| HOME | HELP | FEEDBACK | SUBSCRIPTIONS | ARCHIVE | SEARCH | TABLE OF CONTENTS |
|
|
|
|||||||
|
|||||||||
1Department of Physiology and 2Centre for Nonlinear Dynamics in Physiology and Medicine, McGill University, Montreal, Quebec, Canada; and 3Center of Physiological Medicine, Institute of Biophysics, Medical University Graz, Graz, Austria
Submitted 12 July 2004 ; accepted in final form 3 February 2005
 |
ABSTRACT |
|---|
 TOP ABSTRACT METHODSRESULTSDISCUSSIONAPPENDIXGRANTSREFERENCES |
|---|
100 V/s vs. 10 V/s). On the basis of our voltage- and current-clamp results and data from the literature, we formulated a Hodgkin-Huxley-type ionic model for the electrical activity in these small clusters. The model contains a Ca2+ current (ICa), three K+ currents (IKs, IKr, and IK1), a background current, and a seal-leak current. ICa generates the slow upstroke, whereas IKs, IKr, and IK1 contribute to repolarization. All the currents contribute to spontaneous diastolic depolarization, e.g., removal of the seal-leak current increases the interbeat interval from 392 to 535 ms. The model replicates the spontaneous activity in the clusters as well as the experimental results of application of blockers. Bifurcation analysis and simulations with the model predict that annihilation and single-pulse triggering should occur with partial block of ICa. Embryonic chick ventricular cells have been used as an experimental model to investigate various aspects of spontaneous beating of cardiac cells, e.g., mutual synchronization, regularity of beating, and spontaneous initiation and termination of reentrant rhythms; our model allows investigation of these topics through numerical simulation.
pacemaker; seal-leak current; rapid delayed rectifier potassium current block; slow inward calcium current block; bifurcation analysis
After a couple of days in culture, the electrical activity in an isolated embryonic chick ventricular cell, in a small cluster of a few such cells, or in a sparse monolayer is very different from that in situ or in a reaggregate of hundreds or thousands of cells isolated from the ventricle. For example, when trypsin-dispersed ventricular cells from 7-day embryonic chick hearts are used, the upstroke velocity is much lower in single cells, in small clusters of cells, and in sparse monolayers (17, 49, 51, 95) than in reaggregates (14, 16, 19) or in the intact ventricle (19, 97, 98, 118). Spontaneous beating can be abolished by addition of tetrodotoxin (TTX), a blocker of the fast inward Na+ current (INa), to the medium bathing reaggregates of trypsin-dispersed 7-day ventricular cells (16, 70), but spontaneous activity continues in single cells and monolayers (57, 70, 81, 95). However, spontaneous activity in some TTX-insensitive preparations can be abolished by addition of either of the Ca2+ channel blockers D-600 or diltiazem (49, 51).
We carried out an experimental study to characterize the electrical properties of spontaneously beating clusters of cells isolated from the 7-day embryonic chick ventricle and then assembled a mathematical model of this activity. The goal is to use this model to investigate phenomena seen in experiments, such as irregularity of beating in small clusters (10), mutual synchronization of pairs of cells (18), phase resetting and phase locking (9, 11, 52, 53), current-pulse-induced annihilation of spontaneous activity (90, 94), and spontaneous initiation and termination of spiral-wave reentrant motions in monolayers (6). Although ionic models of reaggregates of embryonic chick atrial cells have been described (9, 13, 52, 89, 90), we are not aware of any models of small clusters of isolated embryonic chick ventricular cells.
|
METHODS |
|---|
|
TOPABSTRACTMETHODSRESULTSDISCUSSIONAPPENDIXGRANTSREFERENCES |
|---|
Ventricular myocytes were isolated from embryonic chick hearts by means of techniques previously described (50) with modifications (79). The hearts of 7-day embryos were removed, and the ventricles were chopped off, minced, and transferred to flasks containing 0.25% trypsin (bovine pancreas; Boehringer Mannheim, Deisenhofen, Germany) in a nominally Ca2+- and Mg2+-free Hanks' balanced salt solution (HBSS; in mM: 137 NaCl, 5.4 KCl, 0.34 Na2HPO4, 0.44 KH2PO4, 4.2 NaHCO3, and 5 glucose, pH 7.4). The flasks were placed in a shaker bath at 37°C for 7 min. The resulting cell suspension was gently agitated with a pipette and filtered through a 100-µm mesh. HBSS, supplemented with fetal calf serum (5% final concentration), was added to stop trypsin activity. The cell suspension was centrifuged at 100 g for 5 min at 4°C, the supernatant was discarded, and the cell pellet was resuspended in fresh trypsin-free HBSS. The centrifugation and resuspension processes were then repeated. The solution was centrifuged for a third time and resuspended in cell culture medium [M199 (Sigma) supplemented with 4% fetal calf serum, 2% horse serum, and 0.7 mM glutamine, pH 7.4] to yield a density of 5 x 105 cells/ml.
The cell suspension was separated into aliquots on nonadhesive plastic culture dishes that were incubated at 37°C in a water-saturated atmosphere of 95% air-5% CO2. To obtain small clusters of cells, 0.6-ml aliquots of cell suspension were removed after 0.5–2 days and placed into the lumen of flexiPERM silicone rings (Heraeus, Hanau, Germany), which were attached to microscope slide coverslips. Coverslips with attached flexiPERM rings were placed in cell culture dishes (Greiner) and stored in the incubator. This procedure allowed the myocytes to adhere to the glass surface, where they could divide and form small clusters of cells. However, we can offer no guarantee that a particular cluster is made up entirely of cells that divided in culture.
