Mathematical model of vertebrate gap junctions derived from electrical measurements on homotypic and heterotypic channels

Authors


Corresponding authors R. Weingart: Department of Physiology, University of Bern, Bühlplatz 5, 3012 Bern, Switzerland. Email: weingart@pyl.unibe.ch

Abstract

  • 1A mathematical model has been developed which describes the conductive and kinetic properties of homotypic and heterotypic gap junction channels of vertebrates.
  • 2The model consists of two submodels connected in series. Each submodel simulates a hemichannel and consists of two conductances corresponding to a high (H) and low (L) conductance state and a switch, which simulates the voltage-dependent channel gating.
  • 3It has been assumed that the conductances of the high state and low state vary exponentially with the voltage across the hemichannel.
  • 4The parameters of the exponentials can be derived from data of heterotypic or homotypic channels. As a result, the behaviour of heterotypic channels can be predicted from homotypic channel data and vice versa.
  • 5The two switches of a channel are governed by the voltage drop across the respective hemichannel. The switches of a channel work independently, thus giving rise to four conformational states, i.e. HH, LH, HL and LL.
  • 6The computations show that the dogma of a constant conductance for homotypic channels results from the limited physiological range of transjunctional voltages (Vj) and the kinetic properties of the channel, so a new fitting procedure is presented.
  • 7Simulation of the kinetic properties at the multichannel level revealed current time courses which are consistent with a contingent gating.
  • 8The calculations have also shown that the channel state LL is rare and of short duration, and hence easy to miss experimentally.
  • 9The design of the model has been kept flexible. It can be easily expanded to include additional features, such as channel substates or a closed state.

In most tissues of vertebrates, adjoining cells communicate with each other via specialized membrane structures called gap junctions. Diffusional and electrical coupling by means of gap junctions is essential for a variety of biological processes including development, growth, secretion and impulse propagation (Bruzzone, White & Paul, 1996). Gap junctions constitute assemblies of intercellular channels. Each channel consists of two hemichannels (connexons) embedded in the cell membranes of adjacent cells. Each hemichannel consists of six transmembrane proteins (connexins) arranged to form an aqueous pore. So far molecular biologists have identified thirteen different vertebrate connexins encoded by a multigene family. The respective cDNAs have been cloned and sequenced. The analysis of amino acid sequences suggests that connexins cross the lipid bilayer four times, thus creating four transmembrane domains, two extracellular loops, one intracellular loop and a cytoplasmic amino and carboxy terminus. The different connexins offer the possibility of forming various types of channels, i.e. homotypic channels (identical hemichannels), heterotypic channels (different hemichannels) and heteromeric channels (hemichannels containing different connexins). This repertoire of different structures may lead to a wide spectrum of channels with distinct functional properties.

Experiments carried out on cell pairs enable one to assess the electrical properties of gap junctions and gap junction channels (Waltzman & Spray, 1995). Studies with homotypic preparations revealed that the channels exhibit several conductance states, a main open state, several substates and a residual state (Bukauskas, Elfgang, Willecke & Weingart, 1995a; Valiunas, Bukauskas & Weingart, 1997). The probability of a channel being in the main open state is controlled by the transjunctional voltage (Vj). Studies on heterotypic preparations gave indirect insight into the conductive properties of hemichannels and their voltage sensitivity (Bukauskas, Elfgang, Willecke & Weingart, 1995b; Weingart, Bukauskas, Valiunas, Vogel, Willecke & Elfgang, 1996; Bukauskas, Vogel & Weingart, 1997). Recent experiments performed on single Xenopus oocytes (Trexler, Bennett, Bargiello & Verselis, 1996) and transfected HeLa cells (Valiunas & Weingart, 1997) revealed direct information on the electrical properties of hemichannels.

The aim of the present project was to develop a generalized mathematical model which describes the electrical behaviour of vertebrate gap junction channels. A model based on macroscopic current measurements (gap junctions consisting of many channels) has been proposed before (Harris, Spray & Bennett, 1981). Our model considers microscopic currents (gap junctions consisting of a single channel) gained primarily from transfected cells expressing different vertebrate connexins. A preliminary version of the model has been presented in abstract form (Weingart et al. 1996).

METHODS

The model was derived from basic principles of electrophysics. All calculations were performed on a Sun system (Sun Microsystems, Mountain View, CA, USA) with SunOS5.4 running MATLAB version 4.2c.1 (The MathWorks Inc., Nattick, MA, USA).

