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1Department of Physiology, Kanazawa Medical University, Ishikawa, and 2Division of Regenerative Medicine and Therapeutics, Tottori University Graduate School of Medical Science, Yonago, Japan
Submitted 28 April 2006 ; accepted in final form 16 September 2006
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ABSTRACT |
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 TOP ABSTRACT THEORY AND METHODSRESULTSDISCUSSIONAPPENDIX A: GLOSSARYAPPENDIX B: MODEL EQUATIONSAPPENDIX C: DETERMINATION OF...GRANTSREFERENCES |
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mathematical model; bifurcation analysis; computer simulation
A first step for creation of the functional BP would be engineering of single BP cells, which requires deep understanding of the BP mechanisms. By bifurcation analyses of a mathematical model for human ventricular myocytes (HVMs), we have elucidated (22) the dynamical mechanisms of BP generation in IK1-downregulated HVMs and the roles of individual sarcolemmal currents in HVM pacemaking. We have suggested that 1) BP activity can be developed by reducing IK1 alone in HVMs as in guinea pig ventricular myocytes (41), 2) the instability of an equilibrium point (EP) with depolarized potentials is essentially important for BP generation, and 3) the dynamical mechanism of ventricular pacemaking is essentially the same as that of natural sinoatrial (SA) node pacemaking as reported by Kurata et al. (21). In the previous study, however, whether BP cells can be developed from HVMs by incorporating "pacemaker currents," which contribute to phase 4 depolarization in SA node pacemaking, was not clearly shown, with the most efficient way to create BP cells remaining unknown; the pacemaker currents include Ih, T-type Ca2+ channel current (ICa,T), sustained inward current (Ist), and low-voltage-activated L-type Ca2+ channel current (ICa,LD). Therefore, we first investigated whether incorporating these pacemaker currents can yield BP activity and how this affects BP generation during IK1 suppression in the model HVM. Our findings suggested that Ih facilitates BP generation during IK1 suppression in combination with the other pacemaker currents, although IK1 downregulation is necessary for constructing BP cells from HVMs.
Our preliminary study also revealed some drawbacks of the IK1-downregulated HVM pacemaker, compared with the SA node pacemaker, to prevent the creation of functional BPs. They include 1) relatively low robustness to hyperpolarizing (electrotonic) loads and 2) low ability to drive adjacent nonpacemaker cells, which may be due to the lack of pacemaker currents. Using a coupled-cell model composed of IK1-deleted BP cells and a nonpacemaking HVM [referred to as "nonpacemaking (NP) cell"], therefore, we further explored the effects of incorporating pacemaker currents on the structural stability of BP cells (robustness of BP activity) to electrotonic modulations and BP cell ability to drive the adjacent NP cell. Our results suggested that ICa,T, Ist, and ICa,LD can, but Ih cannot, significantly improve the structural stability of BPs to electrotonic loads of HVMs and the BP ability to drive the ventricle.
This study indicates that the theoretical approach based on nonlinear dynamics and bifurcation theory may allow us to predict the effects of current modulations on pacemaker cell dynamics accurately and find out how to control the dynamical properties of real myocytes properly. Exploring bifurcation structures of the mathematical model would provide a theoretical background for engineering of BP cells and functional BPs from native cardiomyocytes or embryonic stem (ES) cells, i.e., for gene or cell therapy of bradyarrhythmias (8, 37). This study also provides novel insights into the roles of the pacemaker currents in natural SA node pacemaking. Definitions of terms specific to nonlinear dynamics and bifurcation theory are given at the end of THEORY AND METHODS (see also Refs. 21–23, 30).
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THEORY AND METHODS |
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TOPABSTRACTTHEORY AND METHODSRESULTSDISCUSSIONAPPENDIX A: GLOSSARYAPPENDIX B: MODEL EQUATIONSAPPENDIX C: DETERMINATION OF...GRANTSREFERENCES |
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We used our HVM model (22) as a modified version of the Priebe-Beuckelmann model (33). More elaborate HVM models were recently developed by Iyer et al. (12) and ten Tusscher et al. (42), which are probably superior to the original or modified Priebe-Beuckelmann model in reproducing experimental data. Nevertheless, these models are much more complex or include vector field functions that are not continuous or smooth, being less suitable for bifurcation analyses (for more details, see Ref. 22). We have therefore chosen to use our model, which is more suitable for bifurcation analyses.
