Abstract In this work, the existence of a global minimizer for the previous Lagrangian formulation of nonpolar solvation model proposed in [1] has been proved. One of the proofs involves a construction of a phase field model that converges to the Lagrangian formulation. Moreover, an Eulerian formulation of nonpolar solvation model is proposed and implemented under a similar parameterization scheme to that in [1]. By doing so, the connection, similarity and difference between the Eulerian formulation and its Lagrangian counterpart can be analyzed. It turns out that both of them have a great potential in solvation prediction for nonpolar molecules, while their decompositions of attractive and repulsive parts are different. That indicates a distinction between phase field models of solvation and our Eulerian formulation.
{"title":"Minimization and Eulerian Formulation of Differential Geormetry Based Nonpolar Multiscale Solvation Models","authors":"Zhan Chen","doi":"10.1515/mlbmb-2016-0005","DOIUrl":"https://doi.org/10.1515/mlbmb-2016-0005","url":null,"abstract":"Abstract In this work, the existence of a global minimizer for the previous Lagrangian formulation of nonpolar solvation model proposed in [1] has been proved. One of the proofs involves a construction of a phase field model that converges to the Lagrangian formulation. Moreover, an Eulerian formulation of nonpolar solvation model is proposed and implemented under a similar parameterization scheme to that in [1]. By doing so, the connection, similarity and difference between the Eulerian formulation and its Lagrangian counterpart can be analyzed. It turns out that both of them have a great potential in solvation prediction for nonpolar molecules, while their decompositions of attractive and repulsive parts are different. That indicates a distinction between phase field models of solvation and our Eulerian formulation.","PeriodicalId":34018,"journal":{"name":"Computational and Mathematical Biophysics","volume":"4 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2016-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/mlbmb-2016-0005","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67036071","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Proteins interact to perform biological functions through specific interface residues. Correctly understanding the mechanisms of interface recognition and prediction are important for many aspects of life science studies. Here, we report a novel architecture to study protein interface residues. In our method, multiple dimensional space was built on some meaningful features. Then we divided the space and put all the surface residues into the regions according to their features’ values. Interestingly, interface residues were found to prefer some grids clustered together. We obtained excellent result on a public and verified data benchmark. Our approach not only opens up a new train of thought for interface residue prediction, but also will help to understand proteins interaction more deeply.
{"title":"Multiple dimensional space for protein interface residue characterization","authors":"Tingyi Cao, Yongxiao Yang, Xinqi Gong","doi":"10.1515/mlbmb-2016-0004","DOIUrl":"https://doi.org/10.1515/mlbmb-2016-0004","url":null,"abstract":"Abstract Proteins interact to perform biological functions through specific interface residues. Correctly understanding the mechanisms of interface recognition and prediction are important for many aspects of life science studies. Here, we report a novel architecture to study protein interface residues. In our method, multiple dimensional space was built on some meaningful features. Then we divided the space and put all the surface residues into the regions according to their features’ values. Interestingly, interface residues were found to prefer some grids clustered together. We obtained excellent result on a public and verified data benchmark. Our approach not only opens up a new train of thought for interface residue prediction, but also will help to understand proteins interaction more deeply.","PeriodicalId":34018,"journal":{"name":"Computational and Mathematical Biophysics","volume":"4 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2016-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/mlbmb-2016-0004","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67036547","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Topological data analysis has been recently used to extract meaningful information frombiomolecules. Here we introduce the application of persistent homology, a topological data analysis tool, for computing persistent features (loops) of the RNA folding space. The scaffold of the RNA folding space is a complex graph from which the global features are extracted by completing the graph to a simplicial complex via the notion of clique and Vietoris-Rips complexes. The resulting simplicial complexes are characterised in terms of topological invariants, such as the number of holes in any dimension, i.e. Betti numbers. Our approach discovers persistent structural features, which are the set of smallest components to which the RNA folding space can be reduced. Thanks to this discovery, which in terms of data mining can be considered as a space dimension reduction, it is possible to extract a new insight that is crucial for understanding the mechanism of the RNA folding towards the optimal secondary structure. This structure is composed by the components discovered during the reduction step of the RNA folding space and is characterized by minimum free energy.
