Stochastic Model of Gap Junctions Exhibiting Rectification and Multiple Closed States of Slow Gates

Abstract

Gap-junction (GJ) channels formed from connexin (Cx) proteins provide direct pathways for electrical and metabolic cell-cell communication. Earlier, we developed a stochastic 16-state model (S16SM) of voltage gating of the GJ channel containing two pairs of fast and slow gates, each operating between open (o) and closed (c) states. However, experimental data suggest that gates may in fact contain two or more closed states. We developed a model in which the slow gate operates according to a linear reaction scheme, o↔c₁↔c₂, where c₁ and c₂ are initial-closed and deep-closed states that both close the channel fully, whereas the fast gate operates between the open state and the closed state and exhibits a residual conductance. Thus, we developed a stochastic 36-state model (S36SM) of GJ channel gating that is sensitive to transjunctional voltage (V_j). To accelerate simulation and eliminate noise in simulated junctional conductance (g_j) records, we transformed an S36SM into a Markov chain 36-state model (MC36SM) of GJ channel gating. This model provides an explanation for well-established experimental data, such as delayed g_j recovery after V_j gating, hysteresis of g_j-V_j dependence, and the low ratio of functional channels to the total number of GJ channels clustered in junctional plaques, and it has the potential to describe chemically mediated gating, which cannot be reflected using an S16SM. The MC36SM, when combined with global optimization algorithms, can be used for automated estimation of gating parameters including probabilities of c₁↔c₂ transitions from experimental g_j-time and g_j-V_j dependencies.

Introduction

Connexins (Cxs), a large family of membrane proteins, form gap-junction (GJ) channels that provide a pathway for electrical and metabolic signaling between cells. Each GJ channel is composed of two apposed/docked hemichannels (aHCs), also called connexons, that are hexamers of Cx protein (1, 2, 3). A property that appears to be common to GJ channels formed by any Cx isoform is sensitivity of junctional conductance (g_j) to transjunctional voltage (V_j) (2, 4). Single-channel studies have shown that V_j causes channels to close to a subconductance (residual) state with fast gating transitions (∼1 ms and less) (5, 6). V_j, as well as chemical uncouplers, can also induce slow gating transitions (∼10 ms and more) to the fully closed state (7). Gating to different levels via distinct fast (Fig. 1 D, red arrows) and slow (Fig. 1 D, blue arrows) gating transitions led to the suggestion that there are two distinct V_j-sensitive gates, termed fast and slow, or loop, gating mechanisms (reviewed in (2)) (Fig. 1 A).

Figure 1.

Fast and slow gating mechanisms of the GJ channel. (A) Schematic of fast and slow gates located in series in the GJ channel. (B and C) All possible gating transitions of fast (B) and slow (C) gates. (D) Gating of Cx43 GJs during uncoupling by CO₂ and washout. Open circles in the insets are every 2 ms. Reproduced from Fig. 1 of our previous work (7).

Historically, gating properties of GJ channels were described using a Boltzmann function (4, 8), assuming that each aHC of the GJ channel contains one gate operating between open and fully closed states and it does so only when the aHC is open. Accumulation of more data about the gating of GJs motivated several groups to describe the gating of GJs using gating models of four or more states, which offered a more detailed description of experimentally measured g_j-V_j relationships (9, 10, 11). Initially, we developed a stochastic four-state model (S4SM) in which each aHC contained one gate, fast or slow (12). In the next step, we developed a stochastic 16-state model (S16SM) containing both fast and slow gates in each aHC (13) and used it to evaluate changes of V_j-sensitive gating parameters of homotypic and heterotypic GJs under different pathology-related conditions (14, 15, 16, 17). That model accounted for a contingency in gating when operation of each gate depends on the state of three other gates in series; it also allowed for evaluation of the dynamics of g_j changes or the steady-state value of g_j (g_j,ss) depending on V_j. Additionally, it accounted for the rectification of unitary conductances of fast and slow gates, which has accumulated solid experimental support at the levels of individual GJ channels (18, 19, 20, 21) and undocked/unapposed hemichannels (uHCs) (18, 19, 20, 21, 22, 23, 24, 25). Despite the progress achieved in more detailed simulation of the gating processes by using an S16SM, there still remained some inconsistencies within the experimental data preferentially related to GJ channel relaxation from gating.

Here, we transformed an S16SM of GJ channel gating into a stochastic 36-state model (S36SM) of GJ channel gating in which the fast gate operates between the open and closed states with the reaction scheme o↔c (Fig. 1 B). Following our experimental data and those of others (19, 26), we assumed that the slow gate can be in two closed states, an initial-closed state (c₁) and a deep-closed state (c₂), and operates according to a linear reaction scheme, o↔c₁↔c₂ (Fig. 1 C). We also modified an S36SM into a Markov chain 36-state model (MC36SM) of GJ channel gating, which allowed us to accelerate simulation >100-fold and eliminate noise while maintaining the same g_j-time and g_j-V_j relationships. An inclusion of two, or even more, closed states for the slow gating mechanism allowed us more adequately to reproduce well-documented experimental phenomena such as delayed g_j recovery from V_j gating, hysteresis of g_j-V_j plots, or the low ratio of functional GJ channels to their total number (∼0.1 and below) under physiological conditions and the even lower ratio under various pathology-related conditions.

Materials and Methods

Cells and culture conditions

Experiments were performed using HeLa cells (human cervix carcinoma cells, ATCC CCL2) stably transfected with different Cx isoforms and Novikoff cells (rat hepatoma cells that endogenously express Cx43). HeLa cells were grown in Dulbecco’s modified Eagle’s medium. Novikoff cells were grown in Swim’s S-77 medium. For more details about the DNAs used for transfection and selection of clones stably expressing different Cx isoforms, we refer to (27, 28).

