Molecular permeation through large pore channels: Computational approaches and insights

Abstract

Computational methods such as molecular dynamics (MD) have illuminated how single-atom ions permeate membrane channels and how selectivity among them is achieved. Much less is understood about molecular permeation through eukaryotic channels that mediate flux of small molecules (e.g., connexins, pannexins, LRRC8s, CALHMs). Here we describe computational methods that have been profitably employed to explore the movements of molecules through wide pores, revealing mechanistic insights, guiding experiments and suggesting testable hypotheses. This review illustrates MD techniques such as voltage-driven flux, potential of mean force, and mean first-passage-time calculations, as applied to molecular permeation through wide pores. These techniques have enabled detailed and quantitative modeling of molecular interactions and movement of permeants at the atomic level. We highlight novel contributors to the transit of molecules through these wide pathways. In particular, the flexibility and anisotropic nature of permeant molecules, coupled with the dynamics of pore lining residues, lead to bespoke permeation dynamics. As more eukaryotic large-pore channel structures and functional data become available, these insights and approaches will be important for understanding the physical principles underlying molecular permeation and as guides for experimental design.

Graphical Abstract

Introduction

The functional definition of a wide pore channel is that it is a channel that mediates passive transport of small molecules. The permeation pathways of these channels are not featureless, water-filled right cylinders or truncated cones - each type of wide pore channel has a permeation pathway with unique and complex axial and radial topography, and discrete steric, electrostatic and conformational features composed of amino acid backbones and sidechains. For reasons discussed below, the energetics and mechanisms of molecular permeation and selectivity of wide pores differ from those that govern atomic ion permeation and selectivity in classical ion-selective (“narrow”) pores in which the energetics of permeant dehydration play a major role. This article reviews the computational methods that are being adapted, applied and utilized to investigate, at an atomistic level, the determinants of molecular permeation through these channels. These computational methods can provide insights and predictions about the permeation process ‐ the forces, sites and interactions involved ‐ that are difficult to obtain by other means, providing the basis for testable hypotheses.

The most well-studied family of wide pore channels, the connexin family, exhibits a range of selectivities among molecular permeants across its 21 human isoforms. Channels composed of each connexin isoform exhibit remarkable and differing degrees of selectivity among molecular permeants of similar size and charge (Harris, 2008; Harris & Locke, 2009; Ek-Vitorin & Burt, 2013). Much less is known about molecular selectivity of other wide pores, but presumably analogous interactions with permeants occur within those pores that are amenable to the computational methods applied thus far primarily to connexins. This Special Issue focuses on wide pores formed by alpha-helical transmembrane domains in eukaryotes rather than the beta-barrel pores found in, or originating from, prokaryotes (e.g., porins, VDAC). However, computational methods have been profitably employed to explore molecular permeation of beta-barrel pores, and examples from this work are included.

Most wide pores other than connexins are known primarily for their permeability to ATP, but with the exception of LRRC8 channels, detailed information regarding permeability to other molecules is mostly lacking (Figure 1) (Weber et al., 2004). For example, ATP permeates CALHM channels (Ma et al., 2016), but to our knowledge, permeability to other molecules has not been assessed. In addition to the prominent role of pannexin channels in ATP release, there is indication that caspase-mediated opening of pannexin channels mediates release of certain cytosolic metabolites, and not others, without a clear basis for the distinction (Medina et al., 2020). Innexin channels are permeable to ATP and 6-carboxyfluorescein (Bao et al., 2007; Sangaletti et al., 2014), and unnexin is permeable to DAPI (Guiza et al., 2023). LRRC8 channels mediates permeability to a host of molecules, including amino acids, neurotransmitters, anticancer drugs, inositol, some antibiotics and even a dicyclic nucleotide (cGAMP; 2’3’-cyclic-GMP-AMP) (Jackson & Strange, 1993; Hisadome et al., 2002; Hyzinski-Garcia et al., 2014; Qiu et al., 2014; Planells-Cases et al., 2015; Jentsch, 2016; Lahey et al., 2020). However, there is little quantitative information regarding the absolute or relative permeabilities of these molecules.

Figure 1: Wide pore channels.

(a) The structures of several molecules shown to permeate different wide pore channels are shown. Molecules are depicted in CPK representation, with atoms colored according to their element (carbon=cyan, oxygen=red, nitrogen=blue, phosphorus=brown). Surface representations are shown of cut-through views (b) and top views (c) of each type of wide pore channel, with each subunit uniquely colored. The PDB IDs 2ZW3 (Maeda et al., 2009), 6UIV (Choi et al., 2019), 6LTO (Mou et al., 2020) and 5H1Q (Oshima et al., 2016) are below the structures.

