Semi-quantitative model of the gating of KcsA ion channel. 1. Geometry and energetics of the gating
The aim of this series of papers is to develop the semi-quantitative theory of the gating of KcsA channel. For this purpose the available structural and electrophysiological data and the results of molecular dynamics simulations were used in the context of the concept of dynamical self-organization....
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| Опубліковано в: : | Вiopolymers and Cell |
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| Дата: | 2009 |
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Інститут молекулярної біології і генетики НАН України
2009
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| Цитувати: | Semi-quantitative model of the gating of KcsA ion channel. 1. Geometry and energetics of the gating / V.N. Kharkyanen, S.O. Yesylevskyy, N.M. Berezetskaya, C. Boiteux, Ch. Ramseyer // Вiopolymers and Cell. — 2009. — Т. 25, № 5. — С. 390-397. — Бібліогр.: 13 назв. — англ. |
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Kharkyanen, V.N. Yesylevskyy, S.O. Berezetskaya, N.M. Boiteux, C. Ramseyer, Ch. 2019-06-13T12:12:14Z 2019-06-13T12:12:14Z 2009 Semi-quantitative model of the gating of KcsA ion channel. 1. Geometry and energetics of the gating / V.N. Kharkyanen, S.O. Yesylevskyy, N.M. Berezetskaya, C. Boiteux, Ch. Ramseyer // Вiopolymers and Cell. — 2009. — Т. 25, № 5. — С. 390-397. — Бібліогр.: 13 назв. — англ. 0233-7657 DOI: http://dx.doi.org/10.7124/bc.0007F1 https://nasplib.isofts.kiev.ua/handle/123456789/153000 577.322 The aim of this series of papers is to develop the semi-quantitative theory of the gating of KcsA channel. For this purpose the available structural and electrophysiological data and the results of molecular dynamics simulations were used in the context of the concept of dynamical self-organization. In the first paper we describe the simplified model of the geometry and energetics of the gating process. This work is the first successful attempt of combining the structure and dynamics of a real protein and the general concept of dynamic self-organization. Мета даної серії робіт полягає у розробці напівкількісної теорії воротних процесів в іонному каналі KcsA. Для цього залучено доступні експериментальні дані, а також результати молекулярної динаміки у контексті концепції динамічної самоорганізації. У першій роботі серії представлено спрощену модель геометрії та енергетики воротних процесів. Наведено першу успішну спробу об’єднання даних щодо структури та динаміки реального білка з концепцією динамічної самоорганізації. Целью данной серии работ является разработка полуколи- чественной теории воротных процессов в ионном канале KcsA. Для этого использованы доступные экспериментальные данные, а также результаты молекулярной динамики в контексте концепции динамической самоорганизации. В первой работе серии представлена упрощенная модель геометрии и энергетики воротных процессов. Приведена первая успешная попытка объединения данных о структуре и динамике реального белка с концепцией динамической самоорганизации. en Інститут молекулярної біології і генетики НАН України Вiopolymers and Cell Молекулярна біофізика Semi-quantitative model of the gating of KcsA ion channel. 1. Geometry and energetics of the gating Напівкількісна модель воротних процесів в іонному каналі KcsA. 1. Геометрія та енергетика воротних процесів Полуколичественная модель воротных процессов в ионном канале KcsA. 1. Геометрия и энергетика воротных процессов Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Semi-quantitative model of the gating of KcsA ion channel. 1. Geometry and energetics of the gating |
| spellingShingle |
Semi-quantitative model of the gating of KcsA ion channel. 1. Geometry and energetics of the gating Kharkyanen, V.N. Yesylevskyy, S.O. Berezetskaya, N.M. Boiteux, C. Ramseyer, Ch. Молекулярна біофізика |
| title_short |
Semi-quantitative model of the gating of KcsA ion channel. 1. Geometry and energetics of the gating |
| title_full |
Semi-quantitative model of the gating of KcsA ion channel. 1. Geometry and energetics of the gating |
| title_fullStr |
Semi-quantitative model of the gating of KcsA ion channel. 1. Geometry and energetics of the gating |
| title_full_unstemmed |
