Computational Biology BS123A/MB223 UC-Irvine Ray Luo, MBB, BS

Reading Assignment Leach, Chapter 2 Sec , 5.8.1

Simplistic Molecular Mechanics Force Field Van der WaalsCharge - Charge Bond Angle Improper Dihedral  

Scaling of Nonbonded Terms Scaling of electrostatic energy: charge- charge 1/r; charge-dipole 1/r 2, dipole- dipole 1/r 3. Scaling of van der Waals energy: 1/r 6.

maximasaddle point Potential Energy Surface (PES) Force Field v.s. PES Why EPS is so important? Stable structures of a molecule, such as protein folding, protein- ligand binding Vibrational frequencies The molecular basis of thermodynamics and kinetics.

maximasaddle point Potential Energy Surface (PES)

minimum Potential Energy Surface saddle point maxima

Local minimum vs global minimum Many local minima; only ONE global minimum Energy Minimization

Stationary points: points on a PES with all first energy derivatives (gradients) zero. Minima (local and global): stationary points with all eigenvalues of the Hessian matrix (all second derivatives) positive. Saddle points: stationary points with exactly one negative eigenvalue in the Hessian matrix. Classification of Stationary Points on PES

Energy Minimization: Methods Non-gradient based methods: systematic numeration; simplex; direction set (Powell’s). Gradient based methods: the steepest descents; conjugate gradients Hessian based methods: Newton-Raphson; quasi-Newton

Non-gradient Based Minimization: Powell’s Method 1-d function minimization is easily done. Direction set methods: sequences of 1-d “line minimizations”. How to choose the “direction set”? The set of unit vectors works well in many cases. The best choice is a set of conjugate directions, aka, non-interfering directions.

Conjugate Directions Fact: if we do line min of a function along u, the gradient of it at the line min must be perpendicular to u; otherwise, it is not a line min. We can use Taylor expansion to understand the concept (mathematical derivation provided in class). The condition that motion along v does not spoil our min along u is simply that the gradient stays perpendicular to u, i.e. the change in v has to be perpendicular to u.

Non-gradient Based Minimization: Powell’s Method Initialize the directions, u i = e i, unit vectors; Save the starting point as P 0 ; For i = 1, …, N, move P i-1 to the minimum along direction u i and call this point P i ; Set u i = u i+1 for I = 1, …, N-1; Set u N = P N – P 0 ; Move P N to the minimum along direction u N and call this point P 0.

Non-gradient Based Minimization: Powell’s Method The directions are conjugate, and the method works great for quadratic functions This can be proved, as given by Brent. No need for gradient in the line minimization, just a 1-d function. Therefore the overall method does not rely on gradient.

Gradient Based Minimization: Steepest Descents To start, walk straight downhill along the gradient direction at the initial point. Perform line search along the gradient direction. The next direction to take is orthogonal to the previous direction, and so on. Use adaptive step sizes (small) in the line search. Step size will be reduced if the energy goes up, otherwise it will be increased.

Hessian Based Minimization Newton-Raphson: Idea For any 1-d quadratic function U(x), Taylor expansion at x k gives U(x) = U(x k ) + (x-x k )U’(x k ) + (x-x k ) 2 U”(x k )/2, U’(x k ) = U’(x k ) + (x-x k ) U”(x k ). At the minimum x*: U’(x*) = 0, so that x* = x k - U ’(x k )/U”(x k ) For an n-d quadratic function: x* = x k - U’(x k ) U”(x k ) -1

Hessian Based Minimization Newton-Raphson Note that U”(x k ) -1 is the inverse of a matrix (Hessian), very slow to compute if n is large. Also note that n is 3*no_of_atom for the Hessian matrix. However, the minimization can be performed in one-step for quadratic functions! For force field energy functions, it still takes less steps to minimize than other methods, but each step is much slower. The burden is shifted to the inversion of the Hessian Matrix.

Hessian Based Minimization Newton-Raphson: Pros and Cons Newton-Raphson works well for portion of the PES where the quadratic approximation is good, i.e. near a local minimum. It does not work well far away from a minimum. It does not work if the Hessian matrix has negative eigenvalues.

Which minimization should I use? If gradient is not possible, Powell’s method would be a reasonable choice. If gradient is possible to get, as in force field functions, steepest descents can be used to relax initial bad geometry. This is usually followed by a conjugate gradient method, then a quasi-Newton method. If a small system, less than a few hundred atoms, the Newton-Raphson method may be used if it is sufficiently close to a local minimum.