Nano-scale solution of the Poisson-Nernst-Planck (PNP) equations in a fraction of two neighboring cells reveals the magnitude of intercellular electrochemical waves

Figures

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  Fig 1. Illustration of the computational domain. A. Illustration of the domain. We consider two intracellular domains Ωi (orange), two membrane domains Ωm (green) and an extracellular domain Ωe (pink). The left cell is referred to as the pre-junctional cell and the right cell is referred to as the post-junctional cell. B. Illustration of the channels embedded in the membrane, Ωc,K+ and Ωc,Na+ (purple). The width of the channels is wc in the y- and z-directions. Note that in our simulations, we typically consider clusters of channels, as illustrated in Fig 2. Note also that the illustrations in this figure are not drawn in scale, but the values of the lengths defining the geometry used in the simulations are specified in Table 1.

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  Fig 2. Illustration of the four different Na+ channel cluster sizes considered in our simulations. In the case of 4 channels, the channel cluster width is 2 nm, whereas in the case of 36, 196 and 900 channels, the width of the channel clusters are 6 nm, 14 nm and 30 nm, respectively.

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  Fig 3. Illustration of different parts of the domain boundary involved in the physiologically motivated boundary conditions (see Section 2.3.2) used in most of our simulations. In ∂ΩDD, i.e., in the boundary in the y- and z-directions for the x-values corresponding to the middle third of the extracellular cleft, we apply Dirichlet boundary conditions for both ϕ and all the ionic concentrations. In the simulations used to find the resting state of the system (see Section 2.2), we apply Neumann boundary conditions for ϕ and Dirichlet boundary conditions for the concentrations at ∂ΩN,1D and ∂ΩN,2D. In the simulations with open Na+ channels (see Section 2.2), we apply Neumann boundary conditions for ϕ and Dirichlet boundary conditions for the concentrations at ∂ΩN,1D and Dirichlet boundary conditions for both the concentrations and ϕ at ∂ΩN,2D. In the remaining boundary ∂ΩNN=∂Ω\(∂ΩN,1D∪∂ΩN,2D∪∂ΩDD), we apply Neumann boundary conditions for both ϕ and the ionic concentrations.

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  Fig 4. Illustration of an example mesh for Le = 15 nm and Li = 50 nm in the (x, y)- and the (x, z)-planes. The intracellular domain is colored orange, the extracellular domain is colored pink, the membrane is colored green and the ion channels are colored purple. In this figure, we consider K+ and Na+ channel clusters consisting of 2×2 channels of each type in the membrane of both cells. In the (x, y)-plane, the two types of ion channel are located as different locations (different y-values), whereas in the (x, z)-plane the two types of ion channel overlap because they are located at the same z-values. The mesh in the area close to the ion channels (indicated by dashed lines) is shown in more detail on the right side of the figure.

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  Fig 5. Illustration of the definition of the transmembrane potential, v, and the Na+ current, INa, used in our computations. The illustrations show a small part of the mesh in the (x, y)- or (x, z)-planes close to the Na+ channel (as illustrated on the right side of Fig 4).

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  Fig 6. Stationary solution of the potential, ϕ, the concentration of Na+, K+, Ca2+, and Cl− ions, and the charge density, ρ in the extracellular space between the two cells in simulations with open K+ channels, but closed Na+ channels. The width of the extracellular space, Le, is varied in the columns. The plots show the solution in the (x, y)-plane for the center of the domain in the z-direction at a point in time when steady state is reached. In the y-direction, we focus on the 50 nm closest to the K+ channels. The coordinates on the axes are shifted so that x = 0 marks the end of the membrane of the pre-junctional cell and y = 0 marks the center of the K+ channels. Note that to improve the visibility of the boundary layer, the scaling of the colorbar is different for the different cases.

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  Fig 7. Stationary solution of the potential, ϕ, the concentration of Na+, K+, Ca2+, and Cl− ions, and the charge density, ρ, along lines in the x-direction for open K+ channels and closed Na+ channels. The full green line represents the solution along a line crossing through the K+ channels and the dotted black line represents the solution along a line about 100 nm below the K+ channel cluster. The light green areas mark the membrane. Note that all ion concentrations are zero in the membrane, except for K+ ions in the K+ channel. The coordinates on the axes are shifted so that x = 0 marks the center of the extracellular cleft, and to improve the visibility, the plots only focus on a small part of the intracellular space, closest to the membrane.

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  Fig 8. The PNP model solution in the extracellular space between two cells in a simulation with an Na+ channel cluster of 196 channels on the membrane of the pre-junctional cell (the channels are opened at t = 0). The width of the extracellular space, Le, is 10 nm. The plots show the solution in the (x, y)-plane for the center of the domain in the z-direction at five different points in time (specified in the column titles). In the y-direction, we focus on the 50 nm closest to the Na+ channel clusters. The coordinates on the axes are shifted so that x = 0 marks the end of the membrane of the pre-junctional cell and y = 0 marks the center of the Na+ channel cluster. Note that the short duration of the dynamics is an artefact caused by the small membrane area associated with the Na+ channel cluster in the simulation (see Sections 3.2.3–3.2.5 for more details and an estimation of a more realistic duration).

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  Fig 9. The potential, ϕ, in the extracellular space between the two cells in simulations with open Na+ channel clusters on the membrane of the pre-junctional cell. The width of the extracellular space, Le, and the size of the Na+ channel cluster is varied in the columns and rows of the figure, respectively. The plots show the solution in the (x, y)-plane for the center of the domain in the z-direction at the point in time when the deviation from rest is largest. This time point (defined as the time after the Na+ channel is opened) is specified in the upper right corner of each plot. In the y-direction, we focus on the 50 nm closest to the Na+ channel clusters. The coordinates on the axes are shifted so that x = 0 marks the end of the membrane of the pre-junctional cell and y = 0 marks the center of the Na+ channel cluster. Note that the scaling of the colorbar is different for the different cases.

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  Fig 10. The Na+ concentration in the extracellular space between the two cells in simulations with open Na+ channel clusters on the membrane of the pre-junctional cell. The figure setup is the same as for Fig 9.

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  Fig 11. The K+ concentration in the extracellular space between the two cells in simulations with open Na+ channel clusters on the membrane of the pre-junctional cell. The figure setup is the same as for Fig 9.

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  Fig 12. Largest deviations from rest in the extracellular potential and ion concentrations outside of the Na+ channel clusters in simulations with different widths of the extracellular space between the cells, Le, and different sizes of the Na+ channel clusters. The full lines show the solution outside of the Na+ channel cluster of the pre-junctional cell, which has open Na+ channels. The dotted lines show the solution on the other side of the extracellular space, outside of the post-junctional cell, which has closed Na+ channels.