Modeling and analysis of voltage gating of gap junction channels at a single-channel level

Abstract

The advancement of single-channel-level recording via the patch-clamp technique has provided a powerful means of assessing the detailed behaviors of various types of ion channels in native and exogenously expressed cellular environments. However, such recordings of gap junction (GJ) channels are hampered by unique challenges that are related to their unusual intercellular configuration and natural clustering into densely packed plaques. Thus, the methods for reliable cross-correlation of data recorded at macroscopic and single-channel levels are lacking in studies of GJs. To address this issue, we combined our previously published four-state model (4SM) of GJ channel gating by voltage with maximum likelihood estimation (MLE)-based analyses of electrophysiological recordings of GJ channel currents. First, we consider evaluation of single-channel characteristics and the methods for efficient stochastic simulation of single GJ channels from the kinetic scheme described by 4SM using data obtained from macroscopic recordings. We then present an MLE-based methodology for extraction of information about transition rates for GJ channels and, ultimately, gating parameters defined in 4SM from recordings with visible unitary events. The validity of the proposed methodology is illustrated using stochastic simulations of single GJ channels and is extended to electrophysiological data recorded in cells expressing connexin 43 tagged with enhanced green fluorescent protein.

Significance

Ion channel proteins play important roles in tissue function and their dysfunction is prominent in many human diseases, particularly in the nervous system and the heart. An important experimental approach to understanding of how ion channels work is through analyses of the behaviors of single ion channel proteins. Gap junction (GJ) channels are a special class of ion channel that function to communicate information directly between cells and, because of their physical configuration, present unique challenges that hamper their analyses. Here, we present a methodology for modeling and evaluating the behavior of single GJ channels, which can be applied to assessing how disease mutations affect their function and simulating how altered function can influence electrically connected cell networks.

Introduction

Gap junction (GJ) channels formed of connexin (Cx) protein subunits serve as pathways for direct electrical and chemical signaling between cells. The Cx gene family consists of 21 members that form channels exhibiting distinct biophysical properties, differing widely, in some instances, in their unitary conductances, charge selectivities, and sensitivities to voltage. For GJ channels, the voltage that regulates their opening and closing is the transjunctional voltage (V_j), the voltage difference between two cells, which endows GJs with the ability to sense relative changes in voltage between electrically communicating cells irrespective of their absolute membrane voltages (see reviews in (1,2)). Due to their widespread and tissue-specific expression and evidence of involvement in health and disease, Cx channels are increasingly considered as prospective therapeutic targets (3,4,5,6).

GJ function can be best assessed experimentally using the dual whole-cell patch-clamp technique, which enables direct measurements of transjunctional current (I_j) and hence unequivocal evaluation of gap junctional conductance (g_j). For the most part, such measurements are used to assess the strength and the voltage sensitivity of GJ conductance and how it may be affected by pharmacological agents and by mutations associated with human disease (see reviews in (7,8)). When coupling is sufficiently low or is reduced by pharmacological intervention, unitary events can be visualized to assess single-channel conductance. The analysis and interpretation of such electrophysiological data can be assisted by using mathematical/computational models and methods.

Thus far, the methods and techniques for reliable cross-correlation between the macroscopic and single-channel-level electrophysiological recordings are lacking in mathematical modeling studies of GJ channels. This could be explained by two main reasons. First, until recently, there were no mathematical models that could adequately explain both steady-state and kinetic properties of GJ channel gating. Obtaining a viable kinetic scheme of channel gating is a necessary first step for model-based evaluation of fundamental single-channel-level characteristics, such as mean open and closed dwell times. Second, compared with other types of ion channels, GJ channels present unique challenges when attempting to record single-channel events. Unlike typical ion channels, GJ channels are not accessible to direct patch-clamp recordings, i.e., the recording of channels operating within the area of a patched membrane, due to their intercellular configuration (Fig. 1). Thus, recordings of GJ channels requires whole-cell patching via the dual whole-cell patch-clamp technique, which measures the total current flow between apposing cells mediated by all functional channels present in the junctional/appositional membrane. For GJs, the channels do not exist in isolation, but are clustered into so-called GJ plaques, most likely due to the constraints imposed by the membrane deformation needed to bring two cell membranes containing isolated hemichannels into close enough contact for docking via their extracellular loop domains (Fig. 1 B). GJ plaques typically contain hundreds or even thousands of channels, and even though only a small proportion of these channels appear to be functional (9), a dual whole-cell patch-clamp recording will typically record currents from multiple channels. Thus, the analysis and interpretation of such data can be much more complicated than that obtained from the recording of a single operational channel isolated under a patch pipette. Finally, GJ channels exhibit unusually long open and closed dwell times (10,11) compared with, for example, sodium or potassium channels (12,13). Thus, long duration recordings are necessary to obtain sufficient numbers of gating events, which is technically challenging due to the necessity of maintaining two separate gigaohm-level seals, one in each of the two patched cells, for prolonged periods of time.

Figure 1.

Patch-clamp recording configurations for typical ion channels versus GJ channels. (A) Schematic of a cell-attached patch-clamp recording of an ion channel expressed in the membrane. In this experimental configuration, it is often possible to directly record currents through a single channel located under the area of patched membrane. (B) Recordings of GJ channels require the use of the dual whole-cell patch-clamp technique. In this experimental configuration, GJ channels are not recorded in isolation within the area of patched membrane but rather are recorded as the total number of functional channels located in the appositional membrane area between the coupled cells. Due to this constraint, along with the natural clustering of GJ channels in plaques, recordings from a single functioning GJ channel are unlikely. Created with biorender.com. To see this figure in color, go online.

To address these issues, we combined our recently published four-state model (4SM) of GJ channel gating that accounts for both steady-state and kinetic properties (14), with the maximum likelihood estimation (MLE) analysis of electrophysiological data (15). First, we address the evaluation of the main single-channel characteristics, such as open-state probabilities and dwell times, based on the kinetic scheme of 4SM and parameters previously derived from fits to data obtained from macroscopic recordings using the dual whole-cell patch-clamp technique. Then, we consider the application of 4SM for stochastic simulations of single GJ channels using these parameters. We demonstrate that GJ channel transition rates, obtained from a rapid equilibrium assumption (REA)-based approximation applied to 4SM provides a basis for efficient implementation of the Gillespie method, an algorithm that, together with the methods for evaluating single-channel characteristics from the 4SM kinetic scheme, provide a suitable means of obtaining stochastic simulations and analyses of electrophysiological experiments using long V_j steps. We later apply this methodology to test the validity of the MLE-based analysis of electrophysiological data in which single GJ channel events are visible. In addition, we demonstrate that the exact gating probabilities of GJ channels can be assessed directly from a two-state kinetic scheme that is associated with the gating of each component hemichannel, rather than from a four-state kinetic scheme associated with gating of the whole GJ channel, allowing for much more efficient implementation of the brute-force method for stochastic simulations. To demonstrate the usefulness of the proposed approach, we present an example of its application for simulating the behavior of a pair of electrically excitable cells connected by an electrical synapse formed of either Cx45 or Cx36, both identified as neuronal Cxs.

For the analysis of electrophysiological recordings containing multiple GJ channels, we combine MLE-based analysis (15) together with the REA-based approximation of 4SM. This approach has several advantages. First, our mathematical analysis shows that the estimates of GJ channel opening and closing rate constants have a very simple closed-form solution. Moreover, it is applicable to recordings obtained under nonstationary conditions and, most importantly, allows the pooling of data from separate recordings containing different numbers of functioning channels (10). Thus, this methodology addresses most of the aforementioned technical issues plaguing analyses of GJ channel currents. The validity of the presented methodology is first illustrated by our stochastic simulations and mathematical analyses showing that the 4SM parameters describing wild-type (WT) Cx43 channels can be estimated with high precision from the REA-based estimates of GJ channel opening and closing rates, which in turn are obtained from simulations of single channels. We then apply the presented methodology to real electrophysiological data, obtained from HeLa cells transfected with Cx43, tagged with the enhanced green fluorescent protein (Cx43-EGFP). Previous studies have shown that Cx43-EGFP channels exhibit the loss of the so-called fast gating mechanism involving transitions to residual substates leaving only the slow or loop gating mechanism that gates GJ channels between open and fully closed states (16). The simplification of gating in Cx43-EGFP GJ channels makes for a convenient case study to illustrate and test the proposed methodology.

Methods

Cell lines and culture conditions

Electrophysiological experiments were performed in epithelial human cervical cancer cells, HeLa (ATCC CCL-2), transfected with murine connexin 43 (Cx43) tagged with enhanced green fluorescent protein, Cx43-EGFP. Cells were grown on coverslips in Dulbecco’s modified Eagle’s medium supplemented with 10% fetal calf serum, 100 mg/mL streptomycin and 100 units/mL penicillin, and maintained in a CO₂ incubator (37°C and 5% CO₂).

Electrophysiological recordings

For electrophysiological recording, coverslips were transferred to an experimental chamber, which was placed on the stage of inverted microscope (Olympus, Tokyo, Japan) equipped with a fluorescence imaging system and appropriate filters to enable visualization of EGFP. The cells were bathed in a modified Krebs-Ringer solution containing 140 mM NaCl, 4 mM KCl, 2 mM CaCl₂, 1 mM MgCl₂, 2 mM CsCl, 1 mM BaCl₂, 5 mM glucose, 2 mM sodium pyruvate, 5 mM HEPES (pH 7.4). Recording pipettes were filled with a solution that contained: 130 mM CsCl, 10 mM sodium aspartate, 1 mM MgCl₂, 0.26 mM CaCl₂, 2 mM BAPTA, 5 mM HEPES (pH 7.3). The resistance of the recording pipettes was kept in the 3–5 MΩ range.

