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CTW: Mathematical Challenges in Biomolecular/Biomedical Imaging and Visualization: Titles & Abstracts

Molecular shape and conformation elucidation is the problem of recovering the three-dimensional structure of an individual molecule, a protein or a macromolecular assembly at the highest possible resolution in its natural environment, which could be insitu (the cellular environment) or invitro (aqueous environment). Despite the advances in X-ray imaging and Electron Microscopy (EM), it has been difficult to simultaneously achieve the three goals of recovering shape and conformation (a) at a high resolution, (b) at the larger scale of protein assemblies, and (c) with the particles in their natural environment. In particular, molecular structural feature resolution needs to be fine enough to be useful to (a) visualize conformational changes of proteins upon binding of ligands, (b) study sub-unit composition (c) understand the interaction between an antibody and its antigen, and (d) study protein assembly in cells. In this talk, I shall highlight current progress on several co-mingled computational mathematics algorithms for interpretation, modeling, analyzing and verifying molecular - molecular interactions using multiple imaging modalities, namely X-ray , single particle EM and electron tomography.

Q. Zhang, R. Bettadapura and C. Bajaj Macromolecular Structure Modeling from 3DEM using VOLROVER 2.0 Biopolymers, September, 97(9):709-731, 2012 doi:10.1002/bip.22052.

C. Bajaj, S. Goswami, and Q. Zhang Detection of Secondary and Supersecondary Structures of Proteins for Cryo-Electron Microscopy Journal of Structural Biology, Volume 177, Issue 2, February 2012, Pages 367-381

R. Chowdhury, M. Rasheed, D. Keidel, M. Moussalem, A. Olson, M. Sanner, C. Bajaj Protein-Protein Docking with F2Dock 2.0 and GB-rerank Accepted for Publication in PLOS ONE , 2012

A. Gopinath, G. Xu, D. Ress, O. Oktem, S. Subramaniam, and C. Bajaj Shape-based Regularization of Electron Tomographic Reconstruction IEEE Transactions on Medical Imaging, December, 2012, 31(13):2241-2252, doi: 10.1109/TMI.2012.2214229

C. Bajaj, S.-C. Chen, and A. Rand An Efficient Higher-Order Fast Multipole Boundary Element Solution for Poisson-Boltzmann Based molecular Electrostatics SIAM Journal on Scientific Computing, 33(2) 826-848, 2011, NIHMSID# 266230,doi:10.1137/090764645, PMCID: PMC3110014.

G. Xu, M. Li, A. Gopinath, C. Bajaj Computational Inversion of Electron Tomography Images Using L2-Gradient Flows Journal of Computational Mathematics, 29:501-525, 2011, doi: 10.4208/jcm.1106-m3302.

M. Li, G. Xu, C. Sorzano, F. Sun, and C. Bajaj Single-Particle Reconstruction Using L2-Gradient Flow Journal of Structural Biology (2011) 176 (3): 259-267, NIHMSID 318967,doi:10.1016/j.jsb.2011.08.005, PMID: 21864687,

C. Bajaj, W. Zhao Fast Molecular Solvation Energetics and Force Computation SIAM Journal on Scientific Computing, 31(6): 4524-4552, January 2010, doi:10.1137/090746173,PMCID: PMC2830669.

Jean-Claude Haussmann and Allen Knutson introduced a marvelous description of a certain space of framed space polygons (up to rotation and translation in R^3) as a space which is almost everywhere covered by the Grassmann manifold G_2(C^n) of orthonormal 2-frames in complex n-space. The standard Riemannian metric on the Grassmann manifold then induces a natural metric on this moduli space of framed polygons, within which we can make explicit and useful computations. This description yields a natural setting for many problems of interest in mathematical biology involving curves.

In this talk, we'll cover two applications. The first is predicting the statistical properties of a ring polymer in solution, which is considerably simplified by this framework. The second is curve registration and comparison in 3d, with applications to protein structures (here we are following Younes, Michor, Mumford, Shah, who used the corresponding structure for plane curves).

