Tetrahedral Finite Element Meshing for Biomolecules
Yongjie (Jessica) Zhang*, Chandrajit L. Bajaj*, Zeyun Yu*, Yuhua Song†, Deqiang Zhang‡, Nathan A. Baker†, J. Andrew McCammon‡
†Biochem
*ICES & CS, Univ. of Texas at Austin
1. Introduction
3. Data Acquisition
This poster describes an approach to generate adaptive and
quality tetrahedral meshes for biomolecules from PDB/PQR
or cryoEM data. First we construct electron density maps
from PDB/PQR files, or segment cryoEM data to identify
different subunits. Secondly, we develop a variant of the dual
contouring method to generate interior and exterior meshes.
The mesh adaptivity is determined by specific structural
properties in order to preserve features while minimizing the
number of elements. A parallel algorithm is designed for large
datasets, and different substructures are meshed from
segmented cryoEM data. Thirdly, exterior meshes are
extended to a large boundary. Finally, the mesh quality is
improved to satisfy the requirements of finite element
calculations. Some of our generated meshes, e.g., monomer
and tetrameric mouse acetylcholinesterase (mAChE), have
been successfully used in finite element simulations.
Two methods are adopted to convert PDB/PQR into electron density
maps: the multi-grid method and the blurring method. Based on
properties of cryoEM data, a fast marching method is selected to
distinguish its different subunits.
‡Chem
& Mol Biophy, Washington Univ. in St. Louis
3.1 PDB/PQR Data
The multi-grid method: A characteristic function f(x) is selected to
represent an “inflated” van der Waals-based accessibility.
The blurring method: Blinn modeled electron density maps for molecules
using the summation of Gaussian density distributions:
3.2 cryoEM Data – segmentation [5]
The detection of critical points.
The
detection
of
icosahedral
symmetry axes.
The segmentation of asymmetric
subunits – a variant of the fast marching
method.
4.1 A Variant of Dual Contouring Method
The dual contouring method has been extended to extract tetrahedral
meshes from volumetric scalar field [1][2]. Sign change edge, interior
edge/face in boundary cell and interior cell all need to be analyzed.
Differently, here we only consider two kinds of edges: sign change
edges and interior edges.
4.2 Parallel Meshing Algorithm
First we divide the volumetric data into small ones, then each subvolume is assigned to a processor and is meshed in parallel. For those
sign change edges and interior edges lying on the interface between
subvolumes, we analyze them separately. Finally, we merge the
generated meshes together.
4.3 Subunit Identification
The segmented cryoEM data is divided into several intervals, each of
which contains one subunit. So we make a sequence of these subunits,
and assign a material index to each grid point indicating which subunit
it belongs to. The material index is used to detect the boundary for each
classified or segmented subunit domain.
5. Mesh Extension
& Biochem, UCSD
6. Quality Improvement
Three metrics: edge-ratio, Joe-Liu parameter and min volume
bound, are used to measure mesh quality.
Isolated vertices/components removal – connectivity
Overlap situation detection and removal
Edge-ratio improvement – edge contraction
Joe-Liu parameter and min volume – smoothing
Joe-Liu parameter:
7. Application
•  -- whole domain
•  -- biomolecule domain
•  -- free space in 
• a – reactive region
• r – reflective region
• b – boundary for 
• PDB/PQR – multi-grid method
• PDB/PQR – blurring method
• Segmented cryoEM Data
The generated
meshes of
mAChE have been successfully
used in solving the steady-state
Smoluchowski equaiton [3][4].
Schematic of problem domain [3]


p( r )  pbulk for r  b


(2) p(r )  0 (Dirichlet BC) for r  a
  
 
or n  Jp(r )   (r ) p(r ) (Robin BC)
(1)
(3)
  
n  Jp (r )  0 for x  r
Diffusion rate:
k

a
  
n  Jp( r )dS
pbulk





p (r , t )
   D(r )[p (r )   p (r )U (r )]  0
t
Or in flux operator J:
 
  Jp ( r )  0
 




Jp(r )  D(r )[p(r )   p(r )U (r )]
The next step is to construct tetrahedral meshes gradually from the
sphere S0 to the bounding sphere S1.
(a) mAChE
(b) the cavity
(c) the interior mesh
Fig. 3. Rice Dwarf Virus (RDV)
3.3 Adding an Outer Boundary
First we add a sphere S0 with radius r0 (r0 = 2n/2 = 2n-1) outside the
molecular surface, and generate meshes between the molecular surface
and the outer sphere S0. Then we extend the tetrahedral meshes from the
sphere S0 to the outer bounding sphere S1.
(d) an exterior mesh
within a small sphere
(e) an exterior mesh
within a large sphere
1C2O
(f) an exterior mesh
within a cubic box
1C2B
Ribosome 30S
(3)
Fig. 1. Adaptive tetrahedral meshes of mAChE. The distribution of
electrostatic potential: blue - positive; red - negative; white - neutral.
Int4
Int3
Int2
2. Overview
There are four steps in our biomolecular meshing process:
 Data acquisition – construct volumetric data from PDB/PQR,
or segment cryoEM data to identify its subunits.
Primary mesh extraction – develop a variant of the dual
contouring method to generate tetrahedral meshes, and design a
parallel algorithm for large datasets. For cryoEM data, different
subunits are detected.
Mesh extension – generate the exterior mesh.
Quality improvement – use edge-contraction and smoothing.
(a)
(b)
(c)
Fig. 4. The analysis domain of exterior meshes. (a) – the circum-sphere has the
radius of r. ‘S0’ is the maximum sphere inside the data box, ‘S1’ is an outer sphere
r1 = (20 ~ 40)r. (b) - the diffusion domain. (c) - the boundary is a cubic box.
4 cavities of 1C2B
Ribosome 50S
1C2B
References
4. Tetrahedral Mesh Extraction
Sign change edge
2D triangulation
Fig. 2. Overview of the comprehensive method
13th International Meshing Roundtable, Williamsburg, Virginia, September 19-22, 2004
Parallel 2D triangulation
Fig. 5. 2D/ 3D triangulation.
Fig. 7. Tetrameric mAChE
1.
Interior edge
Microtubule
Hemoglobin
Fig. 6. Meshing examples from PDB/PQR data
Y. Zhang, C. Bajaj, B.-S. Sohn. 3D Finite Element Meshing from Imaging Data. Accepted in
the special issue of CMAME on Unstructured Mesh Generation. 2004.
2. Y. Zhang, C. Bajaj, B.-S. Sohn. Adaptive and Quality 3D Meshing from Imaging Data, ACM
Symposium on Solid Modeling and Applications. pp. 286-291, Seattle, June 2003.
3. Y. Song, Y. Zhang, T. Shen, C. Bajaj, J. McCammon, N. Baker. Finite Element Solution of the
Steady-state Smoluchowski Equation for Rate Constant Calculations. Biophysical Journal,
86(4):2017-2029, 2004.
4. Y. Song, Y. Zhang, C. Bajaj, N. Baker. Continuum Diffusion Reaction Rate Calculations of
Wild Type and Mutant Mouse Acetylcholinesterase: Adaptive Finite Element Analysis.
Biophysical Journal 87(3), 2004.
5. Z. Yu, C. Bajaj. Visualization of Icosahedral Virus Structures from Reconstructed Volumetric
Maps. Techniqical Report, CS Dept., the Univ. of Texas at Austin, 2004.
Acknowledgements: Thank Prof. Wah Chiu for providing the rice dwarf virus (RDV) data.
* Please contact jessica@ices.utexas.edu for further information.