Mathematical Approaches to Molecular Structural Biology

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
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