Cantarella, J., Deguchi, T., & Shonkwiler, C. (2012, June 14). Probability Theory of Random Polygons from the Quaternionic Viewpoint. arXiv.org. http://arxiv.org/abs/1206.3161v2

Cantarella, J., Grosberg, A. Y., Kusner, R. B., & Shonkwiler, C. (2012, October 24). The Expected Total Curvature of Random Polygons. arXiv.org.

Solvation is an elementary process in nature and is of paramount importance to many sophisticated chemical, biological and biomolecular processes. The understanding of solvation is an essential prerequisite for the quantitative description and analysis of biomolecular systems. Implicit solvent models, particularly those based on the Poisson-Boltzmann (PB) equation for electrostatic analysis, are established approaches for solvation analysis. However, ad hoc solvent-solute interfaces are commonly used in the implicit solvent theory and have some severe limitations.

We have introduced differential geometry based multiscale solvation models which allow the solvent-solute interface, electrostatic potential, and even electron densities to be determined by the variation of a total free energy functional. Our models are utilized to evaluate the solvation free energies, protein-protein binding affinities, ion channel charge transport etc. This is primarily joint work with Professor Guowei Wei and Nathan Baker.

Cryo-electron microscopy (cryo-EM) has been successfully applied to determine structures of large macromolecular assemblies (>1 MDa) out to near atomic resolutions (3-5 Å). Most of the reporting structures lacked rigorous and quantitative assessment on the quality of the map and model. I will present case studies as how the maps and models are validated statistically and objectively. On the other spectrum of cryo-EM is to determine the 3-D tomograms of organelles and cells at nanometer resolutions (40-80 Å). Examples will be presented to demonstrate the challenges in delineating the spatial organization of the subcellular components and quantifying their structural parameters.

Molecular structure determines molecular function and geometry is one of the most fundamental aspects of the molecular structure. Despite its importance, the theory for understanding molecular geometry has not been sufficiently developed. In this talk, we will present a unified theory of molecular geometry and demonstrate how the theory can be used for ”accurately“, ”efficiently“, and ”conveniently“ solving molecular problems.

The molecular geometry theory is based on the beta-complex which is a derived structure from the Voronoi diagram of atoms and its dual structure called the quasi-triangulation. Voronoi diagrams are everywhere in nature and are useful for understanding the spatial structure among generators. Unlike the well-known ordinary Voronoi diagram of points, the Voronoi diagram of spherical atoms has been known to be difficult to compute. Once computed, however, it nicely defines the proximity among the atoms in molecules.

This talk will introduce the Voronoi diagram of atoms and its dual structure, the quasi-triangulation, in the three-dimensional space. Based on the quasi-triangulation, we define the beta-complex which concisely yet efficiently represents the correct proximity among all atoms. It turns out that the beta-complex, together with the Voronoi diagram and quasi-triangulation, can be used to accurately, efficiently, and conveniently solve seemingly seemingly unrelated geometry and topology problems for molecules within a single theoretical and computational framework. The correctness and efficiency of solutions can be easily, mathematically guaranteed. Among many application areas, structural molecular biology and noble material design are the most immediate application area.

Application examples include the following: the most efficient/precise computation of van der Waals volume (and area); an efficient docking simulation; the recognition of internal voids and their volume computation; the recognition of molecular tunnels, the comparison (or superposition) of the boundary structures of two molecules, shape reasoning such as measuring the sphericity of molecules, the efficient computation of the optimal side-chain placement for proteins, etc. We anticipate many other important applications will be discovered. In this talk, we will also demonstrate our molecular modeling and analysis software, BetaMol, which is entirely based on the unified, single representation of the beta-complex. Several programs, including the BetaMol, are freely available at the Voronoi Diagram Research Center (VDRC, http://voronoi.hanyang.ac.kr/). The engine software will also be available soon so that researchers can easily create application programs for their own problems using this engine.

