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ORIGINAL PAPER

On the simple random-walk models of ion-channel gate dynamics

reflecting long-term memory

Agata Wawrzkiewicz•Krzysztof Pawelek•

Przemyslaw Borys•Beata Dworakowska•

Zbigniew J. Grzywna

Received: 6 September 2011/Revised: 6 March 2012/Accepted: 13 March 2012/Published online: 7 April 2012

? The Author(s) 2012. This article is published with open access at Springerlink.com

Abstract

elling have been proposed. Although many models describe

the dwell-time distributions correctly, they are incapable of

predicting and explaining the long-term correlations

between the lengths of adjacent openings and closings of a

channel. In this paper we propose two simple random-walk

modelsofthegatingdynamicsofvoltageandCa2?-activated

potassiumchannelswhichqualitativelyreproducethedwell-

timedistributions,anddescribetheexperimentallyobserved

long-term memory quite well. Biological interpretation of

both models is presented. In particular, the origin of the

correlations is associated with fluctuations of channel mass

density.Thelong-termmemoryeffect,asmeasuredbyHurst

R/S analysis of experimental single-channel patch-clamp

recordings,isclosetothebehaviourpredictedbyourmodels.

The flexibility of the models enables their use as templates

for other types of ion channel.

Several approaches to ion-channel gating mod-

Keywords

Conformational diffusion ? Hurst analysis ?

BK channels ? Activation gate

Random walk process ?

Introduction

Theoretical background

Potassium channels are integral proteins that enable rapid,

selective transport of K?ions across the cell membrane

down their electrochemical gradient (Chung et al. 2007). In

general, ion channels are not just simple pores with a

constant permeability—they

changes resulting in transitions between the conducting and

non-conducting (open and closed) states. The probabilities

of these states depend on channel-specific gating stimuli, of

which the most common are membrane potential, ligand

binding, and mechanical force (Hille 2001).

Ion-channel proteins comprise several subunits with

different functions. In particular, distinct structural domains

are responsible for the stimulus sensing and for the channel

activation. Thus, stimulus–response relationships require a

cooperative interaction between the sensor domains and the

gate. When a gating stimulus occurs at the sensor domain,

the conformational change is conveyed to the activation

gate and changes the transition probability between the

open and closed states.

Voltage-dependent,calcium-activatedpotassiumchannels

(BK)arecharacterizedbyalargesingle-channelconductance

(*100–300 pS)(LatorreandMiller1983;Latorreetal.1989;

Marty1983)comparedwiththatofotherpotassiumchannels.

These channels, which can be activated by membrane depo-

larization or by elevation of intracellular calcium concentra-

tion, can be found in many cells and tissues, for example

neurons, chromaffin cells, the inner hair cells of cochlea, and

muscles, and have an important function in several physio-

logical processes (Cui et al. 2008). Irrespective of their

unusually large conductance, BK channels remain highly

selective for K?ions over other cations (Cui et al. 2008).

undergoconformational

A. Wawrzkiewicz (&) ? K. Pawelek ? P. Borys ? Z. J. Grzywna

Department of Physical Chemistry and Technology of Polymers,

Section of Physics and Applied Mathematics, Silesian University

of Technology, Ks. M. Strzody 9, 44-100 Gliwice, Poland

e-mail: agata.wawrzkiewicz@gmail.com

B. Dworakowska

Division of Biophysics, Department of Physics, Warsaw

University of Life Sciences—SGGW, Nowoursynowska 166,

02-787 Warsaw, Poland

123

Eur Biophys J (2012) 41:505–526

DOI 10.1007/s00249-012-0806-8

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In general, the putative structure of BK channels shares

many similarities with that of voltage-dependent Kv chan-

nels (Ma et al. 2006; Liu et al. 2008a, b) and ligand gated

potassium channels, for example MthK (Latorre and

Brauchi 2006; Jiang et al. 2001, 2002; Shi et al. 2002; Xia

et al. 2002; Tang et al. 2003; Yusifov et al. 2008; Kim et al.

2006, 2008). BK channels are tetramers of a channel protein

encoded by the Slo1 gene, composed of seven transmem-

brane domains (S0–S6) and four cytosolic hydrophobic

C-terminal domains (S7–S10) (Cox 2006; Cui et al. 2008;

Latorre and Brauchi 2006). Functionally, segments S1–S4

form a voltage sensor domain (VSD), and segments S5–S6

form a pore-gate domain (PGD). The large cytoplasmic

C-terminal domain serves as the primary ligand sensor. S0

is a segment that is absent from Kv and MthK channels and

causes some problems in homology modelling, directing the

N-terminus to the extracellular side of the membrane (Cui

et al. 2008).

The simplest model of the channel’s gating kinetics can

be described by a two-state system:

C

kO

!

kC

O

ð1Þ

in which the states of the channel switch randomly between

the open and closed conformations with gating-factor-

dependent kinetic rate constants (kO, kC). This kind of

description may belong to a Markov class of model, if the

state transition probabilities depend only on the current

state of the system (Fulin ´ski et al. 1998).

Invention of the patch-clamp technique (Sakmann and

Neher 1995) has enabled experimental observation of the

ionic currents at a single-channel level, which gave the

opportunity to obtain the probability densities of closed

fC(t) and open fO(t) dwell time intervals. For many chan-

nels, the open dwell-time distribution can be reasonably

approximated by a single exponential function, but the

closed dwell-time distribution fits better to the sum of

many exponentials of the form (Goychuk and Ha ¨nggi 2002,

2003):

fCðtÞ ¼

X

N

i¼1

cikiexpð?kitÞ;

ð2Þ

with weight coefficients obeying:

X

N

i¼1

ci¼ 1;

ð3Þ

Quite often Eq. (2) can be replaced by a single stretched

exponential (Millhauser et al. 1988) or a power law

function (Sansom et al. 1989; Blatz and Magleby 1986;

Ring 1986; Mercik and Weron 2001; La ¨uger 1988; Condat

and Ja ¨ckle 1989). When the dwell time distribution can be

described by a sum of exponentials, Markov models with

either a few or many open and closed states have been

proposed. Such models assume the channel protein to be

found in a number of discrete states, separated by relatively

high potential barriers (Sansom et al. 1989).

When multiple open and closed states are fitted to the

model, a possibility arises for allosteric activation of the

channel, as shown for BK channels as an example in

Horrigan et al. (1999), Horrigan and Aldrich (1999), and

Rothberg and Magleby (1999). The HCA model proposed

by Horrigan et al. (1999) and Horrigan and Aldrich (1999)

is a classical, many-state Markov model of BK channel

voltage-dependent gating. It reflects well the channel’s

behaviour at different values of membrane potential in the

absence of any calcium ions and agrees quantitatively with

experimental results. A crucial feature of the HCA model is

the allosteric mechanism of BK channel activation, which

is different from the strict coupling proposed in the liter-

ature to describe gating of the Kv channel (an example of

which is the Hodgkin–Huxley model) (Hille 2001).

Whereas in the strict coupling of Kv channel activation

voltage sensor movement opens a channel, this is not so in

allosteric activation. In the allosteric model (of BK chan-

nels) neither of the sensors directly opens the gate but all of

them affect the gate, and each other, in an allosteric

manner, and gate opening can even precede the sensor’s

activation (Rothberg and Magleby 1999).

The calcium activation of BK channels is characterized

in the voltage-dependent Monod Wyman Changeux

(VD-MWC) model (Cox et al. 1997). Analogously to the

previous case, because of incorporation of an allosteric

mechanism, none of the Ca2?-activated sensor domains

renders the channel conductive in a direct manner. Nev-

ertheless, responding to the ligand binding on any of the

multiple binding sites, the channel undergoes a confor-

mational change that promotes the opening, and (extending

the MWC formalism) indirectly alters the remaining

binding sites by increasing the affinity for Ca2?. In this

model, the open state has a higher Ca2?affinity than a

corresponding closed state with the same number of cal-

cium ions bound. Thus, an open Ca2?-bound state is

energetically preferred over a closed Ca2?-bound state.

The voltage dependence of the gating is embedded in the

equilibrium constants between the closed and open states.

Advanced approaches based on the HCA and MWC

models have been proposed in the literature to account for

both allosteric activation processes in a single model

(Rothberg and Magleby 2000; Horrigan and Aldrich 2002).

