VoroCrust: Voronoi Meshing Without Clipping.

Polyhedral meshes are increasingly becoming an attractive option with particular advantages over traditional meshes for certain applications. What has been missing is a robust polyhedral meshing algorithm that can handle broad classes of domains exhibiting arbitrary curved boundaries and sharp features. In addition, the power of primal-dual mesh pairs, exemplified by Voronoi-Delaunay meshes, has been recognized as an important ingredient in numerous formulations.

Sampling Conditions for Conforming Voronoi Meshing by the VoroCrust Algorithm.

We study the problem of decomposing a volume with a smooth boundary into a collection of Voronoi cells. Unlike the dual problem of conforming Delaunay meshing, a principled solution to this problem for generic smooth surfaces remained elusive. VoroCrust leverages ideas from weighted -shapes and the power crust algorithm to produce unweighted Voronoi cells conforming to the surface, yielding the first provably-correct algorithm for this problem.

Statistical Framework for Uncertainty Quantification in Computational Molecular Modeling.

As computational modeling, simulation, and predictions are becoming integral parts of biomedical pipelines, it behooves us to emphasize the reliability of the computational protocol. For any reported quantity of interest (QOI), one must also compute and report a measure of the uncertainty or error associated with the QOI. This is especially important in molecular modeling, since in most practical applications the inputs to the computational protocol are often noisy, incomplete, or low-resolution.

Viral Capsid Assembly: A Quantified Uncertainty Approach.

Most of the existing research in assembly pathway prediction/analysis of viral capsids makes the simplifying assumption that the configuration of the intermediate states can be extracted directly from the final configuration of the entire capsid. This assumption does not take into account the conformational changes of the constituent proteins as well as minor changes to the binding interfaces that continue throughout the assembly process until stabilization. This article presents a statistical-ensemble-based approach that samples the configurational space for each monomer with the relative local orientation between monomers, to capture the uncertainties in binding and conformations.

Disk Density Tuning of a Maximal Random Packing.

We introduce an algorithmic framework for tuning the spatial density of disks in a maximal random packing, without changing the sizing function or radii of disks. Starting from any maximal random packing such as a Maximal Poisson-disk Sampling (MPS), we iteratively relocate, inject (add), or eject (remove) disks, using a set of three successively more-aggressive local operations. We may achieve a user-defined density, either more dense or more sparse, almost up to the theoretical structured limits.

All-Hex Meshing of Multiple-Region Domains without Cleanup.

In this paper, we present a new algorithm for all-hex meshing of domains with multiple regions without post-processing cleanup. Our method starts with a strongly balanced octree. In contrast to snapping the grid points onto the geometric boundaries, we move points a slight distance away from the common boundaries.

Computational reconstitution of spine calcium transients from individual proteins.

We have built a stochastic model in the program MCell that simulates Ca(2+) transients in spines from the principal molecular components believed to control Ca(2+) entry and exit. Proteins, with their kinetic models, are located within two segments of dendrites containing 88 intact spines, centered in a fully reconstructed 6 × 6 × 5 μm(3) cube of hippocampal neuropil. Protein components include AMPA- and NMDA-type glutamate receptors, L- and R-type voltage-dependent Ca(2+) channels, Na(+)/Ca(2+) exchangers, plasma membrane Ca(2+) ATPases, smooth endoplasmic reticulum Ca(2+) ATPases, immobile Ca(2+) buffers, and calbindin.

Robust All-quad Meshing of Domains with Connected Regions.

In this paper, we present a new algorithm for all-quad meshing of non-convex domains, with connected regions. Our method starts with a strongly balanced quadtree. In contrast to snapping the grid points onto the geometric boundaries, we move points a slight distance away from the common boundaries.

Extracellular sheets and tunnels modulate glutamate diffusion in hippocampal neuropil.

Although the extracellular space in the neuropil of the brain is an important channel for volume communication between cells and has other important functions, its morphology on the micron scale has not been analyzed quantitatively owing to experimental limitations. We used manual and computational techniques to reconstruct the 3D geometry of 180 μm(3) of rat CA1 hippocampal neuropil from serial electron microscopy and corrected for tissue shrinkage to reflect the in vivo state. The reconstruction revealed an interconnected network of 40-80 nm diameter tunnels, formed at the junction of three or more cellular processes, spanned by sheets between pairs of cell surfaces with 10-40 nm width.

Single-particle reconstruction using L(2)-gradient flow.

In this paper, we present an iterative algorithm for reconstructing a three-dimensional density function from a set of two dimensional electron microscopy images. By minimizing an energy functional consisting of a fidelity term and a regularization term, an L(2)-gradient flow is derived. The flow is integrated by a finite element method in the spatial direction and an explicit Euler scheme in the temporal direction.

CRYO-ELECTRON MICROSCOPY DATA DENOISING BASED ON THE GENERALIZED DIGITIZED TOTAL VARIATION METHOD.

The energy functional used in digitalized total variation method is expanded to a general form and a generalized digitized total variation (GDTV) denoising method is obtained. We further expand this method from 2-dimensional (2D) image to 3-dimensional (3D) image processing field. Cryo-electron microscopy (cryo EM) and single particle reconstruction are becoming part of standard collection of structural techniques used for studying macromolecular assemblies.

Quality Multi-domain Meshing for Volumetric Data.

Multi-domain meshing from volumetric data is of great importance in many fields like medicine, biology and geology. This paper proposes a new approach to produce a high quality mesh with separated multiple domains. A point cloud is generated from a preliminary mesh representing the boundary between different domains from the discrete volumetric representation used as input.

An Automatic 3D Mesh Generation Method for Domains with Multiple Materials.

