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Effects of pacemaker currents on creation and modulation of human ventricular pacemaker: theoretical study with application to biological pacemaker engineering

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

	ABSTRACT

TOP ABSTRACT THEORY AND METHODS RESULTS DISCUSSION APPENDIX A: GLOSSARY APPENDIX B: MODEL EQUATIONS APPENDIX C: DETERMINATION OF... GRANTS REFERENCES

 
A cardiac biological pacemaker (BP) has been created by suppression of the inward rectifier K⁺ current (I_K1) or overexpression of the hyperpolarization-activated current (I_h). We theoretically investigated the effects of incorporating I_h, T-type Ca²⁺ current (I_Ca,T), sustained inward current (I_st), and/or low-voltage-activated L-type Ca²⁺ channel current (I_Ca,LD) on 1) creation of BP cells, 2) robustness of BP activity to electrotonic loads of nonpacemaking (NP) cells, and 3) BP cell ability to drive NP cells. We used a single-cell model for human ventricular myocytes (HVMs) and also coupled-cell models composed of BP and NP cells. Bifurcation structures of the model cells were explored during changes in conductance of the currents and gap junction. Incorporating the pacemaker currents did not yield BP activity in HVM with normal I_K1 but increased the critical I_K1 conductance for BP activity to emerge. Expressing I_h appeared to be most helpful in facilitating creation of BP cells via I_K1 suppression. In the coupled-cell model, I_st significantly enlarged the gap conductance (G_C) region where stable BP cell pacemaking and NP cell driving occur, reducing the number of BP cells required for robust pacemaking and driving. In contrast, I_h enlarged the G_C region of pacemaking and driving only when I_K1 of the NP cell was relatively low. I_Ca,T or I_Ca,LD exerted effects similar to those of I_st but caused shrinkage or irregularity of BP oscillations. These findings suggest that expressing I_st most effectively improves the structural stability of BPs to electrotonic loads and the BP ability to drive the ventricle.

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 I_K1-downregulated HVMs and the roles of individual sarcolemmal currents in HVM pacemaking. We have suggested that 1) BP activity can be developed by reducing I_K1 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 I_h, T-type Ca²⁺ channel current (I_Ca,T), sustained inward current (I_st), and low-voltage-activated L-type Ca²⁺ channel current (I_Ca,LD). Therefore, we first investigated whether incorporating these pacemaker currents can yield BP activity and how this affects BP generation during I_K1 suppression in the model HVM. Our findings suggested that I_h facilitates BP generation during I_K1 suppression in combination with the other pacemaker currents, although I_K1 downregulation is necessary for constructing BP cells from HVMs.

Our preliminary study also revealed some drawbacks of the I_K1-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 I_K1-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 I_Ca,T, I_st, and I_Ca,LD can, but I_h 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).

	THEORY AND METHODS

TOP ABSTRACT THEORY AND METHODS RESULTS DISCUSSION APPENDIX A: GLOSSARY APPENDIX B: MODEL EQUATIONS APPENDIX C: DETERMINATION OF... GRANTS REFERENCES

 
Base Mathematical Model for Human Ventricular Myocytes

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 (I_Ca,L), rapid and slow components of delayed rectifier K⁺ currents (denoted I_Kr and I_Ks, respectively), 4-aminopyridine-sensitive transient outward current (I_to), Na⁺ channel current (I_Na), I_K1, background Na⁺ (I_Na,b) and Ca²⁺ (I_Ca,b) currents, Na⁺-K⁺ pump current (I_NaK), Na⁺/Ca²⁺ exchanger current (I_NaCa), and Ca²⁺ pump current (I_pCa). Time-dependent changes in the membrane potential (V) are described by the equation

(1)

where I_stim represents the stimulus current (in pA/pF), being set equal to zero for simulations of BP activity. Details on modifications and expressions of the HVM model are described in our previous article (22).

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 I_h, I_Ca,T, and I_st, 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 I_Ca,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).

View larger version (34K): [in this window] [in a new window]   Fig. 1. Voltage-dependent kinetics of the rabbit sinoatrial (SA) node hyperpolarization-activated current (Ih), T-type Ca2+ current (ICa,T), and sustained inward current (Ist) as well as mouse SA node low-voltage-activated L-type Ca2+ channel current (ICa,LD). Top and middle: voltage dependence of the steady-state probabilities (top) and time constants (, middle) for activation and inactivation gating variables of individual pacemaker currents. Bottom: steady-state (window) currents calculated by the steady-state equations for the gating variables. Numbers in each panel represent the maximum current conductance in nS/pF. See APPENDIX A for symbol definitions.