In contrast, a large number of studies describes the molecular permeability of connexin channels, qualitatively and quantitatively. In general, connexin channels are permeable to molecules in the size range of cellular signaling molecules, including cAMP, cGMP, ATP, IP3, glutamate, and glutathione (Harris, 2007, 2008; Harris & Locke, 2009). Channels formed by each of the human connexin isoforms have strikingly different molecular selectivities, particularly among biological molecules (Veenstra, 1996; Goldberg et al., 2004; Harris, 2007, 2008; Kanaporis et al., 2008; Harris & Locke, 2009; Kanaporis et al., 2011; Hansen et al., 2014; Valiunas et al., 2018; Brink et al., 2020). These connexin-specific permeability properties are not readily inferred from estimated pore widths, differences in permeability to fluorescent tracers or charge selectivity among atomic ions (Veenstra et al., 1995; Bevans et al., 1998; Beltramello et al., 2005; Ayad et al., 2006). In addition, heteromeric connexin channels (composed of more than one isoform) can have molecular selectivities distinct from those of the respective homomeric channels (Bevans et al., 1998; Martinez et al., 2002; Locke et al., 2004; Weber et al., 2004; Sun et al., 2005; Ayad et al., 2006).

Quantitative permeability data, along with high-resolution structural data, provide rich context and basis for computational investigation of mechanisms involved in selective molecular permeation. Of particular importance for computational studies, the connexin literature offers (a) relative permeabilities of different molecules through the same or different types of connexin channel, and (b) in some cases, quantitative data about the absolute permeability of specific permeants (Bedner et al., 2006; Eckert, 2006; Hernandez et al., 2007; Rackauskas et al., 2007; Harris, 2008; Kanaporis et al., 2008; Kanaporis et al., 2011; Valiunas et al., 2019; Valiunas & White, 2020). Availability of this information is essential for validation of computational findings. For these reasons, most of the illustrations of computational approaches described in this article involve connexin channels, most prominently those formed by connexin26 (Cx26).

The starting point for any atomistic calculation is a structural map of the channel protein at sufficient resolution to identify backbone and sidechain positions. In the modern era, these are derived from single-particle cryo-EM studies, structure prediction tools such as AlphaFold2/3 (Jumper et al., 2021; Abramson et al., 2024), RoseTTAFold (Baek et al., 2021), and homology modeling tools such as Modeller (Fiser & Sali, 2003), and SwissModel (Bienert et al., 2017). Primary considerations are whether the structures so obtained are (a) physiologically relevant, and (b) whether they correspond to the functional state of the protein from which its permeability properties have been experimentally determined. Mutations, sample preparation conditions and data processing can result in maps that fail to meet these criteria. In fact, there is no independent way to validate either criterion. As a result, confidence in the ability of computations to reveal mechanistic insights regarding physiological molecular permeation mechanisms relies on a tautology: If the computations produce estimates of functional properties (e.g., absolute or relative permeabilities) that correspond well to those measured experimentally, one may regard both the structure and the computations as reasonably valid for this purpose; the structure and the computations “cross-validate” each other. The rationale is that since the sources of error in a structural map are entirely distinct from the sources of error in a computational strategy, it is unlikely that they would precisely compensate for each other to mimic well experimentally measured physiological properties. Therefore, the primary test for validation of the use of a structure for detailed studies is comparison of a computational result with an experimental one. If a structure satisfies this criterion, one may proceed with a modicum of confidence in the use of that structure for computational studies that explore permeation mechanism, energetics and interactions in greater depth. However, a structure validated solely on the basis of its derived permeability properties should not be presumed to be necessarily valid for study of gating processes.

Computational Considerations

Pore size in the context of molecular permeation

When studying molecular permeation, often the first feature one inspects in a structure is the pore, to assess whether it is sufficiently wide to permit passage of a known molecular permeant (e.g., ATP). If this criterion is satisfied, one may consider the structure to be in an “open” state. However, this determination is often not straightforward. Because a structural map is static, measurement of distance between the most closely apposed atoms (including their van der Waals radii) across the narrowest part of the pore may not correspond well to the limiting diameter experienced by a permeating molecule. Sidechains are flexible and backbones can “breathe”. Aside from these dynamic considerations, calculations of effective (rigid) pore diameter can be problematic, especially when applied to computer-equilibrated structures that have lost radial symmetry that may have been imposed in the initial map. For example, the HOLE algorithm (Smart et al., 1996) and CHAP (Klesse et al., 2019) use a spherical probe of increasing radius, which underestimates the cross-sectional area of the lumen of a radially irregularly-shaped pore. One solution is to employ an ellipsoidal instead of a spherical probe so as to better capture pore asymmetry (David Seiferth. & Biggin, 2024). Alternatives include the grid-based cavity search program trj_cavity (Paramo et al., 2014). For example, the average pore radius of Cx26 obtained using trj_cavity is larger than the radius obtained using the HOLE program (Jiang et al., 2021). Besides the geometry, pore surface hydrophobicity can play an important role in determining ionic permeability (Jia et al., 2018; Seiferth et al., 2022).