Experiments were performed 2–36 h after the cells were plated, on clusters containing two to four cells. Recordings were made in the whole cell mode from clusters that were spontaneously beating before they were patched. We use clusters of a few cells, rather than single cells, because we were unable to obtain recordings of spontaneous activity from single cells. [It is also easier to successfully impale a cell in a cluster, rather than a single cell, if a conventional sharp microelectrode is used (17, 26).] In addition, the effect of the seal-leak current (Iseal; see below) on spontaneous activity is expected to be considerably smaller for a cluster than for a single cell (but see Ref. 78). For electrophysiological recording, the coverslip with attached myocytes was used to form the bottom of the experimental chamber, which was placed on the stage of an inverted microscope (Zeiss, Axiovert). The experimental chamber was perfused with extracellular solution (in mM: 137 NaCl, 5.4 KCl, 1.8 CaCl2, 1.1 MgCl2, 2.2 NaHCO3, 0.4 NaH2PO4, 10 Na-HEPES, and 5.6 glucose, with pH adjusted to 7.4 with NaOH) at 36–37°C with a flow rate of 1.5 ml/min.
Electrophysiological Recording
The transmembrane potential (V) was recorded using the whole cell recording mode of the patch-clamp technique. Patch pipettes (2 M
resistance) were pulled from glass capillary tubes and filled with pipette solution (in mM: 110 KCl, 4.3 K2-ATP, 2 MgCl2, 1 CaCl2, 11 EGTA, and 10 K-HEPES, with pH adjusted to 7.4 with KOH) with estimated free Ca2+ concentration ([Ca2+]) < 10–8 M. Electrode potentials were zeroed before seal formation. After the patch was broken, the transmembrane potential was recorded with an amplifier (model EPC-7, List, Darmstadt, Germany). The membrane capacitance was measured by integration of the capacitive transient in response to a voltage-clamp step from –50 to –60 mV. After capacity compensation, we compensated for series resistance by turning up the series-resistance compensation control (which controls the amount of positive feedback) to just below the value where ringing in the current monitor signal would occur. Usually, compensation could be made for >50% of series resistance. For generation of voltage-clamp protocols and for recording voltage and current, a personal computer equipped with pCLAMP version 5.5.7 software (Axon) and a DigiData 1200 interface (Axon) were used. Signals were also digitized at 44 kHz, pulse-code modulated, and stored on video cassette tape for offline analysis. The signal was played back, sampled at 1 kHz (Axotape, Axon Instruments), and stored on a disk file for computerized analysis.
Action Potential Parameters
Several parameters are measured to characterize spontaneous electrical activity. Interbeat interval (IBI) is the time between consecutive crossings of 0 mV on the upstroke of the action potential. The maximal diastolic potential (MDP) is the most negative voltage recorded during an action potential and the overshoot potential (OS) the most positive. The action potential amplitude (APA) is the difference between MDP and OS. Action potential duration (APD) is the time from the crossing of 0 mV on the upstroke to the time of 50% repolarization (APD50) or 100% repolarization (APD100). The diastolic depolarization rate (DDR) is the slope of the chord joining the point where (MDP + 1 mV) is crossed and the point 70 ms later (108, 114). The maximum rate of rise of the upstroke (
max) is computed using the greatest voltage difference between two consecutive samples on the upstroke (1-ms sampling interval).
Simulation Methods
Numerical integration of the Hodgkin-Huxley-type ionic model was carried out using a forward Euler scheme, with V at time t + 
t calculated as follows
 |
t = 0.1 ms), the change in voltage from time t to t + t was kept to <1 mV. The value of each activation or inactivation variable (
i) at time t + t was obtained from its value at time t using the analytic formula
 |
i(
) is the steady-state or asymptotic value of i at V(t) and 
i is the time constant of i at V(t). L'Hôpital's rule was used to calculate 
n when V came to within ±0.1 mV of the value producing an indeterminate form. The numerical integration routine was written in C, and all variables were double precision (16 significant decimal places). Bifurcation analysis was carried out using AUTO, as incorporated in XPPAUT (25). The model equations file for use with XPPAUT is available as supplemental material (supplemental data for this article may be found at http://ajpheart.physiology.org/cgi/content/full/00683-4.2004.DC1).
Formulation of the Model
Our model consists of six currents: a slow inward Ca2+ current (ICa), a slow delayed K+ current (IKs), a rapid delayed rectifier K+ current (IKr), an inward rectifier K+ current (IK1), a linear time-independent background current (Ib), and a linear nonspecific seal-leak current (Iseal), generated by the leakage of ions through the gigaohm seal of the recording pipette. We now give the rationale for including each of these currents and for the particular formulation that we employ for each of these currents, as well as reasons for using a "first-generation," rather than a "second-generation," model.
Difficulties with second-generation models. In the earlier Hodgkin-Huxley-type ionic models of cardiac cells, all the concentrations of the various ionic species were held fixed, so that no provision had to be made for pumps and exchangers to regulate these concentrations. We refer to models that incorporate both of these refinements as second-generation models, in contrast to the earlier first-generation models. We formulate our model below as a more primitive first-generation model, because there are two major problems with the more physiologically realistic second-generation models: 1) drift, with very slow long-term trends in some of the variables, particularly some ionic concentrations (3, 21, 23, 24, 37, 45, 103, 116), and 2) degeneracy, with nonuniqueness of equilibrium solutions such as steady states and limit cycles (23, 24, 30, 37, 102).
