Sequence-dependent equilibrium distributions of DNA within the cgDNA - PowerPoint PPT Presentation

Theoretical Tools Dimer analysis Clustering Spectral Analysis of the Periodic Stiffness Matrix Sequence-dependent equilibrium distributions of DNA within the cgDNA coarse grain model M´ elissa Nicolier Supervised by J.H. Maddocks and T. Lessinnes LCVM 2 June 16, 2016 1/49

Theoretical Tools Dimer analysis Clustering Spectral Analysis of the Periodic Stiffness Matrix Content overview Theoretical Tools Dimer analysis Clustering Spectral Analysis of the Periodic Stiffness Matrix 2/49

Theoretical Tools Dimer analysis Clustering Spectral Analysis of the Periodic Stiffness Matrix Theoretical Tools 3/49

Theoretical Tools Dimer analysis Clustering Spectral Analysis of the Periodic Stiffness Matrix The cgDNA model 1 : basic idea We assume the probability density function of the configuration w of a given sequence S to be ρ ( w ; S ) = 1 Z exp {− U ( w ; S ) } , where • U ( w ; S ) = 1 2 ( w − � w ( S )) · K ( S )( w − � w ( S )) is the quadratic function approximating the free energy of the sequence in the configuration w , • � w ( S ) is the ground state shape vector, • K ( S ) is the stiffness matrix. 1 O Gonzalez, D Petkeviˇ ci¯ ut˙ e, and J.H Maddocks. “A sequence-dependent rigid-base model of DNA”. . In: The Journal of chemical physics 138.5 (2013), p. 055102. 4/49

Theoretical Tools Dimer analysis Clustering Spectral Analysis of the Periodic Stiffness Matrix Configuration vector 2 w = ( y 1 , z 1 , y 2 , . . . , z n − 1 , y n ) ∈ R 6 n +6( n − 1) = R 12 n − 6 • intra-basepair coordinates y a = ( ν, ξ ) a ∈ R 6 : describe the position of X a with respect to X a . 2 O Gonzalez, D Petkeviˇ ci¯ ut˙ e, and J.H Maddocks. “Supplementary material to A sequence-dependent rigid-base model of DNA”. In: The Journal of chemical physics (). 5/49

Theoretical Tools Dimer analysis Clustering Spectral Analysis of the Periodic Stiffness Matrix Configuration vector 2 w = ( y 1 , z 1 , y 2 , . . . , z n − 1 , y n ) ∈ R 6 n +6( n − 1) = R 12 n − 6 • inter-basepair coordinates z a = ( θ, ζ ) a ∈ R 6 : describe the position of ( X a +1 , X a +1 ) with respect to ( X a , X a ). 2 O Gonzalez, D Petkeviˇ ci¯ ut˙ e, and J.H Maddocks. “Supplementary material to A sequence-dependent rigid-base model of DNA”. In: The Journal of chemical physics (). 5/49

Theoretical Tools Dimer analysis Clustering Spectral Analysis of the Periodic Stiffness Matrix cgDNA model • Parameter set P = { K α , K αβ , σ α , σ αβ : α, β ∈ { A , T , G , C }} , where σ = K � w . • Parameter set + Sequence S ⇒ K ( S ) and σ ( S ) w := K − 1 σ . • � Note: different parameter sets have been constructed, the one used in this project is cgDNAparamset2 . 6/49

Theoretical Tools Dimer analysis Clustering Spectral Analysis of the Periodic Stiffness Matrix cgDNA model 7/49

Theoretical Tools Dimer analysis Clustering Spectral Analysis of the Periodic Stiffness Matrix cgDNA model 7/49

Theoretical Tools Dimer analysis Clustering Spectral Analysis of the Periodic Stiffness Matrix cgDNA model 7/49

Theoretical Tools Dimer analysis Clustering Spectral Analysis of the Periodic Stiffness Matrix cgDNA model 7/49

Theoretical Tools Dimer analysis Clustering Spectral Analysis of the Periodic Stiffness Matrix From Parameter set to Stiffness Matrix. S = X 1 · · · X n + 8/49

Theoretical Tools Dimer analysis Clustering Spectral Analysis of the Periodic Stiffness Matrix From Parameter set to Stiffness Matrix. S = X 1 · · · X n 8/49

Theoretical Tools Dimer analysis Clustering Spectral Analysis of the Periodic Stiffness Matrix Periodic Parameters 3 3 Jaros� law G� lowacki. “Computation and Visualization for Multiscale Modelling of DNA Mechanics”. PhD thesis. Ecole Polytechnique F´ ed´ erale de Lausanne, 2016. 9/49

Theoretical Tools Dimer analysis Clustering Spectral Analysis of the Periodic Stiffness Matrix Periodic Parameters 3 3 Jaros� law G� lowacki. “Computation and Visualization for Multiscale Modelling of DNA Mechanics”. PhD thesis. Ecole Polytechnique F´ ed´ erale de Lausanne, 2016. 9/49

Theoretical Tools Dimer analysis Clustering Spectral Analysis of the Periodic Stiffness Matrix Periodic Parameters 3 3 Jaros� law G� lowacki. “Computation and Visualization for Multiscale Modelling of DNA Mechanics”. PhD thesis. Ecole Polytechnique F´ ed´ erale de Lausanne, 2016. 9/49

