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Ullah SA, Yang X, Jones B, Zhao S, Geng W, Wei GW. Bridging Eulerian and Lagrangian Poisson-Boltzmann solvers by ESES. J Comput Chem 2024; 45:306-320. [PMID: 37830273 PMCID: PMC10993026 DOI: 10.1002/jcc.27239] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/23/2023] [Revised: 08/08/2023] [Accepted: 09/24/2023] [Indexed: 10/14/2023]
Abstract
The Poisson-Boltzmann (PB) model is a widely used electrostatic model for biomolecular solvation analysis. Formulated as an elliptic interface problem, the PB model can be numerically solved on either Eulerian meshes using finite difference/finite element methods or Lagrangian meshes using boundary element methods. Molecular surface generators, which produce the discretized dielectric interfaces between solutes and solvents, are critical factors in determining the accuracy and efficiency of the PB solvers. In this work, we investigate the utility of the Eulerian Solvent Excluded Surface (ESES) software for rendering conjugated Eulerian and Lagrangian surface representations, which enables us to numerically validate and compare the quality of Eulerian PB solvers, such as the MIBPB solver, and the Lagrangian PB solvers, such as the TABI-PB solver. Furthermore, with the ESES software and its associated PB solvers, we are able to numerically validate an interesting and useful but often neglected source-target symmetric property associated with the linearized PB model.
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Affiliation(s)
| | - Xin Yang
- Department of Mathematics, Southern Methodist University, Dallas, Texas, USA
| | - Ben Jones
- Department of Mathematics, Michigan State University, East Lansing, Michigan, USA
| | - Shan Zhao
- Department of Mathematics, University of Alabama, Tuscaloosa, Alabama, USA
| | - Weihua Geng
- Department of Mathematics, Southern Methodist University, Dallas, Texas, USA
| | - Guo-Wei Wei
- Department of Mathematics, Michigan State University, East Lansing, Michigan, USA
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Zhou YC, Argudo D, Marcoline F, Grabe M. A Computational Model of Protein Induced Membrane Morphology with Geodesic Curvature Driven Protein-Membrane Interface. JOURNAL OF COMPUTATIONAL PHYSICS 2020; 422:109755. [PMID: 32921806 PMCID: PMC7480790 DOI: 10.1016/j.jcp.2020.109755] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/11/2023]
Abstract
Continuum or hybrid modeling of bilayer membrane morphological dynamics induced by embedded proteins necessitates the identification of protein-membrane interfaces and coupling of deformations of two surfaces. In this article we developed (i) a minimal total geodesic curvature model to describe these interfaces, and (ii) a numerical one-one mapping between two surface through a conformal mapping of each surface to the common middle annulus. Our work provides the first computational tractable approach for determining the interfaces between bilayer and embedded proteins. The one-one mapping allows a convenient coupling of the morphology of two surfaces. We integrated these two new developments into the energetic model of protein-membrane interactions, and developed the full set of numerical methods for the coupled system. Numerical examples are presented to demonstrate (1) the efficiency and robustness of our methods in locating the curves with minimal total geodesic curvature on highly complicated protein surfaces, (2) the usefulness of these interfaces as interior boundaries for membrane deformation, and (3) the rich morphology of bilayer surfaces for different protein-membrane interfaces.
