Molecular Modeling: Geometry Optimization

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Molecular ModelingGeometry Optimization

• C372
• Introduction to Cheminformatics II
• Kelsey Forsythe

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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)

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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)

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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

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Terminology
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Cyclohexane
Global maxima
Local maxima
Local minima
Global minimum
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Geometry Optimization

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

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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

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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)

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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

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Line Search
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Steepest Descent
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Line Search(1-d)

• Steepest Descent (Gradient Descent Method)

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Global Multidimensional Methods

• Stochastic Tunneling
• Molecular Dynamics
• Monte Carlo
• Simulated Annealing
• Genetic Algorithm

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Second Order MethodsNewtons Method

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

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Modeling Potential energy (1-d)
First Order
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Modeling Potential energy (gt1-d)
Hessian
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Newtons Method
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Newtons Method

• Equivalent to rotating Hessian (coordinate
  transformation, r--gtr) s.t. Hessian diagonal

Gradient projection along ith eigenvector
Eigenvalues from Hessian rotation/diagonalization
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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)

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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

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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

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Simplex(Simplices)
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Simplex Method
Numerical Recipes
Initial Vertices
Reflection
Reflection
Expansion
Contraction
Contraction
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Simplex Methods

• Advantages
• Gradients not required
• Disadvantages
• Time to minimize is long

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Example

• Find minimum of x2y2f(x,y)

Line Search 1 Xnxn-1-.1ex
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Example

• Find minimum of x2y2f(x,y)

Line Search 2 Ynyn-1-.1ey
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Example

• Find minimum of
• x2 xy y2f(x,y)

Line Search 1 xnxn-1-.1ex
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Example

• Find minimum of
• x2 xy y2f(x,y)

Line Search 2 ynyn-1-.1ey
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Example (Spoiling)

• Find minimum of
• x2 xy y2f(x,y)

Line Search 3 xnxn-1-.1ex
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Global-Simulated Annealing

• Crystal Cooling/Heating
• Applications
• Macromolecules (Conformer Searches)
• Traveling Salesman Problem
• Electronic Circuits

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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
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Global-Simulated Annealing

• Uphill moves allowed!!
• Implementation
• Must define T sequence
• Must choose distribution of random numbers

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Global-Monte Carlo Algorithms

• Neumann, Ulam and Metropolis (1940s)
• Fissionable material modeling
• Buffon (1700s)
• Needle drop approximate pi

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Global-Monte Carlo Algorithms

• Approximating p
• Approximating Areas/Integrals with random
  selection of points

C
B
D
0
1
A
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Global-Monte Carlo Algorithms

• Sample Mean Integration
• Consider any uniform density/distribution of
  points, r
• Choose M points at random

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Global-Monte Carlo Algorithms

• Consider any uniform density/distribution of
  points, r

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Global-Monte Carlo Algorithms

• Metropolis et al
• Introduced non-uniform density
• Error a 1/N1/2 (Nsamplings)

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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

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Global-Rugged

• Multi-Resolution
• Graduated Non-Convex

Smoothing
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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

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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

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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

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References

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

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