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Physica A 386 (2007) 573580

Breathing of voltage dependent anion channel as revealed by the

fractal property of its gating

Smarajit Manna1, Jyotirmoy Banerjee1,2, Subhendu Ghosh

Department of Biophysics, University of Delhi South Campus, Benito Juarez Road, New Delhi 110021, India

Received 22 December 2006; received in revised form 19 May 2007

Available online 14 August 2007

Abstract

The gating of voltage dependent anion channel (VDAC) depends on the movement of voltage sensors in the

transmembrane region, but the actual mechanism is still not well understood. With a view to understand the phenomenon

we have analyzed the current recordings of VDAC in lipid bilayer membrane (BLM) and found that the data show self-

similarity and fractal characteristics. We look for the microscopic and molecular basis of fractal behavior of gating of

VDAC. A model describing the oscillatory dynamics of voltage sensors of VDAC in the transmembrane region under

applied potential has been proposed which gives rise to the aforesaid fractal behavior.

r 2007 Published by Elsevier B.V.

Keywords: VDAC; Fractal; Voltage sensor; Bilayer electrophysiology; DFA

1. Introduction

The dynamics of ion channels are traditionally studied through time series data analysis. The latter involves

techniques like calculation of mean, standard deviation, analysis of properties of histogram, and classical

power spectrum analysis [1]. In doing so it is usually considered that the time series are linear, stationary, and

equilibrium in nature [2], whereas in reality these are nonlinear, non-stationary, and non-equilibrium.

Recently a new technique, called fractal analysis, has been introduced widely for analyzing the time series of

various systems. Goldberger et al. [3] used this technique in the time series of heartbeat and gait recordings of

healthy and diseased human beings. Liebovitch et al. [4] used the fractal method to analyze ion channel

kinetics of the time series of a single ATP sensitive potassium channel from rat pancreatic b cells. Ion channel

opening and closing occur due to change in conformational states. The kinetics of transition between theconformational states has a great biological importance as it provides the information on the molecular

structure and the function of the ion channel protein. Fractal model is a more appropriate signature

of ion channel kinetics than the traditional description, i.e. a finite number of discrete states. This fractal

model represents the continuum of states and indicates that the rate constants are not constants in reality.

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0378-4371/$- see front matterr 2007 Published by Elsevier B.V.

doi:10.1016/j.physa.2007.06.049

Corresponding author.

E-mail address: [email protected] (S. Ghosh).1These authors contributed equally to the work.2Present address: EPH Lab, Department of Pharmacology, NDDR, R & D III, Ranbaxy Research Laboratories, Plot No. 20, Sector 18,

Gurgaon, Haryana 122015, India.
http://www.elsevier.com/locate/physahttp://localhost/var/www/apps/conversion/tmp/scratch_3/dx.doi.org/10.1016/j.physa.2007.06.049mailto:[email protected]:[email protected]://localhost/var/www/apps/conversion/tmp/scratch_3/dx.doi.org/10.1016/j.physa.2007.06.049http://www.elsevier.com/locate/physa

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This reflects that the switching of a protein channel occurs in many different time scales. This new emerging

fractal analysis gives a lot of hidden information about the kinetics of the system, which are not extractable

from the conventional methods.

Fractals have fascinating properties that are present in natural objects like airways in the lungs [5], the

distribution of blood flow in the ever-smaller vessels in the heart [6] and the ever-finer infoldings of cellular

membranes [7]. Geometrical objects can be considered as fractals when they satisfy two criteria: self-similarityand fractional dimensionality. Self-similarity means a whole object is composed of subunits under subunits on

multi-levels and each subunit at different levels is equivalent to the whole object [8], i.e. if any small piece

(subunit) of a fractal object is magnified, it appears similar to the whole object. Self-similarity can occur only if

the structures at a small scale are correlated to the structure at a large scale. The second criterion of a fractal

object, distinguishable from Euclidean object, is that it has fractional dimension [9].

Definition of fractal dimension: Fractal dimension is defined as a generic term for dimension that can take

fractional value. As a general concept of empirical dimension, if N is the number of small pieces that go into

the larger one and S is the scale to which the small pieces compare to the larger one, the relation among N, S,

and D is N SD, where D is the dimension. Hence, D log N/log S.