Drift has been managed in several ways: 1) by finely adjusting parameters to achieve flux balance across the membrane (21), 2) by adding an electroneutral Na+ current of a precise size to produce stability of concentrations (74), 3) by monitoring the stimulus current in a paced quiescent cell (37, 45), and 4) by ensuring that certain ionic concentrations remain fixed (3, 55, 103, 121). It is not clear whether strategies 1 and 2 are robust, because a change in some parameter in the model might require further fine adjustment of the stabilizing parameters. (This is reminiscent of a neutrally stable equilibrium.) Strategy 3 is, of course, of no use in an unpaced pacemaker cell. Strategy 4 defeats, at least in part, the initial intent in formulating the model as a second-generation model; e.g., when all the internal and external iconic concentrations are held constant, the Na+-K+ pump current (INaK) and the Na+/Ca2+ exchange current (INaCa) are effectively background currents, and one is left essentially with a first-generation model, in which activity-dependent effects due to changes in certain ionic concentrations are not manifest.
The other major problem noted with second-generation models is degeneracy. In second-generation models of several different types of cardiac cells, the system of differential equations could be rewritten as a system of N – 1 equations in N unknowns (30, 102). The Jacobian is then singular, and there is a continuum of equilibrium points, rather than isolated equilibrium point(s), so that, e.g., the resting potential of a quiescent system depends on the initial conditions (30, 102). A similar finding of degeneracy holds for the limit cycle that corresponds to spontaneous activity (24, 30). It has been suggested that the original N-variable fully differential model should be recast as a differential-algebraic system, with the equation controlling voltage being algebraic and the remaining (N – 1) equations being differential (23, 24, 30, 37, 45, 102). In one report in a sinoatrial (SA) node model in which the differential-algebraic formulation was used, it was stated that there was no long-term drift (23). In earlier work in which drift was abolished by making some ionic concentrations fixed, this also had the unintended benefit of removing the degeneracy, thus allowing the bifurcation analysis of isolated equilibria by means of continuation techniques (103; see also Refs. 55 and 121).
Finally, in situations such as ours in which cells are studied using patch micropipettes, a more realistic model of the experimental situation is one in which internal concentrations are kept fixed as a result of Ca2+ buffering with EGTA and dialysis of the cell contents of a very small cell volume against the much larger pipette volume (55, 121). Making concentrations fixed then also removes degeneracy and drift. Given all the above uncertainties and complications and given that very little information is available about the control of intracellular ionic concentrations in our cells, we decided to use a first-generation model, as have some authors of other quite recent studies (3).
Capacitance. Unless stated otherwise, ventricular and atrial cells will refer to embryonic chick ventricular and atrial cells, respectively, and n-day will refer to a cell isolated from the embryo after n days of incubation. Because the capacitance of a single 7-day ventricular cell in our laboratory is 8–9 pF, we set the capacitance in our three-cell model cluster to 25.5 pF. Our value of the single-cell capacitance agrees with that reported in several whole cell voltage-clamp studies employing single 7-day ventricular cells [e.g., 5–10 pF (29) and 4–7 pF (44)]. We model the cluster, which is a mutually synchronized population oscillator, as an isopotential preparation (27).
ICa. ICa has been described in 7-day reaggregates of ventricular cells (72), in small clusters of ventricular cells (28, 79), in single ventricular cells (13, 28, 42, 44), and at the single-channel level (43, 68). In one laboratory, L-type ICa (ICa,L) was found in 11 of 12 6-day ventricular cells, and T-type ICa (ICa,T) was found in the remaining cell (13). In another laboratory, only ICa,L was found in 7-day ventricular cells (68). In a third study, both currents were found, but in only 45% of 7-day ventricular cells was ICa clearly separable into ICa,L and ICa,T on the basis of the voltage threshold for activation (44). In these cells, ICa,T is half-inactivated at –49 mV, whereas ICa,L is half-inactivated at –27.5 mV. In the 55% of cells with nonseparable ICa, half-inactivation occurs at –42.8 mV.
We have chosen to use a nonseparable description of ICa in our model
 |
The steady-state inactivation (f) curve of the nonseparable current in 7-day ventricular cells, which lies between the inactivation curves for ICa,L and ICa,T, is taken from Fig. 9 of Ref. 44. Because the steady-state activation (d) curve of the nonseparable current was not reported, we assume that it lies between the ICa,L and ICa,T activation curves of Fig. 10 of Ref. 44 and that it has an intermediate slope factor. We have chosen the values of parameters so that the d curve is closer to the ICa,L than to the ICa,T activation curve, in order that the peak-current current-voltage (I-V) curve has a maximum at 0 mV, which is close to the experimental value for the nonseparable current (44). Figure 1A shows the steady-state d and f curves.
|
|
|
d) and inactivation (f) of the nonseparable current. We therefore take the expressions unchanged from Ref. 66, which describes ICa,L in a 37°C guinea pig model. The d curve (Fig. 1B) has a typical bell shape, whereas the f curve (Fig. 1B) has the U shape that is seen in mammalian cells and in 7-day ventricular reaggregates (72).
We have set gCa so that max is 9.5 V/s in the model, which is close to our mean experimental result (8.5 V/s). The peak-current I-V curve then has a maximum value of 30 pA/pF at 0 mV (Fig. 1C), which is within the range seen in our cells [79; see also Fig. 4C of Ref. 44 and Fig. 1 of Ref. 13, scaled for differences in capacitance and temperature (Q10 
3 for peak current amplitude for guinea pig ventricular cells (7))]. Figure 1D shows that the current traces from a voltage-clamp protocol in the model are similar, in magnitude and time course, to the corresponding experimental traces in Ref. 79 (see also Fig. 7B of Ref. 44, scaled for temperature and capacitance).
|
|
Delayed K+ currents. The delayed K+ currents IKs and IKr have been described in reaggregates of ventricular cells (13, 14, 88) and atrial cells (9, 88, 89), in small clusters of ventricular cells (79), and in single ventricular (8) and atrial (12) cells. These currents have also been observed at the single-channel level in ventricular cells (8, 67).