THEORY

The assumptions

In 1981, the first gap junction model was proposed (Harris et al. 1981; see also Baigent, Stark & Warner, 1997). It relied on macroscopic currents measured in pairs of blastomeres isolated from amphibian embryos. To explain their data, the authors postulated that gap junction channels have two voltage-sensitive gates in series, one located in each hemichannel. Because of their symmetrical arrangement, the gates respond to voltages of opposite polarity. In addition, it was assumed that each hemichannel has two conductance states, an open state and a closed state. At small transjunctional voltages (Vj), both gates are open, and at large values of Vj, one of the gates is closed. Hence, a gap junction channel has three conformational states. In the open state, each gate senses half of Vj while in both closed states, the entire Vj develops across the closed gate and none across the open gate. Consequently, the closed gate of a channel must open before the open sister gate can sense a voltage drop and thus may close.

More recently, electrophysiologists have succeeded in studying the properties of single gap junction channels (see Waltzman & Spray, 1995). Experiments in our own laboratory have shown that the channels do not close completely in the presence of large transjunctional voltages (Bukauskas et al. 1995a; Bukauskas & Weingart, 1995; Valiunas & Weingart, 1997). Instead, they flicker between two non-zero conductance states, the main open state γj(main state) and the residual state γj(residual state). To account for this, we assumed that parts of Vj are sensed by both gates irrespective of the conformational state of the channel (first assumption). In the main open state, both gates are in the open position. Hence, each of them detects half of Vj. In the residual state, one gate is in the open position, the other one in the partially closed position. Hence, there is a small voltage drop across the former and a large voltage drop across the latter. These voltage drops govern the gating behaviour of each hemichannel.

Homomeric hemichannels consist of six identical connexins and thus are considered to contain six identical Vj-sensitive subgates. Provided the subgates operate independently, one may expect to see appropriate conductances. Indeed, studies on gap junction channels of vertebrates revealed several interposed states between the main open state and the residual state (Bukauskas et al. 1995a; Bukauskas & Weingart, 1995). However, due to the scarcity of quantitative data, these events were not included in the model (second assumption). Substates are rather rare, i.e. they are preferentially seen early during Vj pulses and at intermediate voltages. Hence, their omission may not seriously impair the validity of the model.

For our model we assumed an exponential relation between the hemichannel conductance and the voltage drop across the hemichannel (third assumption). This assumption was based on the following observations. Experiments on heterotypic gap junction channels indicated that γj(main state) does not obey Ohm's law, i.e. it depends on the amplitude and polarity of Vj (Bukauskas & Weingart, 1995; Weingart et al. 1996). This was discernible in channels consisting of hemichannels with largely different conductances. Recently, this prediction was verified by direct examination of hemichannels. The resulting currents revealed a non-linear relationship between the hemichannel conductance and the membrane potential (Trexler et al. 1996; Valiunas & Weingart, 1997), which was best approximated by a single exponential function.

There are further reasons for using exponential rather than linear functions to model the conductive behaviour of hemichannels. Firstly, biological processes often follow an exponential course. An example is membrane rectification which can be treated with the constant field concept of Goldmann, Hodgkin and Katz or the energy barrier concept of Eyring (Johnston & Wu, 1995). The challenge in the future will be to explain the behaviour of connexons and gap junction channels at the molecular level. This description may involve several exponential processes. Secondly, exponential functions are easy to linearize. Hence, results of computations with exponentials are readily adjustable to linear assumptions.

For kinetic reasons, it is difficult or almost impossible to measure γj(residual state) at small values of Vj (see Discussion). Hence, there is some uncertainty about the course of the conductance of a hemichannel in the residual state. To overcome this deficiency, we assumed that the hemichannel conductance of the partially closed state follows an exponential course as well (fourth assumption). The rationale was that the switching of a hemichannel from the open state to a partially closed state merely alters the multiplier and the decay constant of the exponential without affecting the form of the conductance-voltage relationship of the hemichannel. In other words, our concept of channel gating envisions a change of the electrical and/or geometrical properties of the aqueous pore.

The membrane potential, Vm, of the cells was ignored for this model. This is because the conductive and kinetic properties of vertebrate gap junctions are assumed to be insensitive to Vm (fifth assumption) (Waltzman & Spray, 1995; Bruzzone et al. 1996).