Junctional conductance was measured in cell pairs exhibiting fluorescent GJ plaques in the area of cell-cell contact using the dual whole-cell patch-clamp technique. Each cell within a pair was clamped with a separate patch-clamp amplifier (EPC-8, HEKA Instruments, Holliston, MA). V_j was induced by stepping the voltage in one cell while keeping the other cell constant. The junctional current (I_j) was measured as the change in current in the unstepped cell, and g_j was calculated as g_j = I_j/V_j.

The signals were acquired and analyzed using data acquisition hardware (National Instruments, Austin, TX) and custom-made software (17).

Description of 4SM

Simulation algorithms for single-channel gating were based on our previously published 4SM, which is detailed in (14). Here, we present just a short description of the basic assumptions of the model.

The model describes a GJ channel containing two apposing hemichannels, each containing a voltage-sensitive gate. The gates transit between an open (O) state and a closed (C) state, where the closed state exhibits either a residual conductance or no conductance. Thus, the GJ channel can reside in one of four different states: OO, hemichannel-1 open, hemichannel-2 open; OC, hemichannel-1 open, hemichannel-2 closed; CO, hemichannel-1 closed, hemichannel-2 open; CC, hemichannel-1 closed, hemichannel-2 closed.

Based on the assumption that the free energy difference between system states depends linearly on the voltage across each hemichannel, V, (18), the rates of hemichannel O↔C gating transitions are exponential functions of V:

$$\alpha \left(V\right)=\lambda ·\exp \left(-A_{\alpha }·\Pi ·\left(V-V_0\right)\right)\text{and}\beta \left(V\right)=\lambda ·\exp \left(A_{\beta }·\Pi ·\left(V-V_0\right)\right)\text{.}$$ (1)

Here, A_α and A_β reflect the voltage sensitivities of the respective transition rates; Π describes hemichannel gating polarity (Π = −1, if the hemichannel tends to close at negative voltage, and Π = 1 otherwise); V₀ is voltage, at which hemichannel opening and closing transition rates are equal (i.e., at equilibrium, half of hemichannels are open at V₀); λ is the hemichannel opening and closing rate at V₀.

The exponential functions in Eq. 1 can predict opening rates at high values of V_j opposite in polarity to Π that are unrealistically large, leading to poor predictions when fitting data during V_j polarity reversal experiments. Thus, we included a limiting value to transition rates as was also done in original modeling by Harris et al. (18). We assumed that this limit is Cx specific and can be included as one of the parameters of 4SM.

V_j distributes across apposing hemichannels based on their conductances, and thus depends on their state. The magnitude of the voltage drop across each hemichannel can be estimated from the properties of a simple voltage divider circuit. In this study, we mainly consider a homotypic GJ channel, which is composed of docked hemichannels that are identical and for which the V_j distribution across each of the hemichannels, at each channel state, is evaluated as presented in Table 1.

Table 1.

Voltage distribution across homotypic GJ channels in 4SM

State	OO	OC	CO	CC
Hemichannel-1	$\frac{V_j}{2}$	$\frac{V_j·k}{1+k}$	$\frac{V_j}{1+k}$	$\frac{V_j}{2}$
Hemichannel-2	$-\frac{V_j}{2}$	$-\frac{V_j}{1+k}$	$-\frac{V_j·k}{1+k}$	$-\frac{V_j}{2}$

In Table 1, the ratio between closed and open hemichannel conductances, γ_c/γ_o, is denoted as k ($0\le k\le 1$). Overall, the set of 4SM parameters λ, A_α, A_β, V₀, Π, k, and the limit on the magnitude of transition rates defines V_j gating properties of any Cx isoform in its homotypic GJ channel configuration.

Based on the aforementioned assumptions, we can evaluate the probabilities for the channel to reside in each of four system states at any time t. These states are oo(t), oc(t), co(t), and cc(t). More precisely, the kinetics of these state variables can be described by the linear system of ordinary differential equations, whose solution can be expressed in matrix form:

$$S\left(t\right)=S\left(0\right)·\exp \left(Q\left(V_j\right)·t\right)\text{.}$$ (2)

Here, matrix Q(V_j) defines the transition rates between system states at a given V_j. For a homotypic GJ channel it can be written as follows:

$$\mathrm{Q}\left(V_j\right)=\left(\begin{array}{cc}\begin{array}{cc}\ast & \beta \left(-\frac{V_j}{2}\right)\\ \alpha \left(-\frac{V_j·k}{1+k}\right)& \ast \end{array}& \begin{array}{cc}\beta \left(\frac{V_j}{2}\right)& 0\\ 0& \beta \left(\frac{V_j}{1+k}\right)\end{array}\\ \begin{array}{cc}\alpha \left(\frac{V_j·k}{1+k}\right)& 0\\ 0& \alpha \left(\frac{V_j}{2}\right)\end{array}& \begin{array}{cc}\ast & \beta \left(-\frac{V_j}{1+k}\right)\\ \alpha \left(-\frac{V_j}{2}\right)& \ast \end{array}\end{array}\right)\text{.}$$ (3)

Because diagonal elements of Q(V_j) are equal to the negative sum of the rest of the elements in the same row (i.e., the sum of row elements is equal to 0), for simplicity they are denoted as ∗. The matrix exponential of this transition rate matrix Q(V_j) does not have a closed-form analytical solution and, therefore, must be estimated using some numerical procedure. Multiplying the obtained matrix exponential by the vector of state variables at t = 0, S(0), as described in Eq. 2, gives us a vector S(t) consisting of state variables:

$$S\left(t\right)=\left(\begin{array}{cc}\begin{array}{cc}oo\left(t\right)& oc\left(t\right)\end{array}& \begin{array}{cc}co\left(t\right)& cc\left(t\right)\end{array}\end{array}\right)\text{.}$$ (4)

These state variables oo(t), oc(t), co(t), and cc(t) denote the probabilities for a channel to reside in the respective configuration at time t, thus they must conform to the following constraints: 0 ≤ oo(t), oc(t), co(t), cc(t) ≤ 1, and oo(t) + oc(t) + co(t) + cc(t) = 1.

A steady-state solution $S\left(\infty \right)$ (at a given V_j) can also be obtained using the transition rate matrix Q(V_j) from Eq. 3. More precisely, it can be obtained by solving the following linear system of equations

$$S\left(\infty \right)·\mathrm{Q}\left(V_j\right)=0$$ (5)

with an additional constraint that the sum of the state variables is equal to 1. The solution of this linear system of equations gives the steady-state probability vector $S\left(\infty \right)$:

$$S\left(\infty \right)=\left(\begin{array}{cc}\begin{array}{cc}oo\left(\infty \right)& oc\left(\infty \right)\end{array}& \begin{array}{cc}co\left(\infty \right)& cc\left(\infty \right)\end{array}\end{array}\right)\text{.}$$ (6)

Once state variable vector S(t) is known, junctional conductance (g_j) can be estimated as an averaged conductance of each channel configuration, weighted to its probability. It can be expressed as an inner product of state conductance vector G and system state vector S(t):

$$g_j\left(t\right)=g_{\mathit{max}}·G·S^T\left(t\right)\text{.}$$ (7)

Here, $g_{\mathit{max}}$ denotes the maximum conductance between two cells, which would represent all the functioning channels in all the GJ plaques residing in the open state, and vector G denotes the unitary conductances of each respective GJ channel conformation. For a homotypic GJ channel, it can be estimated from the properties of conductances in series and expressed as follows:

$$G=\left(\begin{array}{cc}\begin{array}{cc}\frac{{\gamma }_o}{2}& \frac{{\gamma }_o·{\gamma }_c}{{\gamma }_o+{\gamma }_c}\end{array}& \begin{array}{cc}\frac{{\gamma }_o·{\gamma }_c}{{\gamma }_o+{\gamma }_c}& \frac{{\gamma }_c}{2}\end{array}\end{array}\right)\text{,}$$ (8)

where γ_o and γ_c denote conductances of open and closed hemichannels, respectively.

Computational modeling

A numerical solution of the computational model was implemented in MATLAB. Examples of the codes are presented in supporting material. For numerical simulations of homotypic Cx45 and Cx43 channels we used the same sets of 4SM parameters obtained in our previous fits to electrophysiological recordings (14). Parameters of Cx36 channels were obtained from yet unpublished fits. For convenience, these parameters are presented in Table 2.

Table 2.

The parameters of 4SM model used in numerical simulations of homotypic GJ channels

	λ (s−1)	Aα (mV−1)	Aβ (mV−1)	|V0| (mV)	k	Limit
Cx36	0.5477	0.0505	0.0357	32.21	0.3015	10.0
Cx43	0.1522	0.0320	0.2150	34.24	0.1285	87.0
Cx45	0.1497	0.1137	0.0777	14.84	0.0626	1.15

Results

Evaluation of single-channel-level gating characteristics of GJ channels using data from macroscopic level recordings

Here, we address evaluation of single-channel properties directly from macroscopic recordings fit to the kinetic scheme of 4SM. The ability to assess such properties would allow cross-correlation of data obtained from macro- and microscopic recordings, which can reduce the amount of technically difficult single-channel-level recordings needed to obtain more detailed assessments of GJ channel behavior.