View larger version (35K): [in this window] [in a new window]   Fig. 2. Changes in stability and dynamics of a single biological pacemaker (BP) cell with increase of each pacemaker current. To induce BP activity, the inward rectifier K+ current (IK1) conductance (gK1) was reduced to 15% of the control. The steady-state (equilibrium point, EP) and stable periodic [maximum diastolic potential (Vmin) and peak overshoot potential (Vmax)] branches (top), as well as the cycle length (CL) (second) and maximum upstroke velocity ([dV/dt]max) (third) of potential oscillations, are shown as functions of the maximum conductance (g). Peak amplitudes of the pacemaker currents during BP oscillations are also plotted (bottom). Steady-state BP dynamics were computed by numerically solving differential equations for 1 min at each conductance value (for details, see Ref. 22), with the initial Na+ concentration ([Na+]i) set equal to 6.55 mM in the absence of pacemaker currents; the conductance value was increased at an interval of 1 pS/pF (for gh, gCa,T, and gCa,LD) or 0.01 pS/pF (for gst) until a bifurcation occurred. Hopf bifurcation points at which stability of an EP reverses (labeled HB at gCa,T = 6.253, gst = 0.04141, and gCa,LD = 1.654 nS/pF) and a saddle-node bifurcation point at which an EP disappears (labeled SNB at gh = 2.982 nS/pF) are shown.

I_Ca,T. The expressions for I_Ca,T were adopted from the rabbit SA node (5, 20) and ventricular (34) models. The effects of these I_Ca,T currents were qualitatively the same, with the effect of the SA node I_Ca,T being more remarkable; thus the data for the rabbit SA node I_Ca,T are shown. Relatively high g_Ca,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 I_Ca,T is relatively small (see Fig. 1). Therefore, the g_Ca,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 g_Ca,T caused shrinkage of BP oscillations and stabilization of an EP via a Hopf bifurcation (see Fig. 2).

I_st. The formulas for I_st were adopted from the rabbit SA node model (20) and the experimental report for the rat SA node (40). The effects of the two I_st currents were essentially the same; thus only data for the rabbit SA node I_st are shown. The g_st value, 0.015 nS/pF in the original SA node model (20), was limited to 0.02 nS/pF; further increases in g_st yielded extremely large window currents leading to Na⁺-overload conditions, because the window region of the model I_st is relatively large (see Fig. 1).

I_Ca,LD. The kinetics of I_Ca,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 I_Ca,LD is 10–20 mV more negative than that of the high-voltage-activated Ca²⁺ channel current (I_Ca,LC), 2) the activation of I_Ca,LD is faster than that of I_Ca,LC, and 3) the inactivation of I_Ca,LD is 1.5–2 times slower than that of I_Ca,LC. According to these data, therefore, we formulated the kinetics of I_Ca,LD as provided in APPENDIX B (Eqs. A38–A42). Because the effects on model cell dynamics of the model I_Ca,LD were relatively small, the g_Ca,LD value was increased up to 1 nS/pF. Further increases in g_Ca,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 I_K1-deleted BP cells were connected to a NP cell with normal or reduced I_K1 via the gap junction conductance (G_C) 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 G_C). Pacemaker currents were incorporated into the BP cells only (not into the NP cell).

View larger version (5K): [in this window] [in a new window]   Fig. 3. Coupled-cell model. NP, nonpacemaking; GC, gap conductance.

(2)

(3)

where C_BP and C_NP represent the membrane capacitance of the BP and NP cells, respectively. I_total(BP) and I_total(NP) are the sum of sarcolemmal ionic currents in the BP and NP cells, with I_GJ denoting the gap junction current. I_stim is the stimulus current applied to the BP cells, usually set equal to zero. The units for V, C, I_total (I_stim), and I_GJ are millivolts, picofarads, picoamperes per picofarad, and picoamperes, respectively. Time-dependent changes in all the other state variables of the BP and NP cells were computed independently, as described for the single HVM in our previous article (22).

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 I_K1 (g_K1) and pacemaker currents (g_h, g_Ca,T, g_st, g_Ca,LD) as well as G_C; g_K1 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 (V_min) and peak overshoot potential (V_max) 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 (V_min, V_max) 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 G_C and two-parameter bifurcation diagrams for G_C and pacemaker current conductance. For construction of one-parameter bifurcation diagrams, the membrane potential at EPs (steady-state branches) and local potential extrema (V_min, V_max) of periodic orbits (periodic branches) were determined and plotted as a function of G_C, 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 G_C 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 g_K1; the primary parameter (G_C) 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 G_C 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 G_C 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 G_C 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.

	RESULTS

TOP ABSTRACT THEORY AND METHODS RESULTS DISCUSSION APPENDIX A: GLOSSARY APPENDIX B: MODEL EQUATIONS APPENDIX C: DETERMINATION OF... GRANTS REFERENCES

 
Effects of Pacemaker Currents on BP Generation in Single HVM Model

Our preceding study (22) suggests that I_K1 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 I_h, I_Ca,T, I_st, and I_Ca,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 I_K1 (g_K1 = 1): BP activity did not appear when g_h, g_Ca,T, g_st, and g_Ca,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 g_Ca,T, g_st, or g_Ca,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 I_K1 suppression by increasing the critical g_K1 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 I_K1 suppression.

