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\begin{abstract}
Na channels \textbf{that produce} the action potentials of nerve and muscle
include a selectivity filter formed by both positively and negatively
charged amino acid residues in a molecular pore. Here we present Monte Carlo
simulations of equilibrium ion absorption in such a system. Ions are treated
as charged hard spheres in a \textbf{uniform} dielectric. Tethered
carboxylate and amino groups known to line the selectivity filter of the Na
channel are represented as charged hard spheres and restricted to \textbf{%
the filter} region of the channel. Consistent with experiments, we find (1)
that absorption of Ca$^{2+}$ into the filter exceeds absorption of Na$^{+}$
only \textbf{when the concentration of } Ca$^{2+}$ is some tenfold larger
than physiological\textbf{\ XXX}; (2) the model channel absorbs smaller
alkali metal ions preferentially \textbf{compared to} larger ones. This
selectivity involves volume exclusion of larger alkali metal ions from the
center of the filter region. In this region, the spontaneous distribution of
structural ions results in a low density of structural net charge, but a
large packing fraction.\textbf{Wolfgang: I did not understand this last
sentence. Do you mean...}

"In this region, the spontaneous distribution of structural ions (that are
part of the protein) results in a low density of structural net charge, but
a high density of matter. That is to say, the structural ions of the protein
fill a large part of the pore of the channel."
\end{abstract}

\begin{center}
\textbf{\LARGE Monte Carlo Simulations of Ion Selectivity in a Biological Na
Channel: Charge Space Competition}

\vspace{1.0cm}

Dezs\H{o} Boda\\[0pt]
\vspace{0.2cm} \textit{Department of Physical Chemistry,}\\[0pt]
\textit{University of Veszpr\'{e}m, H-8201 Veszpr\'{e}m, P.O. Box 158,
Hungary}\\[0pt]
\textit{email:} boda@almos.vein.hu \\[0pt]
\vspace{0.5cm} David D. Busath\\[0pt]
\vspace{0.2cm} \textit{Department of Zoology and Center for Neuroscience,}\\[%
0pt]
\textit{Brigham Young University, Provo, Utah 84602}\\[0pt]
\textit{email:} david\_busath@byu.edu\\[0pt]
\vspace{0.5cm} Bob Eisenberg\\[0pt]
\vspace{0.2cm} \textit{Department of Molecular Biophysics and Physiology,}\\[%
0pt]
\textit{Rush Medical College, Chicago, Illinois 60612}\\[0pt]
\textit{email:} beisenbe@rush.edu\\[0pt]
\vspace{0.5cm} Douglas Henderson\\[0pt]
\vspace{0.2cm} \textit{Department of Chemistry and Biochemistry,}\\[0pt]
\textit{Brigham Young University, Provo, Utah 84602}\\[0pt]
\textit{email:} doug@huey.byu.edu\\[0pt]
\vspace{0.5cm} Wolfgang Nonner*\\[0pt]
\vspace{0.2cm} \textit{Department of Physiology and Biophysics,}\\[0pt]
\textit{University of Miami School of Medicine, Miami, Florida 33101}\\[0pt]
\textit{email:} wnonner@chroma.med.miami.edu\\[0pt]
\vspace{0.5cm} \today
\end{center}

\newpage

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\section{Introduction}

Ion channels are a major class of proteins that bridge cell membranes and
thereby control many biological functions in health \cite{Hille2001} and
disease \cite{Ashcroft1999}. Ion channels regulate the flux of ions into the
cell and, when open, select between ions. The flux is driven by the gradient
of electrochemical potential, and controlled by the physical properties of
the channel protein \cite{HH1952d}. How proteins control selectivity has
long been a central question in biology. Here we examine a mechanism of ion
selectivity for the Na channels of nerve and muscle membranes.

Na channels are part of a large family of structurally related
voltage-controlled channels that also includes Ca and K channels. The
conductive pore in this family is formed between four monomers (K channels)
or four homologous motifs within a single polypeptide (Na and Ca channels).
The ion-selective properties of the channels seem to arise in small segments
of the sequence (the `P loop'); these segments form inserts that locally
restrict the diameter of a wider pore \cite{Doyleetal1998, Guyetal1995}.
Site-directed mutations suggest that the constriction in Na- and Ca-specific
channels is lined by the sidechains of the amino acid residues, whereas in K
channels it is lined by atoms of the polypeptide backbone \cite%
{Doyleetal1998, Terlauetal1991, Yangetal1993, Tsushimaetal1997,
Perez-Garciaetal1997}.

Ionic selectivities of Na and Ca channels seem largely to be determined by
four residues present in corresponding positions of the four P loops. These
are all glutamate (E) in Ca channels (the `EEEE-locus'), but include
aspartate (D), glutamate, lysine (K), and alanine (A) in Na channels
(`DEKA-locus'). Mutating the amino acids in these four positions of the Na
channel from DEKA to DEEE produces a strong shift of selectivity from Na$%
^{+} $ to Ca$^{2+}$, in effect creating a Ca channel \cite{Heinemannetal1992}%
. Further mutagenesis experiments have shown that a positively charged
residue in the locus (such as lysine, arginine, or histidine at pH 6) is
needed to create the low divalent affinity seen in Na channels \cite%
{Favreetal1996, Sunetal1997}.

