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Spatial distribution of ion channel activity in biological membranes: the role of noise

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Spatial distribution of ion channel activity in biological membranes:the role of noise
P. Babinec*, M. Babincova´
Department of Biophysics and Chemical Physics, Comenius University, Mlynska´ dolina F1, 842 48 Bratislava, Slovakia
Received 1 June 2001; received in revised form 19 November 2001; accepted 22 November 2001
Abstract
A new approach is proposed to model a collective ion channel dynamics. We have assumed that ion channels create a two-component spatio-temporal interaction field. Every channel at its current spatial location in membrane contributes permanently to this field with its state(open or closed) and coupling strength to other channels. This field is described by a reaction–diffusion equation, the transition of ionchannel from closed to open state (and vice versa) is described by a master equation, and migration of channels in membrane is described bya set of Langevin equations coupled by the interaction field. Within this model, we have investigated critical conditions for spatialdistribution of ion channel activity.
D
2002 Elsevier Science B.V. All rights reserved.
Keywords:
Ion channels; Collective phenomena; Nonlinear dynamics; Noise
1. Introduction
Voltage-gatedionchannelsareproteinsresponsibleforthegeneration of electrical signals in nerve and other excitablecells. They work by selectively conducting ionic currentsthrough impermeable membranes [1]. Generally, the chan-nels can switch between different conformational states,which are conducting (open) or nonconducting(closed), withvoltage-dependentratesoftransition.Oneconsequenceofthevoltage-dependent transition rates is the intrinsic noise anensembleofchannelsgenerates,whichmayleadtonontrivialdynamics. The suggestion that the noise may be a source of order rather than disorder, and that a biological organismmakes use of energy-driven fluctuations for the purpose of signal and free-energy transduction was put forward almost 10 years ago by Astumian et al. [2]. In a recent study [3], the
fluorescence of a rat neurons stained with the voltage-sen-sitive dye were optically excited in synchrony with electricstimulation of the cell, and recorded with a high spatial re-solution.Duringanactionpotential,thefluorescencepatternsexhibited clusters of different sizes corresponding to a non-homogeneous distribution of electric field across the mem- brane. Togain insight into this and related problems, wehavein this study further developed our model of ion channelcollective dynamics [4].
2. Stochastic model of channel state change andmigration
Let us consider a two-dimensional spatial system withthe total area
A
, with
N
ion channels. Each of them can be inone of two opposite states (open and closed), denoted as
h
i
=
F
1;
i
=1,
. . .
,
N
. Here,
h
i
is considered as a channel parameter, representing an
internal degree of freedom
.Within a stochastic approach, the probability
p
i
(
h
i
,
t
) to findthe channel
i
with the state
h
i
, changes in the course of timedue to the following master equation:dd
t p
i
ð
h
i
;
t
Þ ¼
X
h
i
V
w
ð
h
i
j
h
i
V
Þ
p
i
ð
h
i
V
;
t
Þ
p
i
ð
h
i
;
t
Þ
X
h
i
V
w
ð
h
i
V
j
h
i
Þ
:
ð
1
Þ
Here,
w
(
h
i
V
j
h
i
) means the transition rate to change the state
h
i
into one of the possible states,
h
i
V
, during the next time step,with
w
(
h
i
j
h
i
)=0. In the considered case, there are only two possibilities, either
h
i
= + 1
!
h
i
V
=
1, or
h
i
=
1
!
h
i
V
=+1. We will start with a simple assumption that thechange of states depends on the ‘‘impact factor’’
I
i
, and anoise intensity quantified by a ‘‘
generalized temperature
’’
T
1567-5394/02/$ - see front matter
D
2002 Elsevier Science B.V. All rights reserved.PII: S1567-5394(02)00030-0
*
Corresponding author. Fax: +42-12-6541-2305.