The standard model for the normal activity of single HVMs is described as a nonlinear dynamical system of 15 first-order ordinary differential equations. The membrane current system includes the L-type Ca2+ channel current (ICa,L), rapid and slow components of delayed rectifier K+ currents (denoted IKr and IKs, respectively), 4-aminopyridine-sensitive transient outward current (Ito), Na+ channel current (INa), IK1, background Na+ (INa,b) and Ca2+ (ICa,b) currents, Na+-K+ pump current (INaK), Na+/Ca2+ exchanger current (INaCa), and Ca2+ pump current (IpCa). Time-dependent changes in the membrane potential (V) are described by the equation
 | (1) |
Our full system includes material balance expressions to define the temporal variation in intracellular Ca2+, Na+, and K+ concentrations ([Ca2+]i, [Na+]i, and [K+]i), whereas extracellular Ca2+, Na+, and K+ concentrations ([Ca2+]o, [Na+]o, and [K+]o) were fixed at 2, 140 and 5.4 mM, respectively. As pointed out by Hund et al. (11) and Krogh-Madsen et al. (18), the second-generation model incorporating ion concentration changes exhibits degeneracy (i.e., nonuniqueness of steady-state solutions), not suitable for bifurcation analysis to be applicable to isolated equilibria. One of the ways to remove degeneracy and thus allow bifurcation analyses of isolated equilibria is to make some ionic concentrations fixed (18). Variations in [K+]i during changes in bifurcation parameters were relatively small, not significantly altering the model cell behaviors. For stability and bifurcation analyses, therefore, [K+]i was fixed at 140 mM.
Incorporation of Pacemaker Currents
To investigate how pacemaker currents contributing to phase 4 depolarization in SA node pacemaking affect bifurcation structures of HVM model cells, we incorporated the formulas of Ih, ICa,T, and Ist, which were used for our SA node model (20) or experimentally determined for human cardiomyocytes. The low voltage-activated L-type Ca2+ channel (D-LTCC) was recently reported to activate at pacemaker potential regions, i.e., at potentials 10–20 mV more negative than the high-voltage-activated Ca2+ channel, contributing to mouse SA node pacemaking (16, 27, 31, 51). Therefore, we also tested the effect of the current mediated by D-LTCC (denoted as ICa,LD). The formulas for individual pacemaker currents are given in APPENDIX B. The origins of the expressions and conductance values tested for each current, as well as those chosen for the original SA node models as mean experimental values, are summarized in Table 1 (for details, see below).
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ICa,T. The expressions for ICa,T were adopted from the rabbit SA node (5, 20) and ventricular (34) models. The effects of these ICa,T currents were qualitatively the same, with the effect of the SA node ICa,T being more remarkable; thus the data for the rabbit SA node ICa,T are shown. Relatively high gCa,T was required to affect dynamical properties of the model cell, probably because the window region (i.e., overlap of activation and inactivation curves) of the model ICa,T is relatively small (see Fig. 1). Therefore, the gCa,T value was increased up to 3 nS/pF, although it was 0.458 nS/pF in the original SA node model (5, 20). Further increases in gCa,T caused shrinkage of BP oscillations and stabilization of an EP via a Hopf bifurcation (see Fig. 2).
Ist. The formulas for Ist were adopted from the rabbit SA node model (20) and the experimental report for the rat SA node (40). The effects of the two Ist currents were essentially the same; thus only data for the rabbit SA node Ist are shown. The gst value, 0.015 nS/pF in the original SA node model (20), was limited to 0.02 nS/pF; further increases in gst yielded extremely large window currents leading to Na+-overload conditions, because the window region of the model Ist is relatively large (see Fig. 1).
ICa,LD. The kinetics of ICa,LD was formulated on the basis of experimental data from mouse SA node cells (27, 31, 51) and human embryonic kidney cells expressing D-LTCC (16). These previous studies suggested that 1) the voltage dependence of the activation and inactivation of ICa,LD is 10–20 mV more negative than that of the high-voltage-activated Ca2+ channel current (ICa,LC), 2) the activation of ICa,LD is faster than that of ICa,LC, and 3) the inactivation of ICa,LD is 1.5–2 times slower than that of ICa,LC. According to these data, therefore, we formulated the kinetics of ICa,LD as provided in APPENDIX B (Eqs. A38–A42). Because the effects on model cell dynamics of the model ICa,LD were relatively small, the gCa,LD value was increased up to 1 nS/pF. Further increases in gCa,LD caused Ca2+-overload conditions as well as shrinkage of BP oscillations (see Fig. 2).
Formulation of Coupled-Cell Model
To investigate the electrotonic influences of adjacent HVMs on stability and dynamics of BP cells as well as BP cell ability to drive adjacent HVMs, we used a coupled-cell model (Fig. 3), in which IK1-deleted BP cells were connected to a NP cell with normal or reduced IK1 via the gap junction conductance (GC) of 0–20 nS. One NP cell was connected with one to seven BP cells; more than one BP cells were assumed to be well-coupled enough to synchronize completely and act as a cluster of isopotential cells (or assumed to connect with the NP cell via an identical GC). Pacemaker currents were incorporated into the BP cells only (not into the NP cell).