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Affiliation(s)
- Y. C. Zhou
- Department of Mathematics, Colorado State University, Fort Collins, CO 80523
| | - David Argudo
- Department of Pharmaceutical Chemistry and Cardiovascular Research Institute, University of California, San Francisco, CA 94143
| | - Frank Marcoline
- Department of Pharmaceutical Chemistry and Cardiovascular Research Institute, University of California, San Francisco, CA 94143
| | - Michael Grabe
- Department of Pharmaceutical Chemistry and Cardiovascular Research Institute, University of California, San Francisco, CA 94143
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Borleske G, Zhou Y. Enriched gradient recovery for interface solutions of the Poisson-Boltzmann equation. JOURNAL OF COMPUTATIONAL PHYSICS 2020; 421:109725. [PMID: 32884156 PMCID: PMC7461612 DOI: 10.1016/j.jcp.2020.109725] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/11/2023]
Abstract
Accurate calculation of electrostatic potential and gradient on the molecular surface is highly desirable for the continuum and hybrid modeling of large scale deformation of biomolecules in solvent. In this article a new numerical method is proposed to calculate these quantities on the dielectric interface from the numerical solutions of the Poisson-Boltzmann equation. Our method reconstructs a potential field locally in the least square sense on the polynomial basis enriched with Green's functions, the latter characterize the Coulomb potential induced by charges near the position of reconstruction. This enrichment resembles the decomposition of electrostatic potential into singular Coulomb component and the regular reaction field in the Generalized Born methods. Numerical experiments demonstrate that the enrichment recovery produces drastically more accurate and stable potential gradients on molecular surfaces compared to classical recovery techniques.
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Zhao R, Wang M, Chen J, Tong Y, Wei GW. The de Rham-Hodge Analysis and Modeling of Biomolecules. Bull Math Biol 2020; 82:108. [PMID: 32770408 PMCID: PMC8137271 DOI: 10.1007/s11538-020-00783-2] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/27/2019] [Accepted: 07/20/2020] [Indexed: 12/18/2022]
Abstract
Biological macromolecules have intricate structures that underpin their biological functions. Understanding their structure-function relationships remains a challenge due to their structural complexity and functional variability. Although de Rham-Hodge theory, a landmark of twentieth-century mathematics, has had a tremendous impact on mathematics and physics, it has not been devised for macromolecular modeling and analysis. In this work, we introduce de Rham-Hodge theory as a unified paradigm for analyzing the geometry, topology, flexibility, and Hodge mode analysis of biological macromolecules. Geometric characteristics and topological invariants are obtained either from the Helmholtz-Hodge decomposition of the scalar, vector, and/or tensor fields of a macromolecule or from the spectral analysis of various Laplace-de Rham operators defined on the molecular manifolds. We propose Laplace-de Rham spectral-based models for predicting macromolecular flexibility. We further construct a Laplace-de Rham-Helfrich operator for revealing cryo-EM natural frequencies. Extensive experiments are carried out to demonstrate that the proposed de Rham-Hodge paradigm is one of the most versatile tools for the multiscale modeling and analysis of biological macromolecules and subcellular organelles. Accurate, reliable, and topological structure-preserving algorithms for implementing discrete exterior calculus (DEC) have been developed to facilitate the aforementioned modeling and analysis of biological macromolecules. The proposed de Rham-Hodge paradigm has potential applications to subcellular organelles and the structure construction from medium- or low-resolution cryo-EM maps, and functional predictions from massive biomolecular datasets.
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Affiliation(s)
- Rundong Zhao
- Department of Computer Science and Engineering, Michigan State University, East Lansing, MI, 48824, USA
| | - Menglun Wang
- Department of Mathematics, Michigan State University, East Lansing, MI, 48824, USA
| | - Jiahui Chen
- Department of Mathematics, Michigan State University, East Lansing, MI, 48824, USA
| | - Yiying Tong
- Department of Computer Science and Engineering, Michigan State University, East Lansing, MI, 48824, USA.
| | - Guo-Wei Wei
- Department of Mathematics, Michigan State University, East Lansing, MI, 48824, USA.
- Department of Electrical and Computer Engineering, Michigan State University, East Lansing, MI, 48824, USA.
- Department of Biochemistry and Molecular Biology, Michigan State University, East Lansing, MI, 48824, USA.