Semi-quantitative model of the gating of KcsA ion channel. 1. Geometry and energetics of the gating |
| title_sort |
semi-quantitative model of the gating of kcsa ion channel. 1. geometry and energetics of the gating |
| author |
Kharkyanen, V.N. Yesylevskyy, S.O. Berezetskaya, N.M. Boiteux, C. Ramseyer, Ch. |
| author_facet |
Kharkyanen, V.N. Yesylevskyy, S.O. Berezetskaya, N.M. Boiteux, C. Ramseyer, Ch. |
| topic |
Молекулярна біофізика |
| topic_facet |
Молекулярна біофізика |
| publishDate |
2009 |
| language |
English |
| container_title |
Вiopolymers and Cell |
| publisher |
Інститут молекулярної біології і генетики НАН України |
| format |
Article |
| title_alt |
Напівкількісна модель воротних процесів в іонному каналі KcsA. 1. Геометрія та енергетика воротних процесів Полуколичественная модель воротных процессов в ионном канале KcsA. 1. Геометрия и энергетика воротных процессов |
| description |
The aim of this series of papers is to develop the semi-quantitative theory of the gating of KcsA channel. For this purpose the available structural and electrophysiological data and the results of molecular dynamics simulations were used in the context of the concept of dynamical self-organization. In the first paper we describe the simplified model of the geometry and energetics of the gating process. This work is the first successful attempt of combining the structure and dynamics of a real protein and the general concept of dynamic self-organization.
Мета даної серії робіт полягає у розробці напівкількісної теорії воротних процесів в іонному каналі KcsA. Для цього залучено доступні експериментальні дані, а також результати молекулярної динаміки у контексті концепції динамічної самоорганізації. У першій роботі серії представлено спрощену модель геометрії та енергетики воротних процесів. Наведено першу успішну спробу об’єднання даних щодо структури та динаміки реального білка з концепцією динамічної самоорганізації.
Целью данной серии работ является разработка полуколи- чественной теории воротных процессов в ионном канале KcsA. Для этого использованы доступные экспериментальные данные, а также результаты молекулярной динамики в контексте концепции динамической самоорганизации. В первой работе серии представлена упрощенная модель геометрии и энергетики воротных процессов. Приведена первая успешная попытка объединения данных о структуре и динамике реального белка с концепцией динамической самоорганизации.
|
| issn |
0233-7657 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/153000 |
| citation_txt |
Semi-quantitative model of the gating of KcsA ion channel. 1. Geometry and energetics of the gating / V.N. Kharkyanen, S.O. Yesylevskyy, N.M. Berezetskaya, C. Boiteux, Ch. Ramseyer // Вiopolymers and Cell. — 2009. — Т. 25, № 5. — С. 390-397. — Бібліогр.: 13 назв. — англ. |
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ÌÎËÅÊÓËßÐÍÀ Á²ÎÔ²ÇÈÊÀ
Semi-quantitative model of the gating of KcsA ion
channel. 1. Geometry and energetics of the gating
V. N. Kharkyanen1, S. O. Yesylevskyy1, N. M. Berezetskaya1, C. Boiteux2,
Ch. Ramseyer2
1Institute of Physics of National Academy of Sciences of Ukraine
Prospect Nauky 46, Kyiv, Ukraine, 03680
2UMR CNRS 6624, Faculte des Sciences et Techniques, Universite de Franche-Comte
16 route de Gray, La Bouloie, 25030 Besancon Cedex, France
kharkyanen@iop.kiev.ua
The aim of this series of papers is to develop the semi-quantitative theory of the gating of KcsA channel. For
this purpose the available structural and electrophysiological data and the results of molecular dynamics
simulations were used in the context of the concept of dynamical self-organization. In the first paper we
describe the simplified model of the geometry and energetics of the gating process. This work is the first
successful attempt of combining the structure and dynamics of a real protein and the general concept of
dynamic self-organization.
Keywords: ion channel, KcsA channel, dynamic self-organization, channel gating.
Introduction. The ion channels are classical objects of
biophysics, which are studied both experimentally and
theoretically for more than fifty years now. The
principles of the ion channel functioning were es-
tablished on the phenomenological level well before
the structure of these molecules has become known.
