Biophysical Journal Volume 71 July 1996 19-39
Structure and Dynamics of a Proton Wire: A Theoretical Study of H+
Translocation along the Single-File Water Chain in the Gramicidin
A Channel
Regis Pomes and Benoit Roux
Groupe de Recherche en Transport Membranaire, Departements de Physique et de Chimie, Universit6 de Montr6al, Montr6al,
Qu6bec H3C 3J7 Canada
ABSTRACT The rapid translocation of H+ along a chain of hydrogen-bonded water molecules, or proton wire, is thought to
be an important mechanism for proton permeation through transmembrane channels. Computer simulations are used to
study the properties of the proton wire formed by the single-file waters in the gramicidin A channel. The model includes the
polypeptidic dimer, with 22 water molecules and one excess proton. The dissociation of the water molecules is taken into
account by the "polarization model" of Stillinger and co-workers. The importance of quantum effects due to the light mass
of the hydrogen nuclei is examined with the use of discretized Feynman path integral molecular dynamics simulations. Results
show that the presence of an excess proton in the pore orients the single-file water molecules and affects the geometry of
water-water hydrogen bonding interactions. Rather than a well-defined hydronium ion OH' in the single-file region, the
protonated species is characterized by a strong hydrogen bond resembling that of 02H5 . The quantum dispersion of protons
has a small but significant effect on the equilibrium structure of the hydrogen-bonded water chain. During classical
trajectories, proton transfer between consecutive water molecules is a very fast spontaneous process that takes place in the
subpicosecond time scale. The translocation along extended regions of the chain takes place neither via a totally concerted
mechanism in which the donor-acceptor pattern would flip over the entire chain in a single step, nor via a succession of
incoherent hops between well-defined intermediates. Rather, proton transfer in the wire is a semicollective process that
results from the subtle interplay of rapid hydrogen-bond length fluctuations along the water chain. These rapid structural
fluctuations of the protonated single file of waters around an average position and the slow movements of the average
position of the excess proton along the channel axis occur on two very different time scales. Ultimately, it is the slow
reorganization of hydrogen bonds between single-file water molecules and channel backbone carbonyl groups that, by
affecting the connectivity and the dynamics of the single-file water chain, also limits the translocation of the proton across the
pore.
INTRODUCTION
Unique properties are displayed by proton translocation
phenomena across biological membranes, implying that the
mechanism underlying the conduction of protons is radi-
cally different from that of other ions (Levitt, 1984). Trans-
port through the simple transmembrane channel formed by
the gramicidin A molecule (GA) offers a particularly strik-
ing example of this phenomenon (Hladky and Haydon,
1972). The measured channel conductance to H+ (530
pmho; see Akeson and Deamer, 1991) is more than 15 times
that to potassium (29 pmho). [The maximum conductance
to K+ is approximately 29 pmho, based on previous exper-
imental results (Hladky and Haydon, 1972; Finkelstein and
Andersen, 1981; see also Roux and Karplus, 1991).] As
shown in Table 1, such a difference is much larger than
would be expected simply from the ratio of the mobility of
these ions in bulk water. This is all the more remarkable,
because the size of K+ is similar to that of a hydronium ion
Received for publication S December 1995 and in final form 14 March
1996.
Address reprint requests to Dr. Benoit Roux, Chemistry Department, Univer-
sini de Montr6al, C.P. 6128, succ. A, Montr6al, Qu6bec H3C 3J7 Canada. Tel.:
514-343-7105; Fax: 514-343-7586; E-mail: rouxb@ere.umontreal.ca.
i 1996 by the Biophysical Society
0006-3495/96/07/19/21 $2.00
OH+. In fact, diffusion constants inside the pore deduced
from experimental data suggest that protons move almost
eight times as fast as water molecules themselves (see Table
1). Because of the narrowness of the pore, permeating
waters or ions cannot pass each other inside the GA channel,
and they must move in single file (Finkelstein and
Andersen, 1981). The transport of a cation such as potas-
sium is limited by the displacement of the single file of
water molecules inside the channel; as shown in Table 1, the
estimated diffusion constants of potassium ion or a water
molecule inside the channel are nearly equivalent. In con-
trast, translocation of a proton does not in principle require
the displacement of the single file of water molecules.
Instead, the rapid translocation of protons across the GA
channel is thought to occur through a succession of hops
along the single file of hydrogen-bonded water molecules,
which acts effectively as a proton wire (Hille, 1992).
The concept of proton wires was first introduced by
Nagle and Morowitz to account for the fast conductance of
protons along chains of hydrogen-bonded protonable groups
in biological systems (Nagle and Morowitz, 1978) and
includes the single file of water molecules that fill the
narrow channels of transmembrane pores such as the GA
channel (Akeson and Deamer, 1991). However, the biolog-
ical relevance of water proton wires extends beyond the
19

Volume 71 July 1996
TABLE I Experimental diffusion constants
Diffusion constants (A2/ps)
Species GA channel* Bulk water#
H20 4.4 x 10-3 2.1 x 10-'
K+ 1.9 X 10-3 2.0 x 10-'
H+ 3.4 x lo-2 9.3 x 10-'
*The water diffusion constant inside the GA channel was extracted from
the experimental diffusional water permeability P, = 1.82 X 10- 15 cm3/s
(Finkelstein and Andersen, 1981), using P, = DSIL, with L = 23 A and
S = 7.84 A2. The cation diffusion constant inside the GA channel was
extracted from the experimental maximum conductance using Amax =
De2IkBTL2, with L = 23 A (see Roux and Karplus, 1991).
#From Hille (1992).
realm of transmembrane pores. The mediation of H+ trans-
fer by chains of water molecules is emerging in a wide
range of proton transport phenomena involved in bioener-
getics. It has been proposed that chains of water molecules
could play important roles in the protonation of the Schiff
base of the bacteriorhodopsin of Halobacterium halobium
(Cao et al., 1991) and of the secondary quinone in the
photosynthetic reaction center from Rhodobacter sphae-
roides (Baciou and Michel, manuscript submitted for pub-
lication), and for the proton entry pathway into the cyto-
chrome b6f complex of the plant chloroplast thylakoid
membrane (Martinez et al., 1995). Moreover, the insight
gained from the study of water wires may provide a better
understanding of proton transfer processes in enzymes and
in liquids in general.
Although kinetic models for hydrogen-bonded chains
were proposed in the late 1970s (Nagle and Morowitz,
1978; Knapp et al., 1980), little is known at the present time
about the detailed molecular mechanism of H+ transloca-
tion along proton wires. Scheme 1 summarizes the problem
at hand: (I) an excess proton is incorporated at one end of a
single file of water molecules, and (II) a series of proton
transfers takes place between adjacent water molecules until
(III) a proton is released at the other end. Several questions
H H H
I I I
0%H H 0%H H H H IH H H
I I I
H H H
H H H
I I I
le0% 1,,H
0% H01 0% 11H10 00H 0 0 HaH
~ I I I
H H H
H H
I I
H,.0 0H ,, H
H
H
HI%O H.,
H H
Scheme 1
H+
come to mind. How does this translocation take place? What
are the factors governing the translocation? What are the
various steps and intermediates involved?
In the present work, we propose to study the properties of
the proton wire of the channel formed by the GA molecules
using computer simulations. The gramicidin A molecule is
a synthetic polypeptide chain of 15 residues that associates
as a dimer to form a pore spanning phospholipid bilayers
(Urry, 1971), which has been extensively studied both ex-
perimentally and theoretically. The three-dimensional struc-
ture of the GA channel's ion-conducting state has been
determined to atomic resolution by NMR spectroscopy (Ar-
seniev et al., 1985; Ketchem et al., 1993). Over the past 15
years, the structural simplicity of the GA channel has stim-
ulated the development of numerous theoretical studies (see
the review by Roux and Karplus, 1994, and references
therein), which have provided considerable insight into the
mechanism of ion permeation. Problems ranging from the
properties of the single-file solution filling the pore to the
calculation of the rate of transport of ions have been ad-
dressed in ever-growing detail. In particular, several com-
putational studies of the GA channel have stressed the
important structural and dynamical properties of the linear
chain of water molecules (MacKay et al., 1984; MacKay
and Wilson, 1986; Kim, 1985; Etchebest and Pullman,
1986a, b; Pullman, 1987; Skerra and Brickman, 1987; Chiu
et al., 1989, 1991, 1993; Jordan, 1990; Roux and Karplus,
1991, 1994; Roux, 1995).