Two components of the delayed current, initially termed Ix1 and Ix2, have been seen in atrial reaggregates (89). These two currents correspond to those more recently termed IKr and IKs, respectively, in isolated adult mammalian ventricular cells. Ix2, or IKs, has been described in 7-day ventricular reaggregates (14, 15), in single ventricular cells (8), and in small clusters of such cells (79). However, although Ix1 is robust in experiments carried out using sharp microelectrodes on atrial (89) and ventricular (13) reaggregates, it has not been seen in whole cell clamp experiments carried out using patch pipettes on isolated 7- to 10-day ventricular cells (8) or on single 6- to 11-day atrial cells or small clusters of such cells (12). This might be due to an intrinsic absence of the current [e.g., there is good evidence for cell-cell contact-dependent regulation of expression of two different K+ channels in cultured adult rat ventricular cells (35)] or to rapid washout of this current in the whole cell ruptured-patch recording mode. Indeed, it has been suggested that the main difference in the action potentials of atrial reaggregates and small clusters of atrial cells can be accounted for by the absence of Ix1 in the latter (see Fig. 9 of Ref. 12 and Fig. 17 of Ref. 13). In contrast to the above-mentioned reports, in our 7-day ventricular clusters, the envelope-of-tails test shows two components (79), and application of the specific IKr blocker almokalant (113) removes the more rapidly deactivating component of the tail current. In addition, almokalant produces changes in the action potential consistent with IKr block (see below). At the single-channel level, a K+ channel that activates over a voltage range similar to that over which IKr activates has been described (8). We thus incorporate the IKs and IKr components into our model.
For IKs, we use the formulation previously used in an atrial reaggregate model (52)
 |
n and 
n (see APPENDIX), we have first divided the original equations (52) by a factor of 3, to obtain a n curve consistent with the experimental values in single ventricular cells at room temperature (see Fig. 2C of Ref. 8), and then multiplied n and n by a factor of 2, in correspondence with the reported Q10 (111), to obtain values appropriate for our experimental temperature of 36–37°C (Fig. 2B). The maximal conductance (gKs) was set to give a fully activated I-V curve (Fig. 2C), similar to that seen experimentally (see Fig. 1C of Ref. 8), scaled for capacitance and temperature (Q10 2) (111). The steady-state activation curve (Fig. 2A) and the voltage-clamp currents (Fig. 2D) are similar to those previously reported from our laboratory at 37°C (79), as well as those reported elsewhere (8), when compensated for temperature.
|
 |
|
|
and s curves (Fig. 3, A and B) are very close to the data in Fig. 5, A and B, of Ref. 89. A voltage-clamp protocol (Fig. 3D) gives currents similar to those in Fig. 4 of Ref. 89. We use only one time constant of activation; in SA node cells, two time constants of activation of IKr have been described (76).
|
|
 |
) was originally used (65), we employ the time-dependent description to allow us to later formulate a model with stochastic gating kinetics to investigate beat-to-beat fluctuations in IBI (unpublished observations). However, the time constant of this current is so small (K1 < 0.2 ms over the operative range of voltage; Fig. 4B) that the current is virtually identical in the time-dependent and time-independent descriptions. The maximal conductance (gK1) is reduced from the guinea pig value to reflect the smaller IK1 earlier in development (15, 41). The steady-state I-V curve for the total current (see curve in Fig. 7A) is then very flat between –70 and –30 mV, which agrees with our experimental results (see symbols in Fig. 7A). The IK1 I-V curve (see Fig. 7C) is the main contributor to the positive slope of the total-current I-V curve at very hyperpolarized potentials (see Fig. 7A) and is similar to the Ba2+-sensitive current at hyperpolarized potentials (4, 79).
Ib.
In addition to IK1, which is outward at potentials depolarized to –81 mV, there is inward background current in 7-day ventricular reaggregates (14). This component has been modeled as a Na+ current
 |
|
 |
(20) and Eseal = 0 mV. Figure 7D gives the I-V relation of Iseal. Currents Not Included in the Ionic Model
INa.
There is voltage-clamp evidence for the existence of INa in reaggregates of 7- to 11-day ventricular cells (22, 72), in single 2- to 18-day ventricular cells (29, 40, 82, 83, 112), and at the single-channel level in 7-day ventricular cells (64, 112). Voltage-clamp studies of 7-day ventricular clusters in our laboratory show a fast inward current upon a depolarizing clamp step from potentials more hyperpolarized than about –60 mV. However, our clusters have a very low upstroke velocity (8.5 V/s), suggesting that INa might not contribute appreciably to the upstroke phase, especially because Ca2+ channel blockers abolish spontaneous activity (see Fig. 9, A and B). The MDP (–60 mV in the clusters and –67 mV in the model) is sufficiently depolarized to essentially render INa fully inactivated, because the foot of the INa steady-state inactivation curve lies at about –50 to –60 mV in 7-day ventricular reaggregates (22) and 7-day ventricular cells (29, 82). Indeed, addition of INa to our model, on the basis of the conductance and the activation and inactivation curves from single 7-day ventricular cells (29) and the time constants from 11-day reaggregates (22), slightly increases max from 9.5 to 10.2 V/s. In contrast, reaggregates of trypsin-dissociated 7-day ventricular cells have a TTX-sensitive upstroke velocity of 120 V/s in 1.3 mM K+ and 91 V/s in 4.5 mM K+, presumably due to the more hyperpolarized MDP of about –90 and –76 mV, respectively (16, 19).