Evaluation of steady-state probabilities of GJ channels

Open- (or closed)-state probabilities are a fundamental biophysical characteristics that describe ion channel behaviors. For 4SM, these probabilities can be obtained from the solution of the linear system of equations presented in Eqs. 5 and 6. GJ channels are open only when both hemichannels are open. Thus, open state probability is given by

$$p_o=oo\left(\infty \right)\text{.}$$ (9)

For a homotypic GJ channel, the closed states OC and CO, in which the closed state of each hemichannel can represent a stable conducting substate or a nonconducting state, exhibit the same conductance values. Therefore, the closed state probability is given by

$$p_c=oc\left(\infty \right)+co\left(\infty \right)\text{.}$$ (10)

As we showed in our previous study (14) the CC state in a homotypic GJ channel is energetically unfavorable and the probability $cc\left(\infty \right)$ would be very small. Thus, $cc\left(\infty \right)$ can be disregarded in our analyses or included in p_c. Thus, in a recording containing n identical and independent functioning GJ channels, the probability that k of them are open, can be described using the Bernoulli formula:

$$P_n\left(k\right)=\frac{n!}{k!·\left(n-k\right)!}·p_o^k·p_c^{n-k}\text{.}$$ (11)

In general, Eq. 5 does not have a closed-form solution and GJ channel state probabilities $p_o$ and $p_c$ can only be obtained using numerical procedures (an example of MATLAB code is provided in supporting material). However, at V_j = 0 mV, which is likely a common condition in resting cells due to the tendency of GJ-coupled cells to be isopotential, p_o has a simple analytical expression (see derivation in section 1.1 of supporting material):

$$p_o\left(0\right)=1/\left(1+2\exp \left(-A·\left|V_0\right|\right)+\exp \left(-2A·\left|V_0\right|\right)\right)\text{.}$$ (12)

Here, parameter ${A=A}_{\alpha }+A_{\beta }$ has the same meaning as the Boltzmann parameter A and reflects the steepness of voltage dependence, and hence the gating charge, of a hemichannel. Thus, $p_o\left(0\right)$ should be higher the larger the product of Boltzmann parameters A and V₀. This theoretical estimate and previously published electrophysiological data from a number of Cxs (see Table 1 in review (2)) predict that most GJ channels are expected to have a very high p_o(0) at V_j = 0, where g_j is at a maximum.

Evaluation of average dwell times of GJ channels

Steady-state probabilities, or simply state probabilities, provide information about the long-term behavior of a channel at a given V_j. However, two different channels obviously can have the same open-state probabilities when exhibiting very different gating behaviors due to differing opening and closing rate constants. Thus, measurements of average dwell times spent in the respective states represents an important measurement for understanding channel behavior.

In general, if channel gating behavior is described by a kinetic scheme, which is defined by the transition rate matrix Q, the dwell times are distributed exponentially with a rate parameter equal to a diagonal element of matrix Q multiplied by −1. Thus, for a single GJ channel, described by 4SM and transition rate matrix as in Eq. 3, dwell times in the open state will be distributed exponentially with a transition rate $\beta \left(\frac{V_j}{2}\right)+\beta \left(-\frac{V_j}{2}\right)$. Thus, the average dwell time, at a given V_j, can be expressed as:

$$E\left(t_o\left(V_j\right)\right)=\frac{1}{\left(\beta \left(\frac{V_j}{2}\right)+\beta \left(-\frac{V_j}{2}\right)\right)}\text{.}$$ (13)

For recordings containing multiple identical independent GJ channels, it is convenient to use the REA-based approximation of 4SM. In that case, the average dwell time spent in the state with k out of n GJ channels open will be given by:

$$E\left(t_k\left(V_j\right)\right)=\frac{1}{\left(k·\beta \left(V_j\right)+\left(n-k\right)·\alpha \left(V_j\right)\right)}\text{.}$$ (14)

Note that here, as in the following subsection on the Gillespie algorithm, α and β denote opening and closing rates, respectively, of the whole GJ channel (not an individual hemichannel) at a given V_j, obtained from the REA-based approximation of 4SM.

Similar to the expression for open-state probabilities at V_j = 0 mV, the average open dwell time of a homotypic GJ channel has a simple analytical expression (see section 1.2 in the supporting material for derivation):

$$E\left(t_o\left(0\right)\right)=\frac{\exp \left(A_{\beta }·\left|V_0\right|\right)}{\left(2\lambda \right)}\text{.}$$ (15)

Thus, unlike open-state probability, the average open dwell time depends on the 4SM parameters λ and A_β, which reflect the kinetic properties of GJ channel gating by V_j and cannot be obtained from the Boltzmann function alone.

4SM fits to Cx43 and Cx45 GJ (see Table 2) and Eq. 15 predict very different gating behavior of Cx45 and Cx43 channels at V_j = 0 mV. For example, the estimated $E\left(t_o\left(0\right)\right)$ is ∼10.58 s for the Cx45 channel, and is extraordinarily larger, ∼5172 s, for Cx43 channels.

Model-based evaluation of variation in g_j

One of the advantages of a kinetic model such as 4SM is its ability to predict the magnitude of the variation in g_j. The expression for variance can be written in a matrix form using the vector G of unitary GJ channel conductances as follows:

$$Var\left(g_j\left(t\right)\right)=g_{\mathit{max}}^2·\left(\left(G\circ G\right)·S^T\left(t\right)-{\left(G·S^T\left(t\right)\right)}^2\right)\text{.}$$ (16)

Here, $G\circ G$ denotes the Hadamard, or elementwise product. During analyses of recorded currents it is often convenient to normalize the values of g_j to the maximum conductance g_max. The expression of $Var\left(g_j\left(t\right)\right)$ then simplifies to:

$$Var\left(g_j\left(t\right)\right)=\left(G\circ G\right)·S^T\left(t\right)-{\left(G·S^T\left(t\right)\right)}^2\text{.}$$ (17)

The analysis of variance in g_j can be greatly simplified using the two-state REA approximation of 4SM. In that case, vectors S and G represent probabilities and conductances of just the open and residual states, respectively:

$$S\left(t\right)=\left(\begin{array}{cc}{p}_o\left(t\right)& p_c\left(t\right)\end{array}\right)=\left(\begin{array}{cc}{p}_o\left(t\right)& 1-p_o\left(t\right)\end{array}\right)\text{and}G=\left(\begin{array}{cc}{g}_o& g_c\end{array}\right)\text{.}$$ (18)

Then, the expression of $Var\left(g_j\left(t\right)\right)$ for a GJ plaque containing n channel is as follows (see section 1.3 in supporting material for derivation):

$$Var\left(g_j\left(t\right)\right)=n·{\left(g_o-g_c\right)}^2·p_o\left(t\right)·\left(1-p_o\left(t\right)\right)\text{.}$$ (19)

If g_j is normalized to the maximum conductance, this expression is as follows:

$$Var\left(g_j\left(t\right)\right)={\left(1-\left(g_c/g_o\right)\right)}^2·p_o\left(t\right)·\left(1-p_o\left(t\right)\right)\text{.}$$ (20)

Thus, GJ channels with a higher residual conductance in proportion to the open-state conductance would exhibit smaller V_j gating-induced variation in normalized values of g_j over time. The product $p_o\left(t\right)·\left(1-p_o\left(t\right)\right)$ is largest when $p_o\left(t\right)=0.5$, that is, when the channel opening rate, α, is equal to the closing rate, β.

We suggest that such model-based evaluation of variance can be a valuable tool for evaluating the quality of data obtained from electrophysiological recordings. For example, it can help distinguish fluctuations caused by random V_j gating events from aberrations, such as channel rundown or changes in g_j induced by factors regulating GJ channels (19,20).

4SM and stochastic simulation of GJ channel gating at the single-channel level

In this section, we describe methods for generating stochastic simulations of single GJ channels using 4SM. Simulations of single GJ channels that closely replicate recordings obtained from electrophysiological experiments would be helpful for evaluating the variability that could occur in individual recordings, particularly when recording durations are limited, as they typically are for GJs that require maintaining dual whole-cell patch recordings. Simulations would give a better sense of what errors could be expected from the MLE-based analysis we propose as a solution to obtain sufficient data for assessing single GJ channel properties. Using MLE, segments of electrophysiological data can be collected from different experiments, each of which can be short in duration.

The Gillespie method

The Gillespie algorithm is based on the generation of dwell times spent in the respective system states and the gating transitions between them (21). In general, the durations of the dwell times and the probabilities of gating transitions can be derived from the infinitesimal generator matrix Q of the kinetic scheme. Thus, a Gillespie method-based simulation of a single GJ channel could be implemented using the expression of the matrix Q, which is presented in Eq. 3. However, application of the Gillespie method for realistic simulations of actual electrophysiological recordings of GJs is complicated by the fact that, under typical experimental conditions, using dual whole-cell recording multiple functional GJ channels are typically observed. Thus, it is necessary to derive a general form of the infinitesimal generator Q for multiple channels, which is not a trivial task, but can be greatly simplified using the REA-based approximation that we applied to 4SM (14). A more detailed description, as well as examples of MATLAB implementations of the Gillespie method of 4SM are provided in section 2 of the supporting material.