References

1. Deok-Soo Kim, Youngsong Cho, and Donguk Kim, Euclidean Voronoi diagram of 3D balls and its computation via tracing edges, Computer-Aided Design, Vol. 37, No. 13, pp. 1412-1424, 2005

2. Deok-Soo Kim, Youngsong Cho, and Kokichi Sugihara, Quasi-worlds and Quasi-operators on Quasi-triangulations, Computer-Aided Design, Vol. 42, Issue 10, pp. 874-888, , 2010

3. Deok-Soo Kim, Youngsong Cho, Kokichi Sugihara, Joonghyun Ryu, and Donguk Kim, Three-dimensional Beta-shapes and Beta-complexes via Quasi-triangulation, Computer-Aided Design, Vol. 42, Issue 10, pp. 911-929, , 2010

In this talk, we will introduce multilevel methods for discrete ill-posed problems arising from the discretization of Fredholm integral equations of the first kind. In particular, we will present wavelet-based multilevel methods for signal and image restoration problems as well as for blind deconvolution problems. In these methods, orthogonal wavelet transforms are used to define restriction and prolongation operators within a multigrid-type iteration. The choice of the Haar wavelet operator has the advantage of preserving matrix structure, such as Toeplitz, between grids, which can be exploited to obtain faster solvers on each level where an edge-preserving Tikhonov regularization is applied. We will show results that indicate the promise of these approaches on restoration of signals and images with edges as well as restoration of blurring operator in the case of the blind deconvolution problem.

In this talk I discuss the reconstruction of compactly supported piecewise smooth functions from non-uniform samples of their Fourier transform. This problem is relevant in applications such as magnetic resonance imaging (MRI) and synthetic aperture radar (SAR).

Two standard reconstruction techniques, convolutional gridding (the non-uniform FFT) and uniform resampling, are summarized, and some of the difficulties are discussed. It is then demonstrated how spectral reprojection can be used to mollify both the Gibbs phenomenon and the error due to the non-uniform sampling. It is further shown that incorporating prior information, such as the internal edges of the underlying function, can greatly improve the reconstruction quality. Finally, an alternative approach to the problem that uses Fourier frames is proposed.

When there is no enough observed data or the data contains excessive noise, it is challenging to reconstruct the image of interest. We prent our recent work on how to effectively use prior information to significantly improve the reconstruction. We explore geometric, support and regularity prior etc. and illustrate the ideas in compressive sensing reconstruction and image denoising. All the priors are extracted from the observed data. Various numerical results show the effectivenss of incorporating prior.

In this talk, a semiautomatic extraction algorithm for images of ciliary muscle is presented. The algorithm is based on the region scalable model which draws upon intensity information in local regions at a controllable scale, so that it can segment images with intensity inhomogeneity. The method is used to have morphological assessment of the dimensions of the ciliary muscle in Visante Anterior Segment Optical Coherence Tomography images. Several applications will be discussed.

It is a challenging task to generate quality mesh which can be used for emerging mathematical modeling of biomolecular systems. In the first part, I will present a robust method and software TMSmesh developed recently to generate manifold surface mesh for complex biomolecular surface. A Gaussian surface is used to represent molecular surface. The method computes the surface points by solving a nonlinear equation directly, polygonizes by connecting surface points through a trace technique, and finally outputs a triangulated mesh. To guarantee the generated mesh is a manifold mesh, it is necessary to divide the surface into into single-valued pieces along each x,y,z directions by tracing the extreme points along the fold curves on the surface. TMSmesh is shown be able to generate quality surface mesh for arbitrarily large molecule in PDB. Volume mesh can be generated based on the TMSmesh surface mesh by applying a third-part software such as Tetgen. In the second part, I will present a visualization system MMV developed in our lab. MMV is also specifically designed for continuum modeling community, featured by its multifunction of visualization, modeling and analysis of biomolecule, mesh, and simulation data.

During transcription, both RNA polymerase (RNAP) and the nascent RNA chain need to rotate relative to DNA at a speed of approximately a few hundred rounds per minute. The resulting accumulation of torsional stress cannot be dissipated immediately, and is therefore able to impact the motion of the RNAP. Despite the fact that torque is an essential aspect of transcription, the extent of its impact remains unknown. By performing single molecule measurements using an angular optical trap, we directly measure the torque that RNAP can generate as well as the transcription rate under torque. This approach provides a framework for investigating the influence of torque on various DNA-based translocases.

