Physics Department, Loyola University New Orleans, New Orleans, LA, 70118, USA

Abstract

Background

Fluctuation-induced phenomena caused by both random and deterministic stimuli have been previously studied in a variety of contexts. They are based on the interplay between the spectro-temporal patterns of the signal and the kinetics of the system it is applied to. The aim of this study was to develop a method for designing fluctuating inputs into nonlinear system which would elicit the most desired system output and to implement the method to studies of ion channels.

Results

We describe an algorithm based on constructing the input as a superposition of wavelets and optimizing it according to a selected cost functional. The algorithm is applied to ion channel electrophysiology where the input is the fluctuating voltage delivered through a patch-clamp experimental apparatus and the output is the whole-cell ionic current. The algorithm is optimized to aid selection of Markov models of the gating kinetics of the voltage-gated Shaker K^{+} channel and tested by comparison of numerically obtained ionic currents predicted by different models with experimental data obtained from the Shaker K^{+} channels. Other applications and optimization criteria are also suggested.

Conclusion

The method described in this paper can be useful in development and testing of models of ion channel gating kinetics, developing voltage inputs that optimize certain nonequilibrium phenomena in ion channels, such as the kinetic focusing, and potentially has applications to other fields.

Background

Ion channels in cellular membranes are proteins that form gated pores to allow passive transport of ions down their electrochemical potential gradient

The technique of patch-clamping in ion channel electrophysiology allows measurements of currents flowing through the channels in cellular membranes

These two protocols illustrate the current paradigm on which a vast majority of electrophysiological experiments are based: the voltage protocols are piecewise constant and consist of only few voltage steps at discrete times

Several of these effects can be very promising in investigating and controlling the kinetics of ion channels. For instance, one of the main goals of ion channel research is to develop a model of channel gating kinetics. Most commonly used are the discrete Markov chains, however even the basic features of these models are still disputed, e.g. cooperativity or Markovian character of gating. It has been suggested

A related issue is that of controlling the gating of ion channels. It has an enormous practical importance in biology and medicine. So far the dominant approach is to use pharmacological agents (drugs and toxins) or global electric fields (defibrillator shocks) to modify channel behavior. A much more subtle approach, aimed at forcing ion channels into a specific conformational state has been also proposed

In this paper we describe in detail such a method of designing an input to a system that maximizes a desired output. It is based on constructing a fluctuating signal as a superposition of wavelets in a dyadic wavelet basis and optimizing the wavelet coefficients for a specific system response. We developed the method for use with the experimental technique of patch-clamping but it could be applied in a variety of contexts where fluctuating system inputs are used. We concentrate in this paper on applications to ion channel electrophysiology, in particular since our method represents a radical departure from the current paradigm. We show how one can synthesize a signal with desired properties and implement it in a voltage-clamp experiment with ion channels.

We illustrate the method by applying to one of the two problems mentioned earlier in the introduction, i.e. by developing voltage protocols that maximize the difference between various Markov models for the same ion channels. The numerical results are compared to experimental data obtained from Shaker K^{+} channels. The idea of this method was mentioned in our previous work on the nonequilibrium response spectroscopy technique

Methods

Markov models of ion channels

Ion channel gating results from rearrangements of the tertiary structure of the channel proteins, i.e. transitions between certain meta-stable conformational states of the molecule, in response to changes in trans-membrane potential. These molecular states can be either conducting (open – O) or non-conducting (closed – C or inactivated – I). Examples of Markov models for the Shaker K^{+} channels are shown in Figure

where

Examples of Markov models for Shaker IR K^{+} channels

**Examples of Markov models for Shaker IR K**
^{
+
}
**channels.**

Wavelet decomposition of voltage inputs

The purpose of this paper is to describe and implement a method of constructing fluctuating voltage inputs for ion channel electrophysiology to obtain a certain type of response from ion channels. The method exploits the interplay between spectro-temporal patterns of the signal and the gating kinetics of the channel molecules. It requires simultaneous control over temporal and spectral properties of the voltage pulse. A tool that offers this kind of time-frequency localization is the wavelet analysis, developed in the last two decades

and finite “energy”:

There are many different families of wavelets, some with compact support. In essence, the wavelet analysis is similar to Fourier analysis, however it has certain advantages. In Fourier transform the testing functions (trigonometric) have infinite support hence they are not very well suited to analysis of nonstationary signals. Fourier transform provides spectral information about the signal, however temporal information, although not lost, is hidden in the phases of the sines and cosines. By proper translation and dilation of the testing wavelet, the wavelet transform can provide both spectral and temporal information, i.e. it shows not only what frequencies are present, but also when they are present in a nonstationary signal.

Wavelet analysis begins with a mother wavelet, i.e. a function satisfying conditions (2)-(3). The simplest is the Haar wavelet:

which is also the first member of the Daubechies family of wavelets. By varying the dilation parameter

The continuous wavelet transform CWT is the convolution of the signal

The CWT coefficients ^{
2
}, known as the scalograms. The coefficients _{
0
}

In principle, the CWT is an expansion of a signal _{
a,b
}}. Some wavelets, such as the Daubechies family, have been designed so that a countable subset of dilated and translated wavelets, of the form:

constitutes an orthonormal basis in ^{2}(ℜ). In other words, every signal

can be approximated arbitrarily well by a finite superposition of basis elements _{
m,n
}. Such a basis is called the dyadic wavelet basis. Using wavelets (7) we can define the discrete wavelet transform (DWT) as:

Just like for the CWT, the DWT coefficients can be plotted versus the scale and translation indices (

One of the applications is the so-called multiresolution analysis (MRA). It is an iterative procedure where a given signal is separated into a coarse approximation and the detail. The latter is expressed as a superposition of wavelets at a certain level

The technique we use to construct signals (voltage inputs) with desired properties is based on the inverse DWT (10) and is essentially the opposite of the MRA we just described. For a chosen wavelet type (we tested the Haar wavelets since they most closely resemble currently used voltage step protocols and the Daubechies 8 wavelets for their compact support and the degree of smoothness) we synthesized the signal using (10) and including a finite number of levels. This number of MRA levels was determined by physiological considerations and the bandwidth of the recording apparatus. The latter was typically of the order of 5–10 kHz, and from previous studies

Channel model response and input optimization

In the previous paragraph we described how a signal with desired properties can be synthesized in a dyadic wavelet basis. We should also specify what these “desired” properties are. The voltage input is selected to elicit a certain response from ion channels. The method we are describing in this paper is flexible enough for a variety of purposes. It can be used to synthesize inputs that maximize differences in computed ionic currents between various models, hence aiding model development and testing, or inputs that maximize the parametric sensitivity of the model, as described in

In order to determine the model output, let us consider an **
P
**

where the **
W
**

where **
P
**

with a sufficiently small time step. **
P
**

Here _{
0
}
**
O·P
**

Genetic algorithm

The algorithm for voltage input synthesis is an optimization procedure. Depending on the goals we define the “cost functional” for the model output and then modify the inputs to optimize this functional. We use a version of a genetic algorithm applied in other contexts previously
_{
m,n
}. From this set of coefficients we obtain a number, typically 10–20, of second generation sets by random perturbations according to:

where _{
0
} and _{
0
}) and the rate at which the search narrows down with each generation (_{
0
} = 0.5 and

Channel electrophysiology

As an illustration of this numerical scheme we constructed voltage inputs designed to maximize the difference in model outputs between different Markov models for the Shaker K^{+} channels shown in Figure

We studied a mutant Shaker channel (Shaker K Sk1), stably expressed in tsA201 cells (gift from D. Hanck). The cells were cultured in 35 mm Corning Petri dishes (Corning) in DMEM medium (ATCC, Manassas, VA) supplemented with 10% FBS (ATCC, Manassas, VA), 1% penicillin/streptomycin (Gibco BRL, Gaithersburg, MD) and 200 μg/ml Zeocin (Invitrogen, Carlsbad, CA) at 37°C in 5% CO_{2} in a CO_{2} incubator (Fisher Scientific, Pittsburgh, PA). Patch clamping pipettes were pulled from borosilicate capillary glass (Warner Instruments LLC, Hamden, CT) on a Sutter pipette puller (Sutter Instrument Co., Novato, CA).

