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MEMBRANE TRANSPORTERS, ION CHANNELS, AND PUMPS
1Department of Biophysics and 2Department of Cell Biology, Radboud University Nijmegen, Nijmegen, The Netherlands
Submitted 2 May 2007 ; accepted in final form 13 January 2008
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ABSTRACT |
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gap junctions; calcium waves; pacemaking; electrical coupling; action potential propagation; inositol 1,4,5-trisphosphate receptor; normal rat kidney cell; calcium-induced calcium release
In the cell types mentioned above, the cells are connected by gap junctions, allowing diffusion of IP3 and calcium. Since both IP3 and calcium facilitate intracellular calcium oscillations, diffusion of calcium and IP3 through the gap junctions could provide an effective way for synchronization of intracellular calcium oscillations in neighboring cells and for propagation of waves of intracellular calcium oscillations through the network (see e.g., Refs. 10, 19, 20, 40). The propagation of calcium waves through the network has been the topic of many studies, but the cellular mechanisms involved in the propagation of calcium oscillations can be very different. Most studies refer to the propagation of calcium waves in nonexcitable cells with intracellular IP3-mediated calcium oscillations, where oscillations in cells are coupled by diffusion of IP3 and calcium through gap junctions (see e.g., Refs. 12, 19, 20, 43). In our study, we ignore coupling of oscillations by calcium and IP3 diffusion for good reasons, which are explained in the DISCUSSION.
On the other hand, there is propagation of electrical activity in networks of cells electrically coupled by gap junctions, such as in the ventricular myocardium (17, 18, 23–26). In such types of cell networks, propagation of electrical activity is the result of depolarization of a cell by action potential (AP) firing of its neighbor. For adequate gap junctional coupling, an AP causes a depolarization in neighboring cells, which opens their membrane channels and so triggers an AP.
Some recent studies have focused on cell types that have both IP3-mediated calcium oscillations and APs, such as in interstitial cells of Cajal (ICC) (4), sinoatrial nodal cells in the heart (31), lymphatic smooth muscle cells (22), and NRK fibroblasts (27). These cell types have the interesting property that the mechanisms of IP3-mediated calcium oscillation and AP generation are coupled and interact with each other (28). An AP can trigger a calcium transient, since inflow of calcium during an AP causes CICR. In the other direction, the increase of cytosolic calcium due to release of calcium through the IP3 receptor opens calcium-dependent channels in the membrane, causing a depolarization. In NRK cells, the major calcium-dependent membrane channel type is the calcium-dependent chloride channel (ClCa) with a Nernst potential near –20 mV. This depolarization may then trigger an AP (6, 27). Since electrical coupling through gap junctions is faster than chemical coupling by diffusion of calcium and IP3 through gap junctions (6, 36), the intracellular calcium oscillations between cells are also (indirectly) coupled by the electrical coupling by gap junctions.
Because of the positive interaction between the IP3-mediated calcium oscillator and membrane depolarization, excitable cells with IP3-mediated calcium oscillations may be very robust pacemakers for propagating activity in the network (45). Recently, Imtiaz et al. (22) investigated the various coupling modes of two cells with different amounts of IP3 and, therefore, different intrinsic oscillation frequencies. These authors showed that the chemical and electrical coupling by gap junctions can cause anti-phase or in-phase oscillations of the cell pair, depending on the amount of IP3. Moreover, these authors showed that weak coupling (small conductance of the gap junction) is sufficient to synchronize heterogeneous cell pairs.
Following up on the study by Imtiaz et al. (22) on a pair of cells, we have investigated the initiation and propagation of activity in a network with excitable cells with IP3-mediated calcium oscillators by gap junctional coupling. Imtiaz et al. (22) showed that a pacemaker cell can drive the calcium oscillations in a neighboring cell with a lower IP3 concentration and a correspondingly lower intrinsic oscillation frequency. The questions that we have addressed are, What happens when more cells with a low IP3 concentration are coupled to this single pacemaker? And what happens if a pacemaker is coupled to cells that do not have an intrinsic oscillation frequency, because the IP3 concentration is too small? If that number of coupled follower cells increases, the current from the pacemaker cell to the follower cells will spread through the whole network. When the gap junctional conductance is very small, the current might be too small to depolarize the neighboring cells. However, when the gap junctional conductance is very large, current will spread throughout the network, and if the network is large, the net current into a neighboring cell may also be too small to depolarize the cell. Therefore, in agreement with previous studies on excitable cells without intracellular calcium oscillations (36), we expect an optimal range of gap junctional conductances for initiation and propagation of activity in the network. Because of the positive, reinforcing coupling between the intracellular calcium oscillator and the membrane depolarization, we hypothesize that propagation is more robust in excitable cells with both mechanisms compared with that in cells that lack one of the two.
We address these problems using an experimentally verified model for NRK fibroblast cells (27). Contrary to Imtiaz et al. (22), we did not include voltage-dependent IP3 synthesis. The coupling between cells in our study is by electrical current through the gap junctions. In each individual cell, the electrical phenomena are coupled to the intracellular calcium oscillators by the calcium inflow through the L-type calcium channels and by calcium inflow through the IP3 receptor. In the DISCUSSION we evaluate the consequences of this simplification on the propagation of activity.
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METHODS |
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TOPABSTRACTMETHODSRESULTSDISCUSSIONAPPENDIXGRANTSREFERENCES |
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The model contains two compartments for each cell: the cytosol and the ER. The plasma membrane contains a plasma membrane calcium-ATPase (PMCA) pump, L-type calcium channels, ClCa channels, a nonspecific leak, and inward rectifying potassium channels (14). The ER membrane contains a sarco(endo)plasmic reticulum calcium-ATPase (SERCA) pump, a calcium leak, and an IP3 receptor channel. Cells are electrically coupled by gap junctions.