By extending the assumptions of its templates, these

models reach a large number of states (e.g. 50-state two-

tiered model); their purpose is to give an even more

complete picture of the gating process over a wide range of

506Eur Biophys J (2012) 41:505–526

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voltages and calcium concentrations, assuming all the rate

constants are correctly determined (which is not easy).

What is important here it is that coupling between different

functional parts of the channel may be strong, as for the

sensor–gate interaction, or weak, as for the voltage sensor–

calcium sensor interaction. An allosteric model allows the

channel to open even in the absence of any voltage and

calcium stimuli or to close when all the sensors are acti-

vated. In general, however, the greater the number of

sensors activated the more energetically favoured an open

conformation is. Both the voltage and the ligand sensors

can switch from the resting to the activated state in either

closed or open conformation of a channel.

The multi-exponential dwell-time distribution of the

Markovian systems in some cases reduces to a single

stretched exponential or a power law dependence. This

observation leads to the introduction of the fractal (Lieb-

ovitch et al. 1987) and diffusive models (Millhauser et al.

1988; La ¨uger 1988; Condat and Ja ¨ckle 1989; Goychuk and

Ha ¨nggi 2002, 2003) to address the voltage-dependent

channels (Kv). According to these models, the channel

protein exists in a continuous conformational space with a

rather smooth energy surface, in contrast with a few, well

separated discrete conformations determined by the

potential wells, as expected from the finite state Markovian

models.

A classical model of channel gating by a Markovian

diffusion process over a very (infinitely) large number of

similar energy closed states was introduced by Millhauser

et al. (1988). This model produces the power law depen-

dence in the dwell time distributions. It was generalised by

Ha ¨nggi and Goychuk (2002, 2003) who investigated the

gating dynamics as a continuous diffusion process with part

of the closed state potential being voltage dependent. The

model explains how the closed state dwell time distribution

of a voltage dependent channel can change its character

from a power law to an exponential function, depending on

the membrane polarization. Moreover, the model justifies

the exponential dependence of the Hodgkin–Huxley model

used to describe the experimental relationship between the

voltage and the rate of opening of a single gate in the

potassium channels.

Further generalisations of diffusion modelling lead to

non-Markovian conformational subdiffusion. It has been

shown (Goychuk and Ha ¨nggi 2008) that the subdiffusive

generalization reproduces the main features of the closed

time distribution and predicts the autocorrelation function

of the conductance fluctuations. The non-Markovian nature

of the channel currents suggested by Fulin ´ski et al. (1998)

has also been addressed in different ways (Grzywna and

Siwy 1997; Grzywna et al. 1999; Liebovitch and Sullivan

1987; Liebovitch et al. 1987). Interesting non-Markovian

models were given by Grzywna and Siwy (1997) and

Grzywna et al. (1999), who provide an alternative approach

based on the assumption that the channel behaviour is

governed by a nonlinear deterministic dynamics.

The models described capture many important features

of the gating phenomena but not all of them. Among the

missing features, a very interesting but not completely

understood property of many channel species is the long-

term memory, as seen by Hurst R/S analysis (Campos de

Oliveira et al. 2006; Bandeira et al. 2008). The rescaled

range analysis (R/S analysis, Hurst analysis; Hurst 1951;

Hurst et al. 1965; Feder 1988; Borys 2011), when applied

to the ionic current recordings, measures the correlation of

adjacent dwell times of an ion channel (Varanda et al.

2000), commonly indicative of strong trend-reinforcing

behaviour.

Whether the discrete Markovian models can produce a

long-range memory was the subject of research (Varanda

et al. 2000; Campos de Oliveira et al. 2006) on the 3, 4, and

11-state models. All the tests were negative. Whether

Markovian models with more states can reproduce the

long-range memory, as measured by the Hurst exponent, is,

however, still an open question.

According to the experiments, the long-term memory in

ion channel activity seems to be independent of external

channel activating factors (Barbosa et al. 2007; Varanda

et al. 2000 and the section ‘‘Analysis of experimental data’’

in this paper). It was observed that at each fixed voltage

and calcium concentration (at a macro scale), and thus at

each corresponding average sensor activation level (at a

molecular scale), the long-term correlations in time series

are of similar magnitude. It is, therefore, reasonable to link

the memory with gate fluctuations under given conditions

only. This point of view is also supported by the allosteric

picture of BK channel gating dynamics, with gate fluctu-

ations present even in the absence of gating sensor

activation.

In this paper, we limit our considerations to gate fluc-

tuations, ignoring the full HCA-MWC machinery; this has

the advantage of showing the processes relevant to the

long-term memory without obscuring them by the addi-

tional details of a gate–sensor coupling. For the gate

dynamics we propose two simple random-walk models

belonging to the diffusion class of models (Millhauser et al.

1988; Goychuk and Ha ¨nggi 2002, 2003). The models

generate a dichotomous current time series as found in the

experimental results (the patch-clamp recordings), and

produce the long-term memory of the considered system.

The open and closed-dwell time distributions evaluated for

the simulated series are concurrent, at a qualitative level,

with those observed experimentally. We also propose

biophysical interpretations for both models.

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Description of the models and biological inspirations

It is stated in the ‘‘Theoretical background’’ that the long-

term memory is independent of the sensor state. To check

whether our models agree with such results, we need some

simplified representation of biosystem activation. To pro-

pose a reasonable approach, we briefly review the ideas

which are the basis of BK channel’s voltage and calcium

sensor operation in relation to the gate.

The activation gate is a part of the channel protein

responsible for blocking the ion flux through the pore

(Chung et al. 2007). Because of studies on KcsA and MthK

channels (Jiang et al. 2002) we have physical evidence of

gate-open and gate-closed conformations of these potas-

sium channels. A key difference between these is the

bending of an inner helix segment at a glycine residue close

to the selectivity filter, called the ‘‘Gly-hinge’’.

In many potassium channels, the inner mouth of the pore

(the channel’s entrance) is formed by the hydrophobic

residues on the C-terminus of the inner helix (S6 in the

canonical channel structure with six transmembrane seg-

ments). The residues are arranged in a bundle to block the

potassium ion flux when the channel is closed. Because of

the flexibility of the S6 domain at the Gly-hinge in the

middle of S6 (sometimes assisted with the Pro-kinks sev-

eral residues later), a gate-open conformation can be

reached with inner helices well separated from each other

(Jiang et al. 2002; Long et al. 2005; Ding et al. 2005; Cui

et al. 2008). For the BK channels, it can be shown by

mutations that the Gly-hinges are important but not man-

datory structures to reach the open conformation, which

contrasts with the Kv or MthK channels (Magidovich and

Yifrach 2004). Experiments with quaternary ammonium

ions show further that the BK channel’s selectivity filter

entry is accessible in the gate-closed conformation, which

suggests the existence of a secondary gate in the close

vicinity of the selectivity filter, similarly to several other

ligand-gated ion channels (Wilkens and Aldrich 2006;

Flynn and Zagotta 2001; Bruenning-Wright et al. 2007;

Cui et al. 2008).

The voltage sensor of the BK channel also differs sig-

nificantly from the Kv’s, i.e. the gating charge is not

confined to S4, but rather spreads over the whole voltage-

sensing domain (VSD), and is much smaller than in Kv

channels. Furthermore, the ligand sensor is not exactly

similar to MthK, because the gating ring of a BK channel

has important differences from that of the MthK one (e.g.

the gating ring of MthK is formed by eight identical RCK

domains, whereas in the BK channel it is formed by four

RCK1 and four RCK2 domains).

Ligand sensor operation relies on the interaction

between a C-terminal intracellular RCK domain which

forms a gating ring which expands on calcium binding, and

the inner helix connected to the RCK via an S7-residue

linker, whose mutations reveal a profound effect on the

gating dynamics (Jiang et al. 2002). The voltage depen-

dency of the BK channels is hypothesized on the basis of

homology with the Kv channel gating, which depends on

the motion of the charged S4 segment binding through the

S4–S5 linker to the S6 segment at a Pro-X-Pro motif,

triggering mechanical translations of S6 which result in

gating, probably via Gly-hinge bending (Ding et al. 2005).

