 from uncharged membranes results.

Conclusion

Allowance for the Coulombic electrostatic interactions is a prerequisite for accurate and reliable determination of lipid membrane–water partition coefficients of ionizable drugs from potentiometric titration data. The same conclusion applies to all analytical methods involving drug binding to a surface.

Appendix

The intrinsic (lipid-dependent) membrane surface charge density, σ _mem, is calculated from membrane composition:

$$\sigma _{mem} = z_L e_0 \cdot \frac{{x_C }}{{\left( {1 - x_C } \right) \cdot A_N + x_C \cdot A_C }}{\text{ }} \approx \frac{{z_L e_0 x_C }}{{A_L }}\quad ,$$

(9)

where z _L is the charged lipid valence, e ₀ the elementary electric charge, x _C the molar fraction of the charged lipids with molecular area A _C, and A _N the average surface area of the neutral lipid molecules. In the current study, we used A _N ≈ A _C ≈ 0.65 nm², which is a good approximation for typical fluid-phase phospholipids. During potentiometric titration, the relative proportion of the charged lipids may vary, if the employed pH range overlaps with the lipid titration range. If so, the resulting surface charge density variation must be considered.

The average drug-dependent membrane surface charge density, σ _D, is calculated analogously:

$$\sigma _D = z_D e_0 \frac{{C_{mem}^I }}{{C_L A_L + \left( {C_{mem}^I + C_{mem}^N } \right)A_D }} \approx z_D e_0 \frac{{C_{mem}^I }}{{C_L A_L }}\quad .$$

(10)

z _D is the bound drug valence, C _L the membrane–forming lipid concentration, $C^{{\text{I}}}_{{{\text{mem}}}}$ and $C^{{\text{N}}}_{{{\text{mem}}}}$ the concentrations of the membrane associated ionized and neutral drug forms, respectively. The contribution of the membrane associated drug molecules to the lipid bilayer surface area, A _D, is normally relatively small. It can thus be neglected. We can now calculate σ _D once $C^{{\text{I}}}_{{{\text{mem}}}}$ is known.

To calculate $C^{{\text{I}}}_{{{\text{mem}}}}$, we will start with the total drug concentration C _tot = C _mem + C _aq. Substitution of C _aq from Eq. 2 and rearrangement allow calculation of the total membrane bound drug concentration:

$$C_{mem} = C_{tot} \frac{{P{\text{ }}r}}{{1 + P{\text{ }}r}}\quad .$$

(11)

The ionized drug fraction, α, is:

$$\alpha = \left\{ \begin{aligned}& \frac{{10^{pH - pK_a^{app} } }}{{10^{pH - pK_a^{app} } + 1}}{\text{ for an acid}} \\& \\& \frac{{10^{pK_a^{app} - pH} }}{{10^{pK_a^{app} - pH} + 1}}{\text{ for a base}} \\ \end{aligned} \right.\quad .$$

(12)

Combining Eqs. 6, 11, and 12 yields:

$$C_{mem}^N = C_{tot} \cdot \left( {1 - \alpha {\text{ }}} \right) \cdot \frac{{P_0^N {\text{ }}r}}{{1 + P_0^N {\text{ }}r}}{\text{ }}$$

(13)

$$C_{mem}^I = C_{tot} {\text{ }}\alpha \frac{{P_0^I r exp \left[ { - \Phi \left( {\sigma _{mem} + \sigma _D^\# } \right)} \right]}}{{1 + P_0^I r exp \left[ { - \Phi \left( {\sigma _{mem} + \sigma _D^\# } \right)} \right]}}\;,$$