The key idea of the model (27) is that autocrine and paracrine production of hormones such as PGF2
leads to the production of IP3, which gives rise to IP3-mediated intracellular calcium oscillations. These IP3-mediated calcium oscillations cause periodic calcium transients, which open the ClCa channels. These ClCa channels depolarize the membrane toward the chloride Nernst potential, near –20 mV, thereby causing activation of the L-type calcium channels. Opening of calcium channels generates an influx of calcium in the cells with a concomitant further depolarization toward the equilibrium potential for calcium ions. The ClCa channels remain activated as long as the intracellular calcium level is elevated, resulting in a plateau phase at the chloride Nernst potential at –20 mV. Upon extrusion of calcium from the cytoplasm, the ClCa channels become deactivated, and the cells subsequently repolarize to –70 mV as a result of the activity of inward rectifier potassium channels (14). Just the other way around, calcium APs induce CICR through the IP3 receptors.
The autonomous cell oscillator. The NRK cell model by Kusters et al. (27) that describes the dynamics of the cell has two major components: an IP3-mediated intracellular calcium oscillator and an electrically excitable membrane. In this report we describe the main properties of the NRK cell. For more details, we refer to Ref. 27.
Calcium in the cytosol plays a key role in coupling the dynamics of the IP3-mediated calcium oscillator and the cell membrane (28). The rate of change in the membrane potential due to the currents through inward rectifier potassium channels (IKir), L-type calcium channels (ICaL), ClCa channels [ICl(Ca)], leak channels (Ileak), and store-dependent calcium (SDC) channels (ISDC) is given by Eq. A1 in the APPENDIX, where IKir and Ileak determine the membrane potential of the cell at rest near –70 mV and are specified by Eqs. A2–A6.
In NRK cells, the crucial coupling between electrical events at the excitable membrane and the internal calcium oscillations is controlled by the L-type calcium channel and the ClCa channel. The current through the L-type calcium channels is given by Eq. A7, where Vm refers to the membrane potential and ECaL refers to the Nernst potential of calcium, near 50 mV. The L-type calcium channel has an activation (m) and an inactivation (h) variable. The dynamics of m and h obey a first-order differential equation with steady-state values m
and h given by Eqs. A8 and A10, respectively, and with time constants given by Eqs. A9 and A11, respectively.
The current through the ClCa channel is given by Eq. A12. In addition, the cell membrane has a SDC channel. The conductance of the SDC is inversely related to the calcium concentration in the ER, as given by Eq. A13.
One of the mechanisms for calcium extrusion from the cytosol is the PMCA pump. The flux of calcium ions through the PMCA pump (JPMCA) is described by Eq. A16.
Any changes in the cytosolic calcium concentration ([Ca2+]cyt) are due to buffering of calcium (Eq. A14), to a net flux of calcium through the plasma membrane (JPM; Eq. A15), and to net fluxes through the ER membrane (Eq. A18). The latter has a constant leak of calcium (JleakER; Eq. A19), a flux through the IP3 receptor (JIP3R; Eq. A20), and active transport of calcium into the ER by the SERCA pump (JSERCA; Eq. A24).
The intracellular calcium oscillator is controlled by the intracellular IP3 concentration, which activates the IP3 receptor. The flux of calcium ions through the IP3 channel is described by a Hodgkin-Huxley type formalism (Eq. A20) with activation variable f and inactivation variable w, with the steady-state values f and w given by Eqs. A21 and A22. The time constant for the activation parameter f is considered to be small relative to that of the other processes in the cell. Therefore, we have used f instead of f. The time constant 
w for the inactivation gate is described by Eq. A23. This results in periodic oscillations of calcium flow out of the ER into the cytosol and calcium reuptake in the ER by activity of the SERCA pump. As suggested by Dupont and Goldbeter (8), the flux of calcium ions through the SERCA pump is described by Eq. A24.
The elevated calcium concentration in the cytosol by opening of the IP3 receptor activates the ClCa channels (see Eq. A12), which depolarize the cell membrane to the chloride Nernst potential, near –20 mV. As explained by Kusters et al. (27), this depolarization can open the L-type calcium channels. Opening of the L-type calcium channels gives rise to a further increase in [Ca2+]cyt. As a result, an AP with a plateau phase near –20 mV occurs. Calcium in the cytosol is reduced by reuptake of calcium in the ER by the SERCA and PMCA pumps. When [Ca2+]cyt has been restored to its basal level, the ClCa channels close and the membrane potential repolarizes to the rest potential, near –70 mV.
Parameter modification.
The full set of equations describing the dynamics and the parameter values of this NRK model can be found in Kusters et al. (27). In the present model study, we have had to adjust some parameter values. When calcium in the external medium of NRK cells is replaced by strontium, AP propagation in experiments is more robust (6). The reason is that strontium does not inactivate the L-type calcium channels as calcium does (14). Since many experimental data were obtained using strontium instead of calcium (6, 7, 14), we have omitted the calcium-dependent inactivation factor (vCa) in the model equation for the L-type calcium channel. Another effect is that the current through the L-type calcium channel is much larger for strontium than for calcium, which is why we have used a larger value for the L-type calcium channel conductance (GCaL: 1.6 nS instead of 0.7 nS). Since the L-type calcium channel, ClCa channel, and IP3 receptor are important channels for AP propagation, and since activation of these channels was rather small in our old model (27), we have changed the following parameters: the time constant for inactivation of the L-type calcium channel (h) is a factor 2 longer, the time constant for activation of the L-type calcium channel (m) is a factor 2 smaller, the membrane potential for half-maximal activation of m is set to –10 mV, KCl(Ca) (the half-maximal activation concentration) is set to 18 µM, JPMCA is set to 3 x 10–5 µmol/(s·dm2), and leak conductance (Gleak) is set to 0.058 nS. These changes lie within the range of values from experimental observations. The kon and koff parameters of the buffer are set to 1. Table 1 shows the modified parameter values.
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3 nS, which is in agreement with other experimental data for gap junction coupling between cells where Cx43 subunits are involved (3, 41).