The Gillespie method is computationally very efficient and, thus, is very suitable for simulating channel gating in response to long-duration V_j steps, which are typically necessary experimentally because of the kinetic properties of GJ channels. Fig. S1 shows representative examples of Gillespie method-based simulations of long-duration V_j steps obtained for different numbers of functioning GJ channels.

Brute-force method

One of the main assumptions underlying the Gillespie method is that transition rates do not change much during the simulated time step. However, this would not apply for GJ channel gating under experimental or physiological conditions in which V_j is changing; for example, in experiments using V_j ramp protocols or during the spread of action potentials (APs) between excitable cells where large V_j transients can develop. A more suitable approach in such cases would be the so-called brute-force method, which simulates channel gating transitions in small discrete time steps dt. However, the approach requires evaluating gating transition probabilities for each simulated time step. The exact transition probabilities between system states, which depends on the time step dt and the applied V_j, can be obtained by calculating the exponential matrix exp(Q(V_j)·dt), where Q is an infinitesimal generator matrix that describes the kinetic behavior of the channel. The obtained exponential matrix exp(Q(V_j)·dt) describes the exact transition probabilities between channel states during time step dt and, thus, can be used for stochastic simulations of single channels. For 4SM, which is described by the transition rate matrix Q(V_j) from Eq. 3, exp(Q(V_j)·dt) does not have a closed form solution and can only be estimated using numerical methods, which would be computationally costly. However, due to the V_j gating properties of GJ channels, the brute-force approach can be simplified and implemented much more efficiently. More precisely, the two hemichannels constituting a GJ channel gate separately, but affect each other’s behavior through the changes in the distribution of V_j due to their series arrangement. During a stochastic simulation of a single channel, this V_j distribution can be evaluated implicitly from knowledge about the distinct channel states. Thus, gating probabilities of apposing hemichannels and, consequently, of the simulated GJ channel, can be evaluated directly from a two-state kinetic scheme, O ⇄ C, instead of a more complex four-state scheme, which is used in 4SM to describe the averaged behavior of a GJ channel. A more detailed description, together with MATLAB implementations of the brute-force method of 4SM, are provided in section 2 of the supporting material.

Fig. 2 shows overlays of simulated changes in g_j over time generated using 4SM (gray lines) and using the brute-force implementation (black lines) with the same hemichannel gating probabilities. V_j gating parameters were obtained from 4SM fits to macroscopic junctional currents recorded from Cx45 GJ. Fig. 2 shows that the changes in g_j over time obtained from stochastic simulations converge to the theoretical average with an increasing number of simulated channels, as would be expected for any correct implementation of a stochastic simulation algorithm.

Figure 2.

Illustration of the brute-force implementation of 4SM for stochastic simulations of GJ channels. Upper panel shows the V_j protocol used for the simulations. Lower panels show brute-force approach-based simulations of a GJ composed of Cx45. Black traces show the trajectories for normalized g_j obtained from stochastic simulations. The solid gray lines show the expected theoretical normalized changes in g_j over time obtained from 4SM. Gaussian noise with 0 mean and 6 pS standard deviation was added to all stochastic simulation data to better mimic actual recordings.

Using single-channel-level simulations to test evaluation of GJ channel gating properties using MLE

Dual whole-cell recordings in which single GJ channel transitions are visible can be obtained when cells are weakly coupled, but nonetheless typically contain several channels that are active within the junctional plaque or plaques that are present within the appositional membrane. Often, such dual whole-cell recordings are difficult to maintain for times long enough to achieve reliable estimates of channel properties. Here, we consider the evaluation of single GJ channel gating properties under such conditions using simulations generated using 4SM parameters obtained from macroscopic data. We employ a strategy that combines extraction of idealized current trajectories from the simulated data followed by MLE-based evaluation of channel gating transition rates and fits to 4SM using the REA approximation to obtain gating parameters. Use of MLE allows for the pooling of data from separate relatively short recordings and we use simulations to test the ability of data sets of different lengths generated for different voltages to reliably assess gating parameters. Application of this strategy is illustrated using simulations of WT Cx43 channels. For validation, the obtained set of 4SM parameters from simulated single-channel data should adequately describe both the micro- and macroscopic behaviors of WT Cx43 GJ channels in response to any type of V_j protocol and match the parameters used to generate that simulation data.

Idealization of junctional currents

We applied the Gillespie algorithm to obtain simulated data for WT Cx43 GJ channels. Because noise was not added to the simulation data, it directly generated idealized trajectories of g_j, and the number of open and closed channels was known implicitly at any simulated datapoint. However, even though idealization was not an issue in this case, we limited our simulations to three functional WT Cx43 channels. In theory, this should allow for a reliable reconstruction of idealized trajectories of real electrophysiological data given that each of the channels can be in open or residual subconductance states at any given time. Because the residual subconductance state of the Cx43 channel is ∼1/4 of the open-state conductance (16), it is possible to distinguish current levels for the three channels in various combinations of open and residual conductance states, provided that noise in real electrophysiological recordings is sufficiently low.

Application of MLE for the evaluation of GJ channel gating transition rates from the idealized traces of single-channel-level recordings

For GJ channels, which in 4SM cycles between four states describing combinations of open/closed states of the two series of hemichannels, the MLE approach can be greatly simplified using the REA-based approximation we introduced in 4SM (14), in which GJ channel gating can be approximated by a simpler two-state system. As a result, a GJ containing n independent functional channels can be modeled by a Markov chain with a transition rate matrix as presented in Eq. 11 of supporting material. The MLE-based analysis shows (see section 3 in supporting material for the derivation) that the estimates of channel opening and closing rates, α and β, respectively, have a very simple closed-form expression:

$$\alpha =\frac{N_{opening}}{T_{closed}}\text{and}\beta =\frac{N_{closing}}{T_{open}}\text{.}$$ (21)

Here, N_opening and N_closing denote the overall number of observed opening and closing events, respectively; T_closed and T_open reflect the overall times spent in closed and open states for all functional GJ channels. For example, if we observe a time period of 2.14 s in which three out of five channels were open, it will add 3 × 2.14 = 6.42 s to the total open time and (5–3) × 2.14 = 4.28 s to the total closed time.

In addition to a very simple expression, the ML-based estimates have other important advantages, which makes it a tool for the analysis of single-channel-level recordings that is well-suited for GJs. First, in contrast to dwell-time histograms that are applicable to electrophysiological recordings with the same number of observed channels, MLE-based estimates in Eq. 21 allow the pooling of data from separate electrophysiological recordings, even if the numbers of channels differ. This follows from the fact that the likelihood of observing two (or more) independent experiments is the product of the likelihoods of each individual experiment. The only requirement is that the same (or at least reasonably close) V_j stimulus was applied in all the pooled experiments. Second, MLE takes into consideration the steady-state data as well as the kinetic changes toward a new steady-state that follows upon application of a V_j step.

To test the applicability of MLE-based evaluations of GJ channel gating transition rates, we performed simulation experiments. Specifically, we simulated the idealized traces of g_j for WT Cx43 channels over time for a duration of 1 h. From the simulated recordings we extracted segments of different lengths (in minutes), and estimated the values of GJ channel transition rates using Eq. 21. For convenience, these recordings were extracted from a single simulated trajectory; however, as mentioned previously, MLE-based estimates in Eq. 21 are applicable to data pooled from separate shorter experiments. The extraction of shorter simulated segments allowed us to evaluate the length of an experimental recording needed to reliably assess single-channel characteristics. In addition to channel transition rates α and β, we estimated open-state probabilities and time constants, p_o(V_j) and τ(V_j), respectively, which are given by $p_o=\alpha /\left(\alpha +\beta \right)$ and $\tau =1/\left(\alpha +\beta \right)\text{.}$

Fig. 3A shows the stability plots for α(V_j), β(V_j), p_o(V_j), and τ(V_j) evaluated at three different values of V_j. At V_j = −20 mV, the opening rate of the Cx43 channel is much higher than the closing rate, i.e., $\alpha \left(-20\right)\gg \beta \left(-20\right)$. In contrast, $\alpha \left(-100\right)\ll \beta \left(-100\right)$, while $\alpha \left(-60\right)\approx \beta \left(-60\right)$. For most voltages, a recording length of 20–30 min was sufficient to provide transition rates with reasonable precision. An exception was −20 mV, which can be explained by the very low number of gating transition events occurring at this voltage, even over the course of 20–30 min. Table 3 shows the relative errors of α and β estimates extracted from 20 min of simulated recordings, together with the respective characteristics used in the MLE analysis.

Figure 3.

Evaluation of the gating properties obtained from single-channel-level stochastic simulations of Cx43 GJ channel. (A) Stability plots of ML-based estimates of GJ channel opening (α) and closing (β) rates, open probabilities (p_o), and time constants (τ), evaluated at different V_j values. The estimates of α, β, p_o, and τ (black circles) were obtained from simulated segments of different lengths. Solid gray lines show the respective theoretical values predicted by 4SM. The simulations were performed using the Gillespie method and theoretical transition rates were estimated using 4SM parameters obtained from fits to WT Cx43 GJ. (B) Stability plots of the gating parameters for WT Cx43 obtained from the estimated transition rates α and β. The solid gray lines show the respective theoretical values of 4SM parameters used in simulations. (C) The maximum relative errors (re) resulting from reconstructions of g_j-V_j and τ-V_j relationships of WT Cx43 channels from single-channel-level simulation data of different lengths. (D and E) Examples of such reconstructions obtained from a simulation 20 min in length. The solid gray curves were generated using the real set of parameters that was used in stochastic simulation. The black dashed lines were generated using the reconstructed set of parameters from the stochastic simulation.

