Abstract
The shapes of phospholipid bilayer biomembranes are modeled by the celebrated Canham-Evans-Helfrich model as constrained Willmore minimizers. Several numerical treatments of the model have been proposed in the literature, one of which was used extensively by biophysicists over two decades ago to study real lipid bilayer membranes. While the key ingredients of this algorithm are implemented in Brakke’s well-known surface evolver software, some of its glory details were never explained by either the geometers who invented it or the biophysicists who used it. As such, most of the computational results claimed in the biophysics literature are difficult to reproduce. In this note, we give an exposition of this method, connect it with some related ideas in the literature, and propose a modification of the original method based on replacing mesh smoothing with harmonic energy regularization. We present a theoretical finding and related computational observations explaining why such a smoothing or regularization step is indispensable for the success of the algorithm. A software package called WMINCON is available for reproducing the experiments in this and related articles.
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Notes
- 1.
We assume that the normal of any closed orientable surface points outward. In particular, it means H < 0 for a sphere.
- 2.
The paper [17] derived the same formula based on the characterization of mean curvature by the Laplace-Beltrami operator: Δ S X(x) = H(x), x ∈ S, where \(\mathbf {X}: S \rightarrow {\mathbb {R}}^3\) is the position function of the surface S.
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Acknowledgements
TY thanks Tom Duchamp, Robert Kusner, Shawn Walker and Aaron Yip for extensive discussions. This work is partially supported by NSF grants DMS 1115915 and DMS 1522337. We also thank the support of the Office of the Provost and the Steinbright Career Development Center of Drexel University.
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Brogan, J.P., Yang, Y., Yu, T.P.Y. (2019). Numerical Methods for Biomembranes Based on Piecewise Linear Surfaces. In: Radu, F., Kumar, K., Berre, I., Nordbotten, J., Pop, I. (eds) Numerical Mathematics and Advanced Applications ENUMATH 2017. ENUMATH 2017. Lecture Notes in Computational Science and Engineering, vol 126. Springer, Cham. https://doi.org/10.1007/978-3-319-96415-7_76
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