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

Abstract

1. A mathematical model has been developed which describes the conductive and kinetic properties of homotypic and heterotypic gap junction channels of vertebrates. 2. The 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. 3. It has been assumed that the conductances of the high state and low state vary exponentially with the voltage across the hemichannel. 4. The 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. 5. The 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. 6. The 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. 7. Simulation of the kinetic properties at the multichannel level revealed current time courses which are consistent with a contingent gating. 8. The calculations have also shown that the channel state LL is rare and of short duration, and hence easy to miss experimentally. 9. The 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.

Figure 1. Basic model of the gap junction hemichannel

A, electrical schematic of a hemichannel. γ_H and γ_L 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.

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.

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 V_j. The values γ_H and V_H have been chosen to demonstrate the effects of connecting two identical hemichannels working in state HH. (γ_Hexp(-|V_j|/V_H): dotted curves; γ_HH: continuous curve; γ₁: dash-dotted curve; γ₂: dashed curve).

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. (γ_HH: continuous curve; γ_LH: dash-dotted curve; γ_HL: dashed curve; γ_LL: dotted curve).

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 V_j. The values γ_H1, γ_L2, V_H1 and V_L2 have been chosen to demonstrate the effects of connecting two identical hemichannels working in state HL. (γ_H1exp(V_j/|V_H1|) and γ_L2exp(-V_j/|V_L2|): dotted curves; γ_HL: continuous curve; γ₁: dash-dotted curve; γ₂: dashed curve).

Figure 8. Superposition of kinetic and conductance calculations covering a physiological range of V_j 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. |V_jo1| and |V_jo2| are rate constant equilibrium voltages of cx1 and cx2, respectively. The gap junction shows ‘positive gating’.

Figure 7. Superposition of kinetic and conductance calculations covering a physiological range of V_j 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 V_j to keep them comparable. C, steady-state probabilities versus V_j (state HH: continuous curve; state LH: dotted curve; state HL: dashed curve).