Density-cluster NMA: A new protein decomposition technique for coarse-grained normal mode analysis

Multiscale weighted colored graphs for protein flexibility and rigidity analysis

Protein structural fluctuation, measured by Debye-Waller factors or B-factors, is known to correlate to protein flexibility and function. A variety of methods has been developed for protein Debye-Waller factor prediction and related applications to domain separation, docking pose ranking, entropy calculation, hinge detection, stability analysis, etc. Nevertheless, none of the current methodologies are able to deliver an accuracy of 0.7 in terms of the Pearson correlation coefficients averaged over a large set of proteins. In this work, we introduce a paradigm-shifting geometric graph model, multiscale weighted colored graph (MWCG), to provide a new generation of computational algorithms to significantly change the current status of protein structural fluctuation analysis. Our MWCG model divides a protein graph into multiple subgraphs based on interaction types between graph nodes and represents the protein rigidity by generalized centralities of subgraphs. MWCGs not only predict the B-factors of protein residues but also accurately analyze the flexibility of all atoms in a protein. The MWCG model is validated over a number of protein test sets and compared with many standard methods. An extensive numerical study indicates that the proposed MWCG offers an accuracy of over 0.8 and thus provides perhaps the first reliable method for estimating protein flexibility and B-factors. It also simultaneously predicts all-atom flexibility in a molecule.

NOLB: Nonlinear Rigid Block Normal-Mode Analysis Method

We present a new conceptually simple and computationally efficient method for nonlinear normal-mode analysis called NOLB. It relies on the rotations-translations of blocks (RTB) theoretical basis developed by Y.-H. Sanejouand and colleagues [ Durand et al. Biopolymers 1994 , 34 , 759 - 771 . Tama et al. Proteins: Struct., Funct., Bioinf . 2000 , 41 , 1 - 7 ]. We demonstrate how to physically interpret the eigenvalues computed in the RTB basis in terms of angular and linear velocities applied to the rigid blocks and how to construct a nonlinear extrapolation of motion out of these velocities. The key observation of our method is that the angular velocity of a rigid block can be interpreted as the result of an implicit force, such that the motion of the rigid block can be considered as a pure rotation about a certain center. We demonstrate the motions produced with the NOLB method on three different molecular systems and show that some of the lowest frequency normal modes correspond to the biologically relevant motions. For example, NOLB detects the spiral sliding motion of the TALE protein, which is capable of rapid diffusion along its target DNA. Overall, our method produces better structures compared to the standard approach, especially at large deformation amplitudes, as we demonstrate by visual inspection, energy, and topology analyses and also by the MolProbity service validation. Finally, our method is scalable and can be applied to very large molecular systems, such as ribosomes. Standalone executables of the NOLB normal-mode analysis method are available at https://team.inria.fr/nano-d/software/nolb-normal-modes/ . A graphical user interface created for the SAMSON software platform will be made available at https://www.samson-connect.net .

Generalized flexibility-rigidity index

Flexibility-rigidity index (FRI) has been developed as a robust, accurate, and efficient method for macromolecular thermal fluctuation analysis and B-factor prediction. The performance of FRI depends on its formulations of rigidity index and flexibility index. In this work, we introduce alternative rigidity and flexibility formulations. The structure of the classic Gaussian surface is utilized to construct a new type of rigidity index, which leads to a new class of rigidity densities with the classic Gaussian surface as a special case. Additionally, we introduce a new type of flexibility index based on the domain indicator property of normalized rigidity density. These generalized FRI (gFRI) methods have been extensively validated by the B-factor predictions of 364 proteins. Significantly outperforming the classic Gaussian network model, gFRI is a new generation of methodologies for accurate, robust, and efficient analysis of protein flexibility and fluctuation. Finally, gFRI based molecular surface generation and flexibility visualization are demonstrated.