The model

Based on the assumptions above, we propose the following model for a hemichannel. Figure 1A illustrates the equivalent electrical circuit and Fig. 1B shows an arbitrary set of curves describing the relationships between conductances of a hemichannel and the voltage across the hemichannel. The two conformational states are designated high state (H) and low state (L). The conductances γH and γL are assigned to the high state and low state, respectively. Both conductances are non-Ohmic, i.e. they depend on the actual voltage across the hemichannel. Their characteristics are described by the equations:

display math(1)
display math(2)
Figure 1.

Basic model of the gap junction hemichannel

A, electrical schematic of a hemichannel. γH and γL represent the high and low conductance state, respectively. Switch S models the gating function. The parameters α and β-reflect the life times of the low state and high state, respectively. B, graph describing the low state (lower curve) and high state (upper curve) conductance of the hemichannel as a function of the voltage drop, V, across the hemichannel.

where γH and γL are the multipliers of the respective hemichannel states, V is the voltage across the hemichannel and VH and VL are the decay constants at which γH and γL decline to e−1. γH is always larger than γL. There is no such restriction with regard to VH and VL.

The gating property of the hemichannel, i.e. the transition between the high state and low state, is modelled by the switch S connected in series with γH and γL. This switch is controlled by the voltage V across the hemichannel. The parameters α and β represent the life time of the low state and high state, respectively. They are both functions of V. The mathematical description of a hemichannel can then be summarized as follows:

display math(3)

where γs represents the conductance of the hemichannel. Equation (3) serves as the basis for the computational routines. The multiplier and the decay constant of the exponential function, i.e. γs and Vs, are assigned to the corresponding values depending on the state of the hemichannel.

Two hemichannels, when assembled in series, form a leakproof gap junction channel with no current leak through the channel wall in the docking area. Figure 2A illustrates the equivalent electrical circuit for such a channel. The gating structures S1 and S2 are positioned at the interface between the channel lumen and the cytoplasm, although this may not correspond to their physical location. However, this is irrelevant for the calculations because a displacement of the switches towards the docking area does not affect the electrical properties of the model. The schematic does not intend to make any predictions about the physical nature of the gating structure. The crucial point is that the switches are governed by the voltage that develops across the respective hemichannel, i.e. Vj1 and Vj2.

Figure 2.

Two hemichannels in series form a gap junction channel

A, electrical schematic of a gap junction channel consisting of the hemichannels cx1 and cx2 of cell 1 and cell 2, respectively. B, kinetic behaviour of a gap junction channel modelled by a state machine with four states, i.e. HH, LH, HL and LL. The rate constants β and α refer to the forward reactions high → low and the backward reactions low → high, respectively.

The following equations permit the calculation of the conductances of the gap junction model in Fig. 2A:

display math(4)
display math(5)
display math(6)
display math(7)

V j represents the voltage across the entire gap junction, and the indices of the variables, i.e. 1 and 2, refer to hemichannel cx1 and cx2, respectively. Substitution of Vj1 and Vj2 in eqn (4) and eqn (5) by eqn (6) and eqn (7) leads to a system of two equations in the variables γ1 and γ2:

display math(8)
display math(9)

In order to solve the system, the computational routine has to decide which values to insert for γ1, γ2, V1 and V2. Once the values for γ1 and γ2 are determined, the conductance of the gap junction channel, γtot, can be calculated using the equation:

display math(10)

The diagram in Fig. 2B shows the appropriate state machine of the gap junction model. Since the model has two independent switches, S1 and S2, the channel is expected to have four conformational states, each codified with two capital letters, i.e. HH, LH, HL and LL. The first and second capital letters represent hemichannel cx1 and cx2, respectively.

The four conformational states illustrated in Fig. 2B must have a physical equivalent in reality. State HH corresponds to the conductance γHH, formerly called γj(main state) (Bukauskas et al. 1995a; Bukauskas & Weingart, 1995). States LH and HL correspond to the conductances γLH and γHL. They are caused by closure of the gate in cx1 and cx2, respectively, but formerly generalized to a single residual state with the conductance γj(residual state) (Bukauskas et al. 1995a; Bukauskas & Weingart, 1995). This means that two different conformational states are responsible for γj(residual state). Finally, state LL corresponds to the channel conductance γLL with the gates of cx1 and cx2 both closed. This state has been included in the present model to remain flexible for future expansions. Channel conductances related to this state have not yet been described explicitly in the literature. The reason may be that such events are rare and of short duration (see below).