Results

Models for Shaker K^{+} channels

As an illustration of the proposed method we consider one of the main goals in ion channel studies – the development of kinetic models of channel gating. Such models are developed by fitting to various types of experimental data (mostly various ionic current recordings). If the set of experimental data is small, the number of models compatible with it is typically very large. Model refinement is achieved by introducing new types of data (e.g. gating currents) and by increasing the number of different voltage protocols. The paradigm of electrophysiology is to use voltage protocols consisting of static voltages changing only stepwise at few discrete times. Typical examples are the activation and tail protocols. Very few studies used randomly fluctuating voltages such as the dichotomous noise

Several models have been developed for the Shaker K^{+} channels. We consider here the Bezanilla-Peroso-Stefani (BPS)
_{1} to C_{4} describe the channel conformations with different number of subunits in the A state (0 for C_{1}, 1 for C_{2}, 2 for C_{3}, 3 for C_{4}). The conformation with all four subunits activated is the open (O) state of the channel. The transition rates in the expanded topology follow directly from the rates in the abbreviated version hence the model involves only 1 pair of transition rates (4 model parameters). The ZHA D model introduces two complications. First, the subunit kinetics now involves two resting states, so it is itself a linear chain of three states. Second, there is another closed state (C_{f}) accessible only from the open state. Both corrections were introduced to improve the model fit to a variety of available experimental data. The expanded version of this topology is no longer linear and involves 16 distinct channel states, however the number of model parameters is the same as for the abbreviated version (three pairs of transition rates, i.e. 12 parameters). In that respect the BCS model is significantly more complex and computationally demanding, with 20 parameters. Finally, the SS model is the most complex

For all the models shown in Figure

**A function file which accepts the model parameters of the SS model and the applied voltage as its inputs and returns the transition matrix for this model.**

Click here for file

**A function file which accepts the model parameters of the ZHA D model and the applied voltage as its inputs and returns the transition matrix for this model.**

Click here for file

Model ZHA A:

Model BPS:

Voltage protocols optimized for maximal divergence of model responses

The primary criterion for a model selection is its ability to accurately reproduce various types of data. It is the choice of the type of data and the number of different data sets that determine how good and how unique the model is. The models described in the preceding sections were all developed using mostly different voltage-step protocols. As such they reproduce the experimental ionic current for any similar voltage-step protocol very well. Figure

Comparison of model currents (triangles) and experimental ionic currents (solid line) for activation protocols for four Markov models of Shaker K^{+} channels

**Comparison of model currents (triangles) and experimental ionic currents (solid line) for activation protocols for four Markov models of Shaker K**^{+ }**channels.** The voltage protocol consisted of holding potential of −90 mV followed by a series of steps to potentials from −70 mV to +42 mV in 8 mV increments. Data is shown for models: **A**) ZHA A, **B**) ZHA D, **C**) BPS, **D**) SS.