INa can also be involved in generating the pacemaker potential. There is indeed evidence that INa is necessary in some isolated embryonic cells for the generation of spontaneous activity. After 24 h in culture, 37% of single cells dissociated using trypsin from 7-day hearts (whole hearts, atria, or ventricles) stop beating after the addition of 10–5 g/ml TTX (70), showing that INa is crucial in generating spontaneous activity in these cells. However, the percentage of TTX-insensitive cells increases with time spent in culture: 43% at 4 h, 64% at 24 h, and 100% at 48 h (57). In contrast, reaggregates of trypsin-dissociated 7-day ventricular cells that are cultured for 24–72 h stop beating when exposed to TTX (16, 70). This difference in the response to TTX almost certainly indicates the importance of cell-to-cell interactions (19, 69). In newborn rabbit SA node, a TTX-sensitive current, which gradually disappears within the first 30 days postnatum, has been implicated in the generation of diastolic depolarization (2). This contribution is not due to the INa window current but, rather, is a consequence of relatively slow inactivation of INa in the pacemaker range of potentials. Recently, modeling work has suggested a role for a persistent component of a mutated INa in the generation of diastolic depolarization in long Q-T (LQT3) syndrome (105).
A third role for INa is maintenance of the plateau of the action potential, e.g., via a window current contribution. Application of TTX results in a shortening of the APD in some 7-day ventricular cells before they stop beating (64). Single Na+ channels occasionally (1 of 100 beats) stay open throughout the action potential plateau (64) and burst for >150 ms in 16% of trials during a long voltage-clamp step (40). Because these long openings do not persist into diastole (see Fig. 1 of Ref. 64), they would not contribute to diastolic depolarization. Incorporation of our standard Hodgkin-Huxley-type INa into the model, as described above, results in a slight 3-ms increase in APD50 and a slight 5-ms increase in APD100.
Pacemaker current.
The pacemaker current (If) has been reported in ventricular reaggregates (4, 14, 15, 87, 88) as well as in single atrial and ventricular cells and small clusters of such cells (4, 5, 85). The midpoint of the activation range of If is 30 mV more negative in single ventricular cells and small clusters than in reaggregates, with the foot of the activation curve of this hyperpolarization-activated current lying at –70 mV in single cells and small clusters (5). In our clusters, we find If activated at potentials negative to –70 mV (79). On the basis of the conductance, reversal potential, kinetics, and activation curve described in Ref. 5, we find that addition of If to the model causes only a very slight decrease in IBI from 392 to 390 ms. The MDP in our cells is, hence, too depolarized for If to activate and contribute significantly to pacemaking activity; therefore, we do not include it in our model.
Transient outward current.
Although the size of the transient outward current (Ito) increases with development, the number of isolated ventricular cells possessing Ito is extremely low: 7 of 300 cells at 3 days, 5 of 200 cells at 10 days, and 5 of 100 cells at 17 days (84). At the single-channel level, an early outward channel appeared in only 1 of 80 patches from 7-day ventricular cells (67). Ito was not seen in single atrial cells (12), nor was it "clearly observed" in 7- to 12-day atrial reaggregates (9). Moreover, in our own voltage-clamp experiments, we have also found no evidence of Ito in the clusters. In the SA node, the Ito density is smaller in cells with a smaller capacitance (58, 109). For these reasons, we do not include Ito in our model.
Cl– current. A time-independent Cl–-sensitive current (ICl) has been described in isolated 11-day ventricular cells (63). We do not include ICl explicitly in the model, but we consider it to be a component of Ib.
INaK and INaCa. Currents provided by ion pumps and exchangers, e.g., INaK, INaCa, and the Ca2+ pump, also contribute to V. We employ a first-generation model, which does not have an Na+-K+ pump, an Na+/Ca2+ exchanger, a Ca2+ pump, internal Ca2+ dynamics, and variable ionic concentrations. However, INaK and INaCa are present in 7-day ventricular cells. Although these currents are included in several recent ionic models of cardiac tissue, we do not include them in our model, because this would result in a second-generation model.
In our laboratory, 10 µM ouabain has been used to block INaK in 7- to 10-day ventricular cells or in small clusters of such cells (48). After 1 min of superfusion, the IBI first decreases, due to an increase in DDR, OS and MDP gradually fall, APD rises, and IBI increases, so that within a few minutes, spontaneous activity ceases, with the membrane coming to rest at about –30 mV. (Sharp microelectrodes were used in these experiments, so that artifact due to dialysis and current rundown was minimal.) A similar result is seen in 11-day cells cultured as a confluent layer or polystrand, except the initial effect is seen immediately, presumably because of the use of a perfusion system with rapid perfusate changeover (half-time of 5 s), with the membrane coming to rest at about –40 mV (see Fig. 1 of Ref. 39). It has been estimated that INaK contributes 0.35 pA/pF at –70 mV in spontaneously beating 11-day reaggregates (100). In another report on 11-day reaggregates at an internal [Na+] of 41 mM, the ouabain-sensitive current amounts to 1.7 pA/pF and is independent of voltage over the operating range effective in our clusters (from –60 to +20 mV) (see Fig. 1 of Ref. 99). This value scales to 0.5 pA/pF at an internal [Na+] of 10 mM (see Fig. 7 of Ref. 99), which agrees with SA node modeling work (54).
In our model, we can thus mimic the effect of blocking the electrogenic component of INaK by adding a constant depolarizing current of 10.2 pA (i.e., 0.4 pA/pF). This has the effect of decreasing IBI from 392 to 337 ms and depolarizing the MDP from –67 to –61 mV; both effects are seen immediately upon block of INaK in an experiment (see Fig. 1B of Ref. 39). Because the electrogenic component of INaK is removed within a few seconds of the start of block (Fig. 1 of Ref. 100), some secondary change must be responsible for the cessation of activity that occurs some minutes later (see Fig. 1 of Ref. 39). The most likely candidates are the rises in internal [Na+] and [Ca2+] after INaK block, which occur with a time course on the order of minutes (38, 39).
