# Multifractal Properties of BK Channel Currents in Human

## Transcript Of Multifractal Properties of BK Channel Currents in Human

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Multifractal Properties of BK Channel Currents in Human Glioblastoma Cells

Agata Wawrzkiewicz-Jałowiecka,* Paulina Trybek, Beata Dworakowska, and Łukasz Machura

Cite This: J. Phys. Chem. B 2020, 124, 2382−2391

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ABSTRACT: Potassium channels play an important physiological role in glioma cells. In particular, voltage- and Ca2+-activated large-

conductance BK channels (gBK in gliomas) are involved in the intensive

growth and extensive migrating behavior of the mentioned tumor cells;

thus, they may be considered as a drug target for the therapeutic

treatment of glioblastoma. To enable appropriate drug design, molecular

mechanisms of gBK channel activation by diverse stimuli should be unraveled as well as the way that the speciﬁc conformational states of the

channel relate to its functional properties (conducting/nonconducting).

There is an open debate about the actual mechanism of BK channel gating, including the question of how the channel proteins undergo a range of conformational transitions when they ﬂicker between nonconducting (functionally closed) and conducting (open) states. The details of channel conformational diﬀusion ought to have its

representation in the properties of the experimental signal that describes the ion-channel activity. Nonlinear methods of analysis of

experimental nonstationary series can be useful for observing the changes in the number of channel substates available from

geometrical and energetic points of view at given external conditions. In this work, we analyze whether the multifractal properties of the activity of glioblastoma BK channels depend on membrane potential, and which states, conducting or nonconducting, aﬀect the total signal to a larger extent. With this aim, we carried out patch-clamp experiments at diﬀerent levels of membrane hyper- and depolarization. The obtained time series of single channel currents were analyzed using the multifractal detrended ﬂuctuation

analysis (MFDFA) method in a standard form and incorporating focus-based multifractal (FMF) formalism. Thus, we show the applicability of a modiﬁed MFDFA technique in the analysis of an experimental patch-clamp time series. The obtained results suggest that membrane potential strongly aﬀects the conformational space of the gBK channel proteins and the considered process

has nonlinear multifractal characteristics. These properties are the inherent features of the analyzed signals due to the fact that the main tendencies vanish after shuﬄing the data.

■ INTRODUCTION

The available chemotherapy and radiology treatments turn out to be ineﬀective in the case of gliomas, which are brain tumors arising from glial cells.1,2 Gliomas account for the majority of malignant brain tumors in adults3 and are graded from I to IV, with higher grades being more diﬀerentiated and malignant.2

Grade IV gliomas are called glioblastoma multiforme (GBM)

and exhibit the highest proliferative potential and almost complete resistance to currently available therapies.4,5 The

great majority of patients with GBM do not survive beyond 2

years even when a combination of novel and conventional

therapies, i.e., surgery, chemotherapy, and radiotherapy, was introduced.1,4−6 Focal surgical resection is ineﬀective and

adequate radiotherapy is impossible in glioblastoma due to the

fact that GBM is characterized by extensive invasion, migration, and angiogenesis.7 To enable the development of a more eﬀective therapeutic treatment for glioblastoma, one should

better understand all biological processes taking place at the

molecular and cellular levels in this tumor. Several ion channels

have been implicated in glioblastoma proliferation, migration, and invasion.7

In this work, we focus on the activity of voltage- and Ca2+activated large-conductance K+ channels (BK) obtained from human glioblastoma cells (gBK channels). BK channels are overexpressed in malignant gliomas (in comparison to nonmalignant cortical tissues), and their expression level correlates positively with the malignancy grade of the tumor.8−10 These channels are expressed in speciﬁc isoforms in glioma cells that have slightly diﬀerent characteristics from other BK channel exons (e.g., they are more sensitive to cytosolic concentration of calcium ions11); thus, they are called gBK channels. The gBK

Received: January 21, 2020 Revised: March 3, 2020 Published: March 4, 2020

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channels play an important physiological role in glioma cells:

they program and drive cell growth and extensive migration (also in glioblastoma stemlike cells),7,9,10,12,13 so they

detrimentally facilitate the invasiveness of glioblastoma that

renders them incurable so far.

One can indicate several processes taking place at the

molecular level, in which gBK channels can contribute to the

shape and volume changes of glioma cells during their invasive

migration in a crowded environment (by triggering and motorizing that process).12 First, the distribution of ions aﬀects the water ﬂow across the cell membrane, and consequently, the cell volume, due to the eﬀective osmotic pressure.14 Taking into

account the overexpression of gBK channels in gliomas, these channels can detrimentally aﬀect osmosis and regulation of cell

volume. Second, gBK channels are anticipated to provide the

electrochemical driving force for the ion movement needed for

the release of cytoplasmic water and cell shrinkage, which in turn

facilitates the extensive migrating behavior of glioblastoma cells.

This was indicated by the results provided in the works of Wondergem et al.15,16 Moreover, due to the mechanosensitivity

of gBK channels, they may be involved in the mechanotransduction during cell shape and volume changes.17 Also

deformation or reorganization of the cytoskeleton during an alteration in cell volume or shape may aﬀect the functioning of gBK channels,17 which, as a consequence, should aﬀect the eﬀectiveness of cell migration.

Here, we go beyond the usual description of the changes of

gating kinetics resulting from the electrical stimulation of cell membrane and investigate the eﬀects of membrane depolarization on a channel’s conformational dynamics. Mechanisms of K+ channel activation by diverse stimuli like voltage, Ca2+, Mg2+, H+, HEME, temperature, mechanical strain, etc. still evoke open

discussions among researchers. The authors who model

activation dynamics and gating present many possible scenarios, which diﬀer by the number of available channel states within

both conducting (functionally open) and nonconducting (functionally closed) states’ manifolds; also the mechanics of the state transitions are introduced in the models in a diﬀerent way.18−24 A great breakthrough has been made by resolving the molecular structure of the Ca2+-bound and Ca2+-free BK channels.25−27 It allowed not only for the inference that voltage and Ca2+ sensors are coupled and they can cooperatively inﬂuence the channel pore gate domain, but also these studies

failed to identify a physical gate that could mechanically block the ion ﬂow in the nonconducting state.28 From this point of

view, the pore-forming helices do not form a tight bundle as in some Shaker-like channels to dam K+ transport through the pore. In ref 25, the authors observed four diﬀerent structures for

the BK channel when the channel was neither voltage-activated nor Ca2+-activated. In those terms, the channel would be

expected to be functionally closed. Nevertheless, the inner pore-

forming helices are not so close to each other to prevent potassium ions to access the selectivity ﬁlter. Quite similar

observations are made in the studies on the activation of ligandgated K+ channel from the Slo family.29 Namely, the Na+dependent K+ channel Slo2.2 exhibited eight classes that resemble an open state and two classes being classiﬁed as the closed channel structures at 300 mM Na+. But some open classes

can be nonconductive. The authors also obtained results which

allowed for an inference that stable intermediate conformations

between the closed and open states do not exist. Thus, channel opening is highly concerted and, as a consequence, the open− closed ﬂuctuations occur in a switchlike manner. Due to the fact

that the structures of the Slo 2.2 and Slo 1 channels share some similarities,25,26,29,30 one can expect that at least some of the aforementioned inferences may also refer to the BK channel’s conformational dynamics.

BK channels pass through multiple kinetic states over time during gating. It can be assumed that Hite et al.25,29 have identiﬁed some of the structures corresponding to diﬀerent kinetic states, providing insight into which conformational states’ functionally closed and open states might be adopted.28 Still, some questions arise, among others:

• Where is the actual channel gate that allows for conducting/nonconducting ﬂuctuations at ﬁxed conditions?

• What kind of and how many stable structures of the BK channel protein exist at intermediate Ca2+ and/or voltage levels?

• Are there any diﬀerences in system dynamics in functionally open and closed states, and which ones inﬂuence the system as a whole to a higher extent?

A possible answer for the ﬁrst question is provided by the hydrophobic gating mechanism postulated in ref 31, where the authors conclude that the BK channel does not need a physical gate. Spontaneous ﬂickering between conducting and nonconducting states at ﬁxed conditions can be realized as wetting and dewetting of the channel pore resulting from the fact that the pore can constantly undergo changes in shape and surface hydrophobicity (conformational diﬀusion). The answers for the above questions need broader studies and discussion. However, some clues may be provided here by means of nonlinear analysis of experimental time series describing single channel activity.

As shown in our previous research,17 application of nonlinear methods in the analysis of an experimental nonstationary series describing ion-channel activity can be useful for estimating the changes in the number of channel states available from a geometric and energetic point of view at given external conditions. We claim that channel conformational diﬀusion has its representation in the structure of the signal. The most structurally distant conformations are suspected to aﬀect the complexity of the experimental data at least. In this paper, we would like to continue and broaden the discussion about the conformational diﬀusion of the gBK channel in electrically stimulated membranes based on the results of the multifractal detrended ﬂuctuation analysis (MFDFA).32

For over 30 years, ion-channel recordings have been analyzed paying special attention to their nonlinear, fractal properties.33−38 Here, we not only describe the multifractal nature of the experimental time series of BK channel activity but also present a biological interpretation of the obtained characteristics, which gives an additional insight into the conformational

■dynamics of the channel protein. METHODS

Detrended Fluctuation Analysis (DFA). The DFA method was ﬁrst proposed by Peng in 1994 for investigating the correlation in DNA structure.39 The last years have seen a renewed importance in the application of this method to biological data and also as a technique capable of distinguishing between healthy subjects and heart failure patients.40 This technique relies on the assumption that the signal is inﬂuenced by both short-term and long-term features. For a proper interpretation of the eﬀects hidden behind the internal dynamics, the signal is supposed to be analyzed at multiple

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time scales.41 A brief description of the original DFA algorithm is

presented below. The procedure starts with the calculation of the proﬁle yi as

the cumulative sum of the data xi with the subtracted mean ⟨x ⟩

i

yi = ∑ [xk − ⟨x⟩]

k=1

(1)

Next, the cumulative signal yi is split into ns equal nonoverlapping segments of size s, for which we use powers of 2, s = 2r, r = 4,..., 11. For all segments of size s, v = 1,..., ns, and the local trend yvm,i is calculated. In the standard DFA method, the local trend is calculated by means of the least-squares ﬁt of order m. In

this work, a second-order polynomial m = 2 is used. The variance F2(s, v) as a function of the segment length s is calculated for

each segment v separately.

∑ F2(s, v) ≡ 1 s (ym − y )2

s i=1 v,i v,i

(2)

As the last step, the Hurst exponent H can be determined as the

slope of the regression line of the double-logarithmic dependence log F(s) ∝ H log s of the square root of the average variance

F(s) = 1 ∑Ns F2(s, v) . The exponent H is used as a

Ns v=1

measure of the long-term memory of a time series. Multifractal Detrended Fluctuation Analysis (MFDFA).

