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Internal Coordinate Normal Mode Analysis: A Strategy to Predict Protein Conformational Transitions Elisa Frezza, and Richard Lavery J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.8b11913 • Publication Date (Web): 21 Jan 2019 Downloaded from http://pubs.acs.org on January 23, 2019
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The Journal of Physical Chemistry
Internal Coordinate Normal Mode Analysis: A Strategy
To
Predict
Protein
Conformational
Transitions Elisa Frezza#* and Richard Lavery
MMSB, UMR 5086 CNRS / Univ. Lyon I, Institut de Biologie et Chimie des Protéines, 7 passage du Vercors, Lyon 69367, France
ABSTRACT
We analyse the capacity of normal mode analysis in internal coordinates space to generate large amplitude structural deformations that can describe the conformational changes occurring upon the binding of proteins to other species. We also analyse how many modes need to be studied to capture a given transition and whether a combination of two modes is better than using a single mode. The technique is tested on known unbound-to-bound structural transitions for a set of single or multidomain proteins. The results suggest that this approach is a promising way to generate structures for protein docking or for more refined molecular dynamics simulations.
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1. INTRODUCTION Protein flexibility is crucial for function and underlies not only protein folding, but also the majority of protein interactions with other molecular species1–5 Protein-ligand and protein-protein binding usually involves structural changes that often go beyond local rearrangements. Since so far there is a limited number of protein-protein complexes whose structure has been experimentally determined, predicting structural transitions linked to interactions is very important, particularly in view of the increasing use of relatively lowresolution techniques such as cryo-electron microscopy (cryoEM) and small angle x-ray scattering (SAXS). Predicting changes in protein conformation upon binding of another protein or ligand is still very difficult. This hinders the prediction of the structure of protein-protein complexes which remains a major challenge.6 While molecular dynamics simulations could in principle be used to study protein interactions, including conformational rearrangement of the partners, the millisecond simulations currently attainable8 are computationally expensive and of limited use for protein-protein docking without reliable starting conformations. Therefore, faster, simplified methods are still required. One such method involves normal mode analysis (NMA), which is typically combined with coarse-grain molecular representations and simplified energy models.7–12 Under a harmonic approximation, normal mode analysis (NMA) gives insights on the equilibrium vibrational modes accessible to a system. Depending on the description of the degrees of freedom, normal modes can be computed either using a Cartesian coordinate space (CCS) or an internal coordinate space (ICS)13. The former is nowadays the most widespread computational approach due to its simplicity. It is often combined with coarse-grain protein models based on a simplified protein backbone (i.e. the Cartesian positions of the Cα atoms) where the pseudo-atoms are connected by linear springs within a given cut-off distance. The most common implementation of this choice is provided by the Anisotropic Network Model (ANM) approach.14–17 Other groups would rather employ CCS normal modes including some additional structural information on both the side-chain atoms and the backbone.18–20 CCS NMA (hereafter termed cNMA) has been largely used to understand the protein deformations induced by ligand binding16,21 and also allosteric processes,11,22 while fewer studies are available on the conformational changes occurring between isolated proteins and their complexes.23–28 However, cNMA suffers from the fact that the harmonic approximation is only valid for relatively small movements and the corresponding low frequency modes often (roughly 2/3 of the cases in a recent study13) fail to explore unbound to bound
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The Journal of Physical Chemistry