This paper describes an automatic and efficient approach to construct unstructured tetrahedral and hexahedral meshes for a composite domain made up of heterogeneous materials. The boundaries of these material regions form non-manifold surfaces. In earlier papers, we developed an octree-based isocontouring method to construct unstructured 3D meshes for a single-material (homogeneous) domain with manifold boundary.

A Fast Variational Method for the Construction of Resolution Adaptive C-Smooth Molecular Surfaces.

We present a variational approach to smooth molecular (proteins, nucleic acids) surface constructions, starting from atomic coordinates, as available from the protein and nucleic-acid data banks. Molecular dynamics (MD) simulations traditionally used in understanding protein and nucleic-acid folding processes, are based on molecular force fields, and require smooth models of these molecular surfaces. To accelerate MD simulations, a popular methodology is to employ coarse grained molecular models, which represent clusters of atoms with similar physical properties by psuedo- atoms, resulting in coarser resolution molecular surfaces.

Bio-molecule Surfaces Construction via a Higher-Order Level-Set Method.

We present a general framework for a higher-order spline level-set (HLS) method and apply this to bio-molecule surfaces construction. Starting from a first order energy functional, we obtain a general level set formulation of geometric partial differential equation, and provide an efficient approach to solve this partial differential equation using a C(2) spline basis. We also present a fast cubic spline interpolation algorithm based on convolution and the Z-transform, which exploits the local relationship of interpolatory cubic spline coefficients with respect to given function data values.

Application of new multi-resolution methods for the comparison of biomolecular electrostatic properties in the absence of global structural similarity.

In this paper we present a method for the multi-resolution comparison of biomolecular electrostatic potentials without the need for global structural alignment of the biomolecules. The underlying computational geometry algorithm uses multi-resolution attributed contour trees (MACTs) to compare the topological features of volumetric scalar fields. We apply the MACTs to compute electrostatic similarity metrics for a large set of protein chains with varying degrees of sequence, structure, and function similarity.

Three-dimensional geometric modeling of membrane-bound organelles in ventricular myocytes: bridging the gap between microscopic imaging and mathematical simulation.

A general framework of image-based geometric processing is presented to bridge the gap between three-dimensional (3D) imaging that provides structural details of a biological system and mathematical simulation where high-quality surface or volumetric meshes are required. A 3D density map is processed in the order of image pre-processing (contrast enhancement and anisotropic filtering), feature extraction (boundary segmentation and skeletonization), and high-quality and realistic surface (triangular) and volumetric (tetrahedral) mesh generation. While the tool-chain described is applicable to general types of 3D imaging data, the performance is demonstrated specifically on membrane-bound organelles in ventricular myocytes that are imaged and reconstructed with electron microscopic (EM) tomography and two-photon microscopy (T-PM).

Patient-Specific Vascular NURBS Modeling for Isogeometric Analysis of Blood Flow.

We describe an approach to construct hexahedral solid NURBS (Non-Uniform Rational B-Splines) meshes for patient-specific vascular geometric models from imaging data for use in isogeometric analysis. First, image processing techniques, such as contrast enhancement, filtering, classification, and segmentation, are used to improve the quality of the input imaging data. Then, lumenal surfaces are extracted by isocontouring the preprocessed data, followed by the extraction of vascular skeleton via Voronoi and Delaunay diagrams.

AUTOMATIC STRUCTURE INTERPRETATION OF SINGLE PARTICLE CRYO-ELECTRON MICROSCOPY: FROM IMAGES TO PSUEDO-ATOMIC MODELS.

Three dimensional Electron Microscopy (EM) and in particular single particle reconstruction using cryo-EM, has rapidly advanced over recent years, such that increasingly several macromolecular complexes can be resolved at subnanometer resolution (6-10 Å). This paper reviews some of the main volumetric image and geometric post-processing steps once a three dimensional EM map (henceforth a 3D map) has been reconstructed from single particle Cryo-EM, as essential steps in an enhanced and automated computational structure interpretation pipeline. In particular the paper addresses automated filtering, critical point calculations, symmetric and non-symmetric molecular domain segmentation, molecular surface selection, curation, and protein secondary structure (α- helices and β-sheets) elucidation from 3D maps.

Finite element analysis of the time-dependent Smoluchowski equation for acetylcholinesterase reaction rate calculations.

This article describes the numerical solution of the time-dependent Smoluchowski equation to study diffusion in biomolecular systems. Specifically, finite element methods have been developed to calculate ligand binding rate constants for large biomolecules. The resulting software has been validated and applied to the mouse acetylcholinesterase (mAChE) monomer and several tetramers.

Discrete Surface Modelling Using Partial Differential Equations.

We use various nonlinear partial differential equations to efficiently solve several surface modelling problems, including surface blending, N-sided hole filling and free-form surface fitting. The nonlinear equations used include two second order flows, two fourth order flows and two sixth order flows. These nonlinear equations are discretized based on discrete differential geometry operators.

Continuum diffusion reaction rate calculations of wild-type and mutant mouse acetylcholinesterase: adaptive finite element analysis.

As described previously, continuum models, such as the Smoluchowski equation, offer a scalable framework for studying diffusion in biomolecular systems. This work presents new developments in the efficient solution of the continuum diffusion equation. Specifically, we present methods for adaptively refining finite element solutions of the Smoluchowski equation based on a posteriori error estimates.

Finite element solution of the steady-state Smoluchowski equation for rate constant calculations.

This article describes the development and implementation of algorithms to study diffusion in biomolecular systems using continuum mechanics equations. Specifically, finite element methods have been developed to solve the steady-state Smoluchowski equation to calculate ligand binding rate constants for large biomolecules. The resulting software has been validated and applied to mouse acetylcholinesterase.