\textbf{Although} crystallographic coordinates of Na channels have not yet
been determined, \textbf{many structural properties of the pore }XXXXX have
been deduced by other means. Hille probed the ionic pathway with protons,
alkali metal cations, and organic cations, and postulated a structure for
the `selectivity filter' of the Na channel \cite{Hille1971, Hille1972}. In
this view, the selective portion is a short axial subsegment of the pore
that forms a rectangular frame (aperture $\approx 0.3$ nm $\times $ 0.5 nm)
of oxygen atoms around the ionic pathway, some with charge. \textbf{It is
gratifying that }this structural hypothesis, developed 30 years ago, seems
consistent with possible arrangements of the \textbf{actual primary
structure, namely the }DEKA locus, \textbf{which was unknown and unguessed
at that time, }but the physical principles by which the DEKA locus produces
the observed alkali metal and alkali earth selectivities remain unknown.

Recently, the physical basis of ionic selectivity in the analogous EEEE
locus of Ca channels has been examined using statistical mechanics and a
simple model to capture electrostatic and excluded-volume effects in the
locus \cite{Nonneretal2000, Nonneretal2001, Bodaetal2000, Bodaetal2001a,
Bodaetal2001b}. \textbf{The model is a modest extension and reworking of the
standard models of concentrated salt solutions found in the literature of
physical chemistry in the last decade or two. REF Barthel book or whatever. }%
The four carboxylate groups in the Ca channel are represented by a cluster
of 8 half-charged oxygen ions confined to a small volume (the selectivity
filter) but free to move within that volume. Ca$^{2+}$ accumulates in
preference to Na$^{+}$ or other alkali metal cations (in agreement with
experiments on biological Ca channels) because of Coulombic and hard-sphere
interactions among these confined oxygens and mobile ions. Ca$^{2+}$ is
preferred over Na$^{+}$ because one divalent Ca$^{2+}$ provides the same
amount of charge as two Na$^{+}$ in about one half the ionic volume. The
crowding of charge produces selectivity because of the competition of the
counter\textbf{ions} for space, the CSC (charge/space competition) mechanism 
\cite{Nonneretal2000, Bodaetal2000}.

This paper examines \textbf{how well a} CSC mechanism \textbf{can account for%
} selectivity in the DEKA locus of Na channels. We report Monte Carlo (MC)
simulations of the equilibrium ion absorption that arises when Na$^{+}$
ions, other alkali metal cations, and Ca$^{2+}$ compete \textbf{in} a simple 
\textbf{representation }of the DEKA locus.

\section{Model}

The cylindrical simulation cell is shown in Fig. 1 in an axial
cross-section. The protein (excluding the charged atoms of the pore lining,
see below) and lipid membrane are represented as hard walls inaccessible to
ions (solid lines). \textbf{Ions are reflected off the surface of these
walls and cannot penetrate into the wall itself. }A cylindrical hole through
the protein (radius $R_{f}$, length $H_{f}$) connects two reservoirs (baths)
that contain mobile ions such as Na$^{+}$, K$^{+}$, Ca$^{2+}$, and Cl$^{-}$.
The dimensions assigned to the cylindrical hole ($R_{f}=0.4-0.5$ nm, $%
H_{f}=1 $ nm) represent the narrowest part of the biological channel (the
`selectivity filter'). The baths in this simulation represent both the
actual baths around the protein and the wider portions of the channel.
Earlier simulations of a Ca channel showed that the inclusion of atria at
either end of the pore has only small effects on ion accumulation in the
filter \cite{Bodaetal2001b}. The \textbf{reflecting NOTE reflective is not
an English or American word} radial boundary of the simulation cell is far
enough from the channel ($R=6.25$ nm) so that edge effects can be neglected,
and the axial length of the cell ($H\approx 25$ nm) is sufficient to allow
ionic densities to attain bulk values. Periodic boundary conditions (pbc)
are used in the axial direction, making the two baths regions of a single
bath.

The filter region of the Na channel contains the side chains of the DEKA
locus. Two of these residues (glutamate and aspartate) carry a negatively
charged carboxylate group, one residue carries a positively charged amino
group (lysine), and one residue (alanine) is electrically neutral. These
groups form the ends of side chains and thus are likely to have some
mobility. \textbf{We model these groups as `dissolved' particles that move
without restriction, within the filter volume---as do mobile ions (like Na}$%
^{+}$\textbf{\ or Ca}$^{++}$\textbf{)---but are confined to that volume by
structural constraints (not yet specified in physical detail)}, following
earlier work on the EEEE locus \cite{Nonneretal2000, Nonneretal2001,
Bodaetal2000, Bodaetal2001a, Bodaetal2001b}). \textbf{Note that in this
description the `filter volume' includes atoms that are part of the channel
protein, i.e., atoms that are covalently linked to the protein that makes
the channel wall. }Specifically, the two carboxylate groups are represented
as four \textbf{`dissolved'} half-charged oxygen ions (hard spheres of
diameter 0.28 nm), the amino group is represented as a \textbf{`dissolved'}
ammonium ion (diameter 0.3 nm). The uncharged and short alanine residue is
not represented in the model. The centers of the oxygen and ammonium ions
are restricted to a subvolume of the filter so they cannot protrude from the
filter. Simulations were made with the center of the ammonium ion restricted
to the central 0.75 or 0.25 nm of the filter while the oxygen ions were
restricted to the central 0.75 nm in all simulations.