Electrophysiological measurements

Experiments were performed in modified Krebs-Ringer solution containing (in mM) 140 NaCl, 5 glucose, 5 HEPES, pH 7.4, 4 KCl, 2 CaCl₂, 2 Na pyruvate, and 1 MgCl₂. The cells were perfused with modified Krebs-Ringer solution at room temperature. For electrophysiological recordings, cells were grown on glass coverslips and transferred to an experimental chamber mounted on the stage of an inverted microscope (Olympus IX70, Olympus America, Center Valley, PA). Patch pipettes were filled with a solution containing (in mM) 130 KCl, 10 NaAsp, 5 HEPES, pH 7.2, 3 MgATP, 2 EGTA, 1 MgCl₂, and 0.2 CaCl₂. Junctional conductance was measured in selected cell pairs using a dual whole-cell patch-clamp system (21). In brief, each cell within a pair was voltage clamped with a separate patch-clamp amplifier (EPC-7plus, HEKA, Bellmore, NY). V_j was induced by stepping the voltage in one cell while keeping it constant in the other. Junctional current (I_j) and g_j (g_j = I_j/V_j) were measured as the change in a current in the unstepped cell. Signals were acquired and analyzed using an analog-to-digital converter (National Instruments, Austin, TX) and custom-made software (18).

Results

Origins of fast and slow gates

Simulation of GJ channel gating required consideration of two fast and two slow gates operating in series (Fig. 1 A). The hypothesis about fast and slow gating mechanisms originated from interpretation of recordings of de novo GJ channel formation observed after initiating contact between two patched insect cells (5, 29). Initial opening events of GJs exhibited relatively slow junctional current transitions (∼10–50 ms) from the closed state to the open state with conductance (γ_o) of ∼365 pS followed by V_j-induced fast gating transitions (<∼1 ms) between the open state and the residual state with conductance (γ_res) of ∼64 pS. Furthermore, similar gating transitions were reported in mammalian cells expressing different Cx isoforms (6, 7, 30, 31), and it was concluded that GJs possess two distinct types of V_j-sensitive gating mechanisms, fast and slow, or loop (5). The slow gating mechanism, also associated with chemical gating (7), e.g., acidification, leads to full uncoupling throughout slow gating transitions (Fig. 1 D, insets, blue arrows). Typically, γ_res measured in all Cxs constitute ∼1/4 to 1/5 of γ_o. Two distinct gating mechanisms were also demonstrated in Cx-based uHCs (18). Thus, collected data show that each hemichannel of the GJ channel possesses the fast gate, with conductance of open γ_F,o and residual γ_F,res states, and the slow gate, with conductance of γ_S,o and γ_S,c = 0, operating in series (Fig. 1 A).

Why it should be considered that the slow gate contains multiple closed states

In two independent studies of gating processes in Cx-based uHCs, it was proposed that voltage-sensitive gates most likely can occupy two or more closed states (19, 26) (Fig. 1 C). Below, we present our experimental data that support this view. Fig. 2 B shows the g_j-V_j dependence measured in a pair of HeLaCx43 cells in response to the V_j protocol shown in Fig. 2 A; g_j was relatively low (∼1.5 nS), allowing visualization of unitary gating events of ∼13 channels, each with open-state conductance of ∼115 pS (32). At the beginning of the initial V_j ramp (red arrow), all the channels were open (solid pink line; g_j = γ_o × N, where N is the number of channels), and the number of fully open channels gradually decreased with increasing V_j until they all were in the residual conductance state (pink dashed line; g_j,_ss = γ_res × N). The g_j trace measured in response to the return V_j ramp (green arrow) shows delayed recovery of g_j resulting in a modest degree of hysteresis (double-ended blue arrow). In cells expressing Cx45, the same V_j protocol produced a larger hysteresis and incomplete recovery at the end of the return ramp (Fig. 2 C). We now have observed hysteresis in g_j-V_j plots in all Cxs examined thus far, i.e., Cx26, Cx30.2, Cx36, Cx37, Cx40, Cx43, Cx45, and Cx57.

Figure 2.

Hysteresis of g_j-V_j relationships measured experimentally and simulated using an S16SM. (A–C) g_j-V_j plots measured in Cx43 (B), and Cx45 (C) GJs using the V_j protocol shown in (A). Hysteresis of g_j-V_j plots is indicated by blue arrows. (D) Simulated I_j-time trace obtained using an S16SM shows symmetric I_j responses during the initial (red arrow) and returning (green arrow) V_j ramps, indicating an absence or minimum of hysteresis in g_j dependence on V_j.

Fig. 2 D shows simulated I_j traces obtained in response to the V_j protocol shown in Fig. 2 A using an S16SM; parameters of the model were selected to represent Cx45 GJ gating. Data show that the I_j trace during the returning ramp is a mirror reflection of that during the initial ramp (red arrow). Thus, an S16SM containing one closed state for both fast and slow gates does not adequately reproduce the I_j-time asymmetry or, consequently, hysteresis of the g_j-V_j relationship (see also Fig. 7 C, which shows an absence of significant hysteresis of g_j-V_j dependence, simulated using an MC16SM). Thus, gating models containing one closed state for fast and slow gates cannot replicate hysteresis of g_j-V_j plots.

Figure 7.

Dependence of g_j-V_j hysteresis on probabilities of p_c1→c2 and p_c2→c1 transitions. I_j-time (B) and g_j-V_j (C–E) dependence, simulated using an MC16SM and an MC36SM all obtained using V_j protocol shown in (A). An asymmetry in I_j-time (B) and hysteresis of g_j-V_j (C) plots obtained using an MC16SM are minimal and they increased substantially using an MC36SM. (D) g_j-V_j plots obtained using an MC36SM at various p_c1→c2 and p_c2→c1, but holding the same ratio, k₁ = p_c1→c2/p_c2→c1. (E) g_j-V_j plots obtained using an MC36SM in which p_c1→c2 = 0.01, whereas p_c2→c1 = 0.0001, 0.0002, and 0.0005.