A greater challenge is consideration of the effect of conformational coupling between pore-lining moieties and a permeant as it transits the pore. Computational simulations can yield probability distributions of the positions of each channel-lining moiety and thereby an average and standard deviation of limiting pore width along its length. However, this omits the dynamics that may cause the limiting diameter (and the moieties involved) to fluctuate substantially in response to the presence of the molecular permeant, with consequences for permeation. As mentioned above and illustrated below, permeating (or non-permeating) molecules in the pore can affect these pore dynamics as they interact with pore-lining moieties. For example, computations indicate that the presence of cAMP at different locations inside the pore exerts local and non-local effects on pore radius (Jiang et al., 2021). Since these are elemental features of molecular permeation through wide pores, computational studies that seek to understand the process should endeavor to incorporate them, as much as computationally feasible, since to omit them limits the utility of computations to reveal realistic mechanism.

From continuum theory to molecular dynamics

Continuum theories use a set of partial differential equations to describe ion permeation through channels under the influence of an electrochemical gradient. The equations include the Poisson-Boltzmann equation, which describes the electrostatic potential distribution of the system, and the Nernst-Planck equations, which describe the flux of ions (Corry et al., 2000; Guardiani et al., 2022). However, the continuum approach represents protein, water and membrane as continuous media (i.e., not composed of discrete atomic structures), and neglects fluctuations and correlations between elements in the system, which are particularly important in molecule permeation through large and flexible pores. Various Brownian dynamics (BD) approaches with implicit solvent have been adapted and expanded to explicitly simulate the diffusional motions of individual ions or molecules. In these BD simulations, channel and ions can be described in atomic detail, but membrane, water, and bulk electrolyte usually remain as homogeneous dielectric media. In majority of these applications, only ions are moving while everything else is fixed (Im & Roux, 2002; Egwolf et al., 2010; Kwon et al., 2011). Successful BD simulations of molecular permeation include, but are not limited to, analysis of antibiotic permeabilities through a porin channel (Acharya et al., 2023) and DNA sequencing through nanopores (Comer & Aksimentiev, 2012).

However, as will be described below, when the conformations of the pore elements are coupled to the permeant (e.g., in which local pore-permeant interaction alters the sidechain, backbone and/or permeant conformation) atomistic molecular dynamics (MD) are more desirable. In atomistic MD, every atom in the system is represented explicitly and movements are propagated by integrating the classical equations of motion (Delemotte & Luo, 2023). Therefore, the dynamics and interactions between permeant and environment (protein, lipid, water, ions) can be modeled in detail within the timescale of the simulation. Published studies of atomistic simulations of molecule permeation include ATP transport through VDAC (Noskov et al., 2013; Choudhary et al., 2014), and cAMP and dye molecules (e.g., DAPI, ethidium, calcein) through connexin channels (Zonta et al., 2013); Luo et al. (2016); (Jiang et al., 2021; Gaete et al., 2024). Notably, in both the VDAC and connexin simulations, interactions between the permeating molecule and the corresponding flexible N-termini play critical roles in the permeation process. These considerations make atomistic MD a valuable and appropriate tool for investigation of molecular permeation through wide pores. General instructions for setting up and performing MD simulations of connexin channels are presented in (Zonta et al., 2024). We summarize below MD methods for investigating and quantifying molecular permeation, with a focus on their application to eukaryotic wide pores with alpha-helical transmembrane domains.

The notion of the periodic boundary condition

In standard MD simulation, an ion channel is embedded in a small membrane patch (200~400 nm²) and placed in a solution bath. To mimic the biological environment, the model must incorporate continuity of membrane and solution environment beyond the boundaries of the simulation box to reduce finite-size effects. This is usually done by applying periodic boundary conditions (PBC) in which the system is virtually replicated infinitely in all 3 spatial directions. In this way, the simulation box is surrounded by images of itself instead of by vacuum, and the overall system being modelled is an infinite membrane stack (Figure 2a). The PBC thereby guarantees symmetric ion and solute concentration at the two ends of the channel. For instance, a permeant that exits the solution at the bottom of the simulation box after transit through the pore will re-appear in the solution at the top (Figure 2b). The PBC thus allows simulation of continuous net directional flux through the pore under an external electric field without generating a concentration gradient. If a permeant concentration gradient is desired, a dual-membrane model can be constructed, and grand canonical Monte Carlo applied during the simulations to maintain a desired permeant gradient across the membrane (Kutzner et al., 2011). Alternatively, to avoid a computationally costly dual-membrane system, asymmetric permeant concentration has been realized by application of a constant force to the ions at the edge of the simulation cell (Khalili-Araghi et al., 2013).