E-mail address:
babinec@fmph.uniba.sk (P. Babinec).www.elsevier.com/locate/bioelechemBioelectrochemistry 56 (2002) 167–170
(more precisely, we should write
k
B
T
for the noise intensity, butforsimplicitywehaveputBoltzmannconstant
k
B
=1).At the thermal equilibrium, the generalized temperature is iden-tical with the ordinary Boltzmann temperature, but in bio-membranes may be significant additional stochastic fields of variousorigins,therefore thegeneralizedtemperaturemay besignificantlyhigherthanordinarytemperature.Thisexcessof noise may lead, e.g. to the increase of signal-to-noise ratio,which is known as stochastic resonance effect [5].A possible ansatz for the transition rate reads:
w
ð
h
i
V
j
h
i
Þ ¼
g
exp
f
I
i
=
T
g
:
ð
2
Þ
Here,
g
[1/s] defines the time scale of the transitions.Within a simplified approach, every channel can beascribed a single parameter, the ‘‘strength’’,
s
i
. Furthermore,an ion channel distance
d
ij
may be defined, which measuresthe distance between each two channels (
i
,
j
) in a
biomem-brane space
, which does not necessarily coincide with the physical space. It is assumed that the impact factor betweentwo channels decreases with the distance in a nonlinear manner. The above assumptions are included in the follow-ing ansatz:
I
i
¼
h
i
X
N j
¼
1
;
j
a
i
s
j
h
j
=
d
nij
þ
e
h
i
;
ð
3
Þ
n
>0 is a model constant and
e
is the external influence,which may be regarded as a global preference towards oneof the states. The long-range coupling between ion channelsarises from membrane-mediated direct energy interactions[6], and also from global capacitative coupling [7]. As a
basic element of our theory, a scalar
spatio-temporal inter-action field h
h
(
r
,
t
) is used. Every channel contributes per-manently to this field with its state
h
i
and with its strength
s
i
at its current spatial location
r
i
. The interaction generatedthis way has a certain life time 1/
b
[s], further, it can spreadthroughout the system by a diffusion-like process, where
D
h
[m
2
/s] represents the diffusion constant for interactionexchange. We have to take into account that there are twodifferent states in the system, hence the interaction fieldshould also consist of two components.The spatio-temporal change of the interaction field can be summarized in the following equation:
@ @
t h
h
ð
r
;
t
Þ ¼
X
N i
¼
1
s
i
d
h
;
h
i
d
ð
r
r
i
Þ
b
h
h
ð
r
;
t
Þþ
D
h
D
h
h
ð
r
;
t
Þ
:
ð
4
Þ
Here,
d
h
,
h
i
is the Kronecker Delta, indicating that the chan-nels contribute only to the field component which matchestheir state
h
i
.
d
(
r
r
i
) means Dirac’s Delta function used for continuous variables, which indicates that the channelscontribute to the field only at their current position,
r
i
.The interaction field
h
h
(
r
,
t
) influences the channel
i
asfollows: at a certain location
r
i
, the channel in the state
h
i
=+1 is affected by two kinds of interaction; the inter-action resulting from channels which are in the same state,
h
h
= + 1
(
r
i
,
t
), and the interaction resulting from the channelswith the opposite states
h
h
=
1
(
r
i
,
t
). The diffusion constant
D
h
determines how fast the interaction spreads, and thedecay rate
b
determines, how long a generated interactionwill exist.For the change of states, we can generalize the transition probability, Eq. (2), by replacing the impact factor
I
i
withthe influence of the local interaction field. A possible ansatzreads:
w
ð
h
i
V
j
h
i
Þ ¼
g
exp
f½
h
h
V
ð
r
i
;
t
Þ
h
h
ð
r
i
;
t
Þ
=
T
g
w
ð
h
i
j
h
i
Þ ¼
0
ð
5
Þ
As in Eq. (2), the probability to change state
h
i
is rather small, if the local field
h
h
(
r
i
,
t
), which is related to thesupport of state
h
i
, overcomes the local influence of theopposite state. This effect, however, is scaled again by thegeneralized temperature
T
, which is a measure for therandomness in channel interaction. Note that this temper-ature is measured in units of the interaction field.The complete dynamics of the ensemble of ion channelscan be formulated in terms of the canonical
N
-particledistribution function
P
ð
h
;
r
;
t
Þ ¼
P
ð
h
1
;
r
1
;
...
;
h
N
;
r
N
;
t
Þ
;
ð
6
Þ
which gives the probability to find the
N
channels with thestates
h
1
,
. . .
,
h
N
in the vicinity of
r
1
,
. . .