The standard DFA methodology presented above can be extended to capture the qth statistical moment of the calculated variance in terms of the scaling function42

looooooijjjj 1

2ns

∑

[F 2(s ,

v)]q / 2 yzzzz1 / q ,

q ≠ 0,

S(q, s) = mooookjj 2ns v=1

{zz

ooooooo exp( 1

2ns

∑ ln[F2(s, v)]),

q = 0.

nooo

4ns v=1

(3)

A similar power law S(q, s) ∼ sH(q) can be utilized to calculate the

q-order generalized Hurst exponent H(q). The latter is required to compute the singularity spectrum for a time series. In the ﬁrst step, the mass exponent τ is determined via the relation τ(q) = qH(q) − 1. Next, the qth-order Hölder exponent h(q), a

quantity that characterizes the singularities, is estimated as a

derivative of the mass exponent, h(q) = dτ(q). Finally, the qth-

dq

order multifractal singularity spectrum D(h) (mf-spectrum) can be constructed as a Legendre transform of the mass exponent

D(h) = qh(q) − τ(q) = q[h(q) − H(q)] + 1

(4)

It results in the usual concave-shaped distribution of the singularity strengths.43

Focus-Based Multifractal Formalism. Multifractal properties of a time series are reﬂected in the generalized Hurst

exponent H(q). The standard method for the estimation of

those properties often results in a nonmonotonic characteristic

of this function, which causes the degeneracy of a singularity spectrum.44,45 It is typically a consequence of the ﬁnite length of the series. The scaling function S constructed from diﬀerent

moments q of the measure depends on the scale of observationsee eq 3. However, if one considers the full length

of the time series, the dependency on the moment q disappears.

It means that regardless of the moment, there exists only one value for the ﬂuctuation function S(q, L) = S(L) = SL. It deﬁnes a

single pair of values (log SL, log s) to which the function log S(q, s) for any moment q supposes to converge with a growing scale s.

We will call this pair of values the focus point. The existence of

this point is the central idea of the focus-based multifractal (FMF) formalism.46

The traditional approach to ﬁnd a singularity spectrum is based on ﬁnding a linear representation of the calculated log S

versus log s characteristics for each moment q. An algorithm, however, does not guarantee that the such-determined q−

dependent linear functions will cross with each other at the focus

point or even will not cross at all (see the blue solid lines in Figure 1 for details). Instead of calculating separate ﬁts for

Figure 1. Presentation of the FMF method. The solid gray lines with dots denote the S(q, s) characteristics obtained with the standard DFA technique. The solid blue lines stand for the linear ﬁt calculated with the classic least-squares method. The dashed orange lines represent the set of lines determined using the gradient descent method with a focus point. Note that all of the dashed lines cross at the focus point (L, SL) marked with a circle, while the standard individually ﬁtted lines miss the focus for the majority of moments q.

selected moments q at a time, one can try to ﬁt the whole family of functions S(q, s) keeping the requirement of sustaining one focus point (see Figure 1). The usual way is to construct a cost (loss) function that would capture the properties of both the focus point and the whole family of the qth-order scaling functions. This function can be deduced from the power law

SS((qL, )s) = ikjjj Ls y{zzzH(q) (5)

If we mark an iteratively updated scaling function with a hat, then the total mean squared error cost function MSE to be minimized reads46

MSE =

1

nq ns

∑ ∑ [ log S(̂ q, s) − log S(q, s)]2

nsnq q s

(6)

where ns is the total number of segments of size s and nq stands for the number of scaling factors q. Next, the scaling functions are found for the data. In the subsequent step, the iterative optimization algorithm, e.g., the ﬁrst-order gradient descent, is utilized to ﬁnd the minimum of the cost function.

The newly obtained family of the linear representation of the scaling functions can be later used to calculate a new generalized moment-wise Hurst exponent function Ĥ (q), which in turn directly yields the singular spectrum D(h). From this point onward, the procedure for the calculation of the singular spectrum is the same as that for the classic MFDFA algorithm.42 The analysis by means of the standard MFDFA and focus-based method was performed using our own authorial Python software, where we implemented the methodology developed by Mukli et al.46

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Figure 2. Samples of the original signal of ionic current recorded from a single gBK channel over the range of electric potential U = −60, −40, and −20 mV on the left side (a) and U = +60, +40, +20 mV on the right-hand side (b). The dotted lines indicate functionally open, i.e., conducting (O) and functionally closed, i.e., nonconducting (C) states of the channel.

Figure 3. Histograms of the original signal of ionic current recorded from a single gBK channel over the range of electric potential U = +60, +40, +20 mV (top panels) and U = −60, −40, −20 mV (bottom panels).

The parameters of the singularity spectra (viz., spectral width Δ, half-width Δ1/2, maximum of a spectrum Hmax, spectral symmetry) obtained for the experimental time series of single

channel currents describe the complexity and multifractal

properties of these data. Such characteristics can be, in turn,

interpreted in terms of the process underlying the observed current ﬂuctuations, which is the conformational dynamics of a

channel protein. An appropriate discussion will be provided in

the Results and Discussion section.

■ MATERIAL Cell Line and Solutions. For all of the measurements,

human glioblastoma cells (U-87 MG cell line) were used. The cells were cultured on Petri dishes in Dulbecco’s modiﬁed Eagle’s medium (HyClone) supplemented with 2 mM L-

glutamine (Gibco), 10% fetal bovine serum (Gibco), 100 units/ mL penicillin, and 100 μg/mL streptomycin (Sigma). The cultures were incubated at 37 °C in 5% CO2-enriched air.

Electrophysiology. Experimental results were recorded

from inside-out patches. The measurements were performed at

room temperature (20−23 °C). In all experiments, symmetrical solutions on either side of the cell membrane were used, which contained the following: 130 mM potassium gluconate, 5 mM KCl, 8 mM 4-(2-hydroxyethyl)-1-piperazineethanesulfonic acid (HEPES), 10 mM glucose, 2 mM CaCl2, 1 mM MgCl2, and 2 mM ethylene glycol-bis(β-aminoethyl ether)-N,N,N′,N′-tetraacetic acid (EGTA), and pH was adjusted to 7.3. The ion currents were recorded using an Axopatch 200B ampliﬁer (Axon Instruments). The experimental data were low-pass-ﬁltered at 5 kHz and transferred to a computer at a sampling frequency of 10 kHz using Clampex 7 software (Axon Instruments).

Channel current recordings were analyzed at ﬁxed pipette potentials of −60, −40, −20, 20, 40, and 60 mV. From all (3−7) experimental time series of channel currents, we selected and further analyzed only those where a single active BK channel was present in a patch. The channel current was measured at time intervals of Δt = 10−4 s. The ionic current measurement error was ΔI = 5 × 10−4 pA. Each experimental time series comprised N = 5 × 105 current values at the applied time resolution of the measurement. Figures 2 and 3 present the exemplary data together with the respective histograms. As one can see in Figure

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2, the probability of conducting state (O) increases with membrane depolarization, which is typical for voltage-dependent channels. Single-channel current amplitude increases as the diﬀerence in membrane potential increases; thus, local maxima in Figure 3 become well separated at both deep depolarization and hyperpolarization.

Event Detection. The MFDFA is carried out on a raw experimental data seriestime series of single-channel currents and also on preprocessed resultsseries of ionic currents recorded during the conducting (open) state of a channel and some current ﬂuctuations recorded during the nonconducting state of a channel. The threshold current value used to identify transitions between the subsequent states is evaluated as given in ref 36. The analysis of single-state data sets was performed to indicate the complexity of the signal within currents corresponding to a single manifold of states and any diﬀerences between the properties of the conducting and nonconducting states, as well as to determine which kind of signal corresponding to conducting or nonconducting states determines the characteristics of the total experimental time

■series of single-channel currents. RESULTS AND DISCUSSION The necessity of applying a modiﬁed MFDFA technique in the form of an FMF analysis is presented in Figure 4. The obtained

Figure 4. Comparison of mf-spectra for standard multifractal formalism (blue dots) and modiﬁed focus-based method (orange crosses) calculated for the representative measurement obtained at membrane hyperpolarization, U = −20 mV. The black and red dots mark, respectively, the maximum of the spectrum and the generalized Hurst exponent for the nondegenerated case.

spectra were calculated by both the methods: standard MFDFA (blue dots) and the focus-based modiﬁcation (orange crosses) (see Figure 4). In the standard method, the degeneration of the spectrum (zigzag type) is found. In other words, a large number of cases are characterized by the inversed singularity spectrum. The nonmonotonic relationship of the generalized Hurst exponent as a function of q precedes obtaining this type of spectrum shape. This situation causes diﬃculties in the interpretation of multifractality of the data and also the proper estimation of spectrum parameters such as maximum of spectrum or the spectrum width.

Comparison of mf-Spectra at Diﬀerent Membrane Potentials. A comparison of mf-spectra calculated by the FMF method is presented in Figure 5. For the investigation of the long time series (500 000 data points), the range of scales s ⊂ [24,213] was selected. The comprehensive analysis of the nature of the BK channel’s multifractality requires the calculation of the

spectra for the following cases: (i) raw data and (ii) data after the shuﬄing operation.

Figure 5 summarizes the spectra obtained for all subsequent

channel states (total signal composed of the experimentally

recorded ionic currents), series of potassium currents

corresponding to the conducting state of a channel, and series of some “leak” currents that correspond to the nonconducting states of a channel at diﬀerent stages of depolarization and

hyperpolarization. During the analysis of the total signal, one can note a marked tendency between the diﬀerent values of

membrane potentials in both cases, at hyperpolarization and

depolarization. The average spectra of the data obtained when

the value of membrane potential was closest to zero in each group (20 and −20 mV) are clearly shifted to the smaller values of h(q). In other words, the spectral maximum is most extended to left at this speciﬁc condition, and then along with increasing

applied potential at membrane depolarization and decreasing

applied potential at membrane depolarization successively

moves toward larger values of h(q) (Figure 5a,b). Considering

the results of the multifractal analysis dedicated to the currents

recorded during the open (conducting) and closed (non-

conducting) states of the channel, one can observe that the results obtained during the channel’s closures are completely

consistent with those corresponding to the total signals. The results characterizing open states are the opposite of the

remaining ones. First, the variability of their spectral width is

substantially smaller. Second, the general trend shows an

increase in spectral width when the membrane potential decreases (only for −40 mV, there is a local minimum). Such results suggest that the total signal’s characteristics are mainly

determined by the recordings obtained during the nonconducting states of a channel. The recognized diﬀerences in spectral distributions allow us to infer that the dynamics of

conformational changes within the conducting and nonconducting states’ manifolds diﬀer signiﬁcantly. Roughly

speaking, single-channel currents recorded when the channel

pore exhibits possibly high conductance retain a self-similar

structure over a range of scales regardless of the membrane

potential (Figure 5b) (which is also notable by the trend-

reinforcing behavior, as measured by Hurst exponents). Whereas current ﬂuctuations recorded when the channel is

supposed to not conduct potassium ions as well as the total signal signiﬁcantly lose their multifractal self-similarity (and the

recorded time series become uncorrelated or even anticorrelated, as shown by values of the Hurst exponents) (Figure 5a,c).