Starting from the now famous work of Doyle [1], where
high-resolution three-dimensional structures of the
KcsA channel were determined, a new era in the ion
channel science has started. Rapid advance in the
structural and experimental studies on the ion channels
allows reexamining some theoretical developments at
an entirely new level. One of such theories is the
concept of Dynamic Self-Organization (DSO), which
was initially formulated before the real structure of the
ions channels has become known. The aim of the
present series of works is to reanimate the ideas of the
DSO in application to the ion channels using the best
available data on the channel structure and functioning.
Our goal is to show that the DSO is consistent with the
structure of the KcsA potassium channel and does not
contradict the traditional paradigm of ion channel
gating.
Theory and Methods. The molecular dynamics
(MD) simulations. The general picture of the opening
of KcsA channel is now established. The opening
occurs by concerted complex rotation of four trans-
membrane M2 helices, which leads to the widening of
the intracellular part of the channel pore. In addition to
global rotation the M2 helices seem to rotate around
their long axes and bend near the gating region forming
a kink [2–5].
In the present work the results of MD simulations
of the channel opening, described in details in our
previous papers [4, 5], were used. In these studies the
KcsA channel was driven from the equilibrated closed
state to the hypothetical open state by means of targeted
MD simulations.
390
ISSN 0233-7657. Biopolymers and Cell. 2009. Ò. 25. N 5
Ó Institute of Molecular Biology and Genetics NAS of Ukraine, 2009
The resulted state can only be considered as quasi-
open, since the stability of this state and its similarity to
the real open state are not known. Thus, we will use the
term «quasi-open state» to refer to the results of MD
simulations and the term «open state» to refer to the
real physiological open state. We believe that the latter
state is the same which emerges due to DSO.
First ~2 ns of simulation were used to calculate the
channel average structure in the closed state. Last
~0.5 ns of the trajectory were used to determine the
averaged structure of the quasi-open state. The calcu-
lations of the average structures were performed after
aligning all heavy atoms of the selectivity filter for each
frame with the corresponding atoms of the reference
frame.
The geometry of the model and the structural
coordinate. We model the geometry of opening of
KcsA channel in a simplified way. We assume that only
M2 helices move, while all other parts of the protein
remain fixed. Only one of four symmetric subunits is
considered.
We model the M2 helix as a rigid rod, which goes
approximately through the geometrical axis of the
helix. We subdivided the whole helix into 9 segments
containing 4 residues each starting from the extra-
cellular side (residue THR85). Remaining 3 residues
from the intracellular side were ignored, since they
exhibit large fluctuations and cannot be treated as a
rigid rod. Each segment corresponds approximately to
one turn of the helix, so the geometrical center of each
segment provides good approximation of the position
of the helix axis for this segment (Fig. 1, A).
The analysis of the MD trajectory of the channel
opening shows that each M2 helix rotates approxi-
mately in one plane around the well-defined pivot.
Fig. 1, B shows the distances between the centers of
segments in the closed and open states. It is clearly seen
that the minimal distance along axes Y and Z is
observed for segment 4, but the minimal distance along
X axis is observed for the segments 2 and 3. Thus, we
placed the pivot point between the centers of segments
3 and 4, which seems to be a reasonable approximation.
The rod is positioned in such a way that it goes through
the pivot and the center of the last 9th segment of the
helix. The centers of the last 9th segment in the closed
and quasi-open states (points Tc and To) and the pivot
point P were used to define the plane, in which the M2
helix moves during the opening. We will reference this
plane as an opening plane hereafter (Fig. 2, A). We
assume that the rod is confined to this plane.
The extent of opening is described by the angle j
(calculated in the opening plane) between the current
position of the rod and its position in the closed state.
Thus, the angle j is the structural coordinate in our
model. The values of the opening angle are j c = 0o in
the closed state and jo » 11.4o in the quasi-open state.
The position of the end point for arbitrary opening
angle is defined as T (j).
We assume that the rod can stretch to accommodate
small changes in the length of M2 helix upon opening,
but can not bend. The length of the rod l for any
opening angle can be calculated as linear interpolation
between the lengths in the closed and quasi-open states
l(j) = l(j c) + [l(jo) – l(j c)] j / jo.