The electrical current flowing through the gap junctions between cell i and other cells in the network was incorporated by an extra term at the right-hand side of Eq. A1, which resulted in
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We modeled the AP propagation by electrical coupling of a single pacemaker cell to surrounding follower cells in a one-dimensional strand (cable). Our aim is to understand the electrical load on the single pacemaker cell due to coupling to surrounding cells. Figure 1A shows the equivalent electrical circuit for the passive electrical properties of a one-dimensional strand of follower cells driven by a pacemaker cell (P). Each follower cell is represented by a capacitance C and resistance Rf, and cells are coupled by the gap junctional resistance Rg. This passive model is a good approximation as long as the membrane potential does not reach the threshold for AP firing.
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where Zcell(
) is the impedance of a follower cell in the frequency domain, given by
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f = RfC. The equivalent resistance of an infinitely long one-dimensional strand can be calculated by setting Zn+1() = Zn(), which gives
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For an infinitely large strand of cells, the net current to the first follower cell (see Fig. 1C) is given by
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From Eqs. 5 and 6, it is easy to see that Icell becomes zero for infinitely small values of Rg. If Rg becomes infinitely small, Z becomes zero, and therefore Icell becomes zero. Equation 6 shows that Icell is also zero for large values of Rg (Rg 
, Icell = 0). The optimal value of Rg is found by solving 
Icell/Rg = 0. For the parameter values in our study, the optimal value for Rg is 2.0 x 109 
(Gg = 0.5 nS).
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RESULTS |
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TOPABSTRACTMETHODSRESULTSDISCUSSIONAPPENDIXGRANTSREFERENCES |
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) of the AP or calcium oscillator of a NRK pacemaker cell with intrinsic cycle length (T) as a function of the phase () at which the external input is given. When a current pulse of sufficient amplitude is injected into a cell, the elevated membrane potential results in opening of the L-type calcium channels and, possibly, AP firing. The inflow of calcium causes CICR through the IP3 receptor channel and gives rise to a phase advance of the IP3-mediated calcium oscillator. On the other hand, a calcium injection gives rise to a phase advance of the IP3-mediated calcium oscillator and corresponding calcium transient. The resulting depolarization by the ClCa channels then leads to advanced appearance of the next AP.
The PRC is measured by delivering a precisely timed perturbation and measuring the change in the running cycle duration. The upstroke of the preceding pacemaker AP is chosen as the reference point (phase zero), since it is very sharp compared with the onset of calcium oscillations. Moreover, calcium oscillations change in shape and size. Phase is defined by = tp/T, where tp is the time when the stimulus is applied, relative to the reference point, and the phase change is defined as (T – Tnew)/T, where Tnew is the time of occurrence of the following AP or calcium transient relative to the reference point, i.e., the new cycle length. The PRC quantifies the effect of an input pulse at a given phase on the occurrence of the following AP or calcium transient. If an input pulse does not affect the next AP or calcium transient, T is unchanged and the phase change at that point of the curve is zero. If the input pulse delays the next AP or calcium transient, Tnew > T, and the phase change has a negative value (not observed in our simulations). If the pulse advances the next AP, Tnew < T, and is positive. Figure 2 shows PRCs (bottom panels) generated by injecting a depolarizing current pulse with a 50-ms duration of 10, 15, and 20 pA (A) and a calcium pulse associated with a calcium current injection of 1, 2, and 5 pA for 50 ms (B). The top three subpanels in Fig. 2 show examples of the effect of current (15 pA in A) and calcium pulses (1 pA in B) on the phase advance of the AP and calcium oscillation, respectively (dashed lines), and the unperturbed response (solid lines). The bottom panels of Fig. 2, A and B, show the phase change of the IP3-mediated calcium oscillator as a function of the timing of the current pulse in the cycle of the calcium oscillator. Figure 2A, bottom, shows that a current pulse of 10 pA has no effect on the IP3-mediated calcium oscillator irrespective of the phase in the AP cycle (thick solid line). Depolarizing current pulses of 15 (dashed-dotted line) and 20 pA (thin solid line) injected at phase > 0.2 trigger the next calcium transient almost immediately via an evoked AP. Injection at phase < 0.1 has no effect on the next calcium transient, because this period includes the AP and its refractory period (see Fig. 2A). Note that these current injections are smaller than the current inflow in the cell by the membrane during an AP, which is 25 pA. Figure 2B, bottom, shows the phase change of the AP as a function of the phase of the calcium pulse. The phase of the AP does not change for calcium pulses of 1 (thick solid line), 2 (dashed-dotted line), and 5 pA (thin solid line) at phases < 0.6, < 0.4, and < 0.3, respectively. The phase is maximally advanced for > 0.7, > 0.5 and > 0.4, respectively, when triggered by calcium pulses of 5, 2, and 1 pA, respectively. These calcium injections are small relative to the total inflow of calcium from the ER during a calcium transient (35 x 10–6 µmol) and through the membrane during an AP (100 x 10–6 µmol).
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Entrainment of Calcium Oscillations of Two Cells by Electrical Coupling Since IP3-mediated calcium oscillations and AP generation within a cell are tightly coupled processes (Fig. 2), electrical coupling between cells by gap junctions provides an indirect (electrical) coupling mechanism between IP3-mediated calcium oscillations in two neighboring cells. To investigate the role of gap junctions in the coupling of IP3-mediated calcium oscillations of two neighboring cells, we investigated the entrainment of two pacemaker cells with different intrinsic frequencies (due to different IP3 concentrations, [IP3]) as a function of gap junctional conductance.
Figure 3 shows the major family of entrainment regions, commonly called Arnold tongues (34) (solid lines), as a function of the electrical coupling Gg. Notice that the vertical scale is in picosiemens. The entrainment regions are labeled by the ratio of the frequencies of the calcium oscillations of both cells: f2([IP3 cell 2])/f1([IP3 cell 1]). The [IP3] of cell 1 is set to a value of 1.0 µM (this value causes calcium oscillations at intermediate frequencies: f1 = 1/100 Hz), while the [IP3] of cell 2 is varied in steps of 0.005 µM at a rate of one step per 9,000 s from 0.0 to 8.0 µM.