,
r
N
on the biomem- brane surface
A
at time
t
. Considering both state changesand movement of the channels, the master equation for
P
(
h
,
_ r
,
t
) reads:
@ @
t P
ð
h
;
r
;
t
Þ ¼
X
h
V
a
h
h
w
ð
h
j
h
V
Þ
P
ð
h
V
;
r
;
t
Þ
w
ð
h
j
h
V
Þ
P
ð
h
;
r
;
t
Þ
i
X
N i
¼
1
5
i
ð
a
5
i
h
h
ð
r
;
t
Þ
P
ð
h
;
r
;
t
ÞÞ
D
n
D
i
P
ð
r
;
h
;
t
Þ
:
ð
7
Þ
The first line of the right-hand side of master describes the‘‘gain’’ and ‘‘loss’’ of channels (with the coordinates
r
1
,
. . .
,
r
N
) due to state changes, where
w
(
h
j
h
V
) means any possible transition within the state distribution
h
V
, whichleads to the assumed distribution
h
. The second linedescribes the change of the probability density due to themotion of the channels on the surface. Eq. (7) together withEqs. (4) and (5) forms a complete description of our system.
P. Babinec, M. Babincova´ / Bioelectrochemistry 56 (2002) 167–170
168
3. Critical conditions for spatial separation of ionchannel activity
Starting with the canonical
N
-particle distribution func-tion,
P
(
h
,
r
,
t
), Eq. (7), the spatio-temporal density of chan-nels with state
h
can be obtained as follows:
n
h
ð
r
;
t
Þ ¼
Z X
N i
¼
1
d
h
;
h
i
d
ð
r
r
i
Þ
P
ð
h
1
;
r
1
...
;
h
N
;
r
N
;
t
Þ
d
r
1
...
d
r
N
:
ð
8
Þ
Integrating Eq. (7) according to Eq. (8) and neglectinghigher order correlations, we obtain, using the transitionrates from Eq. (5), the following reaction–diffusion equa-tion for
n
h
(
r
,
t
)
@ @
t n
h
ð
r
;
t
Þ ¼ 5
h
n
h
ð
r
;
t
Þ
a
5
h
h
ð
r
;
t
Þ
i
þ
D
n
D
n
h
ð
r
;
t
Þ
X
h
V
a
h
½
w
ð
h
V
j
h
Þ
n
h
ð
r
;
t
Þþ
w
ð
h
j
h
V
Þ
n
h
V
ð
r
;
t
Þ
:
ð
9
Þ
With
h
={+1,
1}, Eq. (9) is a set of two reaction– diffusion equations, coupled both via
n
h
(
r
,
t
) and
h
h
(
r
,
t
).Inserting the densities
n
h
(
r
,
t
) and neglecting any external preferences, Eq. (4) for the spatial interaction field can betransformed into the linear deterministic equation:
@ @
t h
h
ð
r
;
t
Þ ¼
sn
h
ð
r
;
t
Þ
b
h
h
ð
r
;
t
Þ þ
D
h
D
h
h
ð
r
;
t
Þ
:
ð
10
Þ
The solutions for the spatio-temporal distributions of chan-nels and states are now determined by the four coupledequations, Eqs. (9) and (10). For our further discussion, weassume again that the spatio-temporal interaction fieldrelaxes faster than the related distribution of channels intoa quasi-stationary equilibrium. From Eq. (10), we find with(
D
/
D
t
)
h
h
(
r
,
t
)=0 and
D
h
=0:
h
h
ð
r
;
t
Þ ¼
s
b
n
h
ð
r
;
t
Þ
;
ð
11
Þ
which can now be inserted into Eq. (9), thus reducing the set of coupled equations to two equations.The homogeneous solution for
n
h
(
r
,
t
) is given by themean densities:¯
n
h
¼ h
n
h
ð
r
;
t
Þi ¼
¯
n
2
:
ð
12
Þ
Under certain conditions however, the homogeneous state becomes unstable and a spatial separation of ion channelactivity occurs. In order to investigate these critical con-ditions, we allow small fluctuations around the homoge-neous state
n¯
h
:
n
h
ð
r
;
t
Þ ¼
¯
n
h
þ
d
n
h
;
d
n
h
¯
n
h
K
1
:
ð
13
Þ
Inserting Eq. (13) into Eq. (9), a linearization gives:
@
d
n
h
@
t
¼
D
n
a
s
¯
n
2
b
D
d
n
h
þ
g
s
¯
n
b
T
g
ð
d
n
h
d
n
h
Þ
:
ð
14
Þ
With the ansatz
d
n
h
f
exp
ð
k
t
þ
i
kr
Þ ð
15
Þ
we find the dispersion relation
k
(
k
) for inhomogeneousfluctuations with wave vector
k
:
k
1
ð
k
Þ ¼
k
2
C
þ
2
B
;
k
2
ð
k
Þ ¼
k
2
C B
¼
g
s
¯
n
b
T
g
;
C
¼
D
n
a
s
¯
n
2
b
:
ð
16
Þ
For homogeneous fluctuations, we obtain from Eq. (16)
k
1
¼
2
g
s
¯
n
b
T
2
g
;
k
2
¼
0 for;
k
¼
0