Figure 2: Periodic Boundary Condition (PBC) and dye permeation.

(a) Periodic images of a connexin26 (Cx26) hemichannel (cyan cartoon) within a lipid bilayer. The phosphorus atoms of the lipid head groups are in magenta; purple and green dots are potassium and chloride ions, respectively. The rest of the lipid atoms and water molecules are not shown. (b) Dye (DAPI) permeation through a Cx26 hemichannel. Traces depict the z-coordinate of DAPI molecules during inward flux (downward in the plot) under −200 mV from three replicas (blue, orange, and green). To the right, DAPI (licorice) is shown at four positions, colored by increasing time from red to white to blue. The N-terminal (NT) helices of the connexin are shown in red in cartoon. The C-alpha atoms in vdW are DAPI binding residues (D46/D50 in red, E42/E47 in purple, and D2 in green). The downward blue arrow indicates the direction of DAPI movement in the simulation. The upward curved arrow indicates that, due to the PBC, exit of a permeant from one side of the simulation box (from the extracellular end of the pore in this figure) is followed by its appearance at the other (intracellular) side, maintaining an unchanged concentration of permeant during permeant flux.

Force fields for molecular permeants

For MD simulation, a potential energy function, also referred to as the force field, describes the interactions among all particles in the system. Classical force fields (e.g., CHARMM (Huang et al., 2017) AMBER (Tian et al., 2020), OLPS(Jorgensen et al., 1996) for the atoms and molecules of proteins, lipid, and water have been parameterized (e.g., assigned bond lengths, bond angles, dihedral angles, van der Waals radii, partial charges) and refined over decades. However, the atom types of large pore permeants are often unique and not well-represented in existing force fields. In this case, the penalty score in the CHARMM General Force Field (CGenFF) (Vanommeslaeghe et al., 2010) is a useful measure of how well a particular molecule fits the parameters defined by the CGenFF. For example, the dihedral parameters of cAMP between the purine and ribofuranose groups have a penalty score > 200. Therefore, this dihedral required further optimization to match quantum-mechanical results (Jiang et al., 2021). Dye molecules used as tracers in transport studies of large pores often require an additional parameterization strategy due to their highly delocalized electron density. For instance, the atomic charges of ethidium and DAPI from CGenFF required were further optimization to reproduce the dipole orientations and magnitudes from quantum mechanics calculation (Gaete et al., 2024). Both the FFParam Python package (Kumar et al., 2020) and ffTK plugin in VMD (Mayne et al., 2013) can be used for permeant force field optimization compatible with CHARMM36 (Huang et al., 2017) protein and lipid force fields. Force field development is an active area of research. In particular, polarizable force fields (in which atomic charge distribution varies in response to surrounding electric field)(Cieplak et al., 2001; Patel & Brooks, 2004; Ponder et al., 2010; Lopes et al., 2013; Lemkul et al., 2016; Peng et al., 2016) and neural network potential (Smith et al., 2017) are promising avenues for obtaining more accurate and transferable force fields for small molecules.

Investigation of Molecular Permeation

What are the fundamental differences between molecular permeation of wide pores and atomic ion permeation through “narrow” pores and how can computations address them? In all cases thus far examined, a primary difference is that for single-atom ion conduction in ion-selective pores, the large energy cost of ion dehydration plays a major role in the energetics of permeation, whereas in wide pores there is little or no dehydration of the permeant. The absence of the substantial energy cost of dehydration means that smaller contributors to free energy play more prominent roles; energy terms that are of minor importance in ion selective channels become determining factors. These factors include hydrogen bonds (direct and indirect) and transient polar interactions between the permeant and pore-lining moieties. Molecular permeants can affect permeation energetics in ways that atomic ions cannot, as they:

• are anisotropic in molecular shape and charge distribution

• have rotational degrees of freedom that can be affected by steric and electrostatic interactions with the protein, as well as the voltage within the pore

• may have intrinsic conformational flexibility (e.g., become more compact or extended) as a function of interactions within the pore