Furthermore, Fig. 3 A shows that g_j,norm measured in a HeLaCx43-EGFP cell pair gradually decreased in response to 4% CO₂, which cooperates with a blocking effect of acidification on GJ communication (33, 34). After full recovery of g_j after washout, cells were briefly exposed to heptanol (3 mM), which caused fast uncoupling and rapid recovery (Fig. 3 B). Then, we repeated the same protocol as in Fig. 3 A, but with a brief exposure to heptanol saturated with 4% CO₂. As expected, this caused fast uncoupling, but cells remained uncoupled even after washout of heptanol (Fig. 3 C), which is very different from the prediction shown by the pink curve in Fig. 3 A. We assume that acidification increases the probability of transitions between initial-closed and deep-closed states (p_c1→c2) and/or reduces the probability of c₂→c₁ transitions (p_c2→c1). Transfer of slow gates by heptanol into the c₁ state allows them to transit into the c₂ state, where they are “locked” by reduced p_c2→c1 due to continuous exposure to CO₂. Conversely, Fig. 3 D shows that V_j steps applied during uncoupling by CO₂ in Cx47-expressing cells (blue curve) reduced g_j to g_j,res. After the end of the V_j steps, g_j returned relatively quickly to the predicted level (Fig. 3 D, pink curve). It is well established that Cx40 and Cx47 GJs at V_j values up to ∼80 mV are mainly operated by the fast gating mechanism, evident by a stable g_j,min (31, 35, 36). In contrast, when Cx43-EGFP cells were used instead of Cx47, V_j steps produced a response similar to the response to heptanol, i.e., fully closing the channels significantly reduced g_j recovery (not shown). It is well established that tagged EGFP inactivates the fast gate in Cx43 (32). Thus, it is likely that the c₂ state is a property of the slow gate, whereas the fast gate does not have a c₂ state. Fig. 3 D also shows that octanol (green line) caused full uncoupling with a very slow g_j recovery, as expected from the heptanol/CO₂ results.

Figure 3.

Experimental data supporting the concept of multiple closed states. (A–C) Heptanol augmented CO₂-induced g_j decay and delayed its recovery. (D) Fast V_j-gating does not affect a recovery from CO₂-induced uncoupling.

Furthermore, we reported that in cells expressing Cx36, g_j decreased when [Mg²⁺]_i was increased from control conditions (∼1 mM) to 5 mM (17). Fig. 8 B in (17) and its modified version in the Supporting Material for this article (Fig. S1) show differences in g_j recovery from uncoupling caused by decanol (0.5 mM), which was applied when g_j reached the steady-state at different levels of [Mg²⁺]_i. Recovery of g_j was full at [Mg²⁺]_i = 0.01 mM, ∼50% at [Mg²⁺]_i = 1 mM, and ∼25% at [Mg²⁺]_i = 5 mM. We hypothesize that, similar to acidification, an increase in [Mg²⁺]_i increases p_c1→c2, and/or reduces p_c2→c1, which explains the changes shown in Fig. S1. Chemical uncouplers typically lead to full uncoupling with slow single-channel transitions to the fully closed state like that shown in the left inset of Fig. 1 D, which supports the view that their effect is executed through the slow gating mechanism. In summary, transitions of the slow gate into the c₁ state, whether by voltage or chemical factors, tend to increase the likelihood of gate transfer into the c₂ state.

Development of a stochastic S36SM of GJ channel gating

Thus far, collected data support our contention that the V_j sensitive fast gating mechanism most likely contains one closed state and exhibits o↔c transitions, and the pair of fast gates can be in 2² states, as described in an S4SM (12). Otherwise, the slow gating mechanism operates according to a linear reaction scheme, o↔c₁↔c₂. Therefore, the pair of slow gates can be in 3² states. Consequently, we developed a stochastic 36 (4x9)-state model (S36SM) of GJ channel gating. In this model, as in an S16SM (13), each of four gates is characterized by the unitary conductance of open and closed states, their I-V rectification, and gating polarity, i.e., whether the gate closes at relative negativity or positivity of V_j on its cytoplasmic side. Voltages across the fast gate (V_F) or the slow gate (V_S) define o↔c or o↔c₁ transitions, respectively. Then, V_j = V_F,A + V_S,A + V_F,B + V_S,B, where A and B stand for aHCs A and B. Closing one gate changes voltage across the other three gates in series and this will affect their probability of changing the state over a discrete time interval (Δt). The equilibrium constant between the open and closed states of slow (K_S,o↔c) and fast (K_F,o↔res) gates will be determined through an exponential relationship, as proposed earlier (10). Thus, ${\text{K}}_{\text{F,o}\leftrightarrow \text{res}}={\text{e}}^{{\text{A}}_{\text{F}}\left(-\Pi \times {\text{V}}_{\text{F}}-{\text{V}}_{\text{F,o}}\right)}$ and ${\text{K}}_{\text{S,o}\leftrightarrow c}={\text{e}}^{{\text{A}}_{\text{S}}\left(-\Pi \times {\text{V}}_{\text{S}}-{\text{V}}_{\text{S,o}}\right)}$, where A_F and A_S characterize sensitivity to voltage, V_F,o and V_S,o are voltages at which K_S,o↔c and K_F,o↔res are equal to 1, and Π is a gating polarity (+1 or −1). Then, new sets of V_S and V_F values estimated at the end of Δt_n allow for evaluation of transition probabilities for each gate: ${\text{p}}_{\text{o}\to \text{c}}={\text{P}}_{\tau }\times \text{K/}\left(1+\text{K}\right)$, ${\text{p}}_{\text{o}\to \text{o}}=1-{\text{p}}_{\text{o}\to \text{c}}$, ${\text{p}}_{\text{c}\to \text{o}}={\text{P}}_{\tau }\text{/}\left(1+\text{K}\right)$, and ${\text{p}}_{\text{c}\to \text{c}}=1-{\text{p}}_{\text{c}\to \text{o}}$, where P_τ is a constant that allows one to adjust Δt to real time comparable with that in the experiment. This leads to new states of gates (Fig. 1 B) and new sets of V_S and V_F, along with a new g_j value at the beginning of Δt_n+1, and the process will be repeated.