{"title":"Persistent Homology Analysis of RNA","authors":"A. Mamuye, M. Rucco, L. Tesei, E. Merelli","doi":"10.1515/mlbmb-2016-0002","DOIUrl":"https://doi.org/10.1515/mlbmb-2016-0002","url":null,"abstract":"Abstract Topological data analysis has been recently used to extract meaningful information frombiomolecules. Here we introduce the application of persistent homology, a topological data analysis tool, for computing persistent features (loops) of the RNA folding space. The scaffold of the RNA folding space is a complex graph from which the global features are extracted by completing the graph to a simplicial complex via the notion of clique and Vietoris-Rips complexes. The resulting simplicial complexes are characterised in terms of topological invariants, such as the number of holes in any dimension, i.e. Betti numbers. Our approach discovers persistent structural features, which are the set of smallest components to which the RNA folding space can be reduced. Thanks to this discovery, which in terms of data mining can be considered as a space dimension reduction, it is possible to extract a new insight that is crucial for understanding the mechanism of the RNA folding towards the optimal secondary structure. This structure is composed by the components discovered during the reduction step of the RNA folding space and is characterized by minimum free energy.","PeriodicalId":34018,"journal":{"name":"Computational and Mathematical Biophysics","volume":"4 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2016-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/mlbmb-2016-0002","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67036545","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Novel approaches for generating and comparing flexible (non-rigid) molecular surface meshes are developed. The mesh-generating method is fast and memory-efficient. The resulting meshes are smooth and accurate, and possess high mesh quality. An isometric-invariant shape descriptor based on the Laplace- Beltrami operator is then explored for mesh comparing. The new shape descriptor is more powerful in discriminating different surface shapes but rely only on a small set of signature values. The shape descriptor is applied to shape comparison between molecules with deformed structures. The proposed methods are implemented into a program that can be used as a stand-alone software tool or as a plug-in to other existing molecular modeling tools. Particularly, the code is encapsulated into a software toolkit with a user-friendly graphical interface developed by the authors.
{"title":"Mesh Generation and Flexible Shape Comparisons for Bio-Molecules","authors":"Zhanheng Gao, Reihaneh Rostami, Xiaoli Pang, Zhicheng Fu, Zeyun Yu","doi":"10.1515/mlbmb-2016-0001","DOIUrl":"https://doi.org/10.1515/mlbmb-2016-0001","url":null,"abstract":"Abstract Novel approaches for generating and comparing flexible (non-rigid) molecular surface meshes are developed. The mesh-generating method is fast and memory-efficient. The resulting meshes are smooth and accurate, and possess high mesh quality. An isometric-invariant shape descriptor based on the Laplace- Beltrami operator is then explored for mesh comparing. The new shape descriptor is more powerful in discriminating different surface shapes but rely only on a small set of signature values. The shape descriptor is applied to shape comparison between molecules with deformed structures. The proposed methods are implemented into a program that can be used as a stand-alone software tool or as a plug-in to other existing molecular modeling tools. Particularly, the code is encapsulated into a software toolkit with a user-friendly graphical interface developed by the authors.","PeriodicalId":34018,"journal":{"name":"Computational and Mathematical Biophysics","volume":"4 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2016-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/mlbmb-2016-0001","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67036543","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract A continuum mechanical model based on the Helfrich Hamiltonian is devised to investigate the coupling between lipid composition and membrane curvature. Each monolayer in the bilayer is modeled as a freely deformable surface with a director field for lipid orientation. A scalar field for the mole fraction of two lipid types accounts for local changes in composition. It allows lipids to access monolayer regions favorable to their intrinsic curvature at the expense of increasing entropic free energy. Hemifusion is one of the key fusion intermediates with regions of both positive and negative membrane curvature and where proteins must supply energy in order to bring about large elastic distortions. Using a numerical gradient descent scheme, minimal energy axisymmetric shapes of hemifusion diaphragms are calculated for varying radii. Previous studies assumed a fixed, weighted average for spontaneous curvature. Allowing for local changes in spontaneous curvature yields energies and forces of expansion significantly lower than those obtained from a fixed composition.