In our approach, the conformational dynamics of the

activation gate is described by a discrete random walk

process performed by the RC (reaction coordinate) over the

conformational space. We could assign the RC a meaning

of, e.g., the hinge angle but, because the actual gating

blocks of a BK channel are not yet known, we treat RC as

abstract coordinate in the conformation space. The effect of

the sensors on the gate is allosteric in nature in the BK

channels and is introduced by a drift force acting on the

reaction coordinate. Such a drift force approach to sensor

activation retains the allosteric picture of the system, in

which sensor activation is not mandatory for gate opening.

Furthermore, the drift force is a good candidate to mimic

sensor action, because the sensor seems to operate by

forcing the Gly-hinge. In general, this drift force may

depend nonlinearly on the reaction coordinate, calcium

concentration, and voltage, FD¼ FDð½Ca2þ?;V;RCÞ: In

our case however, the actual transmission of the gating

stimuli is irrelevant (we have assumed that the long-term

memory is independent of these stimuli), and we have only

checked whether the activation bias modifies the Hurst

effect. Having no hints on the RC dependency of the drift

force, it was assumed to be constant, which corresponds to

the linear ramp potential.

In the diffusive modelling we can assume that the open

and closed states of the gate embrace two manifolds of

energetically similar substates. The manifolds observed in

a macroscopic state can be associated with a set of single

protein structure vectors, nearly equal in energy, which

retain the macroscopically observed property of the state.

From such a viewpoint, the shifting between two different

macro-states is realized by a diffusion process between

adjacent conformations.

Because we approximate the conformational dynamics

in terms of the one-dimensional reaction coordinate

x(t) in a conformational potential U(x), as was shown by

(Goychuk and Ha ¨nggi 2002, 2003), the question which

arises is, is this approximation valid for a large 3D system

such as a protein? Research on protein folding suggests this

may often not be true (Dill et al. 1995). Nevertheless, if we

consider the activation gate as a hinge, movement of which

is effectively restricted to one degree of freedom (or the

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other possible conformations that can be projected on it),

the idea of a one-dimensional reaction coordinate may be

valid.

The activation gate’s conformational space is illustrated

in Fig. 1. It is divided into two parts, which correspond to

the open and closed states of the gate, separated by a

threshold point TP. The open and closed states can have

different energies, which reflect the tendency of a channel

to maintain a conducting or non-conducting state under

fixed external conditions. The reaction coordinate performs

a random walk in the conformation space with positions

before the threshold point (TP) indicating a closed state and

those behind the threshold point indicating an open state, as

shown in Fig. 1.

We propose that the long-term memory of the system

may result from synchronized fluctuations of the activation

gate conformational space boundaries (Model 1). The

biophysics of such fluctuating boundaries may be explained

by considering what happens after squeezing of the channel

protein. This may originate from the local fluctuations

in the membrane thickness near the channel location.

Squeezing will increase the mass density of the channel

protein (including the activation gate), and the atoms

(being restricted by their Van der Waals’ volumes) have

less ‘‘space’’ to move, reducing the number of energetically

equivalent substates within a given macro-state. In other

words, the boundaries of the conformational space move to

reduce the maximum conformational difference between

the substates in the closed and open conformations. The

opposite may be observed in case of channel protein

relaxation by the decrease in the local mass density. This

behaviour is schematically depicted in Fig. 2.

Another way of introducing fluctuations of channel

density to the gate dynamics is to add a synchronous drift

force acting on the reaction coordinate (Model 2). In such

picture, lateral compression or relaxation of the channel

does not strictly limit the size of the conformational space,

but rather produces a drift force acting on the reaction

coordinate and favouring the occupancy of substates near

the threshold (by relaxation) or near the boundaries (by

compression). Such an approach reproduces the long-term

memory of the channel system and enables the boundaries

to be kept constant. It is interesting to notice that the effect

of the compressive force is exactly opposite in Model 2

compared with Model 1. In Model 1, the compression

favours rapid state switching whereas in Model 2 this

regime is reached in a relaxed state. This may be important

for experimental validation of these models. Model 2 drops

the assumption of constant energy within the macro-state in

the absence of gate–sensor coupling but energy variation

among the substates is a smooth, slowly increasing or

decreasing function of the RC (Fig. 5).

The conditions for Model 2 may occur if there is a

protein segment, connected with gating, which has a large

amount of space available for bending. In such case,

reduction in the membrane thickness causes the segment to

bend, i.e. a force is generated. If this bending can take

place either toward or away from the open conformation,

Fig. 1 Schematic illustration of a channel gate’s possible state

transitions. One can observe openings (C ? O), closings (O ? C),

and changes among open (O ? O) or closed (C ? C) states. TP

threshold point separating open and closed states, x reaction

coordinate

Fig. 2 Schematic illustration of the changes of local membrane mass

density and its effect on channel protein. The range of channel gate’s

accessible conformations can be reduced or increased as a result of

membrane squeezing and its relaxation, respectively, as denoted by

the frame on the lower ‘‘diamond’’ in the figure

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which is schematically shown in Fig. 3, then it can be

described by the proposed model.

Model 1

The lattice variables of Model 1 are shown in Fig. 4. The

conformational space of the gate was projected to one

dimension and divided into two parts of equal length sep-

arated by a threshold point. The channel is closed if the

reaction coordinate occupies the position below the

threshold point, otherwise it is open. The transition

between the open and the closed conformations requires

overcoming of a transition potential barrier as shown on the

energy landscape in Fig. 5a.

The range of possible reaction coordinate values in the

conformational space is limited by the two moveable

boundaries B1 and B2. The motion of B1 and B2 corre-

sponds to thermal fluctuations in the membrane thickness

Fig. 3 Squeezing of the

membrane will generate a force

acting on the channel segment,

which is related to gating, and

causing its bending. Bending

motion may lead either to the

gate’s closed or open

conformations (a). Stretching of

the membrane results in

occurrence of a force which

enables relaxation of the bent

segment. The force strength

depends on current thickness of

membrane (and consequently,

degree of bending) (b)

Fig. 4 Schematic representation of Model 1. a The one-dimensional

lattice has 20 nodes, among which first 10 represent closed states of

the channel whereas the other nodes correspond to open states.

Boundaries (B1, B) can move, simultaneously reducing (b) or

increasing (c) the space accessible to the reaction coordinate (RC).

Without any external force, no direction of motion is preferred (d),

otherwise a drift in the direction down the potential energy gradient

occurs (e), which changes the probability of finding the system in a

given state

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and internal strains within the protein segments which

restrict the conformational space of the closed and open

states. The boundary fluctuations are synchronized in

direction (e.g. when the conformational space of an open

state shrinks, the same should happen to the conformational

space of the closed state), and the probabilities of these

increasing or reducing the accessible conformational space

were set equal to 0.5.

Because the mass of the fluctuating membrane is much

higher than the mass of the putative activation gate, the

boundaryfluctuationsarerealizedonalargertimescale,i.e.if

weassumethatachangeofthegatestateisrecognizableinone

timestepinsimulation,thenthechangeofboundariesmaybe

performedafteragivennumberoftimestepsaccordingtothe

assumed relationship of the diffusion time scales:

DB¼DRC

600;

ð4Þ

which results from fitting to the experimental data (DBis

the boundary diffusion coefficient and DRCthe reaction

coordinate diffusion coefficient).

To test for the independence of the long-term correla-

tions on sensor activation, a constant drift force from the

channel sensors may be added to the system. The biassed

random walk process can stand for the behaviour of a

channel gate at a fixed value of the activating or inacti-

vating stimulus. For example, in terms of a high value of

the membrane potential (e.g. V = 80 mV) open substates

are preferred, so the additional drift force will favour

movement of the gate toward the open conformation. As a

result, the energy structure of substates may change at

different activation levels, as illustrated in Fig. 5b, c.

