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3. Optimization Methods for Molecular Modeling

3. Optimization Methods for Molecular Modeling

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3. Optimization Methods for Molecular Modeling

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  1. 3. Optimization Methods for Molecular Modeling by Barak Raveh

  2. 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

  3. 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

  4. 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

  5. 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

  6. 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

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

  8. 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

  9. 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

  10. 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

  11. 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

  12. 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

  13. 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

  14. 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

  15. 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

  16. 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

  17. 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

  18. 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:

  19. 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)

  20. 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.

  21. 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)

  22. 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

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

  24. 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

  25. 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)

  26. 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…

  27. 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

  28. (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)

  29. 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

  30. Root Finding – when is f(x) = 0?

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

  32. Taylor’s Approximation: f(x)=ex

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

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

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

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

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

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

  39. 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

  40. 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

  41. 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)

  42. 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

  43. 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)

  44. 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)

  45. 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…

  46. 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

  47. 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

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

  49. iteration 100 *Adapted from slides by Chen Kaeasar, Ben-Gurion University

  50. iteration 200 *Adapted from slides by Chen Kaeasar, Ben-Gurion University