In contrast with the insight gained from these studies, the
detailed mechanism of H+ translocation through the GA
channel has remained an outstanding problem that has not
been addressed at the molecular level. This is due to a
combination of factors. To begin with, the treatment of
proton transfer in water is a challenging theoretical problem,
owing to the difficulty of accurately modeling the potential
energy surface governing the dynamics and the dissociation
of water aggregates. Traditional potential functions with
harmonic bonds, such as the TIP3P and SPC models (Jor-
gensen et al., 1983; Berendsen et al., 1981), cannot account
for these effects. High levels of ab initio calculation have
revealed the complexity of the potential energy surface for
the motion of protonated clusters of water molecules (Schei-
ner, 1985, 1994; DelBene, 1988; Komatsuzaki and Ohmine,
1994). In addition, this difficulty is compounded by the
necessity of accounting for the quantum nature of the proton
nuclei. To this effect, a number of ab initio quantum-
classical methods have been proposed, which are very
promising for the study of proton transfer in biological
systems (see, for example, Bala et al., 1994; Mavri and
Berendsen, 1995). However, these approaches are still too
expensive for the study of proton translocation over the
relatively long distances covered by biological proton wires.
The use of discretized Feynman path integrals in conjunc-
tion with an empirical potential energy function makes it
possible to overcome such a limitation. The path integral
treatment allows the inclusion of the quantum effects arising
from the light mass of protons and has been used by other
20 Biophysical Journal

Structure and Dynamics of a Proton Wire
authors in studies of proton transfer along model hydrogen
bonds (Lobaugh and Voth, 1992; Laria et al., 1994), as well
as in an all-quantum study of small protonated water clus-
ters published recently (Cheng et al., 1995). Moreover,
previous studies have shown that the discretized Feynman
path integral methodology can be used effectively in simu-
lations of large protein systems, particularly to improve the
treatment of hydrogen bonds (Zheng et al., 1989).
In two earlier papers, we described computer simulations
of proton translocations along a linear chain of water mol-
ecules (Pomes and Roux, 1995, 1996). Discretized Feyn-
man path integral molecular dynamics simulations (Feyn-
man and Hibbs, 1965; Chandler and Wolynes, 1980) were
used to account for the quantum nature of all the hydrogen
nuclei of a linear protonated chain of hydrogen-bonded
water molecules in vacuo. The polarization model, version
6 (PM6), developed by Stillinger and co-workers to repro-
duce the structure and dynamics of small water clusters
(Weber and Stillinger, 1982), was employed to account for
the polarization and dissociation of water molecules. The
studies clarified the relative importance of quantum disper-
sion of the protons and of the flexibility of protonated water
chains in vacuo.
In the present paper, we extend that model to study the
single file of water molecules of the gramicidin A channel
and its ability to function as a proton wire. The methodology
used to represent the quantum dispersion of the proton
nuclei and to generate molecular dynamics trajectories of
the water-filled GA channel is reviewed in the next section.
In the subsequent sections, the structure and dynamics of the
single-file water chain with an excess proton are character-
ized. The importance of quantum effects, the interplay of
local and cooperative fluctuations of the wire, and the
influence of the channel environment are examined. The
possibility of describing the proton translocation in terms of
reaction coordinates is analyzed. This study concludes with
the dynamics of proton translocation, and implications are
considered for the molecular mechanism of H+ conduction
by the proton wires of GA and other water-filled biological
channels. Efforts are made to relate the present study to
previous theoretical models describing proton conduction in
biological systems (Nagle and Morowitz, 1978; Knapp et
al., 1980; Akeson and Deamer, 1991).
METHODOLOGY
Microscopic system
The simulation system includes the GA channel together
with 22 water molecules and an excess proton. In all of the
simulations reported here, the initial (reference) configura-
tion was taken from a previous molecular dynamics simu-
lation of the channel embedded in a dimyristoyl phosphati-
dylcholine (DMPC) bilayer (Woolf and Roux, 1994). The
coordinates of the channel were initially derived from the
Arseniev structure (Arseniev et al., 1985). The monomers
(3-helix 20 A in length with an inside diameter of 4 A. The
pore is lined with backbone peptide groups that form 20
intramonomer and six intermolecular hydrogen bonds. The
22 water molecules within 15 A of the center of the channel
from the GA:DMPC configuration were kept. Of these, 10
were located in the single-file region and six were in each
mouth.
The overall translation of the molecular system was
avoided by forcing the center of mass of the dimer to
coincide with the origin by means of a strong harmonic
constraint. Moreover, the center of mass of each monomeric
unit was similarly restrained to remain along the z axis so as
to ensure the alignment of the channel with that axis and
suppress two rotational degrees of freedom of the channel.
Finally, weak harmonic constraints of 0.1 kcal/molIA2 were
also imposed on each heavy atom of the tryptophan side
chains so as to confine them to the vicinity of their position
in the initial, reference structure. These weak energy re-
straints were required to ensure that the channel would
retain its ,B-helical dimer conformation in the absence of an
explicit membrane environment. In a long preliminary sim-
ulation performed without this restraint, the tryptophan side
chains were observed to travel and distort the channel's
mouths, resulting in the unfolding of the channel after
several tens of picoseconds, which is inconsistent with
simulations performed in an explicit membrane environ-
ment (Woolf and Roux, 1994, 1996).
In the reference configuration, 10 water molecules were
located inside the channel in single-file fashion, whereas 6
water molecules made up small solvent caps at either end of
the channel. These caps were restrained to lie in the region
of the mouths of the gramicidin channel by cylindrical
constraints: all of the water oxygen atoms were subjected to
soft quadratic restoring potentials beyond 5 A from the
channel axis, and beyond planes perpendicular to the chan-
nel axis at z = +- iA from the origin at the center of the
pore. These restraints prevented the evaporation of cap
waters or their incursion into regions normally occupied by
the hydrocarbon chains around the channel, without affect-
ing the single-file water molecules.
Potential energy function
The total interaction energy is given by Utw = Uwater +
Uchannel. The force field of the CHARMM program (Brooks
et al., 1983), version 22, was employed for the protein
(Uchannel). The potential energy surface Uwater governing the
motion of the water oxygen and hydrogen nuclei was mod-
eled with the polarization model (PM6) of Stillinger and
co-workers (Stillinger and David, 1978; Stillinger, 1979;
Weber and Stillinger, 1982). This model was initially de-
veloped and parameterized to accurately reproduce the
structure and energy of small hydrogen-bonded cationic and
anionic water clusters. The basic structural elements of PM6
are H+ and 02- atoms, which makes it possible to account
for the full dissociation of water molecules into ionic frag-
Pomes and Roux 21
assemble in head-to-head fashion to form a right-handed

Volume 71 July 1996
ments. For a configuration of the oxygen and hydrogen
constituents with coordinates {ro} {ro,, , roNo } and
{FH} {rH.,* * * rHNH }, and a channel configuration given
by {rx} {rx, rXNX, }, the PM6 potential energy is
Uwater({ro}, {rHJ, {rx})
No NH
= E oo(Iroi rojI) + E OHH(IrH, rHjl)
i<j=l i<j=l
(1)
NO NH No Nx
+ E OOH(Iro, - rHjl) + E E 4ox(Jro, - rxjl)
i=l j=1 i=1 j=1
NH Nx
+ E H(jrHi - rXjI) + ,1({r4}, {rH}, {rx}),
i-i i=l
where 40o, OOH, and HH are the PM6 pairwise radially
symmetric functions (Weber and Stillinger, 1982), and ox
and 4Hx are pairwise radially symmetric functions con-
structed from PM6 and the CHARMM force field. ox and
HX correspond to the nonbonded Lennard-Jones and direct
Coulomb interactions between the channel atoms and the
water nucleus (see below).
{>poI represents a many-body polarization energy contri-
bution resulting from the polarization of the water oxygen
particles,
1 No
D,j({rO}, {rHJ, {rx}) = -22 *GoL) (2)
where
N
G(L) q.r.. [ L(rij)] (3)
j¢6i
is an effective field arising from surrounding particles. In
this equation, N = NO + NH + NX is the total number of
particles in the system, r1j = ri-rj and r1. = lrijI is the
distance between atoms i and j, and L(r) is a parametric
damping function that accounts for the spatial extension of
the electronic cloud surrounding each water oxygen atom
(Stillinger, 1979). The induced atomic dipole moments, go.,
are determined self-consistently from the following set of
equations for the oxygen atoms Oi:
-= aGK),
the motion of the water oxygen and hydrogen, including the
water-channel coupling, involves many-body interactions
due to Eqs. 2-4.