It is also visible in the case of functionally open states that

multifractal spectra have a right truncation. A long left tail

suggests that the time series of channel currents have a multifractal structure that is insensitive to the local ﬂuctuations

with small magnitudes. After the shuﬄing operation (Figure 6), the spectral

distribution is quite diﬀerent, but the most important aspect here is that the spectra are about twice narrower. The eﬀect of

the reduced multifractality after mixing of the data has a large

consequence in the proper interpretation of the source of the

fractal nature of the examined signals. There exist two general sources of multifractality which have

inﬂuence on the shape of the mf-spectrum: (i) the broad

probability density function (pdf), which lies behind the data, and (ii) diﬀerent behaviors of the (auto)correlation function for large and small ﬂuctuations. Furthermore, both situations are possible simultaneously. In case (i), shuﬄing will not change the mf-spectrum; for (ii), it will destroy the eﬀect completely as the

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Figure 5. Raw data calculations: comparison of channel activity characteristics for (a) total recording, (b) functionally open states, and (c) closed states. The black and red dots mark the maximum of the spectrum and the generalized Hurst exponent, respectively.

shuﬄing will erase the possible correlations. When cases (i) and (ii) are present simultaneously, the spectrum will diﬀer from the original one, and the weaker multifractality can be identiﬁed. In our case, we can observe exactly the last mixed situation, and thus we suspect that the multifractality of the data is caused by both the correlation and the broad pdf.

To interpret these results in terms of the considered biological system, one has to note the following facts:

• BK channels are voltage-activated, which means they exhibit more often the conducting state than the functionally closed one at membrane depolarization and tend to retain a nonconducting state at hyperpolarization of the cell membrane. Nevertheless, even at a negative potential, there is a nonzero probability that the channel rapidly opens for a relatively short time (as shown in Figure 2),

• the actual single-channel conductance is determined by two factors. The ﬁrst, and the most detrimental eﬀect, is exerted by the membrane potential. The absolute value of a single-channel current increases as the diﬀerence of the electric potential on both sides of the channel membrane

increases (both toward highly positive values and toward negative ones), as shown in Figure 2a. Higher amplitudes of channel currents at deep-membrane depolarization or hyperpolarization result in broader probability density functions (Figure 2b). Second, the conductance of the channel pore in the open state varies with voltage as a result of the structural changes that a given channel undergoes during voltage activation.25,26,47,48 But these slight changes in geometry can signiﬁcantly inﬂuence the kinetics of switching between conducting and nonconducting states.

Taking into consideration the facts mentioned above, multifractal properties of the analyzed time series of channel currents (total signal) are an inherent feature of the system connected to the dynamics of switching between the channel states, which change with membrane potential but does not depend strictly on the value of current amplitude. The changes in channel currents with voltage pertain mainly to the conducting states of a channel since the nonconducting states form a baseline during the experimental recording and they underlie smaller ﬂuctuations in the examined range of the membrane. Single-channel currents

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Figure 6. Shuﬄed data calculations: comparison of channel activity characteristics for (a) total recording and (b) functionally open states and (c) closed states.

recorded at the conducting states of the channel have multifractal characteristics over a range of scales regardless of the membrane potential (Figure 5b); in contrast, the multifractality within closed states and the total signal vary signiﬁcantly with voltage. As the picture of multifractal properties of the total signal ﬁts to the one obtained for functionally closed states, one can infer that the channel dynamics is mainly inﬂuenced by the dynamics of conformational transitions within the nonconducting states. This inference is compatible with some popular kinetic models of the ion-channel activity,22,49,50 where more kinetic substates correspond to functional closures than openings of a channel. It is possible that the scheme of switching between functionally closed conformations becomes more complex with the increase of the diﬀerence in electric potential on both sides of the membrane, which leads to an eventual widening of the mfspectrum at these conditions.

The conﬁrmed existence of the second source of multi-

fractality of the investigated data, namely, correlations for large and small ﬂuctuations, is also worth noting. The analyzed time series are long-term correlated at all experimental conditions

for channel currents at the conducting states of a channel and at

high diﬀerences of electric potential on both sides of the membranefor nonconducting states, and total signal, as shown by the values of Hurst exponent (Figures 5 and 6).

A strong analogy exists between the multifractal and thermodynamical characteristics. In particular, multifractal spectra can be related to entropy.51−53 In total signal as well as in the series of channel currents in functionally closed states, one can observe shifting of the maximum of multifractal spectra toward higher values when the diﬀerence in electric potential on both sides of the membrane increases (both at depolarization and hyperpolarization). It indicates greater complexity of the signal at highly positive and negative potentials comparing with the data obtained at membrane potentials close to zero, which may suggest an increase in the number of attainable channel substates (mainly within the nonconducting manifold) with absolute value of voltage. The symmetry of the changes in signal multifractality (and, consequently, entropy) occurring both at membrane depolarization and hyperpolarization may be counterintuitive in the case of a voltage-activated channel. The ﬁndings from refs 22, 49, 50, and 54 suggest nonsymmetric nets of conducting and nonconducting states, but there is no information about probabilities of switching between diﬀerent

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substates at diﬀerent voltages and consequently the complexity of conformational switching. Thus, there are no clear presumptions to expect that a channel’s activation should result in a monotonic dependence of multifractal characteristics on the applied voltage (in a whole range of analyzed membrane

■potentials, i.e., from −60 to 60 mV). CONCLUSIONS

In this work, a novel approach of multifractal signal analysis is presented. To the authors’ best knowledge, very few publications can be found that discuss the issue of the multifractal character of a time series by implementing an FMF methodology. An implementation of this unique technique, which is capable of handling empirical signals with a varying degree of heterogeneity, brings a lot of valuable information to the investigation of an ion channel’s activity. This work concludes that the multifractality can be regarded as an inherent feature of the single-channel currents obtained by patch-clamp measurements. The observed multifractal spectra suggest that the characteristics of system dynamics are substantially diﬀerent in functionally open and closed states, and the total signal recorded during experiments is inﬂuenced to a higher extent by the nonconducting states than the conducting ones. It is quite interesting that the symmetric increase of spectrum width and shifting of maximum of mf-spectrum of both the total signal and channel currents recorded during the functionally closed states toward higher values as the diﬀerence in electric potential on both sides of the membrane patch increases. It suggests a higher complexity and entropy of the signal recorded at both strong membrane depolarization and hyperpolarization comparing with the ones obtained at moderate membrane potentials. According to Boltzmann’s deﬁnition of entropy, these results ought to indicate an increase of the attainable substates (stable conformations) mainly in the nonconducting domain with the absolute value of applied voltage. Regardless of the applied voltage, the time series of channel currents recorded at the channel’s conducting conformations are nonrandom but caused by the orderly process exhibiting long-range correlation features.

To sum up, channel dynamics are qualitatively and quantitatively diﬀerent in the case of conducting and nonconducting states of a channel. Assuming that the most distant conformational states from energetic point of view have their representation in the recorded signal (single-channel current), it can be noted that the states obstructing ionic ﬂow through a channel pore are more complex and inﬂuence the multifractality of the total signal to a higher extent than the ones allowing for K+ transport. One should remember that our analysis does not discern between mechanically closed and nonconducting open statesso both groups, physically blocked conformations and suﬃciently narrow ones (implying hydrophobic gating), predominate in shaping the channel’s activity patterns. An interesting task for future investigation can be to carry out a comparative MFDFA analysis of a patch-clamp time series of a single-channel current on a wild-type and genetically modiﬁed BK channel that cannot exhibit relatively narrow conformations of the pore enabling for hydrophobic gating. Such analysis could be used to discriminate between the impacts of both the aforementioned groups of nonconducting states on the total signal. Moreover, the presented multifractal analysis can be a tool of supplemental analysis procedures, e.g., in cases when one should determine to what extent diﬀerent regulatory β subunits can modulate the complexity of channel behavior. The results of

such analysis could help answer the question whether they modulate the relative stabilities of preexisting conformations27 or create new ones.

In cases of glioblastoma, current medical approaches turn out to be almost powerless. Among the challenges to curing primary brain tumors, one can list the development of a precision medicine approach to treating brain tumors. In that aspect, novel approaches should be introduced, which could be based on artiﬁcial intelligence (AI) (e.g., deep learning, neural networks). The AI methods can be used for diagnosing, managing, and designing drugs against gliomas. In the literature, there already exist some reports like ref 55, where the authors present novel AI approaches to predict the grading and genomics from imaging, automate the diagnosis from histopathology, and provide insight into prognosis. Taking into account the therapeutic potential of gBK channel modulators in the treatment of glioblastoma, one could propose some AI methods to determine a group of active substances that could act as a drug against gliomas. Machine learning might be developed with the aim to determine patterns within the experimental data describing ion-channel activity, where some of the classiﬁcation algorithms could be based on the results of an MFDFA analysis. (Our preliminary analyses suggest that multifractal analysis better discriminates singlechannel current from diﬀerent exons of the BK channel than do the kinetic characteristics.) Complexity and multifractality of a signal describing diﬀerent BK channel exons bound or unbound to ligand molecules (speciﬁc modulators) could be one of the factors used in optimizing the structures of potential BK channel modulators used as a drug against glioblastoma.

■ AUTHOR INFORMATION

Corresponding Author Agata Wawrzkiewicz-Jałowiecka − Department of Physical Chemistry and Technology of Polymers, Faculty of Chemistry, Silesian University of Technology, Gliwice 44-100, Poland; orcid.org/0000-0001-6885-3956; Phone: +48 32 237 12 85; Email: [email protected]

Authors Paulina Trybek − Institute of Physics, University of Silesia in Katowice, Katowice 40-007, Poland Beata Dworakowska − Institute of Biology, Department of Physics and Biophysics, Warsaw University of Life SciencesSGGW, Warszawa 02-787, Poland Łukasz Machura − Institute of Physics, University of Silesia in Katowice, Katowice 40-007, Poland

Complete contact information is available at: https://pubs.acs.org/10.1021/acs.jpcb.0c00397

Notes The authors declare no competing ﬁnancial interest.

■ ACKNOWLEDGMENTS

The authors would like to thank Andrew Watson for a kind proofreading of this manuscript. No ﬁnancial support was received for this research.

■ REFERENCES

(1) Aldape, K.; Brindle, K. M.; Chesler, L.; Chopra, R.; Gajjar, A.; Gilbert, M. R.; Gottardo, N.; Gutmann, D. H.; Hargrave, D.; Holland, E. C.; et al. Challenges to curing primary brain tumours. Nat. Rev. Clin. Oncol. 2019, 16, 509−520.

