Computational Challenges in Computing Nearest Neighbor Estimates of - PowerPoint PPT Presentation

Computational Challenges in Computing Nearest Neighbor Estimates of Entropy for Large Molecules Home Page Title Page Contents ◭◭ ◮◮ E. James Harner, Harshinder Singh Shengqiao Li, and Jun Tan ◭ ◮ Page 1 of 40 Research supported by: Biostatistics Branch, National Institute for Occupational Go Back Safety and Health, Morgantown, WV Full Screen September 19, 2003 Close Quit

Home Page Title Page Probabilistic Modelling of Molecular Vibrations Contents ◭◭ ◮◮ ⋆ Modelling random vibrations in molecules is important for studying their prop- erties and functions. ◭ ◮ ⋆ Entropy is a measure of freedom of a system to explore its available configuration space. Page 2 of 40 ⋆ Entropy evaluation is important in order to understand the factors involved in Go Back the stability of a conformation and the change from one conformation to another. Full Screen Close Quit

Home Page Entropy in Protein Folding Title Page Contents ⋆ Proteins are biological molecules that are of primary importance to all living organisms. ◭◭ ◮◮ ⋆ Protein are made up of many amino acids (called residues) linked together. ◭ ◮ ⋆ A human body contains over 30,000 different kinds of proteins. ⋆ Protein misfolding is the cause of protein-folding diseases: Alzheimers disease, Page 3 of 40 mad cow disease, cystic fibrosis and some types of cancer. ⋆ It is important to study the stability of a protein and the key is to find a small Go Back molecule (a drug) that can stabilize the normally folded structure. Full Screen Close Quit

Insulin Protein Home Page Title Page Contents ◭◭ ◮◮ ◭ ◮ Page 4 of 40 Go Back Full Screen Close Quit

Home Page Entropy Title Page ⋆ Entropy of a molecular conformation depends on the coordinates of the confor- Contents mation. These are: ◭◭ ◮◮ – Bond lengths – Bond angles ◭ ◮ – Torsional angles (dihedral or rotational degrees of freedom) ⋆ 1. and 2. are rather hard coordinates, entropy is mainly determined by fluctua- Page 5 of 40 tions in torsional angles. Go Back ⋆ Probability modeling of torsional angles of a molecular system is important for entropy evaluation. Full Screen Close Quit

Methanol Molecule Home Page Title Page Contents ◭◭ ◮◮ ◭ ◮ Page 6 of 40 Go Back Full Screen Close Quit

Home Page Probabilistic Modelling of Torsional Angles Title Page Contents ⋆ In molecular biology literature, torsional angles are assumed to have multivariate Gaussian (Normal) distribution (Karplus and Kushik (1981), Macromolecules, ◭◭ ◮◮ Levy et al (1984), Macromolecules) . The entropy is then given by ◭ ◮ S c = mk B + k B 2 ln[(2 π ) m | Σ | ] 2 Page 7 of 40 ⋆ S c is estimated by using the maximum likelihood estimate of the determinant of the variance-covariance matrix of torsional angles using data on torsional angles Go Back of the molecular system Full Screen Close Quit

Home Page Probability Modeling of Torsional Angles Title Page ⋆ There are common situations where assuming a Gaussian distribution for tor- Contents sional angles is not realistic, e.g., ◭◭ ◮◮ – Modeling a torsional angle which has more than one peak. – Modeling a torsional angle where there is more free movement, e.g., in gases. ◭ ◮ ⋆ In Demchuk and Singh( 2001, Molecular Physics ) Page 8 of 40 – We proposed a circular probability modeling approach for modeling torsional angles. Go Back – The torsional angle of the methanol molecule was modeled by using a von Mises distribution (most commonly used distribution on the circle). Full Screen Close Quit

Probability Modeling of Tosional Angles Home Page ⋆ A circular random variable Θ follows l -mode von Mises distribution if its proba- Title Page bility density function is given by: Contents 1 2 πI 0 ( κ ) e κ cos[ l ( θ − θ 0 )] , f ( θ ) = − π ≤ θ < π. ◭◭ ◮◮ κ = concentration parameter, l = number of modes ◭ ◮ I 0 = Modified Bessel function of order 0 θ 0 = Position of first mode For l > 2, the modes are 2 π/l radians apart. Page 9 of 40 ⋆ For l = 1: Go Back – Mean angle is θ 0 . Full Screen – If κ = 0, it is uniform distribution – For large κ , it is approximately Gaussian dist. Close Quit

von Mises Distribution Home Page Title Page Contents ◭◭ ◮◮ ◭ ◮ Page 10 of 40 Go Back Full Screen Close Quit

von Mises Distribution Home Page Title Page Contents ◭◭ ◮◮ ◭ ◮ Page 11 of 40 Go Back Full Screen Close Quit

Home Page Probability Modeling of Torsional Angles Title Page We assumed independent von Mises distributions for torsional angles. Let Θ i have an Contents l i -mode von Mises distribution, i = 1 , 2 , , m with concentration parameter κ i . Then the entropy of the system is given by: ◭◭ ◮◮ I 1 ( k i ) � � S c = k B [ m ln 2 π + ln I 0 ( k i )] − k i I 0 ( k i ) ◭ ◮ where I 1 is the modified Bessel function of order 1. From the Boltzman Gibbs distri- Page 12 of 40 bution, the potential energy of the system is given by m V ( θ 1 , θ 2 , . . . , θ m ) = 1 Go Back � k i [1 − cos( l i ( θ i − θ i 0 ))] , B i =1 Full Screen Close Quit

