Mathematical Approaches to Molecular Structural Biology

Description

Mathematical Approaches to Molecular Structural Biology offers a comprehensive overview of the mathematical foundations behind the study of biomolecular structure. Initial chapters provide an introduction to the mathematics associated with the study of molecular structure, such as vector spaces and matrices, linear systems, matrix decomposition, vector calculus, probability and statistics. The book then moves on to more advanced areas of molecular structural biology based on the mathematical concepts discussed in earlier chapters. Here, key methods such as X-ray crystallography and cryo-electron microscopy are explored, in addition to biomolecular structure dynamics within the context of mathematics and physics. This book equips readers with an understanding of the fundamental principles behind structural biology, providing researchers with a strong groundwork for further investigation in both this and related fields. Front Cover Mathematical Approaches to Molecular Structural Biology Copyright Page Dedication Contents About the author Preface Acknowledgments Table of symbols 1 Mathematical preliminaries 1.1 Functions 1.1.1 Algebraic functions 1.1.2 Trigonometric functions 1.1.3 Exponential and logarithmic functions 1.1.4 Complex number and functions 1.2 Vectors 1.2.1 Concept of vector in physics 1.2.2 Vector as an ordered set of numbers 1.2.3 Mathematical viewpoint of vector 1.3 Matrices and determinants 1.3.1 Systems of linear equations Gaussian elimination 1.3.2 Matrices 1.3.3 Determinants Definiteness of a symmetric matrix 1.4 Calculus 1.4.1 Differentiation Simple algebraic functions 1.4.2 Integration Integration involving exponential functions Integration involving logarithmic functions Integration by substitution Integration by parts 1.4.3 Multivariate function 1.5 Series and limits 1.5.1 Taylor series 1.5.2 Fourier series Exercise 1 Further reading 2 Vector spaces and matrices 2.1 Linear systems Exercises 2.1 2.2 Sets and subsets 2.2.1 Set Some relevant notations 2.2.2 Subset Exercise 2.2 2.3 Vector spaces and subspaces 2.3.1 Vector space Vector space of m×n matrices 2.3.2 Vector subspaces 2.3.3 Null space/row space/column space Exercise 2.3 2.4 Liner combination/linear independence Generalized concept Exercise 2.4 2.5 Basis vectors The standard basis for m×n matrices Exercise 2.5 2.6 Dimension and rank Exercise 2.6 2.7 Inner product space Norm Distance Dot product Exercise 2.7 2.8 Orthogonality Orthogonal and orthonormal set Coordinates relative to orthogonal basis Orthogonal projection Exercise 2.8 2.9 Mapping and transformation Basic matrix transformations Exercise 2.9 2.10 Change of basis Exercise 2.10 Further reading 3 Matrix decomposition 3.1 Eigensystems from different perspectives 3.1.1 A stable distribution vector 3.1.2 System of linear differential equations Exercise 3.1 3.2 Eigensystem basics Nonuniqueness of eigenvectors Computing eigenvectors Eigenvalues of some special matrices Linear independence of eigenvectors Eigendecomposition Geometric intuition for eigendecomposition Diagonalization Invertibility of matrix P Diagonalizability of a matrix Orthogonal diagonalization Projection matrix and spectral decomposition Exercise 3.2 3.3 Singular value decomposition Eigendecomposition and singular value decomposition compared Exercises 3.3 Further reading 4 Vector calculus 4.1 Derivatives of univariate functions 4.2 Derivatives of multivariate functions Partial derivatives Critical points and local extrema 4.3 Gradients of scalar- and vector-valued functions Vector-valued function expressed as a matrix transformation 4.4 Gradients of matrices 4.5 Higher-order derivates – Hessian Optimization 4.6 Linearization and multivariate Taylor series Exercise 4 Further Reading 5 Integral transform 5.1 Fourier transform 5.2 Dirac delta function Derivative of the δ-function The δ-function in 3D Fourier series and the δ-function Fourier transform and the δ-function Dirac comb 5.3 Convolution and deconvolution 5.4 Discrete Fourier transform 5.5 Laplace transform Exercise 5 Further reading 6 Probability and statistics 6.1 Probability—definitions and properties 6.1.1 Probability function A complement Uniform probability measure 6.1.2 Conditional probability Independence of events Bayes’ theorem 6.2 Random variables and distribution 6.2.1 Discrete random variable The Bernoulli and binomial distributions The Poisson distribution 6.2.2 Continuous random variable Cumulative distribution function The uniform distribution The exponential distribution The normal distribution 6.2.3 Transformation of random variables Linear transformations of random variables 6.2.4 Expectation and variance Expectation of a discrete random variable Expectation of a continuous random variable 6.3 Multivariate distribution 6.3.1 Bivariate distribution Marginal distribution 6.3.2 Generalized multivariate distribution 6.4 Covariance and correlation Covariance matrix Multivariate normal distribution 6.5 Principal component analysis Principal component analysis and singular value decomposition Exercise 6 Further reading 7 X-ray crystallography 7.1 X-ray scattering 7.1.1 Electromagnetic waves 7.1.2 Thomson scattering 7.1.3 Compton scattering 7.2 Scattering by an atom 7.3 Diffraction from a crystal – Laue equations 7.3.1 Lattice and reciprocal lattice 7.3.2 Structure factor 7.3.3 Bragg’s law 7.4 Diffraction and Fourier transform 7.5 Convolution and diffraction 7.6 The electron density equation 7.6.1 Phase problem and the Patterson function 7.6.2 Isomorphous replacement 7.6.3 Electron density sharpening Exercise 7 Further reading 8 Cryo-electron microscopy 8.1 Quantum physics 8.1.1 Wave–particle duality 8.1.2 Schrödinger equation 8.1.3 Hamiltonian 8.2 Wave optics of electrons—scattering 8.3 Theory of image formation 8.3.1 Electrodynamics of lens system 8.3.2 Image formation 8.4 Image processing by multivariate statistical analysis—principal component analysis 8.4.1 Hyperspace and data cloud 8.4.2 Distance metrics Euclidean metric Chi-square metric (χ2–metric) Modulation metric 8.4.3 Data compression 8.5 Clustering 8.5.1 Hierarchical clustering 8.5.2 K-means 8.6 Maximum likelihood Exercise 8 Reference Further reading 9 Biomolecular structure and dynamics 9.1 Comparison of biomolecular structures 9.1.1 Definition of the problem 9.1.2 Quaternions 9.1.3 Quaternion rotation operator 9.1.4 Minimization of residual 9.2 Conformational optimization 9.2.1 Born–Oppenheimer approximation 9.2.2 Biomolecular geometry optimization Newton–Raphson method Conjugate gradient method 9.3 Molecular dynamics 9.3.1 Basic theory 9.3.2 Computation of molecular dynamics trajectory 9.4 Normal mode analysis 9.4.1 Oscillatory systems 9.4.2 Normal mode analysis theory 9.4.3 Elastic network models Gaussian network model Anisotropic network model Exercise 9 References Further reading Index Back Cover