Little information is currently available about kinetic properties of single gap junction channels. In order to introduce dynamic aspects to the model, we had to rely primarily on multichannel data. To start with, it is necessary to have a mathematical description of the channel state machine (see Fig. 2B) adapted to a gap junction consisting of many channels. This can be achieved by introducing rate constants for the transitions between conformational states.

The rate constants β refer to the forward reactions high → low, the rate constants α to the backward reactions low → high. The rate constants are governed by the voltage drop across the respective hemichannel, i.e. Vj1 andI Vj2. These voltage drops are functions of Vj and the actual state of the channel, i.e. HH, LH, HL or LL. Although the model has only two gates, it is necessary to introduce four forward and four backward reaction constants. The reason is that the rate constant of the gate of one hemichannel depends on the states of both hemichannels. For example, for a given Vj, β1 and β3 are not equal although only switch S1 changes the state of cx1 from high to low in both reactions (HH → LH and HL → LL; see Fig. 2B). Both rate constants, β1 and β3, are governed by Vj1. Yet, Vj1 of state HH is not equal to Vj1 of state HL. This is because in the first case cx2 is in the high conductance state and in the second case cx2 is in the low conductance state. The two different states of cx2 do not change per se the course of the rate constants of cx1, but give rise to two different voltage distributions along the hemichannels. Thus, the gate of cx1 senses two different voltage drops across cx1 which results in two different rate constants for the same process of switching S1 from the high state to the low state.

Assuming first order processes, the rate constants for the channel forward reactions (β) and backward reactions (α) can be described as exponential functions governed by the voltage drop across the corresponding hemichannel:

display math(11)
display math(12)
display math(13)
display math(14)
display math(15)
display math(16)
display math(17)
display math(18)

The parameters α01, α02, β01 and β02 are multipliers and Vα1, Vα2, Vβ1 and Vβ2 are decay constants of the rate constants, respectively. These constants allow the fitting to experimental data. The rate constants are functions of the voltage Vssn, where ss indicates the state of the gap junction prior to the next transition. The gate that executes the transition is indicated with n, i.e. n = 1 when gate S1 of hemichannel cx1 switches and n = 2 when gate S2 of hemichannel cx2 switches.

The mathematical treatment of the state machine follows classical reaction kinetics:

display math(19)

The state matrix, Q, and the state vector, n, are defined as:

display math

The parameters n1, n2, n3 and n4 are the fractions of the channels in state HH, LH, HL and LL, respectively. The steady-state value of n can be calculated as follows:

display math

The current from cell 2 to cell 1 (I(t)) as a function of time and the steady-state current (I) are given by the equations:

display math(21)
display math(22)

where N is the total number of gap junction channels between two cells of a cell pair.

RESULTS

The conductive properties

The aim of this section is to give a qualitative picture of the characteristics of the model. We first consider the case of a homotypic gap junction. The parameters were chosen to emphasize the principal properties of the model. For practical reasons, they were kept in relative units.

In a first step, we derived the conductance functions of a gap junction channel working in state HH. This situation prevails when both hemichannels are in the high state. Figure 3A depicts the appropriate electrical schematic. Figure 3B shows the computed conductance functions of a gap junction channel (γHH= f(Vj1), continuous curve) and of its hemichannels (γ1= f(Vj), dash-dotted curve; γ2= f(Vj), dashed curve). The curves are symmetrical with respect to the ordinate because of the ‘antiserial’ arrangement of the hemichannels (Vj1 and Vj2 are of opposite sign). Figure 3B also includes the conductance functions of the two separate hemichannels (γH1= f(Vj1) and γH2= f(Vj2), dotted curves). They were added as a tool to evaluate the limitations of the conductance values. The following considerations serve to elucidate the behaviour of a gap junction channel and its hemichannels and give an insight into the complex interactions between the voltage drops and conductances of the hemichannels.

Figure 3.

Conductance of a homotypic channel working in state HH

A, electrical schematic of state HH with both hemichannels in the high state conductance. B, calculation of channel conductances as a function of Vj. The values γH and VH have been chosen to demonstrate the effects of connecting two identical hemichannels working in state HH. (γHexp(-|Vj|/VH): dotted curves; γHH: continuous curve; γ1: dash-dotted curve; γ2: dashed curve).