Table 3.

Relative errors of estimated transition rates, open probability, and time constants of Cx43 GJ channels

Vj (mV)	re(α)	re(β)	re(po)	re(τ)	Topen (s)	Tclosed (s)	Nopening	Nclosing
−20	−0.81	−0.28	−0.01	4.30	3558.60	41.39	2	2
−40	−0.41	−0.11	−0.02	0.65	3357.70	242.31	21	21
−60	−0.06	−0.21	0.07	0.14	2225.40	1374.60	107	108
−80	−0.04	−0.09	0.04	0.09	314.69	3285.30	148	151
−100	−0.26	−0.12	−0.15	0.14	18.73	3581.30	71	74
−120	−0.36	−0.18	−0.23	0.21	1.19	3598.80	35	38

The estimates indicated were obtained from segments of simulated time courses, 20 min in length, generated by three functional WT Cx43 channels. The simulations were carried out using the Gillespie algorithm. In addition to the relative errors, the table shows the main characteristics of the simulated idealized trajectories, including the overall open and closed times (T_open and T_closed) and opening and closing transitions (N_opening and N_closing).

As expected, the estimated errors of the transition rates, as well as of open-state probabilities and time constants, reach an acceptable level of precision when the number of gating events is sufficiently high. In these examples, the number of events necessary exceeded 100. In addition, these estimates support the validity of the stochastic Gillespie-based simulations, as the errors introduced in the implementation would not allow for the reliable convergence to the theoretical transition rates.

GJ channel V_j gating parameters obtained from the ML-based estimates of transition rates

One of the goals here is to use transition rates obtained from MLE analysis to evaluate 4SM parameters that describe the gating behavior of the GJ channel in response to any type of V_j stimulus. In 4SM, the behavior of a homotypic GJ channel is described by a set of seven parameters. Two of these parameters are known for a number of Cxs from recordings at macroscopic and single-channel levels. One is the gating polarity, Π, which is the voltage polarity that promotes closure of a given hemichannel relative to the cell in which it resides. The other is the ratio of the conductances of the open and the closed (subconductance) states, leaving λ, A_α, A_β, and V₀ as the main parameters to fit. For completeness, we also include a parameter that represents a limit on the transition rates. In 4SM, inclusion of a limit provided for saturation of the transition rates, which improved the ability of 4SM to model complex GJ kinetics, such as that observed with rapid reversals in V_j polarity. For simplicity, the limit parameter can be approximated by the opening rate evaluated at a V_j of 5–10 mV of opposite polarity to Π (14).

Given that there are four 4SM gating parameters to fit, only two pairs of transition rates α and β evaluated at two or three different V_j values may be sufficient for obtaining these parameters. This can be achieved by fitting two or more pairs of estimates of α and β to the respective opening and closing rates obtained from the REA approximation of 4SM. To test this possibility, we used ML-based estimates of WT Cx43 GJ channel transition rates obtained from simulated trajectories of different lengths and V_j values (see Fig. 3 A). Fig. 3 B shows the stability plots for λ, A_α, A_β, and V₀, obtained from fits to ML estimates of α and β evaluated at V_j values of −60, −80, and −100 mV. Of the six values of V_j used in our stochastic simulations, these three had the highest number of gating events and thus the highest precision (see Table 3). The data in Fig. 3 B shows that using simulations for these three V_j values provided estimates of λ, A_α, A_β, and V₀ with very good precision. For example, the estimates obtained from 5 min of simulated single-channel-level recordings were relatively close to the actual values used to generate the simulations (solid gray lines in Fig. 3 B). Fig. 3 C shows the maximum relative errors for steady-state g_j, which would be predicted by the estimates of λ, A_α, A_β, and V₀ within a physiological range of V_j. Interestingly, the relative errors of g_j do not decrease monotonically with increasing length of single-channel recordings, probably due to the stochastic nature of the fitting procedure. However, the maximum relative errors for steady-state g_j are low and do not exceed 10%. In contrast, the maximum relative errors of the time constants, τ, seem to decrease more stably, but their relative errors remain larger in magnitude compared with the respective errors for g_j. Fig. 3, D and E shows the predicted steady-state g_j-V_j and τ-V_j relationships, using the estimated values of the 4SM parameters λ, A_α, A_β, and V₀ obtained from the MLE-based evaluation of the simulated single-channel-level recordings 20 min in duration. The predicted steady-state g_j relationship (solid gray curve in Fig. 3 D) closely matches the expected relationship (dashed black curve in Fig. 3 D). The errors in the predicted time constants (solid gray curve in Fig. 3 E) are somewhat higher, although they are still within a reasonable proximity to the expected values (dashed black curve in Fig. 3 E).

Table 4 shows the relative errors for λ, A_α, A_β, and V₀, as well as the maximum relative errors for g_j and τ, obtained from 20 min of simulated single-channel-level recordings at different combinations of V_j values. Here, we limited ourselves to V_j values highest in amplitude or those that produced the largest number of gating events in the simulated time courses. In any case, the data show that 4SM parameters and steady-state g_j values can be obtained from a single-channel recording with adequate precision using just two or three different values of V_j values.

Table 4.

Relative errors of the estimated 4SM parameters and the main GJ channel gating characteristics using ML-based estimates of gating transition rates obtained from single-channel-level recordings

Vj values	re(λ)	re(Aα)	re(Aβ)	re(V0)	max re(gj)	max re(τ)
−60, −80	−0.0920	−0.0346	0.0660	0.0096	0.0482	0.1758
−100, −120	0.1274	0.2582	−0.0287	0.0216	0.0175	0.3463
−60, −80, −100	0.0710	0.2065	0.0233	0.0326	0.0522	0.2825
−80, −100, −120	0.3250	0.3529	−0.0240	0.0471	0.0489	0.5170
−60, −80, −100, −120	0.1136	0.2518	0.0027	0.0367	0.0486	0.3473
−40, −60, −80, −100, −120	−0.1382	0.0624	−0.0022	−0.0002	0.0194	0.1845
−20 −40, −60, −80, −100, −120	−0.2994	−0.1135	0.0094	−0.0241	0.0388	0.6492

Evaluation of Cx43-EGFP channel gating properties using ML

In the following section, the MLE-based methodology is applied to electrophysiological recordings obtained from cells connected by Cx43-EGFP channels, which possess a simpler gating scheme better described by 4SM. Our data show that the estimates of 4SM parameters obtained from single-channel-level recordings match well with independent validation data recorded from macroscopic junctional currents.

Cx43-EGFP GJs possess a simpler gating scheme to test applicability of extracting 4SM gating parameters from single-channel data

Having demonstrated that 4SM gating parameters that adequately describe the steady-state and kinetic properties of WT Cx43 GJ channels can be obtained from simulated single-channel-level recordings with relatively high precision, the next step is to apply it to actual electrophysiological recordings. However, all Cxs, including Cx43, possess two molecularly distinct gating mechanisms, one in which the channels transit between open and residual (subconducting) states, termed fast or V_j gating, and another in which the channels transit between open and fully closed states, termed slow or loop gating (9,16). Thus, the 4SM parameters that were obtained likely represent some combined average of both gating mechanisms or, perhaps, the property of the gating mechanism that dominated over the V_j range examined experimentally (14). To avoid this complication, we chose to use Cx43-EGFP channels in our analysis of single-channel recordings. Cx43-EGFP channels, i.e., Cx43 with EGFP fused to its C-terminus, was shown to lack gating to the residual conductance state (9,16). This simplified gating behavior makes Cx43-EGFP channels a more appropriate case study to test the suitability of using MLE combined with 4SM to extract gating parameters from actual single-channel-level recordings. Having a single gating mechanism allows a more rigorous test of 4SM, which assumes that the component hemichannels gate according to a simple two-state kinetic scheme. Also, as a matter of practicality, loss of gating to the residual state makes it much easier to deduce the number of open and closed channels from the observed conductance levels.

The upper panel in Fig. 4 A shows a representative example of changes in g_j over time obtained from a dual whole-cell recording of junctional currents after imposition of a V_j of −80 mV. Coupling was low such that gating transitions between distinct conductance levels were clearly evident. The lower panel of Fig. 4 A shows the corresponding all-points amplitude histogram of the recording showing five distinct peaks corresponding to gating transitions between open and fully closed states, including a baseline level of ∼0 pS when all the channels were fully closed.

Figure 4.

Inspection of single-channel-level recordings from cell pairs expressing Cx43-EGFP. (A) Representative example of g_j trace, recorded at V_j = −80 mV (top panel), and the resulting all-points amplitude histogram (bottom panel), which clearly shows distinct levels of g_j, each corresponding to 0–5 open channels. The solid gray line in the amplitude histogram represents an appropriately scaled probability density function of the fitted mixture of normal distributions. The fitted means (0, 114, 221, 328, 438, and 543 pS) indicate that the observed unitary conductances are close to ∼110 pS. (B) Shown are dwell time histograms with log-time binning. Data were pooled from experiments recorded at −80 and −100 mV, in which one of two observed channels was open. In both cases, the observed empirical distributions did not differ significantly from a simple exponential distribution; chi-square test provided p values of 0.3279 and 0.5287 for data recorded at −80 and −100 mV, respectively. Solid gray lines show the appropriately scaled fits.