Since most biological and dye permeants are charged, MD-simulated voltage-driven flux is particularly useful in investigating the aforementioned interactions. As described above, the PBC (Figure 2b) allows simulation of continuous unidirectional flux driven by voltage across the simulation box. To apply voltage, an external electric field, $E=V/\text{Iz}$, is applied to across the simulation system normal to the plane of the membrane, where V is the transmembrane voltage and lz is the height of the simulation box in the z direction (Gumbart et al., 2012). For example, voltage has been used to drive the translocation of ssDNA through various beta-barrel nanopores (Aksimentiev & Schulten, 2005; Mathe et al., 2005; Wells et al., 2007; Zhou et al., 2020). Molecule permeation under “moderate” voltage (< 500 mV) often occurs in the submicro to microsecond timescale. The special-purpose supercomputer Anton2 has made it possible to run simulations reaching micro to millisecond timescale (Shaw et al., 2014). With these longer simulations, multiple permeation events from independent simulations can thus be accumulated to derive the relative occupancy (i.e., molecule positional density) of a permeant along the pore central z-axis. For easier comparison among different permeants, the one-dimensional molecular density profile along the z-coordinate, $P(z)$, is usually converted to a smoothed log-density profile using Boltzmann inversion, $-kTln(P(z))$, where $kT$ is the Boltzmann constant multiplied by the absolute temperature used for the simulation (Figure 3). Because the permeation process is biased by external voltage, log-density profiles do not represent equilibrium free energy profiles. Nevertheless, they provide useful estimates of the locations and relative magnitudes of energetic barriers and binding regions of each permeant, which can be tested by mutagenesis (Jiang et al., 2021).

Figure 3: Log-density profiles.

of cAMP, DAPI and ethidium along the Cx26 hemichannel pore during voltage-driven inward flux (right to left, indicated by blue arrow) simulation. The bulk value is set to zero. The channel is cut through to show the surface of the pore. A single subunit is shown in cartoon mode. Plot reproduced using data from (Jiang et al., 2021; Gaete et al., 2024).

Permeant-pore interactions

Figure 3 shows that at the intracellular and extracellular entrances of the Cx26 hemichannel, the positional density of positively charged DAPI (+2) and ethidium (+1) largely mirrors that of negatively charged cAMP (−1). This suggests that the pore-permeant interactions at these regions are dominated by electrostatics. In particular, it was found that cAMP binds to R99/104 and K103 near z −20 Å, while positively charged dyes interact strongly with D46/50 and E42/47 near z +30 Å (Jiang et al., 2021; Gaete et al., 2024). However, the pore-permeant interactions at the center region (−10 < z < 10 Å) are more dynamic and complex. This is due to the movement of flexible N-terminal (NT) helices as they interact with the molecules as they pass through. DAPI has electrostatic interactions with D2 and tends to align vertically in the pore at the center region (Figure 4c). Ethidium, on the other hand, forms a strong π-stacking interaction with W3 residues, potentially obstructing permeation. Experimental studies showed that mutation of D2 and W3 significantly altered the permeability of both dye molecules in the predicted ways (Gaete et al., 2024). In contrast, cAMP experienced a broad energy barrier while crossing the NT region. The angle between the NT helix and the z-axis changes as DAPI passes through this region (illustrated in Figure 4a and 4b). A clear correlation is seen between DAPI z-position and NT angle at the center region. The electrostatic interactions between DAPI and D2 cause the terminal residues of the NT region to unwind and induce the NT angle change.

Figure 4: Coupling between molecular permeation and pore dynamics.

(a) Snapshot shows the NT angle measurement with respect to the bilayer normal z-axis. Backbone atoms of the first twelve residues are used to measure the NT angle. (b) 2D scatter plot illustrates the time evolution (blue to red) of NT angles as DAPI passes down the pore. (c) Snapshots showing the interaction between DAPI and the D2 side chain that induces the change in NT angle. Experimental mutagenesis studies have confirmed these interactions (Gaete et al., 2024). The Cx26 cut-through surface is in blue, the NT helices are in red cartoon representation. DAPI is in licorice form, and D2 in CPK representation.

Orientation of the permeant

Unlike for atomic ions, rotation of molecular permeants within the pore affects how they interact with pore-lining moieties. The orientation of permeant molecules inside large pores is determined by their molecular shape and charge distribution, in response to local steric and electrostatic interactions. For example, DAPI’s rigid and elongated shape enables formation of simultaneous interactions with charged amino acids from opposing subunits in the extracellular portion of the Cx26 pore. Breaking interactions from one side leads DAPI to align vertically in the pore, facilitating its interaction with the NT region (Figure 4c). In contrast, smaller and more flexible permeants, such as maltotriose (Luo et al., 2016) and cAMP rotate multiple times as they diffuse through Cx26 open pore in the absence of external voltage (Figure 5).

