Lect12_Optimization

= = =Energy Minimization=

Defined a model and the equations that relate energy to geometry Problem is to find the geometry that lowers the energy

This both a local and a global problem Local Problem
 * non-derivative methods
 * first derivative methods
 * second derivative methods

Global Problem
 * Potential Energy Surfaces
 * Complete Methods
 * grid based
 * stochastic
 * Incomplete
 * Molecular Dynamics
 * Monte Carlo

Tamar Schlick, "Molecular Modeling and Simulation", 2002, Springer. Chapter 10. Multivariate Optimization in Computational Chemistry. Good general discussion of optimization methods with emphasis on applications in biomolecule modeling.

Problem Statement:
 * E = f(x)
 * E is a function of coordinates either cartesian or internal
 * At minimum the first derivatives are zero and the second derivatives are all positive

Minimization methods are broken down into non-derivative and derivative methods base upon the difficulty of computing analytical derivatives.

The other important consideration is the cost of evaluating the energy
 * MM methods have analytical derivatives
 * MM energy evaluation is cheap
 * QM may or may not have analytical derivatives
 * QM energy evaluation is expensive

Non Derivative methods
 * Require energy evaluation only and may require many energy evaluations
 * Storage required ~ N2
 * Simplex Method (Nelder and Mead)
 * Powell’s Method (assumes quadratic function)

Derivative Methods Simplex Method
 * Require evaluation of energy and first derivatives
 * Steepest Descent and Conjugate Gradient
 * Quasi-Newton Methods – DFP, BFGS
 * Full Newton-Raphson requires second derivatives
 * Storage requirements vary from 5N to ~N2


 * Simple and robust
 * Slow – requires many function evaluations
 * Simplex is N dimensional geometric figure with N+1 vertices
 * Method
 * Construct initial simplex from N+1 points where N is the number of variables. Usually chose an initial point and generate the remaining vertices by adding constant to each of the variables in turn.
 * Find highest energy point and generate new point by reflecting old point through the centroid of the simplex formed by the N remaining points.
 * f new point is lower than all other points expand simplex by increasing the reflection distance. If this results in still lower energy keep new point, else go back to first reflected point.
 * Else if new point is only better than previous worst point then contract the simplex by reflecting by smaller distance. If this lowers the energy then keep, else contract along a different direction
 * Keep looping until change in energy is below a present level

[|image001.png]

Problems:
 * Slow – requires many function evaluations
 * Does not assume anything about the shape of the surface
 * Can have problems if there is a large difference in variable response
 * Used in parameter optimization
 * Used when there is no analytical functional relationship for variables
 * Not often used for simple geometry optimization

Application:
 * S. Dennis and S. Vajda,”Semiglobal Simplex Optimization and its Application to Determining the Preferred Solvation Sites of Proteins”, J Comp Chem, **23**, 319-334(2002).
 * Were interested in determining solvation sites about small protein, Streptavidin
 * Crystal Structure was available but less than half of the available water sites were filled
 * Charmm to relax the protein, Tip3p Water, and computed electrostatic energy from Poisson-Boltzmann equation
 * Xi are the position of the water molecules, qi are the partial charges, Vexc is the excluded volume, Sconf is the conformational entropy change on binding

[|image002.png]

[|opt_lecture.pdf]