Flexibility-rigidity index for protein-nucleic acid flexibility and fluctuation analysis

Protein-nucleic acid complexes are important for many cellular processes including the most essential functions such as transcription and translation. For many protein-nucleic acid complexes, flexibility of both macromolecules has been shown to be critical for specificity and/or function. The flexibility-rigidity index (FRI) has been proposed as an accurate and efficient approach for protein flexibility analysis. In this article, we introduce FRI for the flexibility analysis of protein-nucleic acid complexes. We demonstrate that a multiscale strategy, which incorporates multiple kernels to capture various length scales in biomolecular collective motions, is able to significantly improve the state of art in the flexibility analysis of protein-nucleic acid complexes. We take the advantage of the high accuracy and O(N) computational complexity of our multiscale FRI method to investigate the flexibility of ribosomal subunits, which are difficult to analyze by alternative approaches. An anisotropic FRI approach, which involves localized Hessian matrices, is utilized to study the translocation dynamics in an RNA polymerase.

Communication: Capturing protein multiscale thermal fluctuations

Existing elastic network models are typically parametrized at a given cutoff distance and often fail to properly predict the thermal fluctuation of many macromolecules that involve multiple characteristic length scales. We introduce a multiscale flexibility-rigidity index (mFRI) method to resolve this problem. The proposed mFRI utilizes two or three correlation kernels parametrized at different length scales to capture protein interactions at corresponding scales. It is about 20% more accurate than the Gaussian network model (GNM) in the B-factor prediction of a set of 364 proteins. Additionally, the present method is able to deliver accurate predictions for some large macromolecules on which GNM fails to produce accurate predictions. Finally, for a protein of N residues, mFRI is of linear scaling (O(N)) in computational complexity, in contrast to the order of O(N(3)) for GNM.

New generation of elastic network models

The intrinsic flexibility of proteins and nucleic acids can be grasped from remarkably simple mechanical models of particles connected by springs. In recent decades, Elastic Network Models (ENMs) combined with Normal Model Analysis widely confirmed their ability to predict biologically relevant motions of biomolecules and soon became a popular methodology to reveal large-scale dynamics in multiple structural biology scenarios. The simplicity, robustness, low computational cost, and relatively high accuracy are the reasons behind the success of ENMs. This review focuses on recent advances in the development and application of ENMs, paying particular attention to combinations with experimental data. Successful application scenarios include large macromolecular machines, structural refinement, docking, and evolutionary conservation.

Computation of the memory functions in the generalized Langevin models for collective dynamics of macromolecules

We present a numerical method to approximate the memory functions in the generalized Langevin models for the collective dynamics of macromolecules. We first derive the exact expressions of the memory functions, obtained from projection to subspaces that correspond to the selection of coarse-grain variables. In particular, the memory functions are expressed in the forms of matrix functions, which will then be approximated by Krylov-subspace methods. It will also be demonstrated that the random noise can be approximated under the same framework, and the second fluctuation-dissipation theorem is automatically satisfied. The accuracy of the method is examined through several numerical examples.

Fast and anisotropic flexibility-rigidity index for protein flexibility and fluctuation analysis