The importance of the endothelium in regulating vessel homeostasis is reflected by its effect on vascular tone and by its role in mediating vasodilatory responses to many physiological stimuli. Purine and pyrimidine nucleotides are important modulator of endothelial function. Purinergic signalling and conducted responses can depend on how flow regulates the interactions between ligand and receptor at the endothelium. Regulatory pathways are also influenced by wall shear stress, via mechanotransduction mechanisms. We review the most relevant computational models that have been proposed to date, and propose a general framework for modelling the responses of the endothelium to alteration in the flow, with a view to understanding the biomechanical processes involved in the pathways to endothelial dysfunction. Simulations are performed on a patient-specific imagebased stenosed carotid artery to investigate the influence of wall shear stress and mass transport phenomena upon the agonist coupling response at the endothelium.

Biology has become accessible to an understanding of processes that span from atom to organism. As such we now have the opportunity to build a spatio-temporal picture of living systems at the molecular level. In our recent work we attempt to create, interact with, and communicate physical representations of complex molecular environments. I will discuss the challenges and demonstrate three levels of interaction with protein interactions: 1) human perceptual and cognitive interaction with complex structural information; 2) interaction and integration of multiple data sources to construct cellular environments at the molecular level; and 3) interaction of software tools that can bridge the disparate disciplines needed to explore, analyze and communicate a holistic molecular view of living systems.

In order to increase our understanding and interaction with complex molecular structural information we have combined two evolving computer technologies, solid printing and augmented reality. We create custom tangible molecular models and track their manipulation with real-time video, superimposing text and graphics onto the models to enhance their information content and to drive interactive computation. We have recently developed automated technologies to construct the crowded molecular environment of living cells from structural information at multiple scales as well as bioinformatics information on levels of protein expression and other data. We can populate cytoplasm, membranes, and organelles within the same structural volume to generate cellular environments that synthesize our current knowledge of such systems.

The communication of complex structural information requires extensive scientific knowledge as well as expertise in creating clear visualizations. We have developed a method of combining scientific modeling environments with professional grade 3D modeling and animation programs such as Maya, Cinema4D and Blender. This gives both scientists and professional illustrators access to the best tools to communicate the science and the art of the molecular cell.

Gillet, A., Sanner, M., Stoffler, D., Olson, A.J. (2005) Tangible interfaces for structural molecular biology. Structure:13:483-491.

Johnson, G.T., Autin, L., Goodsell, D.S., Sanner, M.F., Olson A.J. (2011) ePMV Embeds Molecular Modeling into Professional Animation Software Environments. Structure 19(3):293-303.

Finding the structure of proteins from NMR or x-ray crystallography relies on mathematical methods. Fourier series is used for crystallography, and distance geometry and discrete Frenet frames are used for NMR methods. We summarize these ideas and the connections among them.

The aim of this presentation is to provide an overview of approaches for solving the ill-posed inverse problems associated with signal and image restoration. Spectral decomposition and a Generalized Picard condition analysis highlight the importance of the basis for the solution, which is impacted by regularization choices. Additionally, when introducing regularization the complication of finding appropriate weighting parameters arises. Statistical techniques for finding both the regularization parameters and for filtering the basis are introduced. Total variation regularizers for effective feature extraction are analyzed using the same techniques and verify the significance of appropriately finding the basis and the regularization parameters.

Cryo-electron microscopy is used to acquire two-dimensional projection images of thousands of individual, identical frozen-hydrated macromolecules at random unknown orientations and positions. The goal in single particle reconstruction is to reconstruct the three-dimensional structure of macromolecules with sufficiently high resolution from noisy projection images with unknown pose parameters and poor signal-to-noise ratio (SNR). I will discuss methods for estimating the structure based on common-lines and class averages that do not require any prior model assumption.

The technique of covolutional gridding (CG) has been widely used in applications with non-uniform (Fourier) data such as magnetic resonance imaging (MRI). On the other hand, its error analysis is not fully understood. We consider it as a Fourier frame approximation and present an error analysis accordingly. Moreover, we propose a generalized convolutional gridding (GCG) method as an improved frame approximation.

