Molecular dynamics of the transport of ions in a synthetic channel
D.A. Morton-Blake1a and Conan Kumari-Doyle1b
1
School of Chemistry, Trinity College, Dublin 2, Ireland
a
tblake@tcd.ie, bkumaridc@tcd.ie
Keywords: Molecular dynamics, Ion migration, Synthetic membrane, Electric fields, Oscillating
trajectories, Infra red radiation.
Abstract. Molecular dynamics investigations of ions in certain non-bulk media predict that they
are capable of significantly greater mobilities than in the liquid state. The entries of Li+, Na+, and
K+ ions into a proposed synthetic channel are simulated across a bilayer membrane between two
bulk aqueous salt solutions. The behaviours of the ions are investigated in the presence, and in the
absence of an electric field applied along the channel axis. The ion channel described consists of
twelve 15-crown-ether-5 rings bonded in a stacked conformation. The dynamics of ions as they
encounter this channel is investigated in cases when several channels are embedded as a lattice into
a bilayer membrane and also when a single channel floats freely in an aqueous electrolyte solution.
The frictional forces opposing the ion trajectories are calculated (~10 nN) and found to be
independent of velocity.
Introduction
An understanding is sought of the fundamental principles that underlie the mechanisms occurring in
the motion of ions through matter, including the natural processes that transport them across a cell
membrane. This would supplement the design and synthesis of nanochemical devices [1],
particularly those that can transport small molecules and ions between sites within an entirely
artificial structure. Some of the approaches taken include the emulation of natural processes by
applying molecular dynamics (MD) to a system in which ions in an aqueous medium move inside a
synthetic molecular channel. In some of these studies the cell membrane of the natural system has
been modelled by a simple amphiphilic bilayer, while in others efforts are made to understand the
entry of the ions into the mouth of a single channel, and their subsequent dynamics in the channel.
Molecular systems
The synthetic molecular channel in Fig. 1 consists of fourteen 15-crown-ether-5 (CE) molecules
covalently bonded in a stacked configuration [2], resulting in a channel length of 35.0 Å, which is
close to the thickness of a phospholipid bilayer membrane. At its narrowest point the diameter of
each CE ring is just over 4Å. When the channels were placed in an aqueous ionic solution our
earlier MD investigations [2] showed that while the cavities were too small to admit water
molecules, cations such as Li+, Na+ and K+ could enter; anions however were denied entry even in
the presence of strong electric fields along the axial direction.
Fig. 1 A crown ether ring in (a) is connected to its
neighbour by a CH2 link at the positions ‘*’ in (a) and
(b) to form a 14-ring channel (c). In (d) 25 such
channels are embedded in a bilayer membrane to connect two bulk salt solutions on either side of
the membrane.
The coulombic potential along the channel axis
The Mulliken partial atomic charges on the channel atoms, derived [2] from Hartree-Fock
calculations (using a 311 basis set supplemented by polarization functions) [3] show that the
electrostatic potential on the channel’s cylindrical axis, which has a significant contribution from
the O atoms of the CE rings, has a rather steady value of ca. −13v.
Fig. 2 The coulombic potential along the axis of
the channel in Fig. 1. The potential rises
steeply at the channel ends, but solvent
molecules would present substantial coulomb
energy wells in the two bulk liquid regions
outside the channel. The walls of the potential
well thus constitute an energy barrier to a M+
ion migrating either into or out of the channel.
The electrostatic potential in the axial region of the channel is more negative than at cation
sites near water molecules in the solvent bulk. Consequently, while solvent molecules cannot enter
the channel, a cation may do so by paying an energy penalty when it divests itself of its solvent
atmosphere. As a result, in its trajectory from the solvent bulk into the channel interior the
migrating ion encounters an energy barrier. Despite the negative potential along the channel’s axis,
the entry of a cation is therefore not necessarily a spontaneous process as, in order to surmount the
barrier, the ion may have to await the advent of favourable factors concerned with the ions’
positions and momenta near the mouth of the channel over the time interval considered.
Lattice of channels
In the 25-channel unit-cell microlattice shown in Fig. 1(d) in which the electrolyte solute was
LiF, NaF or KF, we used MD to monitor the motions of the ions and solvent molecules. The
particle trajectories are shown in Fig. 3.
Fig. 3 The trajectories of cations that have entered the channel from aqueous electrolyte solutions
of LiF, NaF and KF. The traces are for the 25 channels in the bilayer.
Only cations (Li+, Na+, K+) can enter the channel. The internal diameter of ~4 Å is too small to
admit water molecules, and F− (whose anion radius is actually smaller than that of K+) is also
denied entry. We conclude that the strongly negative coulombic axial potential (CE rings’ O
atoms) makes the channel a cation trap. After their initial oscillations the M+ migrants come to rest
in specific ‘halting sites’ in the channels. Inspection of their coordinates shows that the sites are
within 1Å of the axis and up to about 1Å from the planes of the ether O atom. In order to determine
the possibility of ion mobility between the sites, a modest electric field of 0.1 v Å−1 was applied to
an aqueous LiF solution. The field displaces some of the Li+ ions to downfield sites.
The behaviour of ions in a single channel
In order to examine details of the entry of the ions in the channel, we now confine our
attention to a single channel which, removed from the membrane, floats freely as a solute in the
aqueous bulk [2]. The energy barrier at the channel end delays the entry of the cation and its
trajectory may not be investigated in a period that is appropriate to the MD timescale. In order to
obtain results in a tractable time we had either to ensure that there were ions near a channel end or
to apply a static electric field along the axis.
