First-principles characterization of the gramicidin A channel in a hydrated lipid membrane: structure selection, tryptophan-mediated modulation of ion permeation, and gating energetics with linear-scaling DFT

Gramicidin A (gA), the paradigm narrow ion channel, has been characterized in vacuum by density-functional theory (Todorović et al., 2013), leaving a precisely stated set of open questions that all hinge on one objective: a first-principles treatment of the complete channel embedded in a hydrated lipid membrane. The vacuum study found that the two experimental channel structures (1MAG, from solid-state nuclear magnetic resonance (NMR) in bilayers of the phospholipid DMPC (dimyristoylphosphatidylcholine), and 1JNO, from solution NMR in micelles of the detergent SDS (sodium dodecyl sulfate)) do not relax to a common geometry (1JNO more stable by 1.8 eV in the gas phase), that the pore electrostatic potential is insensitive to the side chains (screened within about 1 meV), and that the head-to-head dimer is stabilized by 0.88 eV. The open problem is to extend this to the full solvated system (gA + 64 DMPC molecules + 2473 waters, about 16,000 atoms): determine whether bilayer hydrophobic matching reverses the gas-phase stability ordering of the two structures, identify the non-electrostatic (dynamical, lipid- and water-mediated) mechanism by which the tryptophan residues modulate ion permeation, compare the dimer stabilization energy with channel-lipid interaction energetics to establish gating thermodynamics, and establish the accuracy of linear-scaling (order-$N$, including the non-self-consistent variant) DFT for the energetics and energy barriers of such a system. No full membrane-embedded first-principles simulation of this system has been published; progress on these questions since 2013 rests on classical or polarizable force fields or on fragment-level QM/MM, so the problem remains open.

Background

Gramicidin A (gA) is a linear pentadecapeptide antibiotic that inserts in lipid bilayers as a head-to-head dimer of single-stranded $\beta^{6.3}$ helices (about 6.3 residues per turn), forming a narrow pore (~4 Å diameter, ~26 Å long) that conducts monovalent cations and a single-file chain of water molecules to the exclusion of anions and divalent cations; it is a standard model system for ion-channel biophysics. Two experimental atomic structures of the channel exist: 1MAG, determined by solid-state nuclear magnetic resonance (NMR) in hydrated bilayers of the phospholipid DMPC (dimyristoylphosphatidylcholine, a 14-carbon-chain lipid) (Ketchem, Roux and Cross, 1997), and 1JNO, determined by solution NMR in micelles of the detergent SDS (sodium dodecyl sulfate) (Townsley, Tucker, Sham and Hinton, 2001). They are topologically similar but differ in helix parameters — 1MAG has $R = 6.5$ residues per turn and pitch $d = 4.95$ Å versus $R = 6.3$ and $d = 4.79$ Å for 1JNO — and in the orientation of the tryptophan 9 (Trp9) side chain. Classical molecular dynamics (MD) simulation of both structures in a bilayer, combined with solid-state NMR restraints, found the backbones converge to similar conformations with Trp9 spending about 80% of its time in the 1JNO orientation (Allen, Andersen and Roux, 2003).

Each gA monomer carries four tryptophan residues (positions 9, 11, 13, 15) located near the bilayer-water interface. Replacing tryptophans by phenylalanine greatly reduces the single-channel conductance, with the largest effect for Trp9 (Becker, Greathouse, Koeppe and Andersen, 1991), and 5-fluorination of the indole rings alters channel currents (Andersen, Greathouse, Providence, Becker and Koeppe, 1998). One family of proposals attributed these effects to the electrostatic interaction between the permeating ion and the ~2 debye dipole moment of the indole rings (Hu and Cross, 1995); empirical fitting of free-energy profiles to conductance data also implicates the tryptophans in the permeation barrier (Durrant, Caywood and Busath, 2006).

