# Gradient Model Versus Mosaic Model of the Sinoatrial Node

## Abstract

*Background*—A radical reinterpretation (mosaic model) of the makeup of the sinoatrial (SA) node has been proposed to explain the characteristic regional differences in electrical activity between the periphery and center of the SA node. According to the mosaic model, the differences result from a change in the mix of atrial cells and uniform SA node cells from periphery to center, whereas according to the alternative gradient model, there are no atrial cells within the functional SA node, and the differences result from a change in the intrinsic properties of SA node cells from periphery to center.

*Methods and Results*—A mosaic model of peripheral and central tissue has been constructed computationally by use of a coupled ordinary differential equation network (CODE) in a 2D lattice (20×20), with each node of the lattice designated randomly as an atrial cell or SA node cell (in correct proportions for periphery and center). The mosaic model fails to predict the characteristic differences in action potential rate and shape between the periphery and center, whereas the existing gradient model can do so.

*Conclusions*—The mosaic model of the SA node is untenable, and the SA node is adequately described by the gradient model.

In the pages of *Circulation*, Verheijck et al^{1} proposed a radical reinterpretation of the makeup of the sinoatrial (SA) node: the mosaic model. The SA node is a spatially extended structure (≈4×5 mm in rabbit).^{2} The leading pacemaker site is only ≈1% of the area of the SA node and is normally in the center of the SA node.^{2} Normally, the periphery of the SA node serves to conduct the action potential from the center to the surrounding atrial muscle, although in various circumstances it can take over as the leading pacemaker site.^{3} The SA node is heterogeneous, and electrical activity changes from the periphery to the center in a characteristic fashion.^{2} Cell size is known to decrease from the periphery to the center of the SA node,^{2} and Honjo et al^{4} reported that the electrical activity of single SA node cells from the rabbit is heterogeneous and changes from large to small cells in the same manner as from the periphery to the center of the SA node. These findings form the basis of the gradient model. According to this model, the change in electrical activity from periphery to center is the result of a gradual change in the intrinsic properties of SA node cells from periphery to center.

Verheijck et al^{1} reported that 2 cell types are isolated from rabbit SA node: atrial cells and SA node cells. Honjo et al^{4} failed to find a significant number of atrial cells when they isolated cells from the rabbit SA node. In contrast to Honjo et al,^{4} Verheijck et al^{1} failed to find any correlation between electrical activity of SA node cells and cell morphology (an absence of such a correlation was also reported in another study from the same group^{5} ). Verheijck et al^{1} reported that the proportion of atrial cells decreases from 63% to 41% from the periphery to the center of the rabbit SA node. These findings form the basis of the alternative mosaic model. According to the mosaic model, the ratio of intermingling atrial cells to uniform SA node cells decreases from periphery to center, and this is the cause of the change in electrical activity of the SA node from periphery to center. The aim of the present study was to distinguish between the 2 opposing models by computational analysis.

## Methods

In the full implementation of the mosaic model, computational models comprising a coupled ordinary differential equation network (CODE) in a 2D lattice (20×20) were constructed. Each node of a lattice was designated randomly as either an atrial cell or an SA node cell, with atrial cells forming a proportion, p, of the population of cells. p was taken to be 63% for the peripheral lattice and 41% for the central lattice.^{1} Random numbers were generated by a subroutine described by Press et al^{6} with uniform deviates. In a lattice, each cell (except those on the boundary; nonflux boundary condition used) interacts with its 4 neighboring cells with a coupling conductance g_{j} (normally assumed to be 6 nS). The membrane potential of a cell at coordinates i, j [V_{(i,j)}] was calculated from where C is cell capacitance, i_{tot} is total ionic current in a cell, V is membrane potential, and (i−1,j), (i+1,j), (i,j−1), and (i,j+1) are coordinates of surrounding cells. All atrial cells were assumed to have the same electrical properties, as were all SA node cells. Rabbit atrial cells were simulated by use of the Earm-Hilgemann-Noble^{7} or Lindblad et al^{8} equations, and rabbit SA node cells were simulated by use of the Wilders et al,^{9} Oxsoft HEART,^{10} or Zhang et al^{11} equations.

