Owing to their intriguing geometric diversity and importance to cellular phenomena, membranes are a main focus in cellular mechanics. The geometry and shape changes of a cell's many membranes support many critical processes. Complementing the challenge of targeted experimental perturbation of membranes, theoretical modeling gives us insights into the underlying mechanical principles governing membrane systems. Computational experiments have emerged as an important method to test hypotheses for interpreting experimental measurements in a mechanical context. With the recent advances in ultrastructural imaging, segmentation, and surface reconstruction, an extensible 3D membrane simulation framework that uses the discrete surface representation as a starting point is needed.
The most common continuum model to describe membrane deformations is the Helfrich-Canham-Evans Hamiltonian, which describes the lipid bilayer as a two-dimensional fluid shell with a purely geometric Hamiltonian. The theory of discrete differential geometry (DDG) is an emerging field in applied mathematics that studies discrete analogs (e.g., meshes) of smooth geometric objects. The theory has found wide application in the broader field of digital geometry processing and visual computing. Concepts from DDG are also used by Pixar, Disney, and Dreamworks among others [e.g., a scene in Tangled (2010) featuring long hair floating in water] and the broader animated movie industry to bring mesh geometries to life. Rather than viewing the discrete object as an approximation of the smooth object, DDG focuses on constructing a geometric theory directly on discrete polygons. In our work, we take the discrete analog of the Helfrich-Canham-Evans Hamiltonian and derive the corresponding discrete forces by using the formalism of DDG. We find that the final discrete force expressions have a clear correspondence with the well-studied smooth theory leading to a unifying perspective.
The cover image of the September 14 issue of Biophysical Reports is a computer rendering using the software Houdini. The paper in the rendering shows the underlying DDG theory behind our software implementation, Mem3DG, and how the theory contributes to the simulation of vesicular budding driven by protein selective binding and aggregation, depicted by the hand-drawn schematics. A floating metallic ring, casting a shadow upon the end state of vesiculation, symbolizes the realization of the mathematical theory through 3D computer simulation. The large structure is a snapshot of the simulation trajectory. The polygonal wire shows the triangulated mesh used in the simulation, while the contrast between the glassy surface and polygonal cage highlights the unifying perspective from our theory and the idea of formulating multi-physics models on triangulated meshes.
Through our study and this cover, we hope to decrease the gap between our modeling capabilities and the frontier of experimental methods. This initial work establishing the Mem3DG framework will be of immediate interest to purveyors of membrane biomechanics, those who seek to develop new models and methods to facilitate the research. An emerging user will be practitioners of ultrastructural characterization methods such as electron tomography. We envision that Mem3DG can enable them to interrogate hypotheses related to the mechanochemical implications of their scenes of interest. For example, these tools can be used to investigate the mechanics of endocytosis and membrane curvature generation while drawing a direct comparison with 3D microscopy images and reconstructions (see Figure 7 in the paper for budding from vesicles).
If you are interested in finding out more about our studies in membrane mechanics and how cell shape influences cellular functions, please visit https://sites.google.com/eng.ucsd.edu/prangamani.
- Cuncheng Zhu, Christopher T. Lee, and Padmini Rangamani