(14)

which takes into account the Coulombic electrostatic contributions from both σ _D and σ _mem. The superscript I denotes the deprotonated form, X, for an acidic drug and the protonated form, XH, for a basic drug (cf. Eqs. 7 and 8). All concentrations are defined relative to the total suspension volume. Combining Eqs. 10 and 14 finally yields the drug-dependent surface charge density:

$$\sigma _D \approx \frac{{z_D e_0 C_{tot} \alpha }}{{C_L A_L }} \cdot \frac{{P_0^I r exp \left[ { - \Phi \left( {\sigma _{mem} + \sigma _D^\# } \right)} \right]}}{{1 + P_0^I r exp\left[ { - \Phi \left( {\sigma _{mem} + \sigma _D^\# } \right)} \right]}}\quad .$$

(15)

The procedure is applicable at any pH value and ideally should involve the entire titration curve. σ _D is a function of Φ (or ψ) and vice versa, i.e. they are interdependent. The equation must thus be solved in a self-consistent fashion, and typically numerically. (We used Mathcad employing Secant and Mueller method for numerical solving.)

The Debye ion screening length, λ _D, is a property of the electrolyte solution and is given for 1:1 electrolytes by:

$$\lambda _D = \sqrt {\frac{{\varepsilon _0 \varepsilon _r k_B T}}{{2000{\text{ }}e_0 ^2 N_A C_{el} }}} \quad .$$

(16)

ε ₀ is the permittivity of free space (8.8542 × 10⁻¹² As/Vm), ε _r the dielectric constant at the drug binding site (an average value of 40 for the lipid head group was used), k _B the Boltzmann constant (1.38 × 10⁻²³ JK⁻¹), T the absolute temperature, e ₀ the elementary electric charge (1.602 × 10⁻¹⁹ C), N _A Avogadro’s number (6.02205 × 10²³ mol⁻¹), C _el the bulk molar electrolyte concentration.

The electrostatic potential, ψ, of a uniformly charged surface in contact with a 1:1 electrolyte is given within the framework of Gouy–Chapman approximation (38), as a function of the total surface charge density, σ = σ _D + σ _mem, by:

$$\psi \left( \sigma \right) = \frac{{2k_B T}}{{e_0 }}asinh\left( {\frac{{\lambda _D e_0 \sigma }}{{2\varepsilon _0 \varepsilon _r k_B T}}} \right)\quad .$$

(17)

σ is the surface charge density in Cm⁻² and asinh the inverse hyperbolic sine (areasinushyperbolicus). The normalized dimensionless electrostatic potential, Φ, is defined as the ratio of electrostatic potential energy, ze ₀ ψ, and thermal energy, k _B T:

$$\Phi \left( \sigma \right) = \frac{{ze_0 {\text{ }}\psi \left( \sigma \right)}}{{k_{\text{B}} T}}\quad .$$

(18)

Numeric approximations to Eqs. 16 and 18 are given in Table V.

Table V Numeric Approximations to Eqs. 16 and 18 used for Calculating of the Debye–Hückel Screening Length, λ _D, and the Normalized Electrostatic Potential, Φ, in a 1:1 Electrolyte

According to the Gouy–Chapman model, the relationship between the electrostatic potential ψ(x) at distance x from a uniformly charged surface and the electrostatic surface potential ψ(x = 0), is:

$$x = ln \left\{ {\frac{{tanh\left[ {{{e_0 \psi \left( {x = 0,\sigma } \right)} \mathord{\left/{\vphantom {{e_0 \psi \left( {x = 0,\sigma } \right)} {4k_B T}}} \right.\kern-\nulldelimiterspace} {4k_B T}}} \right]}}{{tanh\left[ {{{e_0 \psi \left( {x,\sigma } \right)} \mathord{\left/{\vphantom {{e_0 \psi \left( {x,\sigma } \right)} {4k_B T}}} \right.\kern-\nulldelimiterspace} {4k_B T}}} \right]}}} \right\} \cdot \lambda _D \quad .$$

(19)

This provides means for estimating the effective distance between the lipid and the drug charges in a membrane.

The electrostatic correction described in this article allows only for the Coulombic, i.e. charge–charge interactions. Other contributions, such as hydration (polarity) effects, can be influential as well. If they are not small, such interactions should be considered, following the basic, self-consistent approach described in this work.