Provided Vj= 0 mV, the voltage drop across the hemichannels, Vj1 and Vj2, is zero as well. Under this condition, the conductance of each hemichannel is γH and hence the conductance of the entire gap junction channel is γHHH/2 (see Fig. 3B). According to the conventions used in Fig. 3A, a positive value for Vj gives rise to a positive Vj2 and a negative Vj1. Following eqns (4) and (5), this leads to the conductances γ1≥γH > γ2 and voltage drops VjVj2 > |Vj1|≥ 0. With increasing positive values of Vj, the conductance γ2 progressively decreases and eventually the entire Vj drop occurs across cx2. Thus, for strongly positive values of Vj, Vj2 and γ2 approach Vj and γH2, respectively (dashed curve in Fig. 3B), while Vj1 tends to zero and γ1 to γH (dash-dotted curve in Fig. 3B). Since γ2 is much smaller than γH, for strongly positive Vj the conductance γHH of the gap junction channel is mainly determined by the conductance of cx2 and γHH can be approximated by γ2 or γH2 (see continuous curve and lower right hand dotted curve in Fig. 3B). For the negative range of Vj values, the properties of γ1, γ2 and γHH can be developed in the same way.

This procedure is also useful for reviewing the conductance functions of a gap junction channel working in state LL. Both hemichannels are identical again, but now they are in the low state conformation, i.e. the channel formally resembles a channel in state HH with γL and VL instead of γH and VH. Hence, the functions γLL and γHH behave similarly (see Fig. 5).

Figure 5.

Conductance-voltage relationship of a homotypic channel

The four curves refer to the states HH, LH, HL and LL that a single channel is able to access. (γHH: continuous curve; γLH: dash-dotted curve; γHL: dashed curve; γLL: dotted curve).

In a second step, one has to derive the conductance functions of a gap junction channel operating in states LH and HL. In the case of a homotypic gap junction, these two states are symmetrical with respect to the ordinate. Hence, only one state needs to be explored while the other follows as a mirror image. Figure 4A depicts the electrical schematic of state HL with cx1 in the high state conformation and cx2 in the low state conformation. Using the same procedure as before, the following expressions hold:

display math
Figure 4.

Conductance of a homotypic channel working in state HL

A, electrical schematic of state HL with hemichannel cx1 in high state and cx2 in low state conductance. B, calculation of channel conductances as a function of Vj. The values γH1, γL2, VH1 and VL2 have been chosen to demonstrate the effects of connecting two identical hemichannels working in state HL. (γH1exp(Vj/|VH1|) and γL2exp(-Vj/|VL2|): dotted curves; γHL: continuous curve; γ1: dash-dotted curve; γ2: dashed curve).

In contrast to state HH or LL, it is difficult to find the maximum of the function γHL= f(Vj) for state HL. The calculated plot of γHL in Fig. 4B (continuous curve) has its maximum neither at zero voltage nor at γ12. The position of the maximum would help to identify the state of the hemichannels of a gap junction channel (see below). The following derivation determines the maximum of a series connection of two non-Ohmic conductances. The conductances are defined as:

display math(23)
display math(24)

Thus, the total conductance and its derivative are:

display math(25)
display math(26)

where γ‘tot is dγtot/dVj, f1’ is df1/dVj and f2’ is df2/dVj.

The zero of the derivative leads to the following condition for the maximum:

display math(27)

Let Vjm be the voltage at which γtot reaches its maximum. The maximum of γtot would be located at the intersection f1(Vjm) =f2(Vjm) if the slopes of f1 and f2 at Vjm had the same values and opposite signs. However, at the intersection of γ1 and γ2 only the second condition is true, i.e. the slopes of γ1 and γ2 have opposite signs while the slope of γ1 is greater than that of γ2 (see Fig. 4B). Thus, the maximum of γHL must be located to the right of the intersection where γ1 gets larger and γ2 smaller in order to fulfil eqn (27).

Close inspection of the states HL and LH leads to an important analogy between homotypic and heterotypic gap junction channels. From a structural point of view, a homotypic gap junction channel consists of two identical hemichannels. However, from a functional point of view, the two hemichannels are different (Fig. 4A): one is acting in the high state conformation, the other in the low state conformation. As a result, the model functionally behaves like a heterotypic gap junction channel (see Discussion, The heterotypic gap junction). Therefore, structurally homotypic gap junctions (Fig. 5) exhibit two functionally homotypic states (HH and LL) and two functionally heterotypic states (HL and LH). In contrast, structurally heterotypic gap junctions generally show four functionally heterotypic states (Fig. 8).

Figure 8.

