3. Optimization Methods for Molecular Modeling

3. Optimization Methods for Molecular Modeling by Barak Raveh

Outline • Introduction • Local Minimization Methods (derivative-based) • Gradient (first order) methods • Newton (second order) methods • Monte-Carlo Sampling (MC) • Introduction to MC methods • Markov-chain MC methods (MCMC) • Escaping local-minima

Prerequisites for Tracing the Minimal Energy Conformation I. The energy function:The in-silico energy function should correlate with the (intractable) physical free energy. In particular, they should share the same global energy minimum. II. The sampling strategy:Our sampling strategy should efficiently scan the (enormous) space of protein conformations

The Problem: Find Global Minimum on a Rough One Dimensional Surface rough = has multitude of local minima in a multitude of scales. *Adapted from slides by Chen Kaeasar, Ben-Gurion University

The Problem: Find Global Minimum on a Rough Two Dimensional Surface The landscape is rough because both small pits and the Sea of Galilee are local minima. *Adapted from slides by Chen Kaeasar, Ben-Gurion University

The Problem: Find Global Minimum on a RoughMulti-Dimensional Surface • A protein conformation is defined by the set of Cartesian atom coordinates (x,y,z) or by Internal coordinates (φ /ψ/χ torsion angles ; bond angles ; bond lengths) • The conformation space of a protein with 100 residues has ≈ 3000 dimensions • The X-ray structure of a protein is a point in this space. • A 3000-dimensional space cannot be systematically sampled, visualized or comprehended. *Adapted from slides by Chen Kaeasar, Ben-Gurion University

Characteristics of the Protein Energetic Landscape space of conformations energy smooth? rugged? Images by Ken Dill

Local Minimization Allows the Correction of Minor Local Errors in Structural Models Example: removing clashes from X-ray models *Adapted from slides by Chen Kaeasar, Ben-Gurion University

What kind of minima do we want? The path to the closest local minimum = local minimization *Adapted from slides by Chen Kaeasar, Ben-Gurion University

A Little Math – Gradients and Hessians Gradients and Hessians generalize the first and second derivatives (respectively) of multi-variate scalar functions ( = functions from vectors to scalars) Energy = f(x1, y1, z1, … , xn, yn, zn) Gradient Hessian

Analytical Energy Gradient (i) Cartesian Coordinates E = f(x1, y1 ,z1, … , xn, yn, zn) Energy, work and force: recall that Energy ( = work) is defined as force integrated over distance  Energy gradient in Cartesian coordinates = vector of forces that act upon atoms (but this is not exactly so for statistical energy functions, that aim at the free energy ΔG) Example: Van der-Waals energy between pairs of atoms – O(n2) pairs:

Analytical Energy Gradient (ii) Internal Coordinates (torsions, etc.) Note: For simplicity, bond lengths and bond angles are often ignored E = f(1,1, 1, 11, 12, …) • Enrichment: Transforming a gradient between Cartesian and Internal coordinates(see Abe, Braun, Nogoti and Gö, 1984 ; Wedemeyer and Baker, 2003) • Consider an infinitesimal rotation of a vector r around a unit vector n. From physical mechanics, it can be shown that: n x r n n r • cross product – right hand rule adapted from image by Sunil Singh http://cnx.org/content/m14014/1.9/ Using the fold-tree (previous lesson), we can recursively propagate changes in internal coordinates to the whole structure (see Wedemeyer and Baker 2003)

Gradient Calculations – Cartesian vs. Internal Coordinates For some terms, Gradient computation is simpler and more natural with Cartesian coordinates, but harder for others: • Distance / Cartesian dependent: Van der-Waals term ; Electrostatics ; Solvation • Internal-coordinates dependent: Bond length and angle ; Ramachandran and Dunbrack terms (in Rosetta) • Combination: Hydrogen-bonds (in some force-fields) Reminder: Internal coordinates provide a natural distinction between soft constraints (flexibility of φ/ψ torsion angles) and hard constraints with steep gradient (fixed length of covalent bonds).  Energy landscape of Cartesian coordinates is more rugged.

Analytical vs. Numerical Gradient Calculations • Analytical solutions require a closed-form algebraic formulation of energy score • Numerical solution try to approximate the gradient (or Hessian) • Simple example: f’(x) ≈ f(x+1) – f(x) • Another example: the Secant method (soon)

Gradient Descent Minimization Algorithm Sliding down an energy gradient good ( = global minimum) local minimum Image by Ken Dill

Gradient Descent – System Description • Coordinates vector (Cartesian or Internal coordinates):X=(x1, x2,…,xn) • Differentiable energy function:E(X) • Gradient vector: *Adapted from slides by Chen Kaeasar, Ben-Gurion University

Gradient Descent Minimization Algorithm: Parameters:λ = step size ;  = convergence threshold • x = random starting point • While (x)  >  • Compute (x) • xnew = x + λ(x) • Line search: find the best step size λ in order to minimize E(xnew) (discussion later) • Note on convergence condition: in local minima, the gradient must be zero (but not always the other way around)