As a further test of the gating scheme describing the behavior of single Cx43-EGFP channels, we performed an analysis of the recorded dwell times. Here, idealization was accomplished using a modification of the half-amplitude criterion (22) for multichannel data. That is, we first generated all-points histograms of g_j and fit the mixtures of Gaussians to the obtained histograms. The peaks of the fitted distributions provided the values of idealized g_j levels for each recording. Then, each recorded g_j value was assigned to the closest idealized level. With recordings of just single functional channels, the existence of multiple exponential components in the dwell time histograms, best visualized on a logarithmic time scale, is indicative of multiple open and/or closed states (23). Although multiple Cx43-EGFP channels are observed in this dual whole-cell recordings, as is the case for most recordings of GJs, a simple mathematical analysis shows that the existence of multiple open or closed states should also be evident in recordings where multiple channels are observed. The behavior of two channels, each gating according to a simple two-state scheme with opening and closing transitions α and β, respectively, could be described by a three-state Markov chain (states 0, 1, and 2 would denote the number of open channels) with the following transition rate matrix

$$Q=\left(\begin{array}{ccc}-2\alpha & 2\alpha & 0\\ \beta & -\alpha -\beta & \alpha \\ 0& 2\beta & -2\beta \end{array}\right)\text{.}$$ (22)

Thus, it follows from Markov chain theory that the dwell times spent in each of the three states would be distributed exponentially, with rate parameters equal to the opposite number as the respective diagonal element of the transition rate matrix. For example, dwell times spent in state 1 (that is, one channel out of two is open) would be distributed exponentially with a rate parameter (α + β) or, alternatively, with a mean value of 1/(α + β). In contrast, a more complex channel gating scheme, for example, a linear kinetic scheme with one open and two closed states, would result in a dwell time distribution with more than one exponential component.

Our data show that the dwell time distribution obtained from the indicated multichannel recordings of a Cx43-EGFP GJ are consistent with a simple kinetic scheme transiting between single open and closed states. That is, none of the observed empirical distributions exhibited multiple peaks in logarithmically binned histograms. Fig. 4 B shows representative examples of two such histograms, which show data pooled from a recording containing two functional channels examined at two different V_j values. In both cases, the histograms represent dwell times spent in a state in which one of two channels was open, as was considered in a simple theoretical example above. Both histograms show only a single peak, and the statistical analysis shows that the respective distributions are not significantly different from a single exponential distribution. The solid gray lines show the probability density functions of the appropriately scaled fits (i.e., exponential distributions transformed to logarithmically binned data (23)). These analyses indicate that Cx43-EGFP channels should be well-described by the kinetic scheme described in 4SM.

Evaluation of V_j gating characteristics of Cx43 EGFP channels using data from single-channel-level recordings

Fig. 5A shows the stability plots of the ML estimates of Cx43-EGFP channel gating transition rates at various V_j values obtained from various dual whole-cell recordings in which single-channel transitions were visible. The estimates of all the observed transition rates reach more or less stable values with sufficiently long recording durations (overall durations of pooled recordings were in the range of ∼15–25 min for different V_j values). The converged values of these estimates were then fit to the theoretical expressions of the transition rates, which are provided by the REA approximation of 4SM. We obtained the following values for the main V_j gating parameters of 4SM: λ = 0.7860 s⁻¹, A_α = 0.0104 mV⁻¹, A_β = 0.0600 mV⁻¹, and V₀ = −26.11 mV. We assumed that, for Cx43-EGFP channels, parameter k is very close to zero due to the lack of long-lived residual state. Furthermore, similar to WT Cx43 channels, it was assumed that the limit of the transition rates does not affect the decrease in g_j in response to the applied V_j steps. The gating polarity was assigned as negative, as reported in (24).

Figure 5.

Evaluation of V_j gating parameters for Cx43-EGFP channels obtained from the electrophysiological data recorded at the single-channel-level using MLE and 4SM. (A) Stability plots of Cx43-EGFP gating transition rates, α and β, at different V_j values. Filled circles denote changes in ML-based estimates obtained with increasing numbers of observed gating transition events. Solid gray lines show the final values of the estimates obtained from all the pooled data. (B) Shown is the mean normalized steady-state g_j-V_j relationship (solid gray line) predicted from the estimated parameters obtained for 4SM. V_j gating parameters of Cx43-EGFP channels were obtained by fitting the ML estimates of gating transition rates (A) to the respective theoretical rates predicted by 4SM. Dashed lines show the theoretical standard deviations around the mean g_j, also predicted by 4SM from unitary channel data. Error bars show validation data, which were obtained from macroscopic recordings that were not used in the construction of the predicted g_j-V_j relationship; error bars represent standard deviations (whiskers) around the means (empty squares) obtained from at least 10 experimental datapoints for each V_j value. (C) Mean time courses of decline in g_j elicited in response to V_j steps of different amplitudes. Solid and dashed gray lines show theoretical predictions of mean g_j values and standard deviations, respectively, which were obtained from the same set of V_j gating parameters as in (B). Error bars show the respective normalized means and standard deviations of g_j obtained from at least 15 experimental recordings.

For independent validation, we used datapoints obtained from macroscopic currents (Fig. 5 B) that were not used in the estimation of Cx43-EGFP gating parameters. The predicted mean steady-state g_j-V_j relationship (solid gray line) together with relatively high theoretical standard deviations around the mean (dashed gray lines) fit the independent validation data well. Additional confidence in the obtained set of Cx43-EGFP parameters is provided by independent validation of the kinetic data. Fig. 5 C shows the macroscopic changes in g_j for Cx43-EGFP channels over time obtained with the imposition of the indicated V_j steps. Again, these changes in g_j were not used in the parameter estimation from single-channel-level data. The plot shows averaged normalized changes in g_j and the respective standard deviations over time, obtained from electrophysiological recordings. The solid gray lines show the respective theoretical changes in g_j, which are predicted by the obtained set of the 4SM parameters. As for the steady-state g_j-V_j relationship, the theoretical predictions correspond well to the experimental data. Overall, we view that the proposed methodology, which combines MLE analysis with modeling using 4SM can be successfully applied to evaluate the V_j-dependent gating properties of GJ channels.

Comparison of deterministic and stochastic simulations of GJ channels applied to model a simple neural connection

GJs in excitable cells pose a special case, acting as conduits that serve as pathways for the propagation of APs between cells and the coordination of their electrical activities. Of particular relevance to the simulation of single GJ channel activity we present here, neurons in the CNS are typically weakly coupled, which makes stochastic channel activity potentially relevant to the fidelity of signal transmission between cells. In previous modeling studies, we examined how the transmission of APs between two coupled cells may be affected by V_j transients that arise during sustained AP activity (25,26). However, the models were deterministic and did not take into account the effects of stochastic gating at the single-channel level. Thus, we re-examined the effects of V_j transients on AP propagation in a simple model of two weakly coupled neurons by combining simulations of APs using a Hodgkin-Huxley model (27) with stochastic simulations of GJ channels using brute-force implementation of 4SM. The simulated cells were coupled by Cx45, which is strongly V_j dependent, and Cx36, which is weakly V_j dependent; both these Cxs form GJs between neurons in the mammalian CNS, although Cx36 is considerably more widespread (28). The V_j gating parameters used were obtained from fits to data from these Cxs exogenously expressed in pairs of HeLa and RIN cells (see Table 2).

In the simulations shown in Fig. 6, cell 1 was stimulated by a depolarizing external current, which was sufficient to generate a train of APs with a firing frequency of ∼76 Hz (see spike raster plots of cell 1). Upper and lower panels in Fig. 6 compare the changes in g_j over time after imposition of the depolarizing current using deterministic and stochastic models. Fig. 6, A and B represents different assigned levels of electrical coupling. More precisely, before imposition of a depolarizing current, the model parameters produced conductances of ∼1.90 nS (Fig. 6 A) and ∼1.63 nS (Fig. 6 B) for Cx45 and ∼1.60 nS (Fig. 6 A) and ∼1.46 nS (Fig. 6 B) for Cx36. For better comparison with experimental studies, we evaluated the coupling coefficient, which is typically measured by applying a small hyperpolarizing current to cell 1, and is defined as the ratio of the membrane voltage of cell 2 relative to cell 1, V₂/V₁ (29). In our simulations, these coupling coefficients were ∼0.05–0.07, which is well within the range typically recorded between neurons in various regions of the mammalian CNS (30,31,32). The aforementioned g_j values were generated by assigning g_max values of 2.1 and 1.8 nS for Cx45 and 1.71 and 1.57 ns for Cx36. The values for g_max represent the maximum conductances achievable between the cell pairs if all the modeled GJ channels were open. To achieve these g_max values in stochastic modeling, 70 and 60 Cx45 channels were required using the experimentally measured unitary conductance value of 30 pS (33). Cx36 unitary conductance is much smaller (estimated at ∼10 pS) and, thus, required 171 and 157 channels. Also of note, compared with Cx36, the smaller g_j values relative to g_max calculated for Cx45 by 4SM are due to its steeper voltage dependence and smaller V₀ that results in some channels being closed at V_j = 0.

Figure 6.