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Alimohamadi H, Rangamani P. Modeling Membrane Curvature Generation due to Membrane⁻Protein Interactions. Biomolecules 2018; 8:E120. [PMID: 30360496 PMCID: PMC6316661 DOI: 10.3390/biom8040120] [Citation(s) in RCA: 50] [Impact Index Per Article: 8.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/04/2018] [Revised: 10/15/2018] [Accepted: 10/16/2018] [Indexed: 01/03/2023] Open
Abstract
To alter and adjust the shape of the plasma membrane, cells harness various mechanisms of curvature generation. Many of these curvature generation mechanisms rely on the interactions between peripheral membrane proteins, integral membrane proteins, and lipids in the bilayer membrane. Mathematical and computational modeling of membrane curvature generation has provided great insights into the physics underlying these processes. However, one of the challenges in modeling these processes is identifying the suitable constitutive relationships that describe the membrane free energy including protein distribution and curvature generation capability. Here, we review some of the commonly used continuum elastic membrane models that have been developed for this purpose and discuss their applications. Finally, we address some fundamental challenges that future theoretical methods need to overcome to push the boundaries of current model applications.
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Affiliation(s)
- Haleh Alimohamadi
- Department of Mechanical and Aerospace Engineering, University of California, San Diego, CA 92093, USA.
| | - Padmini Rangamani
- Department of Mechanical and Aerospace Engineering, University of California, San Diego, CA 92093, USA.
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Membrane proteins structures: A review on computational modeling tools. BIOCHIMICA ET BIOPHYSICA ACTA-BIOMEMBRANES 2017; 1859:2021-2039. [DOI: 10.1016/j.bbamem.2017.07.008] [Citation(s) in RCA: 62] [Impact Index Per Article: 8.9] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/23/2017] [Revised: 07/04/2017] [Accepted: 07/13/2017] [Indexed: 01/02/2023]
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Mikucki M, Zhou Y. Fast Simulation of Lipid Vesicle Deformation Using Spherical Harmonic Approximation. COMMUNICATIONS IN COMPUTATIONAL PHYSICS 2017; 21:40-64. [PMID: 28804520 PMCID: PMC5552105 DOI: 10.4208/cicp.oa-2015-0029] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
Lipid vesicles appear ubiquitously in biological systems. Understanding how the mechanical and intermolecular interactions deform vesicle membranes is a fundamental question in biophysics. In this article we develop a fast algorithm to compute the surface configurations of lipid vesicles by introducing surface harmonic functions to approximate the membrane surface. This parameterization allows an analytical computation of the membrane curvature energy and its gradient for the efficient minimization of the curvature energy using a nonlinear conjugate gradient method. Our approach drastically reduces the degrees of freedom for approximating the membrane surfaces compared to the previously developed finite element and finite difference methods. Vesicle deformations with a reduced volume larger than 0.65 can be well approximated by using as small as 49 surface harmonic functions. The method thus has a great potential to reduce the computational expense of tracking multiple vesicles which deform for their interaction with external fields.
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Affiliation(s)
- Michael Mikucki
- Department of Applied Mathematics & Statistics, Colorado
School of Mines, Golden, Colorado, 80401, USA
| | - Yongcheng Zhou
- Department of Mathematics, Colorado State University, Fort Collins,
Colorado, 80523, USA
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Argudo D, Bethel NP, Marcoline FV, Grabe M. Continuum descriptions of membranes and their interaction with proteins: Towards chemically accurate models. BIOCHIMICA ET BIOPHYSICA ACTA 2016; 1858:1619-34. [PMID: 26853937 PMCID: PMC4877259 DOI: 10.1016/j.bbamem.2016.02.003] [Citation(s) in RCA: 30] [Impact Index Per Article: 3.8] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/19/2015] [Revised: 01/30/2016] [Accepted: 02/01/2016] [Indexed: 01/21/2023]
Abstract
Biological membranes deform in response to resident proteins leading to a coupling between membrane shape and protein localization. Additionally, the membrane influences the function of membrane proteins. Here we review contributions to this field from continuum elastic membrane models focusing on the class of models that couple the protein to the membrane. While it has been argued that continuum models cannot reproduce the distortions observed in fully-atomistic molecular dynamics simulations, we suggest that this failure can be overcome by using chemically accurate representations of the protein. We outline our recent advances along these lines with our hybrid continuum-atomistic model, and we show the model is in excellent agreement with fully-atomistic simulations of the nhTMEM16 lipid scramblase. We believe that the speed and accuracy of continuum-atomistic methodologies will make it possible to simulate large scale, slow biological processes, such as membrane morphological changes, that are currently beyond the scope of other computational approaches. This article is part of a Special Issue entitled: Membrane Proteins edited by J.C. Gumbart and Sergei Noskov.