{"title":"Local Changes in Lipid Composition to Match Membrane Curvature","authors":"R. Ryham","doi":"10.1515/mlbmb-2016-0003","DOIUrl":"https://doi.org/10.1515/mlbmb-2016-0003","url":null,"abstract":"Abstract A continuum mechanical model based on the Helfrich Hamiltonian is devised to investigate the coupling between lipid composition and membrane curvature. Each monolayer in the bilayer is modeled as a freely deformable surface with a director field for lipid orientation. A scalar field for the mole fraction of two lipid types accounts for local changes in composition. It allows lipids to access monolayer regions favorable to their intrinsic curvature at the expense of increasing entropic free energy. Hemifusion is one of the key fusion intermediates with regions of both positive and negative membrane curvature and where proteins must supply energy in order to bring about large elastic distortions. Using a numerical gradient descent scheme, minimal energy axisymmetric shapes of hemifusion diaphragms are calculated for varying radii. Previous studies assumed a fixed, weighted average for spontaneous curvature. Allowing for local changes in spontaneous curvature yields energies and forces of expansion significantly lower than those obtained from a fixed composition.","PeriodicalId":34018,"journal":{"name":"Computational and Mathematical Biophysics","volume":"4 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2016-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/mlbmb-2016-0003","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67036546","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Differential geometry (DG) based solvation models have shown their great success in solvation analysis by avoiding the use of ad hoc surface definitions, coupling the polar and nonpolar free energies, and generating solvent-solute boundary in a physically self-consistent fashion. Parameter optimization is a key factor for their accuracy, predictive ability of solvation free energies, and other applications. Recently, a series of efforts have been made to improve the parameterization of these new implicit solvent models. In thiswork, we aim at studying the role of dispersion attraction in the parameterization of our DG based solvation models. To this end, we first investigate the necessity of van derWaals (vdW) dispersion interactions in the model and then carry out systematic parameterization for the model in the absence of electrostatic interactions. In particular, we explore how the changes in Lennard-Jones (L-J) potential expression, its decomposition scheme, and choices of some fixed parameter values affect the optimal values of other parameters as well as the overall modeling error. Our study on nonpolar solvation analysis offers insights into the parameterization of nonpolar components for the full DG based models by eliminating uncertainties from the electrostatic polar component. Therefore, it can be regarded as a step towards better parameterization for the full DG based model.
{"title":"Role of Dispersion Attraction in Differential Geometry Based Nonpolar Solvation Models","authors":"Zhan Chen","doi":"10.1515/mlbmb-2015-0012","DOIUrl":"https://doi.org/10.1515/mlbmb-2015-0012","url":null,"abstract":"Abstract Differential geometry (DG) based solvation models have shown their great success in solvation analysis by avoiding the use of ad hoc surface definitions, coupling the polar and nonpolar free energies, and generating solvent-solute boundary in a physically self-consistent fashion. Parameter optimization is a key factor for their accuracy, predictive ability of solvation free energies, and other applications. Recently, a series of efforts have been made to improve the parameterization of these new implicit solvent models. In thiswork, we aim at studying the role of dispersion attraction in the parameterization of our DG based solvation models. To this end, we first investigate the necessity of van derWaals (vdW) dispersion interactions in the model and then carry out systematic parameterization for the model in the absence of electrostatic interactions. In particular, we explore how the changes in Lennard-Jones (L-J) potential expression, its decomposition scheme, and choices of some fixed parameter values affect the optimal values of other parameters as well as the overall modeling error. Our study on nonpolar solvation analysis offers insights into the parameterization of nonpolar components for the full DG based models by eliminating uncertainties from the electrostatic polar component. Therefore, it can be regarded as a step towards better parameterization for the full DG based model.","PeriodicalId":34018,"journal":{"name":"Computational and Mathematical Biophysics","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2015-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/mlbmb-2015-0012","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67036542","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We present an efficient and reliable algorithm for determining the orientations of noisy images obtained fromprojections of a three-dimensional object. Based on the linear relationship among the common line vectors in one image plane, we construct a sparse matrix, and show that the coordinates of the common line vectors are the eigenvectors of the matrix with respect to the eigenvalue 1. The projection directions and in-plane rotation angles can be determined fromthese coordinates. A robust computation method of common lines in the real space using aweighted cross-correlation function is proposed to increase the robustness of the algorithm against the noise. A small number of good leading images, which have the maximal dissimilarity, are used to increase the reliability of orientations and improve the efficiency for determining the orientations of all the images. Numerical experiments show that the proposed algorithm is effective and efficient.