**Model**

**Rate amplitudes (ms**
^{
-1
}**)**

**Gating charges (units of e)**

**Additional parameters**

**ZHA A**

α(0) = 0.1219

q_{α} = 0.6232

β(0) = 0.0342

q_{β} = −0.0207

**ZHA D**

α(0) = 2.1605

q_{α} = 1.9760

θ = 8.8660

β(0) = 0.0558

q_{β} = −0.0108

γ(0) = 0.2642

q_{γ} = 0.2080

δ(0) = 0.1103

q_{δ} = −0.5311

k_{α}(0) = 0.3505

q_{kα} = 0.0138

k_{β}(0) = 0.5253

q_{kβ} = −0.0203

**BPS**

α_{0}(0) = 0.2761

q_{0} = 0.7467

β_{0}(0) = 1.2647

q_{1} = 1.4896

α_{1}(0) = 1.6095

q_{2} = 0.7891

β_{1}(0) = 0.1061

q_{3} = 0.8988

α_{2}(0) = 1.6809

q_{4} = 0.8376

β_{2}(0) = 0.2955

δ_{0} = 0.2082

α_{3}(0) = 2.0483

δ_{1} = 0.1316

β_{3}(0) = 0.8917

δ_{2} = 2.1390

α_{4}(0) = 1.8216

δ_{3} = 0.3814

β_{4}(0) = 0.3384

δ_{4} = 0.2731

**SS**

α(0) = 1.0859

q_{α} = 0.8288

β(0) = 0.1415

q_{β} = −1.1574

γ(0) = 2.2983

q_{γ} = 0.0160

δ(0) = 1.3948

q_{δ} = −0.1893

ε(0) = 1.3100

q_{ε} = 0.0639

ζ(0) = 0.5676

q_{ζ} = −0.0584

α_{N-1}(0) = 4.5890

q_{αN-1} = 0.0771

β_{N-1}(0) = 0.1965

q_{βN-1} = −0.0779

α_{N}(0) = 1.9438

q_{αN} = 0.6513

β_{N}(0) = 0.1431

q_{βN} = −0.6569

c(0) = 0.0054

q_{c} = 0.0980

d(0) = 0.3853

q_{d} = 0

c_{1}(0) = 0.1680

q_{c1} = 0

d_{1}(0) = 3.9279

q_{d1} = −0.3678

c_{2}(0) = 0.8233

q_{c2} = 0

d_{2}(0) = 9.9927

q_{d2} = 0

We used these models with the parameters optimized for the fit to standard activation and tail protocols. Our goal was to construct new fluctuating voltage waveforms that produce maximal differences in model responses (i.e. model currents) and thus could be used to select one of the models over the others. We used the numerical procedure described in Methods to construct the new voltage input as a superposition of wavelets. After computing the model response to this voltage input, i.e. the model ionic current, using eq. (14), we defined the difference in model response as the χ^{2} error between the corresponding model currents (see Additional file

Sample voltage protocols optimized for model current difference

**Sample voltage protocols optimized for model current difference.** ZHA A vs. SS model with **A**) 8 wavelet levels, **B**) 9 wavelet levels, and **C**) 10 wavelet levels.

**A function file that accepts two discrete time series as its arguments and returns the χ**
^{
2
}** error between the series, normalized to the number of sampling points.**

Click here for file

**Matlab code to optimize a pulse synthesized from wavelets.** The program uses the difference between the computed responses of models SS and ZHA D (computed as χ^{2} error) as the cost functional and optimizes the pulse for maximal value of the cost functional.

Click here for file

Comparison of model currents and experimental currents

Ionic currents were recorded from tSA201 cells stably transfected with the Shaker IR potassium channels, as described in Methods. We used the typical activation and tail protocols. The former were obtained for voltage steps from a holding potential of −90 mV to a series of values from −70 mV to 42 mV in 8 mV intervals. For the latter the protocol consisted of a 32 mV prepulse of 30 ms duration followed by a series of voltage steps from −120 mV to 48 mV in 12 mV intervals. The capacitative transients were removed from the experimental data using the standard P/4 method. The wavelet-based voltage pulses, described in the previous section, were implemented on our recording apparatus as described in the Methods. We monitored the bandwidth by obtaining the cell’s capacitance and the series resistance. Only cells with bandwidth exceeding 5 kHz were considered. For capacitative current transients we used the P/2 method, used also in our earlier paper