Figure 5: Rotation of permeants as they pass through the Cx26 pore.

(a) Overlaid snapshots depict the rotational behavior of a trisaccharide inside the pore (reproduced from (Luo, 2016)). (b) The probability distribution of cAMP dipole angles at each position as it transits the pore is presented as a heatmap. The color bar indicates the probability scale, with darker colors indicating higher probability. Representative dipole angles are indicated on the cAMP molecule depicted above (reproduced from data in (Jiang et al., 2021)).

Forces acting on the permeant in the pore

MD-based methods not only can help to visualize the behavior of the permeant, ions, water and pore-lining residues, they allow the free energy and kinetics of the permeation process to be quantified to reveal mechanistic details that can be tested experimentally. Because a molecule permeation process is usually a mix of small barrier crossing (diffusive) processes and large barrier crossing (binding/unbinding) processes, the overall transit rate is likely dominated by the equilibrium free energy landscape, described by the potential of mean force (PMF), which governs the systematic forces acting on the permeant as it passes through the channel. As a molecular permeant is relatively large relative to pore width, it may be assumed that its diffusion perpendicular to the pore z-axis reaches equilibrium faster than in the z-direction. Therefore, a one dimensional PMF along the z-axis may be sufficient to reflect the intrinsic energetics of the channel-permeant system. In the bulk region of a 1D PMF, a cylindrical restraint (typically a flat-bottom harmonic restraint on the x and y coordinates of the permeant center of mass) is applied to the permeant to confine its diffusion outside the pore. Care needs to be taken to choose the radius of cylindrical restraint larger than the pore radius so that the interaction within the pore is not altered.

Without an external driving force such as voltage or concentration gradient, a permeant will spend most of the simulation time close to a local energy minimum, making it computationally costly to study permeation events under true equilibrium conditions. To address this problem, the PMF can be computed using a set of algorithms called “Enhanced Sampling”. These algorithms often apply computational “tricks” to accelerate the process being studied (molecule permeation in this case), by adding external forces or bias potentials on the molecules or to restrict the sampling to specific regions of the pore. A post-analysis is then used to recover the unbiased Boltzmann ensemble, hence the free energy along the pore-axis. Popular methods, such as umbrella sampling (Allen et al., 2006), metadynamics (Furini & Domene, 2016), adaptive biasing force sampling (Li et al., 2017), their variations (e.g., Hamiltonian replica-exchange umbrella sampling (Luo et al., 2016)), and milestoning (Di Maio et al., 2015; Jiang et al., 2021) have been used to simulate equilibrium PMFs of ion/molecule permeation (Guardiani et al., 2022; Delemotte & Luo, 2023). In certain cases, permeant rotation needs to be incorporated in the enhanced sampling, which will generate a two-dimensional PMF describing the coupling between rotational and translational motions.

To gain further insights from PMF calculations, the total force that contributes to the PMF at any position can be decomposed into individual components, such as the forces acting on the permeant by water, protein, and ions along the z-axis. For example, force decomposition identified a contribution from interactions of the permeant with K⁺ on the PMF in the extracellular side of the Cx26 pore (Luo et al., 2016). The total force acting on the molecule can also be decomposed into individual types such as vdW force and electrostatic force along the permeation axis. It was found that the direction of electrostatic force correlates with the tumbling of the cAMP dipole inside the Cx26 pore (Jiang et al., 2021).

Quantifying Permeability from MD-based Methods

Nonequilibrium steady-state flux

Molecule permeability is often experimentally assessed relative to single-atomic ion permeability. This relative permeability is useful for rigorous comparison between computational prediction and experimental measurements. MD-based methods developed for computing ionic conductance have shown some success at predicting single-channel permeability of charged solutes (Jiang et al., 2021). As described above, application of voltage will generate a nonequilibrium steady-state flux of molecules through an open pore. When a sufficient number of permeation events are obtained from multiple replicas of voltage-driven flux simulations, it is possible to estimate the mean and confidence interval of the transition time by fitting the values to an exponential distribution. With the assumption of low permeant concentration inside the pore, single-channel permeability $\mathit{P}$ can be estimated from the Goldman-Hodgkin-Katz (GHK) flux equation(Hille, 2001), $\mathit{P}=\frac{\mathit{\gamma }\mathit{R}\mathit{T}}{{\mathit{q}}^{\mathbf{2}}{\mathit{F}}^{\mathbf{2}}\mathit{C}}$, where $\mathit{\gamma }$ is is the unitary conductance of a single channel $\mathit{R}$ is the gas constant, $\mathit{q}$ is the charge of permeant, $\mathit{F}$ is the Faraday constant, [$\mathit{C}$] is the bulk concentration of the permeant. From the transition time of cAMP under +/−200 mV, the calculated single-channel permeability of cAMP crossing an open Cx26 channel was within the two experimentally measured permeabilities of 6 and 47 × 10⁻³ μm³ sec⁻¹ (Jiang et al., 2021).