{"title":"Fast and Robust Orientation of Cryo-Electron Microscopy Images","authors":"Guoliang Xu, Xia Wang, Ming Li, Zhucui Jing","doi":"10.1515/mlbmb-2015-0010","DOIUrl":"https://doi.org/10.1515/mlbmb-2015-0010","url":null,"abstract":"Abstract We present an efficient and reliable algorithm for determining the orientations of noisy images obtained fromprojections of a three-dimensional object. Based on the linear relationship among the common line vectors in one image plane, we construct a sparse matrix, and show that the coordinates of the common line vectors are the eigenvectors of the matrix with respect to the eigenvalue 1. The projection directions and in-plane rotation angles can be determined fromthese coordinates. A robust computation method of common lines in the real space using aweighted cross-correlation function is proposed to increase the robustness of the algorithm against the noise. A small number of good leading images, which have the maximal dissimilarity, are used to increase the reliability of orientations and improve the efficiency for determining the orientations of all the images. Numerical experiments show that the proposed algorithm is effective and efficient.","PeriodicalId":34018,"journal":{"name":"Computational and Mathematical Biophysics","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2015-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/mlbmb-2015-0010","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67036539","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Tiantian Liu, Shiyang Bai, Bin Tu, Minxin Chen, B. Lu
Abstract We present a method of constructing the volume meshes of the membrane-channel protein system for finite element simulation of ion channels. The membrane channel system consists of the solvent region and the membrane-protein region. Our method focuses on labeling the tetrahedra in the solvent and membrane-protein regions and collecting the interface triangles between different regions. It contains two stages. Firstly, a volume mesh conforming the surface of the channel protein is generated by the surface and volume mesh generation tools: TMSmesh and TetGen. Then a walk-and-detect algorithm is used to identify the pore region to embed the membrane correctly. This method is shown to be robust because of its independence of the pore structure of the ion channels. In addition, we can also get the information of whether the ion channel is open or closed by the walk-and-detect algorithm. An on-line meshing procedure will be available at our website www.continuummodel.org.
{"title":"Membrane-Channel Protein System Mesh Construction for Finite Element Simulations","authors":"Tiantian Liu, Shiyang Bai, Bin Tu, Minxin Chen, B. Lu","doi":"10.1515/mlbmb-2015-0008","DOIUrl":"https://doi.org/10.1515/mlbmb-2015-0008","url":null,"abstract":"Abstract We present a method of constructing the volume meshes of the membrane-channel protein system for finite element simulation of ion channels. The membrane channel system consists of the solvent region and the membrane-protein region. Our method focuses on labeling the tetrahedra in the solvent and membrane-protein regions and collecting the interface triangles between different regions. It contains two stages. Firstly, a volume mesh conforming the surface of the channel protein is generated by the surface and volume mesh generation tools: TMSmesh and TetGen. Then a walk-and-detect algorithm is used to identify the pore region to embed the membrane correctly. This method is shown to be robust because of its independence of the pore structure of the ion channels. In addition, we can also get the information of whether the ion channel is open or closed by the walk-and-detect algorithm. An on-line meshing procedure will be available at our website www.continuummodel.org.","PeriodicalId":34018,"journal":{"name":"Computational and Mathematical Biophysics","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2015-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/mlbmb-2015-0008","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67036536","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Yi Jiang, Yang Xie, Jinyong Ying, Dexuan Xie, Zeyun Yu
Abstract The Poisson-Boltzmann equation (PBE) is one important implicit solvent continuum model for calculating electrostatics of protein in ionic solvent. We recently developed a PBE solver library, called SDPBS, that incorporates the finite element, finite difference, solution decomposition, domain decomposition, and multigrid methods. To make SDPBS more accessible to the scientific community, we present an SDPBS web server in this paper that allows clients to visualize and manipulate the molecular structure of a biomolecule, and to calculate PBE solutions in a remote and user friendly fashion. The web server is available on the website https://lsextrnprod.uwm.edu/electrostatics/.