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 | (2) |
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Numerical Methods for Dynamic Simulations and Bifurcation Analyses
Dynamic behaviors of the model cells were determined by solving a system of nonlinear ordinary differential equations numerically. Numerical integration as well as bifurcation analyses were performed on Workstation CELSIUS X630 (Fujitsu, Tokyo, Japan) with MATLAB 7 (MathWorks, Natick, MA). We used the numerical algorithms available as MATLAB ODE solvers: 1) a fourth-order adaptive-step Runge-Kutta algorithm that includes an automatic step-size adjustment based on an error estimate (ode45) (6) and 2) a variable time-step numerical differentiation approach selected for its suitability to stiff systems (ode15s) (39). Both solvers usually yielded nearly identical results. The latter, much more efficient than the former, was usually used; the former as the best function for most problems was only sometimes used to confirm the accuracy of calculations. The maximum relative error tolerance for the integration methods was set to 1 x 10–6.
Stability and Bifurcation Analyses
We examined how the stability and dynamics of the model cells alter with changes in bifurcation parameters and constructed bifurcation diagrams for one or two parameters. Bifurcation parameters chosen in this study were the maximum conductance of IK1 (gK1) and pacemaker currents (gh, gCa,T, gst, gCa,LD) as well as GC; gK1 is expressed as a normalized value, i.e., in ratio to the control value of 3.9 nS/pF, unless otherwise stated. The methods for bifurcation analyses of the single BP or NP cell are described in our previous article (22).
Although the methods of bifurcation analyses for the coupled-cell system are essentially the same as for the single cell, the number of state variables increases twice: in the coupled-cell system with fixed [K+]i, 28 state variables define a 28-dimensional state point in the 28-dimensional state space of the system. We calculated EPs, i.e., steady-state values of the variables, and periodic orbits in the state space of the coupled-cell system. Steady-state values of the variables were calculated by Eqs. A53–A58 in APPENDIX C. Asymptotic stability of the EP was also determined by computing 28 eigenvalues of a 28 x 28 Jacobian matrix derived from the linearization of the nonlinear system around the EP (for details, see Ref. 45). Periodic orbits were located with the MATLAB ODE solvers. When spontaneous oscillation (BP activity) appeared, the action potential amplitude (APA) as a voltage difference between the maximum diastolic potential (Vmin) and peak overshoot potential (Vmax) as well as the cycle length (CL) were determined for each calculation of a cycle. Numerical integration was continued until the differences in both APA and CL between the newly calculated cycle and the preceding one became <1 x 10–3 of the preceding APA and CL values. Potential extrema (Vmin, Vmax) and CL of the steady-state oscillation were plotted against bifurcation parameters. When periodic behavior was irregular or unstable, model dynamics were computed for 60 s; all potential extrema and CL values were then plotted.
We constructed one-parameter bifurcation diagrams for GC and two-parameter bifurcation diagrams for GC and pacemaker current conductance. For construction of one-parameter bifurcation diagrams, the membrane potential at EPs (steady-state branches) and local potential extrema (Vmin, Vmax) of periodic orbits (periodic branches) were determined and plotted as a function of GC, which was systematically changed while keeping all other parameters at their standard values. The saddle-node bifurcation point at which two EPs coalesce and disappear with emergence of robust BP activity was determined; the Hopf bifurcation point at which an unstable EP is stabilized with cessation of BP activity was also located. For construction of two-parameter bifurcation diagrams, the critical GC values at bifurcation points were determined as functions of the secondary parameters such as the pacemaker current conductance, the number of BP cells, and NP cell gK1; the primary parameter (GC) was systematically changed, with the secondary parameter fixed at various different values. The path of Hopf and saddle-node bifurcation points was traced in the parameter plane, i.e., bifurcation values for the primary parameter were plotted as a function of the secondary parameter.
We also evaluated the "structural stability" of BP cells, which is defined as the robustness of BP activity to various interventions or modifications that may cause a bifurcation to quiescence or irregular dynamics (21, 22). In this study, we tested the structural stability of BP cells to electrotonic (hyperpolarizing) loads of an adjacent NP cell, using the coupled-cell system, which is also useful in exploring BP cell ability to drive the adjacent NP cell. The method of evaluating the structural stability to electrotonic loads is essentially the same as that for constant bias currents as described in our previous articles (21, 22). In the unstable GC region where the system has no stable EP, the system usually exhibited robust pacemaking and driving without annihilation (9, 24). When the system removes to the stable GC region with a stable EP, it will come to a rest at the stable EP via gradual decline of limit cycles, annihilation, or irregular dynamics, as reported by Guevara and Jongsma (9). In the coupled-cell system, the larger the GC region over which all EPs are unstable (there is no stable EP) and thus BP activity appears is, the more structurally stable the BP system is (for more detail, see Ref. 22).
Definitions of Terms Specific to Nonlinear Dynamics and Bifurcation Theory
Equilibrium point. The equilibrium point (EP) is a time-independent steady-state point at which the vector field vanishes in the state space of a dynamical system, constructing the steady-state branch in one-parameter bifurcation diagrams. This state point corresponds to the zero-current crossing in the steady-state current-voltage (I-V) curve, i.e., a quiescent (resting) state of a cell if it is stable.
