The Gaussian Network Model: Theory and Applications

CONTENTS 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.1.1 Conformational Dynamics: A Bridge Between Structure and Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.1.2 Functional Motions of Proteins Are Cooperative Fluctuations Near the Native State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.2 The Gaussian Network Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.2.1 AMinimalist Model for Fluctuation Dynamics . . . . . . . . . . . . . . . . . 44 3.2.2 GNMAssumes Fluctuations Are Isotropic and Gaussian . . . . . 44 3.2.3 Statistical Mechanics Foundations of the GNM .. . . . . . . . . . . . . . . . 46 3.2.4 Influence of Local Packing Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.3 Method and Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.3.1 Equilibrium Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.3.2 GNMMode Decomposition: Physical Meaning of Slow

and Fast Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.3.3 What Is ANM? How Does GNM Differ fromANM?. . . . . . . . . . . 53 3.3.4 Applicability to Supramolecular Structures . . . . . . . . . . . . . . . . . . . . . 55 3.3.5 iGNM: ADatabase of GNM Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.4 Future Prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

A view that emerges from many studies is that proteins possess a tendency, encoded in their three-dimensional (3D) structures, to reconfigure into

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functional forms, that is, each native structure tends to undergo conformational changes that facilitate its biological function. An efficient method for identifying such functional motions is normal mode analysis (NMA), a method that has foundwidespread use in physical sciences for characterizing molecular fluctuations near a given equilibrium state. The utility of NMA as a physically plausible andmathematically tractable tool for exploring protein dynamics has been recognized for the last 20 years [1, 2]. With recent increase in computational power and speed the application of NMA to proteins has gained renewed interest and popularity. Contributing to this renewed interest in utilizing NMA has been the

introduction of simpler models based on polymer network mechanics. The Gaussiannetworkmodel (GNM) is probably the simplest among these. This is an elastic network (EN) model introduced at the residue level [3, 4], inspired by the full atomic NMA of Tirion with a uniform harmonic potential [5]. Despite its simplicity, the GNM and its extension, the anisotropic network model (ANM) [6], or similar coarse-grained ENmodels combinedwithNMA [7-9], have found widespread use since then for elucidating the dynamics of proteins and their complexes. Significantly, these simplified NMAs with EN models have recently been applied to deduce both the machinery and conformational dynamics of large structures and assemblies including HIV reverse transcriptase [10, 11], hemagglutinin A [12], aspartate transcarbamylase [13], F1 ATPase [14], RNA polymerase [15], an actin segment [16], GroEL-GroES [17], the ribosome [18, 19], and viral capsids [20-22]. Studying proteins with the GNM provides more than the dynamics of

individual biomolecules, such as identifying the common traits among the equilibrium dynamics of proteins [23], the influence of native state topology on stability [24], the localization properties of protein fluctuations [25], or the definition of protein domains [26, 27]. Additionally, GNM has been used to identify residues most protected during hydrogen-deuterium exchange [28, 29], critical for folding [30-34], conserved among members of a given family [35], or involved in ligand binding [36]. The theoretical foundations of the GNM will be presented in this chapter,

alongwith a fewapplications that illustrate its utility. The following questions will be addressed. What is the GNM?What are the underlying assumptions? How is it implemented? Why and how does it work? How does the GNM analysis differ from NMA applied to EN models? What are its advantages and limitations compared to coarse-grained NMA? What are the most significant applications and prospective utilities of the GNM, or the ENmodels in general? To this end, the chapter begins with a brief overview of conformational

dynamics and the relevance of such mechanical motions to biological function. Section 3.2.1 is devoted to explaining the theory and assumptions of the GNM as a simple, purely topological model for protein dynamics. The casual reader may elect to skip over Sections 3.2.2 to 3.2.4 where the derivation of the GNM is presented using fundamental principles from statistical mechanics. In Section 3.3, attention is given to how the GNM is implemented

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(Section 3.3.1), and to what extent it can be or cannot be applied to proteins in general. An interpretation of the physical meaning of both fast and slow modes is presented (Section 3.3.2) with examples for a small enzyme, ribonuclease T1. Section 3.3.3 describes how the GNM differs from the ANM (i.e., from the NMA of simplified EN models) and discusses when the use of one model is preferable to the other. Finally, results are presented in Sections 3.3.4 and 3.3.5 for twowidely different applications: specificmotions of supramolecular structures and classification of motions in general through the iGNM online database of GNM motions. The chapter concludes with a discussion of potential future uses.