conformational transitions. Using CCS also implies that conformational changes defined by the normal modes will quickly deform the valence structure (i.e. bond lengths and bond angles) of the protein. An alternative approach was proposed by Go and co-workers, who noted that ICS (torsion angles, bond angles, and bond lengths) is more advantageous since it extends the validity of the harmonic approximation of the conformational energy hypersurface.9 Thus, larger conformational changes can be modelled by taking into account the eigenvectors of low-frequency modes. In this way, “chemically-relevant” variables are chosen, making it easier to split the degrees of freedom into two categories: “hard” (typically, bond lengths and angles) and “soft” (torsion angles). Moreover, those variables that are unlikely to contribute significantly to large collective movements can be excluded, therefore simplifying and speeding up the computations. For the reasons above, ICS NMA (hereafter named iNMA) is usually performed in torsional angle space, while keeping fixed bond angles and bond lengths, allowing thus to greatly reduce the total number of variables involved. As a consequence, an iNMA analysis can be carried out on any relevant subset of variables without this modifying the molecular representation of the system under exam. Finally, iNMA analysis allows not only to collect important pieces of information regarding those variables at the root of low-frequency collective movements, but, in common with cNMA, it can equally describe spatial movements by a conversion of the modes into Cartesian space. Despite these advantages, applications of iNMA are relatively scarce, although there has been a renewed interest in this technique in recent years.13,17,29 The main obstacle is related to the fact that, in ICS, internal variables must be separated by the overall translations or rotations of the system under study. To do so, the topology of the system under exam is needed, more specifically on which particles are moved by any given variable.30–32 Furthermore, although ICS normal modes represent a suitable tool for the characterization of the variables responsible for low-frequency collective movements, they do not provide a direct description of the movements occurring in Cartesian space. For computing global quantities, this problem can be solved using a second-order expansion13 making it possible to combine the advantages of iNMA with a CCS description of the overall conformational changes.
13,29,33
Bray and co-workers also proposed another approach to project the torsional
modes into a Cartesian representation using a nonlinear optimization of torsional modes in order to minimize the RMSD between the bound and unbound structure.34 However, this method naturally requires prior knowledge of the target structure. In the present study, we use the iNMA approach (coupled with a simplified protein representation and an elastic network energy model) to study large conformational changes of
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a set of proteins for which at least two distinct conformational states are known: an isolated, unbound state and a bound state involving interaction with other proteins, ligands or nucleic acids. In our framework, in order to compare the experimental structures with those predicted by iNMA, we made a conversion from the torsional space into Cartesian space without using any information from the target structure, as proposed earlier.34 The first aim of this work is to test the possibility of using low-frequency modes to describe unbound to bound protein conformational transitions without introducing any deformations in the valence structure (bond lengths or angles) as would be the case with cNMA. This is important in the perspective of using iNMA in docking algorithms, or in hybrid approaches combining NMA with experimental data from multiple sources to generate high-resolution models of protein assemblies. Secondly, one of the limits of cNMA is the difficulty of predicting how many modes are necessary to be able to describe a conformational transition of interest without prior knowledge of the target (namely, the bound structure of the protein). We propose a strategy to determine a very limited number of iNMA modes that will satisfy this condition. The results should help in generating appropriate starting structures for both protein-protein docking and molecular dynamics simulations. 2. METHODS 2.1 Normal Mode Analysis: a brief overview 2.1.1 General Theory Normal modes35 constitute an analytical solution of the classical equations of motion by imposing a harmonic approximation on the potential energy of the system (i.e. by assuming that the energy is a quadratic function of its N coordinates) around the potential energy minimum 𝑞"# :
1 E p = qT Fq 2 !
(1) '(
) where F is the potential energy or Hessian matrix defined by 𝐹"% = '* '* and q is a set of +
,
coordinates. The kinetic energy of the molecule can then be expressed in terms of internal variables:
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The Journal of Physical Chemistry
∂r ∂r 1 1 Ek = q! T Hq! = ∑ maq! T a a q! 2 2 a ∂qi ∂q j !
(2)
'𝐫 '𝐫
where 𝐻"% = ∑1 𝑚1 '*3 '*3 is the kinetic matrix, ra and ma are the position vectors and the +
,
atomic masses of each atom a respectively. The internal coordinates q can be bond lengths, valence angles, torsion or rigid-body rotations and translations. The equations of motion in terms of any set of coordinates are given by Lagrange’s equations, whose solution takes the following form: 𝑞" (𝑡) = 𝑞"# + ∑;9