Superposition of kinetic and conductance calculations covering a physiological range of Vj values for a heterotypic gap junction channel

All rate constants and conductance parameters have been chosen to demonstrate the effects of connecting two different hemichannels. The continuous lines indicate the parts of the curves that would be encountered when collecting data of a gap junction channel with the chosen kinetic parameters. |Vjo1| and |Vjo2| are rate constant equilibrium voltages of cx1 and cx2, respectively. The gap junction shows ‘positive gating’.

For the computations presented so far, it has been assumed that the conductance of a hemichannel decreases exponentially with increasing positive voltage. Let us consider now the case of an increasing exponential. Because of the symmetrical structure of a homotypic gap junction channel, this substitution would not affect the functions γHH= f(Vj) and γLL= f(Vj). Replacement of a decreasing exponential by an increasing one is equivalent to a geometrical reflection at the ordinate, i.e. the conductance function of cx1 would correspond to that of cx2 and vice versa (see dotted curves in Fig. 3). The result of such a procedure is a reproduction of γHH= f(Vj) or γLL= f(Vj). Therefore, conductance data of the states HH or LL provide no hint about the sign of the slope of the underlying exponential functions.

In contrast to HH and LL, the states HL and LH are sensitive to the sign of the slope of the exponentials. The replacement of a decreasing exponential by an increasing one does not give rise to a reproduction of γHL= f(Vj) and γLH= f(Vj) (see pointed curves in Fig. 4), i.e. γHL is reflected at the ordinate to become γLH and vice versa (see Fig. 5). Hence, conductance data of the states HL or LH would help to distinguish between decreasing and increasing exponentials provided it is possible to tell which data are produced by which state. Unfortunately, measured channel data provide no information about the underlying conformational state HL or LH.

In summary, experimental data gained from homotypic gap junction channels do not lead to a definite mathematical description since the distinction between decreasing and increasing exponentials for the hemichannel model remains undetermined. However, this problem can be solved if we consider a heterotypic channel consisting of hemichannels with widely different conductances. In this case, the function γHH= f(Vj) reflects the behaviour of the high state of the electrically narrower hemichannel. Under this condition, it is possible to determine the signs of the exponentials (see Discussion, The heterotypic gap junction).

DISCUSSION

Physiological limits

The plots in Fig. 5 are the result of a complete simulation. They cannot be gained in this way from single channel measurements for the following reasons. Firstly, cell membranes tolerate voltages up to about |150 mV|. Yet, fitting the model to experimental data leads to |VH| values between 100 and 300 mV; for example, |VH|= 240 mV in the case of Cx40 channels, i.e. it is far outside the physiological limits. Hence, only a narrow domain of the bell-shaped relationship γHH= f(Vj) is accessible experimentally (compare Figs 3B and 7A). Since the conductance of state HH varies only little over the physiological range of Vj, experiments led to the impression that γHH is virtually constant (see Fig. 7A). For comparison, the fitting procedure yields |VL| values from 50 to about 1000 mV; e.g. 455 mV in the case of Cx40 channels. Secondly, for kinetic reasons the probability of a channel state varies over the physiological range of Vj values. Figure 6A and B shows a set of hypothetical rate constants plotted versus Vj which apply to a homotypic gap junction channel. In this case, eqns (11)-(18) simplify as follows:

display math
Figure 7.

Superposition of kinetic and conductance calculations covering a physiological range of Vj values for a homotypic gap junction channel

The dotted parts of the curves represent conductance values which are rarely seen during experiments. A, gap junction with ‘positive gating’. B, gap junction with ‘negative gating’. The hemichannels have been modelled with decreasing exponentials.

Figure 6.

Gap junction kinetics

All conductive and kinetic parameters of the channel have been chosen to demonstrate the kinetic behaviour of a channel. A and B, voltage dependence of channel opening (α) and closing (β) rate constants. The rate constants have been plotted versus Vj to keep them comparable. C, steady-state probabilities versus Vj (state HH: continuous curve; state LH: dotted curve; state HL: dashed curve).

Again, the rate constants are governed by the voltage drop across the respective hemichannel. However, since this voltage drop is dependent on Vj, it is reasonable to transform these rate constants and plot them versus Vj. According to Fig. 6A and B, these plots also resemble exponential functions (see eqns (11)-(18)). This is because the relationship between the voltage drop across the hemichannel and Vj is nearly linear in the plotted range. This can be verified by considerations analogous to those that led to the conductance functions (see Results, The conductive properties). The values for the rate constants have been chosen in order to emphasize the kinetic properties, and |VH| has been set to the physiological limit and |Vjo| to |VH|/2. |Vjo| indicates the equilibrium voltage between the states HH and LH or HH and HL (Fig. 6C).

