ABSTRACT
CONTENTS 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 7.2 Methods of NMA.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
7.2.1 Basic Theory of NMA .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 7.2.2 Elastic NMA .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
7.3 Structural Refinement in Cryo-EMMeasurement . . . . . . . . . . . . . . . . . . . . . 140 7.3.1 NMABased on Low-Resolution Density Maps . . . . . . . . . . . . . . . 140 7.3.2 QEDM-Assisted Cryo-EM Structural Refinement . . . . . . . . . . . . 144
7.4 Structural Refinement in Fiber Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 7.4.1 NMA at Length Scales of Several Microns . . . . . . . . . . . . . . . . . . . . . 146 7.4.2 Fiber Diffraction Refinement Based on Long-Range
Normal Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
From several decades of computational research in structural biology and biophysics [1, 2], it is now well established that functions of biological macromolecules involve substantial structural motions, which can occur in a wide range of length scales [1, 2], for example, from vibrations of chemical bonds to global conformational changes of supramolecular complexes. However, molecular motions often impose difficulties in experimental structural determination as they tend to compromise the precision of the measurement. Meanwhile, due to the nature of instruments and molecular systems, experimental structural data are obtained in varying resolution scales, for
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example, from atomic coordinates provided by x-ray crystallography to lowto intermediate-resolution electron density maps from, for example, electron cryomicroscopy (cryo-EM). For certain systems, especially supramolecular complexes, the errors in resolution are further augmented by the motions intrinsic to the complexes. The only way to overcome the errors imposed by motions is to use computational methods to model the structural flexibility in the process of structural refinement against experimentalmeasurement. Therefore, a challenge inmodern computational biophysics is to develop new methods of expanded capacity to efficiently model biomolecular motions in such a wide distribution of length and resolution scales. Harmonic modal analyses are effective ways for analyzing molecular
motions. The most frequently used ones are normal mode analysis (NMA) [3] and quasi-harmonic analysis [4], which is closely related to a method called essential dynamics [5]. Mathematically, all these modal analyses are eigenvalue problems that provide a complete basis set of modes from which any arbitrary molecular deformation can be expressed as a linear combination. Since modal analyses are harmonic approximations, they are particularly effectivewhen elastic (harmonic) properties ofmolecules are concerned. From numerous computational studies, it is known that large-scale elastic deformational motions of biomolecules can be well described by lowfrequency vibrational modes of the structures (typical samples can be found in References 6-21). Therefore, in practice, it is desirable to study biomolecular dynamics by filtering out the less important high-frequency motions and focusing on those dominating low-frequency components. In recent years, significant advance has been made in methodology devel-
opment of NMA [22]. It has provided a substantially enhanced capacity for studying large-scale biomolecular dynamics. In this chapter, a brief outline is given for some of those new methods that were developed to study dynamics at multi-length and multi-resolution scales. Emphases are given to the important applications of the new methods to assisting experimental structural determination. This chapter is a concise reviewwith a focused scope and our apology goes
to the colleagueswhosework is not explicitly referred to in this limited space.
7.2.1 Basic Theory of NMA
