Breathing of voltage dependent anion channel as revealed by the fractal property of its gating

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Physica A 386 (2007) 573–580 www.elsevier.com/locate/physa

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 selfsimilarity 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 the conformational 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. Corresponding author.

E-mail address: [email protected] (S. Ghosh). These authors contributed equally to the work. 2 Present address: EPH Lab, Department of Pharmacology, NDDR, R & D III, Ranbaxy Research Laboratories, Plot No. 20, Sector 18, Gurgaon, Haryana 122015, India. 1

0378-4371/$ - see front matter r 2007 Published by Elsevier B.V. doi:10.1016/j.physa.2007.06.049

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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-similarity and 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 ¼ SD, where D is the dimension. Hence, D ¼ log N/log S. Similarly, the definition of the fractal dimension of a self-similar object is written as fractal dimension ¼ 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) ¼ 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 (2.5–3 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]. VDAC’s crucial role in apoptotic cell death [11–14] and synaptic transmission [15–17] 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) for analyzing 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: Y ðkÞ ¼

k X ½IðtÞ  I ave , t¼1

(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 ¼ 0 to 12,000 is measured. Then the characteristic size of fluctuation F(n) is measured using the formula given below: " #1=2 N X 2 F ðnÞ ¼ ð1=NÞ fY ðkÞ  Y n ðkÞg : (2) k¼1

By the same way F(n) for different box sizes (n ¼ 100, 250, 500, 1000, 1500, 2000, 3000, 4000, 6000, 12,000) are calculated. Then the log–log 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: 1d keff ðteff Þ ¼ Ateff .

(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 of keff and teff, a graph of keff 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 voltagedependent 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 DFA of 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 ¼ 1500) of VDAC gating time trace as in Fig. 1(B) (sampled at frequency 1 kHz). Figs. 3(A) and (B) show the log–log 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 log–log 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 log–log 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 (0–12,000 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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Fig. 1. Continuous current trace of rat brain VDAC where sampling frequency is (A) 100 Hz at +25 mV, (B) 1 kHz at +25 mV, (C) 100 Hz at +15 mV. The medium consisted of 500 mM KC1, 10 mM Hepes, and 5 mM MgCl2 (pH 7.4).

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30000 20000 10000 0 -10000 -20000 -30000 0

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Fig. 2. Integrated time series of the current trace of 12,000 ms at +25 mV, sampling rate 1 ms. Plot of Y(k) versus k divided into boxes of equal length n ¼ 1500 . A least square is drawn in each box.

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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 [22–24]. 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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Fig. 4. Plot of keff vs. teff at +25 mV applied voltage in log–log scale. The linear fit of the plot has the slope value of 0.863970.03. The fractal dimension ‘‘d’’ is 1.8670.03.

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 relatively large 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] VDAC’s 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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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 [25,31–34]. 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 meaningful explanation 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 financial assistance. References [1] L.S. Milescu, G. Akk, F. Sachs, Maximum likelihood estimation of ion channel kinetics from macroscopic currents, Biophys. J. 88 (2005) 2494–2515. [2] S.M. Bezrukov, M. Winterhalter, Examining noise sources at the single-molecule level: 1/f noise of an open maltoporin channel, Phys. Rev. Lett. 85 (2000) 202–205. [3] A.L. Goldberger, L.A.N. Amaral, J.M. Hausdorff, P.C. Ivanov, C.-K. Peng, H.E. Stanley, Fractal dynamics in physiology: alterations with disease and aging, Proc. Nat. Acad. Sci. USA 99 (2002) 2466–2472. [4] L.S. Liebovitch, D. Scheurle, M. Rusek, M. Zochowski, Fractal methods to analyze ion channel kinetics, Methods 24 (2001) 359–375. [5] B.J. West, V. Bhargava, A.L. Goldberger, Beyond the principle of similitude: renormalization in the bronchial tree, J. Appl. Physiol. 60 (1986) 1089–1097. [6] J.B. Bassingthwaighte, J.H.G.M. van Beek, Lightning and the heart: fractal behavior in cardiac function, Proc. IEEE 76 (1988) 693–699. [7] D. Paumgartner, G. Losa, E.R. Weibel, Resolution effect on the stereological estimation of surface and volume and its interpretation in terms of fractal dimensions, J. Microsc. 121 (1981) 51–63. [8] J. Feder, Fractals, Plenum Press, New York, 1988. [9] H. Takayasu, Introduction to fractals, in: H. Takayasu (Ed.), Fractals in the Physical Sciences, Manchester University Press, Manchester and New York, 1990, pp. 1–10. [10] V. De Pinto, M. Tommasino, R. Benz, F. Palmieri, The 35 kDa DCCD-binding protein from pig heart mitochondria is the mitochondrial porin, Biochim. Biophys. Acta 813 (1985) 230–242. [11] J. Banerjee, S. Ghosh, Bax increases the pore size of rat brain mitochondrial VDAC in the presence of tBid, Biochem. Biophys. Res. Commun. 323 (2004) 310–314. [12] S. Shimizu, M. Narita, Y. Tsujimoto, Bcl-2 family proteins regulate the release of apoptogenic cytochrome c by the mitochondrial channel VDAC, Nature 399 (1999) 483–487.

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