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Mathematical approaches to molecular structural biology /

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...

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Bibliographic Details
Main Author: Pal, Subrata
Corporate Author: ScienceDirect (Online service)
Format: eBook
Language:English
Published: London : Academic Press, 2023.
Subjects:
Molecular biology > Mathematics.
Molecular structure > Mathematics.
Molecular Structure
Molecular Biology > methods
Mathematical Concepts
Biologie moléculaire > Mathématiques.
Structure moléculaire > Mathématiques.
molecular structure.
Molecular biology > Mathematics
Molecular structure > Mathematics
Electronic books.
Online Access:Connect to the full text of this electronic book
Table of Contents:
  • 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.
  • 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.