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(2) Abdullaev, I. F.; Rudkouskaya, A.; Mongin, A. A.; Kuo, Y.-H. Calcium-activated potassium channels BK and IK1 are functionally expressed in human gliomas but do not regulate cell proliferation. PLoS One 2010, 5, No. e12304. (3) Goodenberger, M. L.; Jenkins, R. B. Genetics of adult glioma. Cancer Genet. 2012, 205, 613−621. (4) Chinot, O. L.; Wick, W.; Mason, W.; Henriksson, R.; Saran, F.; Nishikawa, R.; Carpentier, A. F.; Hoang-Xuan, K.; Kavan, P.; Cernea, D.; et al. Bevacizumab plus radiotherapy-temozolomide for newly diagnosed glioblastoma. N. Engl. J. Med. 2014, 370, 709−722. (5) Gilbert, M. R.; Dignam, J. J.; Armstrong, T. S.; Wefel, J. S.; Blumenthal, D. T.; Vogelbaum, M. A.; Colman, H.; Chakravarti, A.; Pugh, S.; Won, M.; et al. A randomized trial of bevacizumab for newly diagnosed glioblastoma. N. Engl. J. Med. 2014, 370, 699−708. (6) Wang, H.; Xu, T.; Huang, Q.; Jin, W.; Chen, J. Immunotherapy for malignant glioma: current status and future directions. Trends Pharmacol. Sci. 2020, 41, 123−138. (7) Molenaar, R. J. Ion channels in glioblastoma. ISRN Neurol. 2011, 2011, No. 590249. (8) Liu, X.; Chang, Y.; Reinhart, P. H.; Sontheimer, H. Cloning and characterization of glioma BK, a novel BK channel isoform highly expressed in human glioma cells. J. Neurosci. 2002, 22, 1840−1849. (9) Rosa, P.; Sforna, L.; Carlomagno, S.; Mangino, G.; Miscusi, M.; Pessia, M.; Franciolini, F.; Calogero, A.; Catacuzzeno, L. Overexpression of Large-Conductance Calcium-Activated Potassium Channels in Human Glioblastoma Stem-Like Cells and Their Role in Cell Migration. J. Cell. Physiol. 2017, 232, 2478−2488. (10) Weaver, A. K.; Bomben, V. C.; Sontheimer, H. Expression and function of calcium-activated potassium channels in human glioma cells. Glia 2006, 54, 223−233. (11) Ransom, C. B.; Liu, X.; Sontheimer, H. BK channels in human glioma cells have enhanced calcium sensitivity. Glia 2002, 38, 281−291. (12) Edalat, L.; Stegen, B.; Klumpp, L.; Haehl, E.; Schilbach, K.; Lukowski, R.; Kühnle, M.; Bernhardt, G.; Buschauer, A.; Zips, D.; et al. BK K+ channel blockade inhibits radiation-induced migration/brain infiltration of glioblastoma cells. Oncotarget 2016, 7, 14259. (13) Rosa, P.; Catacuzzeno, L.; Sforna, L.; Mangino, G.; Carlomagno, S.; Mincione, G.; Petrozza, V.; Ragona, G.; Franciolini, F.; Calogero, A. BK channels blockage inhibits hypoxia-induced migration and chemoresistance to cisplatin in human glioblastoma cells. J. Cell. Physiol. 2018, 233, 6866−6877. (14) Baumgarten, C. M.; Feher, J. J. Cell Physiology Source Book; Elsevier, 2001; pp 319−355. (15) Wondergem, R.; Ecay, T. W.; Mahieu, F.; Owsianik, G.; Nilius, B. HGF/SF and menthol increase human glioblastoma cell calcium and migration. Biochem. Biophys. Res. Commun. 2008, 372, 210−215. (16) Wondergem, R.; Bartley, J. W. Menthol increases human glioblastoma intracellular Ca 2+, BK channel activity and cell migration. J. Biomed. Sci. 2009, 16, No. 90. (17) Wawrzkiewicz-Jałowiecka, A.; Trybek, P.; Machura, Ł.; Dworakowska, B.; Grzywna, Z. J. Mechanosensitivity of the BK Channels in Human Glioblastoma Cells: Kinetics and Dynamical Complexity. J. Membr. Biol. 2018, 251, 667−679. (18) Cui, J.; Yang, H.; Lee, U. S. Molecular mechanisms of BK channel activation. Cell. Mol. Life Sci. 2009, 66, 852−875. (19) McManus, O.; Magleby, K. Kinetic states and modes of single large-conductance calcium-activated potassium channels in cultured rat skeletal muscle. J. Physiol. 1988, 402, 79−120. (20) Horrigan, F. T.; Aldrich, R. W. Coupling between voltage sensor activation, Ca2+ binding and channel opening in large conductance (BK) potassium channels. J. Gen. Physiol. 2002, 120, 267−305. (21) Magleby, K. L. Gating mechanism of BK (Slo1) channels: so near, yet so far. J. Gen. Physiol. 2003, 121, 81−96. (22) Geng, Y.; Magleby, K. L. Single-channel kinetics of BK (Slo1) channels. Front. Physiol. 2015, 5, No. 532. (23) Wawrzkiewicz, A.; Pawelek, K.; Borys, P.; Dworakowska, B.; Grzywna, Z. J. On the simple random-walk models of ion-channel gate dynamics reflecting long-term memory. Eur. Biophys. J. 2012, 41, 505− 526.

(24) Wawrzkiewicz-Jałowiecka, A.; Borys, P.; Grzywna, Z. J. On

Application of Langevin Dynamics in Logarithmic Potential to Model Ion Channel Gate Activity. Cell. Mol. Biol. Lett. 2015, 20, 663−684. (25) Hite, R. K.; Tao, X.; MacKinnon, R. Structural basis for gating the

high-conductance Ca 2-activated K+ channel. Nature 2017, 541, 52. (26) Tao, X.; Hite, R. K.; MacKinnon, R. Cryo-EM structure of the

open high-conductance Ca 2.-activated K+ channel. Nature 2017, 541,

46. (27) Tao, X.; MacKinnon, R. Molecular structures of the human Slo1 K+ channel in complex with β4. eLife 2019, 8, No. e51409. (28) Magleby, K. L. Structural biology: Ion-channel mechanisms

revealed. Nature 2017, 541, 33. (29) Hite, R. K.; MacKinnon, R. Structural titration of Slo2. 2, a Na +-dependent K+ channel. Cell 2017, 168, 390−399. (30) Salkoff, L.; Butler, A.; Ferreira, G.; Santi, C.; Wei, A. High-

conductance potassium channels of the SLO family. Nat. Rev. Neurosci.

2006, 7, 921. (31) Jia, Z.; Yazdani, M.; Zhang, G.; Cui, J.; Chen, J. Hydrophobic

gating in BK channels. Nat. Commun. 2018, 9, No. 3408. (32) Kazachenko, V.; Astashev, M.; Grinevich, A. Multifractal analysis of K+ channel activity. Biochem. Suppl. Ser. A 2007, 1, 169−175. (33) Liebovitch, L. S.; Fischbarg, J.; Koniarek, J. P.; Todorova, I.;

Wang, M. Fractal model of ion-channel kinetics. Biochim. Biophys. Acta, Biomembr. 1987, 896, 173−180. (34) Liebovitch, L. S.; Scheurle, D.; Rusek, M.; Zochowski, M. Fractal methods to analyze ion channel kinetics. Methods 2001, 24, 359−375. (35) Fulinś ki, A.; Grzywna, Z.; Mellor, I.; Siwy, Z.; Usherwood, P.

Non-Markovian character of ionic current fluctuations in membrane

channels. Phys. Rev. E 1998, 58, No. 919. (36) Mercik, S.; Weron, K.; Siwy, Z. Statistical analysis of ionic current

fluctuations in membrane channels. Phys. Rev. E 1999, 60, No. 7343. (37) Grzywna, Z. J.; Siwy, Z. Chaos in ionic transport through membranes. Int. J. Bifurcation Chaos 1997, 07, 1115−1123. (38) Siwy, Z.; Ausloos, M.; Ivanova, K. Correlation studies of open

and closed state fluctuations in an ion channel: Analysis of ion current

through a large-conductance locust potassium channel. Phys. Rev. E

2002, 65, No. 031907. (39) Peng, C.-K.; Buldyrev, S. V.; Havlin, S.; Simons, M.; Stanley, H.

E.; Goldberger, A. L. Mosaic organization of DNA nucleotides. Phys.

Rev. E 1994, 49, 1685. (40) Rodriguez, E.; Echeverria, J. C.; Alvarez-Ramirez, J. Detrended

fluctuation analysis of heart intrabeat dynamics. Phys. A 2007, 384, 429−438. (41) Semmlow, J. L.; Griﬀel, B. Biosignal and Medical Image Processing;

CRC Press, 2014. (42) Kantelhardt, J. W.; Zschiegner, S. A.; Koscielny-Bunde, E.;

Havlin, S.; Bunde, A.; Stanley, H. E. Multifractal detrended fluctuation analysis of nonstationary time series. Phys. A 2002, 316, 87−114. (43) Chhabra, A.; Jensen, R. V. Direct determination of the f (α)

singularity spectrum. Phys. Rev. Lett. 1989, 62, No. 1327. (44) Makowiec, D.; Fulinś ki, A. Multifractal Detrended Fluctuation

Analysis as the estimator of long-range dependence. Acta Phys. Pol., B 2010, 41, 1025−1050. (45) Makowiec, D.; Rynkiewicz, A.; Gałaska, R.; Wdowczyk-Szulc, J.; Żarczynś ka-Buchowiecka, M. Reading multifractal spectra: aging by

multifractal analysis of heart rate. Europhys. Lett. 2011, 94, No. 68005. (46) Mukli, P.; Nagy, Z.; Eke, A. Multifractal formalism by enforcing the universal behavior of scaling functions. Phys. A 2015, 417, 150−167. (47) Wawrzkiewicz-Jałowiecka, A.; Borys, P.; Grzywna, Z. J. Impact of

geometry changes in the channel pore by the gating movements on the

channelas conductance. Biochim. Biophys. Acta, Biomembr. 2017, 1859, 446−458. (48) Wawrzkiewicz-Jałowiecka, A.; Grzywna, Z. J. The role of entropic

potential in voltage activation and K+ transport through Kv 1.2

channels. J. Chem. Phys. 2018, 148, No. 115103. (49) Sigg, D.; Bezanilla, F.; Stefani, E. Fast gating in the Shaker K+

channel and the energy landscape of activation. Proc. Natl. Acad. Sci. U.S.A. 2003, 100, 7611−7615.

2390

https://dx.doi.org/10.1021/acs.jpcb.0c00397 J. Phys. Chem. B 2020, 124, 2382−2391

The Journal of Physical Chemistry B

(50) Kim, I.; Warshel, A. Coarse-grained simulations of the gating current in the voltage-activated Kv1. 2 channel. Proc. Natl. Acad. Sci. U.S.A. 2014, 111, 2128−2133. (51) Bunde, A.; Havlin, S. Fractals and Disordered Systems; Springer Science & Business Media, 2012. (52) Chen, Y. Equivalent Relation between Normalized Spatial Entropy and Fractal Dimension, arXiv preprint arXiv:1608.02054. arXiv.org e-Print archive. https://arxiv.org/abs/1608.02054 (submitted Aug 6, 2016). (53) Chen, Y.; Huang, L. Spatial measures of urban systems: From entropy to fractal dimension. Entropy 2018, 20, No. 991. (54) Starek, G.; Freites, J. A.; Bernec̀ he, S.; Tobias, D. J. Gating energetics of a voltage-dependent K+ channel pore domain. J. Comput. Chem. 2017, 38, 1472−1478. (55) Sotoudeh, H.; Shafaat, O.; Sotoudeh, E.; Brooks, M. D.; Bernstock, J. D.; Elsayed, G.; Chen, J.; Szerip, P.; Gazcon, G. C.; Gessler, F.; et al. Artificial intelligence in the management of glioma; Era of personal medicine. Front. Oncol. 2019, 9, No. 768.