The mobile ions are modeled as charged hard spheres with diameters 0.12 nm
(Li$^{+}$), 0.19 nm (Na$^{+}$), 0.266 nm (K$^{+}$), 0.34 nm (Cs$^{+}$),
0.198 nm (Ca$^{2+}$), and 0.362 nm (Cl$^{-}$). The solvent is represented as
a dielectric continuum (dielectric coefficient 78.5) in this so-called
primitive model of electrolytes (PM). Nonner et al. \cite{Nonneretal2000,
Nonneretal2001} have used both the PM and the solvent primitive model (SPM,
in which water is a \textbf{solvent made of uncharged hard spheres}) in
computations \textbf{of} the Ca channel and found that the two models give
qualitatively similar but quantitatively different predictions of ion
accumulation (the SPM gives the same selectivities \textbf{but }in a smaller
filter volume). A similar conclusion can be drawn from the works of Tang et
al. who studied \textbf{XXX} diffusivity and conductivity in a pore using
both the PM \cite{Tangetal2001a} and the SPM \cite{Tangetal2001b} to
represent the electrolyte. We start with the PM because its simulations take 
\textbf{much }less time, allowing \textbf{much }more exploration of
conditions and alternatives.

The dielectric coefficient is assumed to be uniform everywhere (78.5)
although polarization is likely to be less in the protein or lipid than it
is in the aqueous baths. The assumption of a uniform dielectric coefficient
is expected to reduce electrostatic effects in the filter. We accept this
simplification \textbf{temporarily }because the polarizability of the filter
region is presently unknown \textbf{\ and will replace it as soon as the
necessary information is available.}

For the details of the canonical MC simulations the reader is referred to
reference \cite{Bodaetal2001b}. The electrostatic effect of the periodic
images in the axial direction (for which pbc's are applied) was taken into
account using the charged line method \cite{Bodaetal2000}. The accuracy of
this pbc method has been assessed \cite{Yang2etal2002}. An important feature
of the simulation was that equilibration between the small and crowded
filter and the large and dilute bath was accelerated by `preference
sampling'. In this technique, special MC steps, namely particle jumps, are
performed, and the jumps that occur between the filter and the bath are
given preference. The resulting bias is balanced by a factor that scales the
probability of acceptance of MC steps \cite{Bodaetal2001b}. \textbf{This
method has been calibrated and checked and does not introduce significant
error in the domain in which we use it. }Because the canonical ensemble was
used, bath concentrations were an output of the simulation. Each production
run was preceded by a short simulation to obtain an estimate of
concentrations, followed by an adjustment of the dimension H in order to
approximate the desired concentrations in the far bath. \textbf{WOLFGANG
please verify that "H" upper case H is identified in the relevant figure.} A
typical run consisted of $3\times 10^{5}$ Monte Carlo steps. Error bars
shown in Figs. 2, 4, and 6 give the standard errors between local averages
formed about subsets of $10^{4}$ successive steps.

\section{Results and Discussion}

Ca channels contain only carboxylate groups in their selectivity locus and
they strongly prefer Ca$^{2+}$ over Na$^{+}$ under physiological ionic 
\textbf{conditions, in which solutions contain much more Ca}$^{2+\text{ }}$%
\textbf{\ than Na}$^{+}.$Mutated Na channels containing only carboxylate
groups also prefer Ca$^{2+}$. Normal Na channels contain a lysine in their
selectivity locus and chiefly conduct Na$^{+}$ in physiological conditions
(extracellular concentrations [Ca$^{2+}]\approx 10^{-3}$ M; [Na$^{+}]\approx
0.1$ M). Ignoring mobile ions, the net charge in the Na channel filter is $%
-1 $. Therefore, \textbf{XXXX }Ca$^{2+}$ is expected to be excluded\textbf{\
lest it introduce an excess positive charge and violate local
electroneutrality}.

We study the effect of the positively charged group (lysine) present in Na
channels but not Ca channels. In the simulations shown in Fig. 2, CaCl$_{2}$
is added to a 0.1 M NaCl bath solution. Results are shown for $R_{f}=0.4$ nm
and $R_{f}=0.5$ nm. The total number of ions in the channel is substantially
smaller than one (Fig.2). This number of ions was obtained by integrating
average ionic densities over the region defined by $R_{f}$ and $H_{f}$ plus
a `cap' at each pore mouth that extended by one ionic radius in the axial
direction into the bath. With either pore radius, the model of the wild type%
\textbf{---i.e., of the natural, }\emph{not }\textbf{mutated---}Na channel
preferentially absorbs Na$^{+}$ when the Ca$^{2+}$ concentration in the bath
is near the physiological value (0.001 M). More Ca$^{2+}$ than Na$^{+}$ is
accepted into the filter only when the concentration of Ca$^{2+}$ in the
bath is made greater than 30-40 mM. Indeed, \textbf{if we compare this
result to} previous work \cite{Nonneretal2000, Nonneretal2001, Bodaetal2000,
Bodaetal2001a, Bodaetal2001b}, we conclude \textbf{that our} model of the Na
channel is a low quality Ca channel.

