Detailed Syllabus

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This is a first-time offering and so the details of the syllabus will be modified to match student background. Here is an upper bound on what we aim to cover:

Analysis of protein shapes (4 Weeks)

1. Plotting protein shapes (vectors, matrices). Introduction to Matlab. Interactive work and the preparation of batch files. Representation and vectors and matrices in Matlab. Reading external information. Plotting protein backbone. How to find help.

2. Rotations and overlapping structures (least square fits, singular value decompositions). Rotations of structures. Rotation matrices in two dimensions. Rotation matrices in 3D. Overlapping shapes - setting up the least square fit problem. General discussion on Least Squares fit problems. An exact solution for the structure overlap problem.

3. Smoothing and interpolations (linear interpolation, cubic splines). Interpolation of functions. Continuity of function and derivatives. Polynomial interpolation. The need for splines (problems with polynomial interpolation). Setting up the problems in Matlab. Interpolating the backbone of a protein structure.

4. Clustering protein shapes (building RMS matrices, eigenvalue analysis and identification of sub-families). Distances in high dimension. Definition of the RMS. Building the RMS matrix. A symmetric positive definite matrix - diagonalization. Identification of sub-families of protein shapes by examinations of eigenvectors with low eigenvalues.

Dynamics and initial value problems (3 weeks)

1. Integrators: Introduction - why needed? Error analysis. Stability. The simplest Euler method. A sophisticated method -- Runga Kutta. Stable algorithm: Symplectic methods - canonical transformation and the Jacobian. The Verlet algorithm.

2. Applications: The simplest -- Harmonic Oscillator, A “real” problem -- Polymer collapse of a linked chain with van der Waals interactions between monomers, multiple time scale integrators, the need and possible implementation.

3. Fourier transform and analyzing trajectory data. Introduction to Fourier transform. Discrete and continuous Fourier transforms. Filtering our high frequency components and smoothing noisy data. Analysis of polymer collapse trajectory - focusing on global folding motions.

4. Stochastic differential equations - Brownian dynamics (possible project)

5. Brownian dynamics. Master equation and models of evolution. (possible project)

6. Dynamics with constraints (Conjugate gradient and linear constraints) (possible project)

Optimization and protein folding (3 weeks)

1. Conjugate gradient - Introduction. Minimization in high dimension. Steepest Decent paths. Line search methods. Conjugate gradient as the next best thing. Complexity. Memory and CPU requirements.

2. Newton Raphson. Speed of convergence. Radius of convergence. What are the appropriate problems for Newton-like methods versus conjugate gradient methods? Comparing the efficiency of both methods. Truncated Newton Raphson

3. Adjustments of protein shapes. Local minimization versus global optimization. Correcting small experimental “errors” in protein structures by energy minimization. A model system: an optimization of the structure of a collapsed polymer. Measuring RMS as a function of minimization steps. Convergence criteria.

4. Algorithms to smooth potentials. Motivation -- making the local optimization more global. Differential and Integral transform of potentials. Effect of smoothing and recovering of the original minimum. Gaussian transform. The bad derivative method. Example: A one-dimensional double well system. Comparison of different techniques.

5. Simulated annealing with Brownian dynamics (possible project) 6. Folding a protein model (possible project)

Design of score functions and linear programming (2 weeks)

1. The Simplex method. Inequalities, how to solve them. Geometric interpretation. Infeasibilities. Set up of linear programming problems in Matlab.

2. Design of folding potential. The need for an optimal potential. Experimental structures and the protein data bank. Threading and generation of decoy structures. General potential. Contact potential. Setting up linear inequalities and solving them. Analysis of the derived potentials. Eigenvalues and eigenvectors of the contact matrix. The “discovery” of hydrophobicity scale.

Field equations and partial differential equations (2 weeks)

1. Finite difference method. Motivation - partial differential equation in chemistry and biology. Diffusion equation. Setting up the problem on a grid. Examples: diffusion of free particles, harmonic oscillator. Solving with the finite difference method for one-dimensional systems.

2. Fourier transform. The use of Fourier transform in partial differential equation - translating derivatives into an algebraic equation. Boundary conditions. Laplace transform.

3. Asymptotic solutions of diffusion problems. The concept of equilibrium. The Master equation and discrete representation of the diffusion equation. Relaxation times. Transition matrix. Eigenvectors and eigenvalues of the transition matrix. Computing the long time limit from eigenvalue analysis, for time propagation.

4. Application: Ligand diffusion to receptors. (possible project)