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Molecular Modeling: Geometry Optimization

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Molecular Modeling: Geometry Optimization C372 Introduction to Cheminformatics II Kelsey Forsythe Why Extrema? Equilibrium structure/conformer MOST likely observed? – PowerPoint PPT presentation

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Title: Molecular Modeling: Geometry Optimization


1
Molecular ModelingGeometry Optimization
  • C372
  • Introduction to Cheminformatics II
  • Kelsey Forsythe

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

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

4
Local vs. Global?
Conformational Analysis (Equilibrium Conformer)
A conformational analysis is global geometry
optimization which yields multiple structurally
stable conformational geometries (i.e.
equilibrium geometries)
Equilibrium Geometry
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
    wherethe potential gradient is zero)
  • Elocal greater than or equal to Eglobal

5
Terminology
6
Cyclohexane
Global maxima
Local maxima
Local minima
Global minimum
7
Geometry Optimization
  • Basic Scheme
  • Find first derivative (gradient) of potential
    energy
  • Set equal to zero
  • Find value of coordinate(s) which satisfy equation

8
Methods (1-d)
  • No Gradients (No Functional Form for E)
  • Bracketing
  • Golden Section (optimal bracket fractional
    distance (a-b)/(a-c)is Golden Ratio) for agtbgtc
  • Parabolic Interpolation (Brents method)
  • Gradients
  • Steepest Descent

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

10
Methods (n-d)W/Gradients (Frist Order)
  • Steepest Descent
  • Conjugate Gradient (space a N)
  • Fletcher-Reeves
  • Polak-Ribiere
  • Quasi-Newton/Variable Metric (space a N2)
  • Davidon-Fletcher-Powell
  • Broyden-Fletcher-Goldfarb-Shanno

11
Line Search
12
Steepest Descent
13
Line Search(1-d)
  • Steepest Descent (Gradient Descent Method)

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

15
Second Order MethodsNewtons Method
  • Advantages
  • Iterative (fast)
  • Better energy estimate
  • Disadvantages
  • N3
  • Energy involves calculating Hessian
  • Assigning weights to configuration/coordinates

16
Modeling Potential energy (1-d)
First Order
17
Modeling Potential energy (gt1-d)
Hessian
18
Newtons Method
19
Newtons Method
  • Equivalent to rotating Hessian (coordinate
    transformation, r--gtr) s.t. Hessian diagonal

Gradient projection along ith eigenvector
Eigenvalues from Hessian rotation/diagonalization
20
Second Order Methods
  • Advantages
  • Only one iteration for quadratic functions!
  • Efficient (relative to first -order methods)
  • N/N-1 (N-1/N-2)2 (I.e. 10,100,10000 reduction
    in gradient)
  • Better energy estimate
  • Disadvantages
  • N2 storage requirements (compared to N for
    conjugate gradient)
  • N3
  • Involves calculating Hessian (10 times time for
    gradient calculation)
  • Hessian (pseudo-Newton methods)
  • Davidon-Fletcher-Powell
  • Broyden-Fletcher-Goldfarb-Shanno
  • Powell
  • Oft used in transition-structure searches (saddle
    point locator)

21
Second Order MethodsLevenberg-Marquardt
  • Far from minimum (Taylor poor!)
  • r?ro-b/A rro-bb
  • Find beta s.t. move in direction of minimum
  • Given ro,E(ro), pick initial value of l
  • Find A(1l)A
  • Find x s.t. Axb
  • Calculate E(rox), adjust l accordingly to reach
    minimum

22
Simplex Methods
  • Minimization Bounds ? Polygon of N1 vertices
  • Solution is a vertex of N1-d polygon
  • Procedure (Downhill Simplex Method)
  • Begin with simplex for input coordinate values
  • Find lowest point on simplex
  • Find highest point on simplex
  • Reflect (x1-xo)
  • If E(x1)ltE(xo) then expand (xxl)
  • Else
  • Try internediate point
  • If E(xnew)ltE(xo) expand
  • If E(xnew)gtE(xo) contract

23
Simplex(Simplices)
24
Simplex Method
Numerical Recipes
Initial Vertices
Reflection
Reflection
Expansion
Contraction
Contraction
25
Simplex Methods
  • Advantages
  • Gradients not required
  • Disadvantages
  • Time to minimize is long

26
Example
  • Find minimum of x2y2f(x,y)

Line Search 1 Xnxn-1-.1ex
27
Example
  • Find minimum of x2y2f(x,y)

Line Search 2 Ynyn-1-.1ey
28
Example
  • Find minimum of
  • x2 xy y2f(x,y)

Line Search 1 xnxn-1-.1ex
29
Example
  • Find minimum of
  • x2 xy y2f(x,y)

Line Search 2 ynyn-1-.1ey
30
Example (Spoiling)
  • Find minimum of
  • x2 xy y2f(x,y)

Line Search 3 xnxn-1-.1ex
31
Global-Simulated Annealing
  • Crystal Cooling/Heating
  • Applications
  • Macromolecules (Conformer Searches)
  • Traveling Salesman Problem
  • Electronic Circuits

32
Global-Simulated Annealing
  • Uphill moves allowed!!
  • Given configuration Xi and E(Xi)
  • Step in direction DX
  • If
  • E(Xi DX)lt E(Xi) - Move accepted
  • E(Xi DX)lt E(Xi) then
  • Choose 1gtYgt0
  • If
    Accepted

Metropolis et al
33
Global-Simulated Annealing
  • Uphill moves allowed!!
  • Implementation
  • Must define T sequence
  • Must choose distribution of random numbers

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

35
Global-Monte Carlo Algorithms
  • Approximating p
  • Approximating Areas/Integrals with random
    selection of points

C
B
D
0
1
A
36
Global-Monte Carlo Algorithms
  • Sample Mean Integration
  • Consider any uniform density/distribution of
    points, r
  • Choose M points at random

37
Global-Monte Carlo Algorithms
  • Consider any uniform density/distribution of
    points, r

38
Global-Monte Carlo Algorithms
  • Metropolis et al
  • Introduced non-uniform density
  • Error a 1/N1/2 (Nsamplings)

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

40
Global-Rugged
  • Multi-Resolution
  • Graduated Non-Convex

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

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

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

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

45
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