{"title":"SDPBS Web Server for Calculation of Electrostatics of Ionic Solvated Biomolecules","authors":"Yi Jiang, Yang Xie, Jinyong Ying, Dexuan Xie, Zeyun Yu","doi":"10.1515/mlbmb-2015-0011","DOIUrl":"https://doi.org/10.1515/mlbmb-2015-0011","url":null,"abstract":"Abstract The Poisson-Boltzmann equation (PBE) is one important implicit solvent continuum model for calculating electrostatics of protein in ionic solvent. We recently developed a PBE solver library, called SDPBS, that incorporates the finite element, finite difference, solution decomposition, domain decomposition, and multigrid methods. To make SDPBS more accessible to the scientific community, we present an SDPBS web server in this paper that allows clients to visualize and manipulate the molecular structure of a biomolecule, and to calculate PBE solutions in a remote and user friendly fashion. The web server is available on the website https://lsextrnprod.uwm.edu/electrostatics/.","PeriodicalId":34018,"journal":{"name":"Computational and Mathematical Biophysics","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2015-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/mlbmb-2015-0011","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67036541","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract The functionality of biomolecules depends on their flexible structures, which can be characterized by their surface shapes. Tracking the deformation and comparing biomolecular shapes are essential in understanding their mechanisms. In this paper, a new spectral shape correspondence analysis method is introduced for biomolecules based on volumetric eigenfunctions. The eigenfunctions are computed from the joint graph of two given shapes, avoiding the sign flipping and confusion in the order of modes. An initial correspondence is built based on the distribution of a shape diameter, which matches similar surface features in different shapes and guides the eigenfunction computation. A two-step scheme is developed to determine the final correspondence. The first step utilizes volumetric eigenfunctions to correct the assignment of boundary nodes that disobeys the main structures. The second step minimizes the distortion induced by deforming one shape to the other. As a result, a dense point correspondence is constructed between the two given shapes, based on which we approximate and predict the shape deformation, as well as quantitatively measure the detailed shape differences.
{"title":"Shape Correspondence Analysis for Biomolecules Based on Volumetric Eigenfunctions","authors":"Tao Liao, Hao-Chih Lee, Ge Yang, Y. Zhang","doi":"10.1515/mlbmb-2015-0007","DOIUrl":"https://doi.org/10.1515/mlbmb-2015-0007","url":null,"abstract":"Abstract The functionality of biomolecules depends on their flexible structures, which can be characterized by their surface shapes. Tracking the deformation and comparing biomolecular shapes are essential in understanding their mechanisms. In this paper, a new spectral shape correspondence analysis method is introduced for biomolecules based on volumetric eigenfunctions. The eigenfunctions are computed from the joint graph of two given shapes, avoiding the sign flipping and confusion in the order of modes. An initial correspondence is built based on the distribution of a shape diameter, which matches similar surface features in different shapes and guides the eigenfunction computation. A two-step scheme is developed to determine the final correspondence. The first step utilizes volumetric eigenfunctions to correct the assignment of boundary nodes that disobeys the main structures. The second step minimizes the distortion induced by deforming one shape to the other. As a result, a dense point correspondence is constructed between the two given shapes, based on which we approximate and predict the shape deformation, as well as quantitatively measure the detailed shape differences.","PeriodicalId":34018,"journal":{"name":"Computational and Mathematical Biophysics","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2015-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/mlbmb-2015-0007","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67036342","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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