It was proposed that the I-V rectification results from the number and position of charged residues along the channel pore and that it can be described using an electrodiffusive PNP model derived from the Poisson-Nernst-Plank equation (19). Here, we used an exponential function, as proposed earlier (9), to describe dependence of unitary conductance of gates on voltage, e.g., ${\gamma }_{\text{F,res}}={\gamma }_{\text{F},\text{res},0}\times {\text{e}}^{-{\text{V}}_{\text{F}}{\text{/R}}_{\text{F,res}}}$, where γ_F,res,0 is γ_F,res at V_F = 0 and R_F,res is the rectification constant of the residual state. An accounting for the influence of rectification is relatively complex, because voltage drop across the gate affects its conductance, which in turn influences V_F and V_S across all gates. Thus, the process requires several cycles of calculation until a conditional steady-state is approached, as detailed earlier (13) and illustrated in Fig. S2. Recently, we reported that I-V rectification of gates can differ from exponential function and it can be modulated, e.g., it can transform from exponential to hyperbolic function under an increase of [Mg²⁺]_i (37). Thus, in the model, we made it possible to express I-V rectification in any analytical form that follows from experimental data.

In an S36SM, transition probabilities of the fast gate remain the same as in the S16SM, i.e., ${\text{p}}_{\text{o}\to \text{c}}={\text{P}}_{\tau }\times \text{K/}\left(1+\text{K}\right)$ and ${\text{p}}_{\text{o}\to \text{o}}=1-{\text{p}}_{\text{o}\to \text{c}}$. For the slow gate, we made an assumption that p_c1→c2 and p_c2→c1 do not depend on voltage, because c₁ and c₂ represent fully closed states and most likely are exposed to all V_j applied across the GJ channel. Therefore, p_c1→c2 and p_c2→c1 were introduced as voltage-independent variable parameters that can be estimated from experimental g_j-V_j dependences using global optimization (GO) algorithms (see Movies 1 and 2, illustrating GO of g_j-V_j plots, at http://connexons.aecom.yu.edu/Research.htm and http://connexons.aecom.yu.edu/Optimization.htm). Thus, for the slow gate, probabilities of o→o and o→c₁ transitions remain dependent on K in the same fashion as in the S16SM, i.e., ${\text{p}}_{\text{o}\to \text{c}1}={\text{P}}_{\tau }\times \text{K/}\left(1+\text{K}\right)$ and ${\text{p}}_{\text{o}\to \text{o}}=1-{\text{p}}_{\text{o}\to \text{c}1}$. However, p_c1→o and p_c1→c1 cannot be the same as p_c→o and p_c→c used in an S16SM, because all possible transitions related to the c₁ state should satisfy a condition, ${\text{p}}_{\text{c}1\to \text{o}}+{\text{p}}_{\text{c}1\to \text{c}1}+{\text{p}}_{\text{c}1\to \text{c}2}=1$. We assumed that both p_c1→o and p_c1→c1 should decrease proportionally to changes of (1 − p_c1→c2) as follows:

$${\text{p}}_{\text{c}1\to \text{o}}={\text{p}}_{\text{c}\to \text{o}}\times \left(1-{\text{p}}_{\text{c}1\to \text{c}2}\right)={\text{P}}_{\tau }\times \left(1-{\text{p}}_{\text{c}1\to \text{c}2}\right)/\left(1+\text{K}\right)\text{and}{\text{p}}_{\text{c}1\to \text{c}1}=\left(1-{\text{p}}_{\text{c}1\to \text{o}}\right)\times \left(1-{\text{p}}_{\text{c}1\to \text{c}2}\right).$$

Transitions related to the c₂ state are described as p_c2→c1 and p_c2→c2 = 1 – p_c2→c1.

Development of an MC36SM of GJ-channel gating

Due to the stochastic nature of the model, simulated g_j records exhibit noise in g_j-time or g_j-V_j relationships that decreases at higher numbers of GJ channels. However, this increases computation time in proportion to the number of channels. To accelerate the simulation and eliminate noise in g_j records, instead of evaluating gating of each GJ channel, we calculated the probabilities of all 36 states at different V_j values by using a Markov chain approach. Basically, an MC36SM evaluates an expected/averaged conductance of the GJs.

Earlier, we used stationary/steady-state probabilities of the Markov chain to model GJ-channel voltage gating (38, 39). However, this approach was only suitable in evaluating steady-state g_j-V_j dependencies, which can be observed experimentally by measuring g_j at the end of long-lasting V_j steps. To evaluate dynamic changes of g_j in response to V_j of any form, or g_j recovery after V_j returns to zero, we have used transitive rather than steady-state probabilities of MC36SM.

In an MC36SM, g_j values at a given V_j are estimated from probability vectors, p⁽ⁿ⁾, of all 36 states of the GJ channel at the discrete time moment, t_n, that are determined from open probabilities of fast and slow gates. Each of 36 states can be expressed as a 4-tuple, s_i = (s_F,A; s_S,A; s_S,B; s_F,B), where components s_F,A and s_F,B denote o and c states and components s_S,A and s_S,B denote o, c₁, and c₂ states of gates of A and B hemichannels, respectively. The transition probability matrix, P, consists of 36 × 36 entries, each of which describes transition probabilities between states and can be expressed through transition probabilities of individual gates (see Supporting Material). The evaluation of conductances for each of 36 states (γ = γ(s_i)) and their I-V rectification is the same as in an S36SM. Then, the transitive probability vector, p⁽ⁿ⁾, at a discrete time moment, t_n, can be estimated from a following recursive relation, p⁽ⁿ⁾ = p⁽ⁿ⁻¹⁾ P. Finally, the junctional conductance is estimated as an expected value of all possible channel conductances, ${\text{g}}_{\text{j}}={\sum }_{\text{i}}{\text{p}}_{\text{i}}\times \gamma \left({\text{s}}_{\text{i}}\right)$, where p_i is the i^th coordinate of probability vector p⁽ⁿ⁾. If an MC36SM is used to evaluate stationary/steady-state g_j-V_j dependence, then an iterative process is extended until the transitive probability reaches a steady-state, i.e., p⁽ⁿ⁾ = p⁽ⁿ⁻¹⁾.