There is clear evidence for INaCa in 11-day ventricular reaggregates (100) and 11-day cells cultured as a confluent layer or polystrand (38). The maximum amplitude of INaCa in embryonic chick cells is about the same as in guinea pig ventricular cells (62). However, because of the difficulties inherent in interpreting experiments attempting to characterize INaCa, many of its fundamental properties (e.g., stoichiometry) remain uncertain in embryonic chick ventricular cells (62). The extent to which INaCa is involved in generating diastolic depolarization in SA node cells is controversial (56) and is very different in different SA node models, to the extent that although INaCa is inward in most models, it is outward in at least one (see Fig. 7 of Ref. 54). In addition, in a model of spontaneous activity induced by suppression of IK1 in guinea pig ventricular cells, INaCa is inward during the pacemaker potential (91). We are not aware of any studies of the Ca2+ pump in embryonic chick ventricular cells.
Given the above problems, as well as other problems described earlier involving degeneracy and drift in models where pumps and exchangers have been added, we have chosen not to include these currents in our model. Rather, INaK can be thought of as being incorporated into Ib, whereas the time course of ICa in our model very closely resembles the action potential clamp record (i.e., sum of ICa and INaCa and any Ca2+-activated currents) obtained in the SA node when Ca2+ entry is blocked (120).
Other currents. Other currents, such as Ist [a sustained inward current, carried by Na+, insensitive to TTX, and sensitive to Ca2+ channel blockers (71, but see Ref. 107)] and IK(Ca) [a Ca2+-activated K+ current, for which evidence is found only in the perforated-patch configuration (120)], exist in the SA node. Because there are no reports of these currents in ventricular cells, we do not include them in our model.
|
RESULTS |
|---|
|
TOPABSTRACTMETHODSRESULTSDISCUSSIONAPPENDIXGRANTSREFERENCES |
|---|
Action potentials recorded from 17 small clusters clearly show considerable cluster-to-cluster variability (Fig. 5) (17). Figure 6A shows a recording of V obtained from one small cluster (cluster 7 in Fig. 5), whereas Fig. 6B shows the phase-plane trajectory, in which the rate of change of V () is plotted vs. V. For this cluster, the mean values of the parameters, averaged over 100 cycles of activity, were as follows: IBI = 458 ms, MDP = –57 mV, APA = 89 mV, max = 7.5 V/s, DDR = 85 mV/s, APD50 = 124 ms, and APD100 = 224 ms. The action potential parameters (means ± SD) of the 17 clusters are given in Table 1. Because of beat-to-beat variability, the action potential parameters for each cluster were averaged over 100 beats before the population average was taken. Figure 6, C and D, gives the voltage-time series and the phase-plane trajectory for the ionic model, and Table 1 gives the action potential parameters in the model, which are quite close to the mean experimental values.
|
The curve in Fig. 7A gives the steady-state I-V relation for the total current in the model. This curve corresponds closely to the mean I-V data points obtained from five 7-day ventricular clusters in our laboratory (Fig. 7A, symbols); see also Ref. 4. The steady-state I-V relations of the individual currents in the model are shown in Fig. 7, B–D.
Currents Underlying the Action Potential
We previously mentioned that the spontaneous activity in the model (Fig. 6, C and D) has action potential parameters that compare well with the mean experimental values (Table 1). Figure 8 shows V during approximately one cycle of spontaneous activity in the ionic model (A) and the various currents (B–D) and the various activation and inactivation variables (E and F). The upstroke phase is clearly generated by ICa (Fig. 8C), which rapidly activates (d in Fig. 8E, see also Fig. 1B). During the first third of the action potential, the slow activation of IKs (n in Fig. 8F, see Fig. 2B) contributes increasingly to repolarization (Fig. 8C); IKr is small (Fig. 8C), despite rapid activation (s in Fig. 8F, see Fig. 3B) because of its strong inward rectification (z in Fig. 8F, see Fig. 3C); there is also a smaller contribution from Iseal, which is outward but becomes less outward with time (Fig. 8D); Ib is inward and gradually becomes more inward with time (Fig. 8D); IK1 plays no role here (Fig. 8D) because of its strong inward rectification (K1 in Fig. 8E, see Fig. 7C). There is also a secondary increase of ICa (Fig. 8C), which serves to maintain the plateau phase of the action potential, despite decreased activation and increased inactivation of ICa (d and f in Fig. 8E); this is due to an increase in driving force. The overall shape of the waveform of ICa during the action potential resembles that seen in action potential-clamp studies on SA node cells (120) and in some models of such cells (see Fig. 6 of Ref. 54). During the middle part of repolarization, ICa, after its secondary peak (Fig. 8C), falls as a result of inactivation and deactivation (f and d in Fig. 8E), which would per se promote repolarization. Because of a decrease in driving force, IKs decreases (Fig. 8C) and later starts to deactivate (n in Fig. 8F). There is also a fall in the outward Iseal, which eventually becomes an inward current (Fig. 8D). During the final stage of repolarization, there are contributions from IKr (Fig. 8C) and IK1 (Fig. 8D), which are no longer completely rectified (K1 and z in Fig. 8, E and F).