Similarly, the definition of the fractal dimension of a self-similar object is written as fractal dimension

log(number of self-similar pieces)/log(magnification factor). For example, Sierpinski triangle consists of

3 self-similar pieces each with magnification factor 2. Hence, the fractal dimension is (log3/log2) 1.58.Here we have focused our investigations on voltage dependent anion channel (VDAC) from rat brain

mitochondria. VDAC is an abundant protein in the outer mitochondrial membrane, which forms large voltage

gated pore (2.53 nm) on the membrane and act as a pathway for the movement of substances in and out of

the mitochondria by passive diffusion [10]. Recently we have shown that there is an increase in the pore size of

VDAC in the presence of Bax and tBid proteins, which might be a mechanism of cell death [11]. VDACs

crucial role in apoptotic cell death [1114] and synaptic transmission [1517] has made it an important

therapeutic target for various disorders.

In the present work we have analyzed the single channel gating current time trace of VDAC. The amplitude

of fluctuation in time series of VDAC gating does not grow up with time, which means the time series is

bounded. Furthermore, the time series is non-stationary as the statistical parameters of VDAC time trace are

time varying. These are the key points why we were interested to do detrended fluctuation analysis (DFA) foranalyzing self-similarity parameter (fractal behavior) of the time trace. Moreover, in DFA analysis, a bounded

time series is mapped to a self-similar process by integrating the original time series. In the present paper we

demonstrate that the experimental time series data of gating of VDAC at selected membrane potentials have

self-similarity and fractional dimensionality, hence, the time series have fractal behavior. Consequently, we

have looked into the physical or molecular basis of the fractal properties of the single channel VDAC gating,

which is based on the dynamics of voltage sensors of the channel protein.

2. Materials and methods

Diphytanoyl Phosphatidyl Choline (DPhPC) and cholesterol were obtained from Avanti Polar Lipids,

Birmingham, AL, USA. n-Decane, Hepes, and all other chemicals were purchased from Sigma Chemical Co.(St. Louis, MO, USA).

Purification of VDAC: VDAC was purified from rat brain mitochondria using the method of De Pinto et al. [18].

Reconstitution of VDAC in planar lipid bilayers: VDAC was reconstituted into the planar lipid bilayers

according to the method of Roos et al. [19]. Single-channel currents were recorded at sampling frequency of

1 kHz and the data is further sampled at frequency 100 Hz.

Determination of self-similarity parameter: To measure the self-similarity parameter we have first integrated

the VDAC gating time series, which consists of 12,000 data points within an interval of 1 ms according to the

following formula:

Yk Xk

t1

It Iave, (1)

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where I(t) is the value of current in the time trace at time t and Iave the average current value of the total time

trace. Then the integrated time series is divided into equal boxes of size n to measure the vertical

characteristic scale of the integrated series. A least square is drawn in each box. The difference between

integrated data Y(k) and y-coordinates Yn(k) on the linear fit at t 0 to 12,000 is measured. Then the

characteristic size of fluctuation F(n) is measured using the formula given below:

Fn 1=NXNk1

fYk Ynkg2

" #1=2

: (2)

By the same way F(n) for different box sizes (n 100, 250, 500, 1000, 1500, 2000, 3000, 4000, 6000, 12,000) are

calculated. Then the loglog plot of F(n) versus n is drawn followed by a linear fit and the slope value of the

linear fit (a), which characterizes the self-similarity parameter, is measured. This is called the Detrended

Fractal Analysis (DFA) [3].

Determination of fractal dimension: We have used the method proposed by Liebovitch [20] for calculating

fractal dimension of the time trace of VDAC gating. The following relation is used for the determination of

fractal dimension:

keff

teff

At1deff

. (3)

If a channel remains in a state for at least a certain time teff that it would switch to another state, this measure

is referred to as the effective rate constant keff. The effective time resolution at which we measure the data is

denoted by teff. After measurement ofkeff and teff, a graph ofkeff versus teff was plotted on logarithmic scale,

d is the fractal dimension.

3. Results and discussion

Purified rat brain mitochondrial VDAC, when reconstituted in a planar lipid membrane, showed voltage-

dependent gating. Single channel current trace for VDAC at +25 mV and sampling frequencies 100 Hz and

1 kHz are shown in Figs. 1(A) and (B) respectively. Fig. 1(C) shows the single-channel trace of VDAC at

+15 mV (sampling frequency 100 Hz). As described in the Materials and Methods we have carried out DFAof the above-mentioned time series in order to characterize the self-similarity. Fig. 2 shows the integrated time

series (divided into boxes of equal length n 1500) of VDAC gating time trace as in Fig. 1(B) (sampled at

frequency 1 kHz). Figs. 3(A) and (B) show the loglog plot of characteristic size of fluctuation F(n) versus box

size n at +25 and +15 mV respectively. We found a linear relationship between n and F(n) in these loglog

plots, which indicates the fluctuations in small boxes are related to fluctuations in large boxes, i.e. the presence

of scaling (self-similarity). The slopes of the linear fits (which characterizes self-similarity parameter [3]) are

0.779 and 0.743 at +25 and +15 mV respectively.