Effects of incorporating each pacemaker current on BP generation during I_K1 suppression. As clearly shown in Fig. 4, the saddle-node bifurcation point below which BP oscillations occur shifted toward higher g_K1 values with increasing g_h, suggesting that expression of I_h accelerates BP generation during I_K1 suppression by increasing the critical g_K1 value to induce BP activity. To further examine the effects of incorporating the pacemaker currents on BP generation in the I_K1-reduced HVM, we determined the critical g_K1 value for BP generation (i.e., g_K1 value at a saddle-node bifurcation) as functions of g_h, g_Ca,T, g_st, or g_Ca,LD, as well as exploring the influences of the pacemaker currents on the steady-state I-V relation at g_K1 = 1 (Fig. 5). The inward pacemaker currents counteracted the outward component of I_K1, inwardly shifting the steady-state I-V curve. The critical g_K1 value increased with increasing g_h; however, it did not reach the control g_K1 value but reached the maximum of 0.287 (28.7%) at g_h = 0.1039 nS/pF. I_Ca,T at higher densities (g_Ca,T 3 nS/pF) also increased the critical g_K1, with its increase limited to <0.266 (26.6%). The effects of I_st and I_Ca,LD were relatively small within the limited conductance ranges of 0.02 and 1 nS/pF, respectively.

View larger version (25K): [in this window] [in a new window]   Fig. 4. Bifurcation structures during gK1 decreases of the model human ventricular myocyte (HVM) in the absence or presence of the rabbit SA node type Ih. One-parameter bifurcation diagrams for gK1 with the steady-state (EP1–3) and stable periodic (Vmin, Vmax) branches (top), as well as CL of potential oscillations as a function of gK1 (bottom), are shown for gh of 0 (left), 0.02 (center), and 0.05 (right) nS/pF. Steady-state BP dynamics were computed by numerically solving differential equations for 60 min at each gK1 value (for details see APPENDIX B). The gK1 value was reduced at an interval of 0.001, with the initial [Na+]i set equal to 5.75 mM at gK1 = 0.4. The saddle-node bifurcation point at which EP2 and EP3 merge together and disappear is shown (labeled SNB at gK1 = 0.154, 0.219, and 0.260).

View larger version (27K): [in this window] [in a new window]   Fig. 5. Effects of incorporating the rabbit SA node Ih, ICa,T, or Ist or the mouse SA node ICa,LD on the steady-state current-voltage (I-V) relation and BP generation during IK1 inhibition. Top: effects on the steady-state I-V curve for a HVM with normal IK1. [K+]i was fixed at 140 mM. The values of gh, gCa,T, gst, and gCa,LD were increased up to 0.12, 3, 0.02, and 1 nS/pF, respectively. Bottom: 2-parameter bifurcation diagrams for gK1 (y-axis) and gh, gCa,T, gst, or gCa,LD (x-axis), depicting changes in the critical gK1 value, i.e., displacements of saddle-node bifurcation points at which BP activity emerges, with increasing gh, gCa,T, gst, or gCa,LD at an interval of 0.1 (for gh and gst) or 1 (for gCa,T and gCa,LD) pS/pF. The gK1 value was reduced at an interval of 0.001–0.0001 for searching for bifurcation points.

View larger version (37K): [in this window] [in a new window]   Fig. 6. Effects of coexpression of Ih and ICa,T (left), Ist (center), or ICa,LD (right) on BP generation during IK1 inhibition. Top: 2-parameter bifurcation diagrams for gK1 (y-axis) and gh (x-axis) depicting displacements of saddle-node bifurcation points with increasing gh at an interval of 0.1 pS/pF. ICa,T, Ist, or ICa,LD was incorporated with maximum conductance set equal to 1–2, 0.005–0.01, and 0.5–1 nS/pF, respectively. The gK1 value was reduced at an interval of 0.0001 for searching for bifurcation points. Calculations were stopped when [Na+]i exceeded extracellular Na+ concentration ([Na+]o) or when [Ca2+]i exceeded [Ca2+]o. Middle and bottom: steady-state [Na+]i (middle) and [Ca2+]i (bottom) at the bifurcation points are shown as functions of gh.

One of the drawbacks of the I_K1-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 (g_K1 = 0) were connected to a NP cell (g_K1 = 0.2–1) via the G_C ranging from 0 to 20 nS. The coupled-cell system is also useful in exploring BP cell ability to drive adjacent NP cells.

G_C-dependent behaviors of the coupled-cell system. In the system with one BP cell and one normal HVM (g_K1 = 1), an EP was stabilized via a Hopf bifurcation at G_C = 0.689 nS, with the BP cell not exhibiting pacemaker activity at G_C 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 G_C values or 2) inhibition of BP cell pacemaking for higher G_C values. We could not find a G_C 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 I_GJ at various G_C values for the coupled-cell system composed of five BP cells and one normal HVM. With increasing G_C, the BP rate decreased, with the oscillation amplitude getting larger. The BP cells could induce only subthreshold responses of the HVM at G_C = 2.0 nS, whereas they induced driven action potentials at G_C = 2.5–3.5 nS. Further increase in G_C to 5.0 nS abolished BP activity, which was probably due to hyperpolarizing loads of the NP cell as indicated by positive I_GJ.