;
ð
17
Þ
which means that the homogeneous system is marginallystable as long as
k
1
<0, or
s n¯
/
b
T
<1. The condition
B
=0defines a
critical generalized temperatureT
c1
¼
s
¯
n
b
:
ð
18
Þ
For temperatures
T
<
T
1c
, the homogeneous state
n
h
(
r
,
t
)=
n¯
/2, where channels of both states are equally distributed, becomes unstable and the spatial separation process occurs.This is in direct analogy to the phase transition obtainedfrom the Ising model of a ferromagnet. Here, the state with
T
>
T
1c
corresponds to the
paramagnetic
or disordered phase,while the state with
T
<
T
1c
corresponds to
ferromagnetic
ordered phase.The conditions of Eq. (17) denote a
homogeneous
sta- bility condition. To obtain stability against inhomogeneousfluctuations of wave vector
k
, the two conditions
k
1
(
k
)
V
0and
k
2
(
k
)
V
0 have to be satisfied.Taking into account the critical temperature
T
1c
, Eq. (18),we can rewrite these conditions, Eq. (16), as follows:
k
2
ð
D
n
D
c
n
Þ
2
g
T
c1
T
1
0
ð
19
Þ
k
2
ð
D
n
D
c
n
Þ
0
:
Here, a
critical diffusion
coefficient,
D
n
c
results from thecondition
C
=0:
D
c
n
¼
a
2
s
¯
n
b
:
ð
20
Þ
Hence, the condition
D
n
>
D
n
c
denotes a second stabilitycondition. In order to explain its meaning, let us consider that the diffusion coefficient of the channels
D
n
, may be a
P. Babinec, M. Babincova´ / Bioelectrochemistry 56 (2002) 167–170
169
function of the generalized temperature
T
. This is reasonablesince the generalized temperature is a measure of random-ness in channel interaction, and an increase of such arandomness leads to an increase of a random spatial migra-tion. The simplest relation for a function
D
n
(
T
) is the linear one,
D
n
=
l
T
. By assuming this, we may rewrite Eq. (19)using a
second critical temperature
,
T
2c
instead of a criticaldiffusion coefficient
D
n
c
:
k
2
l
ð
T
T
c2
Þ
2
g
T
c1
T
1
0
ð
21
Þ
k
2
l
ð
T
T
c2
Þ
0
:
The second critical temperature,
T
2c
, reads as follows:
T
c2
¼
a
2
l
s
¯
n
b
¼
a
2
l
T
c1
:
ð
22
Þ
The occurrence of two critical temperatures
T
1c
and
T
2c
allows a more detailed discussion of the stability conditions.Therefore, we have to consider two separate cases: (1)
T
1c
>
T
2c
and (2)
T
1c
<
T
2c
. In the first case,
T
1c
>
T
2c
, we candiscuss three ranges of the temperature
T
:(i) For
T
>
T
1c
, both eigenvalues
k
1
(
k
) and
k
2
(
k
), Eq. (16),are nonpositive for all wave vectors,
k
, and the homogenoussolution
n¯
/2 is
completely stable
.(ii) For
T
1c
>
T
>
T
2c
the eigenvalue
k
2
(
k
) is still nonpos-itive for all values of
k
, but the eigenvalue
k
1
(
k
) is negativeonly for wave vectors that are larger than some critical value
k
2
>
k
c
2
:
k
2c
¼
2
gl
T T
c1
T T
T
c2
:
ð
23
Þ
This means that, in the given range of temperatures, thehomogeneous solution
n¯
/2 is
metastable
in an infinitesystem, because it is stable only against fluctuations withlarge wave numbers, i.e. against small-scale fluctuations.Large-scale fluctuations destroy the homogeneous state andresult in a spatial separation process, i.e. instead of ahomogenous distribution of states, channels with the samestate form separated
spatial domains
which coexist.(iii) For
T
<
T
2c
both eigenvalues
k
1
(
k
) and
k
2
(
k
) are positive for all wave vectors
k
, which means that thehomogeneous solution
n¯
/2 is completely unstable. On theother hand, systems with spatial dimension
L
<2
p
/
k
c
arestable in this region.In the second case,