Protein structural fluctuation, typically measured by Debye-Waller factors, or B-factors, is a manifestation of protein flexibility, which strongly correlates to protein function. The flexibility-rigidity index (FRI) is a newly proposed method for the construction of atomic rigidity functions required in the theory of continuum elasticity with atomic rigidity, which is a new multiscale formalism for describing excessively large biomolecular systems. The FRI method analyzes protein rigidity and flexibility and is capable of predicting protein B-factors without resorting to matrix diagonalization. A fundamental assumption used in the FRI is that protein structures are uniquely determined by various internal and external interactions, while the protein functions, such as stability and flexibility, are solely determined by the structure. As such, one can predict protein flexibility without resorting to the protein interaction Hamiltonian. Consequently, bypassing the matrix diagonalization, the original FRI has a computational complexity of O(N(2)). This work introduces a fast FRI (fFRI) algorithm for the flexibility analysis of large macromolecules. The proposed fFRI further reduces the computational complexity to O(N). Additionally, we propose anisotropic FRI (aFRI) algorithms for the analysis of protein collective dynamics. The aFRI algorithms permit adaptive Hessian matrices, from a completely global 3N × 3N matrix to completely local 3 × 3 matrices. These 3 × 3 matrices, despite being calculated locally, also contain non-local correlation information. Eigenvectors obtained from the proposed aFRI algorithms are able to demonstrate collective motions. Moreover, we investigate the performance of FRI by employing four families of radial basis correlation functions. Both parameter optimized and parameter-free FRI methods are explored. Furthermore, we compare the accuracy and efficiency of FRI with some established approaches to flexibility analysis, namely, normal mode analysis and Gaussian network model (GNM). The accuracy of the FRI method is tested using four sets of proteins, three sets of relatively small-, medium-, and large-sized structures and an extended set of 365 proteins. A fifth set of proteins is used to compare the efficiency of the FRI, fFRI, aFRI, and GNM methods. Intensive validation and comparison indicate that the FRI, particularly the fFRI, is orders of magnitude more efficient and about 10% more accurate overall than some of the most popular methods in the field. The proposed fFRI is able to predict B-factors for α-carbons of the HIV virus capsid (313 236 residues) in less than 30 seconds on a single processor using only one core. Finally, we demonstrate the application of FRI and aFRI to protein domain analysis.

Molecular nonlinear dynamics and protein thermal uncertainty quantification

This work introduces molecular nonlinear dynamics (MND) as a new approach for describing protein folding and aggregation. By using a mode system, we show that the MND of disordered proteins is chaotic while that of folded proteins exhibits intrinsically low dimensional manifolds (ILDMs). The stability of ILDMs is found to strongly correlate with protein energies. We propose a novel method for protein thermal uncertainty quantification based on persistently invariant ILDMs. Extensive comparison with experimental data and the state-of-the-art methods in the field validate the proposed new method for protein B-factor prediction.

Multiscale multiphysics and multidomain models--flexibility and rigidity

The emerging complexity of large macromolecules has led to challenges in their full scale theoretical description and computer simulation. Multiscale multiphysics and multidomain models have been introduced to reduce the number of degrees of freedom while maintaining modeling accuracy and achieving computational efficiency. A total energy functional is constructed to put energies for polar and nonpolar solvation, chemical potential, fluid flow, molecular mechanics, and elastic dynamics on an equal footing. The variational principle is utilized to derive coupled governing equations for the above mentioned multiphysical descriptions. Among these governing equations is the Poisson-Boltzmann equation which describes continuum electrostatics with atomic charges. The present work introduces the theory of continuum elasticity with atomic rigidity (CEWAR). The essence of CEWAR is to formulate the shear modulus as a continuous function of atomic rigidity. As a result, the dynamics complexity of a macromolecular system is separated from its static complexity so that the more time-consuming dynamics is handled with continuum elasticity theory, while the less time-consuming static analysis is pursued with atomic approaches. We propose a simple method, flexibility-rigidity index (FRI), to analyze macromolecular flexibility and rigidity in atomic detail. The construction of FRI relies on the fundamental assumption that protein functions, such as flexibility, rigidity, and energy, are entirely determined by the structure of the protein and its environment, although the structure is in turn determined by all the interactions. As such, the FRI measures the topological connectivity of protein atoms or residues and characterizes the geometric compactness of the protein structure. As a consequence, the FRI does not resort to the interaction Hamiltonian and bypasses matrix diagonalization, which underpins most other flexibility analysis methods. FRI's computational complexity is of O(N(2)) at most, where N is the number of atoms or residues, in contrast to O(N(3)) for Hamiltonian based methods. We demonstrate that the proposed FRI gives rise to accurate prediction of protein B-Factor for a set of 263 proteins. We show that a parameter free FRI is able to achieve about 95% accuracy of the parameter optimized FRI. An interpolation algorithm is developed to construct continuous atomic flexibility functions for visualization and use with CEWAR.