View larger version (43K): [in this window] [in a new window]   Fig. 7. Simulated dynamic behaviors of a coupled-cell system at various GC values of 2.0, 2.5, 3.0, 3.5, and 5 nS (from top to bottom). One normal HVM (gK1 = 1) was connected to 5 BP cells (gK1 = 0), which were assumed to be well coupled enough to act as a cluster. Membrane potentials of the BP cells and normal HVM (NP), as well as gap junction currents, were computed by numerically solving differential equations for 60 min at each GC value (for details, see APPENDIX B); the potential and current behaviors during the last 10 s are shown.

Stability and dynamics during G_C 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 G_C 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 G_C. 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 G_C 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 G_C yielded a BP cell ability to drive the normal HVM; however, further increases in G_C led to the cessation of pacemaking and driving via the Hopf bifurcation. The G_C 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).

View larger version (37K): [in this window] [in a new window]   Fig. 8. Stability and dynamics during GC increases of the coupled-cell systems with various size factors (A) or NP cell gK1 (B). One normal HVM was connected to 1, 3, 5, or 7 IK1-deleted BP cells with all the BP cells assumed to be well coupled (A), or 1 IK1-reduced NP cell (gK1 = 0.35, 0.30, 0.25) was connected to 1 BP cell (B). One-parameter bifurcation diagrams for GC depicting EPs, Hopf bifurcation, and saddle-node bifurcation points, as well as extrema of potential oscillations (Vmin, Vmax), of NP (top) and BP (middle) cells are shown; CL of BP oscillations is also shown as functions of GC (bottom). Steady-state model cell dynamics were computed by numerically solving the differential equations for 60–120 min at each GC value. The GC value was increased up to 5 nS at an interval of 0.01 nS. Cm, cell membrane capacitance.

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 G_C-dependent bifurcation structures. Bifurcation structures during G_C increases were first determined at various conductances of each pacemaker current. We constructed two-parameter bifurcation diagrams in which the critical G_C value at Hopf or saddle-node bifurcation points was plotted as functions of the maximum current conductance. Figure 9 shows the effects of incorporating I_h, I_Ca,T, I_st, or I_Ca,LD on the bifurcation structure during G_C 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 I_K1 (g_K1 = 0.35). Stability and dynamics during G_C increases of the coupled-cell systems incorporating each pacemaker current are illustrated in Fig. 10. The results suggest that individual pacemaker currents affect the G_C-dependent bifurcation structure of the system in different ways.

View larger version (28K): [in this window] [in a new window]   Fig. 9. Effects of incorporating pacemaker currents on bifurcation structures during GC increases of the coupled-cell model. Three BP cells were connected to 1 normal HVM (top), or 1 BP cell was connected to 1 NP cell with a reduced gK1 of 0.35 (bottom). The rabbit SA node Ih (gh = 0–0.5 nS/pF), ICa,T (gCa,T = 0–3 nS/pF), or Ist (gst = 0–0.02 nS/pF) or mouse SA node ICa,LD (gCa,LD = 0–1 nS/pF) was incorporated into the BP cells. Two-parameter bifurcation diagrams for GC and the pacemaker current conductance are shown; the critical GC values at Hopf bifurcation and saddle-node bifurcation points were plotted as functions of the maximum current conductance. The GC value was increased up to 10–20 nS at an interval of 0.01 nS for searching for bifurcations. The parameter plane should be divided into 3 areas with different steady states: 1) nonpacemaking and nondriving (P–D–), where BP activity is abolished; 2) pacemaking and driving (P+D+), where BP cells exhibit pacemaker activity and drive the NP cell; and 3) pacemaking and nondriving (P+D–), where BP cells exhibit pacemaker activity but do not drive the NP cell. The loci of Hopf and saddle-node bifurcation points actually divided the parameter plane into 3 areas, P–D–, P+D+, and P+D± (not P+D–). The symbol P+D± was used for the GC region lower than the 1st bifurcation point, because NP cell driving might appear in the vicinity of the 1st bifurcation point (see Fig. 8), but it was not possible to determine the critical GC value at which NP cell driving occurs by this analysis.

View larger version (46K): [in this window] [in a new window]   Fig. 10. Stability and dynamics during GC increases of the coupled-cell systems composed of 3 BP cells and 1 normal HVM (A) or 1 BP cell and 1 NP cell with reduced gK1 of 0.35 (B). Ih (gh = 0.1–0.2 nS/pF), ICa,T (gCa,T = 1–2 nS/pF), Ist (gst = 0.01 nS/pF), or ICa,LD (gCa,LD = 0.5 nS/pF) was incorporated into the BP cells. Bifurcation diagrams depicting EPs, Hopf bifurcation, and/or saddle-node bifurcation points, as well as dynamics (Vmin, Vmax, CL) of the NP and BP cells, are shown as functions of GC. Steady-state dynamics were computed by numerically solving the differential equations for 30–60 min at each GC value, which was increased up to 10 nS at an interval of 0.01 nS.

