Diagonalization in a mixed basis: A method to compute low-frequency normal modes for large macromolecules

Low-frequency collective motions in proteins are generally very important for their biological functions. To study such motions, harmonic dynamics proved most useful since it is a straightforward method; it consists of the diagonalization of the Hessian matrix of the potential energy, yielding the vibrational spectrum and the directions of internal motions. Unfortunately, the diagonalization of this matrix requires a large computer memory, which is a limiting factor when the protein contains several thousand atoms. To circumvent this limitation we have developed three methods that enable us to diagonalize large matrices using much less computer memory than the usual harmonic dynamics. The first method is approximate; it consists of diagonalizing small blocks of the Hessian matrix, followed by the coupling of the low-frequency modes obtained for each block. It yields the low-frequency vibrational spectrum with a maximum error of 20%. The second method consists, after diagonalizing small blocks, of coupling the high- and low-frequency modes using an iterative procedure. It yields the exact low-frequency normal modes, but requires a long computational time with convergence problems. The third method, DIMB (Diagonalization in a Mixed Basis), which has the best performance, consists of coupling the approximate low-frequency modes with the mass-weighted cartesian coordinates, also using an iterative procedure. It reduces significantly the required computer memory and converges rapidly. The eigenvalues and eigenvectors obtained by this method are without significant error in the chosen frequency range. Moreover, it is a general method applicable to any problem of diagonalization of a large matrix. We report the application of these methods to a deca-alanine helix, trypsin inhibitor, a neurotoxin, and lysozyme. © 1993 John Wiley & Sons, Inc.     

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

Normal mode analysis has emerged as a useful technique for investigating protein motions on long time scales. This is largely due to the advent of coarse-graining techniques, particularly Hooke's Law-based potentials and the rotational-translational blocking (RTB) method for reducing the size of the force-constant matrix, the Hessian. Here we present a new method for domain decomposition for use in RTB that is based on hierarchical clustering of atomic density gradients, which we call Density-Cluster RTB (DCRTB). The method reduces the number of degrees of freedom by 85-90% compared with the standard blocking approaches. We compared the normal modes from DCRTB against standard RTB using 1-4 residues in sequence in a single block, with good agreement between the two methods. We also show that Density-Cluster RTB and standard RTB perform well in capturing the experimentally determined direction of conformational change. Significantly, we report superior correlation of DCRTB with B-factors compared with 1-4 residue per block RTB. Finally, we show significant reduction in computational cost for Density-Cluster RTB that is nearly 100-fold for many examples.     

A coarse-grained normal mode approach for macromolecules: an efficient implementation and application to Ca(2+)-ATPase

A block normal mode (BNM) algorithm, originally proposed by Tama et al., (Proteins Struct. Func. Genet. 41:1-7, 2000) was implemented into the simulation program CHARMM. The BNM approach projects the hessian matrix into local translation/rotation basis vectors and, therefore, dramatically reduces the size of the matrix involved in diagonalization. In the current work, by constructing the atomic hessian elements required in the projection operation on the fly, the memory requirement for the BNM approach has been significantly reduced from that of standard normal mode analysis and previous implementation of BNM. As a result, low frequency modes, which are of interest in large-scale conformational changes of large proteins or protein-nucleic acid complexes, can be readily obtained. Comparison of the BNM results with standard normal mode analysis for a number of small proteins and nucleic acids indicates that many properties dominated by low frequency motions are well reproduced by BNM; these include atomic fluctuations, the displacement covariance matrix, vibrational entropies, and involvement coefficients for conformational transitions. Preliminary application to a fairly large system, Ca(2+)-ATPase (994 residues), is described as an example. The structural flexibility of the cytoplasmic domains (especially domain N), correlated motions among residues on domain interfaces and displacement patterns for the transmembrane helices observed in the BNM results are discussed in relation to the function of Ca(2+)-ATPase. The current implementation of the BNM approach has paved the way for developing efficient sampling algorithms with molecular dynamics or Monte Carlo for studying long-time scale dynamics of macromolecules.     