The combination of the water (polarizable-dissociable)
and channel (molecular mechanical) potentials is similar to
that described in hybrid quantum-mechanical/molecular
mechanical (QM/MM) approaches, in which an semiempiri-
cal quantum mechanical potential is coupled to a molecular
mechanical force field (Field et al., 1990; Gao and Xia,
1992). An underlying assumption in the present model is
that proton exchange reactions from the channel to the water
do not play a dominant role in the transport mechanism.
Indeed, dissociation and association events involving the
amide and the carbonyl groups of the channel backbone
cannot occur, because of the choice of the molecular me-
chanical potential function, and are ignored. This approxi-
mation is supported by nuclear magnetic resonance studies
which indicate that the intrinsic rate of exchange of amide
protons that are freely accessible to the solvent is estimated
to vary between 1 and 10 min-' (Wiithrich, 1986), or
between 10-1 and 10-3 S-1 (Jeng and Englander, 1991) in
the pH range 1-6. This is considerably slower than the
estimated rate for proton diffusion in the channel (Table 1).
Therefore, it seems unlikely that proton exchange with the
channel backbone could play a determining role in the
function of the GA proton wire.
The relative strength of water-water, channel-water, and
channel-channel interactions, in particular where hydrogen
bonds are concerned, has been shown to be critical to the
properties of the water single file of the gramicidin channel
(MacKay et al., 1984). These properties are also affected by
the incorporation of polarizability (Jordan, 1990). Lennard-
Jones parameters governing the water-channel pairwise
nonbonded interactions described by 4ox and Hx were
adjusted after the interaction of a single water molecule with
N-methylacetamide (NMA) taken as a model of the channel
backbone. Specifically, a fit of the effective Lennard-Jones
radius o- governing the Lennard-Jones interaction between
water oxygen and backbone hydrogen atoms was performed
so as to approximately reproduce the geometric and ener-
getic properties of the NH OH2 hydrogen bond of the
water-NMA dimer, whereas the Lennard-Jones radius for all
other water-channel interactions was identical to that de-
fined in the TIP3P (Jorgensen et al., 1983) and CHARMM
(Brooks et al., 1983) models. The results for water-NMA
hydrogen bonds are given in Table 2, where they are com-
(4)
where
N
GUK) =- qjr1jI
ji.= rij*i
NoTiA
- K(rij)] + I [1 K(rij)]
k*i
(5)
Here, a is the polarizability of water oxygen atoms, K(r) is
a parametric damping function similar to L(r), and Tik is the
dipole-dipole tensor term for each °i-°k pair (Stillinger,
1979). It should be noted that the potential energy governing
TABLE 2 Water-NMA hydrogen bonds
N-Methylacetamide
C =O N-H
Water model d (A) E (kcal/mol) d (A) E (kcal/mol)
Ab iiitioa 1.88 -7.3 1.99 -5.4
TIP3P 1.77 -7.2 1.93 -5.6
PM6 1.99 -8.6 1.94 -6.5
aResults of HF/6-31G* calculations from Guo and Karplus (1992).
22 Biophysical Joumal

Structure and Dynamics of a Proton Wire
pared to the optimized values obtained with ab initio cal-
culations at the HF/6-31G* level (Guo and Karplus, 1992)
and with the TIP3P model. The energy and distance of the
water-carbonyl hydrogen bond represent a compromise with
respect to results obtained from ab initio calculations. The
PM6 result overestimates both the length (by 0.11 A) and
the absolute energy (by 1 kcal/mol). On the other hand, the
optimized PM6 geometry of the NH ... OH2 hydrogen
bond is comparable to that of TIP3P, although the energy is
overestimated, again by about 1 kcal/mol. A comparison of
the results obtained with the present model and previously
published results will be provided in the Discussion.
Discretized Feynman path integral
The importance of quantum effects was investigated by
exploiting the isomorphism of the discretized Feynman path
integral representation of the density matrix with an effec-
tive classical system obeying Boltzmann statistics (Feyn-
man and Hibbs, 1965; Chandler and Wolynes, 1980). Mo-
lecular dynamics simulations of the effective classical
system are valid for obtaining ensemble averages, although
they do not provide information on the time-dependent
quantum dynamics of the system. Only the quantization of
the water hydrogen nuclei was considered, and the water
oxygens, as well as all the atoms the channel (hydrogen
atoms included), were treated as classical particles. Follow-
ing the path integral approach, each proton was replaced in
the effective classical system by a ring polymer, or neck-
lace, of P fictitious particles with a harmonic spring be-
tween nearest neighbors along the ring. The potential energy
of the effective classical system is
Ueff({ro}, r {r(P)}, {rx}) = Uchannel({rXJ)
ip
+ p E Uwater({ro}, {rH)}, {rx}) (6)
p=l
NH P
+ E 2Kpolymerrp) -r(p+) 2
i=l p=i
where {r(p)} = r(p), . . ., r(p) represents the coordinates
of the pth particle for protons HI through HNH, and
Kpolymer = PMH(kBT/h)2 is the harmonic spring constant
acting between the pth and (p + 1)th nearest neighbors
along the polymer necklace representing each proton Hi
of mass MH. In the third summation of Eq. 6, r(P+ 1) = r(l)
is required to satisfy the closure of the ring polymers. In
the present path integral simulations, each water hydro-
gen nucleus was represented as a polymer necklace of
P = 32 fictitious particles. This number contrasts with
previous path integral simulations of water in which only
three fictitious particles per hydrogen were used to study
the quantization of the rotational and librational motions
of rigid water models (Kuharski and Rossky, 1985). A
much larger number is necessary in the present study,
because high-frequency bond stretching modes are in-
cluded. A discretization of the path integral with 20 to 30
fictitious particles was found to be adequate in recent
path integral studies of proton transfer in model systems
(Lobaugh and Voth, 1992; Azzouz and Borgis, 1992;
Laria et al., 1994). The total number of particles in the
simulation system was 618 and 619, respectively, in the
unprotonated and protonated classical systems (P = 1)
and 2014 in the quantum system (P = 32).
In the quantum simulations (P = 32), the configurational
sampling was performed by generating Langevin molecular
dynamics trajectories of the effective system. The choice of
Langevin dynamics was dictated by the need to avoid the
nonergodicity of path integral molecular dynamics simula-
tions based on the microcanonical ensemble (Allen and
Tildesley, 1987). For all degrees of freedom x, in the
effective classical system, the trajectory was calculated ac-
cording to the Langevin equation of motion:
m&ka = - aXaUeff - 'Ya + f(t), (7)
where y is a friction constant and flt) is a random Gaussian
force obeying the fluctuation-dissipation theorem:
(f(tjf(0)) = 2kBTy6(t). (8)
This simulation ensures that the configurations were gener-
ated according to a Boltzman distribution, exp[- Ueff/kBT],
at temperature T. It should be noted that the resulting
Boltzmann distribution is independent of the choice of
dynamical mass m,, attributed to each degree of freedom.
By contrast, in the simulations performed in the classical
limit (P = 1), the equations of motion were propagated in
the microcanonical ensemble. Deterministic trajectories of-
fer the advantage, over stochastic ones, that dynamical
information, not just thermodynamic averages, can be ex-
tracted from the computer simulation.
During the propagation of the trajectories of the effective
system, the full PM6 potential function, Uwater({ro},
{r(p)}, {rx}), was recalculated for each p-step of the dis-
cretized path integral (Eq. 4). In particular, the interactions
involving the pairwise interactions 4H and O0H were re-
calculated. It is important to note that the polarization en-
ergy contribution, bpOI({rO}, {r(p'}, {rx}), was determined
by solving a set of self-consistent equations (Eq. 4) for each
value of p. In other words, each molecular dynamics time
step involved the determination of 32 polarization states
induced on the set of 22 oxygen atoms. The forces on all
particles due to the many-body polarization were calculated
analytically by solving a similar set of self-consistent equa-
tions derived by Stillinger (1979). For the sake of compu-
tational efficiency, other contributions arising from constant
terms, such as the PM6 oxygen-oxygen 4oo and channel-
water interactions, were calculated only once and stored.
Computational details
Molecular dynamics trajectories were generated using the
integration schemes described above. The time step used for
Pomes and Roux 23

Volume 71 July 1996
the integration of the equations of motion was 0.0005 ps.