Modeling protein conformational transitions by a combination of coarse-grained normal mode analysis and robotics-inspired methods

BACKGROUND

Obtaining atomic-scale information about large-amplitude conformational transitions in proteins is a challenging problem for both experimental and computational methods. Such information is, however, important for understanding the mechanisms of interaction of many proteins.

METHODS

This paper presents a computationally efficient approach, combining methods originating from robotics and computational biophysics, to model protein conformational transitions. The ability of normal mode analysis to predict directions of collective, large-amplitude motions is applied to bias the conformational exploration performed by a motion planning algorithm. To reduce the dimension of the problem, normal modes are computed for a coarse-grained elastic network model built on short fragments of three residues. Nevertheless, the validity of intermediate conformations is checked using the all-atom model, which is accurately reconstructed from the coarse-grained one using closed-form inverse kinematics.

RESULTS

Tests on a set of ten proteins demonstrate the ability of the method to model conformational transitions of proteins within a few hours of computing time on a single processor. These results also show that the computing time scales linearly with the protein size, independently of the protein topology. Further experiments on adenylate kinase show that main features of the transition between the open and closed conformations of this protein are well captured in the computed path.

CONCLUSIONS

The proposed method enables the simulation of large-amplitude conformational transitions in proteins using very few computational resources. The resulting paths are a first approximation that can directly provide important information on the molecular mechanisms involved in the conformational transition. This approximation can be subsequently refined and analyzed using state-of-the-art energy models and molecular modeling methods.

Efficient prediction of protein conformational pathways based on the hybrid elastic network model

Various computational models have gained immense attention by analyzing the dynamic characteristics of proteins. Several models have achieved recognition by fulfilling either theoretical or experimental predictions. Nonetheless, each method possesses limitations, mostly in computational outlay and physical reality. These limitations remind us that a new model or paradigm should advance theoretical principles to elucidate more precisely the biological functions of a protein and should increase computational efficiency. With these critical caveats, we have developed a new computational tool that satisfies both physical reality and computational efficiency. In the proposed hybrid elastic network model (HENM), a protein structure is represented as a mixture of rigid clusters and point masses that are connected with linear springs. Harmonic analyses based on the HENM have been performed to generate normal modes and conformational pathways. The results of the hybrid normal mode analyses give new physical insight to the 70S ribosome. The feasibility of the conformational pathways of hybrid elastic network interpolation (HENI) was quantitatively evaluated by comparing three different overlap values proposed in this paper. A remarkable observation is that the obtained mode shapes and conformational pathways are consistent with each other. Our timing results show that HENM has some advantage in computational efficiency over a coarse-grained model, especially for large proteins, even though it takes longer to construct the HENM. Consequently, the proposed HENM will be one of the best alternatives to the conventional coarse-grained ENMs and all-atom based methods (such as molecular dynamics) without loss of physical reality.

Coarse grained normal mode analysis vs. refined Gaussian Network Model for protein residue-level structural fluctuations

We investigate several approaches to coarse grained normal mode analysis on protein residual-level structural fluctuations by choosing different ways of representing the residues and the forces among them. Single-atom representations using the backbone atoms C(α), C, N, and C(β) are considered. Combinations of some of these atoms are also tested. The force constants between the representative atoms are extracted from the Hessian matrix of the energy function and served as the force constants between the corresponding residues. The residue mean-square-fluctuations and their correlations with the experimental B-factors are calculated for a large set of proteins. The results are compared with all-atom normal mode analysis and the residue-level Gaussian Network Model. The coarse-grained methods perform more efficiently than all-atom normal mode analysis, while their B-factor correlations are also higher. Their B-factor correlations are comparable with those estimated by the Gaussian Network Model and in many cases better. The extracted force constants are surveyed for different pairs of residues with different numbers of separation residues in sequence. The statistical averages are used to build a refined Gaussian Network Model, which is able to predict residue-level structural fluctuations significantly better than the conventional Gaussian Network Model in many test cases.