Application of simulations of GJ channel gating to model a simple neural connection. Simulations of two neurons connected by a GJ, i.e., an electrical synapse, are modeled using deterministic and stochastic versions of 4SM. Parameters for 4SM were obtained from fits to either Cx45 (left panels) or Cx36 (right panels) GJs. Membrane excitability characteristics of both cells were simulated using a Hodgkin-Huxley model. During simulations, one neuron, cell 1, was stimulated by an external current, I_ext, to generate action potentials, which could then be transmitted to the second neuron, cell 2, by means of GJ-mediated coupling (see spike raster plots in the middle panels). (A) Simulations using a level of coupling, which was sufficient to retain a regular firing pattern in cell 2 using deterministic simulations. However, the differences in open-state probabilities and ratios of residual- to open-state conductances predict much higher variance around the mean g_j for Cx45 than for Cx36 synapses (black dashed-dotted lines in middle panels represent 2 SD around the mean). As a result, the drift below threshold level of coupling (gray dotted line in middle and lower panel) can often occur in Cx45, but not Cx36 synapses (see a representative example in the lower panel). In the stochastic simulations, the equivalent maximum conductances of 2.1 and 1.71 nS as were used in deterministic simulations of Cx45 and Cx36 synapse, respectively, were achieved using 70 Cx45 channels, each with 30 pS unitary conductance, and 171 Cx36 channels, each exhibiting 10 pS conductances. (B) Same as in (A), but steady-state g_j was achieved by setting g_max to 1.8 nS for Cx45 and to 1.57 nS for Cx36 synapses, which was not sufficient to induce firing in cell 2 during the deterministic simulation (see spike raster plots). However, in the stochastic simulations, the drift around theoretical average g_j was capable of inducing firing of cell 2 when neurons were coupled through 60 Cx45 but not through 157 Cx36 channels (see representative examples in lower panels).

Following imposition of a depolarizing current and initiation of a train of APs, coupling declined notably in cells coupled by Cx45, but not Cx36. A similar decrease of g_j was previously demonstrated in the electrophysiological data recorded in homotypic Cx45 and heterotypic Cx43/Cx45 channels (34). In our simulations, this reduction in g_j to a new conditional steady state resulted from the V_j transients that developed between the cells during the spread of excitation (detailed in Fig. S2). Evaluations of signal transfer characteristics were done between 4 and 5 s after imposition of the depolarizing pulse when g_j stably resided at the conditional steady-state value. The deterministic model simulates the averaged behavior of GJ channels and, thus, did not vary among simulations given the same set of inputs and applied stimuli. In contrast, the stochastic model, illustrated here using the brute-force method, added the stochastic drift characteristic of GJ channels and, thus, showed variation around the theoretical mean g_j among simulations. In the deterministic simulation with higher coupling (Fig. 6 A), the achieved conditional steady-state g_j exceeded the threshold level of ∼1.5 nS (gray dashed line), which was sufficient to maintain a regular firing frequency of ∼38 Hz in the second cell, i.e., the second cell regularly fired one AP after a series of two APs were generated in the first cell (see spike raster plots of cell 2 in Fig. 6 A). For Cx36, a very similar conditional steady-state g_j was reached after initiation of a train of impulses in cell 1, which similarly resulted in an identical sustained firing pattern in cell 2 of ∼38 Hz. With lower coupling (Fig. 6 B), the mean conditional steady-state g_j for Cx45 and Cx36 lies below this threshold value and firing ceases for both Cxs. However, due to different V_j sensitivities of these two Cxs, 4SM predicts significantly different open-state probabilities, p_o, at the conditional steady states (∼0.73 for Cx45 and ∼0.89 for Cx36 channels). Eq. 19 predicts that these differences in p_o, together with differences in ratios of respective residual and open-state conductances, g_c/g_o, will result in >20-fold higher variance around the conditional steady-state g_j for Cx45 compared with Cx36. As a result, the threshold g_j of ∼1.5 nS, which is sufficient to maintain a regular firing of APs in cell 2, lies within two standard deviations around the mean g_j for Cx45, but not for Cx36 (represented by dashed-dotted gray lines). Thus, the stochastic simulations predict rather different behaviors for Cx45 and Cx36 as illustrated by the representative examples in Fig. 6, A and B. The significant drift around the average g_j for Cx45 results in an erratic firing pattern when g_j drifts between values needed to provide sufficient depolarizing current to cell 2 to reach threshold. This does not happen in Cx36-coupled cells, as the drift in g_j around the mean was small resulting in similar patterns as in the deterministic case. In supporting material (Fig. S3), we illustrate 20 stochastic simulations using different consecutive values of random seed parameter for Cx45-coupled cells containing 70 channels. The irregular firing pattern at the conditional steady-state g_j was observed in half of simulations; estimated average firing frequency was ∼30 Hz with ∼10 Hz standard deviation. In contrast, all 20 stochastic simulations in cells connected by 171 Cx36 channels maintained the same regular firing pattern in cell 2. Likewise, 20 stochastic simulations of 60 Cx45 channels using consecutive values of random seed parameter (Fig. S4) also show that about half of the simulations exhibited alterations from a regular firing pattern; estimated average firing frequency was ∼5 Hz with ∼7 Hz standard deviation. In contrast, firing of cell 2 was not observed in any of the stochastic simulations coupled by Cx36 GJ containing 157 channels.

Discussion

In this study, we address the modeling of GJ channel gating at a single-channel level with the goal of enabling the cross-correlation of data obtained from macro- and microscopic electrophysiological recordings. Analyses of GJs at the single-channel level are made difficult by their unique intercellular positioning and clustering that preclude isolation of single channels under a patch pipette and render recordings subject to dual whole-cell recordings with channel numbers subject to inherent biological variability. To be sure, low levels of coupling with visible unitary currents can occur without pharmacological intervention, particularly when Cxs are exogenously expressed in cultured cells that have made newly established contacts. However, single functional channel recordings generally remain uncommon without the use of uncoupling agents that can effectively reduce levels of channel activity down to a single channel, but have the disadvantage of potentially tainting the natural gating process. The approach we present here uses our previously published model of GJ channel gating, 4SM, which could effectively describe both the kinetic and steady-state properties of GJs, extracting gating parameters from fits to macroscopic recordings of junctional currents. We extend these analyses here and show that 4SM can be applied to evaluate single-channel-level characteristics, such as open-state probabilities and average dwell times. Moreover, we demonstrate that the parameters of 4SM, which we showed could be obtained from a variety of applied V_j paradigms, can be reliably reconstructed from single-channel-level data using MLE analysis. The methodology was tested using a combination of computer simulations and real electrophysiological data recorded from the Cx43-EGFP channels.

Possible sources of errors in the proposed methodology

When assessing V_j gating of GJ channels from single-channel-level recordings, there are, of course, specific sources of error to consider. As with any ion channel, one source of error is in the idealization of single-channel recordings. Noise associated with such recordings can result in a low signal/noise ratio thereby producing errors when attempting to distinguish current levels leading to over- or underestimation of dwell times. Given the necessity of using dual whole-cell recording to measure single GJ channel activity, noise can be especially problematic. However, open-state and substate conductances of GJ channels are typically sufficiently large, allowing for reliable detection. Cx43 falls in this category and made for a good test case for assessing our methodology. Thus, we could utilize a standard method of idealization that assigns a recorded value of g_j to the closest peak of an all-points amplitude histogram. Also, MLE allowed us to pool data from different recordings providing for sufficiently long recording lengths during which under- and overestimation errors of this type can even out. In addition, systematic errors such as missed events can occur, leading to overestimation of dwell times (35). The missing of events arises when they are short-lived and at the limit of detection given the filtering used. Again, this problem is less prevalent in GJs as these channels typically exhibit long residence times in open, closed, and residual conductance states. To further minimize these potential errors and allow better testing of our approach, we only utilized recordings that clearly showed distinct levels of conductance in all-points amplitude histograms. Moreover, since MLE analysis allowed the pooling of data from different recordings, we used relatively short (up to 2 min) individual recordings of applied V_j steps to avoid problems associated with current drift and/or channel rundown.

Use of MLE to extract dwell times is based on the assumption that the maximum number of functioning channels is known (15). As we have indicated, GJ channels are clustered into plaques. Although plaques can contain hundreds or thousands of channels, only a fraction of GJ channels in a plaque appear to be functional (9), with a majority, perhaps, targeted for internalization and rendered nonfunctional (36). Thus, typical junctional recordings will contain multiple functioning GJ channels, but can be at a level in which unitary events are visible when plaques are smaller in size and/or fewer in number. Given the typically high open probability of GJ channels at V_j = 0, we assumed that the number of open channels observed at the beginning of the applied V_j step initiated from V_j = 0 is equal to the maximum number of functional channels. A factor that increases confidence that this assumption was correct is the good correspondence of the obtained estimates with the normalized macroscopic level recordings, in which the exact evaluation of the number of functioning channels is not relevant.

Use of Cx43 and Cx43-EGFP channels to model single-channel data

Fundamentally, the opening and closing of GJ channels occurs by gating of the two component hemichannels. Although the hemichannels gate as separate elements, their gating is linked through the positioning of their voltage sensors in the pore. Thus, the gating state of one hemichannel affects the electric field sensed by the other that lies in series, i.e., their gating is contingent on one another (37). Although 4SM takes contingent gating of the series hemichannels into account, it assumes a simple open/closed gating mechanism for each hemichannel as a starting point for efficient, cost-effective modeling. In actuality, Cx hemichannels exhibit two molecularly distinct gating mechanisms, one termed fast gating, which closes hemichannels to a residual conductance state, and the other termed slow or loop gating, which closes hemichannels fully (1). Thus, single-channel recordings of GJ channels exhibit a mix of gating transitions each associated with one or the other gating mechanism, thereby confounding assessment of gating parameters using 4SM.