Currents Underlying the Pacemaker Potential
Because DDR in a three-cell cluster is 100 mV/s (Table 1), the net current during diastolic depolarization is tiny (2.6 pA); it is not even appreciable on the scale of Fig. 8B. ICa is inward and gradually becomes more inward throughout phase 4 depolarization (Fig. 8C), which agrees with the results from ruptured-patch action potential clamp studies on single SA node cells (120). Ib and Iseal are also inward throughout phase 4 depolarization but gradually become less inward (Fig. 8D). Although IKs, IKr, and IK1 are outward during phase 4 depolarization, IK1 becomes much less outward (Fig. 8D), IKs gradually becomes slightly less outward (not visible on the scale of Fig. 8C) but does not contribute much current, and IKr contributes increasingly less outward current as a result of slow deactivation (Fig. 8, C and F).
IKr deactivates slowly during diastolic depolarization (Fig. 8, C and F), because the time constant for activation (s) is several hundred milliseconds over the pacemaker range of potentials (Fig. 3B). Hence, IKr is not fully deactivated by the beginning of the upstroke of the action potential. However, the increase in voltage during the upstroke rapidly abolishes IKr (Fig. 8C) because of its profound inward rectification (Figs. 3C and 8F). As the membrane then repolarizes, fast recovery from the inactivation of IKr is responsible for its rectification (Fig. 8F). The time course of IKr during spontaneous activity is very different from that seen in an atrial reaggregate model (see Fig. 15 of Ref. 89), where IKr deactivates much more rapidly because of its shorter time constant at the more hyperpolarized MDP of the reaggregate model: about –90 mV (89) vs. –67 mV (present study). However, action potential clamp studies of rabbit SA cells, which are more depolarized than the chick atrial reaggregate, show a time course of IKr very similar to that in our model (see Fig. 1C of Ref. 76), as do SA node models that incorporate a sharply rectifying IKr component (see Fig. 6, D and E, of Ref. 54).
Effect of Ca2+ Channel Blockers on Spontaneous Activity
Application of D-600, a Ca2+ channel blocker, on 7- to 10-day cells and small clusters in our laboratory results in the abolition of spontaneous activity (Fig. 9A), with mean resting potential of –36.2 mV (n = 14) (49). We observed similar results with another Ca2+ channel blocker, diltiazem (51). In Fig. 9A, a sharp microelectrode is used so that the cessation of spontaneous activity is not due to dialysis of the pipette contents against the intracellular medium, leading to effects such as current rundown.
Gradually increasing block of ICa in the model, starting at 45 s in Fig. 9C, gives a time course of the voltage that is similar to the experimentally observed effect of D-600, with the membrane eventually coming to rest at –37 mV at 120 s when 90% of ICa is blocked. Because D-600 blocks ICa,L and because our nonseparable ICa is close to ICa,L, the modeling intervention is similar to the experimental intervention of applying an ICa,L blocker. In the experiment and the model, loss of OS initially proceeds at a slow rate (from just after arrow 1 to just after arrow 3 in Fig. 9C); then the rate of loss accelerates just before spontaneous activity is extinguished (i.e., just after arrow 3 in Fig. 9C). The MDP initially drifts slowly positive and then suddenly depolarizes much more quickly (starting at arrow 2 in Fig. 9C) before spontaneous activity ceases. In the experiment and the model, the phase of more rapid loss of MDP precedes the phase of more rapid loss of OS. The upstroke velocity gradually decreases throughout the course of the block, and APD100 increases (Fig. 9, B and D). Effects in many ways opposite to those described above are seen in our laboratory with administration of a Ca2+ channel agonist (BAY K 8644): there are increases in max, OS, DDR, and APD, as well as a hyperpolarization of MDP and the threshold or take-off potential, and a fall in IBI (28); these changes are also seen in the model.
Figure 9E gives the bifurcation diagram for ICa block, computed using XPPAUT (25). The bifurcation parameter is gCa, and the bifurcation variable is V. The periodic activity of the model corresponds to the existence of a stable limit cycle in the six-dimensional phase space of the system. As gCa is reduced from its nominal value of 30 nS, the limit cycle decreases in size, so that the APA falls: the maximum value of the V coordinate of the limit cycle (i.e., the OS) and its minimum value (i.e., the MDP) are shown in Fig. 9E. The locus of the V coordinate of the unstable steady state, which is also present in the phase space of the system and corresponds to the zero-current crossing of the total-current I-V curve in Fig. 7A, is also shown in Fig. 9E (dashed line). At gCa = 4.2 nS, a subcritical Hopf bifurcation (Fig. 9E) produces an unstable limit cycle oscillation, which grows in amplitude as gCa is reduced further, until the stable and unstable limit cycles collide at gCa = 3.7 nS in a reverse saddle-node bifurcation of limit cycles (31).
Relatively slow ("quasi-static") reduction in gCa from its control value of 30 nS (Fig. 9C) corresponds to moving from right to left along the stable limit-cycle branch of the bifurcation diagram in Fig. 9E. Eventually, at gCa 
3.7 nS, the state point will leave the stable periodic branch and move toward the stable steady state produced in the subcritical Hopf bifurcation. (The resting membrane potential corresponding to this stable steady state is shown in Fig. 9E.) This agrees with the simulation of Fig. 9C, where spontaneous activity is abolished with 90% block of ICa. This "falling off" is responsible for the rapid phase of decline in the OS after arrow 3 just before spontaneous activity is abolished in Fig. 9C. [This is also seen in the experiment (Fig. 9A).] In the model (Fig. 9C), the more rapid phase of decline of MDP starts earlier, before the falling off (just after arrow 2), at gCa 6 nS, which agrees with the change in the slope of MDP in the bifurcation diagram (Fig. 9E). [This pattern is also seen in the experiment (Fig. 9A).] When a trace such as that shown in Fig. 9C, with distinct phases of change of MDP and OS, is seen experimentally, one should begin to think that abolition of spontaneous activity might involve a subcritical, rather than a supercritical, Hopf bifurcation.