The mean cycle length of an infinite number of random peripheral and central lattices can be estimated. In a lattice in which a cell is connected to 4 neighboring cells, on average any cell will be connected to 4p atrial cells and 4(1−p) SA node cells. If it is assumed that all atrial cells have the same electrical properties, all SA node cells have the same electrical properties, and all cells are synchronized, there will be no junctional currents flowing between atrial cells or between SA node cells. The membrane potential of atrial and SA node cells (V_{a} and V_{s}, respectively) within the lattice can be calculated from and where subscripts a and s denote atrial and SA node cell types. Equations 2 and 3 were used in conjunction with the various sets of single-cell equations above.

Ordinary differential equations were solved by the fourth-order Runge-Kutta method with a time step small enough to give an accurate numerical solution of the equations: 0.1 ms (or 0.001 ms when the Lindblad et al^{8} equations are used). Models were written in Fortran 77 and run on MIPS R12000.

## Results

### Regional Differences in Intrinsic Pacemaker Activity

Paradoxically, in the rabbit, although the center of the SA node is the leading pacemaker site in the intact SA node, if tissue is isolated from different regions of the SA node, the electrical activity of peripheral tissue is faster than that of central tissue. Figure 1A⇓ shows cycle length (time between successive spontaneous action potentials) of small tissue samples from the rabbit SA node plotted against the distance of the tissue from the crista terminalis (marks approximate boundary between SA node and atrial muscle): the solid circles show data from Kodama and Boyett^{12} ; solid squares, data from Opthof et al^{13} ; and solid triangles, data from Kodama et al.^{14} These 3 independent data sets (albeit from 2 groups) show an increase in cycle length from peripheral (0 mm) to central (≈1.5 mm) tissue. The example of action potentials from small tissue samples from the periphery and center of the rabbit SA node in Figure 1B⇓ shows the characteristic difference in cycle length. In Figure 1A⇓, the solid diamonds show data from Kirchhoff et al^{15} ; in this case, the cycle length of the rabbit SA node is shown before and after the atrial muscle surrounding the SA node was cut off. These data are relevant, because when the atrial muscle is attached, the center is the leading pacemaker site.^{15} The average cycle length in this case was 348 ms.^{15} After the atrial muscle was cut off, however, the leading pacemaker site shifted from center to periphery (because the periphery was no longer suppressed by atrial muscle).^{15} The average cycle length in this case was 294 ms.^{15} This is again evidence that the intrinsic pacemaker activity of peripheral tissue is faster than that of central tissue. Any model of the SA node has to account for this difference in the intrinsic rate of spontaneous activity between periphery and center.

### Gradient Model

Zhang et al^{11} recently developed equations for peripheral and central cells of the rabbit SA node based on data from large (putative peripheral) and small (putative central) rabbit SA node cells (see Introduction). Together, these equations represent the gradient model. Action potentials generated by large and small rabbit SA node cells are shown in Figure 1C⇑. Note that the cycle length of the large (putative peripheral) cell was less than that of the small (putative central) cell. The action potentials generated by the peripheral and central equations are shown in Figure 1D⇑, and they are similar to those recorded experimentally from peripheral and central rabbit SA node tissue (Figure 1B⇑) as well as large and small rabbit SA node cells (Figure 1C⇑); in particular, the cycle length of the peripheral equations is less than that of the central equations. In Figure 1A⇑, the cycle lengths are shown by inverted solid triangles (large and small cells) and open diamonds (peripheral and central equations).