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|mf1 – nf2| < 
(Gg, m, n), where increases for larger values of Gg. Figure 3 shows only the regions for the major entrainment ratios for f2/f1 (1:2, 2:3, 1:1, 4:3, and 3:2), but in between there are many much smaller regions with other ratios of m/n entrainment. The regions for various modes of entrainment grow with Gg and merge until a value of Gg is reached that gives 1:1 entrainment for all frequencies. For example, when we start at Gg = 35 pS and an IP3 concentration of 0.4 µM for cell 2 (corresponding to an intrinsic oscillation frequency f2 = 1/190 Hz for the cell in isolation), simulations reveal that this cell exhibits calcium transients at one-half the frequency of the calcium oscillations in cell 1. For increasing values of [IP3] in cell 2 at Gg = 35 pS (horizontal dashed line), the following major entrainment regions are observed: 1:2, 2:3, 1:1, 4:3, and 3:2, respectively. In between, there are many much smaller regions with other ratios of m/n entrainment. Figure 3 predicts that electrical coupling near 60 pS or higher between two cells with one cell having an [IP3] of 1.0 µM is sufficient to completely synchronize two heterogeneous NRK cells, irrespective of the IP3 concentration in the second cell. In this range (for gap junctional conductance values above 60 pS), the cell with the lowest oscillation frequency locks to the cell with the highest oscillation frequency. Therefore, in the 1:1 entrainment region left from the ratio f2/f1 = 1.0, the frequency of the calcium oscillations is determined by the reference frequency f1 and on the right side by the variable frequency f2. From the results shown in Fig. 3, we infer that under conditions of a physiological gap junctional coupling strength of 3 nS between NRK cells (13), the intracellular calcium oscillations and APs of two oscillating NRK cells are fully synchronized. The fact that the fastest frequency always determines the synchronized frequency indicates that both calcium oscillators synchronize by phase resetting AP effects and not by continuous interaction (46, 49).
Synchronization of Cells in a Strand by Electrical Coupling To explore the excitation of follower cells by pacemaker cells, we studied entrainment of a strand of follower cells by a terminal pacemaker cell. Since gap junctions allow diffusion of IP3, and because follower cells may be subject to stimulation of subthreshold IP3 production, we assumed non-zero concentrations of IP3 in the follower cells. For most simulations in this study, the follower cells have an [IP3] of 0.1 µM, which does not give rise to spontaneous calcium oscillations. For the pacemaker cell, we chose an [IP3] of 1.0 µM, which causes spontaneous calcium transients and APs (27).
Figure 4 shows the results of such an entrainment simulation. The solid and dashed lines demarcate different entrainment regions for a one-dimensional strand of follower cells with an [IP3] of 0.1 and 0.0 µM, respectively, by a single terminal pacemaker cell ([IP3] = 1.0 µM) as a function of the number of cells and Gg. Figure 4 shows that the mode of entrainment depends on the number of follower cells in the strand as well as the gap junctional conductance. For a single pacemaker cell and one follower cell ([IP3] = 0.1 µM), the minimal gap junctional conductance for full 1:1 entrainment is 0.06 nS. Increasing the number of cells for a fixed value for Gg at 0.1 nS changes the 1:118 entrainment to 1:2 entrainment for three cells and to 1:4 entrainment for four cells. For five or more cells, no synchronization takes place anymore when Gg = 0.1 nS. Coupling in the range between 0.25 and 0.45 nS is sufficient for complete 1:1 synchronization of calcium oscillations and APs in a one-dimensional network of NRK cells with this IP3 level, independent of the network size. If [IP3] in the follower cells is set to 0.0 µM, the same results are obtained for these small values of Gg (Fig. 4, dashed lines superimposed on solid lines). Notice that experimental observations (13) report a gap junctional conductance for NRK cells of 3 nS, which is much larger than the optimal coupling range in our simulations (0.25–0.45 nS). We will come back to this topic in the DISCUSSION.
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20 (solid line). For [IP3] = 0, the IP3 receptor in the follower cell is closed and AP transmission fails for Gg above 0.8 nS when the number of cells exceeds 30 cells (shaded area). Simulations reveal that the area marked with diagonal lines is where the entrainment is 1:3. The range of Gg values that allow 1:1 synchronization for a large number of cells is from 0.25 to 0.4 nS. These results demonstrate that small concentrations of IP3, which do not elicit spontaneous calcium oscillations, support synchronization of activity in networks of cells. For this reason, we used in this study an [IP3] of 0.1 µM for the follower cells to further investigate the interaction between IP3-mediated calcium oscillations and APs.
Summarizing, to completely synchronize a pacemaker and a single follower cell, a Gg near 0.06 nS is sufficient (Fig. 4), whereas for an infinitely long strand of cells, Gg must be in the range between 0.25 and 0.45 nS (Fig. 4). An explanation is given below.
The current from the pacemaker cell through the gap junctions to the follower cells is given by Eq. 2. Increasing the electrical coupling (Gg) increases the leak of current from the pacemaker cell to its neighbor cells. If the net current to a follower cell is large enough and fast enough, the membrane potential might approach the threshold value near –40 mV for L-type calcium channels. If that happens, the L-type calcium channels open, causing an AP and an inward current of calcium ions. The increase of calcium in the cytosol activates the IP3 receptor, leading to a calcium transient.