Fig. 4 The axial trajectories of
three Na+ ions from a 3M
solution and their interactions
in the channel. Their mutual
interactions are observable in
the Figure and are discussed in
the text.
To obtain the trajectories in Fig. 4 several Na+ ions were in the vicinities of the channel ends. While
the Figure shows that a small number of them enter the channel almost immediately, after 100,000
timesteps the remainder still remained outside. Of the three entrants the Figure shows that ion #1
enter from the ‘top’ of the channel (z = 35 Å) and #2 from the ‘bottom’ (z = 0 Å). These two ions
initially repel each other at time 400 fs before reversing their directions; ion #2 then repels ion #3
which has just entered the channel. We find that


No more than three Na+ ions would enter the channel
(even after a million-timestep MD run).
As they enjoy a large negative electrical potential along the axis the cations do not
spontaneously exit the channel.
When the first migrant enters the channel the linear momentum conferred by its ‘drop’ into the axial
low-energy region allows the ion to partially climb the opposite wall of the energy trough. It then
comes to rest in one of the several ‘halting sites’ provided by the crown ether ring units.
Electric fields and temperature
An electric field along the channel axis will affect the migrating cation’s linear momentum.
Fig. 5 shows the ion’s trajectory when such a field of constant magnitude E = 0.3 v Å−1 is applied.
The Na+ initially falls the channel’s 36 Å length until it is reflected by the energy barrier at the
opposite end. The opposing effects of the channel’s internal -13v potential and the applied E = 0.3 v
Å−1 field produces a series of trajectory oscillations whose amplitudes are rapidly damped from an
initial 15 Å, and decay as the Na+ ion is trapped at one of the cavity sites. The system constitutes an
oscillator with frequency ~1012 Hz and a power output of 10-18 Js−1. A 10 cm3 millimolar sample of
such a channel system could generate a power output of 16 W.
Fig. 5 The oscillatory behaviour of a
single Na+ ion after entering the
channel with the aid of an electric
field. It is simultaneously subject to
this field and to the axial one
generated by the atomic coulomb
charges. The channel behaves as a
pulsed infra-red oscillator.
The cusp-like maxima of the oscillations in the Figure are due to the steep walls of the
coulombic potential at the channel ends. Lower intensities of electric field would bring the Na+ ion
to rest at other sites of the channel (the flat portion of the coulomb basin). The damping of the
oscillations is due to the friction between ion and the channel atoms. Damping factors have been
evaluated for the oscillations and related to the sliding frictional forces discussed below.
Can the Na+ ion exit the channel?
Fig. 6 Channel trajectory of a Na+ ion at
various (accessible) temperatures in an
electric field E = 0.3 v Å-1. At the two
highest temperatures the ion exits the
channel.
If the molecular channel is to mimic a membrane channel in a living cell, an ion would need
to exit the channel into the extracellular fluid. To do this it must overcome the energy barrier at the
end of the channel. Simply raising the temperature (except to an unrealistic range) does not
facilitate the exit of the Na+. But Fig. 6 shows that a moderate electric field brings the ion to a
position on the channel axis from which the thermal excitation at temperatures above 360 K can
effect its escape from the into the surrounding solution. At this point it becomes strongly solvated
and is unlikely to re-enter the channel unless the direction of the electric field is reversed.
Owing to the degree of dynamic randomness in the mechanism of the Na+ ion’s desolvation
before it enters the channel, the effect of temperature is not completely regular, but its overall trends
are clear. Fig. 6 shows the coincidence of the initial portions of the trajectory curves (7 < z < 37 Å)
for the different temperatures; the ion channel migration is thus virtually non-activated. Attempts to
monitor velocities on an Arrhenius plots led to an activation energy of less than 1 J mol-1.
Sliding friction of the Na+ channel migration
On atomic scales it is believed that the frictional force f may not obey Amonton’s Law and
may indeed depend on sliding velocity v. But among molecular tribologists there is little consensus
on the dependence of the friction. Forms f  v and f  v2 are commonly assumed; friction force
microscopy (where the velocities are much greater than those here) finds a logarithmic dependence
[6] while work on sliding polymer surfaces concludes that there is no dependence on v [7].
As the forces and velocities are calculated at each timestep of the ion migration the friction
can be tested for a wide range of velocities. These are generated by subjecting the Na+ ion to
electric fields applied in either direction along the channel [-0.3 < E < +0.3 v Å−1]. This imposes a
range of velocities on the ion as may be inferred from Fig. 7(a).
The frictional force at any instant is obtained by subtracting the coulombic force of the
channel atoms on the Na+ ion at that position from the total force calculated, together with the
migration velocity, in the MD. [Noise spikes resulting from velocity corrections from the coupled
Fig. 7 (a) Different values (positive and negative) of electric field E impart different trajectories
and velocities to the migrating Na+. (b) The frictional forces between the ion and the channel seem
to be independent of velocity.
thermostat at high ion velocities (0 to 1000 ts) were minimized by shortening the timestep unit from
1 fs to 0.1 fs (10-16 s).] The forces (nanonewtons) in part (b) of the Figure show that the f  vn
relation is obeyed only for n = 0. Despite the scattered nature of the plots (inadequate statistics) the
friction seems appreciably constant at 8 - 9 nanonewton over the velocity range of 0 to 1103 nm s−1
and we conclude that ionic friction in this molecular system is independent of sliding velocity.
Computational facilities:
An Institúd um Theicneolaíocht Eolais agus Riomhfhorbairt na hÉireann (IITAC).
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