### Mosaic Model

To test the mosaic model, we constructed 2 square lattices of 20×20 cells (roughly 200 to 230×1020 to 1760 μm) to represent small pieces of tissue from the periphery and center of the rabbit SA node (Figure 2A⇓). Each of the 400 cells within a lattice was randomly designated an atrial or SA node cell, with the constraint that the proportion of atrial cells was 63% for the peripheral lattice and 41% for the central lattice (percentages of atrial cells from Verheijck et al^{1}). In Figure 2A⇓, atrial cells are shown in black and SA node cells in white. In a lattice, each cell interacts with its neighboring cells with a coupling conductance g_{j}. g_{j} was initially assumed to be 6 nS; this is between the median (5.3 nS) and mean (7.5 nS) values for rabbit SA node cells (S. Verheule, *Distribution and Physiology of Mammalian Cardiac Gap Junctions*, PhD thesis, University of Utrecht, 1999). Initially, rabbit atrial cells were simulated by use of the Earm-Hilgemann-Noble equations,^{7} and rabbit SA node cells by use of the equations of Wilders et al,^{9} ie, the group who proposed the mosaic model. Within a lattice, all cells were approximately synchronous. Figure 2B⇓ shows action potentials recorded from SA node cells in the peripheral and central lattices. The cycle lengths were 643 and 453 ms for the peripheral and central lattices. For different random distributions of atrial and SA node cells (although always with the same proportions of the 2 cell types), the cycle lengths were found to vary, although the cycle length of the peripheral lattice was always longer than that of the central lattice. The mean±SD values (n=6) are shown in Figure 1A⇑ (open inverted triangle). The mean cycle length of an infinite number of random peripheral and central lattices can be calculated by use of a simplified mosaic model involving Equations 2 and 3 (see Methods for details). Figure 2C⇓ shows action potentials generated by an SA node cell in the simplified mosaic model of the periphery and center. Once again, the cycle length was longer in the peripheral model. The open circles in Figure 1A⇑ show the cycle lengths from the simplified mosaic model; they are close to the mean values from the full implementation of the mosaic model (open inverted triangle in Figure 1A⇑), as expected.

The mosaic model (either in its full implementation or in a simplified form that predicts the average behavior of the mosaic model and using the equations of Earm-Hilgemann-Noble^{7} for rabbit atrial cells and Wilders et al^{9} for rabbit SA node cells) cannot account for the difference in the rate of spontaneous activity observed experimentally between periphery and center. To test whether this is a peculiar feature of the Earm-Hilgemann-Noble^{7} and Wilders et al^{9} equations, calculations were repeated using alternative equations for rabbit atrial and SA node cells. Figure 2D⇑ shows action potentials generated by an SA node cell in the simplified mosaic model of the periphery and center when the Oxsoft HEART equations were used for a rabbit SA node cell.^{10} The recently developed equations for the peripheral and central rabbit SA node cells of Zhang et al^{11} (see above) were also used. The equations for a central SA node cell could not be used, because spontaneous activity failed to develop when the equations were introduced into the peripheral and central mosaic models. This is because the large number of atrial cells in both the peripheral and central mosaic models suppressed the pacemaker activity of the SA node cells. The equations for a peripheral SA node cell could be used, however; Figure 2E⇑ shows action potentials generated by an SA node cell in the simplified mosaic model of the periphery and center when these equations were used. With both the Oxsoft HEART equations^{10} and the peripheral equations of Zhang et al,^{11} the cycle length was once again longer in the peripheral model (Figure 1A⇑: open square, Oxsoft HEART^{10} ; open triangle, Zhang et al^{11} ). The open hexagons in Figure 1A⇑ show the cycle lengths in the simplified mosaic model of the periphery and center when the equations of Lindblad et al^{8} for a rabbit atrial cell were used (Wilders et al^{9} equations for a rabbit SA node cell also used). Once again, the cycle length was longer in the peripheral model. We have checked that the results obtained with the alternative sets of equations for rabbit atrial and SA node cells are qualitatively similar when the full implementation of the mosaic model was used (not shown).