The pore of the channel is aligned with Z axis of the
global coordinate system. The center of coordinates
(point 0) is placed to the center of the pore in XY plane
and to the position of point Tc (intracellular end of M2
helix in the closed state) on Z axis.
The permeating ions are assumed to be confined to
the channel axis. The one-dimensional pore is assumed
to extend from zmin = –10 &A to zmax = 60 &A.
Treatment of the ions. The KcsA channel contains
2–4 K+ ions in the physiological conditions. 2 or 3 ions
are located in the selectivity filter, while the remaining
ion(s) occupies the central chamber of the channel and
the intracellular vestibule [1, 6, 7]. It is clear that
conventional single-particle diffusion theory is not
applicable to the channel pore [8]. A simplified model
was proposed to describe the concerted motion of
strongly interacting ions in the selectivity filter of
KcsA channel [8]. However, this theory is not
applicable to the other parts of the pore.
We are only interested in gating motions, which are
much slower than the characteristic time of ionic
diffusion in the channel pore. Thus, an interaction of
the channel structure with individual ions can be
substituted by the interaction with quasi-equilibrium
density of charges along the channel pore.
The charge density is in local equilibrium with any
current conformation of the channel and changes in a
quasi-static way in accord with the conformational
391
SEMI-QUANTITATIVE MODEL OF THE GATING OF KcsA ION CHANNEL
changes (adiabatic decoupling of slow degrees of
freedom [8–10]). The ion-ion interactions can
influence the time-averaged charge distribution
significantly in the regions, where the ions reside close
to each other i. e. in the selectivity filter. It was shown
that two ions, which occupy the selectivity filter most
of the time, counterbalance deep energy well of the
selectivity filter in highly dynamic way by strong
ion-ion repulsion. As a result, the third permeating ion
initiates the sequence of events known as knock-on
barrier-less conduction [8, 11]. On the longer
timescales this process can be reduced to the motion of
a single quasi-particle, which transfers the unit charge
trough the filter. This means that two ions, which
occupy the filter most of the time, can be considered as
a part of the «permanent» channel structure and the
actual charge distribution of permeating ions is created
by the third ion only [8, 11].
Thus, in order to compute the quasi-equilibrium
charge distribution an approximation of independent
ions can be utilized in any part of the pore without
significant loss of accuracy and the standard metho-
dology of non-equilibrium Focker-Planck equations
[10] can be used
¶ j
¶
¶
¶
c z C C U
t
D
z k T
in ex
B
( , , , , )
[= ´
1
́ +
¶ j
¶
j
V z U
z
c z C C Uion
in ex
( , , )
( , , , , )
+
¶ j
¶
c z C C U
z
in ex( , , , , )
], (1)
where Vion is the energy profile for the permeating ions,
c(j , z, Cin, Cex, U) is quasi-equilibrium one-dimensional
charge distribution in the channel, Cin and Cex are
effective ionic concentrations of internal and external
solutions, U is the transmembrane potential, D is
one-dimensional diffusion coefficient of the ions in the
pore. This equation is subject to boundary conditions
Cin º c(j , zmin, Cin, Cex, U) = pin × Cin
0, Cex = c(j , zmin, Cin,
Cex, U) = pex × Cex
0, where Cin
0 and Cex
0 are real internal
and external ionic concentrations, pin and pex are some
coefficients, which convert real dimensional ionic
392
KHARKYANEN V. N. ET AL.
Fig. 1. Locating the pivot point on M2 helix: A – Centers of the 4-residue segments in the closed (silver balls) and quasi-open (black balls)
states superimposed into the cartoon representation of the single subunit of KcsA channel. The rod, which models the M2 helix, is shown as a
solid line. The indexes of the segments are indicated; B – The distance between the centers of the segments in the closed and the quasi-open
states. The connecting lines are drawn to aid the comparison only
concentrations to dimensionless effective concen-
trations, which have the meaning of probability. We as-
sume further that pin = pex = p. The value of p can be fo-
und by fitting the mean number of ions in the channel to
the experimentally observed value. We use the value
p = 10–4/mM× &A, which leads to the total number of mo-
vable charges in the channel in the range 1–2 for all stu-
died conditions (two «permanent» ions in the selecti-
vity filter are not considered as it is explained above).
