Free-Energy Profiles for Membrane Permeation of Compounds Calculated Using Rare-Event Sampling Methods

Statistical Mechanical Theories of Membrane Permeability

A popular theoretical framework to compute the permeability coefficient of a molecule is provided by the classic Smoluchowski-Kramers treatment of the steady-state diffusive flux across a free-energy barrier. Within this framework, commonly termed "inhomogeneous solubility-diffusion" (ISD), the permeability, P, is expressed in closed form in terms of the potential of mean force and position-dependent diffusivity of the molecule of interest along the membrane normal. In principle, both quantities can be calculated from all-atom MD simulations. Although several methods exist for calculating the position-dependent diffusivity, each of these is at best an estimate. In addition, the ISD model does not account for memory effects along the chosen reaction coordinate. For these reasons, it is important to seek alternative theoretical formulations to determine the permeability coefficient that are able to account for the factors ignored by the ISD approximation. Using Green-Kubo linear response theory, we establish the familiar constitutive relation between the flux density across the membrane and the difference in the concentration of a permeant molecule, j = PΔC. On this basis, we derive a time-correlation function expression for the nonequilibrium flux across a membrane that is reminiscent of the transmission coefficient in the reactive flux formalism treatment of transition rates. An analysis based on the transition path theory framework is exploited to derive alternative expressions for the permeability coefficient. The different strategies are illustrated with stochastic simulations based on the generalized Langevin equation in addition to unbiased molecular dynamics simulations of water permeation of a lipid bilayer.

A methodology of quantifying membrane permeability based on returning probability theory and molecular dynamics simulation

We propose a theoretical approach to estimate the permeability coefficients of substrates (permeants) for crossing membranes from donor (D) phase to acceptor (A) phase by means of molecular dynamics (MD) simulation. A fundamental aspect of our approach involves reformulating the returning probability (RP) theory, a rigorous bimolecular reaction theory, to describe permeation phenomena. This reformulation relies on the parallelism between permeation and bimolecular reaction processes. In the present method, the permeability coefficient is represented in terms of the thermodynamic and kinetic quantities for the reactive (R) phase that exists within the inner region of a membrane. One can evaluate these quantities using multiple MD trajectories starting from phase R. We apply the RP theory to the permeation of ethanol and methylamine at different concentrations (infinitely dilute and 1 mol % conditions of permeants). Under the 1 mol% condition, the present method yields a larger permeability coefficient for ethanol (0.12 ± 0.01 cm s-1) than for methylamine (0.069 ± 0.006 cm s-1), while the values of the permeability coefficient are satisfactorily close to those obtained from the brute-force MD simulations (0.18 ± 0.03 and 0.052 ± 0.005 cm s-1 for ethanol and methylamine, respectively). Moreover, upon analyzing the thermodynamic and kinetic contributions to the permeability, we clarify that a higher concentration dependency of permeability for ethanol, as compared to methylamine, arises from the sensitive nature of ethanol's free-energy barrier within the inner region of the membrane against ethanol concentration.

Leveraging Machine Learning-Guided Molecular Simulations Coupled with Experimental Data to Decipher Membrane Binding Mechanisms of Aminosterols

Understanding the molecular mechanisms of the interactions between specific compounds and cellular membranes is essential for numerous biotechnological applications, including targeted drug delivery, elucidation of the drug mechanism of action, pathogen identification, and novel antibiotic development. However, estimation of the free energy landscape associated with solute binding to realistic biological systems is still a challenging task. In this work, we leverage the Time-lagged Independent Component Analysis (TICA) in combination with neural networks (NN) through the Deep-TICA approach for determining the free energy associated with the membrane insertion processes of two natural aminosterol compounds, trodusquemine (TRO), and squalamine (SQ). These compounds are particularly noteworthy because they interact with the outer layer of neuron membranes, protecting them from the toxic action of misfolded proteins involved in neurodegenerative disorders, in both their monomeric and oligomeric forms. We demonstrate how this strategy could be used to generate an effective collective variable for describing solute absorption in the membrane and for estimating free energy landscape of translocation via on-the-fly probability enhanced sampling (OPES) method. In this context, the computational protocol allowed an exhaustive characterization of the aminosterol entry pathway into a neuron-like lipid bilayer. Furthermore, it provided accurate prediction of membrane binding affinities, in close agreement with the experimental binding data obtained by using fluorescently labeled aminosterols and large unilamellar vesicles (LUVs). The findings contribute significantly to our understanding of aminosterol entry pathways and aminosterol-lipid membrane interactions. Finally, the computational methods deployed in this study further demonstrate considerable potential for investigating membrane binding processes.

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

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).

[Development of Membrane Permeability Coefficient by Means of Novel Molecular Dynamics Methods]

The membrane permeability, and its evaluation, is crucial factor in the process of uptake of compounds from outside to inside the cell and in the inhibition of the activity of disease-causing target proteins. Although molecular dynamics (MD) simulations have been shown to be able to reproduce the conformational changes of compounds occurring during membrane permeation, it is still challenging to extract the membrane permeability at an affordable computational workload solely by conventional MD. Indeed, the time scale accessible by MD is far below the one characterizing the actual permeation process. Phenomena occurring in living organisms escaping the reach of standard MD are generally referred to as biological rare events, and the membrane permeation process is one of them. To overcome this time-scale problem, several enhanced sampling methods have been proposed over the years to improve conformational sampling. In this review, a hybrid sampling method that combines the parallel cascade selection MD (PaCS-MD) and the outlier flooding method (OFLOOD), introduced and developed by our group, is proposed as a tool to study the membrane permeation from structural sampling (rare-event sampling). The obtained trajectories are used to estimate the free energy profiles for the membrane permeation and to compute the membrane permeation coefficients. Moreover, we present an example of application of the free energy reaction network method as a versatile way for incorporating explicitly into reaction coordinates the degrees of freedom related to internal motion.

Recent advances on molecular dynamics-based techniques to address drug membrane permeability with atomistic detail

Several factors affect the passive membrane permeation of small molecules, including size, charge, pH, or the presence of specific chemical groups. Understanding these features is paramount to identifying or designing drug candidates with optimal ADMET properties and this can be achieved through experimental/knowledge-based methodologies or using computational approaches. Empirical methods often lack detailed information about the underlying molecular mechanism. In contrast, Molecular Dynamics-based approaches are a powerful strategy, providing an atomistic description of this process. This technique is continuously growing, featuring new related methodologies. In this work, the recent advances in this research area will be discussed.