Having established the self-similarity of the VDAC gating we calculated its fractal dimension using the

method of Liebovitch [20] as described in Materials and Methods. Fig. 4 shows the loglog plot of keff versus

teff at +25 mV. We found that the logarithm of the effective kinetic rate constant keff as a function of the

effective time scale teff

is linear. It is evident from the figure that the power law as defined in Eq. (3) fits quite

well. A similar result is found for +15 mV. The slope of the straight line gives the measure of fractal

dimension d (1.8670.03). Thus the voltage dependent gating time series of VDAC, as obtained from our

experimental data, satisfies two criteria: self-similarity and fractional dimensionality, hence follows fractal

behavior. It may be mentioned here that the above-mentioned results are obtained from the time series

analysis of segments of the current versus time traces (012,000 ms) at membrane potentials +25 and

+15 mV. However, our observation is that the fractal behavior either breaks or changes the self-similarity

parameter and fractal dimension (multi-fractals) at a different range of segments of the current versus time

trace. Also, a similar property has been proposed earlier for a different ion channel [20].

The discovery of fractal property of single-channel recordings of VDAC suggests a different scenario of the

physical properties of the ion channel protein. Ion channels, like many other proteins, have moving

components that perform useful functions. The channel proteins contain aqueous, ion-selective pore that

crosses the plasma membrane, and they use a number of distinct gating mechanisms to open and close this

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200

2000 4000 6000 8000 10000

80005000 10000

12000

Current(pA)

0

200

Current(pA)

Current(pA)

0

0

Time (ms)

2000 4000 6000 80000

Time (ms)

Time (ms)

Sampling frequency - 100 Hz

Sampling frequency - 1 kHz

Fig. 1. Continuous current trace of rat brain VDAC where sampling frequency is (A) 100Hz at +25 mV, (B) 1 kHz at +25mV,

(C) 100 Hz at +15mV. The medium consisted of 500mM KC1, 10 mM Hepes, and 5 mM MgCl2 (pH 7.4).


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pore in response to biological stimuli, such as the binding of a ligand or a change in the transmembrane

voltage [21]. Voltage dependent ion channel opening follows a very steep dependence on membrane voltage

[22]. In order to allow channels to switch to the open state, gating charges (transmembrane regions containing

charged amino acids on the channel protein) move within the membrane electric field to open the pore [2224].

Voltage gated channels have four-fold symmetry with a central pore domain surrounded by voltage-sensor

regions [25]. Hodgkin and Huxley recorded the steep dependence of channel gating on transmembrane voltage

and argued that it must be due to the movement of some component in the membrane with a substantial

charge or electric dipole moment [24,26], This led to the idea that there is probably a specialized structure, a

voltage sensor that accomplishes the charge movement. It seems unlikely that such a large amount of charge

could be displaced systematically by incidental movements of a few charges here and there in the protein, or by

small angular changes in the dipoles associated with the peptide bonds. Nevertheless, the general principles

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10000 1000100

n

1000

F

(n)

1000

10000

log

F

(n)

log n

Fig. 3. Loglog plot of characteristic size for fluctuation [F(n) vs. box size (n); (A) +25 mV and (B) +15 mV. Slopes of the linear plots

(self-similarity parameter) are (A) 0.779 and (B) 0.743.

0 1500 3000 4500 6000 7500 9000 10500 12000

-30000

-20000

-10000

0

10000

20000

30000

40000

50000

60000

Y(k)

k

Fig. 2. Integrated time series of the current trace of 12,000ms at +25 mV, sampling rate 1 ms. Plot ofY(k) versus kdivided into boxes of

equal length n 1500 . A least square is drawn in each box.


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behind the transmembrane helices upon activation by applied voltage are still under investigation. Here, we

highlight the role of transmembrane movement of the voltage sensors of VDAC in regulation of channel

gating.