Periodic orbit. A periodic orbit is a closed trajectory in the state space of a system, constructing the periodic branch in one-parameter bifurcation diagrams.
Limit cycle. The limit cycle is a periodic limit set onto which a trajectory is asymptotically attracted. A stable limit cycle corresponds to an oscillatory state, i.e., pacemaker activity, of a cell.
Bifurcation. Bifurcation is a qualitative change in a solution of differential equations caused by altering parameters, e.g., a change in the number of EPs or periodic orbits, a change in the stability of an EP or periodic orbit, and a transition from a periodic to a quiescent state. Bifurcation phenomena we can see in cardiac myocytes include generation or cessation of pacemaker activity and occurrence of irregular dynamics such as skipped-beat runs and early afterdepolarizations.
Saddle-node bifurcation. Saddle-node bifurcation is a bifurcation at which two EPs (steady-state branches) or two periodic solutions (periodic branches) emerge or disappear. The saddle-node bifurcation of EPs occurs when one of the eigenvalues of a Jacobian matrix for the EP is zero.
Hopf bifurcation. The Hopf bifurcation is a bifurcation at which the stability of an EP reverses with emergence or disappearance of a limit cycle, occurring when eigenvalues of a Jacobian matrix have a single complex conjugate pair and its real part reverses the sign through zero.
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RESULTS |
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TOPABSTRACTTHEORY AND METHODSRESULTSDISCUSSIONAPPENDIX A: GLOSSARYAPPENDIX B: MODEL EQUATIONSAPPENDIX C: DETERMINATION OF...GRANTSREFERENCES |
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Our preceding study (22) suggests that IK1 downregulation, ensuring both the occurrence of a saddle-node bifurcation and instability of the EP with depolarized potentials, yields spontaneous oscillations (BP activity) in HVMs without incorporating any pacemaker current. However, a saddle-node bifurcation to induce BP activity may also be yielded by incorporating pacemaker currents such as Ih, ICa,T, Ist, and ICa,LD, which are abundant in SA node primary pacemaker cells but absent or very small in native HVMs. In the preliminary study, we found that these pacemaker currents could not yield BP generation in the model HVM with normal IK1 (gK1 = 1): BP activity did not appear when gh, gCa,T, gst, and gCa,LD were increased up to the limited values of 0.12, 3, 0.02 and 1 nS/pF, respectively (see THEORY AND METHODS) or even when gCa,T, gst, or gCa,LD was further increased to the critical value above which an EP at depolarized potentials would be stable and thus BP oscillations could not occur (see Fig. 2). Nevertheless, incorporating the pacemaker currents is expected to accelerate BP generation during IK1 suppression by increasing the critical gK1 value at which a saddle-node bifurcation occurs and thus BP oscillations emerge. We therefore investigated how these currents affect the saddle-node bifurcation and BP generation during IK1 suppression.
Effects of incorporating each pacemaker current on BP generation during IK1 suppression.
As clearly shown in Fig. 4, the saddle-node bifurcation point below which BP oscillations occur shifted toward higher gK1 values with increasing gh, suggesting that expression of Ih accelerates BP generation during IK1 suppression by increasing the critical gK1 value to induce BP activity. To further examine the effects of incorporating the pacemaker currents on BP generation in the IK1-reduced HVM, we determined the critical gK1 value for BP generation (i.e., gK1 value at a saddle-node bifurcation) as functions of gh, gCa,T, gst, or gCa,LD, as well as exploring the influences of the pacemaker currents on the steady-state I-V relation at gK1 = 1 (Fig. 5). The inward pacemaker currents counteracted the outward component of IK1, inwardly shifting the steady-state I-V curve. The critical gK1 value increased with increasing gh; however, it did not reach the control gK1 value but reached the maximum of 0.287 (28.7%) at gh = 0.1039 nS/pF. ICa,T at higher densities (gCa,T 
3 nS/pF) also increased the critical gK1, with its increase limited to <0.266 (26.6%). The effects of Ist and ICa,LD were relatively small within the limited conductance ranges of 0.02 and 1 nS/pF, respectively.
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One of the drawbacks of the IK1-downregulated HVM pacemaker may be the relatively low structural stability to hyperpolarizing loads such as electrotonic modulations of surrounding nonpacemaker cells, because it lacks the pacemaker currents that are abundant in SA node primary pacemaker cells and contribute to pacemaker depolarization. We therefore examined the structural stability of BP cells to electrotonic loads of the adjacent NP cell, using a coupled-cell model in which BP cells (gK1 = 0) were connected to a NP cell (gK1 = 0.2–1) via the GC ranging from 0 to 20 nS. The coupled-cell system is also useful in exploring BP cell ability to drive adjacent NP cells.