Stationary solution of eq. (1) is
c(f, z, r, U) = Cin × exp [–Vion(j , z, U)] ́
́ {exp [Vion (j , zmin, U)] × [1 – h (j , z, U)] +
+ r exp [Vion(j , zmax, U)] × h (j , z, U)}, (2)
where
h (j , z, U) = exp[ ( , , )] /
min
V z U dz
ion
z
z
jò
/ exp[ ( , , )] ,
min
max
V z U dz
ion
z
z
jò (3)
r = Cin/Cex is the concentration ratio. We assume that Cin
does not change and the only concentration-dependent
parameter in our model is r. Expression is used to com-
pute the interaction between the ionic charge distribu-
tion and the channel structure as it is described below.
Interaction between the structure and the ions. The
charge density in the channel pore interacts with the
charges localized in the movable M2 helix. In addition
there are short-range Van-der-Waals interactions in the
narrow region of the gate, which are responsible for
steric closure of the pore in the closed state [2, 3, 6]. We
assume that there is only one effective charged group
with charge q on the rod, which model the M2 helix.
The charged group is positioned at the distance xql from
the intracellular end of the rod.
We also assume that there is only one effective
particle, located in the narrowest part of the pore, which
interacts with the ions by Van-der-Waals forces (the
VDW group). The VDW group is positioned on the
«stem» of length lw. The stem is oriented in such a way
that it is directed precisely toward the channel axis in
the closed state. The stem itself is positioned at the
distance xwl from the intracellular end of the rod.
The positions of the effective groups xq and xW serve
as adjustable parameters. The value of effective charge
q is computed by summing up all atomic charges of the
M2 helix as defined in AMBER94 force field [12] used
in our MD simulations. It appears that q = 0.99. The
arrangement of the effective particles is shown in Fig.
2, A.
393
SEMI-QUANTITATIVE MODEL OF THE GATING OF KcsA ION CHANNEL
Fig. 2. A – The model of M2 helix
motion superimposed on the car-
toon representation of the single
subunit of KcsA channel (aligned
closed and open states are shown).
The opening plane is shaded. The Z
axis is shifted in XY plane from its
real position for clarity. The scale
on Z axis is shown; B – The nomen-
clature of the rotating coordinate
system in the opening plane
In order to define the forces, which act on the M2
helix and the permeating ions it is convenient to
introduce new moving coordinate system linked to the
M2 helix with the orts {
r
i(j),
r
j(j),
r
k(j)}, which depend
on the opening angle j . The ort
r
i(j) is normal to the
opening plane and oriented from the channel axis, the
ort
r
j(j) lies in the opening plane and shows the
instantaneous direction of motion of the end point of
the helix during rotation, the ort
r
k(j) is directed from
the current position of the end of the helix T(j) to the
hinge point P (Fig. 2, B). Let us also define the position
of the point G, which lie on the rod at the distance l(j)×x
from its end (0 < x < 1):
r
G (j x j j, ) ( ) ( ) (= - × × -
r r
P k l 1
-x).
We also need to find the distance between the point
G and any point z on the channel axis
r
R(j , z, x) =
r
G(j ,
x) – z ×
r
k. This vector determines the direction of the
Coulombic force acting between the ion and the char-
ged group.
The projection of this vector on the direction of
motion
r
j( )j is:
R z R z jj j j x j( , ) ( , , ) ( ).= ×
r r
(4)
The projection (4) is used below for the qualitative
analysis.
The Coulomb force fq, which acts between the
charged group and the ion located in point z of the
channel axis, is shielded by the water molecules in the
pore and various polarizable groups, which lie between
the ion and the effective charge. This shielding is
distance-dependent and can be very complex. We ap-
proximate it with the exponential Debye-like term as it
was done in our previous work [8]
r
r
r
r
f z q
R z
R z
R z
d
q ( , , )
( , , )
( , , )
exp(
( , , )
),j x c
j x
j x
j x
= -
3
where q is the effective charge of the charged group, is
dimensional constant, which converts the force to kBT/
&A units, d is the shielding constant. The value of d is an
adjustable parameter in our model.