The probabilities of the value of the reaction coordinate

(RC) decreasing (q) or increasing (p) were evaluated

according to the formulas (Berg 1983):

p ¼1

q ¼1

2?DU

2þDU

4kT;

ð5Þ

4kT;

ð6Þ

where k is the Boltzmann constant, T denotes absolute

temperature and DU is a potential energy difference within

a lattice step centred around the reaction coordinate. The

potential U(x) is postulated to take the following form

(depicted in Fig. 5):

Fig. 5 Schematic

representation of the potential

function associated with Model

1. a No force arising from

sensor activity is acting, in

particular, on the direction of

the gate. The corresponding

conformational potential energy

U(x) of the reaction coordinate

is ‘‘flat’’ and symmetric around

the threshold point separating

open and closed substates. If a

nonzero drift force is present,

it is represented by the

conformational potential energy

U(x) sketched on b or c. In

b external potential field is

expressed by a decreasing ramp

function. As a result, open states

will be preferred. Conversely, in

a linearly increasing potential

field closed states will be

preferred (c)

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UðxÞ¼ðx?B1Þ?AþUB1; x2hB1;TP?1:5Þ

UðxÞ¼ðx?ðTP?1:5ÞÞ?BþUTP?1:5; x2hTP?1:5;TPÞ

UðxÞ¼ðx?TPÞ?ð?BÞþUTP; x2hTP;TPþ1:5i

UðxÞ¼ðx?ðTPþ1:5ÞÞ?AþUTP?1:5; x2ðTPþ1:5;B2i

A¼UTP?1:5?UB1

B¼UTP?UTP?1:5

TP?1:5?B1

TP?ðTP?1:5Þ

8

>

>

>

>

>

>

>

>

>

>

>

>

<

:

where B1 and B2 are the locations of the left-hand and

right-hand boundaries, respectively, TP is the threshold

point position, UB1, UTP, and UTP-1.5denote the potential

values at given points, and A and B are the potential slopes

between points B1 and TP - 1.5 and between points

TP - 1.5 and TP, respectively. According to Eq. (7) the

constant A represents the drift force toward the open or

closed states of the system and the constant B describes the

repulsive force originating from the potential barrier

between the open and closed states.

The lattice size was set to 2BMAXnodes, where BMAX

determines the maximum size of the open (and closed)

state space. This space is further restricted by the reflecting

boundary positions (B1, B2), which were initially set to

-BMAX/2 and BMAX/2. The starting RC position was set to

-1 (a closed substate nearest the threshold TR = 0). The

time step length to move the reaction coordinate tRCwas

equal to 1 and the time step length to move the boundaries

was set to tB= 600 (i.e. DB¼DRC

The model variables were estimated by the gradient

optimization technique. The values were adjusted in the

direction of decreasing error up to a point when the relative

errors of the Hurst exponent (EH), the open state proba-

bility (Epop), and the open and closed state dwell time

histograms fitting (Edw,open) and (Edw,closed) were less than

5 %, and their sum (E) was less than 10 %. The overall

error was given by the formula:

;

ð7Þ

600).

E ¼ EHþ Epopþ Edw;openþ Edw;closed

where:

??

pop;exp? pop;sim

pop;exp

???

v2

ð8Þ

EH¼

Hexp? Hsim

Hexp

??

v2

??

;

ð9Þ

Epop¼

??

;

ð10Þ

Edw;open¼

exp;open? v2

v2

exp;open

sim;open

???

;

ð11Þ

Edw;closed¼

exp;closed? v2

v2

exp;closed

sim;closed

??? ???

:

ð12Þ

where Hexp is the Hurst exponent of the experimental

series’ subsequent open and closed dwell times, Hsimis the

Hurst exponent of a series of subsequent open and closed

dwell times obtained from the simulated data, pop,expis the

open state probability calculated from the experimental

data, pop,simis the open state probability calculated from

the simulated data, and vexpand vsimare obtained on the

basis of an appropriate normalized dwell time histogram.

vexpis calculated according to the formula:

v2

exp¼1

N

X

N

i¼1

d2

i

ð13Þ

where N is the number of the dwell time histogram bins and

diis the standard deviation of a given bin (the standard

deviation corresponds to the five different patch-clamp

traces obtained under the same conditions of voltage and

calcium concentration).

vsimis described by the equation below:

v2

sim¼1

N

X

N

i¼1

ðpi;exp? pi;simÞ2

ð14Þ

where N is the number of the dwell time histogram bins and

piis the probability corresponding to the ith histogram bin.

The sensitivity of the total error (E) to the terms of

Model 1 is given in Appendix 3.

The optimum values of the terms in Model 1 are pre-

sented in Table 1.

The simulation details of Model 1 (and Model 2) are

summarized in Appendix 2.

Model 2

Model 2 differs in the way the channel protein density

fluctuations are introduced to the gate dynamics. The

boundary positions (B1, B2) are kept constant, and addi-

tional, random drift force is added to the system which

makes synchronized changes to the energy structure of the

open and closed states, as presented in Fig. 6. This force

causes the reaction coordinate to fluctuate near the

Table 1 Values of the terms used in the Model 1 simulation

TermValue

BMAX

BMIN

TP

14 rcu

-14 rcu

0 rcu

104rcu2/s

Reaction coordinate diffusion

coefficient DRC

Barriers diffusion coefficient DB

UTP- UTP-1.5

Initial position of RC

DRC/600

1.0 kT

-1 rcu

Initial position of B1, B2Floor (BMIN/2), Floor

(BMAX/2) rcua

arcu reaction coordinate unit

512 Eur Biophys J (2012) 41:505–526

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Page 9

threshold point (TP) or conversely, far away from it. The

force fluctuations follow their own random walk, which is

slower than the gating dynamics. The appropriate potential

is depicted in Fig. 6 and is given by:

8

>

>

UðxÞ¼ðx?B1Þ?AþUB1; x2hB1;TP?1:5Þ

UðxÞ¼ðx?ðTP?1:5ÞÞ?BþUTP?1:5; x2hTP?1:5;TPÞ

UðxÞ¼ðx?TPÞ?ð?BÞþUTP; x2hTP;TPþ1:5i

UðxÞ¼ðx?ðTPþ1:5ÞÞ?ð?AÞþUTP?1:5; x2ðTPþ1:5;B2i

A¼UTP?1:5?UB1

B¼UTP?UTP?1:5

TP?1:5?B1

TP?ðTP?1:5Þ

>

>

>

>

>

>

>

>

>

>

<

:

where B1 and B2 are the locations of the left-hand and

right-hand boundaries, respectively, TP is the threshold

point position, UB1, UTP, and UTP-1.5denote the potential

values in given points, and A and B are the potential slopes

between points B1 and TP - 1.5, and between points

TP - 1.5, and TP, respectively. The A constant represents

the drift force which facilitates the RC locations near the

threshold point or near the boundaries. The B constant

describes the repulsive force originating from the potential

barrier between open and closed states.

The effect of the sensors on gate operation is reflected

by the threshold point location. The conformational space

is divided into two parts not necessarily equal, and the TP

position is chosen such that the probability of finding the

gate in an open state resembles the channel open proba-

bility under the given experimental conditions (Fig. 7).

Such a mechanism can be justified by considering the

effect of the sensing domains (e.g. the S5–S6 linker) on the

gate, where they move the part of protein responsible for

;

ð15Þ

gating to increase the number of accessible closed con-

formations and reduce the number of open ones, or the

opposite. The allosteric picture of the channel’s activation

is conserved within such approach.

The jump probabilities to the threshold point (pT), and to

the boundaries (qT), are evaluated by use of Eqs. (5) and

(6) (with pT= p and qT= q when RC\TP, and

pT= q and qT= p when RC[TP), and the potential

given by Eq. (15). The random walk performed by the RC

is illustrated in Fig. 8.

The landscape potential slope (the drift force):

Fd¼ ?UTP?1:5? UB1

TP ? 1:5 ? B1

is controlled by a separate unbiased random walk of UB1,

taking place in a time scale a factor of 1,200 slower than

the time scale of the reaction coordinate. The potential

slopes cannot exceed minimum and maximum values, as

specified in Table 2, with the other simulation values.

