Mathematical Approaches to Molecular Structural Biology

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