The potential created by this force
V z f z dzq q q q
z
z
( , , ) ( , ' , ) '
min
j x j x= - ò
r
(5)
can be computed analytically, but the expression is too
complicated to be given here.
The VDW group is positioned at the point
r
RW(
j x, , )z
W
= l i z k G
W W
× + × -
r r r
( , ).j x We are mainly
interested in the strong VDW repulsion, which allows
steric closure of the pore in the gating region, thus we
neglect weak attractive term and the VDW force fw is
written as
r
r
rf z w
R z
R z
W
W W
W W
( , )
( , , )
( , , )
,j
j x
j x
= -
0 13
where w0 is empirical constant.
The potential created by the VDW force is
V z f z dz
W W W W
z
z
( , , ) ( , ' , ) '
min
j x j x= - =ò
r
=
w
R z
W W
0
12r
( , , )
.
j x
(6)
The total force, which acts on the M2 helix from the
ion localized at point z, is
r
f (j x) = j x), , ( , ,z f zq
r
+
+ j
r
f z
W
( , ).
The momentum of this force projected on the
moving ort
r
j( )f is
m z m z m zq w q q W w( , , , ) ( , , ) ( , , ),j x x j x j x= =
where
m z f z j lq q q q q( , , ) ( ( , , ) ( )) ( ) ( )j x j x j j x= × × × -
r r
1
and
m z f z j l
W q W W
( , , ) ( ( , ) ( )) ( ) ( )j x j j j x= × × × -
r r
1 are
momenta of each force component. Only this pro-
jection can rotate the helix, while two other projections
are counterbalanced by the reaction forces (the rod is
confined to the plane) and thus neglected.
One can compute the average momentum created
by the quasi-equilibrium distribution of the ions in the
channel given by eq. (2)
M r U q w( , , , , )j x x =
= ×ò m z c z r U dzq w
z
z
( , , , ) ( , , , ) .
min
max
j x x j (7)
394
KHARKYANEN V. N. ET AL.
This equation (7) is only valid in the adiabatic
approximation, which was used to calculate the
distribution c. According to this approximation the
influence of individual ions is averaged out and the M2
helix only «feels» the average quasi-equilibrium
charge density in the channel pore.
Modeling the rest of the channel – an effective
energy profile for the permeating ions. Since only the
M2 helices are modeled explicitly, the rest of the
channel is taken into account by introducing the
effective energy profile (EEP) for the permeating ions
created by all remaining parts of the channel structure.
This profile includes implicitly not only the
interactions with the channel walls, but also with other
ions and water molecules. We presume that EEP in the
selectivity filter does not change significantly upon
opening and remains flat, thus all subsequent consi-
derations are applied to the chamber, gate and intra-
cellular part of the pore.
Let us express the effective energy profile for the
ion Vion as Vion = VM2+ VEEP + U, where VM2 is the profile
created by the M2 helices, VEEP is the profile created by
the rest of the channel, water molecules and the ion-ion
interactions, U is the transmembrane electrochemical
potential. At this stage we assume that transmembrane
potential is zero U = 0. The VM2 term is determined by
the geometry of our model and the positions of charged
and VDW groups. Using (5) and (6) we can write
VM2(j j x j x, ) [ ( , , ) ( , , )],z V z V zq q W W
= +4 (8)
where the coefficient 4 accounts for existence of four
identical subunits in KcsA channel.
In order to estimate VEEP one can use the following
considerations. In the open state the KcsA channel
conducts the ions at the rate, which is close to the dif-
fusion limit, which means that there are no large energy
barriers or deep wells on the effective energy profile.
This means that the forces, which act on the permeating
ion from the M2 helices, are counterbalanced almost
completely by other forces, which contribute to VEEP:
Vrest » –VM2(jo).
It is not known whether such compensation occurs
in the closed state, because the channel conducts no
current in this state. However, it seems to be estab-
lished that the total number of ions accommodated by
the channel depends on the opening state only slightly.
This means that no additional deep energy wells appear
in the closed state of the channel. Thus, one can assume
that the rest of the channel compensates the potential
profile created by the M2 helix almost completely at
any stage of the opening and VEEP can be considered
independent on the opening angle.