ð16Þ

Fig. 6 Schematic

representation of the potential

function associated with Model

2. a The random drift force

facilitates movement toward

boundaries. As a result, the

conformational potential energy

U(x) of the reaction coordinate

increases from the boundary B1

to the threshold point TP and

decreases from TP to the second

boundary B2. In the opposite

case, because of the effect of

random force, positions around

the threshold are preferred. The

appropriate potential energy

U(x) is sketched in b

Fig. 7 Schematic representation of the diffusive space used in Model

2. Boundaries (B1, B2) and the threshold position (TP) are fixed

during the simulation. a If no gating stimulus is present in the system,

the manifolds of closed and open states are of the same length. b The

different numbers of possible open and closed substates account for

preference of the channel for a particular macroscopic state. Among

20 nodes of the lattice used as the diffusive space first six indicate the

closed substates of the gate, the other 14 nodes indicate open substates

Eur Biophys J (2012) 41:505–526 513

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Page 10

The model terms are estimated, analogously as for

Model 1, by the gradient optimization technique.

Materials and methods

Experimental

Cell line and solutions

Human bronchial epithelial cells were cultured on Petri

dishes in minimum essential Eagle’s medium (Sigma) with

10 % fetal calf serum, 100 units/ml penicillin, and 100 lg/

ml(PAA)at37 ?Cin5 %CO2.Forpatch-clamprecordings,

the culture medium was exchanged for an extracellular

solution containing 4 mM CaCl2, 2 mM EGTA, 10 mM

HEPES, and 135 mM potassium gluconate, pH 7.3.

Electrophysiology

Experimental results were recorded from outside-out pat-

ches of human bronchial epithelial cells. The measurements

were performed at room-temperature (20–21 ?C). In all

experiments symmetrical solutions on either side of the cell

membrane were used. Patch pipettes were pulled from

borosilicate capillary tubes of 1.2 mm diameter (Clark

Electromedical Instruments) on a Narishige micro puller.

The tips of the pipettes were smoothed in a Narishige micro

forge. The ion currents were recorded using an Axopatch

200B amplifier (Axon Instruments). The experimental data

were low-pass filtered at 10 kHz and transferred to a com-

puter at a sampling frequency of 20 kHz using Clampex 7

software (Axon Instruments). Single currents were initially

analysed by use of pClamp 7 software (Axon Instruments).

The single-channel recordings were analysed at a fixed

pipette potential. The experiments were carried out at: -80,

-60,-40,-20,20,40,60,and80 mVandfiveindependent

measurements were recorded for each voltage. Each exper-

iment was performed on a different patch (so we obtained

8 9 5 = 40 different patch-clamp recordings in total). The

channel current was measured at time intervals of

Dt = 5 9 10-5s. The ionic current measurement error was

DI = 5 9 10-4pA. Each of patch-clamp recordings lasted

300 s, thus the experimental time series comprised

N = 6 9 106currentvaluesattheappliedtimeresolutionof

the measurement.

Data analysis

Event detection

Investigating experimental time series, we considered two

modes of ion-channel conduction, here called the open

(conducting) and closed (non-conducting) states. The

threshold current value used to identify transitions between

following states was evaluated by use of a procedure

described elsewhere (Mercik et al. 1999). The ion current

Fig. 8 Schematic representation of Model 2. The reaction coordinate

(RC) shifts either toward the threshold position (TP) with probability

pTrepresented as an arrow pointing toward TP (a, b, pT[qT) or

away from it (arrow pointing toward B1 or B2) (c, d, qT[pT). The

potential slope and, consequently, the values of pTand qTevolve

randomly during the simulation (at a slower rate than evolution of the

reaction coordinate)

Table 2 Values of the terms used in simulation of Model 2

Term Value

B1-18 rcu

B2 18 rcu

104rcu2/s

Reaction coordinate diffusion coefficient DRC

Drift force diffusion coefficient DDF

UTP- UTP-1.5

Initial position of RC

DRC/1,200

0.2 kT

-1 rcu

Initial drift force

Drift force’s increment |Dk|

Max value of Fd

Min value of Fd

0.00 kT/rcu

0.005 kT/rcu

0.20 kT/rcu

-0.20 kT/rcu

514Eur Biophys J (2012) 41:505–526

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probability density function (PDF) approximated by

the nonparametric kernel density estimate with the

Epanechnikov kernel was plotted on a log–log scale. The

resulting graph reflects the bimodal nature of the analysed

data set. The estimate obtained may be regarded as a

mixture of two unimodal densities that satisfy power laws.

By means of linear regression we obtained the intervals in

both component distributions where the power laws and the

corresponding formulas expressing scaling relationships

were valid. A point of intersection of the power law plots

indicates a threshold current value, in relation to which we

identify open and closed states.

It is well worth noticing that if one applies the double

Gaussian method of threshold current determination, the

difference between the values obtained and those estimated

by the kernel density method is less than 5 % (for each

patch-clamp time series obtained in our experiments). The

nonparametric technique of the PDF approximation avoids

the assumption that the analysed data belong to any par-

ticular distribution, so we prefer this approach.

The R/S Hurst analysis

The rescaled range analysis (R/S Hurst analysis) is an

important statistical method used for testing whether a

system under consideration has long-term memory (Hurst

1951; Varanda et al. 2000). The Hurst exponent (H)

describes the time scaling of a range (R) in a random

motion, normalized to zero mean increments, R / tH: It

can take values from 0 to 1, and provides a means of

classification of temporal series in terms of predictability.

A Hurst exponent of 0.5 is indicative of purely random

behaviour of the system, and is an easily derived charac-

teristic of the scaling of the standard deviation in Brownian

motion (Risken 1996) (it is also characteristic of the scaling

of R in the Brownian motion, as shown with somewhat

larger effort by Borys 2011). In such a case there is no

correlation among any element of the temporal series. If

0\H\0.5 the system is said to be anti-persistent, i.e. the

range of corresponding anomalous Brownian motion grows

slower than randomly. This means that a positive increment

will tend to be followed by a negative one (and vice versa).

The time series with a Hurst exponent value from 0.5 to 1

is persistent, or trending, i.e. the range of the corresponding

Brownian motion grows faster than randomly. This means

that a positive increment will tend to be followed by a

positive one and a negative increment will tend to be fol-

lowed by a negative one. The larger the H value, the

stronger the trend, and the easier to predict the future

behaviour of the system. In this paper we carry out a Hurst

analysis for a sequence of adjacent open and closed times

for both the BK channel recordings available, and the

modelled data.

The procedure for evaluating the Hurst exponent can be

found in many references (Hurst 1951; Hurst et al. 1965;

Feder 1988; Varanda et al. 2000; Borys 2011). Details are

also included in Appendix 1.

Results and discussion

Analysis of experimental data

The samples of original single-channel patch-clamp

recordings, and the corresponding current–voltage relation-

ship, are shown in Fig. 9. As one can see in Fig. 9a, channel

opening varies with the potential value in a way typical of

voltage-dependent ion channels, i.e. membrane depolarisa-

tionleadstoanincreaseinthefrequencyofoccurrenceofthe

open state. According to Fig 9b, the reversal potential was

near 0 mV. Under the experimental conditions, the mean

conductance was estimated to be 235.6 pS.

Table 3 shows the values of the Hurst exponent, H, and

the channel opening probability, po, for the single-channel

recordings obtained at different voltages V at fixed calcium

concentration[Ca2?]equalto2 mM.Itrevealsnosignificant

effect of membrane depolarization on long-term correla-

tions. There is just small variability around the mean of H,

which is expected because H is a random variable with a

standard deviation of 0.1 (Hurst et al. 1965). The mean r2

coefficient oflinear regression(usedby evaluatingH) forall

experimental dwell time series was equal to 0.992.

Concurrent results were reported by Varanda et al.

(2000). Analogously, the calcium ion concentration does

not affect values of the Hurst exponent significantly

(Barbosa et al. 2007). This is consistent with our modelling

approach in which we neglect the details of operation of the

gating sensors.

Analysis of the generated time series

From both of our models we generated n = 5 time series

with N = 6 9 106samples for five different values of the

variable responsible for sensor–gate coupling. For each of

the time series, we evaluated the open state probability (po)

and mean open and closed dwell times; we then performed

Hurst analysis of the corresponding dwell-time series of

subsequent openings and closings. To determine whether

the long-range memory found for the simulated data is its

intrinsic feature, the simulated data were shuffled, and the

Hurst exponent was calculated for a randomized time ser-

ies. Table 4 shows the values of the drift-regulating term,

the mean values of open state probability, mean open and

closed dwell times, the Hurst exponent corresponding to

generated time series H, and the Hurst exponent obtained

after shuffling (Hsh).