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Multifractal Properties of BK Channel Currents in Human Glioblastoma Cells

Agata Wawrzkiewicz-Jałowiecka,* Paulina Trybek, Beata Dworakowska, and Łukasz Machura

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ABSTRACT: Potassium channels play an important physiological role in glioma cells. In particular, voltage- and Ca2+-activated large-

conductance BK channels (gBK in gliomas) are involved in the intensive

growth and extensive migrating behavior of the mentioned tumor cells;

thus, they may be considered as a drug target for the therapeutic

treatment of glioblastoma. To enable appropriate drug design, molecular

mechanisms of gBK channel activation by diverse stimuli should be unraveled as well as the way that the speciﬁc conformational states of the

channel relate to its functional properties (conducting/nonconducting).

There is an open debate about the actual mechanism of BK channel gating, including the question of how the channel proteins undergo a range of conformational transitions when they ﬂicker between nonconducting (functionally closed) and conducting (open) states. The details of channel conformational diﬀusion ought to have its

representation in the properties of the experimental signal that describes the ion-channel activity. Nonlinear methods of analysis of

experimental nonstationary series can be useful for observing the changes in the number of channel substates available from

geometrical and energetic points of view at given external conditions. In this work, we analyze whether the multifractal properties of the activity of glioblastoma BK channels depend on membrane potential, and which states, conducting or nonconducting, aﬀect the total signal to a larger extent. With this aim, we carried out patch-clamp experiments at diﬀerent levels of membrane hyper- and depolarization. The obtained time series of single channel currents were analyzed using the multifractal detrended ﬂuctuation

analysis (MFDFA) method in a standard form and incorporating focus-based multifractal (FMF) formalism. Thus, we show the applicability of a modiﬁed MFDFA technique in the analysis of an experimental patch-clamp time series. The obtained results suggest that membrane potential strongly aﬀects the conformational space of the gBK channel proteins and the considered process

has nonlinear multifractal characteristics. These properties are the inherent features of the analyzed signals due to the fact that the main tendencies vanish after shuﬄing the data.

■ INTRODUCTION

The available chemotherapy and radiology treatments turn out to be ineﬀective in the case of gliomas, which are brain tumors arising from glial cells.1,2 Gliomas account for the majority of malignant brain tumors in adults3 and are graded from I to IV, with higher grades being more diﬀerentiated and malignant.2

Grade IV gliomas are called glioblastoma multiforme (GBM)

and exhibit the highest proliferative potential and almost complete resistance to currently available therapies.4,5 The

great majority of patients with GBM do not survive beyond 2

years even when a combination of novel and conventional

therapies, i.e., surgery, chemotherapy, and radiotherapy, was introduced.1,4−6 Focal surgical resection is ineﬀective and

adequate radiotherapy is impossible in glioblastoma due to the

fact that GBM is characterized by extensive invasion, migration, and angiogenesis.7 To enable the development of a more eﬀective therapeutic treatment for glioblastoma, one should

better understand all biological processes taking place at the

molecular and cellular levels in this tumor. Several ion channels

have been implicated in glioblastoma proliferation, migration, and invasion.7

In this work, we focus on the activity of voltage- and Ca2+activated large-conductance K+ channels (BK) obtained from human glioblastoma cells (gBK channels). BK channels are overexpressed in malignant gliomas (in comparison to nonmalignant cortical tissues), and their expression level correlates positively with the malignancy grade of the tumor.8−10 These channels are expressed in speciﬁc isoforms in glioma cells that have slightly diﬀerent characteristics from other BK channel exons (e.g., they are more sensitive to cytosolic concentration of calcium ions11); thus, they are called gBK channels. The gBK

Received: January 21, 2020 Revised: March 3, 2020 Published: March 4, 2020

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channels play an important physiological role in glioma cells:

they program and drive cell growth and extensive migration (also in glioblastoma stemlike cells),7,9,10,12,13 so they

detrimentally facilitate the invasiveness of glioblastoma that

renders them incurable so far.

One can indicate several processes taking place at the

molecular level, in which gBK channels can contribute to the

shape and volume changes of glioma cells during their invasive

migration in a crowded environment (by triggering and motorizing that process).12 First, the distribution of ions aﬀects the water ﬂow across the cell membrane, and consequently, the cell volume, due to the eﬀective osmotic pressure.14 Taking into

account the overexpression of gBK channels in gliomas, these channels can detrimentally aﬀect osmosis and regulation of cell

volume. Second, gBK channels are anticipated to provide the

electrochemical driving force for the ion movement needed for

the release of cytoplasmic water and cell shrinkage, which in turn

facilitates the extensive migrating behavior of glioblastoma cells.

This was indicated by the results provided in the works of Wondergem et al.15,16 Moreover, due to the mechanosensitivity

of gBK channels, they may be involved in the mechanotransduction during cell shape and volume changes.17 Also

deformation or reorganization of the cytoskeleton during an alteration in cell volume or shape may aﬀect the functioning of gBK channels,17 which, as a consequence, should aﬀect the eﬀectiveness of cell migration.

Here, we go beyond the usual description of the changes of

gating kinetics resulting from the electrical stimulation of cell membrane and investigate the eﬀects of membrane depolarization on a channel’s conformational dynamics. Mechanisms of K+ channel activation by diverse stimuli like voltage, Ca2+, Mg2+, H+, HEME, temperature, mechanical strain, etc. still evoke open

discussions among researchers. The authors who model

activation dynamics and gating present many possible scenarios, which diﬀer by the number of available channel states within

both conducting (functionally open) and nonconducting (functionally closed) states’ manifolds; also the mechanics of the state transitions are introduced in the models in a diﬀerent way.18−24 A great breakthrough has been made by resolving the molecular structure of the Ca2+-bound and Ca2+-free BK channels.25−27 It allowed not only for the inference that voltage and Ca2+ sensors are coupled and they can cooperatively inﬂuence the channel pore gate domain, but also these studies

failed to identify a physical gate that could mechanically block the ion ﬂow in the nonconducting state.28 From this point of

view, the pore-forming helices do not form a tight bundle as in some Shaker-like channels to dam K+ transport through the pore. In ref 25, the authors observed four diﬀerent structures for

the BK channel when the channel was neither voltage-activated nor Ca2+-activated. In those terms, the channel would be

expected to be functionally closed. Nevertheless, the inner pore-

forming helices are not so close to each other to prevent potassium ions to access the selectivity ﬁlter. Quite similar

observations are made in the studies on the activation of ligandgated K+ channel from the Slo family.29 Namely, the Na+dependent K+ channel Slo2.2 exhibited eight classes that resemble an open state and two classes being classiﬁed as the closed channel structures at 300 mM Na+. But some open classes

can be nonconductive. The authors also obtained results which

allowed for an inference that stable intermediate conformations

between the closed and open states do not exist. Thus, channel opening is highly concerted and, as a consequence, the open− closed ﬂuctuations occur in a switchlike manner. Due to the fact

that the structures of the Slo 2.2 and Slo 1 channels share some similarities,25,26,29,30 one can expect that at least some of the aforementioned inferences may also refer to the BK channel’s conformational dynamics.

BK channels pass through multiple kinetic states over time during gating. It can be assumed that Hite et al.25,29 have identiﬁed some of the structures corresponding to diﬀerent kinetic states, providing insight into which conformational states’ functionally closed and open states might be adopted.28 Still, some questions arise, among others:

• Where is the actual channel gate that allows for conducting/nonconducting ﬂuctuations at ﬁxed conditions?

• What kind of and how many stable structures of the BK channel protein exist at intermediate Ca2+ and/or voltage levels?

• Are there any diﬀerences in system dynamics in functionally open and closed states, and which ones inﬂuence the system as a whole to a higher extent?

A possible answer for the ﬁrst question is provided by the hydrophobic gating mechanism postulated in ref 31, where the authors conclude that the BK channel does not need a physical gate. Spontaneous ﬂickering between conducting and nonconducting states at ﬁxed conditions can be realized as wetting and dewetting of the channel pore resulting from the fact that the pore can constantly undergo changes in shape and surface hydrophobicity (conformational diﬀusion). The answers for the above questions need broader studies and discussion. However, some clues may be provided here by means of nonlinear analysis of experimental time series describing single channel activity.

As shown in our previous research,17 application of nonlinear methods in the analysis of an experimental nonstationary series describing ion-channel activity can be useful for estimating the changes in the number of channel states available from a geometric and energetic point of view at given external conditions. We claim that channel conformational diﬀusion has its representation in the structure of the signal. The most structurally distant conformations are suspected to aﬀect the complexity of the experimental data at least. In this paper, we would like to continue and broaden the discussion about the conformational diﬀusion of the gBK channel in electrically stimulated membranes based on the results of the multifractal detrended ﬂuctuation analysis (MFDFA).32

For over 30 years, ion-channel recordings have been analyzed paying special attention to their nonlinear, fractal properties.33−38 Here, we not only describe the multifractal nature of the experimental time series of BK channel activity but also present a biological interpretation of the obtained characteristics, which gives an additional insight into the conformational

■dynamics of the channel protein. METHODS

Detrended Fluctuation Analysis (DFA). The DFA method was ﬁrst proposed by Peng in 1994 for investigating the correlation in DNA structure.39 The last years have seen a renewed importance in the application of this method to biological data and also as a technique capable of distinguishing between healthy subjects and heart failure patients.40 This technique relies on the assumption that the signal is inﬂuenced by both short-term and long-term features. For a proper interpretation of the eﬀects hidden behind the internal dynamics, the signal is supposed to be analyzed at multiple

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time scales.41 A brief description of the original DFA algorithm is

presented below. The procedure starts with the calculation of the proﬁle yi as

the cumulative sum of the data xi with the subtracted mean ⟨x ⟩

i

yi = ∑ [xk − ⟨x⟩]

k=1

(1)

Next, the cumulative signal yi is split into ns equal nonoverlapping segments of size s, for which we use powers of 2, s = 2r, r = 4,..., 11. For all segments of size s, v = 1,..., ns, and the local trend yvm,i is calculated. In the standard DFA method, the local trend is calculated by means of the least-squares ﬁt of order m. In

this work, a second-order polynomial m = 2 is used. The variance F2(s, v) as a function of the segment length s is calculated for

each segment v separately.

∑ F2(s, v) ≡ 1 s (ym − y )2

s i=1 v,i v,i

(2)

As the last step, the Hurst exponent H can be determined as the

slope of the regression line of the double-logarithmic dependence log F(s) ∝ H log s of the square root of the average variance

F(s) = 1 ∑Ns F2(s, v) . The exponent H is used as a

Ns v=1

measure of the long-term memory of a time series. Multifractal Detrended Fluctuation Analysis (MFDFA).