I_CA,T EFFECT. Similar to I_h, I_Ca,T (3 nS/pF) did not significantly change the critical G_C 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 I_Ca,T yielded a saddle-node bifurcation and thus enlarged the P⁺D⁺ region; I_Ca,T at 2 nS/pF enlarged the G_C region where all EPs are unstable and BP activity appears (Fig. 10A). When the NP cell I_K1 was relatively small, I_Ca,T dramatically enlarged the P⁺D⁺ region by producing a saddle-node bifurcation (Fig. 9, bottom; Fig. 10B). With relatively high conductance of I_Ca,T, the amplitudes of BP oscillations and driven action potentials of the NP cell were relatively small.

I_ST EFFECT. In contrast to I_h or I_Ca,T, incorporation of I_st (0.02 nS/pF) dramatically shifted Hopf bifurcation points toward higher G_C values and further produced a saddle-node bifurcation to ensure robust pacemaking and driving, i.e., enlarged the P⁺D⁺ region (Fig. 9). I_st at 0.01 nS/pF dramatically enlarged the G_C region where BP oscillations occur, with CL of BP oscillations being relatively stable during increases in G_C (Fig. 10).

I_CA,LD EFFECT. The effects of I_Ca,LD were similar to those of I_st, although much larger conductance was required for I_Ca,LD to exert the effects as dramatically as I_st. Incorporation of I_Ca,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, I_Ca,LD (0.5 nS/pF) enlarged the G_C region where BP oscillations occur. However, I_Ca,LD at high densities of 0.5–1 nS/pF caused unstable CL and irregular dynamics during increases in G_C.

Influences of altering pacemaker current kinetics on their modulation of bifurcation structures. Individual pacemaker currents affected the G_C-dependent bifurcation structure in different ways, as well as with different potencies. The effects of I_h 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 I_Ca,T, I_st, and I_Ca,LD were apparently different in the system with three BP cells and one normal HVM: the locus of Hopf bifurcation points at lower G_C was separated from that of saddle-node bifurcation points with I_Ca,T, but not with I_st or I_Ca,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 I_Ca,T or I_st with slowed or accelerated inactivation on G_C-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 I_Ca,T with the inactivation time constant (_fT) 10 times higher than the control yielded a bifurcation diagram similar to that for the original I_st; in contrast, the modified I_st 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 I_Ca,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.

View larger version (38K): [in this window] [in a new window]   Fig. 11. Effects of the pacemaker currents with different inactivation kinetics on GC-dependent bifurcation structures of the coupled-cell system composed of 3 BP cells and 1 normal HVM. ICa,T (gCa,T = 0–3 nS/pF) or Ist (gst = 0–0.02 nS/pF) was incorporated into the BP cells, with their inactivation time constants (fT, qi) increased or decreased to the values shown as ratios to the control values. Two-parameter bifurcation diagrams for GC and the pacemaker current conductance were constructed as for Fig. 9.

View larger version (27K): [in this window] [in a new window]   Fig. 12. Effects of incorporating the pacemaker currents on size factor-dependent bifurcation structures of the coupled-cell systems. One to six BP cells were connected to 1 normal HVM. Rabbit SA node Ih (gh = 0.1–0.2 nS/pF), ICa,T (gCa,T = 1–2 nS/pF), or Ist (gst=0.01–0.02 nS/pF) or mouse SA node ICa,LD (gCa,LD = 0.5–1 nS/pF) was incorporated into the BP cells. Two-parameter bifurcation diagrams depicting the displacement of saddle-node bifurcation and Hopf bifurcation points are shown; the critical GC value at the bifurcation points was plotted as functions of the size factor (i.e., the number of BP cells) increased at an interval of 0.01. The GC value was increased up to 20 nS at an interval of 0.01 nS for searching for bifurcations.

View larger version (27K): [in this window] [in a new window]   Fig. 13. Effects of incorporating the pacemaker currents on NP cell gK1-dependent bifurcation structures of the coupled-cell systems. One BP cell was connected to 1 NP cell with gK1 = 0.2–0.7. Ih (gh = 0.1–0.2 nS/pF), ICa,T (gCa,T = 1–2 nS/pF), Ist (gst = 0.01–0.02 nS/pF), or ICa,LD (gCa,LD = 0.5–1 nS/pF) was incorporated into the BP cell. The critical GC value at the bifurcation points was plotted as functions of the normalized gK1 of the NP cell decreased at an interval of 0.001. The GC value was increased up to 10 nS at an interval of 0.01 nS for searching for bifurcations.

