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彩色图像增强的向前和向后适应性贝尔特拉米流-color image enhancement by a forward and backward adaptive beltrami maswatikepor.gq
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Careers and apprenticeships Equal opportunities Vacancies Apprenticeships. Advanced Search Watchlist Search history Search help. Limit the search to the library catalogue. Direct access to the library catalogue. Bates, Wei and Zhao incorporated the energy minimization principle into macromolecular surface generation, and proposed one of the first variational frameworks for biomolecular surfaces [ 5 , 6 ].
Basically, a free energy functional of the biomolecular surface model is defined. Through the Euler-Lagrange equation of surface free energy minimization, a generalized mean curvature flow equation is attained. The resulting molecular surface, called the minimal molecular surface MMS , is then constructed by the mean curvature flow [ 7 ].
PDE algorithms for biomolecular surface generation have become a popular topic in theoretical biology [ 75 , 79 , 4 ]. Both aforementioned arbitrarily high-order geometric PDEs and PDE transform have been applied to biomolecular surface construction [ 4 , 82 ]. Similar approaches were employed by Cheng et al. Consequently, almost all the biological processes in cell, such as signal transduction, transcription, translation, protein folding, protein ligand binding, and charge and mass transport, occur in aqueous surroundings.
Therefore, the understanding of the solvation is of fundamental importance for quantitative modeling and analysis of all the above-mentioned processes. Explicit solvent models and implicit solvent models are two major approaches for solvation analysis [ 48 , 52 , 54 ].
For explicit solvent models, both the solvent and the solute are described in atomic detail and extensive sampling is required. Implicit solvent models are designed to reduce the number of degrees of freedom by using a dielectric continuum to describe the solvent while admitting a microscopic atomic description for the biomolecules [ 74 , 3 , 8 ].
Due to their fewer degree of freedom, implicit solvent models, such as the Poisson-Boltzmann PB model or the Poisson equation PE model when there is no salt in the solvent, are widely used [ 1 , 2 , 45 ]. The coupling of the PB or PE with the generalized Laplace-Beltrami flow has the potential of describing the formation of molecular surface in realistic solvation environments. Conceptually, a solvation free energy can be divided into two major parts: a nonpolar part associated with inserting an uncharged solute into the solvent [ 14 ] and a polar part associated with charging the solute in vacuum and solvent [ 38 , 11 ].
The nonpolar free energy and polar free energy can be represented by a total free energy functional [ 63 ]. By using the variational principle, a new geometric flow equation is generated that controls the biomolecular surface formation and evolution via curvature and potential driven [ 4 , 11 , 12 , 13 ]. This model takes into consideration of the surface energy minimization and also the solvent-solute interaction, and gives a multiresolution representation of biomolecular surfaces in their native environment.
Additionally, the external potential term can be used to incorporate different kinds of effects, such as chemical reaction, fluid flow, and elastic description of marcomolecules [ 68 , 65 ].
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The objective of the present paper is to provide an expository investigation and summary of tools, algorithms and methodologies for geometric modeling of biomolecules. We particularly focus on tools, algorithms and methodologies required for biophysical models in the Eulerian representation.
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Although Eulerian formation [ 11 ] and Lagrangian formation [ 12 ] of biomolecular surfaces can be formally equivalent, they depend on different tools, algorithms and methodologies. For the latter, the high order PDEs are introduced to perform noise reduction. Geometric features, such as Gaussian curvatures, mean curvatures, and shape index, are employed to describe the geometric properties of biomolecular multiresolution surfaces generated by generalized geometric flows and from the EMDB for the first time.
The rest of this paper is organized as follows. Section 2 is devoted to computational algorithms. We discuss in great detail data sources, related softwares, and computational techniques for surface construction, quality improvement, and geometric characterization. We provide advanced interface methods for the evaluation of surface area and surface enclosed volume in the Cartesian representation.
Efficient algorithms for calculating various curvature properties, such as Gaussian curvature, mean curvature, maximum and minimum principal curvatures, shape index, and curvedness are developed. The performance of these algorithms is compared.
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Mathematical models are presented in Appendix A. Two variational models for macromolecular surface generation from PDB data are introduced. We make use of the differential geometry theory of surfaces and geometric measure theory to formulate protein surfaces in conjugation with electrostatic analysis. Additionally, geometric flow based methods are utilized to render high quality macromolecular surfaces from EMDB data.
Finally, methods for characterizing geometric features of macromolecules are also proposed. This paper ends with concluding remarks. Most data downloaded from the PDB need to be processed for preparing structures used in theoretical analysis and modeling [ 10 ]. Visualization is of great importance to our understanding and conceptualization of the biomolecular systems. Many softwares can be employed to generate triangular surface meshes for biomolecules. An example is the MSMS package.
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However, the MSMS surface cannot be directly used in Cartesian domain modeling and computation as discussed below. In order to generate the Cartesian representation for finite difference type of methods, one needs to carry out the transformation from Lagrangian to Eulerian representation, i. In this process, one needs to extract interface information from the triangle mesh representation, including the coordinates of intersecting points between the surface and Cartesian mesh lines, and surface normal directions at these intersecting points.
For example, if we have a surface mesh in. The bounded box to encompass the protein can be constructed by expanding the tightest axis-aligned bounding box, i. One can specify the mesh spacing, i. Similar relations exist for y and z coordinates. As the goal is to find the intersection points of each triangle with grid lines, we first find the plane equation. For each mesh triangle, one has the coordinates for its three vertices v 1 , v 2 and v 3. The 2D plane that the triangle belongs to is.
We find the intersection points by testing grid edges within the bounding box of the triangle. It is easy to find the coordinate ranges for all the relevant grid edges, e. For all the points within the triangle, the values of x coordinate should fall in the range [ x s , x b ]. Thus, to find an intersection between a surface triangle and a grid line along x direction, one can choose two arbitrary index j , k within the corresponded ranges, with their associated coordinates defined as y j , z k , and calculate the related point in the triangle plane.
The related coordinates are denoted as x o , y j , z k , and evaluated from. The intersecting points form a set, which is the collection of only three possible types of points:. The only task left to do is to determine whether the planar point we calculate falls within the triangle. The points located outside the triangle are discarded. If the point located on the boundary edges or the interior of the triangle, it is indeed a point where the interface intersects with the Cartesian grid lines.
The normal vector for this interface intersecting point is defined to be the same as that of the triangle, or for efficiency, it can be computed as the linear interpolations of vertex normals.