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Unformatted text preview: Molecular Modeling:
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 Use in other simulations (e.g. dynamics calculation) Used in reaction rate calculations (e.g. 1 / saddle reaction time ) Characteristics of transition state PES interpolation (Collins et al) Nomenclature PES equivalent to BornOppenheimer surface Point on surface corresponds to position of nuclei Minimum and Maximum
Local Global Saddle point (min and max) 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 where the potential gradient is zero) Elocal greater than or equal to Eglobal Terminology Cyclohexane
Global maxima Local maxima Local minima Global minimum 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 (ab)/(ac)is Golden Ratio) for a>b>c Parabolic Interpolation (Brent's method) Steepest Descent Gradients Methods (n-d) W/O Gradients (Zeroth Order) NO GRADIENTS = ZEROTH ORDER Line Search Simplex/Downhill Simplex (Useful for rough surfaces) FletcherPowell (Faster than simplex) Methods (n-d) W/Gradients (Frist Order) Steepest Descent Conjugate Gradient (space N) FletcherReeves PolakRibiere QuasiNewton/Variable Metric (space N2) DavidonFletcherPowell BroydenFletcherGoldfarbShanno Line Search
v xn +1 = xn + u v v u =e Steepest Descent v xn +1 = xn + lu uf v u =- f =- ux
xn Line Search(1-d) Steepest Descent (Gradient Descent Method)
( ) = 3 - 2 2 + 2 = 0.1
df dx xi +1 = xi + u u=
xi Global Multidimensional Methods Stochastic Tunneling Molecular Dynamics Monte Carlo Simulated Annealing Genetic Algorithm Second Order Methods Newton's Method Advantages Iterative (fast) Better energy estimate N3 Energy involves calculating Hessian Assigning weights to configuration/coordinates Disadvantages Modeling Potential energy (1-d)
dU U(r) = U(req ) dr 1 d 2U (r - req ) + 2 dr 2 r= req
First Order (r - req ) 2
r= req 1 d 3U - 3 dr r= req 1 d nU (r - req ) 3 ....+ n! dr n
N-1th Order (r - req ) n
r= req Modeling Potential energy (>1-d)
v v U( ra + dr ) = U(ra ) i dU dr dri +
r= ra i, j i j 1 d 2U dri 2 dri dr j drj + .....
r= req v v 1 vT v c - bd r + d r A d r 2
Hessian Newton's Method
v v U( ra + dr ) = U(ra ) i dU dr dri +
r= ra i, j i j 1 d 2U dri 2 dri dr j drj + .....
r= req v v 1 vT v c - bd r + d r A d r 2 v v dU( ra + dr ) =0 v dr v v b dr = A Newton's Method
v v dU( ra + dr ) =0 v dr v v b dr = A Equivalent to rotating Hessian (coordinate transformation, r>r') s.t. Hessian diagonal ' '
r = Dri l fi ' Dri = ei i Gradient projection along ith eigenvector Eigenvalues from Hessian rotation/diagonalization Second Order Methods Advantages Only one iteration for quadratic functions! Efficient (relative to first order methods) Disadvantages Better energy estimate N/N1 = (N1/N2)2 (I.e. 10,100,10000 reduction in gradient) N2 storage requirements (compared to N for conjugate gradient) Involves calculating Hessian (~10 times time for gradient calculation) N3 ~Hessian (pseudoNewton methods) Oft used in transitionstructure searches (saddle point locator) DavidonFletcherPowell BroydenFletcherGoldfarbShanno Powell Second Order Methods
Levenberg-Marquardt Far from minimum (Taylor poor!) rrob/A r=ro *b Find beta s.t. move in direction of minimum Given ro,E(ro), pick initial value of Find A'=(1+) A Find x s.t. A'x=b Calculate E(ro+x), adjust accordingly to reach minimum Simplex Methods Minimization Bounds Polygon of N+1 vertices Procedure (Downhill Simplex Method) Solution is a vertex of N+1d polygon 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+ ) Else Try internediate point If E(x new)<E(xo) expand If E(x new)>E(xo) contract Simplex (Simplices) Simplex Method
Initial Vertices Numerical Recipes v v v Pi = P0 + e
Reflection Reflection v v Pnew = - Phigh Expansion Contraction Contraction Simplex Methods Advantages Gradients not required Time to minimize is long Disadvantages Example Find minimum of x2+y2=f(x,y)