Despite the presence of two gating mechanisms, 4SM was able to capably describe both the kinetic and steady-state macroscopic properties of homotypic Cx43 GJ channels. It turns out that the vast majority of gating events observed in single-channel-level recordings of Cx43 with V_j steps ranging from ±10 to 90 mV occurs between open and residual states. Accordingly, the macroscopic g_j-V_j relationship in this voltage range shows symmetric declines in g_j that appear to plateau to nonzero conductances corresponding to a majority of channels residing in the residual conductance state (16). A decline in g_j beyond the residual plateau value occurs at V_j values exceeding ±100 mV and at the single-channel level is accompanied by transitions to the fully closed state indicative of the engagement of the slow or loop gating mechanism. Thus, the fast gating mechanism dominates over a large voltage range, allowing for extraction of 4SM parameters that well describe the Cx43 g_j-V_j relationship as it relates to fast V_j gating. For other Cxs, the sensitivities and kinetic properties of the two gating mechanisms relative to one another can cause dominance of loop gating over a given V_j range, or a mix of both mechanisms. In the latter case, 4SM parameters would describe some averaged behavior of the two gating mechanisms.

To avoid errors associated with having two gating mechanisms, we utilized Cx43-EGFP GJ channels to model single-channel recordings using MLE. Tagging the C-terminus of Cx43 with a fluorescent protein such as EGFP was found to selectively eliminate the fast gating mechanism, postulated to occur through altered interactions among cytoplasmic domains, most notably the N-terminal domain that participates in the movements associated with fast gating (1,16). Accordingly, Cx43-EGFP channels only show full closures at the single-channel level and the macroscopic steady-state g_j-V_j relationship shows a decline in g_j toward zero rather than tending toward a residual conductance. This difference between WT Cx43 and Cx43-EGFP GJs is evident in the macroscopic g_j-V_j relationships predicted from fits to data using 4SM and MLE from single-level recordings, providing validation for the approach we present here (Figs. 3 and 5).

Gating schemes and models for GJ channels

Due to the complexities related to obtaining single GJ channel recordings without pharmacological intervention, there is little data regarding analyses of dwell times that could address the number of open or closed states that may be associated with GJ channel gating. Most mathematical modeling studies of GJs assumed a single gating mechanism in which gating occurs between one open state, defined by both hemichannels being fully open, and one closed state, defined by either hemichannel being in a residual conductance state (18,38,39,40,41,42). The residual conductance was considered, in essence, the closed state that did not completely occlude ion flow, thereby leaving a “residual” conductance. In some mathematical models, multiple substates were considered. For example, the models presented for the gating of Cx30 GJ channels (43) proposed a kinetic scheme with multiple residual substates, each exhibiting a different conductance, while the model for Cx37 GJ channels included two residual substates of equal conductances for each hemichannel (44). Additional models, for GJ channels or undocked hemichannels, have incorporated gating induced by both voltage and by divalent cations, considering kinetic schemes with two fully closed states (45,46); both of these models assumed an allosteric effect of divalent cations, acting on voltage-sensitive gating that drives the channels fully closed to a second or “deep” closed state with a long dwell time.

We took advantage of the simplified gating properties of Cx43-EGFP and single-channel-level recordings obtained without pharmacological intervention to examine whether there was evidence for multiple open and/or closed states associated with loop gating. In our analyses here, dwell times exhibited single peaks when assessed using logarithmically binned histograms (Fig. 4 B). Accordingly, the data could be fit by single exponential distributions, suggesting that loop gating can be described by a simple open/closed scheme, at least under the recording conditions we used. A study of undocked Cx50 hemichannels also showed that dwell times of open and closed states at hyperpolarizing voltages could be fit by single exponentials (47). For these undocked hemichannels, hyperpolarizing voltages only elicit loop gating. In contrast, a recent study on the influence of the N-terminal domain on the properties of Cx46 and Cx50 channels showed that distributions of open-state dwell times exhibited two exponential components (10), while an aforementioned study of Cx37 GJ channel gating (44) observed two exponential components in the distribution of residual-state dwell times. However, two exponential components in the residual-state dwell-time distribution can, in theory, be observed due to contingent gating even assuming a single gating mechanism operating with a simple open/closed gating scheme. 4SM predicts that, for a given applied V_j, gating transitions can occur from the OO to either the OC or the CO, although the transition to the closed state would be energetically unfavorable in the hemichannel that experiences a V_j that is opposite in polarity that promotes its closure. Nonetheless, these transitions can occur in sufficiently long recordings and might not have manifested clearly in the limited sets of data we used here for Cx43-EGFP channels. The same can be true for undocked hemichannels, such as Cx50, in which loop gating and fast gating operate at opposite voltage polarities. In principle, this possibility can be tested using fits to data using 4SM along with simulations of single channels as we present here.

Previously, we developed a more complex gating model, termed 36SM, which incorporated a three-state linear kinetic scheme for each hemichannel containing a single open state and two fully closed states, with the latter closed state representing the deep closed state (25). Although 36SM was better at describing GJ channel kinetics for some electrophysiological recordings than a simpler 16SM version, which lacked the second closed state (42), both models used a theoretical parameter, P_t, to calibrate timescales of electrophysiological recordings to describe gating kinetics. Given that P_t does not reflect any biophysical property of a GJ channel and does not depend on V_j, these models ultimately failed to adequately describe GJ channel gating kinetics. Also, the existence of multiple closed states was not validated by single-channel-level recordings in that study.

Application of MLE analysis, current limitations, and future directions

Given that application of MLE analyses to single GJ channel data represents a viable approach for evaluating transition rates associated with voltage-dependent GJ channel gating, such analyses can be applied to formulate statistically testable hypotheses about changes in GJ channel gating resulting from mutations or the application of pharmacological agents. For example, a comparison of the estimated likelihoods of the observed kinetic changes in g_j allows a more rigorous testing of effects on GJ channel opening and closing rates. Additional applications include assessing gating properties of a given Cx in homotypic and heterotypic configurations. That is, our modeling studies of macroscopic currents showed that parameter fits were different when the same Cx hemichannel resided in a heterotypic versus a homotypic configuration. This difference may reflect real changes in V_j gating properties caused by structural perturbations resulting from the docking of dissimilar hemichannels (14,40). Our model fitting of macroscopic data using 4SM could not provide a clear confirmation of this possibility. MLE analysis can provide a much simpler and “cleaner” way for testing whether docking affects Cx hemichannel gating, because the extracted opening and closing rates can almost be assumed to be model-independent biophysical characteristics. Moreover, there are statistical tests (e.g., the likelihood ratio test) that can assess whether the observed differences in gating rates are significantly different.

Applying simulations to two Cxs, Cx45 and Cx36, expressed in neurons also proved useful in providing insights into beneficial properties of Cx36 as the dominant Cx in mammalian CNS. Simulations indicated that electrical synapses composed of Cx36 would exhibit lower variations in g_j, despite low levels of coupling. This property held even during fairly intense simulated neural activity. Low levels of coupling between neurons in the CNS may have evolved to achieve connectivity that is modifiable and capable of readily promoting or inhibiting signal transmission and the coordination of electrical activity without leaving neurons vulnerable to epileptic activity. However, such low levels of coupling may be susceptible to noise, which is inherent in neural signaling systems of living organisms. Cx36, with its weak voltage dependence coupled with a high residual conductance results in g_j that exhibits low variance providing for more stable signal transmission. Thus, Cx36 may be the dominant Cx in mammalian CNS, perhaps because of these beneficial properties that are important for reliable information processing in finely tuned neural networks (48).

The main limitation of the proposed methodology is, of course, the inclusion of only one gating mechanism in each hemichannel in 4SM. With the modeling of more Cxs, 4SM may continue to capably reproduce macroscopic kinetic and steady-state properties due to dominance of one mechanism over the voltage range examined or by approximating the averaged behavior of the two gating mechanisms. Clearly, 4SM modeling of single-channel behavior would fail to reproduce two distinct gating mechanisms in the same simulated time course. However, evaluation of open-state probabilities, averaged dwell times and predictions of theoretical error ranges, are useful for practical purposes even without the inclusion of the second type of gating mechanism. Of course, the methodology presented can be extended to a more complete gating model that includes two gating mechanisms. In that case, the MLE analysis would require using a more complex estimation procedure and the use of numerical methods, which would be less computationally efficient. We view that such an extension of the proposed methodology is most likely to succeed initially in the modeling of undocked Cx hemichannels, because they could be described by a simpler kinetic scheme than a GJ channel. Successful development of a hemichannel gating model using MLE analysis would likely provide direct information about transition rates of the two gating mechanisms, which could be obscured in macroscopic-level recordings. Thus, we view the presented study as an important intermediate step in the further development of GJ channel and hemichannel gating models.

Author contributions

M.S. and V.K.V. contributed to the concept of the manuscript, supervised the study, and wrote the manuscript. M.S. performed mathematical analyses and numerical simulations. L.G., L.K., and T.K. performed cell culturing, electrophysiological experiments, and preliminary analysis of electrophysiological data. All authors contributed to the final version of the manuscript.

Editor: Carlos Alberto Villalba-Galea.

Supporting citations

References (14,15,27) appear in the supporting material.