Geometric modeling of subcellular structures, organelles, and multiprotein complexes

Abstract

SUMMARY Recently, the structure, function, stability, and dynamics of subcellular structures, organelles, and multi-protein complexes have emerged as a leading interest in structural biology. Geometric modeling not only provides visualizations of shapes for large biomolecular complexes but also fills the gap between structural information and theoretical modeling, and enables the understanding of function, stability, and dynamics. This paper introduces a suite of computational tools for volumetric data processing, information extraction, surface mesh rendering, geometric measurement, and curvature estimation of biomolecular complexes. Particular emphasis is given to the modeling of cryo-electron microscopy data. Lagrangian-triangle meshes are employed for the surface presentation. On the basis of this representation, algorithms are developed for surface area and surface-enclosed volume calculation, and curvature estimation. Methods for volumetric meshing have also been presented. Because the technological development in computer science and mathematics has led to multiple choices at each stage of the geometric modeling, we discuss the rationales in the design and selection of various algorithms. Analytical models are designed to test the computational accuracy and convergence of proposed algorithms. Finally, we select a set of six cryo-electron microscopy data representing typical subcellular complexes to demonstrate the efficacy of the proposed algorithms in handling biomolecular surfaces and explore their capability of geometric characterization of binding targets. This paper offers a comprehensive protocol for the geometric modeling of subcellular structures, organelles, and multiprotein complexes.

Multiscale geometric modeling of macromolecules I: Cartesian representation

“…To set up the initial conditions, two domains are defined, one is D χ representing the domain enclosed by the van der Waals surface; the other is an enlarged domain D : $$D_{\chi }={\cup }_{i=1}^{n}false\{\mathrm{boldx}:false|\mathrm{boldx}-{\mathbf{x}}_{i}false|<r_{i}false\};$$ $$D={\cup }_{i=1}^{n}false\{\mathrm{boldx}:false|\mathrm{boldx}-{\mathbf{x}}_{i}false|<1.3r_{i}false\}.$$ Here we choose a factor of 1.3 to guarantee the formation of properly connected surfaces. In fact, this special parameter can be adjusted to give different scales of the molecular surface, which may lead to dramatically different geometric features [22]. We denote S as the Cartesian representation of the hypersurface.…”

“…We can control the process by three parameters, the integration time t , the PDE order q , and the external term P ( u , |∇ u |). For the protein surface generation from the PDB, a proper combination of the PDE order and integration time is required to deliver high-quality surface [81, 22, 23]. If the solvation process is considered, the potential driven term should incorporate both the nonpolar and polar effects.…”

“…In our MMS and surface generated from geometric and potential driven geometric flows, the geometric singularities are removed, and the surface is smooth with less local curvature uctuations. Further, a multiresolution model is proposed in our recent work [22]. Obtained with an adjustable parameter, a family of multiresolution surfaces can be designed to reduce local curvature variations.…”

“…Geometric modeling not only bridges the gap between biomolecular data and biological conceptualization and interpretation, but also provides a basis for mathematical modeling, analysis and computation [78, 22]. In 1953, Corey and Pauling proposed the atom and bond model of molecules [18], which has since become a cornerstone in physical science.…”

New software developments for quality mesh generation and optimization from biomedical imaging data

“…One of the challenges in using the FEM, however, is to generate high-quality meshes on which the simulations are performed. In order to obtain more accurate simulation results, it is also practically important to incorporate realistic biological structures into geometric mesh models [11, 14, 29, 48, 49, 56]. To this end, the goal of the present paper is to construct high-quality geometric mesh models from 3D biomedical imaging data.…”