An opposite effect of increasing the electrical coupling is that it decreases the equivalent impedance of the network. Since the equivalent impedance Z for an infinitely large strand of follower cells decreases for increasing values of Gg (Z/ Rg > 0 for all Rg; see Eq. 5), decreasing Rg (increasing Gg) implies a smaller value for Z. If the equivalent impedance of the network decreases, the available current from the pacemaker cell spreads to a larger number of follower cells in the network, which makes it harder for the pacemaker cell to depolarize its neighboring follower cell to the threshold of the L-type calcium channels for generation of an AP. In other words, a larger gap junction conductance leads to a decrease in the effective impedance and to a smaller net current from the pacemaker cell to its neighboring follower cell, which explains the shift from 1:1 to 1:2 entrainment for strong coupling (Gg > 0.45 nS) in Fig. 4 and to 1:4 entrainment for Gg > 1.0 nS for network sizes in the range of 20 cells and more.
In conclusion, small coupling conductances prohibit a sufficiently large current from the pacemaker to the follower cell to reach the membrane threshold for excitation. For a large conductance, the threshold for AP generation cannot be reached, since a large part of the current from the pacemaker to its neighboring follower cell flows to other follower cells, limiting the net current from pacemaker to its neighbor follower cell.
Entrainment and Propagation in a Strand of Electrically Well-Coupled NRK Cells Failing AP transmission during strand entrainment. Figure 5 shows the results of a simulation of the membrane potential behavior of a pacemaker cell in isolation (A, thick solid line) and that of a pacemaker cell (B, thick solid line) coupled to 100 follower cells (thin solid lines) in a strand with a Gg of 3 nS. Note that this value for Gg is much larger than the values of Gg in Fig. 4. For Gg = 3 nS, the entrainment for a network of 20 cells or less is 1:1 and is 1:4 when the number of cells exceeds 30.
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Figure 5, right, shows the results when the pacemaker is coupled to 100 follower cells. Comparing the left and right panels reveals some important differences between the results for an uncoupled pacemaker cell and for a pacemaker cell coupled to a strand of follower cells. The main difference relates to the slow and small increase of [Ca2+]cyt in the pacemaker cell (compare Fig. 5, C and D, thick solid line) and the corresponding slow and small depolarization of the membrane potential (compare Fig. 5, A and B) in the coupled situation for the pacemaker cell. Coupling the pacemaker cell to its neighboring follower cells by gap junctions causes the current from the pacemaker cell to leak away to the follower cells. As a result, depolarization of the membrane potential of the pacemaker cell by the ClCa channels reaches a lower peak value (Fig. 5B), and the level of activation of the L-type calcium channels of the pacemaker cell is not as high as in the case of the isolated pacemaker cell. This becomes clear by comparing the data in Fig. 5, I and J, which show the activation gate (m; thick dashed-dotted line) and inactivation gate (h; thick solid line) for the uncoupled and coupled situation, respectively. For the uncoupled pacemaker cell, the m gates open (Fig. 5I), which does not happen for the coupled pacemaker cell (Fig. 5J), leading to AP failure in the strand. As a consequence of the small activation of the L-type calcium channels in the coupled pacemaker cell, there is hardly any calcium inflow through the L-type calcium channels into the cytosol (compare Fig. 5, G and H, and notice the different scales of the vertical axes). This also affects the boosting of CICR by activation of the IP3 receptor, because the smaller inflow of calcium through the L-type calcium channel weakens CICR and results in a slower and smaller calcium flow through the IP3 receptor (compare Fig. 5, E and F) for the coupled pacemaker.
For the isolated pacemaker cell, the fraction of open activation gates (f; thick dashed dotted line) increases rapidly, followed by a slow closure of the inactivation gates (w; thick solid line in Fig. 5K). When the pacemaker is coupled to a strand of 100 follower cells the fraction of open activation gates does not reach as high values as for the isolated pacemaker (compare thick dashed-dotted lines in Fig. 5, K and L). Notice that the fraction of open inactivation gates (w) is near 0.4 for the follower cells (thin solid lines in L, which all superimpose). The gate is not yet completely deinactivated after the previous AP, which has implications for the propagation of the AP, as we will show when we compare Figs. 5 and 6.
In summary, when the pacemaker cell is coupled to a linear strand of follower cells, the membrane potential of the pacemaker cell increases much more slowly than in the uncoupled case and does not reach values as high during the plateau phase due to the decreased flux of calcium ions through the L-type calcium channels and IP3 receptor of the pacemaker cell. The depolarization of the follower cells due to electrical coupling activates the L-type calcium channels in the follower cells only to a very small extent (Fig. 5J) and causes only a very small inflow of calcium ions (Fig. 5H). This inflow is too small to induce a significant CICR through the IP3 receptor in the follower cells (thin solid line in Fig. 5F), because w is not yet sufficiently deinactivated. As a consequence, a single pacemaker cell is not powerful enough to supply 1:1 entrainment between AP propagation and calcium oscillations in a strand of NRK cells with [IP3] = 0.1 µM.
Entrained AP transmission. Figure 6 shows the propagation of activity for a pacemaker cell coupled to 100 follower cells with a Gg of 3 nS when the pacemaker cell succeeds in triggering AP propagation in the follower cells. In Fig. 6A, the coupled pacemaker cell (thick solid line) depolarizes to –20 mV. This depolarization is triggered by calcium release from the store through the IP3 receptor (Fig. 6C). The increased [Ca2+]cyt causes depolarization of the cell by activation of the ClCa channels. Note that the rise of the membrane potential for the pacemaker cell and its neighboring follower cells is much slower than that for distant follower cells. This causes a gradual start of the inactivation of the L-type calcium channel (decreasing h; solid lines in Fig. 6E) before activation m increases (dashed-dotted lines). Since the activation m in the pacemaker cell hardly increases above the value zero (thick dashed-dotted line in Fig. 6E), the L-type calcium channels in the pacemaker cell hardly open (Fig. 6D). For more distant follower cells, the rise of the membrane potential is much faster and activation m of the L-type calcium channel increases rapidly, resulting in an AP.
