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Molecular Modeling: Geometry Optimization. C372 Introduction to Cheminformatics II Kelsey Forsythe. Why Extrema?. Equilibrium structure/conformer MOST likely observed? Once geometrically optimum structure found can calculate energy, frequencies etc. to compare with experiment

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Molecular modeling geometry optimization

Molecular Modeling:Geometry Optimization

C372

Introduction to Cheminformatics II

Kelsey Forsythe


Why extrema
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)


Nomenclature
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)


Local vs global
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



Cyclohexane
Cyclohexane

Global maxima

Local maxima

Local minima

Global minimum


Geometry optimization
Geometry Optimization

  • Basic Scheme

    • Find first derivative (gradient) of potential energy

    • Set equal to zero

    • Find value of coordinate(s) which satisfy equation


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 a>b>c

Parabolic Interpolation (Brent’s method)

Gradients

Steepest Descent

Methods (1-d)


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)

Methods (n-d)W/O Gradients (Zeroth Order)


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

Methods (n-d)W/Gradients (Frist Order)




Line search 1 d
Line Search(1-d)

  • Steepest Descent (Gradient Descent Method)


Global multidimensional methods

Stochastic Tunneling

Molecular Dynamics

Monte Carlo

Simulated Annealing

Genetic Algorithm

Global Multidimensional Methods


Second order methods newton s method
Second Order MethodsNewton’s Method

  • Advantages

    • Iterative (fast)

    • Better energy estimate

  • Disadvantages

    • N3

    • Energy involves calculating Hessian

    • Assigning weights to configuration/coordinates


Modeling potential energy 1 d

N-1th Order

Modeling Potential energy (1-d)

First Order




Newton s method1
Newton’s Method

  • Equivalent to rotating Hessian (coordinate transformation, r-->r’) s.t. Hessian diagonal

Gradient projection along ith eigenvector

Eigenvalues from Hessian rotation/diagonalization


Second order methods
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)


Second order methods levenberg marquardt
Second Order MethodsLevenberg-Marquardt

  • Far from minimum (Taylor poor!)

    r≠ro-b/A r=ro-b*b

    • Find beta s.t. move in direction of minimum

      • Given ro,E(ro), pick initial value of l

      • Find A’=(1+l)A

      • Find x s.t. A’x=b

      • Calculate E(ro+x), adjust l accordingly to reach minimum


Simplex methods
Simplex Methods

  • Minimization Bounds  Polygon of N+1 vertices

    • Solution is a vertex of N+1-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)<E(xo) then expand (x=x+l)

        • Else

        • Try internediate point

          • If E(xnew)<E(xo) expand

          • If E(xnew)>E(xo) contract


Simplex simplices
Simplex(Simplices)


Simplex method
Simplex Method

Numerical Recipes

Initial Vertices

Reflection

Reflection

Expansion

Contraction

Contraction


Simplex methods1
Simplex Methods

  • Advantages

    • Gradients not required

  • Disadvantages

    • Time to minimize is long


Example
Example

  • Find minimum of x2+y2=f(x,y)

Line Search #1

Xn=xn-1-.1ex


Example1
Example

  • Find minimum of x2+y2=f(x,y)

Line Search #2

Yn=yn-1-.1ey


Example2
Example

  • Find minimum of

    • x2 + xy +y2=f(x,y)

Line Search #1

xn=xn-1-.1ex


Example3
Example

  • Find minimum of

    • x2 + xy +y2=f(x,y)

Line Search #2

yn=yn-1-.1ey


Example spoiling
Example (Spoiling)

  • Find minimum of

    • x2 + xy +y2=f(x,y)

Line Search #3

xn=xn-1-.1ex


Global simulated annealing
Global-Simulated Annealing

  • Crystal Cooling/Heating

  • Applications

    • Macromolecules (Conformer Searches)

    • Traveling Salesman Problem

    • Electronic Circuits


Global simulated annealing1
Global-Simulated Annealing

  • Uphill moves allowed!!

    • Given configuration Xi and E(Xi)

    • Step in direction DX

    • If

      • E(Xi+ DX)< E(Xi) - Move accepted

      • E(Xi+ DX)< E(Xi) then

        • Choose 1>Y>0

        • If Accepted

Metropolis et al


Global simulated annealing2
Global-Simulated Annealing

  • Uphill moves allowed!!

  • Implementation

    • Must define T – sequence

    • Must choose distribution of random numbers


Global monte carlo algorithms
Global-Monte Carlo Algorithms

  • Neumann, Ulam and Metropolis (1940s)

    • Fissionable material modeling

  • Buffon (1700s)

    • Needle drop – approximate pi


Global monte carlo algorithms1
Global-Monte Carlo Algorithms

  • Approximating p

  • Approximating Areas/Integrals with random selection of points

C

B

D

0

1

A


Global monte carlo algorithms2
Global-Monte Carlo Algorithms

  • Sample Mean Integration

  • Consider any uniform density/distribution of points, r

  • Choose M points at random


Global monte carlo algorithms3
Global-Monte Carlo Algorithms

  • Consider any uniform density/distribution of points, r


Global monte carlo algorithms4
Global-Monte Carlo Algorithms

  • Metropolis et al

    • Introduced non-uniform density

    • Error a 1/N1/2 (N=#samplings)


Global genetic algorithms
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


Global rugged
Global-Rugged

  • Multi-Resolution

  • Graduated Non-Convex

Smoothing


Others
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


Geometry optimization summary
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/beauty/difficulty


Geometry optimization summary1
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 (Powell’s)

        • W/ derivatives

          • Conjugate gradient

          • Newton or variable metric methods

      • N-d Global

        • Monte Carlo

        • Simulated Annealing

        • Genetic Algoritms

    • Form of energy

      • Analytic

      • Not analytic


References
References

  • Computer Simulation of Liquids, Allen, M. P. and Tildesley, D. J.

  • Numerical Recipes:The Art of Scientific Computing Press, W. H. et. Al.