Intercalated disk nanoscale structure regulates cardiac conduction

Figure 1.. ID structure and FEM representation.(A) Scanning electron microscopy (SEM) image of the ID from monkey ventricular myocyte. S and R indicate “steps” and “risers,” denoting the plicate and interplicate regions, respectively. Reproduced from Zhang et al. (1996) with permission. (B) TEM image of mouse ventricular myocyte ID, illustrating the plicate and the interplicate regions. (C) View of the 3-D FEM model, with plicate and interplicate regions and GJs highlighted. (D) Slice through the FEM mesh, illustrating a corresponding view to the 2-D TEM image. The sections in the gray boxes are enlarged in E. (E) Magnified view of TEM images and FEM mesh slices, showing structural details for the plicate folds (top) and the GJs (bottom).

Figure S1.. Illustration of the mesh generation (left) and cleft conductance (right) calculation algorithms.

Figure 2.. Outline of FEM mesh generation algorithm. The ID map defines two levels of plicate regions, such that the interplicate region mesh is defined by the region separating the two levels. The cell–cell junction distribution determines a GJ map for both plicate and interplicate regions, which defines the locations of the two membranes connecting. A 2-D sine wave function replicates the folds in the plicate membranes. A series of connecting meshes join the plicate and the interplicate region membranes, ensuring that the full mesh encloses a continuous volume (i.e., the mesh is “waterproof”). Combining all of these mesh components generates the cleft mesh.

Figure S2.. Map generation algorithm and GJ distribution calculations.(A) Maps are generated by drawing a 2-D array of Gaussian random noise (left). The noise is then smoothed (center) and thresholded (right) to generate a map (for either the ID or GJ distribution). (B) Plicate and interplicate GJ distributions. Left: GJ cluster examples, illustrating GJs (yellow) and membrane (blue). Right: Cluster size distributions for the plicate and interplicate regions. (C) Parameter study for GJ map algorithm parameters (μ and σ, see supplemental text at the bottom of the PDF for details), illustrating the parameter dependence of cluster membrane area fractions, median cluster area, and mean cluster area. The red + and X denote the plicate and interplicate distribution parameters, respectively.

Figure S3.. Fitting cluster length measurements to 2-D maps. In both left panels, each of the red lines represent a slice, or a 1-D sample, of the 2-D map, with the associated 1-D slices plotted on the right. (A) GJ length measurements are obtained by averaging the length of all continuous GJ segments over all slices (yellow, value of 1). (B) Plicate length measurements are obtained by averaging the length of all continuous plicate segments over all slices, from both levels (blue or yellow, h = 0 or 1).

Figure 3.. Intercellular cleft conductivity and tissue model with ID nanodomains.(A) Partitioning of the FEM mesh into 200 partitions or compartments, with different colors denoting different compartments. Overlaid on the mesh is a partial sketch of the reduced electrical network. (B) Example of adjacent partitions remeshing for equivalent conductance calculation. Partition centroids (orange circles), connecting line (solid black line), and cut lines (dashed black lines) are shown. (C) Solution to Laplace’s equation (Eq. 1) on the shared volume. (D) Resulting reduced electrical network for 200 nodes and the corresponding connections between adjacent partitions. Connection thickness is proportional to the conductance value, and edges in plicate or interplicate regions are shown as orange and blue, respectively. (E) Solution to Laplace’s equation on the full-cleft FEM mesh. (F) Comparison between voltages on the full-cleft FEM mesh and reduced network results. (G) Schematic of the electrical circuit (Eq. S2) and illustration of the corresponding electrical network at the ID, equivalent to the network shown in D.

Figure S4.. Illustration of Laplace’s equation on a regular geometry and uniform material. The problem has two Dirichlet (fixed) boundary conditions on apposing faces, with area A and separated by distance L: 0 V on the left face (highlighted in blue) and 1 V on the right face (red). All other boundaries have no-flux conditions. The gray arrows represent the direction of the electric field or current flux.

Figure S5.. Error of the cleft network approximation of the Laplace’s equation solution on the full 3-D cleft FEM mesh. The RMSD between the voltage from the cleft network and 3-D FEM mesh is shown as a function of the number of partitions (see also Fig. 3, D–F). Each data point is the mean ± SEM of five different reduced cleft network/FEM mesh comparisons.

Figure S6.. Schematic of the electrical circuits for the tissue models compared in the study.(A) Standard monodomain model; the cells are directly connected through the GJ resistor ggap. (B) Single-cleft model: In addition to GJ coupling, a single intercellular cleft node mediated ephaptic coupling between adjacent cells at the ID. (C) The extracellular/intercellular cleft space is modeled as a network of nodes, coupled via electrical resistors with conductance values derived from a FEM mesh (Fig. 3).

Figure 4.. Illustration of multiple-cleft FEM meshes and cleft conductance calculations.(A) Five example ID maps, illustrating variable but similar overall geometries. (B and C) Corresponding FEM meshes (B) and slices through the meshes above (C), illustrating variation in the lengths of plicate and interplicate regions, and GJs. (D) Distribution of cleft conductances (gC) in each mesh, for M = 200 partitions. The black line represents the mean conductance for each mesh. Parameters: plicate membrane distance dP = 30 nm, interplicate membrane distance dIP = 15 nm.

Figure 5.. Interplicate and plicate region conductances.(A) Distribution of conductances in the interplicate region (gIp), with values from the five meshes in Fig. 4. (B) Interplicate conductances in the presence and absence of GJs. (C) Interplicate conductance distributions for increasing interplicate intermembrane distance dIP. (D) Mean interplicate conductance (gIp) and mean plicate conductance (gP) are shown as a function of dIp. Note that the blue line values correspond to the vertical lines in C. (E) Distribution of conductances in the plicate region (gP), with values from the five meshes in Fig. 4. (F) Plicate conductances in the presence and absence of plicate folds. (G) Plicate conductance distributions for increasing plicate intermembrane distance dP. (H) gIp and gP are shown as a function of dP. Note that the orange line values correspond to the vertical lines in G. In A–C and E–G, vertical lines denote distribution means. Parameters: A–D, dP = 30 nm; E–H, dIp = 15 nm.

Figure S7.. Interplicate and plicate region conductances for different FEM mesh variants.(A) Interplicate conductance gIp distribution for varying plicate distances dP. (B) g _ Ip distribution for the presence or absence of plicate folds. (C) Plicate conductance gP distribution for varying interplicate distances dIp. (D) gP distribution for the presence or absence of GJs.