GC-dependent behaviors of the coupled-cell system. In the system with one BP cell and one normal HVM (gK1 = 1), an EP was stabilized via a Hopf bifurcation at GC = 0.689 nS, with the BP cell not exhibiting pacemaker activity at GC values higher than the bifurcation value. Thus the coupling of one BP cell to one normal HVM produced either 1) BP cell pacemaking at slower rates without driving the NP cell for lower GC values or 2) inhibition of BP cell pacemaking for higher GC values. We could not find a GC value for which one BP cell could exhibit pacemaking itself and drive the normal HVM but found that three or more BP cells, assumed to be well coupled to form a cluster, were required for driving the normal HVM.
Figure 7 shows the simulated membrane potentials and IGJ at various GC values for the coupled-cell system composed of five BP cells and one normal HVM. With increasing GC, the BP rate decreased, with the oscillation amplitude getting larger. The BP cells could induce only subthreshold responses of the HVM at GC = 2.0 nS, whereas they induced driven action potentials at GC = 2.5–3.5 nS. Further increase in GC to 5.0 nS abolished BP activity, which was probably due to hyperpolarizing loads of the NP cell as indicated by positive IGJ.
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Stability and dynamics during GC increases of the coupled-cell system composed of one to seven BP cells and one normal HVM are shown in Fig. 8A. EPs and potential extrema were determined for both the BP and NP cells, plotted as functions of GC values. Hopf and saddle-node bifurcation points were also determined and located in the bifurcation diagrams. In the systems with one to five BP cells, hyperpolarizing loads of the NP cell caused 1) prolongation of CL with an increase in oscillation amplitude, 2) irregular dynamics, and then 3) a bifurcation to quiescence (Hopf bifurcation), with increasing GC. These behaviors are essentially the same as those reported for interactions of the SA node cell and the atrial or ventricular myocyte (13, 44, 47, 48). The critical GC value at the Hopf bifurcation to cause stabilization of EPs and cessation of BP activity increased with increasing number of BP cells. One or two BP cells could not drive the normal HVM. With three to five BP cells, the increase in GC yielded a BP cell ability to drive the normal HVM; however, further increases in GC led to the cessation of pacemaking and driving via the Hopf bifurcation. The GC region of driving the normal HVM was very limited with three to five BP cells, whereas it dramatically enlarged when the number of BP cells increased to more than five. With six or more BP cells, a saddle-node bifurcation at which two EPs (EP2, EP3) disappear occurred to ensure the robust pacemaking and driving without annihilation (see Fig. 8A, far right, for 7 BP cells).
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Effects of Pacemaker Currents on Structural Stability and Driving Ability of HVM Pacemaker
Using the coupled-cell model, we examined how incorporating the pacemaker currents affects the structural stability of BP cells to electrotonic loads of the adjacent NP cell, as well as BP cell ability to drive the NP cell. Figure 8 suggests that a Hopf bifurcation point is just at or close to the point at which BP activity is abolished by hyperpolarizing loads of the NP cell, whereas a saddle-node bifurcation point is the point above which robust pacemaking and driving occur. To assess the effects of the pacemaker currents on the structural stability and driving ability of BP cells, therefore, we focused on the shift in the Hopf bifurcation point, as well as emergence of a saddle-node bifurcation.
Effects of incorporating pacemaker currents on GC-dependent bifurcation structures. Bifurcation structures during GC increases were first determined at various conductances of each pacemaker current. We constructed two-parameter bifurcation diagrams in which the critical GC value at Hopf or saddle-node bifurcation points was plotted as functions of the maximum current conductance. Figure 9 shows the effects of incorporating Ih, ICa,T, Ist, or ICa,LD on the bifurcation structure during GC increases of the coupled-cell system in which three BP cells were connected to one normal HVM or one BP cell was connected to one NP cell with reduced IK1 (gK1 = 0.35). Stability and dynamics during GC increases of the coupled-cell systems incorporating each pacemaker current are illustrated in Fig. 10. The results suggest that individual pacemaker currents affect the GC-dependent bifurcation structure of the system in different ways.
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0.5 nS/pF) did not significantly change the critical GC value at Hopf bifurcations where an EP is stabilized and BP activity is abolished, not enlarging the region of pacemaking labeled "P+D±" (Fig. 9, top). As shown in Fig. 10A, Ih (0.2 nS/pF) did not enlarge the GC region where all EPs are unstable and BP activity appears. When IK1 of the NP cell was relatively small (gK1 = 0.35), however, Ih enlarged the pacemaking and driving region labeled P+D+ by producing a saddle-node bifurcation (Fig. 9, bottom; Fig. 10B). Overexpression of Ih (gh > 0.2043 nS/pF) caused cessation of BP activity at higher GC values, with a stable resting potential dramatically hyperpolarized probably via drastic increases in [Na+]i and [Ca2+]i.
ICA,T EFFECT.