Nonequilibrium steady-state flux

Molecule permeability is often experimentally assessed relative to single-atomic ion permeability. This relative permeability is useful for rigorous comparison between computational prediction and experimental measurements. MD-based methods developed for computing ionic conductance have shown some success at predicting single-channel permeability of charged solutes (Jiang et al., 2021). As described above, application of voltage will generate a nonequilibrium steady-state flux of molecules through an open pore. When a sufficient number of permeation events are obtained from multiple replicas of voltage-driven flux simulations, it is possible to estimate the mean and confidence interval of the transition time by fitting the values to an exponential distribution. With the assumption of low permeant concentration inside the pore, single-channel permeability P can be estimated from the Goldman-Hodgkin-Katz (GHK) flux equation(Hille, 2001), $P=\frac{\text{γ}RT}{q^2{F}^{2}C}$, where $\gamma$ is the unitary conductance of a single channel $R$ is the gas constant, $q$ is the charge of permeant, $F$ is the Faraday constant, $\left[C\right]$ is the bulk concentration of the permeant. From the transition time of cAMP under +/−200 mV, the calculated single-channel permeability of cAMP crossing an open Cx26 channel was within the two experimentally measured permeabilities of 6 and 47 × 10⁻³ μm³ sec⁻¹ (Jiang et al., 2021).

Equilibrium PMF-based methods

The net flux of a permeant under an imposed voltage is by definition nonequilibrium since there is an external driving force. While accelerating molecule permeation, voltage also imposes forces on charged and polar residues within the protein. To eliminate these effects of voltage and examine the unperturbed character of interactions between the permeant and the pore, it is desirable to compute molecule permeability under an equilibrium condition (no external force, so no net flux). There are multiple ways to use the equilibrium PMF computed from enhanced sampling to estimate permeability. For example, the relative transition rates of two sugar molecules through Cx26 was estimated using transition state theory (TST) based on the PMF of each molecule (Luo et al., 2016). However, TST requires assumptions such as a single dominant free energy barrier that are unlikely to be satisfied in many channels. Alternatively, permeability can be derived from a modified inhomogeneous solubility-diffusion equation (ISD) $p=\text{π}{r}^2{\left(\underset{z_1}{\overset{z_2}{\int }}\frac{e^{w\left(z\right)/k_{B}T}}{D\left(z\right)}dz\right)}^{-1}$, where $w\left(z\right)$ is the PMF, $T$ is temperature and $k_B$ is the Boltzmann’s constant (Allen et al., 2004; Zhu & Hummer, 2012). $D\left(z\right)$ is the position-dependent diffusion constant of the studied permeant along the z-axis, which can be computed using a Laplace transformation of the velocity autocorrelation function (Berne et al., 1988; Woolf & Roux, 1994; Hummer, 2005). However, due to the computational cost, $D\left(z\right)$ of a flexible and anisotropic permeant is often simplified to a constant value (Noskov et al., 2013). $r$ is the radius of the cylindrical restraint used to confine the permeant in the bulk region. The effective cross-sectional area is $\text{π}{r}^2$ in a homogenous bulk. The interval of the integration $[z_1,z_2]$ is defined as the lower and upper boundaries of the channel pore, beyond which PMF reaches the bulk value of zero. The ISD theory has been useful in quantifying permeability of antibiotics through porin channels (Acharya et al., 2023).

Mean first passage time

A class of rare-event sampling techniques allows one to extract kinetic quantities directly from multiple short MD trajectories without the need to compute D\left(z\right) These include Markov state models (MSM) (Bowman et al.), milestoning (Faradjian & Elber, 2004), and weighted ensemble path sampling (Huber & Kim, 1996; Bhatt et al., 2010; Zhang et al., 2010). For example, MSM was used to compute the mean-first-passage-time (MFPT; the average time for a given molecule to diffuse through the entire length of the pore for the first time) of ATP permeation through VDAC (Choudhary et al., 2014). Markovian milestoning was used to compute the MFPT of cAMP through Cx26, from which the permeability was calculated and compared with experimental data (Figure 6) (Jiang et al., 2021). The relation between permeability, PMF or $w\left(z\right)$ and MFPT <t> has been derived from the fluctuation-dissipation theorem as $P=\frac{\text{π}{r}^{2{\int }_{z_1}^{z_2}{e}^{-\text{β}w\left(z\right)}}dz}{2\langle t\rangle }$ (Votapka et al., 2016), and validated recently against the ISD and GHK flux equations using a carbon nanotube model (Lin & Luo, 2022).