The depolarization of the pacemaker cell to the Nernst potential of the ClCa channels near –20 mV activates the L-type calcium channels in the neighboring follower cell slightly (hardly visible in Fig. 6D, see arrow), causing a small inflow of calcium ions. Although the inflow of calcium in the follower cells is small and hardly visible in Fig. 6D, the fraction of open inactivation gates (w) of the IP3 receptor (solid lines in Fig. 6F) is large enough to allow a significant CICR through the IP3 receptor (Fig. 6C, see arrow). Moreover, the [Ca2+]cyt (Fig. 6B) due to small influx through the L-type calcium channel (Fig. 6D) and through the IP3 receptor (Fig. 6C) is large enough to cause full depolarization to the Nernst potential of the ClCa channels (Fig. 6A). The CICR through the IP3 receptor in each follower cell reinforces the speed of their depolarization and, therefore, contributes to a better and stronger AP propagation in the one-dimensional array of cells.
The fractions of open activation (f; thick dashed-dotted lines) and inactivation gates of the IP3 receptor for a pacemaker cell (w; thick solid line) coupled to 100 follower cells (w; thin solid lines) are shown in Fig. 6F. Notice that the fraction of open w gates for the neighboring follower cells is a little higher than in Fig. 5K (thin solid lines). This is due to the small [IP3] of 0.1 µM (see Eq. A23), which gives a long time constant w for deinactivation. It takes 300 s to reopen the inactivation gate w of the follower cells completely. Since the pacemaker cell generates an AP every 90 s, the inactivation gate w of the follower cells after a calcium transient has not recovered sufficiently at the next AP of the pacemaker cell. This is clear in Fig. 5F, where the inactivation gate of the neighboring follower cell is 0.4, whereas it reaches values near 0.5 and 0.6 in Fig. 6F (thin solid lines).
The main difference between Figs. 5 and 6 is that the inactivation gate w (thin solid lines) of the IP3 receptor in the follower cells has recovered to higher values in Fig. 6F than in Fig. 5K. Therefore, the relatively small inflow of calcium ions through the L-type calcium channels is large enough to activate the IP3 receptor by CICR in the first follower cell (see arrow, Fig. 6C). The same occurs in the other follower cells. The membrane potential exerts a positive feedback on the calcium oscillator through calcium influx through L-type calcium channels. On the other hand, the release of calcium through the IP3 receptor exerts a positive feedback on the depolarization of the membrane. As a result, AP propagation with underlying calcium oscillations is generated in the follower cells. This positive interaction between membrane excitability and IP3 receptor explains why a small amount of IP3 in the cell supports synchronization and propagation of activity in a network of cells.
We can understand the 1:4 entrainment of AP propagation (Figs. 5 and 6) by having a closer look at the deinactivation time constant w for the w gate. The time constant w (Eq. A23) determines the time for deinactivation (w) of the IP3 receptor, which depends among other things on [IP3]. For low [IP3] values (near 0.1 µM) in the follower cells, the time constant w of the inactivation gate w of the IP3 receptor is long (in the order of 300 s). For high [IP3] values (near 1.0 µM) in the pacemaker cell, w is shorter and about 90 s. Since the time constant w in the follower cells is long (about 300 s) relative to the time interval between APs generated by the pacemaker cell (about 90 s), the deinactivation of the IP3 receptor in the follower cells has not yet recovered after an AP and IP3-mediated calcium oscillation when the pacemaker cell generates the next AP. This explains why 1:1 synchronization between pacemakers and followers is not possible in this case and why synchronization becomes harder for smaller amounts of IP3 in the cell.
Propagation in a strand of cells. We now address the experimental observation that APs and calcium oscillations are propagated with a Gg that is much larger (3 nS) than the predicted optimal gap junctional coupling range for synchronization of a pacemaker to followers (range 0.25–0.4 nS in Fig. 4). We have to keep the following in mind: if the gap junction conductance is very small, the current from the pacemaker to the follower cell is too small to depolarize its neighbor cell rapidly and to a sufficiently high membrane potential. If the gap junction conductance is very large, the pacemaker cell is not able to provide enough current for depolarization of its neighboring cell, since most of the current flows through the network to other cells. As we will show, robust AP propagation and excitation of follower cells in the latter case can be achieved in the model in two ways: by increasing the conductance of the L-type calcium channels and by increasing the number of pacemaker cells.
The first solution is to increase the conductance GCaL of the L-type calcium channel, which is helpful for both the pacemaker and the follower cells. With increased values of GCaL, depolarization of the membrane potential of the pacemaker cell leads to a larger current through the L-type calcium channels and to more current through the gap junctions. This generates a larger current from pacemaker to follower cells and makes it easier to reach the threshold level for opening of L-type calcium channels in the follower cells. For follower cells, a larger opening of their L-type calcium channels leads to better facilitation of CICR through the IP3 receptor. However, since it is known that the maximal value for GCaL is close 1.6 nS (15), it would not be realistic to set GCaL to values higher than 1.6 nS, used in this study.
The second solution is to increase the number of terminal pacemaker cells in the one-dimensional network, which contributes to a larger current to the follower cells. When enough pacemakers are placed in the network, 1:1 AP propagation is observed. This is illustrated in Fig. 7, which shows the result of a simulation of a coupled strand with three terminal pacemakers ([IP3] = 1.0 µM) and 100 follower cells ([IP3] = 0.1 µM). The release of calcium from the ER to cytosol by the IP3 receptor in the pacemaker cells (Fig. 7C) activates the ClCa channels, causing a depolarization to –20 mV (Fig. 7A). This depolarization activates the L-type calcium channels (Fig. 7D). Note the delay of the fluxes through the L-type calcium channels (Fig. 7D) relative to the fluxes through the IP3 receptor (Fig. 7C). The depolarization of the three pacemaker cells causes a depolarization of the follower cells (Fig. 7A) and APs by activation of the L-type calcium channels of the followers (Fig. 7D). The three pacemakers cause 1:1 entrainment for the full network with 100 followers.
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In Fig. 4 we showed that a single pacemaker could not induce propagating activity in a linear strand with a large number of follower cells for large gap junctional conductance. With three pacemakers, 1:1 entrainment is obtained for gap junctional conductances Gg above 0.25 nS.