Density-functional theory (DFT), the standard first-principles electronic-structure method, conventionally scales as the cube of the system size; linear-scaling (order-$N$) DFT achieves roughly linear cost by truncating the spatial range of the density matrix (controlled by an auxiliary density-matrix cutoff radius), which permits calculations on $10^4$-$10^5$ atoms. The CONQUEST code implements this with pseudo-atomic orbital basis sets and a stable non-self-consistent (NSC) option that avoids iterative self-consistent charge-density updates and is therefore cheaper, but whose accuracy for total energies, forces and energy barriers was untested. Todorović, Bowler, Gillan and Miyazaki (2013) used the CONQUEST code with the PBE (Perdew-Burke-Ernzerhof) semilocal exchange-correlation functional to treat the entire isolated gA molecule (552 atoms) with DFT for the first time. Their findings define the open problem: (i) geometry optimization from 1MAG and 1JNO in the gas phase reinforces their structural differences, and the total energy of the optimized 1JNO structure is lower than that of 1MAG by 1.8 eV, but because lipid bilayers are usually thicker than the gA length, the longer 1MAG helix has the smaller hydrophobic mismatch and may be stabilized by lipid-channel interactions, possibly reversing the ordering; (ii) an alanine-tube control (the 1MAG backbone with all side chains replaced by alanine) shows the pore electrostatic potential of gA is identical to that of the side-chain-free tube to within ~1 meV, so the side chains' dipoles are electrostatically screened and cannot explain the conductance sensitivity — dynamical effects involving interactions of the side chains with water molecules and lipid polar head groups were proposed instead and left for future work; (iii) the stabilization energy of the head-to-head dimer (six intermonomer hydrogen bonds) is 0.88 eV in the gas phase, and this value was never compared with lipid-channel and lipid-lipid interaction energies, which is what governs the association/dissociation gating of the channel; (iv) convergence of the linear-scaling total energy with the density-matrix cutoff was demonstrated, and the NSC method was checked against the self-consistent method only for the optimized gas-phase structures, so its accuracy for the energetics and dynamics of the full system (for example, energy barriers for ion permeation) remained untested. The authors stated they had begun the full all-atom DFT simulation of gA embedded in 64 DMPC molecules with 2473 water molecules (about 16,000 atoms) and would report it in a future publication.

Neighboring quantum-level work treats only fragments of the problem: QM/MM (quantum mechanics/molecular mechanics, in which only a fragment of the system is treated quantum-mechanically) ab initio MD of polarization effects in the pore (Timko and Kuyucak, 2012), DFT-based MD and metadynamics of proton transport in gA embedded in a solvated DMPC membrane of about 2,000 atoms (Dreyer, Zhang, Ippoliti and Carloni, 2013), and DFT calculations on MD snapshots used to interpret oxygen-17 NMR of the channel water wire (Paulino et al., 2020). Reviews of large-scale DFT in biology (Cole and Hine, 2016) and of the CONQUEST code (Nakata et al., 2020) describe the 2013 vacuum study as the state of the art for gA and confirm that no full membrane-embedded first-principles calculation of the system has been reported.

Problem Statement

Deliver a first-principles (DFT-level) characterization of the complete gramicidin A channel embedded in a hydrated lipid membrane with counterions — the system prepared by Todorović et al. (2013) as gA plus 64 DMPC molecules and 2473 water molecules, roughly 16,000 atoms — such that the following four linked questions, which together constitute the source's stated open problem, are answered within one coherent study:

1. Structure selection under hydrophobic matching (the energetic penalty incurred when the hydrophobic length of the channel does not match the hydrocarbon thickness of the bilayer): determine, at the first-principles level, the relative stability in the bilayer environment of the two experimental channel structures 1MAG (solid-state NMR in DMPC) and 1JNO (solution NMR in SDS micelles) — in particular whether the interactions of the longer 1MAG helix with a bilayer thicker than the channel reverse the gas-phase ordering in which 1JNO is more stable by 1.8 eV.

2. Tryptophan mechanism: given the vacuum finding that the pore electrostatic potential is insensitive to the side chains (screened to within about 1 meV), identify and quantify the mechanism by which the four tryptophan residues modulate ion permeation — the dynamical effects involving interactions of the side chains with water molecules and lipid polar head groups proposed by the source — with an explicit decomposition sufficient to rule the competing electrostatic-dipole picture (Hu and Cross, 1995; Andersen et al., 1998) in or out in the membrane environment.

3. Gating energetics: compare the head-to-head dimer stabilization energy (0.88 eV in the gas phase) with the lipid-channel (and lipid-lipid) interaction energies in the same first-principles framework, so as to establish the thermodynamics of channel formation and dissociation (gating).

4. Method validation: establish the accuracy of linear-scaling DFT — including the non-self-consistent variant — for total energies, forces, and energy barriers (for example, for ion permeation) for the roughly $10^4$-atom solvated system studied here, to the standard needed for the conclusions of items 1-3 to be trusted.

Items 1-4 are the jointly necessary deliverables of the single overarching objective — a first-principles account of the functioning channel in its membrane environment — and an answer that resolves any subset without the rest does not close the problem.

References

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