T
1c
<
T
2c
, which corresponds to
a
>2
l
,already small inhomogeneous fluctuations result in aninstability of the homogenous state for
T
<
T
2c
, i.e. we havea direct transition from the completely stable to the com- pletely unstable regime at the critical temperature
T
=
T
2c
.In conclusion, we would like to note that our model of collective ion channel dynamics only sketches some basicfeatures of structure formation in excitable biomembranesystems. There is no doubt, that in the real living cells, amore complex behavior among the ion channels occurs, andmay depend on numerous influences beyond a quantitativedescription. One especially interesting direction of the fur-ther development is analysis of the possible orchestration of ion channel fluctuations by a cytoskeleton in a cooperativemanner [8,9] in a squid’s giant axons. The first step is to
analyse subthreshold dynamics in periodically stimulatedaxons [10] in the framework of our model. Our next stepwould be the study of channels noise in neurons of thesuperficial medial entorhinal cortex, which are responsiblefor delivering information to the hippocampus via the perforant path. These neurons exhibit subthreshold oscilla-tions in membrane potential at a frequency 8 Hz [11].We hope that our model may give rise to further inves-tigations in this exciting field of bioelectrochemistry.
Acknowledgements
The authors would like to thank VEGA grant 1/8310/01and 1/9179/02 for the financial support.
References
[1] B. Hille, Ionic Channels of Excitable Membranes, Sinauer Associates,Sunderland, 1992.[2] R.D. Astumian, P.B. Chock, T.Y. Tsong, H.V. Westerhof, Effects of oscillations and energy-driven fluctuations on the dynamics of enzymecatalysis and free energy transduction, Phys. Rev. A 39 (1989) 6416– 6435.[3] P. Gogan, I. Schmiedel-Jakob, Y. Chitti, S. Tycˇ-Dumont, Fluorescenceimaging of local membrane fields during the excitation of single neu-rons in culture, Biophys. J. 69 (1995) 299–310.[4] P. Babinec, M. Babincova´, Collective dynamics of ion channels in biological membranes, Gen. Physiol. Biophys. 15 (1996) 65–69.[5] R.D. Astumian, J.C. Weaver, R.K. Adair, Rectification and signalaveraging of weak electric fields by biological cells, Proc. Natl. Acad.Sci. U. S. A. 92 1995, pp. 3740–3743.[6] U. Hianik, V.A. Buckin, B. Piknova´, Can a single bacteriorhodopsinmolecule change the structural state of one liposome?
Gen. Physiol.Biophys. 13 (1994) 493–501.
[7] R.F. Fox, Y.N. Lu, Emergent collective behavior in large numbers of globally coupled independently stochastic ion channels, Phys. Rev. E49 (1994) 3421–3431.[8] Y. Hanyu, G. Matsumoto, Spatial long-range interactions in squidgiant axons, Physica D 49 (1991) 198–213.[9] W.D. Griggs, Y. Hanyu, G. Matsumoto, Cultured giant fiber lobe of squid expresses three distinct potassium channel activities in selectivecombinations, J. Membr. Biol. 152 (1996) 25–37.[10] D.T. Kaplan, J.R. Clay, T. Manning, L. Glas, M.R. Guevara, A. Shrier,Subtreshold dynamics in periodically stimulated squid giant axons,Phys. Rev. Lett. 76 (1996) 4074–4077.[11] J.A. White, R. Klink, A. Alonso, A.R. Kay, Noise from voltage-gatedion channels may influence neuronal dynamics in the entorhinal cor-tex, J. Neurophysiol. 80 (1998) 262–269.
P. Babinec, M. Babincova´ / Bioelectrochemistry 56 (2002) 167–170
170

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