Adopting the conventions for the polarity of Vj used during experiments (see Fig. 2), the model also allows us to elaborate on the gating polarity. ‘Positive gating’ means that the gate of the hemichannel which experiences a positive voltage drop is involved in the gating process. Thus, a positive Vj forces a gap junction channel with ‘positive gating’ to change from HH to HL since S2 of cx2 senses a positive voltage drop (see Figs 2A and B, and 7A). Conversely, a negative Vj forces this gap junction channel to change from HH to LH since S1 of cx1 senses a positive voltage drop. This behaviour can be verified by examining the Vj dependence of the rate constants (see Fig. 6A): as Vj grows more positive (negative), α1 and β2 become larger (smaller) while α2 and β1 become smaller (larger). Hence, the equilibrium tends towards HL (LH). ‘Negative gating’ can be defined accordingly, i.e. a positive Vj forces a gap junction channel with ‘negative gating’ to change from HH to LH since S1 of cx1 senses a negative voltage drop (see Figs 2A and B, and 7B).

The approximation of the rate constants by exponential functions causes a problem. In reality, the growth of the forward rate constants β is limited by the physiological voltage boundaries of the cells. This is not the case for the backward rate constants α; e.g. α1 grows with increasing positive Vj to very high and unphysiological values (see Fig. 6). However, since this process is expected to saturate in nature, we introduced an upper limit. Simulations revealed that a limit of 2α0 is reasonable because it does not impair the accuracy of the calculations. Furthermore, the numerical solution of the model becomes difficult without limits on the rate constants. This is because large rate constants lead to large integration steps Δn(t) when solving eqn (19):

display math(28)

Large integration steps Δn(t) lead to inaccurate numerical solutions of the system. The value of Δn(t) can be kept small by choosing very small integration steps, Δt. While this increases the precision of the calculation, it disproportionately extends the calculation time. The numerical simulation of kinetics could be improved by more sophisticated integration schemes such as variable time-stepping methods or stiff integrators.

Figure 6 C illustrates the dependence of the probability that a channel is in a certain state on Vj. The probability n4 of state LL has been omitted because it is very close to zero over the entire range of Vj. There would be a high probability of accessing state LL if the gap junction channel were operating in state LH (HL) and β43) were large at the same time (compare Figs 2 and 6). However, this coincidence never occurs because the gap junction is in state LH (HL) at negative (positive) Vj, while β43) is large at positive (negative) Vj. Therefore, in essence the model behaves like a three state system which produces contingent curves as previously proposed (Harris et al. 1981).

The conductance and kinetic computations can now be combined with the physiological boundary conditions to obtain the steady-state properties of a gap junction (see Fig. 7). The plots in Figs 6C and 7 show that experimental γHH data are gained with high probability at Vj < |Vjo|, and data of γHL and γLH at Vj > |Vjo|, respectively. Furthermore, there are transition zones around where HH, LH or HL occur with intermediate probability. They also demonstrate that the former γj(residual state) results from two different channel states (LH and HL). The plots in Fig. 7A assume decreasing exponentials for the hemichannel conductances, ‘positive gating’ and a physiological Vj range of about -VH to VH. In Fig. 7B, the gating property has been changed to ‘negative gating’. As explained before (see Results, The conductive properties), the conductance of the hemichannels can also be described with increasing exponentials. In this case, the course of the functions γLH= f(Vj) and γHL= f(Vj) in Fig. 7A would be exchanged. Provided the assumption of ‘positive gating’ is maintained, i.e. γHL becomes ‘visible’ at VjVjo, the conductance plots resemble those shown in Fig. 7B.

In summary, experimental data gained from homotypic gap junction channels can always be modelled in two ways. Hemichannels with increasing exponential conductance functions and ‘negative gating’ or decreasing exponential functions and ‘positive gating’ lead to the same result. This is also true for increasing exponentials and ‘positive gating’ as compared with decreasing exponentials and ‘negative gating’. Therefore, it is not possible to distinguish between two ‘different’ models which produce the same results. This can only be achieved with additional data from heterotypic gap junctions.