VDACs could fold as a 12 b-strand barrel with an a-helix at the N-terminus protruding outside or

interacting with the membrane surface [27]. There lies a lot of controversy over the gating mechanism of

VDAC with respect to its voltage sensors. Although the amino acid residues affecting the voltage sensitivity

of VDAC has been identified using site-directed mutagenesis [28], the mechanism of the voltage sensor activity

of VDAC needs to be studied in detail. The voltage-sensing domain in VDAC is distributed over a relativelylarge region of the protein and the mechanism for voltage gating of this channel requires the movement of a

major fraction of the protein mass across the membrane [28]. These changes might be due to the changes in

position of the voltage sensor regions of VDAC in response to a particular voltage. It is important to note here

that as per the existing model [28] VDACs gating is controlled by the movement of the voltage-sensing

residues in four different transmembrane domains of the channel and not by the gating particles. Keeping in

view that the voltage sensor regions of VDAC are embedded in the transmembrane region [28] and their

movement is responsible for the voltage-dependent gating we propose a mechanism to describe the changes in

the conformational states of VDAC due to the movement of voltage sensors in the transmembrane region.

In VDAC the movement of voltage sensor occurs because they are made up of charged amino acid residues,

thus sensitive for voltage dependent ion channel gating. Hence voltage sensor regions start moving from their

initial positions as soon as the external voltage is applied. As a result, charged voltage sensor regions

experience an imbalance in electrostatic force due to their mutual interactions [29]. Hence fluctuations of

different amplitudes are taking place in the voltage sensor regions depending on the initial conditions of

conformational state. If the kinetic energy of fluctuation is sufficient to overcome the energy barrier between

any two conformational states the transition takes place leading to channel gating. To begin with let us

consider the movement of only one voltage sensor. When a voltage sensor region tends to move from its

equilibrium position, a number of forces are acting on it as follows: (i) a restoring force that acts in the

opposite direction of its displacement due to elastic property of the protein molecule, (ii) a damping force

arising due to viscosity of the medium in which the voltage sensor region is moving, and (iii) a driving

electrostatic force due to the mutual interactions of the voltage sensor regions. The movement of a voltage

sensor, as per our understanding, is analogous to a forced oscillator and that of a set of voltage sensors are like

coupled oscillators. At this juncture, we would like to mention that coupled oscillators in biological systems

have been linked to fractal behavior [30]. Hence, it is expected that the movement of the voltage sensors of

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10010

0.01

0.1

logK

log T

eff

eff

Fig. 4. Plot ofkeff vs. teff at +25 mV applied voltage in loglog scale. The linear fit of the plot has the slope value of0.863970.03. The

fractal dimension d is 1.8670.03.


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VDAC at a particular applied potential would give rise to fractal gating behavior. The self-similarity in the

movement of sensor implies that there is a kind of breathing movement in the channel. This appears to be

an inherent property of VDAC and can be used to characterize the ion channel protein.

Despite tremendous progress in the investigation of voltage-gated ion channels the molecular mechanism

underlying voltage sensing has remained a matter of debate. On the basis of electrophysiological studies, a

number of structural and functional models of the voltage-gated ion channels have been proposed [25,3134].But the process by which gating charges are repositioned has been a subject of intense controversy. It may be

mentioned again that our investigations are limited to selected membrane potentials. Hence, our analysis of

fractal behavior of VDAC gating is not a generalized conclusion, the latter being pending for future.

Use of fractal as a tool to investigate the voltage sensor activity in voltage gated ion channel is easier and

much more effective than the other conventional approaches like X-ray diffraction, NMR, etc., where ion

channel structure is used to determine its functioning. Here we have studied qualitatively the structural

dynamics of VDAC using knowledge of its functional properties. Although the present studies were carried

out with VDAC at selective membrane potentials, these, if found true for other voltages and various ion

channels, will throw light on the mechanism of the voltage dependent gating of voltage gated ion channels in

general. Since there is a lot of controversy on the role of voltage sensor in the gating of VDAC, the present

work will give a structural (dynamical) insight into the movement of voltage sensors, which is analogous to

forced oscillator movement. The coupled oscillator model of the voltage sensors gives a meaningfulexplanation of the functioning of the voltage gated ion channels in the light of large number of conformational

states, i.e. the existence of its fractal behavior. In addition, this work will give us an idea about the kind of

electric field required for its movement in the transmembrane region. Thus it will help in determining the ways

to control the voltage sensor movements in the transmembrane region and hence the regulation of VDAC

gating. That is expected to give insight to various cellular processes like functioning of mitochondria, cell

death, synaptic transmission, etc. Experimental studies on the fractal behavior of VDAC at other membrane

potentials and of similar voltage sensitive ion channels are in progress.

Acknowledgement

Authors thank the Department of Science and Technology, Government of India, for the financialassistance.

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