This short time step is made necessary by the high fre-
quency of 0-H vibrations in PM6 waters. No SHAKE
restriction was applied on chemical bond vibrations. Calcu-
lations were performed on a Silicon Graphics R4400 work-
station. The propagation of the classical trajectories re-
quired about 15 min of cpu time per picosecond, and that of
the quantum trajectory, about 6 h per picosecond. The
temperature of the bath was 300K. Initial velocities were
assigned at random from a Gaussian distribution. In the
equilibration (early) phase of the simulations, the velocities
were reassigned every 200 steps, whereas in the production
phase the temperature simply fluctuated within 10K of the
300K mean in the classical (Verlet) simulations.
The equilibrations were performed as follows. First, PM6
water molecules were substituted for the original TIP3P
(Jorgensen et al., 1983) water molecules used in the refer-
ence configuration (Woolf and Roux, 1994). The position of
the water hydrogen nuclei were optimized by steepest-
descent energy minimization performed with all other atoms
fixed. The minimization procedure was repeated with all
water atoms allowed to move, and the solvent was thermal-
ized at 300K for 5 ps. The polypeptide channel atoms were
then allowed to move, and the whole system was equili-
brated for a period of 20 ps. From this configuration, four
distinct trajectories were created. These trajectories are
summarized in Table 3 and will be referred to henceforth as
trajectories A, B, C, and D. The production stage of trajec-
tory A was generated directly without an additional proton,
whereas the other simulations included an excess hydrogen
nucleus. In addition, trajectory C included a quantum treat-
ment of all water hydrogen nuclei (P = 32), and trajectory
D was propagated in the absence of the electrostatic field
created by the partial charges {qx} of channel atoms X in
the calculation of FPOI({rO}, {rH}, {rx}) and of the direct
nonbonded coulomb interaction in #ox and Hx (Eqs. 1-5).
The water-channel nonbonded Lennard-Jones interactions
were kept unchanged. This artificial model maintains the
GA structure but removes the electrostatic interactions giv-
ing rise to water-channel hydrogen bonding. For this reason,
the simulation D can be thought of as a proton wire in the
interior of a "hydrophobic channel" that is both structurally
and dynamically equivalent to the GA channel.
To construct trajectories B, C, and D, an excess proton
was inserted into the single file of water molecules. A site
was selected in the middle of a monomer between water
oxygen atoms 07 and 08, at z = 5.29 A. Two energy
minimization cycles were then imposed, first with only the
hydrogen nuclei being allowed to move, and again with
TABLE 3 Description of the computer simulations
Code P Excess proton? Ue.i. off? Total time (ps)
A 1 No No 200
B 1 Yes No 400
C 32 Yes No 100
D 1 Yes Yes 200
rigid constraints on the channel only. The two-stage equil-
ibration process was then repeated, once for the water and
again for the whole system. The equilibration of system C
was performed after superposition of 31 extra p-particles to
each of the classical water hydrogen nuclei. The production
stages followed, for up to 400 ps for the classical simulation
B, and 100 ps for the quantum simulation C. The fourth
trajectory (D) was performed to examine the importance of
water-channel interactions on the translocation mechanism.
From the equilibrated configuration obtained with simula-
tion B, the nonbonded channel-water coulomb interactions
and the channel-water components of the polarization func-
tion were turned off, and a 200-ps trajectory was produced
after an additional two-stage equilibration.
RESULTS AND DISCUSSION
Structure of the gramicidin channel
The channel retained its (3-helix structure over the hundreds
of picoseconds spanned by each of the four molecular
dynamics trajectories. Moreover, the average (4, /i) values
of the Ca angles were very similar for trajectories A, B, and
C, indicating that the integrity of the channel is altered
neither by the inclusion of an excess proton, nor by a
quantum treatment of water hydrogen nuclei. On the other
hand, the structure of the channel was slightly affected by
the disappearance of electrostatic interactions with the lin-
ear water chain. Minor differences in the channel confor-
mation (100 to 150) arose in trajectory D, where no hydro-
gen bonds could be formed between the channel and water
molecules. A statistically representative configuration of the
trajectory C is shown in Fig. 1.
In all of the simulations, the near-symmetry of the chan-
nel with respect to its center at the dimer junction was
evident in both the average conformation and backbone
fluctuations. The average rms fluctuations of the 4 and qi
torsions were of about 8° or 90 near the center of each
pentadecapeptide monomer. The channel mouths, and above
all the region of the dimer junction (residues 1 to 5 of each
gramicidin monomer), were significantly more mobile, with
backbone torsions fluctuating up to twice as much. A good
agreement was observed among the various trajectories,
although the absence of water-channel hydrogen bonds in
simulation D resulted in somewhat more uniform fluctua-
tions of 100 to 120 throughout the channel. This analysis
suggests that the fine structure of water-channel interactions
can modulate the local flexibility of the backbone, not only
at the mouths of the pore but also in the single-file region.
In turn, the librational motion of the backbone CO groups
has been shown to have a strong influence on the dynamics
of the single file (Chiu et al., 1991).
The lipid bilayer has a profound influence on the stability
and the dynamics of the channel. To date, only one com-
putational study of the GA channel included the lipid ma-
trix, together with a large number of water molecules and
periodic boundary conditions (Woolf and Roux, 1994,
24 Biophysical Joumal

Structure and Dynamics of a Proton Wire
FIGURE 1 Cross section of the gramicidin A channel with 22 water molecules and an excess proton. The axis of the channel, z, runs from left to right.
This snapshot was taken from a quantum Feynman path integral molecular dynamics simulation, in which the water hydrogen nuclei are modeled by flexible
ring polymers (light blue). In this configuration, H+ is solvated by two water molecules in a 02H1 ion located near the center of the channel.
1996). The starting configuration for the present study was
chosen from the equilibrated part of that trajectory and
reflects the influence of the lipid bilayer. It is not possible to
preserve the long-term integrity of the channel structure
during a simulation in vacuo without incorporating the
influence of the membrane, if only approximately. To com-
pensate for the omission of the membrane, weak harmonic
constraints were imposed on the position of the tryptophan
side-chain atoms, so as to preserve the long-term integrity of
the channel. Although in principle our positional restraints
on Trp side chains necessarily limit the extent of backbone
torsional freedom, they do not by any means prevent the
fluctuations of channel backbone groups, or the rearrange-
ment of channel-water hydrogen bonds, as we will see in the
analysis of the trajectories.
Structure of the single file
Statistically representative configurations of the water mol-
ecules are shown in Figs. 1 and 2, and the average atomic
distributions of water 0 and H particles along the channel
axis are shown in Fig. 3. The distributions of single-file
water oxygen atoms obtained from trajectories A through C
are very similar. There are on average 10 water molecules in
the single-file region in all three simulations A, B, and C
(Fig. 2). In addition, two water molecules mark the ends of
the channel at z - + 12 A; these two water molecules are off
the axis of the channel and provide an interface with the
mouth region rather than forming part of the single file per
se. Five water molecules complete the solvent "caps" at
each end of the channel.
In the unprotonated case A, Fig. 2 shows that there is a
continuous and monodirectional donor-acceptor pattern of
hydrogen bonds linking single-file water molecules 1
through 9. This pattern breaks down at molecule 9, which
does not form a good hydrogen bond with water 10. These
observations hold over the entire trajectory A, as the struc-
ture displayed in the average density of oxygen and hydro-
gen particles shows (Fig. 3 A). Moreover, the discrete nature
of the water oxygen density in the single-file region indi-
cates that there is no significant diffusion of waters during
the simulation. Rather, the regular shape and spacing of the
oxygen peaks suggest that each oxygen atom fluctuates
around a well-defined mean position on the z axis. We note
two exceptions to the regularity of oxygen density peaks:
water number 9 is less mobile (sharper distribution of the
oxygen peak), whereas water molecules 1 and 2 are more
mobile, with possibly two nearly spaced favored locations
along the z axis for each of them. The asymmetry of the
distributions reflects the lack of convergence due to the
finite length of the simulations.
The quasiperiodic distribution of the single-file water
oxygen atoms arises from both water-water and water-
channel interactions. Hydrogen bonding between adjacent
water molecules is marked by the relative positions of0 and
H peaks in the single-file region. The regular donor-
acceptor pattern linking waters 1 through 9 noted above is
evidenced by the alternance of the peaks. Additionally, OH
bonds not involved in the water-water hydrogen-bonded
chain are oriented roughly perpendicularly to the channel
axis (resulting in overlapping 0 and H densities in Fig. 3)
and engage in hydrogen bonds with the channel backbone
Pornes and Roux 25

Volume 71 July 1996
K
Vi
I
F e~ r
V
(w s4f
A
B
I
C
A
D
FIGURE 2 Snapshots of water configurations obtained, respectively, from simulations A, B, C, and D. The single-file region of the pore contains a
hydrogen-bonded chain of 10 water molecules. In configurations B, C, and D, the excess proton is solvated in OH'+, 03H', and 02H1 ions, respectively.