Fig. 4 shows that g_j-time (Fig. 4, B and C) and g_j-V_j (Fig. 4, E and F) plots simulated in response to V_j protocols shown in Fig. 4, A and D, by using an S36SM (black) and an MC36SM (gray) overlap. However, a data set attained using an S36SM exhibited significant noise when accounting for the presence of 100 channels (Fig. 4, B and E) and noise was smaller at 10,000 channels (see Fig. 4, C and F, insets). All g_j-time and g_j-V_j plots obtained using an MC36SM were noise free. Also, g_j records shown in Fig. 4, C and F, took ∼1 s when simulated using an MC36SM and ∼180 s when simulated using an S36SM (see also Table S1). Thus, a Markov chain approach fully eliminates noise and significantly reduces computation time. However, if there is a need to simulate gating processes at the single-channel level, as shown in Fig. 5, then an S36SM must be used.

Figure 4.

g_j-time and g_j-V_j plots simulated using an S36SM and an MC36SM. (B and C) Normalized g_j-time traces obtained in response to the V_j step (A) using an S36SM encompassing 100 (B) and 10,000 (C) channels (black). Also shown are the response traces using an MC36SM (gray). (E and F) The same as in (B)–(C), but with the normalized g_j-V_j plots obtained using the V_j ramp protocol shown in (D). The insets in (C) and (F) show that noise in the g_j recordings is reduced using an MC36SM. In contrast, noise is substantial using an S36SM but is diminished by increasing the number of GJ channels.

Figure 5.

Simulated and experimental records of g_j relaxation at the single-channel level. (A) V_j protocol used for simulation of g_j-time plots shown in (B)–(E). (B) g_j-time trace simulated using an S16SM in the junction containing four channels. (C–E) g_j-time traces simulated using the same parameters as in (B), but using an S36SM with different values of p_c1→c2 and p_c2→c1. (F) Experimental g_j-time trace measured in the cell pair expressing Cx43-EGFP (gray) obtained in response to the V_j protocol shown in (A); the black line is a fitting curve obtained using an MC36SM and the Exkor GO algorithm.

V_j-gating models are multiparametric, and it is impractical to evaluate their parameters manually from experimental data. To automate this process, we have used GO algorithms (13). The resulting fitting curves are shown in Fig. 5 (see Fig. 8). However, the GO procedure requires us to perform numerous simulations of the model at different values of gating parameters, which can last several hours if an S36SM is used. In performing GO of the experimental records shown in Fig. 6 (see Fig. 8), we combined an MC36SM with GO algorithms, preferentially Exkor (40), which allowed us to reduce the GO time from hours to several minutes. Additionally, GO time can be significantly shortened by restricting the search space to smaller ranges of parameter values. This was achieved by data mining of the collected parameter values for different Cx isoforms.

Figure 8.

Dependence of g_j-V_j hysteresis on duration of V_j-protocol. Hysteresis of g_j-V_j plots measured in cell pairs expressing Cx45 (B) and Cx36 (C) (gray traces) in response to V_j protocol shown in (A). Black lines are the fitting curves obtained using an MC36SM combined with the Exkor GO algorithm.

Figure 6.

Simulated and experimental records of V_j-gating and g_j relaxation at the macroscopic level. (A) V_j protocol used for simulation of g_j and probability of c₂ state changes. (B) g_j recovery after application of −100 mV steps of different duration (A). The recovery of g_j consists of two distinct phases indicated by circles. Initially, g_j recovers fast through c₁→o transitions followed by slower recovery through c₂→c₁→o transitions, which is reflected in changes of c₁ (C) and c₂ (D) probabilities. Plots in (B)–(D) show that the process reaches a steady-state if the V_j step is longer than 1000 a.u.; p_c1→c2 and p_c2→c1 are equal to 0.01 and 0.001, respectively. (E) The g_j recording measured in response to a V_j step and repeated ramps in Novikoff cells endogenously expressing Cx43. The blue and red traces show fitting of the experimental g_j-time curve using an MC16SM and an MC36SM, respectively, combined with the Exkor GO algorithm.

Below, we apply the S36SM and the MC36SM to simulate several experimentally observed phenomena that cannot be adequately reproduced using a 16-state model.

Delayed recovery of g_j after V_j-gating

Fig. 5 B shows that g_j relaxation after a V_j-step (Fig. 5 A) was fast when the simulation was performed using an S16SM; the cell pair contained four GJ channels with open-channel conductance of 10 pS. Fig. 5, C–E, shows that an S36SM allows the relaxation of g_j to be modified in a broad range by changing p_c1→c2 and p_c2→c1. Experimental data measured in cells expressing different Cx isoforms show that g_j relaxation is significantly slower than that shown in Fig. 5 B. Fig. 5 F shows g_j changes in HeLa cells expressing Cx43-EGFP in response to a V_j step of −110 mV. Stepwise transitions of g_j (gray) allow us to predict that the cell pair expressed nine GJ channels. At the end of the V_j step, all channels closed. Repeated bipolar pulses applied after the V_j step (Fig. 5 F, inset) allowed us to measure g_j recovery, which was slow. The fitting of the experimental g_j-time trace using an MC36SM and the Exkor GO algorithm (black curve) revealed that p_c1→c2 = 0.11 and p_c2→c1 = 0.004. Also, we assumed that the fast gate was inactive, as reported previously for Cx43-EGFP GJs (32).