A new approach for determining low-frequency normal modes in macromolecules

A new method for calculating a set of low-frequency normal modes in macromolecules is proposed and applied to the case of proteins. In a first step, the protein chain is partitioned into blocks of one or more residues and the low-frequency modes are evaluated at a low-resolution level by combining the local translations and rotations of each block. In a second step, these low-resolution modes are perturbed by high-frequency modes explicitly calculated in each block, thus leading to the exact low-frequency modes. The procedure is tested for three cases–decaalanine, icosaleucin, and crambin–using a perturbation-iteration scheme in the second step. Convergence properties and numerical accuracy are assessed and tested for various partitions. The low-resolution modes obtained in the first step are always found to be good starting approximations. Potential advantages of the method include a central processing unit time roughly N² dependent on the size of the problem (N being the number of degrees of freedom), the possibility of using parallel processing, the nonrequirement for loading the complete mass-weighted second-derivative input matrix into central memory, and the possibility of introducing in the procedure further structural hierarchy, such as secondary structures or motifs. In addition, any improvement or refinement of the algorithm benefits from the efficient formalism of the effective Hamiltonian theory. © 1994 John Wiley & Sons, Inc.     

Analyzing the normal mode dynamics of macromolecules by the component synthesis method: Residue clustering and multiple-component approach

A study of the component synthesis method (CSM) for analyzing the normal mode dynamics of macromolecules is reported. The procedure involves a reduction of the dimensions of the normal mode problems for large molecular systems and the accurate extraction of the low-frequency modes. A macromolecule is divided into small components based on a hierarchical clustering of the residues in the structure. Interactions between coupled components are treated by the method of static correlation. The normal modes of the components are obtained first, and a fraction of the low-frequency normal modes of the components under mutual correlations are then used as a reduced basis for solving for the normal modes of the whole molecule. Multiple components are introduced for large macromolecules so that the dimensions of the eigenvalue problems at the component level are small. The method is applied to the protein crambin. In test calculations in which the dimensions of the eigenvalue equations are reduced to 1/6 of their natural size, the errors in the normal mode frequencies calculated by the CSM procedure are only about 1–2% when compared with the exact values. The rms fluctuations of all atoms in crambin calculated by the CSM procedure are basically identical to the exact results. The CSM procedure is shown to be accurate for calculating the normal modes of large macromolecules with a significant reduction of the size of the problem. © 1994 John Wiley & Sons, Inc.     

Influence of the Amino-Acid Sequence on the Inverse Temperature Transition of Elastin-Like Polymers

This work explores the dependence of the inverse temperature transition of elastin-like polymers (ELPs) on the amino-acid sequence, i.e., the amino-acid arrangement along the macromolecule and the resulting linear distribution of the physical properties (mainly polarity) derived from it. The hypothesis of this work is that, in addition to mean polarity and molecular mass, the given amino-acid sequence, or its equivalent—the way in which polarity is arranged along the molecule—is also relevant for determining the transition temperature and the latent heat of that transition. To test this hypothesis, a set of linear and di- and triblock ELP copolymers were designed and produced as recombinant proteins. The absolute sequence control provided by recombinant technologies allows the effect of the amino-acid arrangement to be isolated while keeping the molecular mass or mean polarity under strict control. The selected block copolymers were made of two different ELPs: one exhibiting temperature and pH responsiveness, and one exhibiting temperature responsiveness only. By changing the arrangement and length of the blocks while keeping other parameters, such as the molecular mass or mean polarity, constant, we were able to show that the sequence plays a key role in the smart behavior of ELPs.     

Normal-mode refinement of anisotropic thermal parameters for potassium channel KcsA at 3.2 A crystallographic resolution

We report a normal-mode method for anisotropic refinement of membrane-protein structures, based on a hypothesis that the global near-native-state disordering of membrane proteins in crystals follows low-frequency normal modes. Thus, a small set of modes is sufficient to represent the anisotropic thermal motions in X-ray crystallographic refinement. By applying the method to potassium channel KcsA at 3.2 A, we obtained a structural model with an improved fit with the diffraction data. Moreover, the improved electron density maps allowed for large structural adjustments for 12 residues in each subunit, including the rebuilding of 3 missing side chains. Overall, the anisotropic KcsA structure at 3.2 A was systematically closer to a 2.0 A KcsA structure, especially in the selectivity filter. Furthermore, the anisotropic thermal ellipsoids from the refinement revealed functionally relevant structural flexibility. We expect this method to be a valuable tool for structural refinement of many membrane proteins with moderate-resolution diffraction data.     

