Mathematical Approaches to Molecular Structural Biology
- 1st Edition - November 19, 2022
- Imprint: Academic Press
- Author: Subrata Pal
- Language: English
- Paperback ISBN:9 7 8 - 0 - 3 2 3 - 9 0 3 9 7 - 4
- eBook ISBN:9 7 8 - 0 - 3 2 3 - 9 0 6 6 3 - 0
Mathematical Approaches to Molecular Structural Biology offers a comprehensive overview of the mathematical foundations behind the study of biomolecular structure. Initial c… Read more
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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.
- Includes a detailed introduction to key mathematical principles and their application to molecular structural biology
- Explores the mathematical underpinnings behind advanced techniques such as X-ray crystallography and Cryo-electron microscopy
- Features step-by-step protocols that illustrate mathematical and statistical principles for studying molecular structure and dynamics
- Provides a basis for further investigation into the field of computational molecular biology
- Includes figures and graphs throughout to visually demonstrate the concepts discussed
Researchers working in the areas of structural biology, molecular biology, biophysics, and related areas. Graduate and PhD students involved in structural biology, computational molecular biology, mathematics or statistics of molecular and structural biology and related disciplines
1. Mathematical Preliminaries
1.1 Algebraic functions
1.2 Trigonometric functions
1.3 Complex number and functions
1.4 Vectors
1.5 Matrices and determinants
1.6 Calculus
1.7 Series and limits
2. Vector Spaces and Matrices
2.1 Vector space
2.2 Mapping and transformation
2.3 Operators
2.4 Linear independence
2.5 Orthogonality
2.6 Linear systems / Systems of linear equations
2.7 Rotation
3. Matrix Decomposition
3.1 Eigenvalues and eigenvectors
3.2 Eigen decomposition and diagonalization
3.3 Symmetric and Hermitian matrices
4. Vector Calculus
4.1 Differentiation of univariate functions
4.2 Differentiation of multivariate functions
4.3 Gradients of vector-valued functions
4.4 Gradients of matrices
4.5 Higher-order derivatives – Hessian
4.6 Linearization and multivariate Taylor series
5. Integral Transform
5.1 Fourier transform
5.2 Dirac ö-function
5.3 Convolution and deconvolution
5.4 Discrete Fourier transform
5.5 Laplace transform (including convolution)
6. Probability and Statistics
6.1 Probability – definitions and properties
6.2 Random variables and distributions
6.3 Binomial distribution
6.4 Poisson distribution
6.5 Gauss’ normal distribution
6.6 Transformation of variables
6.7 Multivariate distribution
6.8 Multivariate normal distribution
6.9 Covariance and correlation – covariance matrix
6.10 Maximum likelihood
6.11 Principal component analysis
7. X-ray Crystallography
7.1 Scattering of X-rays – Thomson scattering – Compton scattering – Scattering by atoms
7.2 Diffraction from a crystal
7.3 Reciprocal lattice
7.4 Bragg’s law
7.5 Structure factor
7.6 Diffraction and Fourier transform
7.7 Convolution
7.8 Electron density equation
8. Cryo-Electron Microscopy
8.1 Particle optics of electrons – motion in electromagnetic field
8.2 Wave optics of electrons – scattering
8.3 Theory of image formation
8.4 Image processing by multivariate statistical analysis – principal component analysis
8.5 Hierarchical clustering – K-means clustering and maximum likelihood method
8.6 3D reconstruction – back projection
8.7 Conformational variability revealed by electron microscopy
9. Biomolecular Structure and Dynamics
9.1 Comparison of biomolecular structures – rotation matrix
9.2 Protein flexibility and dynamics
9.3 Molecular dynamics
9.4 Conformational optimization
9.5 Normal mode analysis
9.6 Elastic network model
9.7 An example with antibodies
1.1 Algebraic functions
1.2 Trigonometric functions
1.3 Complex number and functions
1.4 Vectors
1.5 Matrices and determinants
1.6 Calculus
1.7 Series and limits
2. Vector Spaces and Matrices
2.1 Vector space
2.2 Mapping and transformation
2.3 Operators
2.4 Linear independence
2.5 Orthogonality
2.6 Linear systems / Systems of linear equations
2.7 Rotation
3. Matrix Decomposition
3.1 Eigenvalues and eigenvectors
3.2 Eigen decomposition and diagonalization
3.3 Symmetric and Hermitian matrices
4. Vector Calculus
4.1 Differentiation of univariate functions
4.2 Differentiation of multivariate functions
4.3 Gradients of vector-valued functions
4.4 Gradients of matrices
4.5 Higher-order derivatives – Hessian
4.6 Linearization and multivariate Taylor series
5. Integral Transform
5.1 Fourier transform
5.2 Dirac ö-function
5.3 Convolution and deconvolution
5.4 Discrete Fourier transform
5.5 Laplace transform (including convolution)
6. Probability and Statistics
6.1 Probability – definitions and properties
6.2 Random variables and distributions
6.3 Binomial distribution
6.4 Poisson distribution
6.5 Gauss’ normal distribution
6.6 Transformation of variables
6.7 Multivariate distribution
6.8 Multivariate normal distribution
6.9 Covariance and correlation – covariance matrix
6.10 Maximum likelihood
6.11 Principal component analysis
7. X-ray Crystallography
7.1 Scattering of X-rays – Thomson scattering – Compton scattering – Scattering by atoms
7.2 Diffraction from a crystal
7.3 Reciprocal lattice
7.4 Bragg’s law
7.5 Structure factor
7.6 Diffraction and Fourier transform
7.7 Convolution
7.8 Electron density equation
8. Cryo-Electron Microscopy
8.1 Particle optics of electrons – motion in electromagnetic field
8.2 Wave optics of electrons – scattering
8.3 Theory of image formation
8.4 Image processing by multivariate statistical analysis – principal component analysis
8.5 Hierarchical clustering – K-means clustering and maximum likelihood method
8.6 3D reconstruction – back projection
8.7 Conformational variability revealed by electron microscopy
9. Biomolecular Structure and Dynamics
9.1 Comparison of biomolecular structures – rotation matrix
9.2 Protein flexibility and dynamics
9.3 Molecular dynamics
9.4 Conformational optimization
9.5 Normal mode analysis
9.6 Elastic network model
9.7 An example with antibodies
- Edition: 1
- Published: November 19, 2022
- Imprint: Academic Press
- Language: English
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Subrata Pal
Subrata Pal obtained his bachelor’s and master’s degrees, both in physics, from Calcutta University. Subsequently, he pursued his predoctoral research in molecular biology and received a PhD degree from the same university in 1982. He carried out his postdoctoral research in DNA replication and gene expression at two of the premiere institutions in the USA: the National Institutes of Health, Bethesda, Maryland and Harvard Medical School, Boston Massachusetts. At Harvard, he received a Claudia Adams Barr special investigator award for basic contribution to cancer research. Professor Pal has a long teaching experience at the undergraduate and graduate levels – the areas of his teaching include physics, molecular biology and genomics.
Affiliations and expertise
Professor (retired), Jadavpur University, Kolkata, IndiaRead Mathematical Approaches to Molecular Structural Biology on ScienceDirect