Minimal value for GCaL. As explained above, L-type calcium channels are necessary for propagation of activity in a network of NRK fibroblasts (7). Figure 8 shows the simulation results where we tried to find the minimum number of pacemaker cells as a function of GCaL at a constant gap junction conductance of 3 nS. The minimal number of pacemaker cells required for stable 1:1 propagation of APs in a linear array of follower cells decreases for increasing GCaL. The data points of the simulations in Fig. 8 reveal that no AP propagation is possible for GCaL below 1.45 nS. Therefore, in our model, a critical minimal value for GCaL is necessary for AP propagation in a network.
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DISCUSSION |
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Role of Electrical Coupling Strength The main result of this study is that it emphasizes the important functional role of the coupling between the excitable membrane and CICR from intracellular stores for calcium APs and the important role of electrical coupling between cells for the initiation and propagation of calcium APs. These results provide a better understanding of empirical results by Yao and Parker (48), who concluded that "electrical transmission may provide a means to ‘leapfrog’ slow chemical wave transmission and rapidly synchronize Ca2+ release within large individual cells and across populations of electrically coupled cells." A similar idea was reached by Sanders et al. (37), who concluded, on the basis of a large set of empirical data, that "voltage-dependent Ca2+ entry that increases Ca2+ activity in pacemaker units near IP3 receptors may be responsible for coordination of Ca2+ release events and entrainment of unitary currents within a network of ICC."
Figure 3 shows entrainment for two heterogeneous pacemaker cells, which display full synchronization of intracellular calcium oscillations for a coupling conductance (Gg) above 60 pS, which is well in agreement with Imtiaz et al. (22), who found that weak electrical coupling is sufficient to synchronize heterogeneous cells of cell pairs. This result explains that at the physiological gap junctional coupling strength (3 nS) of NRK cells (14), the intracellular calcium oscillations and APs of two oscillating cells are completely synchronized in phase. In the modeling study by Imtiaz et al. (22), it was reported that anti-phase behavior could occur for two weakly coupled cells for high oscillation frequencies, which is a well known result for interacting oscillators (9). This anti-phase coupling in their study is the result of the time constants involved in voltage-dependent IP3 synthesis. Since it takes some time before changes in the membrane potential affect the production of IP3, the effective coupling between cells by voltage-dependent IP3 production has a frequency-dependent delay. Therefore, we conclude that voltage-dependent IP3 synthesis cannot play an important role, since it would disable synchronization between cells to form a pacemaker cluster.
For low-frequency intracellular calcium oscillations (typically 1 cycle/min or lower), this delay is small relative to the period of the calcium oscillations, which results in in-phase behavior. For high-frequency intracellular calcium oscillations (2 cycles/min or higher), the period of calcium oscillations is small relative to the time constant for production of IP3, resulting in out-of-phase oscillations. In NRK cells, the highest oscillation frequencies are near 1 cycle/min, but in general, the oscillation frequency is much lower. Therefore, out-of-phase synchronization due to voltage-dependent IP3 production does not happen in NRK cells, and therefore, we did not include voltage-dependent IP3 production.
Figure 4 shows that synchronization of follower cells by a pacemaker cell is easier if the follower cells have a non-zero concentration of IP3, allowing calcium transients by CICR. The CICR through the IP3 receptor in each follower cell reinforces the speed of its depolarization and, therefore, contributes to a better and stronger AP propagation in the one-dimensional array of cells. Moreover, Fig. 4 shows that entrainment and synchronization is optimal for a coupling between 0.25 and 0.45 nS, whereby 1:1 propagation of APs in the network is facilitated. However, the actual conductance of gap junctions between NRK cells is 3 nS, which is much larger than the optimal coupling range that follows from Fig. 4. In our view, the relatively high conductance of 3 nS has at least two consequences.
The first one is that it prevents initiation of activity by a single pacemaker but allows synchronized activity of a small cluster of pacemaker cells. It prevents spontaneous random activity by a single pacemaker cell and ensures robust initiation by a small cluster. For a linear strand, at least three pacemakers are sufficient for propagation. Pilot studies for a two-dimensional network show that roughly 300 pacemaker cells are necessary to initiate propagating activity.
The second aspect related to the gap junctional conductance relates to the velocity of propagation. A larger gap junctional conductance gives a larger propagation velocity (36). Previous studies on propagation of activity in a network of nonexcitable cells with IP3-mediated calcium oscillations (10, 19, 20, 40) have shown that propagation may occur via diffusion of calcium and/or IP3 through the gap junctions. Since diffusion is relatively slow relative to electrical coupling, the propagation velocity in such a network is typically in the range from 5 to 50 µm/s (19, 35, 38), which is much slower than propagation in excitable cells (typically 0.5–100 cm/s) and in our NRK cells (6, 16), where the propagation velocity is approximately a few millimeters per second.
Required Number of Pacemaker Cells for Exciting the Strand The analysis of the one-dimensional network in Figs. 5 and 6 with physiological electrical coupling (Gg = 3 nS) reveals that one terminal pacemaker cell cannot deliver sufficient current to the follower cells to obtain 1:1 synchronization and AP propagation in the cell strand. Increasing the value of L-type calcium conductance (GCaL) alone would not help (see Fig. 8). The minimal number of pacemaker cells required to initiate AP propagation depends on the [IP3] of the follower cells (compare results for [IP3] = 0 and [IP3] = 0.1 µM in follower cells in Fig. 4) and critically depends on GCaL, which needs to be larger than 1.45 nS to facilitate propagation of APs at all (Fig. 8). In this study we have fixed GCaL at 1.6 nS, which results in a requirement of three pacemaker cells for 1:1 propagation of APs in the one-dimensional network.