In a system in which the dielectric coefficient in the membrane protein and
filter is less than that \textbf{the value of 78.5 }assumed here \textbf{XXX}%
, one expects the Ca$^{2+}$ to be absorbed even less,\textbf{\ because less
polarization charge would be present to screen the divalent cation } \cite%
{Moyetal2000, Corryetal2000}. Na channels in which the DEKA locus is mutated
to DAAA, AEAA, or AAEA, \textbf{are known, }however, \textbf{to }conduct Ca$%
^{2+}$ ion \cite{Favreetal1996, Sunetal1997}. The filter of these \textbf{%
mutated }channels contains a single negative net charge like the wild type
channel that we model. \textbf{The experimental finding of }Ca$^{2+}$
conduction \textbf{in a channel containing a single negative charge, XXX}
indicates that charge neutrality \textbf{is} not \textbf{XX} strictly
preserved within the selectivity locus of \textbf{real or mutated }Na
channels. \textbf{This finding agrees with the properties of our model.
Properties of the channel other than net charge }seem to be important for
reducing Ca$^{2+}$ access. One such factor might be the \textbf{spatial }%
arrangement of positive and negative groups in the locus\textbf{, i.e. some
details of the atomic structure of the channel}.

The locations of the sidechains of the DEKA locus are unknown and so, 
\textbf{in our model, we allow} the charged groups of the sidechains be
anywhere in the filter. \textbf{XXX }Fig. 3 \textbf{shows the profiles of}
axial concentration\textbf{\ XXX} plotted for the channel of Fig. 2 \textbf{%
with width }$R_{f}=0.4$ nm. \textbf{Note that t}he four O$^{-1/2}$ ions find
their way to the two ends of the filter\textbf{\ even though the }the entry
and exit \textbf{ends }of the filter are \textbf{identical in our model}. 
\textbf{NOTE: Wolfgang, the previous sentence is wonderful. }We emphasize
that this is a \textsl{spontaneous nonuniform }arrangement of the tethered
ions (\emph{not} imposed by structural constraints from the protein). 
\textbf{Wolfgang: note rearrangement of sentence and extra emphasis on
"spontaneous nonuniform". This is a crucial part of the paper and has to be
written just so, in my view. }The distribution of the single positive 
\textbf{structural} ion\textbf{---the tethered NH}$_{4}^{+}$\textbf{---}is
much more uniform \textbf{than the distribution of the structural anions,
the tethered O}$^{-1/2}$\textbf{\ ions}.

The spontaneous \textbf{nonuniform }spatial arrangement of tethered charges
has important effects on the distributions of mobile species, e.g., Na$^{+}$
and Ca$^{2+}$\textbf{\ and is responsible for some of the differences
between calcium and sodium channels.} In the EEEE locus of the Ca channel, Na%
$^{+}$ and Ca$^{2+}$ accumulate in the center of the filter. In the DEKA
locus of the Na channel, mobile ions are excluded from the center of the
filter and are present in much smaller density altogether. In fact, Na$^{+}$%
, the native ion of this channel\textbf{---which is} \textbf{responsible for
the biological function of the wild type Na}$^{+}$\textbf{\ channel---}has
nearly the same density in the bath and filter, although it is somewhat more
concentrated in the baths just outside the filter. \textbf{NOTE: Wolfgang
you might want to quote Frankenhaeuser and Hodgkin here who, if I remember
correctly, concluded from the (relative) lack of depletion of Na during Na
currents, that the "effective" space available to Na was larger than that
available to K.}

The distribution of Na$^{+}$ is consistent with earlier models of Na
permeation and with several types of experiments. For instance, the
biological channel is known \cite{Greenetal1987, Garberetal1987,
NaranjoandLatorre1993} to maintain substantial conductance even when the
baths are diluted to less than 1 mM NaCl, as if Na$^{+}$ density near the
channel were buffered by a negative surface charge at the channel mouths.
Unidirectional tracer fluxes in Na channels indicate that there is no
significant correlation among the passages of individual Na$^{+}$ ions \cite%
{BegenesichandBusath1981, Rakowskietal2001}, as if the (mean time averaged)
density of Na$^{+}$ in the channel were not very large (corresponding to
less than one ion present on average). \textbf{NOTE: Wolfgang, the previous
sentence is wonderful. }Our model gives a distribution of Na$^{+}$ rather
like that in chemical kinetic models that have two weak `binding sites'
flanking an activation barrier \cite{Hille1975, Ravindranetal1992,
NaranjoandLatorre1993}.

\textbf{Our simulations give this distribution of structural charge as a
result, not as an assumption. The simulations }show that the energetically
most favorable distribution of structural charge in the DEKA locus implies%
\textbf{\ XXX} a distribution of Na$^{+}$ \textbf{like that found when two
weak binding sites flank a barrier.} On the other hand, the density of Na$%
^{+}$ in the center of the filter \textbf{in our simulation turns out to be}
similar to \textbf{the density of Na}$^{+}$ in the bath, so an energy
`barrier' in the sense of transition state theory does not exist for the 
\textbf{native} ion \textbf{that carries large currents in the physiological
situation}. \textbf{Some workers have doubted that a channel designed to
conduct maximal currents would contain large energy barriers. Note: Wolfgang
if the last sentence is too much, cut it out. But some of us did make this
point clearly at least ten years ago.}