Note how the donor-acceptor pattern of hydrogen bonds inverts around these ions.
oxygen atoms lining the interior of the pore. The helicity of
the backbone is recovered in the spacing of oxygen density
peaks along the channel axis. A 4.5-A spacing along the
axis corresponds to six residues in the channel backbone,
approximately one turn in the helix, and two single-file
water molecules.
It is desirable to compare the results obtained here on the
unprotonated system A to previous computational studies of
the gramicidin channel. These studies have been reviewed
recently (Roux and Karplus, 1994), so that here we only
emphasize the results pertinent to the properties of the
proton wire. Previous studies concluded that eight to ten
water molecules are present in the single-file region of the
GA channel, in accordance with experimental results and
with the present study. Numerous hydrogen bonds were
observed between backbone carbonyl oxygen atoms and
water molecules. Furthermore, the single-file water mole-
cules were found to be oriented the same way (i.e., in a
consistent donor-acceptor pattern) in most studies. How-
ever, gaps and defects in the hydrogen-bonded chain were
observed, very consistently in the most realistic study (Roux
and Karplus, 1994) and in another study including polariza-
tion of the water molecules (Jordan, 1990), and more infre-
quently with two other models (Chiu et al., 1989; Fornili et
al., 1984). The restriction imposed to prevent the evapora-
tion of the small water caps also prevents the possibility of
net water diffusion over the course of the simulations.
Although this would constitute a serious limitation in an
investigation of long-term transport properties, previous
studies indicate that the structure of the single file can be
reasonably well characterized over the few hundreds of
picoseconds spanned by each trajectory. Because the exper-
26 Biophysical Journal
"..I /P"" r-
O."
402 AO
11%0
VIO
*p0)

Structure and Dynamics of a Proton Wire
0.1s
0.12LiiiA
0.09
0.06 ,, ,
0.03'
Bl
B
0.12
0.09 '
0.06:
0.03 S~
0.12
0.09 ~
oSt X~~~~~~~~:0.06
-12 -lo -8 -6 -4 -2 0 2 4 6 8 lo 12
Z (X)
FIGURE 3 Distribution of water oxygen (bold) and hydrogen (dashed)
particles along the channel axis obtained, respectively, from simulations A,
B, and C. In all three cases, the 10 single-file water molecules are located
between z = -11.5 and z = 1 1.5. Hydrogen bonding between water
molecules is characterized by alterating B and H peaks, whereas water-
channel hydrogen bonds correspond to nearly overlapping O and H den-
sities. In cases B and C, the presence of an excess proton induces strong
hydrogen bonds in which the proton is located halfway between two
oxygen atom, notably at z = -2.
imentally determined mobility of water molecules (see Ta-
ble 1 ) corresponds to water displacements of about 2 A over
100 ps, much longer trajectories would be required to ob-
serve net diffusion.
Influence of an excess proton
In the following analysis we examine the effect of adding of
a proton to the single file. In all three simulations B, C, and
D, the presence of an excess H+ affects hydrogen bonding
among the single-file water molecules. In the particular
configuration of Fig. 2 B, the protonated species appears to
be a hydronium ion H30+, around which the donor-
acceptor pattern of the chain inverts, with OH bonds point-
27
chain will be discussed below, but we note here that in spite
of the differences in charge and hydrogen-bonded structure
of the single-file water chain, the average distribution of
water 0 atoms along the z axis is remarkably similar in
simulations A and B (compare Fig. 3, A and B). The
somewhat broader shape of the peaks in Fig. 3 B arises in
part from a longer time of simulation with respect to sim-
ulation A (400 ps versus 200 ps), and the mean position of
most oxygen atoms is nearly identical in the two simula-
tions. There is, however, significantly more mobility of
oxygen atoms 01 through 05, with two distinct peaks for
01, corresponding to alternate hydrogen bonding with wa-
ter 2 (right peak) and with the mouth water at z =-12 A
(left peak). In the single file of B, there are thus two defects
in the linear hydrogen-bonded chain, one between waters 9
and 10, and the other alternatively between 1 and 2 and
between 1 and the water at the channel mouth.
The analysis of single-file hydrogen atom distributions in
Fig. 3 B provides further information on the influence of an
excess proton. Again, the hydrogen bond donor-acceptor
organization is reflected in the relative locations of 0 and H
peaks. Remarkably, the hydrogen distribution between wa-
ters 4 and 5 is located halfway between the oxygen peaks,
indicating that sampling might be dominated by cases in
which the proton is shared by two water molecules in a
O2H' cluster. On either side of that cluster, the hydrogen
bonding pattern inverts, as noted earlier from Fig. 2 B.
In the isomorphic path integral treatment of simulation C,
the hydrogen nuclei are represented by flexible ring poly-
mers or necklaces of beads or p-particles. The average
distribution of the p-particles corresponds to the quantum
dispersion of the hydrogen nuclei. From the snapshots
shown in Figs. 1 and 2 C, it is observed that the spatial
dispersion of hydrogen nuclei in the water chain ranges
from 0.2 to 0.5 A. These conformations suggest tight sol-
vation of the excess proton by two water molecules in
O2H+, or in the less likely arrangement of a linear 03HL
cluster, in which two hydrogen nuclei are shared by three
oxygen atoms.
The defect in the linear hydrogen-bonded chain observed
in the previous cases subsists, with the orientation of the
plane of molecule 9 perpendicular to the axis of the channel.
As before, hydrogen-bonding features within the proton
wire are visible in the density of single-file atoms (Fig. 3 C).
The oxygen peaks are sharper than in Fig. 3 B because of the
shorter simulation time (100 ps versus 400 ps), but they are
located at the same positions along the z axis as in cases A
and B. Delocalization in the position of hydrogen nuclei
halfway between two oxygen peaks is evident for the
04-05 hydrogen bond, and to a lesser extent for 05-06.
This suggests that the presence of an excess proton together
with quantum treatment favors the emergence of tight
O2H+, and perhaps O3H', clusters inside the channel.
Finally, in the artificial "hydrophobic channel" simula-
tion D, where the electrostatic interactions between the
channel and water molecules were turned off, the absence of
ing away from the ion. The fine structure of the protonated
Pomes and Roux
hydrogen bonds between the single-file water molecules

Volume 71 July 1996
and the channel backbone results in a better connectivity of
the hydrogen-bonded water chain. In the configuration
shown in Fig. 2 D, the protonated water cluster resembles an
02Hz ion, with the excess proton located halfway between
two water molecules, in a strong hydrogen bond. Unlike the
results obtained from the other simulations, there are no
interruptions in the hydrogen-bonded chain across the
single-file region. In addition, the absence of water-channel
hydrogen bond results in the greater translational freedom
of individual water molecules, whose density along the
channel axis (not shown) displays practically no structure
compared to those obtained from trajectories A through C.
The importance of water-channel hydrogen bonds is exam-
ined in more detail below.
Channel-water hydrogen bonds
Hydrogen bonding among water molecules and between
channel polar groups and water molecules is illustrated in
Fig. 4. This configuration was obtained from trajectory C.
Each of the two water molecules forming a protonated
dimer forms hydrogen bonds with its water neighbors in the
single file, and one with a backbone carbonyl atom. Similar
interactions are also shown for the adjoining water mole-
cules. Although water-channel hydrogen bonds limit the
location of water molecules along the channel axis over the
time of the simulation, they do not preclude local deforma-
tions in the proton wire, as the irregularity of water-water
hydrogen-bonding distances shows.
A systematic characterization of channel-water and
water-water interactions can be gained through the study of
atom-atom radial distribution functions. Because the water
is in single file, no radial normalization was used; the i-j
atom-atom function simply measures the relative probabil-
ity of finding any atom of type j at a distance ri from any
atom i, averaged over the number of type i atoms and over
the total number of configurations. The atom-atom distri-
butions of the distances between channel backbone carbonyl
oxygen atoms and single-file water oxygen and hydrogen
particles obtained from simulations A through C are shown
in Fig. 5. The hydrogen bonds formed between the channel
and the single-file water molecules are notably consistent in
all three cases. All three 0-0 distributions have the same
structure and peak sharply at 2.95 A. The presence of an
excess proton results in a better-defined 0-0 peak in cases
B and C, compared to case A. The structure of 0-H distri-
butions is similar; the first peak at 1.9 to 2.0 A matches the
0-0 peak in area, and together they define well-aligned
FIGURE 4 Hydrogen bonds involving single-file water molecules in the GA channel. This configuration is identical to the one shown in Fig. 1. The
average dispersion of the Feynman path integral polymers (light blue) is about 0.3 A. In general, each water molecule forms two hydrogen bonds with its
neighbors in the single file, and one with a carbonyl oxygen atom on the backbone.