To study transitions of slow gates between o, c₁, and c₂ states during a V_j step and after returning of V_j to zero, we examined the dynamics of their probabilities. Fig. 6 shows changes in the probabilities of c₁ (Fig. 6 C) and c₂ (Fig. 6 D) states during V_j steps of different duration. Fig. 6 B shows that g_j recovery curves consist of two distinct phases (open circles on g_j recordings). At first, the recovery is fast and resembles that obtained using an MC16SM model, which mainly represents opening of slow gates residing in the c₁ state. Afterward, there was a much slower recovery of g_j, which we expect is due to the opening of slow gates from the c₂ to the c₁ state and then to the open state. At longer V_j pulses, an amplitude of the fast phase decreased along with a decrease in probability of the c₁ state (Fig. 6 C), whereas the amplitude of the slow phase increased along with an increase in probability of the c₂ states (Fig. 6 D). These changes approached steady-state at equilibrium between the c₁→c₂ and c₂→c₁ transitions.

Fast and slow phases in g_j recovery after V_j-induced gating were observed in cells expressing different Cx isoforms, including those reported for Cx43 in Figs. 1 and 8 of (32). Fig. 6 E shows a representative record demonstrating fast and slow g_j recovery after a V_j step in Novikoff cells endogenously expressing Cx43. The fast and slow phases of g_j recovery were also reported in Cx57-EGFP gap junction channels (see Fig. 2 E in (14)). Fig. 6 E shows that an MC36SM fits the experimental data pretty well, whereas the MC16SM fails to capture the slow recovery phase of g_j.

Hysteresis of g_j-V_j plots

Data shown in Fig. 2, B and C, demonstrate that GJ channels exhibit well-expressed hysteresis of g_j-V_j plots. Fig. 7 B shows no visible asymmetry in the I_j-time recording simulated using an MC16SM during initial and returning ramps (Fig. 7 A). Correspondingly, there was very small hysteresis in g_j-V_j plots (Fig. 7 C). Simulations using an MC36SM show significantly reduced I_j (Fig. 7 B) during an initial ramp and even stronger I_j reduction during a returning V_j ramp, which induced an asymmetry in the g_j-time recording and, correspondingly, hysteresis in the g_j-V_j plots (Fig. 7 C). All parameters of gates were the same, but in an MC36SM, p_c1→c2 = 0.1 and p_c2→c1 = 0.01.

Fig. 7, D and E, shows that changing p_c1→c2 and p_c2→c1 allows for modulating the magnitude of hysteresis in the g_j-V_j plots in a broad range. In Fig. 7 D, all g_j-V_j plots were simulated at the same ratio, k₁ = p_c1→c2/p_c2→c1 =10. Decay in p_c1→c2 and p_c2→c1 to the same degree delayed g_j recovery and increased hysteresis. Fig. 7 E shows that at constant p_c1→c2 (0.01), a decay in p_c2→c1 resulting in an increase of k₁ from 20 to 100 increased hysteresis of g_j-V_j plots severalfold. Thus, hysteresis in g_j-V_j plots can be increased broadly by reducing p_c1→c2 at the same k₁ or by increasing k₁ at the same p_c1→c2.

Moreover, hysteresis of g_j-V_j plots at given values of p_c1→c2 and p_c2→c1 is not constant and decreases with an increase in duration of the V_j ramps used. Fig. 8 B shows that hysteresis decreased when the duration of rising and declining ramps in the V_j protocol (Fig. 8 A) increased from 10 to 25 s. The g_j-V_j plots (gray traces) were recorded in HeLa cells expressing Cx45. Importantly, we observed hysteresis of g_j-V_j plots in all examined Cxs so far, such as Cx36, Cx43, Cx47, and Cx57. Fitting of the experimental g_j-V_j plots of Cx45 GJs using an MC36SM combined with the Exkor GO algorithm (Fig. 8 B, black curves) revealed that p_c1→c2 = 0.04 and p_c2→c1 = 0.003. Fig. 8 C shows hysteresis in g_j-V_j plots (gray traces) obtained in cells expressing Cx36-EGFP; the duration of V_j ramps was 20 s. GO of experimental records (black traces) revealed that p_c1→c2 = 0.015 and p_c2→c1 = 0.001 (see also Table S2).

Functional efficiency of GJ channels

Previously, we demonstrated that the ratio of functional or open GJ channels at a given time (N_O) to their total number (N_T), N_O/N_T, under normal physiological conditions was ∼0.1 for Cx43 (27), ∼0.04 for Cx45 (15), ∼0.01 for Cx57 (14), and ∼0.008 for Cx36 (41), all expressed in HeLa cells. In MesV neurons expressing mainly Cx36, this ratio was ∼0.001 (42). To test the degree at which a deep-closed state can account for a low functional efficiency of GJs, we performed simulations using an MC36SM (Fig. 9). In all simulations, initially V_j = 0 and all GJ channels are open. Over time, g_j decayed due to stochastic transitions of gates until steady-state values of g_j (g_j,ss,Vj=0) were reached. They reflect an averaged distribution of channels between fully open, partially open (fast gate(s) closed), and fully closed (at least one slow gate is closed). Then, the V_j step (Fig. 9 A) was applied to measure V_j-gating.

Figure 9.

g_j-time and g_j-V_j dependencies simulated using an MC36SM at various p_c1→c2 and p_c2→c1. (B) A family of normalized g_j traces at p_c1→c2 = 0.02 while p_c2→c1 varied from 0.01 (k₁ = 2) to 0.0005 (k₁ = 20). All records were obtained using a V_j protocol shown in (A). The left and right insets show that the dynamics of g_j decay are similar independent of p_c1→c2 and p_c2→c1 values, whereas g_j decelerated substantially at smaller k₁. (D) Normalized g_j-V_j dependence simulated at the same MC36SM parameters as in (B), but using a V_j protocol shown in (C). Data shown demonstrate that p_c1→c2 and p_c2→c1 very significantly suppress g_j values over the entire V_j range, but to a lesser degree increased the sensitivity to V_j-gating that is reflected in the g_j-V_j plot shown in the inset.