Although the \textbf{XXX} location of the O$^{-1/2}$ ions \textbf{in the
computation reported} in Fig. 3 seems \textbf{reasonable and }consistent
with experiment, we were interested in \textbf{studying the effects of
confining NH}$_{4}^{+}$ \textbf{in a smaller region in the center of the
filter. Such confinement in the center might produce produce an
electropositive potential barrier in the center of the filter and thus
enhance selectivity.} Fig. 4 examines the \textbf{XXX }Na$^{+}$/Ca$^{2+}$
selectivity \textbf{of such a channel, with the NH}$_{4}^{+}$\textbf{\ more
tightly tethered.} Here, we shortened the NH$_{4}^{+}$ confinement region
from a length 0.375 nm on either side of the channel center to a length of
0.125 nm while $R_{f}=0.4$ nm. Forcing the NH$_{4}^{+}$ to the center of the
filter reduces \textbf{the absolute selectivity (i.e., }the apparent
affinity) of the filter region for Na$^{+}$, but \textbf{it increases the
relative selectivity because it reduces the absolute selectivity of }Ca$%
^{2+} $ \textbf{more so. That is to say, the }channel has become an even
less efficient Ca channel and more of a Na channel. Density profiles for the
constricted NH$_{4}^{+}$ case are shown in Fig. 5.

\underline{Calcium Block of Sodium Current. }\textbf{\ NOTE: an underline or
separator is needed here I think. }In biological Na channels, \textbf{%
raising the (extracellular) concentration of Ca}$^{2+},$\textbf{\
substantially reduces the Na}$^{+}$\textbf{\ current flowing into the cell }%
(see \cite{Ravindranetal1991} and references therein). \textbf{If the
density of Ca}$^{2+}$\textbf{\ in the filter were found to be raised, when Ca%
}$^{+2}$\textbf{\ is raised in the bath, and the density of Na}$^{+}$\textbf{%
\ in the filter was found to be concomitantly reduced, the current of Na
would be reduced. }

Such depletion of Na$^{+}$ by Ca$^{2+}$ is not seen in \textbf{our} model,%
\textbf{\ when Ca}$^{+2}$ is raised in the bath: as Ca$^{2+}$ is added%
\textbf{\ XXX}, more Ca$^{2+}$ is accepted into the pore \textbf{but} only
small amounts of Na$^{+}$ are removed from the filter (Fig. 2). Furthermore,$%
\mathbf{XXXX}$ \textbf{the }overall small (average) densities in the filter%
\textbf{\ imply a small} probability for a Ca$^{2+}$ and a Na$^{+}$ to meet
each other\textbf{\ (in the filter)}. \textbf{The positively charged group
in the model of the DEKA locus reduces Ca}$^{2+}$\textbf{\ affinity,
compared to the Ca affinity in a Ca channel without the positively charged
group by preventing Ca}$^{2+}$\textbf{\ from displacing Na}$^{+}$\textbf{.} 
\textbf{Our (}self-consistent) calculation \textbf{suggests that } a
different explanation of the Ca$^{2+}$ block of Na$^{+}$ channels needs to
be considered\textbf{, one that may extend beyond the domain of our model
considered here.} For instance, Na$^{+}$ might be displaced by Ca$^{2+}$
from a region outside the selectivity filter. \textbf{In that case, when
large currents flowed, Na}$^{+}$\textbf{\ might be depleted outside and
inside the channel, and Na}$^{+}$\textbf{\ current might then be blocked.} 
\textbf{Nonequilibrium computations are needed to examine this possibility.}

\underline{Anion-cation selectivity}. \textbf{NOTE: Wolfgang, we need an
underline here in my view. }The model of the filter excludes the anion, Cl$%
^{-}$, regardless of whether the NH$_{4}^{+}$ are restricted to the central
0.75 or 0.25 nm of the filter (Figs. 3, 5). \textbf{The central region of
the filter actually contains net positive structural charge in the
restricted case, even though this is a cation channel. } Cl$^{-}$, however,
is excluded throughout \textbf{the filter, even in the positive central
region of the restricted case}; \textbf{the forces arising from the excluded
volume of Cl}$^{-}$ \textbf{are so large--due to its large diameter (NOTE:
PLEASE PUT A NUMBER HERE)---overwhelm the electrical forces in this case}.

\underline{Selectivity among monovalent cations}. \textbf{Fig. 6 shows
selectivity among monovalent cations by plotting the number of Li}$^{+}$%
\textbf{, K}$^{+}$\textbf{, and Cs}$^{+}$\textbf{\ ions absorbed compared to 
}the number of Na$^{+}$ ions absorbed (when the Na$^{+}$ channel of radius $%
R_{f}=0.4$ nm is \textbf{exposed to} mixed solutions. We begin with 0.1 M
NaCl and gradually replace Na$^{+}$ ions with one of these cations, holding
the concentration of Cl$^{-}$ constant. Li$^{+}$, K$^{+}$, and Cs$^{+}$ ions
replace Na$^{+}$ ions in the bath as well as in the filter. Selectivity is
determined by observing which species predominates in the filter when the
test cation concentration equals that of Na$^{+}$ in the bath, i.e., at $%
\left[ \text{Na}^{+}\right] =0.05$ M. As would be expected from\textbf{\ the
CSC of crowded charges}, the ions of smaller diameter are \textbf{XXX }%
accepted into the channel \textbf{selectively in preference to larger ions}.
Unlike the Ca channel \textbf{(i.e., unlike the EEEE locus)}, this \textbf{%
DEKA} Na channel is, indeed, reluctant to accept a high density of \textsl{%
any} \textbf{NOTE: Wolfgang the previous word should be emphasized in my
view }counterion, \textbf{but smaller alkali metal ions are excluded less
than the large ones}. The non-physiological ion Li$^{+}$ is preferred over Na%
$^{+}$, as has been observed \textit{in vitro}.

