1. Molecular Modeling: Geometry Optimization C372 Introduction to Cheminformatics II Kelsey Forsythe

2. Why Extrema? Equilibrium structure/conformer MOST likely observed? Equilibrium structure/conformer MOST likely observed? Once geometrically optimum structure found can calculate energy, frequencies etc. to compare with experiment Once geometrically optimum structure found can calculate energy, frequencies etc. to compare with experiment Use in other simulations (e.g. dynamics calculation) Use in other simulations (e.g. dynamics calculation) Used in reaction rate calculations (e.g.  saddle  reaction time ) Used in reaction rate calculations (e.g.  saddle  reaction time ) Characteristics of transition state Characteristics of transition state PES interpolation (Collins et al) PES interpolation (Collins et al)

3. Nomenclature PES equivalent to Born-Oppenheimer surface PES equivalent to Born-Oppenheimer surface Point on surface corresponds to position of nuclei Point on surface corresponds to position of nuclei Minimum and Maximum Minimum and Maximum Local Local Global Global Saddle point (min and max) Saddle point (min and max)

4. Local vs. Global? Conformational Analysis (Equilibrium Conformer) Equilibrium Geometry A conformational analysis is global geometry optimization which yields multiple structurally stable conformational geometries (i.e. equilibrium geometries) An equilibrium geometry may be a local geometry optimization which finds the closest minimum for a given structure (conformer) or an equilibrium conformer BOTH are geometry optimizations (i.e. finding where the potential gradient is zero) E local greater than or equal to E global

5. Terminology

6. Cyclohexane Local maxima Global minimum Global maxima Local minima

7. Geometry Optimization Basic Scheme Basic Scheme Find first derivative (gradient) of potential energy Find first derivative (gradient) of potential energy Set equal to zero Set equal to zero Find value of coordinate(s) which satisfy equation Find value of coordinate(s) which satisfy equation

8. Methods (1-d) No Gradients (No Functional Form for E) No Gradients (No Functional Form for E) Bracketing Bracketing Golden Section (optimal bracket fractional distance (a-b)/(a-c)is Golden Ratio) for a>b>c Golden Section (optimal bracket fractional distance (a-b)/(a-c)is Golden Ratio) for a>b>c Parabolic Interpolation (Brent’s method) Parabolic Interpolation (Brent’s method) Gradients Gradients Steepest Descent Steepest Descent

9. Methods (n-d) W/O Gradients (Zeroth Order) NO GRADIENTS = ZEROTH ORDER Line Search Line Search Simplex/Downhill Simplex (Useful for rough surfaces) Simplex/Downhill Simplex (Useful for rough surfaces) Fletcher-Powell (Faster than simplex) Fletcher-Powell (Faster than simplex)

10. Methods (n-d) W/Gradients (Frist Order) Steepest Descent Steepest Descent Conjugate Gradient (space  N) Conjugate Gradient (space  N) Fletcher-Reeves Fletcher-Reeves Polak-Ribiere Polak-Ribiere Quasi-Newton/Variable Metric (space  N 2 ) Quasi-Newton/Variable Metric (space  N 2 ) Davidon-Fletcher-Powell Davidon-Fletcher-Powell Broyden-Fletcher-Goldfarb-Shanno Broyden-Fletcher-Goldfarb-Shanno

11. Line Search

12. Steepest Descent

13. Line Search(1-d) Steepest Descent (Gradient Descent Method) Steepest Descent (Gradient Descent Method)

14. Global Multidimensional Methods Stochastic Tunneling Stochastic Tunneling Molecular Dynamics Molecular Dynamics Monte Carlo Monte Carlo Simulated Annealing Simulated Annealing Genetic Algorithm Genetic Algorithm

15. Second Order Methods Newton’s Method Advantages Advantages Iterative (fast) Iterative (fast) Better energy estimate Better energy estimate Disadvantages Disadvantages N 3 N 3 Energy involves calculating Hessian Energy involves calculating Hessian Assigning weights to configuration/coordinates Assigning weights to configuration/coordinates

16. Modeling Potential energy (1-d) First Order N-1th Order

17. Modeling Potential energy (>1-d) Hessian

18. Newton’s Method

19. Equivalent to rotating Hessian (coordinate transformation, r-->r’) s.t. Hessian diagonal Equivalent to rotating Hessian (coordinate transformation, r-->r’) s.t. Hessian diagonal Gradient projection along i th eigenvector Eigenvalues from Hessian rotation/diagonalization

