Fig 1.
Error in Helfrich energy computation on meshes with variable quality.
(A) Percentage difference ΔH between mesh calculated Helfrich energy and theoretical Helfrich energy on a sphere with optimal mesh quality (mesh generated by using [57]). (B) ΔH under minor distortion to the mesh. (C) ΔH under significant distortion. (D) Histogram of ΔH for (A-C). Percentage error ΔH = [H(i) − H(i)]/H(i) × 100% comparing the computed, discrete value of the Helfrich energy H(i) at each vertex i (see Methods) to the exact value H(i) = 8πκ/Nv, where 8πκ is the total Helfrich energy of an ideal sphere and Nv is the number of vertices.
Fig 2.
Remeshing controlled by barrier crossings in a double-barrier potential.
(A) Remeshing via flipping procedures. (B) Remeshing via splitting and merging procedures controlled by barrier crossings in a potential Vin at indicated lengths. (C) Initial triangular meshes and high-quality triangular mesh after relaxation. (D) Triangular mesh element: right-hand rule. (E) Definition of 1-ring neighbors nb(i) and Voronoi area Av in purple [30]. ri and rj are the vectors defined by vertex i and j, and θij and ϕij are the two angles for computing the discrete Helfrich energy at i Eq (5).
Table 1.
Parameter values.
Fig 3.
Morphodynamics of a biconcave red blood cell and a vesicle fusion.
(A) Biconcave morphology at equilibrium (5 × 104 iterations). Insert: down-sized side view. The size of the simulated cell is comparable to the typical size of a red blood cell at 6−8μm in width. See also S1 Video. (B) Vesicle fusion at three stages. 5 × 104 iterations at equilibirum stage. See also S2 Video. (C) Number of remeshing for geometry- (Geo) and free energy- (FE) based method performing vesicle fusion. (D) Standard deviation of coordinates of remeshing events for Geo- and FE- based method for 2 × 104 iterations. See also S3 Video.
Fig 4.
Applications of the proposed method to generate high-quality meshes of salient geometries under external forces.
(A) Definition of external control points attracting the nearest mesh vertex i and i’s 1-ring neighbors nb(i). (B) Formation of a filopodium by placement of a control point at the tip and control points holding the remainder of the globular cell in place. (C) Formation of a lamellipodium by placement of control points in a circular rim and control points holding the remainder of the globular cell in place. (D) Formation of an invagination by placement of a control point 1/3 into the globular cell and control points holding the remainder of the cell in place. Matching videos are provided in S4–S6 Videos (2 × 104 iterations).
Fig 5.
Applications of the proposed method to study mechanical coupling/decoupling of membrane tethers.
(A) Association of splitting and merging to redistributing lipids, and lipid diffusion between neighbors. (B) Initiation and coupling of two tethers under naive membrane condition. (C) Initiation and coupling of two tethers with diffusion barrier. (D) Initiation and coupling of two tethers with diffusion barrier and membrane reservoir. (E) Fourth-order polynomial fitting of the tether pulling under the three membrane conditions. Radius is normalized to the value of the initial morphology. Circles: last values of radius of left and right tethers obtained from the fitted curves. (F) Scan results of last value of the right tether under six different kd for the three membrane conditions. (G) Potential topological defect and possible solution.
Table 2.
Computational cost.