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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