How is Membrane Permeation of Small Ionizable Molecules Affected by Protonation Kinetics?

Abstract

According to the pH-partition hypothesis, the aqueous solution adjacent to a membrane is a mixture of the ionization states of the permeating molecule at fixed Henderson-Hasselbalch concentrations, such that each state passes through the membrane in parallel with its own specific permeability. An alternative view, based on the assumption that the rate of switching ionization states is instantaneous, represents the permeation of ionizable molecules via an effective Boltzmann-weighted average potential (BWAP). Such an assumption is used in constant-pH molecular dynamics simulations. The inhomogeneous solubility-diffusion framework can be used to compute the pH-dependent membrane permeability for each of these two limiting treatments. With biased WTM-eABF molecular dynamics simulations, we computed the potential of mean force and diffusivity of each ionization state of two weakly basic small molecules: nicotine, an addictive drug, and varenicline, a therapeutic for treating nicotine addiction. At pH = 7, the BWAP effective permeability is greater than that determined by pH-partitioning by a factor of 2.5 for nicotine and 5 for varenicline. To assess the importance of ionization kinetics, we present a Smoluchowski master equation that includes explicitly the protonation and deprotonation processes coupled with the diffusive motion across the membrane. At pH = 7, the increase in permeability due to the explicit ionization kinetics is negligible for both nicotine and varenicline. This finding is reaffirmed by combined Brownian dynamics and Markov state model simulations for estimating the permeability of nicotine while allowing changes in its ionization state. We conclude that for these molecules the pH-partition hypothesis correctly captures the physics of the permeation process. The small free energy barriers for the permeation of nicotine and varenicline in their deprotonated neutral forms play a crucial role in establishing the validity of the pH-partitioning mechanism. Essentially, BWAP fails because ionization kinetics are too slow on the time scale of membrane crossing to affect the permeation of small ionizable molecules such as nicotine and varenicline. For the singly protonated state of nicotine, the computational results agree well with experimental measurements (P₁ = 1.29 × 10^-7 cm/s), but the agreement for neutral (P₀ = 6.12 cm/s) and doubly protonated nicotine (P₂ = 3.70 × 10^-13 cm/s) is slightly worse, likely due to factors associated with the aqueous boundary layer (neutral form) or leaks through paracellular pathways (doubly protonated form).

Figure 1:

Chemical structures and Henderson–Hasselbalch fractions of each protonation state ($f_0=\text{neutral}$, $f_1=\text{singly}\text{protonated}$, $f_2=\text{doubly}\text{protonated}$) based on aqueous $\text{p}{K}_{\text{a}}$ over a range of pH for A. nicotine ($\text{p}{K}_{\text{a}}=3.22$ for $s=2$, and $\text{p}{K}_{\text{a}}=7.72$ for $s=1$) and B. varenicline ($\text{p}{K}_{\text{a}}=9.22$ for $s=1$)

Figure 2:

Symmetrized PMF (top) and diffusivity (bottom) of A. nicotine and B. varenicline for each protonation state (0 = neutral, 1 = singly protonated, 2 = doubly protonated). The PMF was computed by WTM-eABF enhanced sampling with the C36 additive force field. The diffusivity was computed by MSD along $z$ with a lag time of 20 ps via eq 27. The PMF and diffusivity correspond to a POPC bilayer at 308 K.

Figure 3:

Offset PMFs (top three rows) over pH = 4, 7, and 10 for A. nicotine and B. varenicline in each ionization state, where in each case the singly protonated state is arbitrarily set to 0. The offset PMFs are are equal to $W_s\left(z\right)+\Delta G_{s\to s^{\prime }}$, where $s^{\prime }=s+1$, and $\text{Δ}{G}_{s\to s^{\prime }}=\ln \left[10\right]k_{\text{B}}T\left(\text{p}{K}_{\text{a}}\left(s^{\prime }\right)-\text{pH}\right)$ is the free energy of protonation in aqueous solution.^, The offsets are equivalent to that obtained by the normalized equilibrium probability density (eq 7). Resulting BWAP (bottom row), $W_{\text{m}}$, for the same range of pH values, computed via eq 16.

Figure 4:

Effective permeability results based on the BWAP and pH-partitioning treatments for A. nicotine and B. varenicline. Logarithm of permeability (top) over a range of pH using the BWAP (eq 16) and pH-partitioning (eq 2) treatments. Ratio (bottom) of the permeability computed for the fast and slow limits.

Figure 5:

Committors (top) of each protonation state across the membrane based on eq 25, and the components $\delta J_{\text{off}}$ and $\delta J_{\text{on}}$ (bottom) of the deprotonation/protonation contribution to the overall steady-state flux for A. nicotine and B. varenicline. The value of the committor represents the probability that the molecule will cross to the right before returning to the left, given its current position. The component $\delta J_{\text{on}}$ is calculated with eq 34, with $k_{\text{on}}={10}^3{\text{s}}^{-1}$, corresponding to the protonation rate in bulk solution at $\text{pH}=7$, and the component $\delta J_{\text{off}}$ is calculated with eq 33. Note that the doubly protonated state of nicotine was not included in the committor flux model for the sake of simplicity.

Figure 6:

Logarithm of permeability versus pH for each component of the total reactive flux (top) and ratio of the permeability derived from the BWAP and perturbed reactive flux to the pH-partitioning treatment for A. nicotine and B. varenicline. Top: The perturbed reactive flux permeability (solid, eq 22) is shown in its individual components, where the diffusive component (dashed, eq 23) tracks the pH-partition permeability, and the state transition component (dotted, eq 24) reaches a constant value at low pH. Bottom: The permeability ratio relative to the pH-partition treatment is shown for the perturbed reactive flux approach (black line) and BWAP (green line). A ratio close to 1 indicates agreement with pH-partitioning. For nicotine, the ratio is displayed as a dotted line when the perturbative approach becomes invalid because the protonation rate is too large (below pH = 4).

Figure 7:

A. Typical permeation trajectories of nicotine at $\text{pH}=7$ generated by combining Brownian dynamics with Markov state model (BD-MSM) for allowing transitions between different discrete states. The segments of the trajectory corresponding to a neutral deprotonated and charged protonated nicotine are shown in blue and red, respectively. The protonation/deprotonation rates were multiplied by a factor of 100 to increase the probability of capturing the events depicted in the figure. From top to bottom: a neutral crossing, deprotonation and subsequent neutral crossing, protonation following neutral crossing, and sequential deprotonation–neutral crossing–protonation. B. Permeability computed by counting crossings in the combined BD and Markov model for several ionization kinetics accelerating factors (fac) at $\text{pH}=7$ (fac = 0 does not allow switching states, fac = 1 uses the experimental rate). Each permeability is the mean of 100 trajectories, for which 16 were initiated in the neutral state and 84 were initiated in the protonated state. The BD was run with a timestep of 0.01 ps in 1-dimension with periodic boundaries at 40 Å. The position dependent forces, diffusivity and its derivative, and local $k_{\text{off}}$ were computed by linear interpolation from the computed PMFs and diffusivity profiles.