Theoretical Tools Dimer analysis Clustering Spectral Analysis of the Periodic Stiffness Matrix Comparing probability density functions: Kullback-Leibler Divergence Let E be a space of probability density functions ρ , with ρ ( x ) ≥ 0 � and Ω ρ ( x ) dx = 1. A divergence is a function D : E × E → R such that D ( x , y ) ≥ 0 and D ( x , y ) = 0 ⇐ ⇒ x = y . Define ρ i as the probability density function associated with a sequence S i . It has a ground state shape vector � w i and a stiffness matrix K i , i = 1 , 2. Then, the Kullback-Leibler divergence, or symmetrised relative entropy is defined as follows: � � ρ 1 � � ρ 2 �� � 1 D KL ( ρ 1 , ρ 2 ) = ρ 1 ln + ρ 2 ln dx 2 ρ 2 ρ 1 = 1 4[ K − 1 : K 2 + K − 1 : K 1 − 2 I : I ] 1 2 + 1 4( � w 1 − � w 2 ) · ( K 1 + K 2 )( � w 1 − � w 2 ) . 10/49

Theoretical Tools Dimer analysis Clustering Spectral Analysis of the Periodic Stiffness Matrix Kullback-Leibler divergence in 1D Corresponding density of the Two different PDFs Kullback-Leibler integrand 11/49

Theoretical Tools Dimer analysis Clustering Spectral Analysis of the Periodic Stiffness Matrix Kullback-Leibler divergence in 1D Corresponding density of the Two different PDFs Kullback-Leibler integrand 11/49

Theoretical Tools Dimer analysis Clustering Spectral Analysis of the Periodic Stiffness Matrix Stiffness contribution of the Kullback-Leibler divergence For K 1 = K ⊺ 1 ≥ 0 and K 2 = K ⊺ 2 ≥ 0, ∃ n real eigenvalues µ i > 0 for the generalized eigenvalue problem 1 K 2 x i = 1 K − 1 K − 1 K 1 x i = µ i K 2 x i ⇐ ⇒ ⇐ ⇒ 2 K 1 x i = µ i x i . x i µ i 1 K 2 ) = � 1 2 K 1 ) = � µ i . ⇒ trace( K − 1 µ i , and trace( K − 1 And hence 1 4( K − 1 + K − 1 D stiff ( ρ 1 , ρ 2 ) = : K 2 : K 1 − 2 I : I ) 1 2 � �� � � �� � trace( K − 1 trace( K − 1 K 2 ) K 1 ) 1 2 � � √ µ i − � 2 � 1 ( 1 + µ i − 2) = 1 1 = √ µ i 4 µ i 4 12/49

Theoretical Tools Dimer analysis Clustering Spectral Analysis of the Periodic Stiffness Matrix Linkage 4 : Definition through an example 4 mathworks. Hierarchical Clustering . http://ch.mathworks.com/help/stats/hierarchical-clustering.html . 2016. 13/49

Theoretical Tools Dimer analysis Clustering Spectral Analysis of the Periodic Stiffness Matrix Linkage 4 : Definition through an example 4 mathworks. Hierarchical Clustering . http://ch.mathworks.com/help/stats/hierarchical-clustering.html . 2016. 13/49

Theoretical Tools Dimer analysis Clustering Spectral Analysis of the Periodic Stiffness Matrix Linkage 4 : Definition through an example 4 mathworks. Hierarchical Clustering . http://ch.mathworks.com/help/stats/hierarchical-clustering.html . 2016. 13/49

Theoretical Tools Dimer analysis Clustering Spectral Analysis of the Periodic Stiffness Matrix From linkage to clustering 14/49

Theoretical Tools Dimer analysis Clustering Spectral Analysis of the Periodic Stiffness Matrix Dimer analysis We have • 16 dimers X m , m = 1 , . . . , 16 (ten independent under Crick-Watson symmetry). • For each dimer, a sequence S k m = X m X m · · · X m , � �� � k times k = 1 , 10 , 20 , . . . • For each sequence, we compute the stiffness matrix K k m and w k the ground state shape vector � m using cgDNA , • And construct the ”Divergence matrix” D : D ij = D KL ( S k i , S k j ) . • Recall D KL ( S 1 , S 2 ) = 1 4[ K − 1 : K 2 + K − 1 : K 1 − 2 I : I ] 1 2 + 1 4( � w 1 − � w 2 ) · ( K 1 + K 2 )( � w 1 − � w 2 ) . 15/49

Theoretical Tools Dimer analysis Clustering Spectral Analysis of the Periodic Stiffness Matrix Divergence matrix, with an arbitrary ordering of the dimers 16/49

Theoretical Tools Dimer analysis Clustering Spectral Analysis of the Periodic Stiffness Matrix Divergence matrix manually ordered The ordering has been obtained in two steps • I first reordered the dimers to sort the first line in ascending order. This gave the 4 × 4 corner blocks. • The 8 × 8 inner block was still complicated, so I repeated the first step to order it. 17/49

Theoretical Tools Dimer analysis Clustering Spectral Analysis of the Periodic Stiffness Matrix Divergence matrix of the manually ordered trimers in YR-alphabet 18/49

Theoretical Tools Dimer analysis Clustering Spectral Analysis of the Periodic Stiffness Matrix Clustering 19/49

Theoretical Tools Dimer analysis Clustering Spectral Analysis of the Periodic Stiffness Matrix Kullback-Leibler Divergence: Dimers, in ATGC-alphabet 20/49

Theoretical Tools Dimer analysis Clustering Spectral Analysis of the Periodic Stiffness Matrix Kullback-Leibler Divergence: Dimers, in YR-alphabet 21/49

Theoretical Tools Dimer analysis Clustering Spectral Analysis of the Periodic Stiffness Matrix Kullback-Leibler Divergence: Trimers, in ATGC-alphabet 22/49