Similar to Ih, ICa,T (3 nS/pF) did not significantly change the critical GC value at Hopf bifurcations, not enlarging the P+D± region, in the coupled-cell system with three BP cells and one normal HVM (Fig. 9, top). However, higher densities of ICa,T yielded a saddle-node bifurcation and thus enlarged the P+D+ region; ICa,T at 2 nS/pF enlarged the GC region where all EPs are unstable and BP activity appears (Fig. 10A). When the NP cell IK1 was relatively small, ICa,T dramatically enlarged the P+D+ region by producing a saddle-node bifurcation (Fig. 9, bottom; Fig. 10B). With relatively high conductance of ICa,T, the amplitudes of BP oscillations and driven action potentials of the NP cell were relatively small.
IST EFFECT.
In contrast to Ih or ICa,T, incorporation of Ist (0.02 nS/pF) dramatically shifted Hopf bifurcation points toward higher GC values and further produced a saddle-node bifurcation to ensure robust pacemaking and driving, i.e., enlarged the P+D+ region (Fig. 9). Ist at 0.01 nS/pF dramatically enlarged the GC region where BP oscillations occur, with CL of BP oscillations being relatively stable during increases in GC (Fig. 10).
ICA,LD EFFECT.
The effects of ICa,LD were similar to those of Ist, although much larger conductance was required for ICa,LD to exert the effects as dramatically as Ist. Incorporation of ICa,LD (1 nS/pF) increased the Hopf bifurcation value and produced a saddle-node bifurcation to enlarge the P+D+ region (Fig. 9). As shown in Fig. 10, ICa,LD (0.5 nS/pF) enlarged the GC region where BP oscillations occur. However, ICa,LD at high densities of 0.5–1 nS/pF caused unstable CL and irregular dynamics during increases in GC.
Influences of altering pacemaker current kinetics on their modulation of bifurcation structures.
Individual pacemaker currents affected the GC-dependent bifurcation structure in different ways, as well as with different potencies. The effects of Ih were completely different from those of the other pacemaker currents, which probably resulted from the difference in the voltage range of activation. Furthermore, the effects of ICa,T, Ist, and ICa,LD were apparently different in the system with three BP cells and one normal HVM: the locus of Hopf bifurcation points at lower GC was separated from that of saddle-node bifurcation points with ICa,T, but not with Ist or ICa,LD. The differences in their effects may result from the large differences in their inactivation kinetics, whereas the different potencies are probably due to the different degrees of the overlap of activation and inactivation curves (see Fig. 1). Thus we examined the effects of incorporating the modified ICa,T or Ist with slowed or accelerated inactivation on GC-dependent bifurcation structures of the system with three BP cells and one normal HVM (Fig. 11). Changing the inactivation kinetics altered the loci of Hopf bifurcations: the modified ICa,T with the inactivation time constant (
fT) 10 times higher than the control yielded a bifurcation diagram similar to that for the original Ist; in contrast, the modified Ist with the inactivation time constant (qi) decreased to 0.01 of the control yielded a locus of Hopf bifurcations separated from that of saddle-node bifurcations, like the original ICa,T. Thus the different effects of the pacemaker currents are at least in part ascribable to the difference in their inactivation kinetics, which appeared to affect the Hopf bifurcation, i.e., stability of EPs, but not the saddle-node bifurcation.
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DISCUSSION |
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TOPABSTRACTTHEORY AND METHODSRESULTSDISCUSSIONAPPENDIX A: GLOSSARYAPPENDIX B: MODEL EQUATIONSAPPENDIX C: DETERMINATION OF...GRANTSREFERENCES |
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We first investigated whether incorporating pacemaker currents exerts beneficial effects on creation of BP cells by constructing two-parameter bifurcation diagrams in which the critical gK1 value (saddle-node bifurcation point) to produce BP activity was plotted as functions of the maximum current conductance.
Pacemaker currents facilitate BP generation during IK1 suppression. Our results suggest that the pacemaker currents, especially Ih, can facilitate BP generation during IK1 suppression, although their effects were limited (Fig. 5). The mechanism underlying the facilitation of BP generation appears to be relatively simple: the inward pacemaker currents counteract the outward component of IK1, inwardly shift the steady-state I-V curve, and thus lead to a saddle-node bifurcation at a higher gK1 value (see Fig. 5, top). Nevertheless, the critical gK1 value never reached the control value of 1, suggesting that any pacemaker currents cannot yield BP activity in the HVM with normal IK1.