Experimental evaluation of the ratio N_O/N_T, called functional efficiency (27), was based on an assumption that the measured g_j can be transformed into N_O from the ratio N_O = g_j/γ_j, where γ_j is a single-channel conductance, whereas N_T was evaluated from fluorescence of EGFP tagged to Cx43, as reported in (32). Parameters of gates were selected so that GJs roughly reflect highly V_j-sensitive Cxs, such as Cx45 or Cx57. Fig. 9 B shows that at p_c1→c2 = 0 that corresponds to an MC16SM, g_j,ss,Vj=0 was ∼0.52, which is well above expected values (14, 15). Color traces in Fig. 9 B were obtained using an MC36SM. All parameters were the same as in an MC16SM, but p_c1→c2 was constant and equal to 0.02, whereas k₁ = p_c1→c2/p_c2→c1 varied. At k₁ = 20, g_j,ss,Vj=0 decayed, reaching ∼0.06. This allows one to explain to a major extent the experimentally measured low functional efficiency of GJ channels, which cannot be achieved using an MC16SM. The left and right insets show that the dynamics of g_j decay during the V_j step are similar independent of the p_c1→c2 and p_c2→c1 values, whereas g_j relaxation was delayed substantially at smaller k₁.

Furthermore, we explored how changes in p_c1→c2 and p_c2→c1 affect V_j-gating in response to negative and positive V_j ramps (Fig. 9 C). Fig. 9 D demonstrates a significant decline in the magnitude of g_j-V_j plots when k₁ increases. A family of g_j-V_j plots shown in the inset of Fig. 9 D were additionally normalized to their maximum values at V_j = 0. These data demonstrate that an increase in k₁ slightly raised sensitivity to V_j-gating. However, the difference among g_j-V_j plots at different values of k₁ is not significant in contrast to changes in the g_j relaxation processes, which is reflected in the right inset of Fig. 9 B.

Discussion

Two fast and two slow gates arranged in series in an S16SM (13) allowed us to describe voltage-sensitive gating properties of homotypic and heterotypic GJ channels more precisely than when an S4SM was used (12). However, recently accumulated data on V_j- and chemically mediated gating in GJs formed of various Cx isoforms revealed several phenomena, e.g., those shown in Figs. 2 and 3 in this article or Fig. 8 B in (17), which were difficult or impossible to explain by an S16SM. In light of these data, a hypothesis was generated that the slow gate can be in more than one closed state. Thus, we developed an S36SM and an MC36SM enclosing two closed states for the slow gate, c₁ and c₂. If p_c1→c2 = 0, then an S36SM shows results identical to those produced by an S16SM, i.e., 16-state models can be viewed as a separate case of more general 36-state models. We believe that an S36SM is the simplest extension of an S16SM, allowing replication of several experimentally proven phenomena, such as slow g_j recovery, hysteresis in g_j-V_j plots, low functional efficiency of GJs, and others related to chemically mediated gating, that cannot be reflected adequately using a 16-state model.

Delayed g_j recovery and hysteresis of the g_j-V_j relationship

According to an S16SM, g_j recovery after V_j returns to 0 occurs relatively quickly and with the same kinetics independent of the “history” of gating (i.e., it does not depend on the magnitude and duration of the V_j values used). Typically, in a physiological experiment at higher and longer-lasting V_j, the kinetics of g_j recovery are slower and can take tens of seconds. A recovery of g_j after chemically mediated gating can take tens of minutes. Fig. 5 shows that the time constant of g_j recovery can be modulated in a broad range using 36-state gating models by varying values of p_c1→c2 and p_c2→c1. Furthermore, Fig. 6 shows that g_j recovery has two distinct phases, fast and slow, associated with the numbers of channels in which slow gates reside in the c₁ and c₂ states, respectively. All these observations can be reproduced using an S36SM at the single-channel level and an MC36SM macroscopically. Fitting of g_j-time or g_j-V_j plots exhibiting hysteresis by using an MC36SM allows us to predict p_c1→c2 and p_c2→c1 values. Furthermore, we show in Fig. 8 that the magnitude of hysteresis decreases with an increase in the duration of V_j ramps and theoretically can approach zero at very long V_j ramps, which for some Cx isoforms can exceed the duration of experiments lasting on average ∼30 min.

Does a deep-closed state explain low functional efficiency of GJ channels?

Previously, it was demonstrated that only a small fraction of GJ channels formed of various Cx isoforms and assembled into junctional plaques are functional (14, 15, 41, 43). The remaining channels were assumed to be nonfunctional and were defined as those that cannot be gated by V_j (15). Various hypotheses were proposed to explain the existence of nonfunctional or permanently silent channels, related, for example, to the lag time necessary for channel maturation after docking of HCs, the peripheral versus central location of channels in junctional plaques, differences in phosphorylation, etc. Our studies show that channels that were previously believed to be nonfunctional could remain functional but probabilistically dwell in the c₂ state or exhibit c₁↔c₂ transitions. This assumption puts all channels into the framework of the gating model and eliminates the need to differentiate them into functional and nonfunctional categories. However, it is possible that some of the channels are indeed permanently silent, but most likely not as much as 90% or more. Our data show that N_O/N_T can be modulated substantially by chemical factors. For example, alkalization and acidification of Cx45 causes increases and decreases of N_O/N_T, respectively, by severalfold (15). Similarly, low [Mg²⁺]_i increases N_O/N_T in Cx36 (17), which can be explained by a reduction in p_c1→c2 and/or an increase in p_c2→c1, whereas elevated [Mg²⁺]_i reduces N_O/N_T, which can be explained by an increase in p_c1→c2 and/or a decrease in p_c2→c1. Thus, introduction of a deep-closed state sheds more light on the possible mechanisms of low functional efficiency of GJs.

Are there one or more deep-closed states?