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Affiliation(s)
- David Argudo
- Cardiovascular Research Institute, Department of Pharmaceutical Chemistry, University of California San Francisco, San Francisco, CA 94158, United States
| | - Neville P Bethel
- Cardiovascular Research Institute, Department of Pharmaceutical Chemistry, University of California San Francisco, San Francisco, CA 94158, United States
| | - Frank V Marcoline
- Cardiovascular Research Institute, Department of Pharmaceutical Chemistry, University of California San Francisco, San Francisco, CA 94158, United States
| | - Michael Grabe
- Cardiovascular Research Institute, Department of Pharmaceutical Chemistry, University of California San Francisco, San Francisco, CA 94158, United States.
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Cheng X, Jo S, Qi Y, Marassi FM, Im W. Solid-State NMR-Restrained Ensemble Dynamics of a Membrane Protein in Explicit Membranes. Biophys J 2016; 108:1954-62. [PMID: 25902435 DOI: 10.1016/j.bpj.2015.03.012] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/17/2014] [Revised: 02/05/2015] [Accepted: 03/10/2015] [Indexed: 10/23/2022] Open
Abstract
Solid-state NMR has been used to determine the structures of membrane proteins in native-like lipid bilayer environments. Most structure calculations based on solid-state NMR observables are performed using simulated annealing with restrained molecular dynamics and an energy function, where all nonbonded interactions are represented by a single, purely repulsive term with no contributions from van der Waals attractive, electrostatic, or solvation energy. To our knowledge, this is the first application of an ensemble dynamics technique performed in explicit membranes that uses experimental solid-state NMR observables to obtain the refined structure of a membrane protein together with information about its dynamics and its interactions with lipids. Using the membrane-bound form of the fd coat protein as a model membrane protein and its experimental solid-state NMR data, we performed restrained ensemble dynamics simulations with different ensemble sizes in explicit membranes. For comparison, a molecular dynamics simulation of fd coat protein was also performed without any restraints. The average orientation of each protein helix is similar to a structure determined by traditional single-conformer approaches. However, their variations are limited in the resulting ensemble of structures with one or two replicas, as they are under the strong influence of solid-state NMR restraints. Although highly consistent with all solid-state NMR observables, the ensembles of more than two replicas show larger orientational variations similar to those observed in the molecular dynamics simulation without restraints. In particular, in these explicit membrane simulations, Lys(40), residing at the C-terminal side of the transmembrane helix, is observed to cause local membrane curvature. Therefore, compared to traditional single-conformer approaches in implicit environments, solid-state NMR restrained ensemble simulations in explicit membranes readily characterize not only protein dynamics but also protein-lipid interactions in detail.
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Affiliation(s)
- Xi Cheng
- Department of Molecular Biosciences and Center for Computational Biology, The University of Kansas, Lawrence, Kansas
| | - Sunhwan Jo
- Department of Molecular Biosciences and Center for Computational Biology, The University of Kansas, Lawrence, Kansas
| | - Yifei Qi
- Department of Molecular Biosciences and Center for Computational Biology, The University of Kansas, Lawrence, Kansas
| | | | - Wonpil Im
- Department of Molecular Biosciences and Center for Computational Biology, The University of Kansas, Lawrence, Kansas.