Nuclear magnetic resonance spectroscopy (NMR) is heavily employed by chemists and biochemists to study the structures and properties of chemical compounds. The measured data however often contain mixtures of chemicals, subject to changing background and environmental noise. A mathematical problem is to unmix or decompose the measured data into a set of pure or source spectra without knowing the mixing process, a so called blind source separation problem.

In the talk, the speaker shall present algorithms for blind separation of spectral mixtures in noisy conditions. The approach combines geometrical and statistical analysis of the data, the geometric approach is based on the vertexes and facets identification of cone structures of the data, while the statistical approach is on decomposing fitting errors when partial knowledge of the source spectra is available. Computational results on data from NMR, Raman spectroscopy and differential optical absorption spectroscopy show the applicability of the methods. This is joint work with Jack Xin from UC Irvine.

We present a method for computing "choking" loops - a set of surface loops that describe the narrowing of the volumes inside/outside of the surface and extend the notion of surface homology and homotopy loops. The intuition behind their definition is that a choking loop represents the region where an offset of the original surface would get pinched. Our generalized loops naturally include the usual $2g$ handles/tunnels computed based on the topology of the genus-$g$ surface, but also include loops that identify chokepoints or bottlenecks, i.e., boundaries of small membranes separating the inside or outside volume of the surface into disconnected regions. Based on persistent homology theory, our definition builds on a measure to topological structures, thus providing resilience to noise and a well-defined way to determine topological feature size.

Fourier reconstruction of piecewise-smooth functions suffers from the familiar Gibbs phenomenon. In Fourier imaging modalities such as magnetic resonance imaging (MRI), this manifests as a loss of detail in the neighborhood of tissue boundaries (due to the Gibbs ringing artifact) as well as long scan times necessitated by the collection of a large number of spectral coefficients (due to the poor convergence properties of the reconstruction method). We present a framework for incorporating edge information - for example, the locations and values of jump discontinuities of a function - in the reconstruction. We show that a simple relationship exists between global Fourier data and local edge information. Knowledge of such edges, either a priori or estimated, enables the synthesis of high-mode spectral coefficients beyond those collected by the MR scanner. Incorporating these synthesized coefficients in an augmented Fourier partial sum reconstruction allows for the generation of scans with significantly improved effective resolution. Further use of spectral re-projection schemes can result in the elimination of all Gibbs artifacts. Numerical results showing accelerated convergence and improved reconstruction quality will be presented.

Electron tomography is the most widely applicable method for obtaining 3D information by electron microscopy. It has been realized that electron tomography is capable of providing a complete, molecular resolution three-dimensional mapping of entire proteomes including their detailed interactions. However, to realize this goal, information needs to be extracted efficiently form these tomograms. Owing to low SNR, this task is currently mostly carried out manually. Apart from the subjectivity of the process, its time consuming nature precludes the prospects of high throughput processing. Standard template matching approaches rely on Ômatched filteringÕ, which can be shown to be a Bayesian classifier as long as the template and the target are nearly identical and the noise is independent and identically distributed, Gaussian, and additive. These conditions are not very well met for electron tomographic reconstructions because the noise is spatially correlated by the reconstruction process and the point spread function. As a consequence, many false hits are generated by this method in areas of high density such as membranes or dense vesicles.

In order to address this challenge, we are developing an alternative method for feature recognition in electron tomography, which is based on the use of reduced representation templates. Reduced representations approximate the target by a small number of anchor points. These anchor points are then used to calculate the scoring function within the search volume. This strategy makes the approach robust against noise and against local variations such as those expected from uneven staining. We recently completed a number of proof-of-concept application of this algorithm for detecting ribosomes in electron tomograms of high-pressure-frozen plastic embedded as well as cryo-sectioned mammalian cell sections. The percentage of false hits for this data drops dramatically from ~50% to under 20% if compared to the matched filter approach. Here, I will describe the principles underlying our approach and will present results obtained from its application.