Home Page Modeling Torsional Angle of Methanol Title Page As a case study, we considered the torsional angle of a methanol molecule. We Contents assumed a 3-mode von Mises distribution for its torsional angle Θ i.e.: ◭◭ ◮◮ 1 2 πI 0 ( κ ) e κ cos[3( θ − θ 0 )] , f ( θ ) = − π ≤ θ < π. ◭ ◮ The potential energy Page 13 of 40 V (Θ) = k B [1 − cos(3( θ − θ 0 ))] = V 0 2 [1 − cos 3( θ − θ 0 )] Go Back where V 0 = maximum potential energy. Full Screen Close Quit

Home Page A Bathtub Shaped Distribution for Potential Energy Title Page For methanol molecule, the potential energy is Contents V = V 0 ◭◭ ◮◮ 2 [1 − cos 3( θ − θ 0 )] Assuming Θ to have a 3-mode von Mises distribution, we derived the following p.d.f. ◭ ◮ for V : 1 πI 0 ( κ ) e κ (1 − 2 v v 0 ) v − 1 2 ( v 0 − v ) − 1 g ( v ) = 2 Page 14 of 40 This is a bathtub shaped probability distribution. For κ = 0 , V/V 0 has beta(1/2, Go Back 1/2) distribution Full Screen Close Quit

A Bath-tub Shaped Distribution Home Page Title Page Contents ◭◭ ◮◮ ◭ ◮ Page 15 of 40 Go Back Full Screen Close Quit

Histograms of Torsional Angle and Energy Home Page Title Page Contents ◭◭ ◮◮ ◭ ◮ Page 16 of 40 Go Back Full Screen Close Quit

Fitting von Mises and Bath-tub Shaped Distributions Home Page Title Page Contents ◭◭ ◮◮ ◭ ◮ Page 17 of 40 Go Back Full Screen Close Quit

A Bivariate Circular Model(Singh et al, 2002, Biometrika) Home Page Title Page ⋆ Let Θ 1 and Θ 2 be the two circular random variables. We introduced a joint probability distribution for Θ 1 and Θ 2 with pdf given by Contents ◭◭ ◮◮ f ( θ 1 , θ 2 ) = Ce κ 1 cos( θ 1 − µ 1 )+ κ 2 cos( θ 2 − µ 2 )+ λ sin( θ 1 − µ 1 ) sin( θ 2 − µ 2 ) , − π = θ 1 , θ 2 < π, ◭ ◮ where κ 1 , κ 2 ≥ 0 , −∞ < λ < ∞ , − π ≤ µ 1 , µ 2 < π and C is normalizing constant Page 18 of 40 ⋆ If fluctuations in Θ 1 and Θ 2 are sufficiently small, then (Θ 1 , Θ 2 ) follows approxi- mately a bivariate normal distribution with Go Back κ 2 κ 1 λ σ 2 κ 1 κ 2 − λ 2 , σ 2 1 = 2 = κ 1 κ 2 − λ 2 , ρ = . √ κ 1 κ 2 Full Screen Close Quit

A Bivariate Circular Model Home Page Title Page ⋆ The normalizing constant C is given by � � λ 2 Contents ∞ � m 1 � 2 m � C = 4 π 2 I m ( κ 1 ) I m ( κ 2 ) m 4 κ 1 κ 2 ◭◭ ◮◮ m =0 where I m is a modified Bessel function of order m . ◭ ◮ ⋆ E [sin(Θ i − µ i )] = 0 , i = 1 , 2 implies that ?i is the circular mean of Θ i . Page 19 of 40 ⋆ Circular variance of Θ 1 is given by Go Back � � λ 2 ∞ � m � 2 m � 1 − E [cos( θ 1 − µ 1 )] = 1 − 4 Cπ 2 I m +1 ( κ 1 ) I m ( κ 2 ) Full Screen m 4 κ 1 κ 2 m =0 Close Quit

A Bivariate Circular Model Home Page Title Page ⋆ The conditional distributions of Θ 1 and Θ 2 are von Mises Contents ⋆ The marginal distribution of Θ 1 is symmetric around θ 1 = µ 1 and unimodal (bimodal) when ◭◭ ◮◮ A ( κ 2 ) = I 1 ( κ 2 ) I 0 ( κ 2 ) ≤ ( ≥ ) κ 1 κ 2 ◭ ◮ λ 2 ⋆ A generalization which allows multiple peaks in marginal distributions Page 20 of 40 Go Back f ( θ 1 , θ 2 ) = Ce κ 1 cos( l 1 ( θ 1 − µ 1 ))+ κ 2 cos( l 2 ( θ 2 − µ 2 ))+ λ sin( l 1 ( θ 1 − µ 1 )) sin( l 2 ( θ 2 − µ 2 )) , − π ≤ θ 1 , θ 2 < π, Full Screen where l 1 , l 2 are positive integers. Close Quit

Home Page Nearest Neighbor Estimates of Entropy(Singh et al., 2002) Title Page ⋆ Let X 1 , X 2 , .., X n be a random sample from a population with pdf f ( x ). Contents ⋆ R i,k = Euclidean distance from X i to its k th closest neighbor. ◭◭ ◮◮ ⋆ Then a reasonable estimate of f ( X i ) is given by R p i,k π p/ 2 ◭ ◮ k Γ( p/ 2 + 1) = k ˆ f ( X i ) n Page 21 of 40 ⋆ The above equation gives Go Back f ( X i ) = k Γ( p/ 2 + 1) ˆ i,k π p/ 2 , i = 1 , 2 , 3 , . . . , n, nR p Full Screen Close Quit