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Lee KI, Pastor RW, Andersen OS, Im W. Assessing smectic liquid-crystal continuum models for elastic bilayer deformations. Chem Phys Lipids 2013; 169:19-26. [PMID: 23348553 DOI: 10.1016/j.chemphyslip.2013.01.005] [Citation(s) in RCA: 14] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/14/2012] [Revised: 01/08/2013] [Accepted: 01/10/2013] [Indexed: 10/27/2022]
Abstract
For four decades, since W. Helfrich's pioneering study of smectic A liquid crystals in 1973, continuum elastic models (CEMs) have been employed as tools to understand the energetics of protein-induced lipid bilayer deformations. Among the assumptions underlying this use is that all relevant protein-lipid interactions can be included in the continuum representation of the protein-bilayer interactions through the physical parameters determined for protein-free bilayers and the choice of boundary conditions at the protein/bilayer interface. To better understand this assumption, we review the general structure of CEMs, examine how different choices of boundary conditions and physical moduli profiles alter the predicted bilayer thickness profiles around gramicidin A (gA) and mitochondrial voltage-dependent anion channels (VDAC), respectively, and compare these profiles with those obtained from all-atom molecular dynamics simulations. We find that the profiles differ qualitatively in the first lipid shell around the channels, indicating that the CEMs do not capture accurately the consequences of the protein-induced local changes in lipid bilayer dynamics. Therefore, one needs to be careful when interpreting the results of CEM-based analyses of lipid bilayer-membrane protein interactions.
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Affiliation(s)
- Kyu Ii Lee
- Department of Molecular Biosciences and Center for Bioinformatics, The University of Kansas, Lawrence, KS 66047, USA
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Callenberg KM, Latorraca NR, Grabe M. Membrane bending is critical for the stability of voltage sensor segments in the membrane. ACTA ACUST UNITED AC 2012; 140:55-68. [PMID: 22732310 PMCID: PMC3382720 DOI: 10.1085/jgp.201110766] [Citation(s) in RCA: 28] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/29/2022]
Abstract
The interaction between membrane proteins and the surrounding membrane is becoming increasingly appreciated for its role in regulating protein function, protein localization, and membrane morphology. In particular, recent studies have suggested that membrane deformation is needed to stably accommodate proteins harboring charged amino acids in their transmembrane (TM) region, as it is energetically prohibitive to bury charge in the hydrophobic core of the bilayer. Unfortunately, current computational methods are poorly equipped for describing such deformations, as atomistic simulations are often too short to observe large-scale membrane reorganization and most continuum approaches assume a flat membrane. Previously, we developed a method that overcomes these shortcomings by using elasticity theory to characterize equilibrium membrane distortions in the presence of a TM protein, while using traditional continuum electrostatic and nonpolar energy models to determine the energy of the protein in the membrane. Here, we linked the elastostatics, electrostatics, and nonpolar numeric solvers to permit the calculation of energies for nontrivial membrane deformations. We then coupled this procedure to a robust search algorithm that identifies optimal membrane shapes for a TM protein of arbitrary chemical composition. This advance now permits us to explore a host of biological phenomena that were beyond the scope of our original method. We show that the energy required to embed charged residues in the membrane can be highly nonadditive, and our model provides a simple mechanical explanation for this nonadditivity. Our results also predict that isolated voltage sensor segments do not insert into rigid membranes, but membrane bending dramatically stabilizes these proteins in the bilayer despite their high charge content. Additionally, we use the model to explore hydrophobic mismatch with regard to nonpolar peptides and mechanosensitive channels. Our method is in quantitative agreement with molecular dynamics simulations at a tiny fraction of the computational cost.
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Affiliation(s)
- Keith M Callenberg
- Joint Carnegie Mellon University-University of Pittsburgh PhD Program in Computational Biology, Pittsburgh, PA 15213, USA
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