Fig. 7 plots axial densities of Na$^{+}$ and K$^{+}$ in one simulation in
which the bath contained Na$^{+}$ and K$^{+}$ in the ratio $\approx 1:3$.
The distributions of K$^{+}$ and Na$^{+}$ \textbf{differ from those in the
bath}. Whereas Na$^{+}$ has similar concentrations in the bath and in the
filter, K$^{+}$ is less concentrated in the filter than in the bath; K$^{+}$
has a larger diameter than Na$^{+}$ and thus experiences stronger repulsion
from the excluded volume effects that arise in the filter: the K$^{+}$ ions
attracted to the channel mostly dwell just outside the mouths of the
channel. \textbf{NOTE: Wolfgang note the transposition of sentences. I hope
you agree. }Since K$^{+}$ is repelled from the center of the filter, K$^{+}$
is expected to be conducted to a lesser extent than the overall count of 
\textbf{ALL???? }ions in the filter (Fig. 6) would suggest.\textbf{\ NOTE
IMPORTANT IMPORTANT: I do not understand the last sentence. Please explain
why the average concentration is not the (approximate) determinant of
conductance. K conductance could be decreased even more by reducing} the
volume in which the structural ions are confined. Nevertheless, \textbf{even
when K is `penalized', significant amounts of }K$^{+}$ are absorbed by the
channel when the ratio of K$^{+}$/Na$^{+}$ in the bath is made large enough
(Fig. 6).\textbf{\ Garber et al \cite{Garberetal1987} have shown that Na
channels conduct K}$^{+}$\textbf{\ under such ionic conditions:} \textbf{%
when Na}$^{+}$ is\textbf{\ removed altogether from the baths}, the filter of
the Na channel will accept other \textbf{cations to maintain overall
electroneutrality.}

\underline{Dielectric Coefficient}. \textbf{NOTE: Wolfgang, we need an
underline here in my view. }Our simulations use a uniform dielectric
coefficient everywhere, which is that of bulk water (78.5). If a smaller
dielectric coefficient were applied to the filter and/or protein, the need
for charge balance within the filter itself would be \textbf{greater}, and
so a larger density of counterion would be found in the filter \textbf{itself%
}. \textbf{Given the intrinsic charge of }$-1$\textbf{\ of the DEKA locus, }%
the countercharge more closely approaches +1\textbf{, when the dielectric
coefficient is smaller than \symbol{126}79}. On the other hand,\textbf{\
ions entering a filter with reduced dielectric coefficient might} have to be
desolvated to some degree,\textbf{\ and that} would reduce \textbf{entry into%
} the filter. If the counterion density were larger than that in our
simulations, \textbf{as the net effect of a reduction in dielectric
coefficient, }excluded volume effects would be stronger. \textbf{Thus, a
channel with a (spatially) uniform reduced dielectric coefficient} is
expected to exclude large ions more strongly than the channel with the
uniformly large dielectric coefficient. \textbf{Predictions of this sort are
no substitute for calculation, however, since many terms of similar
magnitude are involved in selectivity, and sometimes coupling effects are
important as well.}

\underline{Experimental Measures of Selectivity}. \textbf{NOTE: Wolfgang, we
need an underline here in my view. }Most experimental assessments of Na$^{+}$%
/K$^{+}$ selectivity are based on \textbf{measurements of the zero-current
potential in bi-ionic conditions, NOTE comma} when one cation species is
present in only one bath, and the other species \textbf{is present} only in
the other bath. Typically, a permeability ratio $\approx 10:1$ has been
found (reviewed in \cite{Hille2001})\textbf{, reflecting} both \textbf{XXX}
equilibrium binding and frictional\textbf{\ factors} of permeation \cite%
{Gillespie2002}. \textbf{Selectivity has also been estimated from
measurements} of current \textbf{XXX} made by varying the mole fractions of
Na$^{+}$ and K$^{+}$ in both baths, \textbf{NOTE\ COMMA }while keeping the
total salt concentration constant\textbf{\ in both baths}. This \textbf{%
`mole fraction experiment'} more directly assesses the partitioning factor
of selectivity \cite{Gillespie2002}. Ravindran et al. \cite%
{Ravindranetal1991} \textbf{found }that current in batrachotoxin treated Na
channels is reduced \textbf{by} about one-half when \textbf{XXX} Na$^{+}$
and K$^{+}$ mole fractions are equal, which suggests \textbf{that
partitioning is not the prime determinant of selectivity in this case.}
However, these experiments \textbf{are clearly performed away from
equilibrium}, and so quantitative predictions of their results require
non-equilibrium simulations or theory, \textbf{NOTE PREVIOUS\ COMMA} beyond
those reported here. We expect to deal with this issue in the future using
density functional theory in a drift/diffusion model of transport \textbf{%
NOTE: PUT IN REFERENCE TO DFT STUFF}. For the time being, we note that the
CSC mechanism gives equilibrium ionic affinities that agree with the order
of permeabilities determined from zero-current potentials for alkali metal
ions.