Eur Biophys J (2012) 41:505–526515

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The mean r2coefficient of linear regression used to

evaluate H for all dwell time series generated by Model 1

was equal to 0.994 (0.996 for the corresponding Hsh) and

for those generated by Model 2—0.987 (0.995 for the

corresponding Hsh).

The values obtained indicate that our models enable

estimation of approximate ionic current characteristics in

quite good agreement with those obtained experimentally

under different external conditions. By tuning the regula-

tory variables Fd(Model 1) or TP (Model 2) one obtains

different open probabilities in the generated time series,

which mimic the behaviour of the experimental time series

under different gate-activating conditions (e.g. membrane

depolarization).

The mean open dwell times generated by Model 2 are

in quantitative agreement with experimental values, and

the opening probability grows with open interval length,

as expected. This tendency is conserved at a qualitative

level in the data generated by Model 1. The long-term

memory in the simulated data is exhibited independently

of po(as expected), and the values obtained for the Hurst

exponent are close to those corresponding to the experi-

mental open and closed dwell time series. After a shuf-

fling procedure, the long-term memory has disappeared

Fig. 9 a Samples of the

original ion current signal

recorded from a single BK

channel over the range of

membrane potentials in fixed

calcium concentration

([Ca2?] = 2 mM). Dashed line

indicates closed state of the

channel. b Current–voltage

relationship for the BK channel

recordings presented in a

Table 3 Mean experimental

values of the Hurst exponent

(H), the open state probability

(po), and mean open and closed

dwell time (Sopenand Sclosed)

obtained at [Ca2?] = 2 mM

under different voltage

conditions

Errors are given as standard

deviations

V (mV)

H ± DHpo± Dpo

Sopen± DSopen(ms)

Sclosed± DSclosed(ms)

-80 0.70 ± 0.050.17 ± 0.020.70 ± 0.100.46 ± 0.10

-600.76 ± 0.020.33 ± 0.071.00 ± 0.300.60 ± 0.20

-40 0.63 ± 0.030.63 ± 0.031.30 ± 0.150.80 ± 0.10

-200.73 ± 0.04 0.78 ± 0.071.20 ± 0.200.25 ± 0.10

20 0.82 ± 0.050.86 ± 0.07 1.90 ± 0.350.25 ± 0.10

400.63 ± 0.04 0.92 ± 0.01 2.80 ± 0.15 0.20 ± 0.05

600.75 ± 0.08 0.94 ± 0.04 2.85 ± 0.850.65 ± 0.35

800.77 ± 0.040.95 ± 0.06 3.10 ± 0.400.25 ± 0.10

516 Eur Biophys J (2012) 41:505–526

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(Hsh& 0.5), which means that it is an inherent property

of the considered system. The open and closed state

dwell-time distributions obtained for the time series

generated by Models 1 and 2 were plotted on a log–log

scale in Figs. 10 and 11, respectively, and compared with

the experimental distribution.

As follows from Fig. 10 Model 1 generates a power law

dependence in the closed dwell-time distribution. This

result coincides qualitatively with the empirical data and

the literature suggestions that this kind of dependence

should be expected at least in some cases (Goychuk and

Ha ¨nggi 2002, 2003). In contrast, open dwell time distri-

bution has an exponential tail. In both cases short dwell

times are overestimated, and long dwell times are under-

estimated in comparison with experimental data.

Model 2 enables reconstruction of a correct open dwell

time distribution, but the closed dwell time distribution is

no longer power law, and is characterized by an expo-

nential tail.

Values of the potential barrier energy separating open

and closed states (UTP- UTP-1.5) equal to 1.0 kT (Model 1)

and 0.2 kT (Model 2), and the diffusion coefficient of the

density fluctuations equal to 600 (Model 1) and 1,200

(Model 2) times the diffusion coefficient of reaction coor-

dinate were set to generate a time series which gave the

current characteristics similar to the experimental charac-

teristics and had long-term memory as observed in the

measurements. The assumed height of the activation barrier

is not very large, which concurs with the diffusive nature of

the gate dynamics, in which a rather smooth transition from

open to closed conformation is expected. Moreover, the

energy structure of the conformational space expressed by a

smooth function without significantly differing peaks looks

quite reasonable from a biological viewpoint, taking into

account the existence of a large number of similar protein

allosteric states with similar energy.

Conclusion and outlook

In this work we have proposed two random-walk models of

BK channel gate dynamics with long-term memory. Their

construction was motivated by Hurst analysis applied to the

experimental data measured in the voltage range from -80

to 80 mV and for a Ca2?concentration of approximately

2 mM, supported by the results obtained by other authors

under different experimental conditions (Bandeira et al.

2008; Barbosa et al. 2007; Varanda et al. 2000), which

suggest that intrinsic long-term memory exists in the series

of subsequent single channel state dwell-times.

Accordingtotheexperiments,themechanismresponsible

for the long-term memory seems not to be directly related to

thevoltageandcalciumsensors’activation(butmay,still,be

related to their activity). Taking into account the biological

context of the BK channel system, it is reasonable to assume

that thermal fluctuations of the activation gate machinery by

itself could be responsible for the spontaneous transitions

betweentheopenandclosedstatesofthechannelunderfixed

conditions of the activating voltage and [Ca2?]. Neverthe-

less, the simple random fluctuations of an activation gate are

not enough to produce the long-term memory, and a degree

of synchronicity is required.

Table 4 The values of characteristics arising from the gating stimulus, and the mean values and standard deviations (SD) of open state

probability (po)

Fd(kT/rcu)

po± SD

H ± SD

Sopen± DSopen(ms)

Sclosed± DSclosed(ms)

Hsh± SD

Model 1

0.400.15 ± 0.010.74 ± 0.020.32 ± 0.011.75 ± 0.15 0.51 ± 0.01

0.20 0.25 ± 0.010.79 ± 0.01 0.46 ± 0.021.38 ± 0.110.52 ± 0.01

0.000.50 ± 0.010.82 ± 0.01 0.74 ± 0.07 0.74 ± 0.070.52 ± 0.01

-0.200.74 ± 0.01 0.79 ± 0.01 1.25 ± 0.09 0.43 ± 0.200.51 ± 0.01

-0.40 0.85 ± 0.01 0.73 ± 0.011.62 ± 0.04 0.31 ± 0.01 0.53 ± 0.01

TP

po± SD

H ± SD

Sopen± DSopen(ms)

Sclosed± DSclosed(ms)

Hsh± SD

Model 2

14 0.16 ± 0.020.69 ± 0.010.63 ± 0.013.25 ± 0.440.51 ± 0.01

7 0.32 ± 0.020.71 ± 0.01 1.35 ± 0.082.93 ± 0.46 0.53 ± 0.01

00.50 ± 0.010.72 ± 0.02 2.16 ± 0.282.13 ± 0.25 0.52 ± 0.01

-7 0.68 ± 0.02 0.71 ± 0.01 2.87 ± 0.45 1.32 ± 0.080.52 ± 0.01

-140.85 ± 0.020.68 ± 0.013.79 ± 0.660.64 ± 0.020.51 ± 0.01

Hurst exponents: H for original, and Hshfor shuffled simulated time series. The aforesaid values were calculated as means from five time series

(Model 1, Model 2) for each stimulus-regulatory value

Eur Biophys J (2012) 41:505–526517

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Fig. 10 Closed (a) and open

(b) residence time distribution

for simulated time by use of

Model 1, and recorded

experimentally at

[Ca2?] = 2 mM and

V = 80 mV. Simulation data

are given in Table 1, and the

drift coefficient was chosen

appropriately to obtain the open

state probability possibly close

to experimental data. The time

unit for modelled data was

rescaled to the experimental

value. The fitting procedure was

performed for experimental

data, first, in linear coordinates;

the results were then

transformed to the logarithmic

scale

Fig. 11 Closed (a) and open

(b) residence time distribution

for simulated time by use

of Model 2, and recorded

experimentally at

[Ca2?] = 2 mM and

V = 80 mV. Simulation data

are given in Table 2, and the

threshold point was chosen

appropriately to obtain the open

state probability possibly close

to experimental data. The time

unit for modelled data was

rescaled to the experimental

value. The fitting procedure was

performed for experimental

data, first, in linear coordinates;

the results were then

transformed to the logarithmic

scale

518 Eur Biophys J (2012) 41:505–526

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We have shown how to reproduce the long-term corre-

lations in the gate dynamics by membrane thickness fluc-

tuations, which affect the available size of conformational

space and/or induce stress on the gating segments. The

slow thermal fluctuation of the membrane density sur-

rounding the activation gate was incorporated by intro-

duction of the synchronous diffusion space boundary

fluctuations in Model 1, and the drift force fluctuations in

Model 2. The synchronicity is thought to originate from a

channel reaction to squeezing or stretching. For example, if

the cell membrane underlies squeezing, a part of the

membrane (here: the channel’s surrounding) increases its

density. This also affects the channel gate by narrowing the

distribution of its conformational states.