20. Second Order Methods Advantages Advantages Only one iteration for quadratic functions! Only one iteration for quadratic functions! Efficient (relative to first -order methods) Efficient (relative to first -order methods) N/N-1 = (N-1/N-2) 2 (I.e. 10,100,10000 reduction in gradient) N/N-1 = (N-1/N-2) 2 (I.e. 10,100,10000 reduction in gradient) Better energy estimate Better energy estimate Disadvantages Disadvantages N 2 storage requirements (compared to N for conjugate gradient) N 2 storage requirements (compared to N for conjugate gradient) N 3 N 3 Involves calculating Hessian (~10 times time for gradient calculation) Involves calculating Hessian (~10 times time for gradient calculation) ~Hessian (pseudo-Newton methods) ~Hessian (pseudo-Newton methods) –Davidon-Fletcher-Powell –Broyden-Fletcher-Goldfarb-Shanno –Powell Oft used in transition-structure searches (saddle point locator) Oft used in transition-structure searches (saddle point locator)

21. Second Order Methods Levenberg-Marquardt Far from minimum (Taylor poor!) Far from minimum (Taylor poor!) r≠r o -b/A r=r o -  * b Find beta s.t. move in direction of minimum Find beta s.t. move in direction of minimum –Given r o,E(r o ), pick initial value of –Given r o,E(r o ), pick initial value of –Find A’=(1+  A –Find x s.t. A ’ x=b –Calculate E(r o +x), adjust accordingly to reach minimum

22. Simplex Methods Minimization Bounds  Polygon of N+1 vertices Minimization Bounds  Polygon of N+1 vertices Solution is a vertex of N+1-d polygon Solution is a vertex of N+1-d polygon Procedure (Downhill Simplex Method) Procedure (Downhill Simplex Method) Begin with simplex for input coordinate values Begin with simplex for input coordinate values Find lowest point on simplex Find lowest point on simplex Find highest point on simplex Find highest point on simplex Reflect (x 1 =-x o ) Reflect (x 1 =-x o ) If E(x 1 )<E(x o ) then expand (x=x+ ) If E(x 1 )<E(x o ) then expand (x=x+ ) –Else –Try internediate point If E(x new )<E(x o ) expand If E(x new )<E(x o ) expand If E(x new )>E(x o ) contract If E(x new )>E(x o ) contract

23. Simplex (Simplices)

24. Simplex Method Numerical Recipes Reflection Expansion Contraction Initial Vertices Reflection

25. Simplex Methods Advantages Advantages Gradients not required Gradients not required Disadvantages Disadvantages Time to minimize is long Time to minimize is long

26. Example Find minimum of x 2 +y 2 =f(x,y) Find minimum of x 2 +y 2 =f(x,y) 2-d parabolaxyf(x,y) 112 1.112.21 0.911.81 0.811.64 0.711.49 0.611.36 0.511.25 0.411.16 0.311.09 0.211.04 0.111.01 1.39E-1611 -0.111.01 -0.211.04 -0.311.09 Line Search #1 X n =x n-1 -.1e x

27. Example Find minimum of x 2 +y 2 =f(x,y) Find minimum of x 2 +y 2 =f(x,y) Line Search #2 Y n =y n-1 -.1e y 011 01.11.21 00.90.81 00.80.64 00.70.49 00.60.36 00.50.25 00.40.16 00.30.09 00.20.04 00.10.01 01.39E-161.93E-32 0-0.10.01 0-0.20.04 0-0.30.09

28. Example Find minimum of Find minimum of x 2 + xy +y 2 =f(x,y) x 2 + xy +y 2 =f(x,y) Line Search #1 x n =x n-1 -.1e x xy 113 0.912.71 0.812.44 0.712.19 0.611.96 0.511.75 0.411.56 0.311.39 0.211.24 0.111.11 1.39E-1611 -0.110.91 -0.210.84 -0.310.79 -0.410.76 -0.510.75 -0.610.76

29. Example Find minimum of Find minimum of x 2 + xy +y 2 =f(x,y) x 2 + xy +y 2 =f(x,y) Line Search #2 y n =y n-1 -.1e y -0.510.75 -0.50.90.61 -0.50.80.49 -0.50.70.39 -0.50.60.31 -0.50.50.25 -0.50.40.21 -0.50.30.19 -0.50.20.19 -0.50.10.21

30. Example (Spoiling) Find minimum of Find minimum of x 2 + xy +y 2 =f(x,y) x 2 + xy +y 2 =f(x,y) Line Search #3 x n =x n-1 -.1e x -0.50.30.19 -0.40.30.13 -0.30.30.09 -0.20.30.07 -0.10.30.07