Line Search Methods – Solving argminλE[x + λ(x)]: • This is also an optimization problem, but in one-dimension… • Inexact solutions are probably sufficient Interval bracketing – (e.g., golden section, parabolic interpolation, Brent’s search) • Bracketing the local minimum by intervals of decreasing length • Always finds a local minimum Backtracking (e.g., with Armijo / Wolfe conditions): • Multiply step-size λ by c<1, until some condition is met • Variations: λ can also increase 1-D Newton and Secant methods We will talk about this soon…

2-D Rosenbrock’s Function: a Banana Shaped ValleyPathologically Slow Convergence for Gradient Descent 0 iterations 1000 iterations 100 iterations 10 iterations The (very common) problem: a narrow, winding “valley” in the energy landscape  The narrow valley results in miniscule, zigzag steps

(One) Solution: Conjugate Gradient Descent • Use a (smart) linear combination of gradients from previous iterations to prevent zigzag motion Conjugated gradient descent Parameters:λ = step size ;  = convergence threshold • x0 = random starting point • Λ0 = (x0) • While Λi  >  • Λi+1 = (xi) + βi∙Λi • choice of βi is important • Xi+1 = xi + λ ∙ Λi • Line search: adjust step size λ to minimize E(Xi+1) gradient descent • The new gradient is “A-orthogonal” to all previous search direction, for exact line search • Works best when the surface is approximately quadratic near the minimum (convergence in N iterations), otherwise need to reset the search every N steps (N = dimension of space)

Root Finding – when is f(x) = 0?

Taylor’s Series First order approximation: Second order approximation: The full Series: = Example: (a=0)

Taylor’s Approximation: f(x)=ex

Taylor’s Approximation of f(x) = sin(x)2x at x=1.5

From Taylor’s Series to Root Finding(one-dimension) First order approximation: Root finding by Taylor’s approximation:

Newton-Raphson Method for Root Finding(one-dimension) • Start from a random x0 • While not converged, update x with Taylor’s series:

Newton-Raphson: Quadratic Convergence Rate THEOREM: Let xroot be a “nice” root of f(x). There exists a “neighborhood” of some size Δ around xroot , in which Newton method will converge towards xrootquadratically( = the error decreases quadratically in each round) Image from http://www.codecogs.com/d-ox/maths/rootfinding/newton.php

The Secant Method(one-dimension) • Just like Newton-Raphson, but approximate the derivative by drawing a secant line between two previous points: Secant algorithm: Start from two random points: x0, x1 While not converged: • Theoretical convergence rate: golden-ratio (~1.62) • Often faster in practice: no gradient calculations

Newton’s Method:from Root Finding to Minimization Second order approximation of f(x): take derivative (by X) Minimum is reached when derivative of approximation is zero: • So… this is just root finding over the derivative (which makes sense since in local minima, the gradient is zero)

Newton’s Method for Minimization: • Start from a random vector x=x0 • While not converged, update x with Taylor’s series: • Notes: • if f’’(x)>0, then x is surely a local minimum point • We can choose a different step size than one

Newton’s Method for Minimization:Higher Dimensions • Start from a random vector x=x0 • While not converged, update x with Taylor’s series: • Notes: • H is the Hessian matrix (generalization of second derivative to high dimensions) • We can choose a different step size using Line Search (see previous slides)

Generalizing the Secant Method to High Dimensions: Quasi-Newton Methods • Calculating the Hessian (2nd derivative) is expensive  numerical calculation of Hessian • Popular methods: • DFP (Davidson – Fletcher – Powell) • BFGS (Broyden – Fletcher – Goldfarb – Shanno) • Combinations • Timeline: • Newton-Raphson (17th century) Secant method  DFP (1959, 1963)  Broyden Method for roots (1965)  BFGS (1970)

Some more Resources on Gradient and Newton Methods • Conjugate Gradient Descenthttp://www.cs.cmu.edu/~quake-papers/painless-conjugate-gradient.pdf • Quasi-Newton Methods: http://www.srl.gatech.edu/education/ME6103/Quasi-Newton.ppt • HUJI course on non-linear optimization by Benjamin Yakir http://pluto.huji.ac.il/~msby/opt-files/optimization.html • Line search: • http://pluto.huji.ac.il/~msby/opt-files/opt04.pdf • http://www.physiol.ox.ac.uk/Computing/Online_Documentation/Matlab/toolbox/nnet/backpr59.html • Wikipedia…

Harder Goal: Move from an Arbitrary Model to a Correct One Example: predict protein structure from its AA sequence. Arbitrary starting point *Adapted from slides by Chen Kaeasar, Ben-Gurion University

iteration 10 *Adapted from slides by Chen Kaeasar, Ben-Gurion University

3. Optimization Methods for Molecular Modeling

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