28 Biophysical Journal

Structure and Dynamics of a Proton Wire
25 50 75 100 125 150 175 200
Time (ps)
FIGURE 11 Time evolution of the z coordinate of all water oxygen
atoms from simulation A. Note the presence of correlated motions.
ygen atoms of the channel backbone. As one moves down
the single-file region, the water chain becomes increasingly
mobile, and there is a marked collective character to the
fluctuations in oxygen positions along the channel axis.
15 _
Wim MlTie'ps
10 IVn ;
5D
-lo
-15_
50 100 150 200 250 300 350 400
Time (ps)
FIGURE 12 Time evolution of the z coordinate of all water oxygen
atoms (plain) and of the hydronium oxygen coordinate (bold) from simu-
lation B.
Strong correlations in the dynamics of the single-file water
chain were also noted and studied by other authors (Chiu et
al., 1991, 1993). In this simulation, the magnitude of cor-
related motions culminates with water 1, which appears to
oscillate between two favored channel positions near z
-10.5 A.
Fig. 12 shows analogous time series obtained from the
protonated classical wire of simulation B, with the addition
(in bold) of the oxygen of the OH+ coordinate defined in the
previous section. Over the course of the simulation, the
proton translocation coordinate hopped many times between
water molecules within -7.5 A ' z ' 3.5 A, with preferred
residence in the -6 A z ' -1 A region, as noted before
from Fig. 10 B. As much as is visible from this graph, the
lifetime of transient H30 ionic species appears to be
comparable to the rapid fluctuations of the water molecules
in the channel. First, we note that the two water molecules
at z +12 A are less mobile than their single-file coun-
terparts, which reflects the fact that they form hydrogen
bonds both with the channel mouths and with the solvent
caps, the motions of which are restrained. Overall, the
magnitude of fluctuations in the positions of water oxygen
atoms in the protonated single file (Fig. 12) is comparable to
that observed in the unprotonated case (Fig. 1 1). Again, the
relative mobility of water molecules increases as z de-
creases, a tendency that culminates in the cases of 01, 02,
and 03, respectively, at z -10 A, z -8 A, and z -
-5.5 A. Each of these three atoms underwent a relatively
large displacement of about 1 A between two preferential
positions along the z axis. These displacements appear to be
correlated with occasional displacements of the "hydronium
coordinate" to water oxygen atoms 02 and 03. The increas-
ing mobility of oxygen atoms with diminishing z arises from
the combined effects of hydrogen bond defects near the
ends of the single file with opposite consequences: at
the "top" of the channel, the defect induces restrictions in
the displacement of 09, whereas at the "bottom" of the
channel, the defect allows 01 to alternately bind two sites in
the channel. The larger flexibility that results near the bot-
tom of the proton wire, in turn, might help to explain the
preferred presence of the excess proton in the lower region
of the channel. This consideration implies a strong coupling
of the proton translocation to the dynamics of the proton
wire, which is now considered in more detail.
Fig. 13 shows the water oxygen and hydronium z coor-
dinates over a smaller, 25-ps window in simulation B. That
time range is sufficiently short that individual translocations
between adjacent single-file oxygen atoms can be distin-
guished, yet is long enough for a number of representative
features to be observed. Most of the time, the proton trans-
location coordinate is either fluctuating very rapidly be-
tween two adjacent water molecules (e.g., at 76 ps t 78
ps, 82 ps ' t ' 84 ps) or between three adjacent water
molecules (e.g., 90 ps ' t ' 94 ps), or is stabilized in a
particular H30+-like ion for periods of up to 1 or 2 ps (e.g.,
78 ps ' t ' 80 ps). Occasionally, H+ translocation occurs
very rapidly over as many as six oxygen atoms (see 88 ps '
r- -w
Pornes and Roux 33
.A-.
10
v 4- q-.! $0.
5
PAO
0
-5
-10
-15

Volume 71 July 1996
trajectory B. The z coordinate of the midpoint of the shortest
15 AAA_ 0-0 separation is shown in bold, and that of the hydronium
oxygen atom in dashed lines. The two coordinates follow
each other very closely over the 25-ps window considered
1o here. At times, the "short0-0" coordinate appears to fluc-
tuate less than its hydronium counterpart (e.g., at 76 ps'
t, _.t 77.5 ps and 82 ps' t' 84 ps), which suggests that at
s these times the protonated species may look more like a
02H5 ion than like H30 . Generally, the two coordinates
are within half a hydrogen bond of each other, although
discrepancies can occur transiently (e.g., at t - 78.5 ps or
mt 90.3 ps), indicating that neither of them is totally
-5 w foolproof whenever the geometry of the chain is interme-
diate between well-defined states. At most times, however,
the two coordinates appear to be equally valid depictions of
the H+ translocation coordinate, a result that is reflected in
the equivalence of the two distributions of Fig. 10 B. Be-
-15 _ cause of the strong coupling between 0-0 separations andH+ translocation, it is appropriate to study the dynamics of
80 85 90 95 14 o fluctuations in 0-0 separations along the single file.
This is done in Fig. 15, where the successive 0-0 sepa-
Time (ps)
FIGURE 13 Same as Fig. 12 for a shorter time window.
3
2.8
t ' 90 ps), with very short-lived intermediates. From Fig. 2.6 1
13, proton translocation is clearly seen to occur on the time 2.8 Id 1l
scale of small fluctuations in the position of oxygen atoms 2.6
along the channel axis. Furthermore, although some coop-
erativity in the position of oxygen atoms is evident, it does 2.8 9
not consist in concerted, collective fluctuations of the sin-
gle-file water molecules, contrary to the unprotonated chain. 2.8
A detailed understanding of the coupling of H+ transloca- 2.6 8
tion to the dynamics of other atoms in the proton wire as a
whole, consequently, necessitates a closer look at the dy- 2.8 7
namics and fluctuations in 0-0 separations along the chain. 2,U
The equivalence of the time evolution of the two H+ 2.8
translocation coordinates defined earlier, i.e., the "hydro- 2.6 6
nium 0" and "shortest0-0" coordinates, can be assessed
from Fig. 14 for the 75-100-ps window in the time series of 0 2.8
62.6
4 2.8
2 2.8
0 ~~~~~~~~~~~~~~~~~~~~~~~2.6
2.8
2.6
-6
~~~~~~~~~~~~~~~~~~~~~~2.6
80 85 90 95 100 75 80 85 90 95 100
Time (ps) time (ps)
FIGURE 14 Time evolution of the hydronium oxygen (dashed) and FIGURE 15 Time evolution of all water-water hydrogen bond lengths in
shortest 0-0 (bold) coordinates from simulation B. The time window is the the proton wire obtained from simulation B. Bond numbers appear at the
same as in Fig. 13. right of the figure. The time window is the same as in Fig. 13.
34 Biophysical Journal

Structure and Dynamics of a Proton Wire
rations in the single-file region are shown as a function of
time for the same window in time as in Figs. 13 and 14. As
before, 0-0 separations labeled 1, 2 through 11 correspond
to distances between the water at the mouth of the channel
and 01, 01 and 02, etc.... through the 010-mouth water
separation. The 0-0 distances fall into three categories: 1)
intermittent or defectuous (1 and 10), 2) regular (2, 9, and
1 1), and 3) those alternatively strong and regular (3 through
8); in the latter category, there is a core of four hydrogen
bonds that are often short (4 through 7), whereas the outer
two are short and strong very occasionally, and then merely
in a transient way (3 and 8).
