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Triangular mesh of the RBC drawn to scale in its biconcave-disc, using Eq. 1 in Mathematica. The triangularization was with BoundaryDiscretizationRegion, applied to ImplicitRegion (Supplementary Information). Number of edges = 120,042 and MeshCoordinates gave 40,136 points and 80,268 triangles. Green colouring was chosen over red (the natural choice for an RBC) for ease of visualizing the mesh. Notebook 2 has the Mathematica script used to generate this graphic.
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The highly deformable red blood cell (erythrocyte; RBC) responds to mechanically imposed shape changes with enhanced glycolytic flux and cation transport. Such morphological changes are produced experimentally by suspending the cells in a gelatin gel, which is then elongated or compressed in a special apparatus inside an NMR spectrometer. However,...
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... mathematical expression for the RBC discocyte used here is close to those in [20], and it is a continuous degree-4 surface that can be written either in Cartesian or disc-cyclide coordinates, making it versatile for numerical exploration [15]. The shape is constrained by three principal distances, the main diameter, d, the thickness at the centre of the dimples, b, and the maximum thickness (height) near the periphery of the cell, h (see Figure 1 for ...
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... coordinates of the nodes of the mesh, and the list of all triangles, were obtained with the functions MeshCoordinates and MeshPrimitives. Finally, MeshTriangles was plotted by using Graphics3D (Figure 1). ...
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... shapes of these surfaces can be illustrated with the particular example of the hyperbolic paraboloid (saddle) [16]. Such a graphical rendering is shown in Supplementary Information, Figure S1. The observation of, in general, two Principal Curvatures motivates the implementation of expressions that describe the curvature of the surface at a given point on the RBC. ...
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... at the three vertices: The next step after triangularization (as shown in Figure 1) was to assign the values of curvature to each triangular face. This was done by applying Eqs. ...
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... length: The next step was to compute the distribution of edge lengths, in order to study how these edges, which could be thought of as modelling the spectrin network, might imply that the spectrin is either stretched or compressed. Figure 2 shows the distribution of sides in Figure 1, first at a resolution of 20 linearly space bins. The distribution is slightly skewed to the left, but it is unimodal; however, the inset which was based on 200 bins now appears to be at least trimodal (see Discussion). of the total bin contents and spanned 60 -65 nm; while the two most abundant bins spanning 60 -70 nm contained 75% of the edge lengths. ...
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... area distribution: Another feature of the surface triangularization, that is of biophysical importance, is the area of each triangle and whether this is consistent with the known span of membrane proteins that are corralled in the network. Figure S2a shows the separate triangularization of the RBC which (as for Figure 1) clearly indicates a range of shapes and sizes of the triangles. Sorting the triangles according to area showed a span from 0.0094 to 3431 nm 2 . ...
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... area of each triangle was computed (Theory of Methods; and Notebook 2) so the total area occupied by triangles in a given bin was able to be expressed as a fraction of the total RBC area, 128 m 2 . For speed of computation the triangularization was made with fewer mesh points than for Figure 1, specifically 13640 triangles and 6822 mesh points. ...
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... correlations could be made with experimental measurements like those already reported on stretched/compressed gels [1,2], and in electrophysiological measurements on whole RBCs. Figure 1 Triangular mesh of the RBC drawn to scale in its biconcave-disc, using Eq. 1 in Mathematica. The triangularization was with BoundaryDiscretizationRegion, applied to ImplicitRegion (Supplementary Information). ...