Although the value of 6 nS used for g_{j} is close to the median and mean values for rabbit SA node cells, g_{j} for rabbit SA node cells is reported to vary between 1 and 25 nS (S. Verheule, *Distribution and Physiology of Mammalian Cardiac Gap Junctions*, cited above). To check that the conclusions concerning the mosaic model are not sensitive to the value of g_{j}, we tested this range of g_{j}. Figure 3⇓ shows the cycle lengths of the simplified mosaic model of the periphery (open symbols) and center (solid symbols) as a function of g_{j}. Data obtained with the equations of Wilders et al^{9} (circles), Oxsoft HEART^{10} (squares), and Zhang et al^{11} (triangles; peripheral equations used) for a rabbit SA node cell are shown (Earm-Hilgemann-Noble equations^{7} for a rabbit atrial cell were also used). With all values of g_{j} and all sets of single-cell equations, the cycle length was longer in the peripheral model (Figure 3⇓). The difference in cycle length between the peripheral and central models increased to a maximum with an increase in g_{j} from 1 to 25 nS (Figure 3⇓). We have checked that similar results are obtained when the full implementation of the mosaic model was used (not shown).

## Discussion

We conclude that the mosaic model of the SA node cannot account for the characteristic regional difference in intrinsic pacemaker activity between the periphery and center of the SA node, whereas the existing gradient model can. For this reason, we conclude that the mosaic model is untenable and the SA node is adequately represented by the gradient model. This conclusion is valid for 2 different sets of equations for rabbit atrial cells, 3 different sets of equations for rabbit SA node cells, and a range of g_{j}. The mosaic model also fails in that the action potential configurations predicted (Figure 2B⇑ through 2E) are unlike those observed experimentally (Figure 1B⇑). For example, in the mosaic model, the action potential overshoot was less in the peripheral lattice (Figure 2B⇑ through 2E), whereas experiments show it to be less in the center (Figure 1B⇑). In addition, within a lattice, except at the highest g_{j}, the action potential profile differed markedly between neighboring atrial and SA node cells (not shown). Such cell-to-cell differences are not seen experimentally. With the mosaic model, the slower pacemaker activity of the peripheral lattice can be readily explained by the well-known electrotonic suppression of the pacemaker activity of the SA node cells by the nonpacemaking atrial cells.^{15} ^{16} There may be a gradient in coupling in the SA node: in the periphery of the SA node, g_{j} may be higher than in the center.^{17} ^{18} Figure 3⇑ shows that even in this case (peripheral lattice with high g_{j} and central lattice with low g_{j}), the mosaic model cannot explain the difference in cycle length between peripheral and central tissue. The mosaic model is based on the assumptions that there are a substantial number of atrial cells in the SA node and that the properties of SA node cells are uniform. These assumptions may not be valid, however, because (1) atrial muscle abuts the SA node but is not connected to it,^{18} and it is important to remove this atrial muscle when isolating cells, and (2) in the study of Wilders et al,^{5} in which no relationship was found between electrical activity and cell size, “single” SA node cells with a capacitance (measure of cell size) >115 pF are included. Such high values indicate cell clusters and invalidate any conclusion. Unlike the mosaic model, the gradient model can account for the differences in the electrical activity of tissue isolated from the periphery and center of the SA node (Figure 1⇑). The gradient model can also account for the electrical activity and behavior of the intact SA node: Zhang et al^{11} ^{19} used their equations for peripheral and central rabbit SA node action potentials to construct a model of the intact SA node. In this model, the action potential was initiated in the center and propagated to the periphery and then onto the atrial muscle (as observed experimentally), and in response to various interventions, there was a shift of the leading pacemaker site (as observed experimentally).

- Received May 5, 2000.
- Revision received August 7, 2000.
- Accepted August 7, 2000.

- Copyright © 2001 by American Heart Association

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- Gradient Model Versus Mosaic Model of the Sinoatrial NodeH. Zhang, A. V. Holden and M. R. BoyettCirculation. 2001;103:584-588, originally published January 30, 2001https://doi.org/10.1161/01.CIR.103.4.584
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