\underline{Mechanism of Selectivity.} \textbf{NOTE: Wolfgang, we need an
underline here in my view. }Our simulations reveal a simple mechanism for
alkali metal selectivity in the Na channel that depends on the \textbf{%
crowding of} charges in the channel. \textbf{The mechanism of selectivity in
the Na channel is} a different twist on the competition between charge and
space found in the Ca channel. In the Na channel, the presence of a
positively charged lysine residue greatly diminishes affinity for divalent
cations.

The structure underlying \textbf{our} proposed mechanism is not the
structure that has previously been ascribed to the \textbf{selectivity }%
filter \textbf{of the Na}$^{+}$ channel, in which oxygen atoms (some
charged) \textbf{were said to }form the rigid wall of a 0.3 $\times $ 0.5 nm
wide hole through the protein \cite{Hille1971, Hille1972}. The latter
structure has been widely accepted -- it is found in textbooks -- but its 
\textbf{ability to }select Na$^{+}$ ion has, \textbf{to }our knowledge, not
been tested in self-consistent computations.

Following Eisenman \cite{Eisenman}, Hille proposed that a channel of this
structure could use an anionic `high field-strength site' to select small
alkali metal ions over larger ones by electrostatics. In contrast to this
view, the alkali metal selectivity that we have computed in Monte Carlo
simulations arises chiefly from the effects of excluded volume, \textbf{not
from electric field strength}. \textbf{In our calculations, the chief effect
of the electric field is to force overall electroneutrality. In our model,}
larger ions are \emph{excluded} more than small ions, whereas in the
electrostatic hypothesis smaller cations are \emph{attracted} more than
larger cations from the aqueous baths. The exclusion of K$^{+}$ in our model
is evident in the density profiles of Fig. 7: K$^{+}$ density is minimal
(and below bulk density) in the center of the pore. Thus the region of
strongest selectivity is where the density of structural net charge is
minimal (see densities of O$^{-1/2}$ and NH$_{4}^{+}$ in Fig. 3). Recent
experiments have shown that Na channels whose DEKA locus is mutated to DAAA,
AEAA, or AAEA (i.e., a locus containing a single carboxylate) select for Ca$%
^{2+}$ over Na$^{+}$ \cite{Favreetal1996, Sunetal1997}. Thus, creating a
`high field-strength site' in the sense of Eisenman yields a Ca channel, not
a Na channel.\textbf{\ Wolfgang: This is a wonderful paragraph.
Congratulations!}

Goulding et al. (\cite{Gouldingetal2000, Gouldingetal2001}) have used
density functional theory and simulations to study `entropic selectivity of
microporous materials.' They specifically examine absorption \textbf{into a
hard-wall pore }of an uncharged hard-sphere fluid \textbf{made of }a solvent
and two solutes, with diameters corresponding to water, Na$^{+}$ and K$^{+}$%
. The results reveal selective \emph{excess absorption} of one of the
solutes over the other depending on certain small ratios of pore:particle
diameters. Little selectivity is observed for the diameter we assign to the
Na channel ($R_{f}/R_{Na}\approx 4$). On the other hand, our simulations
indicate substantial excluded-volume selectivity that is \emph{repulsive}
(small ions are repelled less than large ions)\textbf{\ under these
conditions.} Our simulations \textbf{differ from those of Goulding et al.
because we }include a high density of particle species that are restricted
to the pore (the tethered oxygen and nitrogen ions) and we use a
non-particulate description of water. Thus, \textbf{in our model} the pore
contains a fluid-like density of confined and mobile ions, whereas the bath
is represented as an ion gas. The asymmetry in densities is the basis of the
observed repulsion of ions from the filter. This repulsion increases
monotonically with ionic diameter (Fig. 6), whereas the absorption observed
by Goulding et al. can select for either smaller or larger ions dependent on
small variations of channel diameter.

\textsl{Repulsive phenomena as observed in our simulations can in principle
occur in two bulk phases of different particle densities and do not require
a particular wall structure or channel diameter}. \textbf{NOTE: Wolfgang, I
think the previous sentence should be italicized. }In fact, Nonner et al. 
\cite{Nonneretal2000, Nonneretal2001} have studied bulk models of a Ca
channel and assessed entropic repulsive effects. In these models, water can
be conveniently described in both the particulate and non-particulate
models, so it is possible to assess the role of a particulate solvent. It
was found that comparable repulsive selectivities can arise from excluded
volume effects when appropriate filter volumes are chosen for \textbf{either}
model: if water is described as particulate, a smaller filter volume is
needed. In this case \textbf{of particulate water}, the confined particles
in the filter are more compressed \textbf{compared to calculations with
non-particulate water. In both cases, there} exists an asymmetry of density
between the filter and bath fluids. Thus,\textbf{\ we imagine that} the
repulsive selectivity observed in our simulations of the Na channel would
occur also when water is described as particulate, if an appropriate
reduction is made to the filter volume.