The exact shape of the effective energy profile in
the presence of compensation is still unknown, but the
depths of any energy wells or heights of any barriers
can not exceed 1–2 kBT. It is usually believed that the
chamber of the channel contains substantial density of
the ions most of the time, in particular, an additional
ion is usually placed into the chamber in most of MD si-
mulations [2, 6, 13]. The effective energy profile sho-
uld have a shallow energy well in the chamber region to
account for this effect. We modeled this well by an
inverted Gaussian curve centered in the chamber
region
V A
z z
well
c= - -
-
exp(
( )
),
2
2s
where s is the half-width of the energy well, A is the
depth of the well, zc is the position of well. This well is
assumed to be independent on the opening state as it is
described above.
Structural potential for the M2 helix. Slow diffusi-
ve rotational motion of the M2 helix can be described
by the stochastic Langevin equation. Using (7) one can
write
& ( ( , , , ) ( , , , )
min
max
j j x x j
j
= - × +ò
D
k T
m z c z r U dz
B
q w
z
z
+ +
¶ j
¶j
xj
V
D tstr ( )
) ( ),2 (9)
where Dj is diffusion coefficient of rotational motion,
x(t) is the white noise, Vstr is so-called structural
potential. Structural potential is the potential of mean
force, which acts on the moving helix from the other
parts of the channel but not from the ions in the pore.
The computation of the potential of mean force Vstr
for such complex moving object as the long M2 helix is
not possible using standard MD protocols, thus we
used simplified semi-quantitative approach to estimate
Vstr. The rotation of the M2 helices is the dominant
395
SEMI-QUANTITATIVE MODEL OF THE GATING OF KcsA ION CHANNEL
motion in the system during opening, thus it is possible
to assume that this rotation makes the major con-
tribution to the change of total energy Etotal. If this is
true, one can write as a first approximation
V Estr total
( ) ( ) .j b j» + const (10)
This allows to monitor the total energy of the sys-
tem as a function of the opening angle instead of com-
puting the potential of mean force explicitly, but resul-
ting Vstr can only be considered as semi-quantitative
approximation.
We analyzed the trajectories of opening by plotting
the total energy of the system against the opening angle
(Fig. 3). We approximated it as a sum of two compo-
nents. The first one, Vwall describes sharp «wall» at the
right and similar wall at the left, which is not sampled
in MD due to the direction of targeted pulling. The se-
cond linear component, Vlin approximates the bottom of
the potential. The final expression for Vstr can be written
using as
V V Vstr wall lin
(j b j j b j) ( ( ) ( )) ( . (= + = × -0005
- - × + × +45 433 10 45 206 4) . ),j
where coefficient b is identical to the corresponding co-
efficient in (10) and incorporates the conversion factor
from dimensional MD units to dimensionless model
units. We use the value b = 0017. k T
B
in all subsequent
calculations.
Results and Discussion. In the present work we
have established a simplified framework describing the
geometry and the energetics of the KcsA channel
gating based on the concept of dynamical self-
organization. We used the geometry of the channel
opening obtained from experiments and MD
simulations. The structural potential for the motion of
the M2 helix is derived in the semi-qualitative way
from the MD results for single forced opening of the
channel. Once the opening pathway of KcsA is
determined with sufficient degree of confidence in
atomic details, the all-atom model can be built easily
using the principles described in this work. In the next
paper the obtained data will be used to develop a closed
DSO-based theory of the channel gating.
Acknowledgements. V. Kh. thanks the University
of French-Comte for providing the invited professor
grant, which stimulated this work. Prof. L. N. Chris-
tophorov is acknowledged for critical reading of the
manuscript.