	DISCUSSION

TOP ABSTRACT THEORY AND METHODS RESULTS DISCUSSION APPENDIX A: GLOSSARY APPENDIX B: MODEL EQUATIONS APPENDIX C: DETERMINATION OF... GRANTS REFERENCES

 
Possible Roles of Pacemaker Currents in Creation of BP Cells

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 g_K1 value (saddle-node bifurcation point) to produce BP activity was plotted as functions of the maximum current conductance.

Pacemaker currents facilitate BP generation during I_K1 suppression. Our results suggest that the pacemaker currents, especially I_h, can facilitate BP generation during I_K1 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 I_K1, inwardly shift the steady-state I-V curve, and thus lead to a saddle-node bifurcation at a higher g_K1 value (see Fig. 5, top). Nevertheless, the critical g_K1 value never reached the control value of 1, suggesting that any pacemaker currents cannot yield BP activity in the HVM with normal I_K1.

I_K1 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 g_K1 = 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 I_K1, albeit with the requirement of very high conductance (e.g., g_h > 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 I_h and/or the other pacemaker currents would not yield BP generation in HVMs with normal I_K1, and thus I_K1 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 I_h could induce BP activity in the Luo-Rudy guinea pig ventricular model without reducing I_K1. We are not sure about the reason for this inconsistency. Further studies are required to clarify whether the effects of I_h and other pacemaker currents depend on species (human vs. guinea pig) or on the property of a model itself as well as whether expressing I_h can actually induce BP activity in real NP myocytes of the ventricle.

Overexpression of I_h has been reported to induce BP activity (or increase BP rates) in canine atrial or Purkinje myocytes (32, 36). Figure 5 indicates that increasing I_h alone can induce BP activity via a saddle-node bifurcation if g_K1 is smaller than the control value of 3.9 nS/pF, e.g., when a g_K1 value is 25% of the control value (0.975 nS/pF), increasing g_h 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 I_K1 (0.134 nS/pF) is only 18.3% of that of the human ventricular I_K1 (0.732 nS/pF). As suggested by previous reports (14, 32, 36), expression of I_h alone would induce pacemaker activity in myocytes with relatively low density of I_K1, 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.

I_h effect is relatively small and NP cell g_K1 dependent. The effects of I_h on G_C-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. I_h 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 I_K1 was relatively small, however, I_h improved the structural stability and driving ability of BP cells by producing a saddle-node bifurcation (Figs. 9 and 10B). Moreover, the effects of I_h were shown to depend on I_K1 density of the NP cell, as well as the size factor (Figs. 12 and 13). These results suggest that I_h can support the robust pacemaking and driving of BPs only when I_K1 density in adjacent NP cells is relatively small. I_K1 density is much larger in the human ventricle than in the human atrium (17); thus I_h may support the robust pacemaking and driving of BPs (as well as the SA node) in the atrium with relatively small I_K1, but not in the ventricle with relatively large I_K1. Possible roles of I_h 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.

I_Ca,T effect is NP cell g_K1 dependent and characterized by shrinkage of voltage oscillations. Similar to I_h, I_Ca,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 G_C values (Fig. 9, top); the low efficacy of I_Ca,T appeared to be due to the rapid inactivation kinetics, because the modified I_Ca,T with slowed inactivation significantly increased the Hopf bifurcation value (Fig. 11). Nevertheless, higher densities of I_Ca,T improved the structural stability and driving ability of BP cells by producing a saddle-node bifurcation. The effects of I_Ca,T also depend on the NP cell I_K1, as well as the size factor (Figs. 9, 12, and 13); I_Ca,T may effectively improve the functions of BPs in the atrium, whereas much larger I_Ca,T is required for the ventricle. However, higher densities of I_Ca,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 I_Ca,T inactivation and is not desirable for functional BPs.

I_st most dramatically improves structural stability and driving ability of BP cells. In contrast to I_h or I_Ca,T, I_st at relatively low conductance dramatically improved the structural stability and driving ability of BP cells by shifting Hopf bifurcation points toward higher G_C values and yielding a saddle-node bifurcation (Fig. 9), producing BP oscillations with stable frequency over the broad G_C range (Fig. 10). Furthermore, I_st, as well as I_Ca,T and I_Ca,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 I_st 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 I_st, exhibiting relatively slow inactivation kinetics, most dramatically improves the structural stability and driving ability of the HVM pacemaker. I_st may be indispensable for robust pacemaking and driving of ventricular BPs.