In the "core" of four hydrogen bonds in which the proton
is often shared, there are clearly two distinct states corre-
sponding to long and short hydrogen bonds, respectively, at
2.7 A and 2.45 A separations. A close look reveals that there
is exactly one short hydrogen bond in the proton wire at a
given time. Following the propagation of the "shortest 0-0"
reaction coordinate yields information on the mechanism of
H+ translocation in the proton wire. At t = 75 ps, hydrogen
bond number 5 is short for about 0.3 ps, then there is a large
fluctuation of about 0.2 A that is anticorrelated with an
inverse fluctuation in bond 4, which thereafter becomes
short and stays so for about a third of a picosecond; then
there is a large fluctuation in the 0-0 separation that is
anticorrelated to bond 5. The latter remains short for about
1.5 ps, and the translocation coordinate then hops succes-
sively to bonds 6, 7, 6, 7, 8, 7, etc. ..., in the manner best
depicted in Fig. 14. It is important to note that the succes-
sive hops always take place through successive vibrations in
the 0-0 distance of generally adjacent hydrogen bonds. A
corollary is that the lifetime of a short 0-0 bond is limited
by thermal fluctuations in the proton wire, and ranges from
transient lifetimes of under 0.05 ps to stable ones lasting up
to a couple of picoseconds (see bond 4 at 82 ps ' t ' 84
ps). Another consequence is that whenever the arrangement
of oxygen atoms in the single-file water chain allows it,
there are rapid cascades of translocations occurring in a
coherent fashion over up to five or six hydrogen bonds, e.g.,
at 88 ps ' t s 90 ps between bonds 3 and 8.
As one moves up and down along the hydrogen-bonded
chain, away from the translocation coordinate, the fluctua-
tions in 0-0 separations become less influenced by the
dynamics of proton translocation. The proximity to the short
0-0 bond means shorter 0-0 distances on average (recall
Fig. 7), and the transient formation of a tight cluster of water
molecules around the excess proton often means a looser
connectivity between water molecules a few A away in the
single file, as seen in Fig. 15 for bond 4 at t = 79 ps. In the
regions never or seldom visited by the translocation coor-
dinate, water-water hydrogen bonds are looser on average
and, at the extremities of the single file, they may even be
broken. Thus, in Fig. 15 hydrogen bonds labeled 3 and 8 are
seldom strong, bonds 2 and 9 are moderately good, whereas
bonds 1 and 10 are intermittent. At sufficiently long dis-
tances from the net proton charge, the electrostatic forces
are supplanted by the need for water molecules to form
good hydrogen bonds with the channel backbone. Near the
channel mouth, the mobility of the single-file water mole-
cules is affected by the reduced number of configurations
that they can assume to form good hydrogen bonds with a
reduced number of peptide bonds. In addition, the mobility
of water molecules at the mouth is further limited by the
greater accessibility of peptide groups in the last turn of the
gramicidin monomer. Conversely, the reduced mobility of
the water molecules near the mouths of the channel hinders
their full participation in the long-range cooperative thermal
motions that help determine the position of the translocation
coordinate. The case of bond number 1 at the "bottom" of
the single file offers a good illustration (Fig. 15). Both
bonds 1 and 2 are intermittent up to t = 88 ps, a situation
reflected by the distribution of 01 alternatively near the
mouth water molecule and near 02. At t = 88 ps, bond 1 is
suddenly broken, so that bond 2 becomes stronger. Because
of the enhanced connectivity and cooperativity at the bot-
tom of the single file, the proton translocation coordinate,
which had been confined to regions further up, is then able
to visit neighboring hydrogen bond 3 several times in the
ensuing 12 ps.
Mechanism of proton translocation
The mechanism of proton translocation along a linear
hydrogen-bonded chain has been conceptualized in terms of
hopping and turning defects, primarily by Nagle and other
workers (Nagle and Morowitz, 1978; Knapp et al., 1980;
Nagle and Tristam-Nagle, 1983; Nagle, 1987), based on the
theory of proton translocation in ice crystals. Before we
consider how the results of this study compare to that
model, we briefly review the hop-and-turn mechanism here.
Assuming an oriented chain (Scheme 1) in which all hy-
drogen atoms involved in water-water hydrogen bonds are
bound to the oxygen atoms on their left (I), the net trans-
location of a proton from left to right occurs through suc-
cessive hypothetical "hops" which, once the proton has
exited the chain from the right, leave the chain in an
inverted donor-acceptor pattern, with hydrogen atoms bind-
ing the oxygen atom to their right (III). This chain reaction
has been called diffusion of an "ionic defect" (Nagle and
Morowitz, 1978). The passage of another proton down the
chain from left to right requires the preliminary inversion of
the donor-acceptor pattern in each of the hydrogen bonds in
the single file. This second chain reaction, which requires
the sequential reorientation (or "turn") of each water mol-
ecule, has been described in terms of propagation of a
"bonding defect." It is important to note after other authors
(Nagle and Morowitz, 1978; Nagle, 1987; Deamer, 1987)
that in those channels which mediate the successive trans-
location of several protons, both stages (hop and turn) are
needed for the net translocation of each successive H+ by
the proton wire. Accordingly, each of the hop-and-turn
chain reactions can be viewed as the translocation of a
35Pornes and Roux
that pull the water chain together nearer the excess charge

Volume 71 July 1996
partial charge, corresponding, respectively, to the ionic de-
fect and the bonding defect (Nagle and Morowitz, 1978).
In the gramicidin channel, the proton translocation does
not take place through a hypothetical, totally concerted
mechanism in which all hydrogen-bonding protons of the
chain would hop in a single step. But it does not happen
either via a succession of incoherent hops between well-
defined hydronium ion intermediates. Rather, proton trans-
fer in the wire is a semidelocalized process that results from
the complex interplay of rapid hydrogen-bond length fluc-
tuations along the chain. For lack of a more precise descrip-
tion of the translocation coordinate, the species hosting the
excess proton may be described as OH2++1, a protonated
cluster of n single-file water molecules forming a tightly
hydrogen-bonded chain within which correlated motions are
strong. The nature of this cluster is transient, with n and the
precise location of the transferring proton near the cluster's
center both varying with rapid thermal fluctuations, and
with more infrequent events involving hydrogen bond reor-
ganizations with the channel backbone. This cluster may be
identified with the tight "core" described in the analysis of
the dynamics of the single file, with 2 ' n ' 5.
The dynamics of important cooperative effects within the
single file, like the dynamics of the hydrogen-bonded chain,
can be decomposed into two groups. On the one hand,
motions occurring in the short-time range (t s 1 ps) include
the rapid fluctuations involving H+ motions within the tight
protonated chain of n water molecules. Hopping of the
translocation coordinate occurs through anticorrelated fluc-
tuations in the length of adjacent hydrogen bonds (short/
long-long/short isomerization). The cooperativity decreases
for 0-0 bonds lying further away because of decreasing
electrostatic interactions along the single file. Importantly,
the translocation of proton along the single file does not
require large motions of the water molecules. Indeed, fluc-
tuations of 0.2 A in 0-0 separations suffice to displace the
proton coordinate by about 2 A along the channel axis.
Thus, over short periods of time (on the order of 1 ps),
translocation can occur over large portions of the channel
without net translation of the water molecules in the single-
file region.
On the other hand, the dynamics of water molecules in
outlying regions of the single file that are relevant to proton
translocation reflect the influence of the channel more di-
rectly. Whereas the excess proton is tightly solvated by
water molecules in a linear OH2++1 cluster, the dynamics
of water molecules further removed in the single file is less
sensitive to electrostatic forces arising from the net charge,
and is modulated instead by a delicate balance of hydrogen-
bonding interactions with other water molecules and with
the channel. Thus, the importance of the balance between
water-water and water-channel interactions, noted in previ-
ous investigations of ion transport through gramicidin (see
Roux and Karplus, 1994, and references therein), appears to
be particularly relevant in the case of proton translocation.
Occasional exchange between these hydrogen bonds occurs
tances in the tight protonated cluster, because it is linked
perhaps to slow conformational fluctuations in the backbone
of the channel dimer, and certainly to the unfrequent, ther-
mally activated net translation of water molecules in the
channel. Such motions lead to local reorganizations in the
hydrogen-bonding network involving the single-file region
of the proton wire, and they occur much less frequently than
those governing proton translocation within the protonated
cluster. Indeed, the dynamical interconversion that accom-
panies the making and breaking of a hydrogen bond be-
tween water 1 and the water molecule at the mouth of the
channel typically occurred every 25 ps or so in trajectory B
(see the frequency of I-A translations of O1 in Fig. 12).
Furthermore, no significant change in the relative strength
of the hydrogen bonds involving water 9, another site of
disruption in the hydrogen-bonded chain, was evident from
the analysis of the entire 400-ps simulation. Yet such mod-
ulations in the connectivity of the single file appear to limit
the extent of H+ translocation, as seen in the case of water
1, where the formation of the 01-02 hydrogen bond cor-
related with the sharing of a proton by 02 and 03. In-
versely, the lack of a good hydrogen bond between 09 and
010 and the subsequent restraints on the dynamics of water
molecules at the top of the hydrogen-bonded chain is prob-
ably the reason why the proton translocation does not reach
the top of the channel over the course of simulation B. The
long time scale involved in the reorganization of the hydro-
gen bonds in the single file is notably consistent with the
very slow diffusion of waters inside the channel; as seen
from Table 1, the diffusion of waters inside the GA channel
is reduced by a factor of 50 compared to the bulk. In effect,
the relative mobility of water molecules and H+ differs
more markedly inside the channel than in the bulk.