In applications such as functional Magnetic Resonance Imaging (fMRI), full, uniformly-sampled Cartesian Fourier (frequency space) measurements are acquired. In order to reduce scan time and increase temporal resolution for fMRI studies, one would like to accurately reconstruct these images from a highly reduced set of Fourier measurements. Compressed Sensing (CS) has given rise to techniques that can provide exact and stable recovery of sparse images from a relatively small set of Fourier measurements. For example, if the images are sparse with respect to their gradient, total-variation minimization techniques can be used to recover those images from a highly incomplete set of Fourier measurements. In this discussion, we propose a new algorithm to further reduce the number of Fourier measurements required for exact or stable recovery by utilizing prior edge information from a high resolution reference image. This reference image (routinely acquired during fMRI studies for anatomic landmarking of activations), or more precisely, the fully sampled Fourier measurements of this reference image, can also be used to provide approximate edge information. By combining this edge information with CS techniques for sparse gradient images, numerical experiments show that we can further reduce the number of Fourier measurements required for exact or stable recovery by a factor of 1.6 – 3, compared with CS techniques alone, without edge information.

The classical phase retrieval problem concerns the reconstruction of a function from the magnitude of its Fourier transform. Phase retrieval is an important problem in many applications such as molecular imaging and signal processing. Today the phase retrieval problem has been extended to include more general reconstructions for magnitudes of samples. In this talk I will present an overview of this field, including some of the latest advances both in the theoretical and computational aspects of phase retrieval.

Recently, the structure, function, stability, and dynamics of subcellular structures, organelles, and multiprotein 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. We introduce 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. We offer a comprehensive protocol for the geometric modeling of subcellular structures, organelles, and multiprotein complexes.

Calculation of electrostatic potential energy for proteins in ionic solvent is a fundamental task in the simulation study of the structure and biological function of proteins, catalytic activity, and ligand association. To reflect the polarization correlations among water molecules, several nonlocal dielectric models have been developed for a wide range of dielectric materials and dipolar liquids in the last thirty years. However, the study of a nonlocal model has been limited to the case of pure water solvent so far due to modeling and algorithmic complications. Most significantly, none of the current ionic models incorporate nonlocal dielectric effects.

As the first step toward the direction of changing this situation, we recently proposed a nonlocal dielectric model for protein in ionic solvent, alongside a new efficient numerical algorithm and program package for solving the model. In this talk, I will give our recent progresses a short review. I then will describe our finite element program package to show how we combine biomolecular tetrahedral mesh generation and visualization tools with our fast finite element solvers. With this package, we now can calculate and visualize the numerical solution of the nonlocal model by only inputing in a protein data bank (PDB) file of a protein immersed in salt solution.

Biological structures provide rich information on their functions. Depending on the size and imaging resolution, a biological structure is often given as one of two common forms: (1) bio-molecules at atomic-resolutions typically obtained by X-ray crystallography or NMR, and (2) raster-scan images taken with various imaging devices such as optical or electron microscopes. Over the last decade, we have seen a rapidly increasing number of biological structures that are freely available through public databases such as the Protein Data Bank (PDB) and Electron Microscopy Data Bank (EMDB). Three-dimensional (3D) geometric modeling of these structures becomes critical not only for shape visualization, quantification, and analysis of the structures themselves, but also for mathematical simulations of their functions.

In this talk, I shall present computational approaches to constructing high-quality surface mesh models from both bio-molecules and imaging data. In addition, I shall briefly discuss how volumetric (tetrahedral) meshes can be generated and optimized for finite element based numerical simulations. A software toolkit we recently developed will be demonstrated to support the pipeline of geometric modeling and visualization from bio-molecules and imaging data.

We present a new approach to solve the inverse source problem arising in Fluorescence Tomography (FT). In general, the solution is non-unique and the problem is severely ill-posed. It poses tremendous challenges in image reconstructions. In practice, the most widely used methods are based on Tikhonov-type regularizations, which minimize a cost function consisting of a regularization term and a data fitting term. We propose an alternative method, which overcomes the major difficulties, namely the non-uniqueness of the solution and noisy data fitting, in two separated steps. First we find a particular solution called orthogonal solution that satisfies the data fitting term. Then we add to it a correction function in the kernel space so that the final solution fulfills the regularization and other physical requirements, such as positivity. Moreover, there is no parameter needed to balance the data fitting and regularization terms. In addition, we use an efficient basis to represent the source function to form a hybrid strategy using spectral methods and finite element methods in the algorithm. The resulting algorithm can drastically improve the computation speed over the existing methods. And it is also robust against noise and can greatly improve the image resolutions. This is a joint work with Shui-Nee Chow (Math), Ke Yin (Math) and Ali Behrooz (ECE) from Georgia Tech.