Comparing model currents and ionic currents from electrophysiological experiments is the ultimate test for any model of channel gating. In most studies the fit is limited to the activation and tail ionic currents. Sometimes other voltage protocols are added but they typically consist of few voltage steps at discrete times. As noted earlier (see Figure

Model currents for wavelet-based voltage pulses

**Model currents for wavelet-based voltage pulses.** Responses of the four models to voltage pulses from Figure

Model currents for wavelet-based voltage pulses

**Model currents for wavelet-based voltage pulses.** Responses of the four models to voltage pulses from Figure

Discussion

The goal of this paper is to describe a new method for developing voltage protocols for ion channel patch-clamping experiments. It is expected that in order to obtain new information about channel gating kinetics or to explore new phenomena that ion channels may exhibit, one must go beyond the typical set of stepped voltage protocols. The method we develop allows tailoring the voltage inputs in a patch-clamping experiment to maximize a desired output. The design is based on the inverse discrete wavelet transform where the voltage pulse is constructed as a superposition of wavelets in a dyadic wavelet basis. Each such pulse is characterized by the corresponding set of wavelet coefficients and it can be synthesized using a variety of available numerical packages. The desired outcome of the patch-clamping experiment needs to be quantified through a suitably defined cost functional and then the design algorithm is essentially an optimization procedure where the set of wavelet coefficients is optimized to maximize (or minimize) the cost functional. The optimization procedure we used is a search of the parameter space through a random genetic algorithm. We used a method where the “daughter” parameter sets are generated by randomly perturbing the parameters of the “parent” set, with a range of random fluctuations decreasing through the generations. This ensures convergence but also avoids trapping in a local extremum of the cost functional. We used this method previously e.g. for fitting of kinetic models of channel gating. Other algorithms are possible, for instance

This method has many potential uses but we apply it to a specific example: testing of kinetic models of ion channel gating. It is known that the models are notoriously ambiguous. Different research groups propose very different models that match available data equally well, hence there is no way to decide which model (or maybe none) is the best approximation of reality. It is accepted that in order to improve the model design and testing process, we need to include more data and that new data should be substantially different from all existing data. Just adding more of the same will not improve the process. There are two approaches to this. One is to get new data that is of different physical nature. The best illustration would be to add gating currents to previously used ionic currents, or to use single channel data in addition to whole-cell data. This is not the focus of this manuscript. We consider the second approach where the new data is of the same physical nature as previously (i.e. the same physical quantity is measured) but the input to the system has been substantially altered. In recent years there has been interest in using fluctuating voltages as opposed to piecewise constant voltages
^{2} error. Then we constructed several pulses that maximize that cost functional, i.e. pulses which when input into these models generate maximally divergent outputs. These model currents were then compared to the experimental currents recorded in patch-clamping experiments, in response to the same voltage inputs. This facilitated the selection of the best models, and the process can be done iteratively. A preliminary version of this method was previously applied to models of a human heart sodium channel

Conclusions

In this article we describe a new method that represents a radical departure from the current paradigm in ion channel electrophysiology. We propose using fluctuating voltages through a patch-clamp apparatus. While the use of fluctuating inputs to nonlinear systems to elicit new types of system responses is well known and researched in different areas of physics, our method has the added advantage of allowing tailoring inputs to achieve a particular, most desired outcome. If we quantify the “desirability” of outcome through a suitable “cost functional”, through our method one can construct an input that optimizes this cost functional. The pulse design is based on the wavelet decomposition. In particular, we implement the method in ion channel electrophysiology where the input is the voltage applied in patch-clamp experiments, the output – the measured whole-cell ionic current and the “cost functional” – the fit of various Markov models to the experimental data. Our method can be used to aids selection and testing of models of channel gating kinetics. We also comment on other possible applications of this method to electrophysiology and other fields.

Competing interests

The author declares that he no competing interests exist.

Authors’ contributions

AK performed the all experimental and numerical work presented in this paper and designed the method presented.

Acknowledgment

The author acknowledges the support of Loyola University New Orleans where he is a Faculty member.