It should be noted that for the time being, the existence of a deep-closed state is still hypothetical. Unlike a residual state of the fast gate, it cannot be directly measurable, because conductance of the c₁ and c₂ states virtually equals zero. It can be hypothesized that 1) the c₂ state represents synchronous conformation of six Cxs, which form an aHC, resulting in a higher energetic barrier for c₂→c₁ transitions, or 2) conformation of one of the six Cxs leads to full channel closure, whereas closing conformations of the other five Cxs lead stepwise to deeper closing states. For example, if the second Cx transits into a deep-closed state, which we name the c₃ state, then we can assume that the slow gate will operate according to linear reaction scheme o↔c₁↔c₂↔c₃ and the model would transform from 36 states into 64 states.

Fig. 10 shows that at p_c1→c2 = p_c2→c3 = 0.05 and p_c2→c1 = p_c3→c2 = 0.01, the time constant of g_j recovery using an MC64SM is around four times longer than that estimated using an MC36SM. Our data show that recovery of g_j can be delayed by increasing p_c1→c2 and/or reducing p_c2→c1 in the MC36SM, but the introduction of more than one deep-closed state of the slow gate extends the range of the time constant of g_j relaxation. Also, Fig. 10 B demonstrates that when an MC64SM is used, g_j drops to a lower level under V_j=0, referred to as g_j,ss,Vj=0, which in the example shown explains an approximately twofold reduction of the functional efficiency of GJ channels. Moreover, g_j responds similarly to a V_j step in both models (left inset), but g_j recovery is significantly slower (right inset) when the second deep-closed state is included. The width of hysteresis in g_j-V_j plots under the same V_j protocol was also wider in the model with two deep-closed states (not shown). Thus, inclusion of multiple deep-closed states could be considered in explaining a very long g_j relaxation time after V_j- or chemically mediated gating, large hysteresis of g_j-V_j relationships, or functional efficiency of GJ channels well below 1%.

Figure 10.

Comparison of g_j recovery using models with a single and two deep-closed states. (A) V_j step of −100 mV. (B) Simulated g_j records in response to a V_j step using an MC36SM containing a single (gray) and two (black) deep-closed states. Normalized g_j traces show that the kinetics of g_j decay during initiation of gating does not differ between the two models (left inset), whereas g_j recovery from gating is significantly slower using the model with two deep-closed states (right inset).

Potential novelties, limitations, and future directions

Fig. 11 summarizes major characteristics of the gating processes that were used in developing stochastic and Markov chain 36-state models. Initial ideas related to the multiple closed states of GJ channels were raised by two other groups based on data obtained in studies of Cx32 and Cx46 hemichannels (19, 26). The data presented above allowed us to suggest that multiple closed states can be attributed to the slow gating mechanism. Nevertheless, at this stage, we cannot eliminate the possibility that this property can be applied also to fast gating.

Figure 11.

Summary of the principal aspects of gating processes reflected in stochastic and Markov chain 36-state gating models of the GJ channel. Operation of each gate depends on the fraction of V_j across it. Changes in the state of one gate influence operation of the other three gates.

Figs. 9 B and 10 show that kinetics of g_j decay during V_j-gating are the same whether 16-state or 36-state models are used. Furthermore, normalized steady-state g_j-V_j plots show a relatively insignificant increase in V_j-gating, even though p_c1→c2 and p_c2→c1 varied in a broad range (Fig. 9 D). These data demonstrate that gating parameters estimated in previous studies using an S16SM model (13, 15, 16, 17, 37) should be close to those evaluated using an MC36SM. Instead, the major differences are in channel-reopening kinetics, which are significantly slower due to c₁↔c₂ transitions. Relaxation of GJ channels from gating remained out of the focus of research for a long time because major attention was being paid to measurements of steady-state g_j-V_j dependence. Thus, we believe that the proposed 36-state model better reflects the latest knowledge on the physiology of GJ channels.

Furthermore, we view that developed 36-state models open an avenue allowing one to describe chemical gating and, most importantly, to integrate voltage- and chemically mediated gating into a single model. We assume, as proposed earlier (2), that chemical coupling modulators execute their effect through the gating element of the slow gating mechanism triggered by different sensorial elements, including those sensitive to V_j and, for some Cxs, to transmembrane voltage, V_m (3). This hypothesis is supported by data showing that uncoupling induced by long-chain alkanols, arachidonic acid, or high [Ca²⁺]_i or [H⁺]_i can be reversed by changing V_j in Cx32-expressing cells (44) or by hyperpolarization in cells that demonstrate V_m-sensitive gating (45, 46). This view is supported also by well-established findings showing that chemical uncouplers reduce g_j to zero and that single-channel events demonstrate transitions from the open/main state to the fully closed state, as illustrated in Fig. 1 D, under CO₂-induced acidification. If action of chemical uncouplers would be executed through the fast gate, then we should see g_j decay to a residual conductance. Thus, a deep-closed state of the slow gate could offer an explanation for chemically mediated gating. For example, it may be suggested that acidification-induced uncoupling, documented in various Cx isoforms (15, 34, 47), generally can be explained through an increase in p_c1→c2 and/or reduction in p_c2→c1. Similarly, high-[Mg²⁺]_i-mediated uncoupling in Cx36 GJs (17), which is determined by the hexametric ring formed of D47 residues in the first extracellular loop (37), may also act through changes in p_c1→c2 and/or reduction in p_c2→c1. Thus, we think that the deep-closed state(s) is important in accounting for g_j modulation by chemical factors, and the models presented integrate currently existing knowledge on the gating mechanisms of GJ channels. We expect that structure-function studies will bring new views on regulation of Cx-based channels, which will be a new challenge in advancing gating models of GJs.

Author Contributions

F.F.B. designed and performed the experiments and coordinated the study. M.S, T.K., N.P., and K.M. performed the experiments, analyzed the data, and critically revised the manuscript.

Editor: James Keener.