The part of the channel protein responsible for chan-

nel gating has notably smaller mass than the whole

channel protein immersed in the cell membrane. This

difference manifests itself in the fluctuation time scales

of the channel gate and its surrounding. To illustrate a

connection between the physical and theoretical proper-

ties of the channel system consider the following

reasoning:

Within a random walk framework, the diffusion coeffi-

cient D can be described by the formula:

D / v ? d ¼ v2? s

where v is the velocity of a considered object, d denotes the

mean free path, and s is the mean time between subsequent

collisions with the environment.

The frequency of collisions is proportional to the

object’s surface (S), so s is inversely proportional to S.

Taking this into account, and the proportionality between

the mass and volume (m * V * S3/2), the following

relationship is valid:

ð17Þ

s / m?2=3:

According to the equipartition theorem:

ð18Þ

kT /1

2mv2;

ð19Þ

where k is the Boltzmann constant and T is temperature.

By combining Eqs. (17)–(19), one obtains the relation-

ship between the mass and diffusion coefficient:

?

Thus, the relationship between the diffusion coefficients

(and time scales) used in our models corresponds to the

ratio of the masses of the considered channel elements:

?

(subscripts: S, ‘‘surrounding’’; CG, channel gate).

m /

2kT

D

?3=5

:

ð20Þ

mS

mCG¼

DCG

DS

?3=5

ð21Þ

The models predict a ratio between the gate’s diffusion

coefficient and diffusion coefficient of the surroundings

expressed by

DB¼ 600 and

Model 1 and Model 2, respectively. The problem, however,

is that such a gate diffusion coefficient may not necessarily

be the ‘‘true’’ gate diffusion coefficient, and it may be an

apparent one observed because of a low sampling rate in a

self-similar gating process (Liebovitch and Toth 1990).

Considering the S6-helix of a BK channel of r = 2.5 A˚

radius, its diffusion coefficient in the cell membrane should

take a value similar to D & 4 9 10-8cm2/s = 4 9

10-12m2/s (Almeida and Vaz 1995). To estimate our

apparentgatingdiffusioncoefficientweneedtoapproximate

ourreactioncoordinateunit(rcu)bysomephysicalquantity.

ItseemsreasonabletoassumethattheS6-helixmayfunction

as the channel gate. The channel vestibule diameter is

approximatelyequaltod & 20 A˚(Cuietal.2008).Fromthe

symmetry of the BK channel system (which is a tetrameric

protein)oneS6-helixhasad/2 = 10 A˚rangeofmovements.

In our simulation, this distance is projected on a lattice of

length equal to 2BMAXlength (Model 1). Thus:

DS

DCG¼DRC

DS

DCG¼DRC

DDF¼ 1,200 for

1rcu ?

d

2 ? 2BMAX

¼ d ¼ 0:25˚ A

ð22Þ

i.e. 1 rcu corresponds to 0.25 A˚.

The simulation time step length is equal to s = 5 9

10-5s (because of the experimental sampling frequency of

20 kHz), which renders the gate diffusion coefficient DCG

estimate:

DCG¼d2

6s¼ 1:04 ? 10?17m2

s

??

:

ð23Þ

This value differs by six orders or magnitude from the

expected value of DCG. This difference arises from the s

value restricted by the sampling frequency. Although more

rapid channel gate fluctuations are not recorded, one cannot

preclude their existence. According to Liebovitch and Toth

(1990), the kinetic properties of patch-clamp recordings

may have a self-similar nature. So, it is justified to consider

the DCGas an apparent diffusion coefficient resulting from

the observation time scale s, which should be higher (lower

s, fixed d in Eq. (23)) on the thermal fluctuation scale.

The diffusion coefficient of the channel’s surrounding

DSwas assumed in the simulation to satisfy Eq. (4) (Model

1), thus DSmay be estimated as DCG/600 = 1.73 9 10-20

h i

between the actual mass of the fluctuating membrane (mm)

and the channel gate (m), which is:

m2

s

: From Eq. (21), we can estimate the relationship

mm

m¼

D

DS

??3=5

¼ 1:04 ? 105

ð24Þ

Eur Biophys J (2012) 41:505–526 519

123

Page 16

To check whether this large ratio is acceptable

physically,one may estimate

membrane patch used in the patch-clamp experiment,

which could act as a ‘‘fluctuating surrounding’’ of the

assumed channel gate. The volume of an S6-helix V

(channel gate) may be approximated as a cylindrical

volume spanning the membrane, which is:

the volumeof the

V ¼ p ? r2? dm? p ? ð2:5Þ2? 70˚ A3¼ 1;374˚ A3

where dm= 70 A˚, is the average membrane thickness.

The volume of the whole-cell membrane patch Vwmis

described as:

ð25Þ

Vwm¼ Swm? dm¼ 108? 70˚ A3¼ 7 ? 109˚ A3

where Swm denotes the surface area of the whole cell

membrane patch, Swm= 108A˚2(Sakmann and Neher

1995).

The ratio of volumes yields Vwm/V = 5.09 9 106. By

use of this result, and the ratio of masses mm/m = 1.04 9

105, one can infer that 2 % of the whole patch volume

should be sufficient to act as the fluctuating ‘‘channel

surrounding’’. Such a result seems reasonable, and it may

confirm, in some sense, the validity of our models.

The models can be further verified experimentally by

measuring the membrane thickness fluctuations (possibly

by changes in the membrane capacitance) simultaneously

with the current recordings. In such an experiment, our

model predicts a correlation between membrane thickness

and local dwell time. A positive correlation between

membrane thickness and dwell time would indicate

Model 1, whereas a negative correlation would indicate

Model 2.

Table 4 shows that the characteristics of the generated

time series remain in a good agreement with those obtained

experimentally (Table 3), in particular:

ð26Þ

1. Our models enable generation of time series with

different open state probabilities, and mean open dwell

times which are in reasonable agreement with exper-

imental values, recorded under different conditions of

channel activating stimuli.

TheHurstanalysisappliedtothesequenceoftheopening

and closing times in the simulated data furnishes mean

values of 0.77 and 0.70 for Model 1, and Model 2,

respectively. This result of the rescaled range analysis

suggest that investigated data are long-term correlated,

and the exponent remains in the allowed error range for

the data (±0.1) (Hurst et al. 1965).

2.

The Hurst exponents for the shuffled series show no

long memory effects. In consequence, it is reasonable to

assume that the trend-reinforcing behaviour is an inherent

feature of the modelled systems.

Considering the experimental data obtained under the

conditions described in the ‘‘Electrophysiology’’ section, it

can be stated that the closed dwell-time distributions have

power law dependence, and the open dwell-time distribu-

tions are characterized by an exponential tail. By use of

Model 1 one can obtain the distributions which qualita-

tively reflect the empirical dependencies. By use of Model

2, qualitative and quantitative agreement in the open dwell

time distributions is reached, but the experimental and

simulated closed dwell time distribution types do not match

exactly.

Details of the voltage and of the calcium sensors’

behaviour, and their effect on channel gate activity, were

reduced in our models. The coupling between sensors and

the gate was introduced in the simplest way possible: a

constant drift acting on the reaction coordinate (Model 1)

or a moveable threshold position (Model 2). As a conse-

quence, our models provide only an approximate descrip-

tion of the dependence of gating on [Ca2?] and voltage, but

the ideas presented can be introduced to more sophisticated

models, for example MWC or HCA; this will, however,

substantially increase their complexity.