The heterotypic gap junction

As mentioned before (see Results, The conductive properties), heterotypic gap junctions exhibit four distinct functional heterotypic states. They can be treated in the same way as the conductances of state LH or HL of a homotypic gap junction channel (see above). Hence, a complete set of conductance plots includes four asymmetrical curves (see Fig. 8). The curves of γLH and γHL or the linear approximations in their experimentally accessible range (see continuous segments of γLH and γHL in Fig. 8) do not intersect at Vj= 0 mV. This happens because two different states, i.e. L1H2 and H1L2, produce these curves. Again, there is not a single state that represents the former residual state. This fact explains the discontinuity of residual conductance data at Vj= 0 mV that has been encountered in experiments on heterotypic channels with hemichannels exhibiting widely different conductances (Bukauskas et al. 1997).

The following considerations help to determine the sign of the exponentials and the polarity of gating. The slopes of the exponentials of cx1 and cx2 must have the same sign; otherwise the conductance plot would not be bell shaped. Furthermore, the values of γH1 and γH2 have been determined from data of homotypic gap junctions consisting of cx1 and cx2. Provided γH1 < γH2 and the exponentials have negative slopes, the function γHH= f(Vj) has its maximum on the right hand side of the ordinate. Alternatively, the function γHH= f(Vj) has its maximum on the left hand side of the ordinate when γH1 > γH2 with the same slope (see continuous and dotted curves in Fig. 4B; keep in mind that the state HH of heterotypic channels corresponds to state LH of a homotypic channel). In Fig. 8, γH1 is about three times as large as γH2 while the maximum of γHH is located on the left hand side of the ordinate. Therefore, the conductance function of the hemichannels must follow a decreasing exponential, i.e. their slopes have negative signs. Since γL2 < γH1, the maximum of γLH must be located on the left hand side of the ordinate, too. Thus, the function γHL= f(Vj) can be easily identified. Since the probability of state HL is high with positive Vj (see Fig. 8), a ‘positive gating’ mechanism can be postulated for the gap junction depicted.

The state LL

The state LL is associated with a fourth channel conductance smaller than the conductance of states HH (the former γj(main state)), LH and HL (previously summarized as γj(residual state)), but larger than zero. The probability of reaching the state LL depends on the experimental protocol, as can be shown by simulations with data of Cx43-Cx43 (Banach & Weingart, 1996). The probability remains almost zero for every constant voltage, Vj. The largest probability is obtained with large voltage steps Vj, e.g. starting from -80 mV and ending at 80 mV in the case of Cx43-Cx43. In such a simulation, about one gap junction channel out of 1000 reaches the state LL for a few milliseconds. Careful inspection of our own data showed a state with a conductance smaller than γj(residual state) and larger than zero (V. Valiunas & R. Weingart, unpublished observation). However, these events were rare and of very short duration. Hence, new investigations should be done with an experimental protocol designed to detect state LL with a realistic probability. To do so, it is helpful to know the kinetic data of the gap junction examined. In this way, it is possible to plan experiments using our new model.

Conclusions

The calculations show that two hemichannels with exponential conductance-voltage relationships connected in series reproduce a quasi-constant conductance of homotypic gap junction channels. This impression arises from the limited voltage range accessible under physiological conditions. Data of future experiments should be fitted with eqns (8) and (9). The fitting process can be simplified when V1= V2 (see eqns (8) and (9)). In general this is the case for state HH and LL. Thus, conductance data of γj(main state) can be fitted with the formula:

display math

A fourth state, i.e. state LL, has newly been introduced. Kinetic calculations revealed that it occurs rarely and is of short duration. Conceivably, specific experimental protocols have to be considered to verify its existence.

The model can easily be adapted to take into account the results of further investigations, including substates or complete closure of gap junction channels. Substates can be introduced by adding new conductances, a completely closed state by adding a zero conductance in parallel to γh and γl. This model will help to plan future experiments aimed at further elucidating the electrical properties of specific homotypic and heterotypic gap junction channels, e.g. detection of state LL with a realistic probability, assessment of kinetics and conductances of substates and search for heterotypic channels consisting of hemichannels with opposite gating characteristics (positive and negative gating in series) or opposite functions of conductance (decreasing and increasing exponential functions in series).

Acknowledgements

We wish to thank V. Valiunas for useful discussions, H. P. Clamann for critical comments on the manuscript and the Braintool Group at the Institute for Informatics and Applied Mathematics, University of Bern, for providing computing facilities. This work was supported by the Swiss National Science Foundation (grant 31-45554.95).

Ancillary

Advertisement