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# Molecular Modeling: Geometry Optimization - PowerPoint PPT Presentation

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

C372

Introduction to Cheminformatics II

Kelsey Forsythe

• 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)

• PES equivalent to Born-Oppenheimer surface

• Point on surface corresponds to position of nuclei

• Minimum and Maximum

• Local

• Global

• Saddle point (min and max)

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

Global maxima

Local maxima

Local minima

Global minimum

• Basic Scheme

• Find first derivative (gradient) of potential energy

• Set equal to zero

• Find value of coordinate(s) which satisfy equation

Bracketing

Golden Section (optimal bracket fractional distance (a-b)/(a-c)is Golden Ratio) for a>b>c

Parabolic Interpolation (Brent’s method)

Steepest Descent

Methods (1-d)

Line Search

Simplex/Downhill Simplex (Useful for rough surfaces)

Fletcher-Powell (Faster than simplex)

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

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)

• Steepest Descent (Gradient Descent Method)

Molecular Dynamics

Monte Carlo

Simulated Annealing

Genetic Algorithm

Global Multidimensional Methods

Second Order MethodsNewton’s Method

• Iterative (fast)

• Better energy estimate

• N3

• Energy involves calculating Hessian

• Assigning weights to configuration/coordinates

Modeling Potential energy (1-d)

First Order

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

Gradient projection along ith eigenvector

Eigenvalues from Hessian rotation/diagonalization

• 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

• 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 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

• 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)

Numerical Recipes

Initial Vertices

Reflection

Reflection

Expansion

Contraction

Contraction

• Gradients not required

• Time to minimize is long

• Find minimum of x2+y2=f(x,y)

Line Search #1

Xn=xn-1-.1ex

• Find minimum of x2+y2=f(x,y)

Line Search #2

Yn=yn-1-.1ey

• Find minimum of

• x2 + xy +y2=f(x,y)

Line Search #1

xn=xn-1-.1ex

• Find minimum of

• x2 + xy +y2=f(x,y)

Line Search #2

yn=yn-1-.1ey

• Find minimum of

• x2 + xy +y2=f(x,y)

Line Search #3

xn=xn-1-.1ex

• Crystal Cooling/Heating

• Applications

• Macromolecules (Conformer Searches)

• Traveling Salesman Problem

• Electronic Circuits

• 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

• Uphill moves allowed!!

• Implementation

• Must define T – sequence

• Must choose distribution of random numbers

• Neumann, Ulam and Metropolis (1940s)

• Fissionable material modeling

• Buffon (1700s)

• Needle drop – approximate pi

• Approximating p

• Approximating Areas/Integrals with random selection of points

C

B

D

0

1

A

• Sample Mean Integration

• Consider any uniform density/distribution of points, r

• Choose M points at random

• Consider any uniform density/distribution of points, r

• Metropolis et al

• Introduced non-uniform density

• Error a 1/N1/2 (N=#samplings)

• “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

• Multi-Resolution

Smoothing

• 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)

• 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(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

• Newton or variable metric methods

• N-d Global

• Monte Carlo

• Simulated Annealing

• Genetic Algoritms

• Form of energy

• Analytic

• Not analytic

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

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