It is also interesting to consider the consequences if water itself were
attracted or repelled \textbf{into or out of} the selectivity filter. In the
case of attraction, water would compete more strongly with ions for the
space in the filter, and so the selective effects of excluded volume would
be enhanced. If water were repelled from the filter, less repulsive
selectivity would be expected. Again,\textbf{\ (experimentally) }appropriate
selectivities \textbf{could be produced by adjusting} the volume of the
filter of the model.

Our simulations were made using a specific representation of charged
tethered groups and a\textbf{\ XXX} \textbf{particular estimate of the }%
volume accessible to these groups but the\textbf{\ simulations} do not hinge
on these \textbf{XXX} choices. For instance, we could have represented
larger parts of the DEKA sidechains by including neutral confined particles
in the simulation, and increased the volume in which these groups are
confined. Specifically, the relatively bulky sidechain of lysine could be
modeled more realistically this way \cite{Sunetal1997}.

\textsl{The crucial parameter for a CSC mechanism as described here is the
packing fraction of the ensemble of all atoms, mobile and structural, in the
filter}. The crowding of ions and protein atoms and the need for
(approximate) electroneutrality allow the finite size of the ions to
determine selectivity. \textbf{NOTE: Wolfgang, I think the last sentence
needs emphasis. }The mechanism does not require a specific geometry in which
atomic positions and dimensions \textbf{NOTE: note transposition of the
previous clauses} are closely maintained.

\section{Conclusions}

Ion absorption \textbf{has been computed} for a model of the DEKA locus of
biological Na channels in which mobile ions and charged structural groups
are represented as charged hard spheres. The model \textbf{channel }selects
Na$^{+}$ essentially by \emph{excluding} larger monovalent cations while
having a relatively weak affinity for Ca$^{2+}$. The presence of the
positively charged lysine residue in the locus is essential for making the
Na channel a `reluctant' Ca channel. Excluded volume effects \textbf{of
crowded charges }are essential for selecting small monovalent cations over
large ones (CSC mechanism).

\subsection*{Acknowledgments}

This work was supported by grants from the Hungarian National Research Fund
(OTKA-F035222, to D.B.), DARPA (B.E., W.N.), NIH (Al 23007, to D.D.B.), and
NSF (CHE98-13729, to D.H.). The authors thank Dr. D. Gillespie for many
useful comments and to Dr. R.F. Rakowski for sharing results submitted for
publication.

\bigskip

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\subsection*{Figure captions}

\begin{description}
\item[Figure 1] Schematic diagram of the simulation cell. Ions are excluded
from the membrane (including the channel protein) but not from the bath and
channel. Dimensions are specified in the text.

\item[Figure 2] Numbers of Na$^{+}$ and Ca$^{2+}$ absorbed in the channel
volume from a bath containing 0.1 M NaCl and the concentrations of CaCl$_{2}$
indicated on the abscissa. Simulation results (symbols) shown for two filter
radii $R_{f}$. The lines are the fit used in Refs \cite{Bodaetal2000,
Bodaetal2001a} and are meant only to aid the reader in connecting the
symbols. The number of ions was determined by integrating ionic
concentrations in the region of the filter and a `cap' at each channel mouth
that had the radius $R_{f}$ and extended one ionic radius in the axial
direction of the bath. Note that many error bars are smaller than the
symbols. The structural charge in the filter was -1e in all computations.

\item[Figure 3] Axial concentration profiles. The ordinate gives
concentrations averaged over the radial coordinate $0 \leq r \leq R_{f}$
(channel) and $0 \leq r \leq R - R_{ion}$ (bath), in units of moles/liter.
The volume accessible to the ion centers was used for the denominator. Note
that the abscissa $z=0$ corresponds to the center of the simulation cell and
channel; the profiles shown are averages of both the left and right half of
the symmetrical simulation cell. The arrow marks the position of the mouth
of the filter.

\item[Figure 4] Selectivity of a Na channel using a model with $R_{f}=0.4$
nm and $H_{f}=1$ nm for different NH$^{+}_{4}$ confinements. In the two sets
of simulations, the NH$^{+}_{4}$ ion center is confined to a length of 0.375
nm and 0.125 nm on either side of the channel center, and the structural
charge in the filter is -1e. The symbols give the simulation results (the
lines are meant only to aid the reader in connecting the symbols).

\item[Figure 5] Axial concentration profiles obtained with the Na channel
model in which the NH$^{+}_{4}$ ion center is confined to a length of 0.125
nm on either side of the channel center. Profiles are computed as described
in Fig. 3.

\item[Figure 6] Selectivity of Li$^{+}$, K$^{+}$, and Cs$^{+}$ vs. Na$^{+}$
in a Na channel. The filter radius in the model is $R_{f}=0.4$ nm, and the NH%
$^{+}_{4}$ is restricted to the central 0.75 nm of the filter. The
structural charge in the filter is -1e. In these simulations we start with a
0.1 M NaCl solution in the bath and gradually replace the Na$^{+}$ ions with
different cations, holding the concentration of Cl$^{-}$ constant at 0.1 M.
The straight lines are meant only to aid the reader in connecting the points.

\item[Figure 7] Axial concentration profiles of the Na channel model with $%
R_{f}=0.4$ nm. In this simulation, the bath contained a 0.1 M mixture of
NaCl and KCl in the ratio $\approx 1:3$. Profiles are computed as described
in Fig. 3.
\end{description}

\end{document}