I_Ca,LD effect is similar to I_st effect but characterized by generation of irregular dynamics. The effects of I_Ca,LD were qualitatively similar to those of I_st, although its potency was much smaller (Figs. 9, 12, and 13). However, I_Ca,LD at high densities caused unstable CL and irregular dynamics during increases in G_C, which is possibly a consequence of producing Ca²⁺ overload conditions, not desirable for functional BPs. We suppose that the role of I_Ca,LD in the working of pacemaker systems is essentially the same as that of I_Ca,LC: I_Ca,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 I_h (or the other pacemaker currents) is not indispensable but is helpful for creating BP cells from HVMs, 2) expression of I_h improves structural stability and driving ability of BPs when I_K1 density in adjacent NP cells is relatively small, as in the atrium, and 3) expression of I_st, 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 create BP systems with robust pacemaking and driving. In the coupled-cell model, the critical size factor (the critical number of BP cells) for stable driving of a normal HVM was quite large, e.g., at least six BP cells were required in the absence of a pacemaker current (see Figs. 8A and 12). A similar size effect, a size factor of 5, was previously reported for other coupled-cell systems composed of SA node cells and ventricular myocytes (19, 47). However, the coupled-cell model cannot predict the critical size of a BP to be required for stable driving of the ventricle. Using a two-dimensional system in which real atrioventricular node or model SA node cells were coupled to ventricular or atrial model cells, Joyner et al. (14) determined the critical size of the pacemaker cells for driving the surrounding quiescent cells and suggested that the critical size depends on the scale of the systems, I_K1 density in surrounding NP cells, G_C values, and many other factors. Moreover, the roles of pacemaker currents in pacemaking and driving of BPs may also change depending on the dimension of the systems, as well as the number of BP cells, I_K1 of adjacent NP cells, and G_C values (see Figs. 12 and 13). Further studies using multicellular (2 or 3 dimensional) models are required to determine the critical BP size for driving the ventricle and to determine how incorporating pacemaker currents affects critical BP size and BP functions.

	APPENDIX A: GLOSSARY

TOP ABSTRACT THEORY AND METHODS RESULTS DISCUSSION APPENDIX A: GLOSSARY APPENDIX B: MODEL EQUATIONS APPENDIX C: DETERMINATION OF... GRANTS REFERENCES

 
General

F

Faraday constant

V

Membrane potential (mV)

Cell Geometry

C_m

Cell membrane capacitance

C_BP

C_m of BP cells

C_NP

C_m of NP cells

V_i

Myoplasmic volume available for Ca²⁺ diffusion

V_rel

Volume of junctional SR (Ca²⁺ release store)

V_up

Volume of network SR (Ca²⁺ uptake store)

Ionic Concentrations

[Na⁺]_o

Extracellular Na⁺ concentration

[K⁺]_o

Extracellular K⁺ concentration

[Ca²⁺]_o

Extracellular Ca²⁺ concentration

[Na⁺]_i

Intracellular Na⁺ concentration

[K⁺]_i

Intracellular K⁺ concentration

[Ca²⁺]_i

Myoplasmic Ca²⁺ concentration

[Ca²⁺]_rel

Ca²⁺ concentration in junctional SR

[Ca²⁺]_up

Ca²⁺ concentration in network SR

Equilibrium Potentials

E_Na

Equilibrium (Nernst) potential for Na⁺

E_K

Equilibrium (Nernst) potential for K⁺

Sarcolemmal Ionic Currents

g_h

Maximum I_h conductance

g_Ca,T

Maximum I_Ca,T conductance

g_st

Maximum I_st conductance

g_Ca,LD

Maximum I_Ca,LD conductance

d_L

Activation gating variable for I_Ca,L

f_L

Voltage-dependent inactivation gating variable for I_Ca,L

d_T

Activation gating variable for I_Ca,T

f_T

Inactivation gating variable for I_Ca,T

p_a

Activation gating variable for I_Kr

n

Activation gating variable for I_Ks

q

Inactivation gating variable for I_to

y

Activation gating variable for I_h

q_a

Activation gating variable for I_st

q_i

Inactivation gating variable for I_st

_qa

Opening rate constant for q_a

_qa

Closing rate constant for q_a

_qi

Opening rate constant for q_i

_qi

Closing rate constant for q_i

d_LD

Activation gating variable for I_Ca,LD

f_LD

Voltage-dependent inactivation gating variable for I_Ca,LD

f_Ca,

Ca²⁺-dependent inactivation gating variable for I_Ca,LD

h

Inactivation gating variable for I_Na

x

Steady-state value of a gating variable x

_x

Time constant for a gating variable x

J_Ca,net

Net Ca²⁺ flux through the sarcolemmal membrane

J_Na,net

Net Na⁺ flux through the sarcolemmal membrane

Intracellular Ca²⁺ Dynamics (SR Function and Ca²⁺ Buffering)

J_rel

Ca²⁺ release flux from junctional SR to subspace

J_up

Ca²⁺ uptake flux from myoplasm to network SR

J_tr

Ca²⁺ transfer flux from network SR to junctional SR

J_leak

Ca²⁺ leak flux from network SR to myoplasm

[CM]_tot

Total calmodulin concentration

[CQ]_tot

Total calsequestrin concentration

[TC]_tot

Total concentration of troponin-Ca complex

f_TC

Fractional occupancy of troponin-Ca complex by Ca²⁺

f_CM

Fractional occupancy of calmodulin by Ca²⁺ in myoplasm

f_CQ

Fractional occupancy of calsequestrin by Ca²⁺

K_dCM

Ca²⁺-binding constant for calmodulin

K_dCQ

Ca²⁺-binding constant for calsequestrin

B_CM

Scaling factor for fast Ca²⁺ buffering by calmodulin

B_CQ

Scaling factor for fast Ca²⁺ buffering by calsequestrin

	APPENDIX B: MODEL EQUATIONS

TOP ABSTRACT THEORY AND METHODS RESULTS DISCUSSION APPENDIX A: GLOSSARY APPENDIX B: MODEL EQUATIONS APPENDIX C: DETERMINATION OF... GRANTS REFERENCES