Acknowledgments

the Warsaw University of Life Sciences, Warsaw, Poland, for

enabling us to carry out all patch-clamp experiments to obtain the

original time series of ionic current recorded from BK potassium

channels. The authors would like to thank The Ministry of Science

and Higher Education for providing financial support under project

N N508 409137.

We are grateful to Professor K. Dołowy from

Open Access

Creative Commons Attribution License which permits any use, dis-

tribution, and reproduction in any medium, provided the original

author(s) and the source are credited.

This article is distributed under the terms of the

Appendix 1

Steps of the Hurst exponent evaluation procedure:

1.The given time series T1, T2,…,TN of length N is

divided into d subseries of n elements, such that d ?

n ¼ N: The subseries obtained are denoted Im

(m = 1,…,d) and their elements as tm,k(k = 1,…,n).

The mean value Emand the standard deviation Smfor

each subseries Imare evaluated.

For each Im an appropriate mean adjusted series

Xm= xm,1,…,xm,dis found, in which the elements are

defined as xm,k= tm,k- Em.

The cumulativedeviate

ym;k¼Pk

…,Yn,m) - min(Y1,m,…,Yn,m) are found.

2.

3.

4.series

Ym

ofelements

j¼1xm;kare calculated.

ranges

Rm

5. The

described as:

Rm= max(Y1,m,

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Page 17

6. The rescaled range

Rm

Smfor each of the subseries is

Rm

Sm

calculated. Then, the mean value

D E

for the current

division of the original time series is evaluated.

All aforementioned steps are then repeated for different

lengths (n) of subseries. (What is often seen in the literature

is that the lengths of subseries are chosen to be equal to the

powers of 2 (n = 2p, where p = 1, 2, 3,…)).

The resulting relationship between

Rm

Sm

D E

and n is

described by power law scaling of the form:

?

where H denotes the Hurst exponent and c is a constant.

Taking the logarithm from Eq. (27) one obtains:

?

The Hurst exponent may be evaluated by use of the

least-squares linear regression method, and its value is

equal to the slope of the plot of Eq. (28) as illustrated in

Fig. 12.

When one time series is analysed the goodness of fit (R2)

may be used as the error of the estimated Hurst exponent

value. However, in the case when one has at least a few

repetitions of the experiment, e.g. a series of independent

measurements carried out under the same conditions (in

our studies these are the patch-clamp time series for five

patches on the same voltage values) it is better to use a

standard deviation from the mean value obtained for the

repetitions as a measure of the Hurst coefficient error. A

fitting error for one series of finite length could not

resemble the properties of a set of analogous series,

because of the possibility of over or underestimation,

owing to the poorly averaged terms in the fitting at large

Rm

Sm

?

n

¼ c ? nH

ð27Þ

log

Rm

Sm

?

n

??

¼ H ? logðnÞ þ logðcÞð28Þ

n values which are likely to perturb randomly the linearity

of the data.

Appendix 2

The Model 1 simulation procedure:

1. The lattice with the maximum number of nodes equal

to 2BMAXis set.

The initial positions of the reaction coordinate x and

the fluctuation boundaries B1 and B2 are set. The

threshold point (TP) is equal to 0. The B1 and B2 are

symmetrically located around the TP.

The potential function (U(x)) described by Eq. (7) is

associated with the lattice.

The position of the reaction coordinate is randomly

changed for one step length in one time step, with the

probability of movement to the right p and to the left

q described by Eqs. (5) and (6), respectively.

The position of the reaction coordinate relative to the

TP is checked. If the RC is at the right-hand side of the

TP, the open state is recognized. Otherwise the closed

state is stated.

If the RC reaches the B1 or B2 positions during its

random walk, it is reflected to its previous position.

Steps 4, 5, 6 are repeated for a number of time steps,

which is determined by Eq. (4).

The boundaries B1 and B2 are randomly and synchro-

nously moved for one step length toward or away from

the TP with equal probability. (If B1 reaches BMINor

TP - 1 positions, it is reflected to its previous position.

Analogously,ifB2reachesBMAXorTP ? 1positions,it

is reflected to its previous position.)

Steps 4–8 are repeated for a desired time series length.

2.

3.

4.

5.

6.

7.

8.

9.

The Model 2 simulation procedure:

Fig. 12 The Hurst exponent

value (H) may be obtained by

plotting Eq. (28) and evaluating

its slope. In the figure, H is

equal to 0.77

Eur Biophys J (2012) 41:505–526521

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Page 18

1. The lattice with the maximum number of nodes equal

to 2B2 and the position of the threshold point TP are

set.

The initial position of the reaction coordinate x is set.

The potential function (U(x)) described by Eq. (15) is

associated with the lattice. The initial ‘‘synchronous’’

drift is set to be equal to 0.

The position of the reaction coordinate is randomly

changed for one step length in one time step with the

probability of movement toward the TP (pT) and away

from it (qT) described by Eqs. (5) and (6).

The position of the reaction coordinate relative to the

TP is checked. If the RC is at the right-hand side of the

TP, the open state is recognized. Otherwise the closed

state is stated.

If the RC reaches the B1 or B2 positions during its

random walk, it is reflected to its previous position.

Steps 4, 5, 6 are repeated for a number of time steps,

which is determined by Eq. (4).

2.

3.

4.

5.

6.

7.

8. The potential slopes (and, as a consequence, the value

of the synchronous drift force described by Eq. (16)) is

randomly changed by a constant ±|Dk| with equal

probability.

Steps 4–8 are repeated for a desired time series length.9.

Appendix 3

The model variables are determined by use of the gradient

optimization technique. Optimization was performed until

the sum of relative errors is less than 10 %. The values

chosen are provided in Tables 1 and 2.

To support the quality of the values found we provide

the dependencies of the total error in the model variables

when one of them is being changed and the rest are kept

constant and equal to the optimum values (Figs. 13, 14, 15,

16, 17, 18, 19). The error plots, being realizations of a

stochastic process (they are evaluated after a simulation

Fig. 13 The total error as a

function of BMAX(Model 1).

The values of all other variables

are given in Table 1

Fig. 14 The total error as a

function of the boundaries and

reaction coordinate time scales

ratio (Model 1). The values of

all other variables are given in

Table 1

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Fig. 15 The total error

(E) (values on the left Y axis) as

a function the height of the

potential barrier (Model 1). The

appropriate error components

corresponding to the Hurst

exponent (EH) are represented

by the dashed line (values on

the right Y axis). The values of

all other variables are given in

Table 1

Fig. 16 The total error as a

function of the drift force’s

increment |Dk| (Model 2). The

values of all other variables are

given in Table 2

Fig. 17 The total error as a

function of the boundaries and

reaction coordinate time scales’

ratio (Model 2). The values of

all other variables are given in

Table 2

Eur Biophys J (2012) 41:505–526523

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run), reveal some variability between realizations of the

simulation, which can be characterized by a standard

deviation of ±2 %.

On each of the figures, the chosen value is denoted by a

dot.

From Fig. 15, it could be roughly stated that the total

errors in the range of potential barriers from 0.2 to 1.0 kT

maintain quite similar values. In the simulations, we have

chosen its value equal to 1.0, because for lower values the

error component corresponding to the Hurst exponent (EH)

was relatively higher, as shown by the dashed line in

Fig. 15. Minimizing the EHis thought to be important,

taking into account the main objectives of the model.

Considering the errors connected with |Dk| estimation,

although the total error was lower for |Dk| = 0.010 than for

|Dk| = 0.005 (Fig. 16), we have regarded the second value

as optimum, because of the corresponding Hurst exponent

errors, which were equal to 0.066 and 0.053 for

|Dk| = 0.010 and 0.005, respectively.

Analysing the total error dependency on the time scales’

ratio (Fig. 17), one can state, that the ratios below 600 are

characterized by relatively high total errors. In the range up

to 1,800 one can observe errors of similar magnitude, which

average to about 13 %. Then, for ratios larger than 1,800 the

error grows again. Because the exact plot in range between

600 and 1,800 has variability depending on the simulation

realization (it is a random variable), we have chosen the

middle value of this interval, i.e.600þ1;800

2

¼ 1,200.

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