The coexistence of a stable limit cycle and a stable steady state for 3.7 nS < gCa < 4.2 nS in Fig. 9E implies that, over this range, one should be able to trigger activity from the resting state by injecting a stimulus and that this activity should be annihilated by injection of a single well-timed stimulus (31). We have indeed observed single-pulse triggering and annihilation in the model at gCa = 3.9 nS. Annihilation has been seen in isolated ventricular cells (94) and in reaggregates of atrial cells exposed to TTX (90). We do not know whether the bistable range in Fig. 9E would be wide enough in the experiment to allow observation of single-pulse triggering and annihilation in these clusters, because this would necessitate adjustment of the D-600 concentration to a value within a rather narrow range, which will be different from cluster to cluster and will be unknown a priori. However, other experimental evidence supports the scenario of Fig. 9E.
The existence of a saddle-node bifurcation in Fig. 9E is consistent with three prior observations from our laboratory: 1) During washout of D-600, transient flurries of action potentials occur spontaneously before spontaneous activity is permanently reestablished. The amplitude of the first action potential in each flurry is relatively large, with the amplitude of the following action potentials gradually declining during the course of each episode of transient triggered activity (see Fig. 2C of Ref. 49). 2) Once beating has stopped under the influence of diltiazem, injection of a hyperpolarizing bias current can provoke an episode of transient triggered activity, with the first action potential being an anodal-break response (see Fig. 2 of Ref. 51). As time proceeds and the degree of block continues to increase during quiescence, the number of action potentials in an episode decreases. This "critical slowing-down" behavior is consistent with the existence of a saddle-node bifurcation of limit cycles and can be seen in simulations with the model. 3) In some cells that are initially found to be not spontaneously active, injection of a single hyperpolarizing current pulse again elicits a flurry of triggered action potentials, with the action potential amplitude gradually decreasing during each flurry (see Figs. 4 and 6 of Ref. 49). During ongoing superfusion of these cells with D-600, the number of nondriven action potentials in any one trial gradually decreases from tens of action potentials, then the membrane does not produce triggered action potentials, and finally the membrane becomes inexcitable (see Fig. 4 of Ref. 49).
A response similar to that shown in Fig. 9, A–D, is seen with ICa,L block in the SA node in experiments (see references in Ref. 31) and in several ionic models (31, 54, 55). However, in some of these SA node models, a supercritical Hopf bifurcation occurs, so that annihilation and single-pulse triggering cannot occur.
Effect of Almokalant on Spontaneous Activity
Addition of 1 or 2 mM almokalant, a specific blocker of IKr (113), to the bath results in a slowing of the terminal rate of repolarization, a small depolarization of the MDP, and a slight loss of OS (Fig. 10A).
In the model, 100% block of IKr (Fig. 10B shows 50%, as well as 100%, block of IKr) results in a marked depolarization of MDP, a slight slowing of the terminal rate of repolarization, a small increase in APD100, a decrease in APD50, a decrease in IBI, a fall in max, and a decrease in OS. The slowing of the terminal rate of repolarization and the depolarization of the MDP are due to the absence of the IKr contribution normally present (see IKr trace in Fig. 8C during control activity). The resultant relative depolarization then causes IKs to remain considerably more outward during the entire pacemaker potential and even during the early part of the action potential (Fig. 10C shows IKs time course during 100% IKr block), which is the major cause of the decrease in APD50 in Fig. 10B. This secondary increase in IKs also accounts for the relatively small effect of the loss of IKr on the rate of terminal repolarization in Fig. 10B. In cells that do not have IKs, this indirect effect of IKr block on IKs would not occur, and one would expect a prolongation of APD, as observed in SA node cells in which IKs was not found (106). [In later studies, however, IKs was clearly found in SA node cells (59, 110).] The fall in max and OS is due to a decrease in ICa, with a fall in peak value from 263 to 207 pA during 100% IKr block as a result of greater inactivation of ICa during diastolic depolarization. Similarly, in experiments on the SA node, effects on action potential parameters due to selective block of IKr with the compound E-4031 have been found to be due to "a combination of direct and indirect effects on various ionic currents" (106).
The shortening of APD50 in the model with IKr block (Fig. 10B) does not occur in the experiment (Fig. 10A). One reason that might account for this discrepancy is the rundown of IKs that occurs during the 10-min period between rupture of the patch (to enter whole cell recording mode) and initiation of the recording of the effect of almokalant (79). Figure 10D shows the combined effect in the model of 50% block of IKr (to simulate almokalant) and 20% block of IKs [to simulate the degree of rundown seen experimentally (79)]: APD50 is no longer decreased.
To avoid the above-mentioned artifact due to dialysis of cytoplasm against the pipette contents and rundown of currents, we also used visual monitoring of mechanical beating to assess the effect of almokalant. Although three clusters stopped beating on exposure to almokalant (and the effect was reversible on washout of almokalant from the bath), another four clusters did not stop beating: IBI was increased in three of four clusters, and there was no change in the remaining cluster. In response to 0.1 µM E-4031, another specific IKr blocker, half of the single SA node cells stopped beating in one study (106), whereas none stopped in another study (76). When the concentration was raised to 1.0 µM, at which there are still negligible effects on other currents, all cells ceased spontaneous activity in both studies. A similar response was found in small balls of tissue isolated from the SA node: with 1.0 µM E-4031, activity was abolished in all balls; with 0.1 µM E-4031, activity was sometimes extinguished in balls from the central area, but not in those from more peripheral areas (46). The fact that 0.1 µM E-4031 abolishes spontaneous activity in smaller, but not in larger, single SA node cells agrees with this observation (59), provided that small cells do indeed stem from the central area of the node and larger cells stem from the periphery.
Complete blo
, stable; 
, unstable; HB, Hopf bifurcation.