The standard DFA methodology presented above can be extended to capture the qth statistical moment of the calculated variance in terms of the scaling function42

looooooijjjj 1

2ns

∑

[F 2(s ,

v)]q / 2 yzzzz1 / q ,

q ≠ 0,

S(q, s) = mooookjj 2ns v=1

{zz

ooooooo exp( 1

2ns

∑ ln[F2(s, v)]),

q = 0.

nooo

4ns v=1

(3)

A similar power law S(q, s) ∼ sH(q) can be utilized to calculate the

q-order generalized Hurst exponent H(q). The latter is required to compute the singularity spectrum for a time series. In the ﬁrst step, the mass exponent τ is determined via the relation τ(q) = qH(q) − 1. Next, the qth-order Hölder exponent h(q), a

quantity that characterizes the singularities, is estimated as a

derivative of the mass exponent, h(q) = dτ(q). Finally, the qth-

dq

order multifractal singularity spectrum D(h) (mf-spectrum) can be constructed as a Legendre transform of the mass exponent

D(h) = qh(q) − τ(q) = q[h(q) − H(q)] + 1

(4)

It results in the usual concave-shaped distribution of the singularity strengths.43

Focus-Based Multifractal Formalism. Multifractal properties of a time series are reﬂected in the generalized Hurst

exponent H(q). The standard method for the estimation of

those properties often results in a nonmonotonic characteristic

of this function, which causes the degeneracy of a singularity spectrum.44,45 It is typically a consequence of the ﬁnite length of the series. The scaling function S constructed from diﬀerent

moments q of the measure depends on the scale of observationsee eq 3. However, if one considers the full length

of the time series, the dependency on the moment q disappears.

It means that regardless of the moment, there exists only one value for the ﬂuctuation function S(q, L) = S(L) = SL. It deﬁnes a

single pair of values (log SL, log s) to which the function log S(q, s) for any moment q supposes to converge with a growing scale s.

We will call this pair of values the focus point. The existence of

this point is the central idea of the focus-based multifractal (FMF) formalism.46

The traditional approach to ﬁnd a singularity spectrum is based on ﬁnding a linear representation of the calculated log S

versus log s characteristics for each moment q. An algorithm, however, does not guarantee that the such-determined q−

dependent linear functions will cross with each other at the focus

point or even will not cross at all (see the blue solid lines in Figure 1 for details). Instead of calculating separate ﬁts for

Figure 1. Presentation of the FMF method. The solid gray lines with dots denote the S(q, s) characteristics obtained with the standard DFA technique. The solid blue lines stand for the linear ﬁt calculated with the classic least-squares method. The dashed orange lines represent the set of lines determined using the gradient descent method with a focus point. Note that all of the dashed lines cross at the focus point (L, SL) marked with a circle, while the standard individually ﬁtted lines miss the focus for the majority of moments q.

selected moments q at a time, one can try to ﬁt the whole family of functions S(q, s) keeping the requirement of sustaining one focus point (see Figure 1). The usual way is to construct a cost (loss) function that would capture the properties of both the focus point and the whole family of the qth-order scaling functions. This function can be deduced from the power law

SS((qL, )s) = ikjjj Ls y{zzzH(q) (5)

If we mark an iteratively updated scaling function with a hat, then the total mean squared error cost function MSE to be minimized reads46

MSE =

1

nq ns

∑ ∑ [ log S(̂ q, s) − log S(q, s)]2

nsnq q s

(6)

where ns is the total number of segments of size s and nq stands for the number of scaling factors q. Next, the scaling functions are found for the data. In the subsequent step, the iterative optimization algorithm, e.g., the ﬁrst-order gradient descent, is utilized to ﬁnd the minimum of the cost function.

The newly obtained family of the linear representation of the scaling functions can be later used to calculate a new generalized moment-wise Hurst exponent function Ĥ (q), which in turn directly yields the singular spectrum D(h). From this point onward, the procedure for the calculation of the singular spectrum is the same as that for the classic MFDFA algorithm.42 The analysis by means of the standard MFDFA and focus-based method was performed using our own authorial Python software, where we implemented the methodology developed by Mukli et al.46

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Figure 2. Samples of the original signal of ionic current recorded from a single gBK channel over the range of electric potential U = −60, −40, and −20 mV on the left side (a) and U = +60, +40, +20 mV on the right-hand side (b). The dotted lines indicate functionally open, i.e., conducting (O) and functionally closed, i.e., nonconducting (C) states of the channel.

Figure 3. Histograms of the original signal of ionic current recorded from a single gBK channel over the range of electric potential U = +60, +40, +20 mV (top panels) and U = −60, −40, −20 mV (bottom panels).

The parameters of the singularity spectra (viz., spectral width Δ, half-width Δ1/2, maximum of a spectrum Hmax, spectral symmetry) obtained for the experimental time series of single

channel currents describe the complexity and multifractal

properties of these data. Such characteristics can be, in turn,

interpreted in terms of the process underlying the observed current ﬂuctuations, which is the conformational dynamics of a

channel protein. An appropriate discussion will be provided in

the Results and Discussion section.

■ MATERIAL Cell Line and Solutions. For all of the measurements,

human glioblastoma cells (U-87 MG cell line) were used. The cells were cultured on Petri dishes in Dulbecco’s modiﬁed Eagle’s medium (HyClone) supplemented with 2 mM L-

glutamine (Gibco), 10% fetal bovine serum (Gibco), 100 units/ mL penicillin, and 100 μg/mL streptomycin (Sigma). The cultures were incubated at 37 °C in 5% CO2-enriched air.

Electrophysiology. Experimental results were recorded

from inside-out patches. The measurements were performed at

room temperature (20−23 °C). In all experiments, symmetrical solutions on either side of the cell membrane were used, which contained the following: 130 mM potassium gluconate, 5 mM KCl, 8 mM 4-(2-hydroxyethyl)-1-piperazineethanesulfonic acid (HEPES), 10 mM glucose, 2 mM CaCl2, 1 mM MgCl2, and 2 mM ethylene glycol-bis(β-aminoethyl ether)-N,N,N′,N′-tetraacetic acid (EGTA), and pH was adjusted to 7.3. The ion currents were recorded using an Axopatch 200B ampliﬁer (Axon Instruments). The experimental data were low-pass-ﬁltered at 5 kHz and transferred to a computer at a sampling frequency of 10 kHz using Clampex 7 software (Axon Instruments).

Channel current recordings were analyzed at ﬁxed pipette potentials of −60, −40, −20, 20, 40, and 60 mV. From all (3−7) experimental time series of channel currents, we selected and further analyzed only those where a single active BK channel was present in a patch. The channel current was measured at time intervals of Δt = 10−4 s. The ionic current measurement error was ΔI = 5 × 10−4 pA. Each experimental time series comprised N = 5 × 105 current values at the applied time resolution of the measurement. Figures 2 and 3 present the exemplary data together with the respective histograms. As one can see in Figure

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2, the probability of conducting state (O) increases with membrane depolarization, which is typical for voltage-dependent channels. Single-channel current amplitude increases as the diﬀerence in membrane potential increases; thus, local maxima in Figure 3 become well separated at both deep depolarization and hyperpolarization.

Event Detection. The MFDFA is carried out on a raw experimental data seriestime series of single-channel currents and also on preprocessed resultsseries of ionic currents recorded during the conducting (open) state of a channel and some current ﬂuctuations recorded during the nonconducting state of a channel. The threshold current value used to identify transitions between the subsequent states is evaluated as given in ref 36. The analysis of single-state data sets was performed to indicate the complexity of the signal within currents corresponding to a single manifold of states and any diﬀerences between the properties of the conducting and nonconducting states, as well as to determine which kind of signal corresponding to conducting or nonconducting states determines the characteristics of the total experimental time

■series of single-channel currents. RESULTS AND DISCUSSION The necessity of applying a modiﬁed MFDFA technique in the form of an FMF analysis is presented in Figure 4. The obtained

Figure 4. Comparison of mf-spectra for standard multifractal formalism (blue dots) and modiﬁed focus-based method (orange crosses) calculated for the representative measurement obtained at membrane hyperpolarization, U = −20 mV. The black and red dots mark, respectively, the maximum of the spectrum and the generalized Hurst exponent for the nondegenerated case.

spectra were calculated by both the methods: standard MFDFA (blue dots) and the focus-based modiﬁcation (orange crosses) (see Figure 4). In the standard method, the degeneration of the spectrum (zigzag type) is found. In other words, a large number of cases are characterized by the inversed singularity spectrum. The nonmonotonic relationship of the generalized Hurst exponent as a function of q precedes obtaining this type of spectrum shape. This situation causes diﬃculties in the interpretation of multifractality of the data and also the proper estimation of spectrum parameters such as maximum of spectrum or the spectrum width.

Comparison of mf-Spectra at Diﬀerent Membrane Potentials. A comparison of mf-spectra calculated by the FMF method is presented in Figure 5. For the investigation of the long time series (500 000 data points), the range of scales s ⊂ [24,213] was selected. The comprehensive analysis of the nature of the BK channel’s multifractality requires the calculation of the

spectra for the following cases: (i) raw data and (ii) data after the shuﬄing operation.

Figure 5 summarizes the spectra obtained for all subsequent

channel states (total signal composed of the experimentally

recorded ionic currents), series of potassium currents

corresponding to the conducting state of a channel, and series of some “leak” currents that correspond to the nonconducting states of a channel at diﬀerent stages of depolarization and

hyperpolarization. During the analysis of the total signal, one can note a marked tendency between the diﬀerent values of

membrane potentials in both cases, at hyperpolarization and

depolarization. The average spectra of the data obtained when

the value of membrane potential was closest to zero in each group (20 and −20 mV) are clearly shifted to the smaller values of h(q). In other words, the spectral maximum is most extended to left at this speciﬁc condition, and then along with increasing

applied potential at membrane depolarization and decreasing

applied potential at membrane depolarization successively

moves toward larger values of h(q) (Figure 5a,b). Considering

the results of the multifractal analysis dedicated to the currents

recorded during the open (conducting) and closed (non-

conducting) states of the channel, one can observe that the results obtained during the channel’s closures are completely

consistent with those corresponding to the total signals. The results characterizing open states are the opposite of the

remaining ones. First, the variability of their spectral width is

substantially smaller. Second, the general trend shows an

increase in spectral width when the membrane potential decreases (only for −40 mV, there is a local minimum). Such results suggest that the total signal’s characteristics are mainly

determined by the recordings obtained during the nonconducting states of a channel. The recognized diﬀerences in spectral distributions allow us to infer that the dynamics of

conformational changes within the conducting and nonconducting states’ manifolds diﬀer signiﬁcantly. Roughly

speaking, single-channel currents recorded when the channel

pore exhibits possibly high conductance retain a self-similar

structure over a range of scales regardless of the membrane

potential (Figure 5b) (which is also notable by the trend-

reinforcing behavior, as measured by Hurst exponents). Whereas current ﬂuctuations recorded when the channel is

supposed to not conduct potassium ions as well as the total signal signiﬁcantly lose their multifractal self-similarity (and the

recorded time series become uncorrelated or even anticorrelated, as shown by values of the Hurst exponents) (Figure 5a,c).