x 1 y 1 f(x,y) 2 1.1 1 2.21 2-d parabola Line Search #1 Xn=xn-1-.1ex 0.9 1 1.81 0.8 1 1.64 0.7 1 1.49 0.6 1 1.36 0.5 1 1.25 0.4 1 1.16 Example Find minimum of x2+y2=f(x,y)
0 0 0 0 1 1.1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 1.39E-16 -0.1 -0.2 -0.3 1 1.21 0.81 0.64 0.49 0.36 0.25 0.16 0.09 0.04 0.01 1.93E-32 0.01 0.04 0.09 Line Search #2 Yn=yn-1-.1ey 0 0 0 0 0 0 0 0 0 0 0 Example Find minimum of x y x2 + xy +y2=f(x,y) 1 0.9 0.8 0.7 0.6 0.5 0.4 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 2.71 2.44 2.19 1.96 1.75 1.56 1.39 1.24 1.11 1 0.91 0.84 0.79 0.76 0.75 0.76 Line Search #1 xn=xn-1-.1ex 0.3 0.2 0.1 1.39E-16 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 Example Find minimum of -0.5 -0.5 -0.5 -0.5 -0.5 -0.5 -0.5 -0.5 -0.5 -0.5 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.75 0.61 0.49 0.39 0.31 0.25 0.21 0.19 0.19 0.21 x2 + xy +y2=f(x,y) Line Search #2 yn=yn-1-.1ey Example (Spoiling) Find minimum of -0.5 0.3 0.19 x2 + xy +y2=f(x,y) -0.4 0.3 0.13 -0.3 0.3 0.09 Line Search #3 xn=xn-1-.1ex -0.2 0.3 0.07 -0.1 0.3 0.07 Global-Simulated Annealing Crystal Cooling/Heating Applications Macromolecules (Conformer Searches) Traveling Salesman Problem Electronic Circuits Global-Simulated Annealing Uphill moves allowed!! Given configuration Xi and E(Xi) Step in direction X If E(Xi+ X)< E(Xi) Move accepted E(Xi+ X)< E(Xi) then Choose 1>Y>0 + X ) - E ( X )) (E( Xi i - Metropolis k bT If Accepted Y >e et al Global-Simulated Annealing Uphill moves allowed!! Y >e - ( E ( X i + X ) - E ( X i )) k bT Implementation Must define T sequence Must choose distribution of random numbers Global-Monte Carlo Algorithms Neumann, Ulam and Metropolis (1940s) Fissionable material modeling Needle drop approximate pi Buffon (1700s) Global-Monte Carlo Algorithms Approximating h >x it:1 ,y>0 m s:x> o y> is 1r 1 C B 4 Area under curve ABC 4 hit = Area of square shot D 0 1 A Approximating Areas/Integrals with random selection of points Global-Monte Carlo Algorithms Sample Mean Integration Consider any uniform density/distribution x- x A f(x d i) x= 2 1 = ) x x f(x , N x i 1
x 2 N1 - of points, Choose M points at random
f(x ) 1 M f(x) A = ) x i (x d ) M i i) (x x (x 1
x 2 Global-Monte Carlo Algorithms Consider any uniform density/distribution of points, x 2 f(x ) 1 M f(x) A = ) x i (x d ) M i i) (x x (x 1 1 Ltx e ( )= ,t e hn x -x 2 1
1 f(x ) x -x A= f(x)dx i 2 1 f (xi) M i (xi) M i x1
x2 M M f(xi), for M=N-1 x
i M Global-Monte Carlo Algorithms Metropolis et al Introduced nonuniform density Error 1/N1/2 (N=#samplings) 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 MultiResolution Graduated NonConvex Smoothing Others Fragment Approach Fix/Constrain part while optimizing other Proteins RuleBased 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
u dV (r ) u =0 dr u d 2V (r ) u2 > 0, min dr u d 2V (r ) u2 < 0, max As N increases so does dimensionality/complexity/beauty/ dr difficulty Geometry Optimization (Summary) Method used depends on System size 1d (line search, bracketing, steepest descent) Nd local (Downhill) W/o derivatives Simplex Direction set methods (Powell's) W/ derivatives Conjugate gradient Newton or variable metric methods Monte Carlo Simulated Annealing Genetic Algoritms Nd Global Form of energy Analytic Not analytic References Computer Simulation of Liquids, Allen, M. P. and Tildesley, D. J. Numerical Recipes:The Art of Scientific Computing Press, W. H. et. Al. Next Time
Energy Calculation Optimization Calculation Properties Calculation Vibrations Rotations ...
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This note was uploaded on 06/28/2011 for the course C 372 taught by Professor Yoonsuplee during the Spring '11 term at Korea Advanced Institute of Science and Technology.
- Spring '11