Figure 6: Computing molecule permeability through a wide pore.

(a) PMF and MFPT of cAMP permeating through a Cx26 hemichannel under 200 mV and 0 mV. PMF and MFPT at 0 mV were computed from milestoning simulations. The PMF at 200 mV was computed as the sum of the equilibrium PMF and the additional potential introduced by the external field. The later includes the sum of the constant electric field and the reaction potential due to the voltage-induced changes of the charge distributions and dipoles (Jiang et al., 2021). The MFPT plot indicates the time needed for a permeant to diffuse from the extracellular side (right) to each z-position in the pore without an external driving force. The MFPT for the full transit is the final value at z= −40 Å. (b) Single molecule permeability of cAMP estimated from voltage-driven flux and milestoning simulations are consistent and within experimental values. Reproduced from data in (Jiang et al., 2021).

Conclusions and Perspectives

This review presents a summary of MD methods that have been used to gain insights into molecule permeation through large-pore channels. Among them, voltage-driven molecule flux is a non-equilibrium MD approach that allows direct observation of charged molecules permeating under an external driving force, from which the binding sites and permeation barriers of the molecules can be inferred. Semi-quantitative pair-wise interactions (Genheden & Ryde, 2015) between pore-lining channel residues and permeant molecules from the MD simulation trajectories can then be used to guide mutagenesis experiments. It has been shown that with sufficient numbers of permeation events simulated on the supercomputer Anton2, molecular permeability can be estimated using the GHK flux equation with results in good agreement with experimental data (Jiang et al., 2021). On the other hand, equilibrium approaches require more computational resources and enhanced sampling techniques, but quantitative and intrinsic free energy and kinetics can be computed. Molecule permeability can be estimated from equilibrium PMF and MFPT. With equilibrium simulations, the molecule conformation and orientation inside the pore can be analyzed without external perturbation. The forces acting on the molecule from protein, water, and ions can be analyzed individually, thus providing in depth mechanistic insights. When the computational resources permit, using both nonequilibrium flux and equilibrium PMF methods can cross-validate results and provide complimentary insights (Jiang et al., 2021) (Lin & Luo, 2022).

The fact that there are highly specific and idiosyncratic interactions between a permeant and the flexible moieties that determine whether the molecule passes a point in the pore means that a molecule may itself transiently alter what would otherwise be a barrier to permit passage. This is essentially a gating process at a permeability filter, in which the “gating” is driven by interaction with a “ligand” (the permeant molecule), analogous to the filter gating known to occur in some ion-selective channels (Schewe et al., 2016; Natale et al., 2021). After a successful interaction with a permeating molecule, the “barrier” resets, similar to a swing barrier turnstile gate, ready for the next potential molecular permeant. It remains challenging to quantitatively compute the coupling between two slow processes, “gating” and “permeation”, within the large-pore channels. It is certain that the computational work outlined in this review serves as merely a starting point for further exploration. Advanced MD sampling algorithms and machine-learning techniques will play an important role in capturing higher dimensional free energy landscapes that delineate the complex and dynamic interactions by which a molecule mediates its transit through a protein pore.

ABBREVIATIONS

AMBER

Assisted Model Building with Energy Refinement

ATP

Adenosine Triphosphate

BD

Brownian Dynamics

CALHM

Calcium Homeostasis Modulator

cAMP

Cyclic Adenosine Monophosphate

CHARMM

Chemistry at Harvard Macromolecular Mechanics

CHAP

Channel Annotation Package

cGMP

Cyclic Guanosine 3’,5’-monophosphate

DAPI

4’,6-diamino-2-phenylindole

GHK

Goldman-Hodgkin-Katz

IP3

Inositol Triphosphate

ISD

Inhomogeneous Solubility-Diffusion

LRRC8

Leucine-Rich Repeat-Containing Protein

MFPT

Mean-First Passage-Time

MSM

Markovian State Model

OLPS

Optimized Potentials for Liquid Simulations

PBC

Periodic Boundary Conditions

PMF

Potential of Mean Force

TST

Transition State Theory

VDAC

Voltage-Dependent Anion Channel

VMD

Visual Molecular Dynamics