31. Global-Simulated Annealing Crystal Cooling/Heating Crystal Cooling/Heating Applications Applications Macromolecules (Conformer Searches) Macromolecules (Conformer Searches) Traveling Salesman Problem Traveling Salesman Problem Electronic Circuits Electronic Circuits

32. Global-Simulated Annealing Uphill moves allowed!! Uphill moves allowed!! Given configuration X i and E(X i ) Given configuration X i and E(X i ) Step in direction  X Step in direction  X If If E(X i +  X)< E(X i ) - Move accepted E(X i +  X)< E(X i ) - Move accepted E(X i +  X)< E(X i ) then E(X i +  X)< E(X i ) then –Choose 1>Y>0 –If Accepted Metropolis et al

33. Global-Simulated Annealing Uphill moves allowed!! Uphill moves allowed!! Implementation Implementation Must define T – sequence Must define T – sequence Must choose distribution of random numbers Must choose distribution of random numbers

34. Global-Monte Carlo Algorithms Neumann, Ulam and Metropolis (1940s) Neumann, Ulam and Metropolis (1940s) Fissionable material modeling Fissionable material modeling Buffon (1700s) Buffon (1700s) Needle drop – approximate pi Needle drop – approximate pi

35. Global-Monte Carlo Algorithms Approximating  Approximating  Approximating Areas/Integrals with random selection of points Approximating Areas/Integrals with random selection of points A C B D 0 1

36. Global-Monte Carlo Algorithms Sample Mean Integration Sample Mean Integration Consider any uniform density/distribution of points,  Consider any uniform density/distribution of points,  Choose M points at random Choose M points at random

37. Global-Monte Carlo Algorithms Consider any uniform density/distribution of points,  Consider any uniform density/distribution of points, 

38. Global-Monte Carlo Algorithms Metropolis et al Metropolis et al Introduced non-uniform density Introduced non-uniform density Error  1/N 1/2 (N=#samplings) Error  1/N 1/2 (N=#samplings)

39. Global-Genetic Algorithms “Population” of conformations/structures “Population” of conformations/structures Each “parent” conformer comprised of “genes” Each “parent” conformer comprised of “genes” “Offspring” generated from mixtures of “genes” “Offspring” generated from mixtures of “genes” “mutations” allowed “mutations” allowed Most fit “offspring” kept for next “generation” Most fit “offspring” kept for next “generation” “Fitness” = low energy “Fitness” = low energy

40. Global-Rugged Multi-Resolution Multi-Resolution Graduated Non-Convex Graduated Non-Convex Smoothing

41. Others Fragment Approach Fragment Approach Fix/Constrain part while optimizing other Fix/Constrain part while optimizing other Rule-Based Rule-Based Proteins Proteins Fix tertiary structure according to statistically likelihood of amino acid sequence to adopt such a structure Fix tertiary structure according to statistically likelihood of amino acid sequence to adopt such a structure Homology modeling Homology modeling Use geometry of similar molecules as start for aforementioned methods Use geometry of similar molecules as start for aforementioned methods

42. Geometry Optimization (Summary) Optimum structure gives useful information Optimum structure gives useful information First Derivative is Zero - At minimum/maximum First Derivative is Zero - At minimum/maximum Use Second Derivative to establish minimum/maximum Use Second Derivative to establish minimum/maximum As N increases so does dimensionality/complexity/beauty/difficulty As N increases so does dimensionality/complexity/beauty/difficulty

43. Geometry Optimization (Summary) Method used depends on Method used depends on System size System size 1-d (line search, bracketing, steepest descent) 1-d (line search, bracketing, steepest descent) N-d local (Downhill) N-d local (Downhill) –W/o derivatives Simplex Simplex Direction set methods (Powell’s) Direction set methods (Powell’s) –W/ derivatives Conjugate gradient Conjugate gradient Newton or variable metric methods Newton or variable metric methods N-d Global N-d Global –Monte Carlo –Simulated Annealing –Genetic Algoritms Form of energy Form of energy Analytic Analytic Not analytic Not analytic

44. References Computer Simulation of Liquids, Allen, M. P. and Tildesley, D. J. Computer Simulation of Liquids, Allen, M. P. and Tildesley, D. J. Numerical Recipes:The Art of Scientific Computing Press, W. H. et. Al. Numerical Recipes:The Art of Scientific Computing Press, W. H. et. Al.

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