Chemical Coupling Versus Electrical Coupling We have shown that calcium waves and propagation of APs can be achieved by a mechanism where depolarization by AP firing and calcium-triggered opening of chloride channels cause an AP and intracellular calcium transient in its neighbor cell. Such a coupling mechanism is significantly more effective than that of the chemical coupling-based class of models, since a membrane potential change has a quick coupling effect over distances several orders of magnitude greater than diffusion of either calcium or IP3 through gap junctions (40).
Both calcium and IP3 have been shown to permeate through gap junctions by diffusion. This mechanism plays a crucial role in the propagation of calcium waves in networks with nonexcitable cells (10, 19, 20, 40). In our study with excitable cells, the electrical coupling and the relatively fast dynamics of the L-type calcium channel provide a much faster propagation than in nonexcitable cells. Since diffusion of calcium and IP3 takes place on a time scale much longer than the time scale of propagation by electrical coupling and activation of the L-type calcium channels, we have not incorporated diffusion of calcium and IP3 in our study.
In a recent study, Tsaneva-Atanasova et al. (43) suggested that intercellular calcium diffusion is necessary and sufficient to synchronize the oscillations in neighboring cells with different intrinsic oscillation frequencies. The results of our study indicate that intercellular calcium diffusion may be sufficient but is not necessary, since coupling of intracellular calcium oscillations by the excitable membrane and electrical intercellular coupling also achieves synchronization of pacemaker cells with different intrinsic oscillation frequencies.
Several studies (see e.g., Ref. 19) calculated the minimal gap junctional permeability for calcium, which is required for calcium wave propagation, as a function of the diffusion coefficient for calcium. The minimal value is found to be 0.05 µm/s, which gives an inflow of about 0.25 x 10–6 µmol in our cell. The results of our model study show that the total change of [Ca2+]cyt due to inflow through the L-type calcium channels is about 100 x 10–6 µmol per AP. This is much larger than the change due to diffusion of calcium, which explains why the propagation of calcium waves mediated by the L-type calcium channels is faster and more robust. In the present model study, agonist or IP3 diffusion may improve local synchronization of the surrounding follower cells by smoothing the sensitivities of the CICR mechanism.
Voltage-Dependent Gap junctional Conductance
In our study, the gap junctional conductance is assumed to be independent of the voltage of the membrane. However, it is well known that the gap junctional conductance is not constant but voltage dependent. The gap junctional conductance between two cells may decrease up to 20% during an AP compared with the steady-state conductance without any voltage difference across the gap junction. In a recent study using transfected neuroblastoma cells, inactivation kinetics of Cx43 were studied by imposing an AP clamp instead of a rectangular voltage pulse on one of the cells (30). These experiments showed that following the peak of the AP, the junctional conductance decreased within 25 ms to 58% of control. These relatively slow time constants are in agreement with experimental observations (2), which indicate that the transition rates for the gap junction channels are significantly longer than the time constant of the cell membrane, which is 1 ms. Comparison of these inactivation times to transjunctional conduction times observed during steady-state propagation under conditions of severe uncoupling suggests that gap junctional gating has only a minor effect on overall conduction velocities (36).
Functional Implications The results of this study show that the coupling of intracellular calcium oscillations and AP firing causes propagation of activity through a network of cells, which is robust and much faster than propagation of calcium waves in a network of nonexcitable cells (12, 19, 20, 43).
CICR through the IP3 receptor, triggered by calcium inflow through L-type calcium channels during an AP, supports cell depolarization by activation of ClCa channels. This boosting of propagation of activity by CICR provides a robust mechanism that is also found in gastrointestinal cells (44), urethral cells (4, 22), and heart pacemaker cells (31). In all these cell types, robust pacemaking and propagation of activity is crucially important for the function of the cell network in the organism.
In smooth muscle cells, oscillatory release of calcium through IP3 receptors and voltage-dependent calcium influx through L-type calcium channels underlie rhythmic vasomotion (1, 50). Spontaneous calcium waves occurring after a long AP plateau may also modulate the removal of voltage-dependent inactivation of L-type calcium channels and affect the likelihood of the occurrence of early afterdepolarizations (48). Spontaneous calcium oscillations may be implicated in diverse manifestations of heart failure-impaired systolic performance, increased diastolic tonus, and an increased probability of the occurrence of arrhythmias (48). Therefore, the outcomes of this model study are also of interest for understanding mechanisms of pacemaker synchronization and AP propagation in many other systems.
Conclusion Consistent with experimental observations for NRK cells (30), we have shown that electrical intercellular coupling is sufficient for synchronizing calcium oscillations of pacemaker cells and for propagation of action AP-coupled calcium waves over a linear network of cells. For NRK cells it has become clear that membrane excitation can evoke and enhance release of calcium from the ER store via voltage-dependent calcium inflow through L-type calcium channels. Our general message is that some form of CICR interaction with or caused by calcium inflow through voltage-dependent calcium channels can boost propagation of electrical excitation and its continuous calcium oscillations. Any form of CICR (ryanodine-mediated receptors are another example) would engage a similar interaction.
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APPENDIX |
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TOPABSTRACTMETHODSRESULTSDISCUSSIONAPPENDIXGRANTSREFERENCES |
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 | (A1) |
 | (A2) |
 | (A3) |
 | (A4) |
 | (A5) |
 | (A6) |
 | (A7) |
 | (A8) |
 | (A9) |
 | (A10) |
 | (A11) |
 | (A12) |
 | (A13) |
 | (A14) |
 | (A15) |
 | (A16) |
The equations defining the properties of the IP3-mediated intracellular calcium dynamics are as follows:
 | (A17) |
 | (A18) |
 | (A19) |
 | (A20) |
 | (A21) |
 | (A22) |
 | (A23) |
 | (A24) |
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GRANTS |
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FOOTNOTES |
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The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
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REFERENCES |
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) for 1:1 synchronization and propagation of APs as a function of the conductance of the L-type calcium channel (GCaL) and number of pacemaker cells for a constant Gg of 3 nS. No propagation takes place for GCaL < 1.4 nS whatever the number of pacemaker cells.