Recently, we have introduced a pseudo-time coupled nonlinear partial differential equation (PDE) model for biomolecular solvation analysis. Based on a free energy optimization, a boundary value system is derived to couple a nonlinear Poisson-Boltzmann (NPB) equation for electrostatic potential with a generalized Laplace-Beltrami (GLB) equation defining the biomolecular surface. By introducing a pseudo-time in both processes, a more efficient coupling is achieved through the steady state solution of two nonlinear parabolic PDEs. For the GLB equation, the pseudo-transient continuation is attained via a potential driven geometric flow PDE, which defines a smooth biomolecular surface to characterize the dielectric boundary between biomolecules and the surrounding aqueous environment. The resulting smooth dielectric profile, however, introduce some instability issue in solving the time-dependent NPB equation. This motivates us to develop an operator splitting alternating direction implicit (ADI) scheme, in which the nonlinear instability is completely avoided through analytical integration. To speed up the computation of molecular surface, a new fully implicit ADI scheme is developed too for solving the geometric flow equation. Unconditional stability can be realized by both ADI schemes in solving unsteady NPB and GLB equations separately. In solving a coupled system for real biomolecules and chemical compounds, the proposed numerical schemes are found to be conditionally stable. Nevertheless, the time stability can be maintained by using very large time increments, so that the present biomolecular simulation becomes much faster.

Posters

A new finite-difference time-domain (FDTD) algorithm is introduced to solve two dimensional (2D) transverse magnetic (TM) modes with a straight dispersive interface. Driven by the consideration of simplifying interface jump conditions, the auxiliary differential equation of the Debye constitution model is rewritten to form a new Debye-Maxwell TM system. Supplementary differential equations are utilized to describe the transient changes in the regularities of electromagnetic fields across a dispersive interface. The resulting time dependent jump conditions are rigorously enforced in the FDTD discretization by means of the matched interface and boundary (MIB) scheme. Higher order convergences are numerically achieved for the first time in the literature in the 2D FDTD simulations of dispersive inhomogeneous media.

The high prevalence of heart failure is largely due to our lack of accurate understanding of the complex pathology including abnormal excitation¿contraction (EC) coupling in cardiac myocytes. Ca2+ movement plays a central role in studying EC coupling in both normal and diseased cardiac myocytes. The elementary events of Ca2+ movements in heart muscle cells are calcium sparks. A nonlinear system of diffusion-reaction equations model is developed in [1] to simulate the Ca2+ movements including Ca2+ sparks in cardiac myocytes. The inherent nonlinearity of the model implies their high sensitivity to parameter changes. A parameter sensitivity study reveals that the concentration of the indicator are essential determinants of the shape of Ca2+ sparks, whereas the stationary buffers and pumps are less influential

The high prevalence of heart failure is largely due to our lack of accurate understanding of the complex pathology including abnormal excitation¿contraction (EC) coupling in cardiac myocytes. Ca2+ movement plays a central role in studying EC coupling in both normal and diseased cardiac myocytes. The elementary events of Ca2+ movements in heart muscle cells are calcium sparks. A nonlinear system of diffusion-reaction equations model is developed in [1] to simulate the Ca2+ movements including Ca2+ sparks in cardiac myocytes. The inherent nonlinearity of the model implies their high sensitivity to parameter changes. A parameter sensitivity study reveals that the concentration of the indicator are essential determinants of the shape of Ca2+ sparks, whereas the stationary buffers and pumps are less influential

A robust version of Principal Component Analysis (PCA) can be constructed via a decomposition of a data matrix into low rank and sparse components, the former representing a low-dimensional linear model of the data, and the latter representing sparse deviations from the low-dimensional subspace. This decomposition has been shown to be highly effective, but the underlying model is not appropriate when the data are not modeled well by a single low-dimensional subspace. We construct a new decomposition corresponding to a more general underlying model consisting of a union of low-dimensional subspaces, and demonstrate the performance on a video background removal problem.