 
The mathematical expressions used for the pacemaker currents (I_h, I_Ca,T, I_st, I_Ca,LD), as well as differential equations for state variables of the coupled-cell system, are given below. Units are millivolts, picoamperes, nanosiemens, picofarads, milliseconds, millimolar, and liters. The temperature assumed for the model is 37°C. The symbols used and their definitions are the same as those in our rabbit SA node model (20). Expressions for other currents and dynamics of SR Ca²⁺ handling, as well as standard parameter values and initial conditions for computations, are given in our previous article (22). See APPENDIX A for symbol definitions.

Hyperpolarization-Activated Current (I_h)

Rabbit SA node I_h from Kurata et al. (20) model [adopted from Wilders et al. (49) model].

(A1)

(A2)

(A3)

(A4)

(A5)

Human ventricle I_h from Cerbai et al. (3).

(A6)

(A7)

(A8)

(A9)

(A10)

Human HCN2 current from Moroni et al. (29).

(A11)

(A12)

(A13)

(A14)

(A15)

T-type Ca²⁺ Channel Current (I_Ca,T)

Rabbit SA node I_Ca,T from Kurata et al. (20) model [adopted from Demir et al. (5) model].

(A16)

(A17)

(A18)

(A19)

Rabbit ventricle I_Ca,T from Pugligi-Bers (34) model.

(A20)

(A21)

(A22)

(A23)

Sustained Inward Current (I_st)

Rabbit SA node I_st from Kurata et al. (20) model.

(A24)

(A25)

(A26)

(A27)

(A28)

(A29)

Rat SA node I_st from Shinagawa et al. (40).

(A30)

(A31)

(A32)

(A33)

(A34)

(A35)

(A36)

(A37)

Low-Voltage-Activated L-Type Ca²⁺ Channel Current (I_Ca,LD)

(A38)

(A39)

(A40)

(A41)

(A42)

Differential Equations for State Variables

Membrane potential (V).

(A43)

(A44)

(A45)

Gating variables.

(A46)

Intracellular ion concentrations.

(A47)

(A48)

(A49)

(A50)

(A51)

(A52)

(A53)

(A54)

	APPENDIX C: DETERMINATION OF EPS AS INITIAL CONDITIONS

TOP ABSTRACT THEORY AND METHODS RESULTS DISCUSSION APPENDIX A: GLOSSARY APPENDIX B: MODEL EQUATIONS APPENDIX C: DETERMINATION OF... GRANTS REFERENCES

 
The methods for determination of EPs as initial conditions for the single cell were described previously (22). To determine an EP of the coupled-cell system, steady-state values of the state variables V, [Ca²⁺]_i, [Na⁺]_i, [Ca²⁺]_rel, and [Ca²⁺]_up for BP and NP cells were calculated numerically by the algebraic method with a nonlinear equation solver available in MATLAB 7. For bifurcation analyses and simulations using the [K⁺]_i-fixed system, steady-state values of the state variables for BP and NP cells were computed by the following algebraic equations derived from the differential equations.

(A55)

(A56)

(A57)

(A58)

(A59)

(A60)

Equations A57–A60 are for either the BP or the NP cell; in total, 10 equations are required for calculating the steady state of the coupled-cell system.

	GRANTS

TOP ABSTRACT THEORY AND METHODS RESULTS DISCUSSION APPENDIX A: GLOSSARY APPENDIX B: MODEL EQUATIONS APPENDIX C: DETERMINATION OF... GRANTS REFERENCES

 
This work was supported in part by Ministry for Education, Science, Sports and Culture of Japan Grant-in-Aid for Scientific Research (C) 17590192 (to Y. Kurata and T. Shibamoto), Kanazawa Medical University Grant for Collaborative Research C2005-1 (to Y. Kurata and T. Shibamoto), and Kanazawa Medical University Grant for Promoted Research S2006-6 (to Y. Kurata).

	ACKNOWLEDGMENTS

 
Present address of H. Matsuda: Group in Leading Project for Biostimulation, Kyoto Univ., Kyoto, Japan.

	FOOTNOTES

 

Address for reprint requests and other correspondence: Y. Kurata, Dept. of Physiology, Kanazawa Medical Univ., 1-1 Daigaku, Uchinada-machi, Kahoku-gun, Ishikawa 920-0293, Japan (e-mail: yasu{at}kanazawa-med.ac.jp)

The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.

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