IK1 downregulation is a requisite for BP generation in HVM. The limited effects of the pacemaker currents appear to be due to the induction of intracellular Na+ and Ca2+ overloads: as shown in Fig. 6, overexpressions of the pacemaker currents cause increases in [Na+]i and [Ca2+]i, which enhance the outward Na+-K+ pump current, attenuate the inward background Na+ and Ca2+ currents (by decreasing the driving force for Na+ and Ca2+), and thus counteract the inward shift of I-V curves that leads to a saddle-node bifurcation. When [Na+]i, which is a state variable in the standard system, was fixed at the control value of 6.14 mM (for gK1 = 1) to prevent the pacemaker currents causing Na+ and Ca2+ overloads, increasing the pacemaker currents led to BP generation via a saddle-node bifurcation without inhibition of IK1, albeit with the requirement of very high conductance (e.g., gh > 0.996 nS/pF) (data not shown). Thus Na+ and Ca2+ overloads would prevent a bifurcation to BP generation during enhancements of the pacemaker currents. In conclusion, incorporating Ih and/or the other pacemaker currents would not yield BP generation in HVMs with normal IK1, and thus IK1 downregulation would be necessary for constructing a BP cell system from HVMs. Inconsistent with our result, Viswanathan et al. (46) reported that the rabbit SA node Ih could induce BP activity in the Luo-Rudy guinea pig ventricular model without reducing IK1. We are not sure about the reason for this inconsistency. Further studies are required to clarify whether the effects of Ih and other pacemaker currents depend on species (human vs. guinea pig) or on the property of a model itself as well as whether expressing Ih can actually induce BP activity in real NP myocytes of the ventricle.
Overexpression of Ih has been reported to induce BP activity (or increase BP rates) in canine atrial or Purkinje myocytes (32, 36). Figure 5 indicates that increasing Ih alone can induce BP activity via a saddle-node bifurcation if gK1 is smaller than the control value of 3.9 nS/pF, e.g., when a gK1 value is 25% of the control value (0.975 nS/pF), increasing gh to 0.0403 nS/pF or more yields BP generation. Koumi et al. (17) reported that a slope conductance at a reversal (K+ equilibrium) potential of the human atrial IK1 (0.134 nS/pF) is only 18.3% of that of the human ventricular IK1 (0.732 nS/pF). As suggested by previous reports (14, 32, 36), expression of Ih alone would induce pacemaker activity in myocytes with relatively low density of IK1, such as atrial and Purkinje myocytes.
Possible Roles of Pacemaker Currents in Robust Pacemaking and Driving of Functional BP
Using the coupled-cell model, we examined whether incorporation of pacemaker currents exerts beneficial effects on BP functions, i.e., whether it improves the structural stability of BPs to electrotonic loads of adjacent NP cells and the BP ability to drive the ventricle. The results suggest that individual pacemaker currents affect the BP functions in different ways.
Ih effect is relatively small and NP cell gK1 dependent. The effects of Ih on GC-dependent bifurcation structures were quite different from those of the other pacemaker currents, which is probably due to the difference in the voltage range of activation. Ih did not significantly improve the structural stability of BP cells to electrotonic loads of the normal HVM or the BP cell ability to drive the normal HVM (Figs. 9 and 10A). When NP cell IK1 was relatively small, however, Ih improved the structural stability and driving ability of BP cells by producing a saddle-node bifurcation (Figs. 9 and 10B). Moreover, the effects of Ih were shown to depend on IK1 density of the NP cell, as well as the size factor (Figs. 12 and 13). These results suggest that Ih can support the robust pacemaking and driving of BPs only when IK1 density in adjacent NP cells is relatively small. IK1 density is much larger in the human ventricle than in the human atrium (17); thus Ih may support the robust pacemaking and driving of BPs (as well as the SA node) in the atrium with relatively small IK1, but not in the ventricle with relatively large IK1. Possible roles of Ih in the working of ventricular BPs could be to modulate pacemaker frequency in response to autonomic regulations, as well as to prevent excess hyperpolarization, rather than to improve the structural stability and driving ability by destabilizing EPs or yielding saddle-node bifurcations.
ICa,T effect is NP cell gK1 dependent and characterized by shrinkage of voltage oscillations. Similar to Ih, ICa,T did not improve the structural stability of BP cells to electrotonic loads of the normal HVM by shifting the Hopf bifurcation point at lower GC values (Fig. 9, top); the low efficacy of ICa,T appeared to be due to the rapid inactivation kinetics, because the modified ICa,T with slowed inactivation significantly increased the Hopf bifurcation value (Fig. 11). Nevertheless, higher densities of ICa,T improved the structural stability and driving ability of BP cells by producing a saddle-node bifurcation. The effects of ICa,T also depend on the NP cell IK1, as well as the size factor (Figs. 9, 12, and 13); ICa,T may effectively improve the functions of BPs in the atrium, whereas much larger ICa,T is required for the ventricle. However, higher densities of ICa,T caused shrinkage in BP oscillations and driven action potentials of the NP cell (Fig. 10), which is probably due to the rapid kinetics of ICa,T inactivation and is not desirable for functional BPs.