Â. M. Õàð êÿ íåí, Ñ. Î. ªñå ëå âñüêèé, Í. Ì. Áå ðå çåöü êà, Ñ. Áó à òî,
×. Ðàì çåºð
Íàï³âê³ëüê³ñíà ìî äåëü âî ðîò íèõ ïðî öåñ³â â ³îííîìó êà íàë³
KcsA. 1. Ãå îìåòð³ÿ òà åíåð ãå òè êà âî ðîò íèõ ïðî öåñ³â
Ðå çþ ìå
Ìåòà äà íî¿ ñå𳿠ðîá³ò ïî ëÿ ãຠó ðîç ðîáö³ íàï³âê³ëüê³ñíî¿ òåî-
𳿠âî ðîò íèõ ïðî öåñ³â â ³îí íî ìó êà íàë³ KcsA. Äëÿ öüî ãî çà ëó ÷å íî
äîñ òóïí³ åê ñïå ðè ìåí òàëüí³ äàí³, à òàêîæ ðå çóëü òà òè ìî ëå êó -
ëÿð íî¿ äè íàì³êè ó êîí òåêñò³ êîí öåïö³¿ äè íàì³÷íî¿ ñà ìî îð -
ãàí³çàö³¿. Ó ïåðø³é ðî áîò³ ñå𳿠ïðåä ñòàâ ëå íî ñïðî ùå íó ìî äåëü
ãå î ìåò𳿠òà åíåð ãå òè êè âî ðîò íèõ ïðî öåñ³â. Íà âå äå íî ïåð øó
óñï³øíó ñïðî áó îá’ºäíàí íÿ äà íèõ ùîäî ñòðóê òó ðè òà äè íàì³êè
ðå àëü íî ãî á³ëêà ç êîí öåïö³ºþ äè íàì³÷íî¿ ñà ìî îð ãàí³çàö³¿.
Êëþ ÷îâ³ ñëî âà: iîíí³ êà íà ëè, êà íàë KcsA, äè íàì³÷íà ñà ìî îð -
ãàí³çàö³ÿ, âî ðîòí³ ïðî öå ñè.
Â. Í. Õàð êÿ íåí, Ñ. À. Åñå ëåâ ñêèé, Í. Ì. Áå ðå çåö êàÿ, Ñ. Áó à òî,
×. Ðàì çååð
Ïî ëó êî ëè ÷åñ òâåí íàÿ ìî äåëü âî ðîò íûõ ïðî öåñ ñîâ â èîí íîì
êà íàëå KcsA. 1. Ãå î ìåò ðèÿ è ýíåð ãå òè êà âî ðîò íûõ ïðî öåññîâ
Ðå çþ ìå
Öåëüþ äàí íîé ñå ðèè ðà áîò ÿâ ëÿ åò ñÿ ðàç ðà áîò êà ïî ëó êî ëè-
÷åñ òâåí íîé òå î ðèè âî ðîò íûõ ïðî öåñ ñîâ â èîí íîì êà íà ëå KcsA.
Äëÿ ýòî ãî èñ ïîëü çî âà íû äîñ òóï íûå ýêñ ïå ðè ìåí òàëü íûå äàí -
íûå, à òàê æå ðå çóëü òà òû ìî ëå êó ëÿð íîé äè íà ìè êè â êîí òåê -
ñòå êîí öåï öèè äè íà ìè ÷åñ êîé ñà ìî îð ãà íè çà öèè.  ïåð âîé ðà-
396
KHARKYANEN V. N. ET AL.
Fig. 3. Structural potential for the M2 helix obtained in MD
simulations and the fit used in the model. Each dot corresponds to a
single frame of the targeted MD trajectory. See text for details
áîòå ñå ðèè ïðåä ñòàâ ëå íà óïðî ùåí íàÿ ìî äåëü ãå î ìåò ðèè è
ýíåð ãå òè êè âî ðîò íûõ ïðî öåñ ñîâ. Ïðè âå äå íà ïåð âàÿ óñïåø íàÿ
ïî ïûò êà îá ú å äè íå íèÿ äàí íûõ î ñòðóê òó ðå è äè íà ìè êå ðå àëü -
íî ãî áåë êà ñ êîí öåï öè åé äè íà ìè ÷åñ êîé ñà ìî îð ãà íè çà öèè.
Êëþ ÷å âûå ñëî âà: èîí íûå êà íà ëû, êà íàë KcsA, äè íà ìè ÷åñ êàÿ
ñà ìî îð ãà íè çà öèÿ, âî ðîò íûå ïðî öåñ ñû.
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ÓÄÊ 577.322
Íàä³éøëà äî ðå äàêö³¿ 12.03.09
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SEMI-QUANTITATIVE MODEL OF THE GATING OF KcsA ION CHANNEL
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