It is also visible in the case of functionally open states that

multifractal spectra have a right truncation. A long left tail

suggests that the time series of channel currents have a multifractal structure that is insensitive to the local ﬂuctuations

with small magnitudes. After the shuﬄing operation (Figure 6), the spectral

distribution is quite diﬀerent, but the most important aspect here is that the spectra are about twice narrower. The eﬀect of

the reduced multifractality after mixing of the data has a large

consequence in the proper interpretation of the source of the

fractal nature of the examined signals. There exist two general sources of multifractality which have

inﬂuence on the shape of the mf-spectrum: (i) the broad

probability density function (pdf), which lies behind the data, and (ii) diﬀerent behaviors of the (auto)correlation function for large and small ﬂuctuations. Furthermore, both situations are possible simultaneously. In case (i), shuﬄing will not change the mf-spectrum; for (ii), it will destroy the eﬀect completely as the

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Figure 5. Raw data calculations: comparison of channel activity characteristics for (a) total recording, (b) functionally open states, and (c) closed states. The black and red dots mark the maximum of the spectrum and the generalized Hurst exponent, respectively.

shuﬄing will erase the possible correlations. When cases (i) and (ii) are present simultaneously, the spectrum will diﬀer from the original one, and the weaker multifractality can be identiﬁed. In our case, we can observe exactly the last mixed situation, and thus we suspect that the multifractality of the data is caused by both the correlation and the broad pdf.

To interpret these results in terms of the considered biological system, one has to note the following facts:

• BK channels are voltage-activated, which means they exhibit more often the conducting state than the functionally closed one at membrane depolarization and tend to retain a nonconducting state at hyperpolarization of the cell membrane. Nevertheless, even at a negative potential, there is a nonzero probability that the channel rapidly opens for a relatively short time (as shown in Figure 2),

• the actual single-channel conductance is determined by two factors. The ﬁrst, and the most detrimental eﬀect, is exerted by the membrane potential. The absolute value of a single-channel current increases as the diﬀerence of the electric potential on both sides of the channel membrane

increases (both toward highly positive values and toward negative ones), as shown in Figure 2a. Higher amplitudes of channel currents at deep-membrane depolarization or hyperpolarization result in broader probability density functions (Figure 2b). Second, the conductance of the channel pore in the open state varies with voltage as a result of the structural changes that a given channel undergoes during voltage activation.25,26,47,48 But these slight changes in geometry can signiﬁcantly inﬂuence the kinetics of switching between conducting and nonconducting states.

Taking into consideration the facts mentioned above, multifractal properties of the analyzed time series of channel currents (total signal) are an inherent feature of the system connected to the dynamics of switching between the channel states, which change with membrane potential but does not depend strictly on the value of current amplitude. The changes in channel currents with voltage pertain mainly to the conducting states of a channel since the nonconducting states form a baseline during the experimental recording and they underlie smaller ﬂuctuations in the examined range of the membrane. Single-channel currents

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Figure 6. Shuﬄed data calculations: comparison of channel activity characteristics for (a) total recording and (b) functionally open states and (c) closed states.

recorded at the conducting states of the channel have multifractal characteristics over a range of scales regardless of the membrane potential (Figure 5b); in contrast, the multifractality within closed states and the total signal vary signiﬁcantly with voltage. As the picture of multifractal properties of the total signal ﬁts to the one obtained for functionally closed states, one can infer that the channel dynamics is mainly inﬂuenced by the dynamics of conformational transitions within the nonconducting states. This inference is compatible with some popular kinetic models of the ion-channel activity,22,49,50 where more kinetic substates correspond to functional closures than openings of a channel. It is possible that the scheme of switching between functionally closed conformations becomes more complex with the increase of the diﬀerence in electric potential on both sides of the membrane, which leads to an eventual widening of the mfspectrum at these conditions.

The conﬁrmed existence of the second source of multi-

fractality of the investigated data, namely, correlations for large and small ﬂuctuations, is also worth noting. The analyzed time series are long-term correlated at all experimental conditions

for channel currents at the conducting states of a channel and at

high diﬀerences of electric potential on both sides of the membranefor nonconducting states, and total signal, as shown by the values of Hurst exponent (Figures 5 and 6).

A strong analogy exists between the multifractal and thermodynamical characteristics. In particular, multifractal spectra can be related to entropy.51−53 In total signal as well as in the series of channel currents in functionally closed states, one can observe shifting of the maximum of multifractal spectra toward higher values when the diﬀerence in electric potential on both sides of the membrane increases (both at depolarization and hyperpolarization). It indicates greater complexity of the signal at highly positive and negative potentials comparing with the data obtained at membrane potentials close to zero, which may suggest an increase in the number of attainable channel substates (mainly within the nonconducting manifold) with absolute value of voltage. The symmetry of the changes in signal multifractality (and, consequently, entropy) occurring both at membrane depolarization and hyperpolarization may be counterintuitive in the case of a voltage-activated channel. The ﬁndings from refs 22, 49, 50, and 54 suggest nonsymmetric nets of conducting and nonconducting states, but there is no information about probabilities of switching between diﬀerent

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substates at diﬀerent voltages and consequently the complexity of conformational switching. Thus, there are no clear presumptions to expect that a channel’s activation should result in a monotonic dependence of multifractal characteristics on the applied voltage (in a whole range of analyzed membrane

■potentials, i.e., from −60 to 60 mV). CONCLUSIONS

In this work, a novel approach of multifractal signal analysis is presented. To the authors’ best knowledge, very few publications can be found that discuss the issue of the multifractal character of a time series by implementing an FMF methodology. An implementation of this unique technique, which is capable of handling empirical signals with a varying degree of heterogeneity, brings a lot of valuable information to the investigation of an ion channel’s activity. This work concludes that the multifractality can be regarded as an inherent feature of the single-channel currents obtained by patch-clamp measurements. The observed multifractal spectra suggest that the characteristics of system dynamics are substantially diﬀerent in functionally open and closed states, and the total signal recorded during experiments is inﬂuenced to a higher extent by the nonconducting states than the conducting ones. It is quite interesting that the symmetric increase of spectrum width and shifting of maximum of mf-spectrum of both the total signal and channel currents recorded during the functionally closed states toward higher values as the diﬀerence in electric potential on both sides of the membrane patch increases. It suggests a higher complexity and entropy of the signal recorded at both strong membrane depolarization and hyperpolarization comparing with the ones obtained at moderate membrane potentials. According to Boltzmann’s deﬁnition of entropy, these results ought to indicate an increase of the attainable substates (stable conformations) mainly in the nonconducting domain with the absolute value of applied voltage. Regardless of the applied voltage, the time series of channel currents recorded at the channel’s conducting conformations are nonrandom but caused by the orderly process exhibiting long-range correlation features.

To sum up, channel dynamics are qualitatively and quantitatively diﬀerent in the case of conducting and nonconducting states of a channel. Assuming that the most distant conformational states from energetic point of view have their representation in the recorded signal (single-channel current), it can be noted that the states obstructing ionic ﬂow through a channel pore are more complex and inﬂuence the multifractality of the total signal to a higher extent than the ones allowing for K+ transport. One should remember that our analysis does not discern between mechanically closed and nonconducting open statesso both groups, physically blocked conformations and suﬃciently narrow ones (implying hydrophobic gating), predominate in shaping the channel’s activity patterns. An interesting task for future investigation can be to carry out a comparative MFDFA analysis of a patch-clamp time series of a single-channel current on a wild-type and genetically modiﬁed BK channel that cannot exhibit relatively narrow conformations of the pore enabling for hydrophobic gating. Such analysis could be used to discriminate between the impacts of both the aforementioned groups of nonconducting states on the total signal. Moreover, the presented multifractal analysis can be a tool of supplemental analysis procedures, e.g., in cases when one should determine to what extent diﬀerent regulatory β subunits can modulate the complexity of channel behavior. The results of

such analysis could help answer the question whether they modulate the relative stabilities of preexisting conformations27 or create new ones.

In cases of glioblastoma, current medical approaches turn out to be almost powerless. Among the challenges to curing primary brain tumors, one can list the development of a precision medicine approach to treating brain tumors. In that aspect, novel approaches should be introduced, which could be based on artiﬁcial intelligence (AI) (e.g., deep learning, neural networks). The AI methods can be used for diagnosing, managing, and designing drugs against gliomas. In the literature, there already exist some reports like ref 55, where the authors present novel AI approaches to predict the grading and genomics from imaging, automate the diagnosis from histopathology, and provide insight into prognosis. Taking into account the therapeutic potential of gBK channel modulators in the treatment of glioblastoma, one could propose some AI methods to determine a group of active substances that could act as a drug against gliomas. Machine learning might be developed with the aim to determine patterns within the experimental data describing ion-channel activity, where some of the classiﬁcation algorithms could be based on the results of an MFDFA analysis. (Our preliminary analyses suggest that multifractal analysis better discriminates singlechannel current from diﬀerent exons of the BK channel than do the kinetic characteristics.) Complexity and multifractality of a signal describing diﬀerent BK channel exons bound or unbound to ligand molecules (speciﬁc modulators) could be one of the factors used in optimizing the structures of potential BK channel modulators used as a drug against glioblastoma.

■ AUTHOR INFORMATION

Corresponding Author Agata Wawrzkiewicz-Jałowiecka − Department of Physical Chemistry and Technology of Polymers, Faculty of Chemistry, Silesian University of Technology, Gliwice 44-100, Poland; orcid.org/0000-0001-6885-3956; Phone: +48 32 237 12 85; Email: [email protected]

Authors Paulina Trybek − Institute of Physics, University of Silesia in Katowice, Katowice 40-007, Poland Beata Dworakowska − Institute of Biology, Department of Physics and Biophysics, Warsaw University of Life SciencesSGGW, Warszawa 02-787, Poland Łukasz Machura − Institute of Physics, University of Silesia in Katowice, Katowice 40-007, Poland

Complete contact information is available at: https://pubs.acs.org/10.1021/acs.jpcb.0c00397

Notes The authors declare no competing ﬁnancial interest.

■ ACKNOWLEDGMENTS

The authors would like to thank Andrew Watson for a kind proofreading of this manuscript. No ﬁnancial support was received for this research.

■ REFERENCES

(1) Aldape, K.; Brindle, K. M.; Chesler, L.; Chopra, R.; Gajjar, A.; Gilbert, M. R.; Gottardo, N.; Gutmann, D. H.; Hargrave, D.; Holland, E. C.; et al. Challenges to curing primary brain tumours. Nat. Rev. Clin. Oncol. 2019, 16, 509−520.