Ist most dramatically improves structural stability and driving ability of BP cells. In contrast to Ih or ICa,T, Ist at relatively low conductance dramatically improved the structural stability and driving ability of BP cells by shifting Hopf bifurcation points toward higher GC values and yielding a saddle-node bifurcation (Fig. 9), producing BP oscillations with stable frequency over the broad GC range (Fig. 10). Furthermore, Ist, as well as ICa,T and ICa,LD, may contribute to reducing the critical size of a BP system (i.e., the number of BP cells) to be required for driving the ventricle: with Ist at 0.01 nS/pF, four BP cells are enough to yield robust pacemaking and driving of a normal HVM via creation of a saddle-node bifurcation, whereas more than five BP cells are required without a pacemaker current (see Fig. 12). Our results suggest that Ist, exhibiting relatively slow inactivation kinetics, most dramatically improves the structural stability and driving ability of the HVM pacemaker. Ist may be indispensable for robust pacemaking and driving of ventricular BPs.
ICa,LD effect is similar to Ist effect but characterized by generation of irregular dynamics. The effects of ICa,LD were qualitatively similar to those of Ist, although its potency was much smaller (Figs. 9, 12, and 13). However, ICa,LD at high densities caused unstable CL and irregular dynamics during increases in GC, which is possibly a consequence of producing Ca2+ overload conditions, not desirable for functional BPs. We suppose that the role of ICa,LD in the working of pacemaker systems is essentially the same as that of ICa,LC: ICa,LD may play a major role in producing the action potential upstroke as well as excitation-contraction coupling, rather than in supporting robust pacemaking and driving.
Implications and Significance of Study
In the present study, we used stability and bifurcation analyses to investigate the effects of pacemaker currents on creation and modulations of the HVM pacemaker. Previous reports indicate that exploring bifurcation structures of SA node or ventricular models is useful for general understanding and systematic descriptions of the dynamical mechanisms of normal and abnormal pacemaker activities, e.g., the roles of individual currents in pacemaking (4, 9, 21, 22, 24, 45). Pacemaker currents are not indispensable for basal pacemaking of SA node cells (20, 21) or BP cells (22, 46). As shown in this study for BP cells, however, the pacemaker currents may be necessary for constructing a pacemaker system with high structural stability and high driving ability, as well as for finer regulations of pacemaking rates.
Bifurcation analyses of model cell systems may also allow us to accurately predict and properly control the dynamics and bifurcation structures of real cells, possibly applicable to cell system engineering to develop functional BPs from native cardiomyocytes or ES cells for gene or cell therapy of bradyarrhythmias (8, 10, 15, 25, 28, 32, 36, 50). Pacemaker currents are not necessarily required for creating BP cells (22, 46). Nevertheless, stable transfection of pacemaker channel genes or transplantation of BP cells expressing pacemaker channels (e.g., ES cell-derived SA node cells) to host tissues may be helpful or even necessary for creating functional BPs with robust pacemaking and driving, as well as responsiveness to autonomic regulations. This study suggests that 1) expression of Ih (or the other pacemaker currents) is not indispensable but is helpful for creating BP cells from HVMs, 2) expression of Ih improves structural stability and driving ability of BPs when IK1 density in adjacent NP cells is relatively small, as in the atrium, and 3) expression of Ist, i.e., a pacemaker current with relatively slow inactivation kinetics, may especially be effective in yielding robust BP activity with stable frequency.
Limitations and Perspectives of Study
As mentioned in our previous article (22), the limitations of our study include incompleteness of the HVM model due to the lack of experimental data from HVMs as well as the lack of experimental evidence for bifurcation structures of real myocytes. More elaborate HVM models, such as recently published ones (12, 42), with refinements of formulas to be suitable for bifurcation analyses would have to be developed for more detailed investigations using the theoretical approach. In addition, the pacemaker current formulas based on experiments for human cardiomyocytes should be used in future studies.
As suggested by Silva and Rudy (41), a potential advantage of the BP over the electronic pacemaker is responsiveness to 
-adrenergic stimulation (-AS). BP responsiveness to -AS was very limited in the guinea pig ventricular model (41). We also found in the simulations according to their method that the HVM pacemaker did not exhibit a significant rate increase in response to -AS, suggesting that the ventricular pacemaker is less sensitive to -AS than the SA node pacemaker. Such low sensitivity to -AS of the ventricular pacemaker would be due at least in part to the lack of the pacemaker currents that are abundant in SA node primary pacemaker cells (20, 21, 27, 31, 51) and are known to be enhanced by -AS (26, 35, 43). Nevertheless, we did not show the data on this issue, because it is very important but beyond the aim of this study and because our model lacks the intracellular modulating factors such as cAMP and protein kinases. Further sophisticated models incorporating the modulating factors, as developed by Saucerman et al. (38), would be required.
We used the simple coupled-cell system to investigate the structural stability and driving ability of BP cells, believing that this study is valuable as a first step toward development of a theoretical background for engineering of functional BPs. Nevertheless, a real pacemaker system such as the intact SA node has much more complex architectures to facilitate optimization of the electrical loading by surrounding atrial or ventricular tissues (1, 2, 14, 44). Thus more elaborate multicellular models are required for investigation of the structural stability to electrotonic loads and ability to drive the heart of a BP system in vivo and of how to cr