To summarize the above analysis, it appears that the
cooperativity among water molecules, which facilitates a
rapid translocation in the single file, is affected by limits in
the flexibility of the linear water chain imposed by hydro-
gen bonding with the channel. Whereas the competition is
not significant at the center of the channel, where H+
translocation is fast, it is sufficient to reduce the cooperat-
ivity between water motions near the mouths of the GA
channel, so that the dynamics of H+ translocation, ulti-
mately, is limited by the dynamics of bonding "defects" in
the connectivity of the proton wire. In the present system, as
in ice (Nagle and Morowitz, 1978), propagation of an ionic
defect appears to occur on a time scale at least one order of
magnitude shorter than that of bonding defects. This is in
qualitative agreement with the conclusions reached by Ake-
son and Deamer (1991) in an extensive experimental study
of the GA conductance to proton.
The last result presented here illustrates the importance of
the bonding defect on the dynamics of H+ translocation in
the GA channel. Fig. 16 depicts the time evolution of
single-file waters and of the hydronium coordinate along the
channel axis computed from the "hydrophobic channel"
simulation D, in which the computation of electrostatic
on time scales much longer than fluctuations in 0-0 dis-
36 Biophysical Joumal
interactions between the channel and water atoms was omit-

Structure and Dynamics of a Proton Wire
25 50 75 100 125 150 175 200
Time (ps)
FIGURE 16 Time evolution of the z coordinate of all water oxygen
atoms (plain) and of the hydronium oxygen coordinate (bold) from the
"hydrophobic channel" simulation D.
ted. These results show a dramatic increase in the mobility
of the water molecules and, above all, of the hydronium z
coordinate. Because there are no more water-channel hy-
drogen bonds, there is now no preferred position for the
single-file water molecules along the channel axis. Better
water-water hydrogen bonds can therefore be formed and
maintained, so that the proton wire also displays a very
strong cooperativity, with water displacements correlating
from end to end. As a consequence, H+ mobility is strik-
ingly enhanced, with very rapid hops spanning the entire
length of the channel within a few picoseconds. In this line
of thought, it is interesting to note that the higher proton
mobility exhibited by our model "hydrophobic channel"
simulation D is consistent with recent experimental obser-
vations of very fast proton translocation across lipid bilay-
ers, which was attributed to transient transmembrane chan-
nels formed by hydrophobic polyamino acid a-helices
(Oliver and Deamer, 1994).
CONCLUSION
In this work we have studied the properties of both unpro-
tonated and protonated linear chains of water molecules in
the pore of the gramicidin channel. To our knowledge, this
is the first attempt to investigate a proton wire in a biolog-
ically relevant system using a molecular model. Whereas
the presence of an excess proton in the single file does not
affect the conformation of the channel, it strongly influences
the geometry of the water wire by inducing a strong hydro-
gen bond in which the proton is shared by two water
molecules. Additionally, the net charge pulls much of the
rest of the hydrogen-bonded chain closer together and ori-
ents it. The properties of the water chain are also affected by
the quantization of the proton motions, with the quantum
dispersion of hydrogen nuclei resisting compression in short
hydrogen bonds. The atomic distribution functions along
the proton wire suggest that the transfer of a proton between
hydrogen-bonded water molecules is facilitated by the in-
clusion of quantum effects. In a previous study of water
wires in vacuo, this effect was attributed primarily to the
zero-point energy of the protons, and more marginally to
tunneling (Pomes and Roux, 1995).
As for the mechanism of the gramicidin proton wire, the
transfer was seen to be strongly coupled to the geometry and
the dynamics of the water chain. The translocation of a
proton between adjacent water molecules in the chain oc-
curs spontaneously, with successive thermal fluctuations of
0-0 separations. The process does not take place via a
totally concerted mechanism in which the donor-acceptor
pattern would flip over the entire chain in a single step, nor
does it happen via a succession of incoherent hops between
well-defined hydronium ion intermediates. Rather, proton
transfer in the wire is a semicollective process that results
from the subtle interplay of rapid hydrogen-bond-length
fluctuations along the water chain. Thus, the proton transfer
coordinate hovers back and forth and may travel over sev-
eral consecutive hydrogen bonds within a few picoseconds.
These observations have important implications for con-
structing theoretical models of proton wires. For instance, it
may not be appropriate to describe the rapid dynamics of the
ionic defect in terms of an incoherent stochastic Markov
rate process (Knapp et al., 1980). On a short time scale, the
dynamics of the protonated linear cluster exhibits frequent
coherent and strongly correlated microscopic events. In
addition, because of the significant delocalization of the
protonated cluster, a meaningful theoretical model must
allow the transfer to take place over the full extent of the
single file of water molecules. This warns against theoreti-
cal approaches in which proton dissociation would be taken
into account within a pair of water molecules, whereas the
other waters in the channel would be treated with a classical
nondissociable model.
Importantly, the results obtained from our simulations
indicate that hydrogen bonds between single-file waters and
the channel stabilize the local structure of the proton wire
and thereby diminish the integrity of the hydrogen-bonded
water chain. In turn, the properties governing proton trans-
fer along the chain are modified. This is especially notable
near the mouths of the channel, where the greater accessi-
bility of the protein backbone allows for a better binding of
the water molecules, thus generating defects, or weak links
in the water chain, that prevented full H+ translocation
across the channel over the course of our simulations. More-
over, the closing of a defect, or its migration in one direc-
tion, causes the excess proton to quickly translocate in the
same direction. Based on this study, one may conclude that
the full translocation represented in Scheme 1, step II, could
Pornes and Roux 37

38 Biophysical Journal Volume 71 July 1996
be two orders of magnitude faster than the complete reori-
entation of the hydrogen-bonded chain of the single-file
water molecules necessary to go from structure III to struc-
ture I. These observations suggest that the formation and
breaking of hydrogen bonds with the channel constitute the
limiting factor for the translocation of protons across the
membrane.
Despite rapid excursions along the single file of water
molecule, the excess proton remains localized on average
around z = -2 A, resulting in no net transport over the
length of the simulations. This observation should be not
interpreted as an indication that a proton binding site exists
near the center of the dimer at monomer-monomer junc-
tions. Test simulations in which the proton was equilibrated
at different locations along the channel axis demonstrate
that a large number of stable positions can be found else-
where in the system. The rapid structural fluctuations of the
protonated single-file of waters around an average position
and the slow movements of the average position along the
channel axis occur on two very different time scales. Al-
though this is difficult to prove, it appears that the translo-
cation mechanism is controlled by processes involving the
slow rearrangement of the complex hydrogen bond network
in the system. The significantly increased proton mobility
observed in simulation D strongly suggests that water-
channel hydrogen bonds are particularly important and de-
termine the rate of transport.
Perhaps a lesson of general importance to proton transfer
along chains of water molecules in protein cavities is that
proton conductance requires the presence of a well-
connected chain of hydrogen bonds. Furthermore, this chain
has to be flexible enough to undergo cooperative motions
that facilitate the translocation, which occurs via a succes-
sion of rapid transfers between nearest water neighbors
coupled to the donor-acceptor separation. However, the
flexibility of the water chain is to be balanced against the
need for interactions with hydrophilic groups that allow the
very presence of water in the pore.
These conclusions help delineate future directions for
theoretical research. For instance, a quantitative evaluation
of the rate limitation may require a more realistic channel
mouth environment. Molecular dynamics simulations per-
formed on larger systems that include the lipid bilayer and
large solvent caps would make it possible to address im-
portant questions, such as the influence of bulk to channel
translocation on the mechanism, and perhaps help in the
definition of a collective reaction coordinate. In turn, such a
parameter would open the way to the calculation of free
energy profiles for proton translocation in ways similar to
what has been done with other cations (Roux and Karplus,
1994), and could ultimately make possible a quantitative
comparison with experimental measurements of conduc-
tance through the refinement of kinetic models that take into
account the presence of an external field, such as the model
proposed by Prokop and Skaila (1994). The computational
study of proton wires in this and other transmembrane
systems should continue to offer useful insight into the
mechanisms controlling biological H+ translocation.
We thank Dr. T. B. Woolf for providing the initial solvated GA structure.
This work was supported by a grant from the Medical Research Council of
Canada. BR is a FRSQ research fellow.
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