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(2) Abdullaev, I. F.; Rudkouskaya, A.; Mongin, A. A.; Kuo, Y.-H. Calcium-activated potassium channels BK and IK1 are functionally expressed in human gliomas but do not regulate cell proliferation. PLoS One 2010, 5, No. e12304. (3) Goodenberger, M. L.; Jenkins, R. B. Genetics of adult glioma. Cancer Genet. 2012, 205, 613−621. (4) Chinot, O. L.; Wick, W.; Mason, W.; Henriksson, R.; Saran, F.; Nishikawa, R.; Carpentier, A. F.; Hoang-Xuan, K.; Kavan, P.; Cernea, D.; et al. Bevacizumab plus radiotherapy-temozolomide for newly diagnosed glioblastoma. N. Engl. J. Med. 2014, 370, 709−722. (5) Gilbert, M. R.; Dignam, J. J.; Armstrong, T. S.; Wefel, J. S.; Blumenthal, D. T.; Vogelbaum, M. A.; Colman, H.; Chakravarti, A.; Pugh, S.; Won, M.; et al. A randomized trial of bevacizumab for newly diagnosed glioblastoma. N. Engl. J. Med. 2014, 370, 699−708. (6) Wang, H.; Xu, T.; Huang, Q.; Jin, W.; Chen, J. Immunotherapy for malignant glioma: current status and future directions. Trends Pharmacol. Sci. 2020, 41, 123−138. (7) Molenaar, R. J. Ion channels in glioblastoma. ISRN Neurol. 2011, 2011, No. 590249. (8) Liu, X.; Chang, Y.; Reinhart, P. H.; Sontheimer, H. Cloning and characterization of glioma BK, a novel BK channel isoform highly expressed in human glioma cells. J. Neurosci. 2002, 22, 1840−1849. (9) Rosa, P.; Sforna, L.; Carlomagno, S.; Mangino, G.; Miscusi, M.; Pessia, M.; Franciolini, F.; Calogero, A.; Catacuzzeno, L. Overexpression of Large-Conductance Calcium-Activated Potassium Channels in Human Glioblastoma Stem-Like Cells and Their Role in Cell Migration. J. Cell. Physiol. 2017, 232, 2478−2488. (10) Weaver, A. K.; Bomben, V. C.; Sontheimer, H. Expression and function of calcium-activated potassium channels in human glioma cells. Glia 2006, 54, 223−233. (11) Ransom, C. B.; Liu, X.; Sontheimer, H. BK channels in human glioma cells have enhanced calcium sensitivity. Glia 2002, 38, 281−291. (12) Edalat, L.; Stegen, B.; Klumpp, L.; Haehl, E.; Schilbach, K.; Lukowski, R.; Kühnle, M.; Bernhardt, G.; Buschauer, A.; Zips, D.; et al. BK K+ channel blockade inhibits radiation-induced migration/brain infiltration of glioblastoma cells. Oncotarget 2016, 7, 14259. (13) Rosa, P.; Catacuzzeno, L.; Sforna, L.; Mangino, G.; Carlomagno, S.; Mincione, G.; Petrozza, V.; Ragona, G.; Franciolini, F.; Calogero, A. BK channels blockage inhibits hypoxia-induced migration and chemoresistance to cisplatin in human glioblastoma cells. J. Cell. Physiol. 2018, 233, 6866−6877. (14) Baumgarten, C. M.; Feher, J. J. Cell Physiology Source Book; Elsevier, 2001; pp 319−355. (15) Wondergem, R.; Ecay, T. W.; Mahieu, F.; Owsianik, G.; Nilius, B. HGF/SF and menthol increase human glioblastoma cell calcium and migration. Biochem. Biophys. Res. Commun. 2008, 372, 210−215. (16) Wondergem, R.; Bartley, J. W. Menthol increases human glioblastoma intracellular Ca 2+, BK channel activity and cell migration. J. Biomed. Sci. 2009, 16, No. 90. (17) Wawrzkiewicz-Jałowiecka, A.; Trybek, P.; Machura, Ł.; Dworakowska, B.; Grzywna, Z. J. Mechanosensitivity of the BK Channels in Human Glioblastoma Cells: Kinetics and Dynamical Complexity. J. Membr. Biol. 2018, 251, 667−679. (18) Cui, J.; Yang, H.; Lee, U. S. Molecular mechanisms of BK channel activation. Cell. Mol. Life Sci. 2009, 66, 852−875. (19) McManus, O.; Magleby, K. Kinetic states and modes of single large-conductance calcium-activated potassium channels in cultured rat skeletal muscle. J. Physiol. 1988, 402, 79−120. (20) Horrigan, F. T.; Aldrich, R. W. Coupling between voltage sensor activation, Ca2+ binding and channel opening in large conductance (BK) potassium channels. J. Gen. Physiol. 2002, 120, 267−305. (21) Magleby, K. L. Gating mechanism of BK (Slo1) channels: so near, yet so far. J. Gen. Physiol. 2003, 121, 81−96. (22) Geng, Y.; Magleby, K. L. Single-channel kinetics of BK (Slo1) channels. Front. Physiol. 2015, 5, No. 532. (23) Wawrzkiewicz, A.; Pawelek, K.; Borys, P.; Dworakowska, B.; Grzywna, Z. J. On the simple random-walk models of ion-channel gate dynamics reflecting long-term memory. Eur. Biophys. J. 2012, 41, 505− 526.

(24) Wawrzkiewicz-Jałowiecka, A.; Borys, P.; Grzywna, Z. J. On

Application of Langevin Dynamics in Logarithmic Potential to Model Ion Channel Gate Activity. Cell. Mol. Biol. Lett. 2015, 20, 663−684. (25) Hite, R. K.; Tao, X.; MacKinnon, R. Structural basis for gating the

high-conductance Ca 2-activated K+ channel. Nature 2017, 541, 52. (26) Tao, X.; Hite, R. K.; MacKinnon, R. Cryo-EM structure of the

open high-conductance Ca 2.-activated K+ channel. Nature 2017, 541,

46. (27) Tao, X.; MacKinnon, R. Molecular structures of the human Slo1 K+ channel in complex with β4. eLife 2019, 8, No. e51409. (28) Magleby, K. L. Structural biology: Ion-channel mechanisms

revealed. Nature 2017, 541, 33. (29) Hite, R. K.; MacKinnon, R. Structural titration of Slo2. 2, a Na +-dependent K+ channel. Cell 2017, 168, 390−399. (30) Salkoff, L.; Butler, A.; Ferreira, G.; Santi, C.; Wei, A. High-

conductance potassium channels of the SLO family. Nat. Rev. Neurosci.

2006, 7, 921. (31) Jia, Z.; Yazdani, M.; Zhang, G.; Cui, J.; Chen, J. Hydrophobic

gating in BK channels. Nat. Commun. 2018, 9, No. 3408. (32) Kazachenko, V.; Astashev, M.; Grinevich, A. Multifractal analysis of K+ channel activity. Biochem. Suppl. Ser. A 2007, 1, 169−175. (33) Liebovitch, L. S.; Fischbarg, J.; Koniarek, J. P.; Todorova, I.;

Wang, M. Fractal model of ion-channel kinetics. Biochim. Biophys. Acta, Biomembr. 1987, 896, 173−180. (34) Liebovitch, L. S.; Scheurle, D.; Rusek, M.; Zochowski, M. Fractal methods to analyze ion channel kinetics. Methods 2001, 24, 359−375. (35) Fulinś ki, A.; Grzywna, Z.; Mellor, I.; Siwy, Z.; Usherwood, P.

Non-Markovian character of ionic current fluctuations in membrane

channels. Phys. Rev. E 1998, 58, No. 919. (36) Mercik, S.; Weron, K.; Siwy, Z. Statistical analysis of ionic current

fluctuations in membrane channels. Phys. Rev. E 1999, 60, No. 7343. (37) Grzywna, Z. J.; Siwy, Z. Chaos in ionic transport through membranes. Int. J. Bifurcation Chaos 1997, 07, 1115−1123. (38) Siwy, Z.; Ausloos, M.; Ivanova, K. Correlation studies of open

and closed state fluctuations in an ion channel: Analysis of ion current

through a large-conductance locust potassium channel. Phys. Rev. E

2002, 65, No. 031907. (39) Peng, C.-K.; Buldyrev, S. V.; Havlin, S.; Simons, M.; Stanley, H.

E.; Goldberger, A. L. Mosaic organization of DNA nucleotides. Phys.

Rev. E 1994, 49, 1685. (40) Rodriguez, E.; Echeverria, J. C.; Alvarez-Ramirez, J. Detrended

fluctuation analysis of heart intrabeat dynamics. Phys. A 2007, 384, 429−438. (41) Semmlow, J. L.; Griﬀel, B. Biosignal and Medical Image Processing;

CRC Press, 2014. (42) Kantelhardt, J. W.; Zschiegner, S. A.; Koscielny-Bunde, E.;

Havlin, S.; Bunde, A.; Stanley, H. E. Multifractal detrended fluctuation analysis of nonstationary time series. Phys. A 2002, 316, 87−114. (43) Chhabra, A.; Jensen, R. V. Direct determination of the f (α)

singularity spectrum. Phys. Rev. Lett. 1989, 62, No. 1327. (44) Makowiec, D.; Fulinś ki, A. Multifractal Detrended Fluctuation

Analysis as the estimator of long-range dependence. Acta Phys. Pol., B 2010, 41, 1025−1050. (45) Makowiec, D.; Rynkiewicz, A.; Gałaska, R.; Wdowczyk-Szulc, J.; Żarczynś ka-Buchowiecka, M. Reading multifractal spectra: aging by

multifractal analysis of heart rate. Europhys. Lett. 2011, 94, No. 68005. (46) Mukli, P.; Nagy, Z.; Eke, A. Multifractal formalism by enforcing the universal behavior of scaling functions. Phys. A 2015, 417, 150−167. (47) Wawrzkiewicz-Jałowiecka, A.; Borys, P.; Grzywna, Z. J. Impact of

geometry changes in the channel pore by the gating movements on the

channelas conductance. Biochim. Biophys. Acta, Biomembr. 2017, 1859, 446−458. (48) Wawrzkiewicz-Jałowiecka, A.; Grzywna, Z. J. The role of entropic

potential in voltage activation and K+ transport through Kv 1.2

channels. J. Chem. Phys. 2018, 148, No. 115103. (49) Sigg, D.; Bezanilla, F.; Stefani, E. Fast gating in the Shaker K+

channel and the energy landscape of activation. Proc. Natl. Acad. Sci. U.S.A. 2003, 100, 7611−7615.

2390

https://dx.doi.org/10.1021/acs.jpcb.0c00397 J. Phys. Chem. B 2020, 124, 2382−2391

The Journal of Physical Chemistry B

(50) Kim, I.; Warshel, A. Coarse-grained simulations of the gating current in the voltage-activated Kv1. 2 channel. Proc. Natl. Acad. Sci. U.S.A. 2014, 111, 2128−2133. (51) Bunde, A.; Havlin, S. Fractals and Disordered Systems; Springer Science & Business Media, 2012. (52) Chen, Y. Equivalent Relation between Normalized Spatial Entropy and Fractal Dimension, arXiv preprint arXiv:1608.02054. arXiv.org e-Print archive. https://arxiv.org/abs/1608.02054 (submitted Aug 6, 2016). (53) Chen, Y.; Huang, L. Spatial measures of urban systems: From entropy to fractal dimension. Entropy 2018, 20, No. 991. (54) Starek, G.; Freites, J. A.; Bernec̀ he, S.; Tobias, D. J. Gating energetics of a voltage-dependent K+ channel pore domain. J. Comput. Chem. 2017, 38, 1472−1478. (55) Sotoudeh, H.; Shafaat, O.; Sotoudeh, E.; Brooks, M. D.; Bernstock, J. D.; Elsayed, G.; Chen, J.; Szerip, P.; Gazcon, G. C.; Gessler, F.; et al. Artificial intelligence in the management of glioma; Era of personal medicine. Front. Oncol. 2019, 9, No. 768.

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2391

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