Weak Intra-Ring Allosteric Communications of the Archaeal Chaperonin Thermosome Revealed by Normal Mode Analysis

Chaperonins are molecular machines that use ATP-driven cycles to assist misfolded substrate proteins to reach the native state. During the functional cycle, these machines adopt distinct nucleotide-dependent conformational states, which reflect large-scale allosteric changes in individual subunits. Distinct allosteric kinetics has been described for the two chaperonin classes. Bacterial (group I) chaperonins, such as GroEL, undergo concerted subunit motions within each ring, whereas archaeal and eukaryotic chaperonins (group II) undergo sequential subunit motions. We study these distinct mechanisms through a comparative normal mode analysis of monomer and double-ring structures of the archaeal chaperonin thermosome and GroEL. We find that thermosome monomers of each type exhibit common low-frequency behavior of normal modes. The observed distinct higher-frequency modes are attributed to functional specialization of these subunit types. The thermosome double-ring structure has larger contribution from higher-frequency modes, as it is found in the GroEL case. We find that long-range intersubunit correlation of amino-acid pairs is weaker in the thermosome ring than in GroEL. Overall, our results indicate that distinct allosteric behavior of the two chaperonin classes originates from different wiring of individual subunits as well as of the intersubunit communications.     

A eukaryotic expression vector for the study of nuclear localization signals

We describe a new vector designed to produce β-galactosidase fusion proteins which can be used to assess subcellular localization of target peptide fragments or proteins in eukaryotic cells. The vector was constructed in such a way as to produce the peptide of interest in fusion via a short linker of proline residues to the N terminus of the reporter protein. Efficiency of the transport machinery is optimized using this particular protein fusion construction. This vector has potential uses for readily testing putative nuclear localization sequences and identifying their crucial amino-acid residues.     

Structural Analysis of Toxin-Neutralizing,Single-Domain Antibodies that Bridge Ricin’s A-B Subunit Interface

Ricin toxin kills mammalian cells with notorious efficiency. The toxin’s B subunit (RTB) is a Gal/GalNAc-specific lectin that attaches to cell surfaces and promotes retrograde transport of ricin’s A subunit (RTA) to the trans Golgi network (TGN) and endoplasmic reticulum (ER). RTA is liberated from RTB in the ER and translocated into the cell cytoplasm, where it functions as a ribosome-inactivating protein. While antibodies against ricin’s individual subunits have been reported, we now describe seven alpaca-derived, single-domain antibodies (V_HHs) that span the RTA-RTB interface, including four Tier 1 V_HHs with IC₅₀ values <1 nM. Crystal structures of each V_HH bound to native ricin holotoxin revealed three different binding modes, based on contact with RTA’s F-G loop (mode 1), RTB’s subdomain 2γ (mode 2) or both (mode 3). V_HHs in modes 2 and 3 were highly effective at blocking ricin attachment to HeLa cells and immobilized asialofetuin, due to framework residues (FR3) that occupied the 2γ Gal/GalNAc-binding pocket and mimic ligand. The four Tier 1 V_HHs also interfered with intracellular functions of RTB, as they neutralized ricin in a post-attachment cytotoxicity assay (e.g., the toxin was bound to cell surfaces before antibody addition) and reduced the efficiency of toxin transport to the TGN. We conclude that the RTA-RTB interface is a target of potent toxin-neutralizing antibodies that interfere with both extracellular and intracellular events in ricin’s cytotoxic pathway.     

Ligand-induced conformational change of a protein reproduced by a linear combination of displacement vectors obtained from normal mode analysis

The conformational change of a protein upon ligand binding was examined by normal mode analysis (NMA) based on an elastic-network model (ENM) for a full-atom system using dihedral angles as independent variables. Specifically, we investigated the extent to which conformational change vectors of atoms from an apo form to a holo form of a protein can be represented by a linear combination of the displacement vectors of atoms in the apo form calculated for the lowest-frequency m normal modes (m=1, 2,…, 20). In this analysis, the latter vectors were best fitted to the former ones by the least-squares method. Twenty-two paired proteins in the holo and apo forms, including three dimer pairs, were examined. The results showed that, in most cases, the conformational change vectors were reproduced well by a linear combination of the displacement vectors of a small number of low-frequency normal modes. The conformational change around an active site was reproduced as well as the entire conformational change, except for some proteins that only undergo significant conformational changes around active sites. The weighting factors for 20 normal modes optimized by the least-squares fitting characterize the conformational changes upon ligand binding for these proteins. The conformational changes sampled around the apo form of a protein by the linear combination of the displacement vectors obtained by ENM-based NMA may help solve the flexible-docking problem of a protein with another molecule because the results presented herein suggest that they have a relatively high probability of being involved in an actual conformational change.     

