## All-Atom Normal Mode Calculations of Large Molecular Systems Using Iterative Methods

CONTENTS 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2 Normal Mode Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.3 Iterative Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3.1 Methods Based on the Rayleigh Quotient. . . . . . . . . . . . . . . . . . . . . . . . 21 2.3.2 Perturbation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.3.3 Mixed Basis Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.4 The DIMB Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.4.1 Initial Guess Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.4.2 Iterative Procedure for Obtaining the NMs . . . . . . . . . . . . . . . . . . . . . 24 2.4.3 Convergence Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.5 Applications of DIMB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.5.1 Neocarzinostatin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.5.1.1 Comparison between DIMB and the SM . . . . . . . . . . . . . . 28 2.5.1.2 Utilization of DIMB with a Different Partition. . . . . . . . 29 2.5.1.3 Coupling Between Backbone Collective Motions

and Side-Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.5.2 Hemoglobin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Appendix A Detailed Description of the DIMB Method . . . . . . . . . . . . . . . . . . . 34

A.1 Initial Guess Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 A.2 Iterative Procedure for Obtaining the NMs . . . . . . . . . . . . . . . . . . . . . . . 36

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

The internal motions of proteins play an important role in their biological function, especially large amplitude motions that may be necessary for

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enzymatic activity, allosteric transitions, and various biological processes such as signal transduction, etc.; characterizing these conformational changes is of general concern [1-4]. Molecular dynamics (MD) simulations have played a dominant role since the late 1970s to reveal the internal motions of proteins, however, due to the complex nature of the energy surface and the existence of a tremendously large number of energyminima, this method was not well-adapted to apprehend collective motions. Indeed, the conformational jumps from one minimum to another, taking place on the picosecond time scale [5], correspond to local motions that propagate into larger amplitude ones in a much longer time, as in liquids. On the contrary normal mode (NM) analysis appeared to be the most direct way to obtain the large amplitudemotions [6,7]. It showed that theymayoccur ina correlated fashion, all atoms moving together along given directions, without the requirement of important local conformational changes of residues, as in solids. These two types of dynamics (MD and NM) raised the question whether the proteins have rather liquid-or solid-like properties [8-10]. Many experimental observations indicated that the predominance of one or the other type depends on the temperature of the system, the harmonic dynamics (i.e., NM) being important at low temperatures (lower than 200 K) and a diffusive type dynamics (i.e., MD), at higher temperatures [11-14]. The NM analysis is based on the harmonic approximation of the poten-

tial energy (see Section 2.2), which a priori represents a severe restriction for the protein to undertake large movements. Despite this approximation, low-frequency modes obtained by this method appeared to describe well the wide conformational changes that are observed experimentally [15-19]. NM analysis consists of diagonalizing the Hessian matrix whose elements are the secondderivatives of the potentialwith respect to coordinates. The size of this matrix increases as the square of the number of atoms, which constituted for a long time a serious limitation for this method. Since the 1980s many efforts were devoted to overcome this difﬁculty by reducing the number of degrees of freedom, which can be done by several ways. The ﬁrst approach [9,20] was based on the use of dihedral angle space, neglecting the other degrees of freedom, which results inmore than tenfold reduction of theHessianmatrix’ size. This approach yields a frequency spectrum and directions ofmotion in agreementwith experiments, but still the sizeof thematrix increases as the squareof the dihedral angle’s number, which may represent a limitation for its application to very large proteins. Recently several authors [21, 22] considered each residue (or set of residues) as a rigid block having six translation-rotation degrees of freedom (RTB or BNM methods), which reduces drastically the size of the Hessian matrix and consequently the computational time. The quality of the modes depends on the number of residues taken in a block, one residue per block yielding low-frequency modes comparable to those obtained for all degrees of freedom, while increasing this number results in a rapid deterioration of the mode quality. Another approach is the elastic network model (ENM) proposed originally

by Tirion [23]; it is based on a simpliﬁed potential where the structure is only maintained by springs between neighboring atoms, which is consistent with

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the solid-like nature of proteins. This potential is constructed in a way that corresponds to theminimum energy. Thus, its principal advantage is to avoid energy minimization. However, to calculate the NM for large proteins, the size of the matrix still should be reduced. For that, one may only consider the coordinates of the Cα atoms of the protein as a reduced basis set. Several variants of this model were proposed by Bahar et al. [24, 25], Hinsen [26], and Tama and Sanejouand [19]; they all yield low-frequency modes in a rather good agreement with observed conformational changes, but give unrealistic frequency spectra. All themethods presented above neglect the coupling between the retained

degrees of freedom and those that were ignored, although such a coupling may be important for a detailed description of the conformational change mechanism. Thus, it is important to consider other approaches that take into account all the degrees of freedom. The only way to do so is to use iterative schemes in order to diagonalize the Hessian matrix [27-29]. The interest of suchapproaches is thatnomoreapproximation than theharmonicone isdone. These methods will constitute the object of this chapter, but beforehand let us take a look at the theory of NMs.