Transconformations of the SERCA1 Ca-ATPase: a normal mode study

The transport of Ca(2+) by Ca-ATPase across the sarcoplasmic reticulum membrane is accompanied by several transconformations of the protein. Relying on the already established functional importance of low-frequency modes in dynamics of proteins, we report here a normal mode analysis of the Ca(2+)-ATPase based on the crystallographic structures of the E1Ca(2) and E2TG forms. The lowest-frequency modes reveal that the N and A(+Nter) domains undergo the largest amplitude movements. The dynamical domain analysis performed with the DomainFinder program suggests that they behave as rigid bodies, unlike the highly flexible P domain. We highlight two types of movements of the transmembrane helices: i), a concerted movement around an axis perpendicular to the membrane which "twists open" the lumenal side of the protein and ii), an individual translational and rotational mobility which is of lower amplitude for the helices hosting the calcium binding sites. Among all modes calculated for E1Ca, only three are enough to describe the transition to E2TG; the associated movements involve almost exclusively the A and N domains, reflecting the closure of the cytoplasmic headpiece and high displacement of the L7-8 lumenal loop. Subsequently, we discuss the potential contribution of the remaining low-frequency normal modes to the transconformations occurring within the overall calcium transport cycle.     

Low-frequency normal modes in horse liver alcohol dehydrogenase and motions of residues involved in the enzymatic reaction

Normal mode analysis using the elastic network model has provided characteristics and directions of the low-frequency large domain motions of horse liver alcohol dehydrogenase. Three normal modes (mode 1, mode 7, and mode 8) were identified as representative domain motions that may promote the onset of Near Attack Conformers or facilitate the product to be released. The pattern of the atomic displacement for some key residues (such as Val292 and Val203) revealed in this study is in line with experimental structural and kinetic studies and theoretical simulations.     

Analyzing the normal mode dynamics of macromolecules by the component synthesis method.

This paper presents a general method for studying the harmonic dynamics of large biomolecules and molecular complexes. The performance and accuracy of the method applied to a number of molecules are also reported. The basic approach of the method is to divide a macromolecule into a number of smaller components. The local normal modes of the components are first calculated by treating individual components and the interactions between nearest neighboring components. The physical displacements of all atoms are then represented in the local normal mode space, in which a selected range of high-frequency local modes is neglected. The equation of motion of the molecule in the local normal mode space will then have a smaller dimension, and consequently the normal modes of the whole structure, particularly for large molecules, can be solved much more easily. The normal modes of two polypeptides--(Ala)6 and (Ala)12--and a double-helical DNA--d(ATATA).d(TATAT)--are analyzed with this method. Reductions on the dimensions of harmonic dynamic equations for these molecules have been made, with the fraction of the deleted high-frequency modes ranging from 1/2 to 5/6. The calculated low-frequency normal modes are found to be very accurate as compared to the exact solutions by standard procedure. The major advantage of the present approach on macromolecule harmonic dynamics is that the reduction on the dimensionality of the eigenvalue problems can be varied according to the size of molecules, so the method can be easily applied to large macromolecules with controlled accuracy.     

A theorem on amplitudes of thermal atomic fluctuations in large molecules assuming specific conformations calculated by normal mode analysis

An exact theorem is proved and its implication is discussed. The theorem states that, if a large molecule, typically biological macromolecules such as proteins, undergoes small-amplitude conformational fluctuations around its native conformation in such a way that within the range of conformational fluctuations at thermal equilibrium the conformational energy surface can be approximated by a multidimensional parabola, then the mass-weighted mean-square displacement of constituent atoms is given by the sum of the contributions from each normal mode of conformational vibration, which in turn is proportional to the inverse of the square of its frequency. This theorem provides a firm theoretical basis for the fact hitherto empirically recognized in the conformational dynamics of, for instance, native proteins that very-low-frequency normal modes make dominant contributions to the conformational fluctuations at thermal equilibrium. Discussion is given on the implication of this theorem, especially on the importance of the concept of the low-frequency normal modes, even in the case where the basic assumption of the harmonicity of the energy surface does not hold.