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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

LCVM2

June 16, 2016

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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

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

Theoretical Tools

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

The cgDNA model1: basic idea

We assume the probability density function of the configuration w

• f 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.

1O 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.

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

Configuration vector2

w = (y1, z1, y2, . . . , zn−1, yn) ∈ R6n+6(n−1) = R12n−6

• intra-basepair coordinates ya = (ν, ξ)a ∈ R6:

describe the position of Xa with respect to X a.

2O 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 ().

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

Configuration vector2

w = (y1, z1, y2, . . . , zn−1, yn) ∈ R6n+6(n−1) = R12n−6

• inter-basepair coordinates za = (θ, ζ)a ∈ R6:

describe the position of (Xa+1, X a+1) with respect to (Xa, X a).

2O 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 ().

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

cgDNA model

• Parameter set

P = {K α, K αβ, σα, σαβ : α, β ∈ {A, T, G, C}}, where σ = K w.

• Parameter set + Sequence S ⇒ K(S) and σ(S)

w := K −1σ. Note: different parameter sets have been constructed, the one used in this project is cgDNAparamset2.

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

cgDNA model

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

cgDNA model

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

cgDNA model

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

cgDNA model

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

From Parameter set to Stiffness Matrix. S = X1 · · · Xn

+

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

From Parameter set to Stiffness Matrix. S = X1 · · · Xn

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

Periodic Parameters3

3Jaros

law G

• lowacki. “Computation and Visualization for Multiscale

Modelling of DNA Mechanics”. PhD thesis. Ecole Polytechnique F´ ed´ erale de Lausanne, 2016.

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

Periodic Parameters3

3Jaros

law G

• lowacki. “Computation and Visualization for Multiscale

Modelling of DNA Mechanics”. PhD thesis. Ecole Polytechnique F´ ed´ erale de Lausanne, 2016.

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

Periodic Parameters3

3Jaros

law G

• lowacki. “Computation and Visualization for Multiscale

Modelling of DNA Mechanics”. PhD thesis. Ecole Polytechnique F´ ed´ erale de Lausanne, 2016.

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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 ρi as the probability density function associated with a sequence Si. It has a ground state shape vector wi and a stiffness matrix Ki, i = 1, 2. Then, the Kullback-Leibler divergence, or symmetrised relative entropy is defined as follows: DKL(ρ1, ρ2) = 1 2

• ρ1 ln

ρ1 ρ2

• + ρ2 ln

ρ2 ρ1

• dx

= 1 4[K −1

1

: K2 + K −1

2

: K1 − 2I : I] + 1 4( w1 − w2) · (K1 + K2)( w1 − w2).

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

Kullback-Leibler divergence in 1D

Two different PDFs Corresponding density of the Kullback-Leibler integrand

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

Kullback-Leibler divergence in 1D

Two different PDFs Corresponding density of the Kullback-Leibler integrand

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

Stiffness contribution of the Kullback-Leibler divergence

For K1 = K ⊺

1 ≥ 0 and K2 = K ⊺ 2 ≥ 0, ∃ n real eigenvalues µi > 0

for the generalized eigenvalue problem K1xi = µiK2xi ⇐ ⇒ K −1

1 K2xi = 1

µi xi ⇐ ⇒ K −1

2 K1xi = µixi.

⇒ trace(K −1

1 K2) = 1 µi , and trace(K −1 2 K1) = µi.

And hence Dstiff(ρ1, ρ2) = 1 4( K −1

1

: K2

• trace(K −1

1

K2)

+ K −1

2

: K1

• trace(K −1

2

K1)

−2I : I) = 1 4

• ( 1

µi + µi − 2) = 1 4 √µi − 1 √µi 2

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

Linkage4: Definition through an example

• 4mathworks. Hierarchical Clustering.

http://ch.mathworks.com/help/stats/hierarchical-clustering.html. 2016.

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

Linkage4: Definition through an example

• 4mathworks. Hierarchical Clustering.

http://ch.mathworks.com/help/stats/hierarchical-clustering.html. 2016.

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

Linkage4: Definition through an example

• 4mathworks. Hierarchical Clustering.

http://ch.mathworks.com/help/stats/hierarchical-clustering.html. 2016.

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

From linkage to clustering

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

Dimer analysis

We have

• 16 dimers Xm, m = 1, . . . , 16 (ten independent under

Crick-Watson symmetry).

• For each dimer, a sequence Sk

m = XmXm · · · Xm

• k times

, k = 1, 10, 20, . . .

• For each sequence, we compute the stiffness matrix K k

m and

the ground state shape vector wk

m using cgDNA,

• And construct the ”Divergence matrix” D :

Dij = DKL(Sk

i , Sk j ).

• Recall

DKL(S1, S2) = 1 4[K −1

1

: K2 + K −1

2

: K1 − 2I : I] + 1 4( w1 − w2) · (K1 + K2)( w1 − w2).

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

Divergence matrix, with an arbitrary ordering of the dimers

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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

• rder. This gave the 4 × 4

corner blocks.

• The 8 × 8 inner block was still

complicated, so I repeated the first step to order it.

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

Divergence matrix of the manually ordered trimers in YR-alphabet

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

Clustering

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

Kullback-Leibler Divergence: Dimers, in ATGC-alphabet

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

Kullback-Leibler Divergence: Dimers, in YR-alphabet

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

Kullback-Leibler Divergence: Trimers, in ATGC-alphabet

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

Kullback-Leibler Divergence: Trimers, in YR-alphabet

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

Kullback-Leibler Divergence: Tetramers, in YR-alphabet

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

Dimers

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

KL Divergence with Equivalence Classes

• Physically, ρ(S) ∼

= ρ(S), under change of reading strand.

• ρ(poly − XY ) should be close to ρ(poly − YX) Therefore, we

define the shift operator Tk : Tk(X1 · · · Xn) = Xn−k+1 · · · XnX1 · · · Xn−k. Example: T1(ATGC) = CATG. For convenience, we set T0(S) := S.

• We then introduce the following equivalence class of

sequences: S1 ∼ S2 ⇐ ⇒ ∃k ≥ 0 s.t S1 = Tk(S2) or S1 = Tk(S2). Then define the Kullback-Leibler divergence with equivalence classes as Dequiv

KL

(S1, S2) = min{DKL([S1], [S2])}, where [S] is the equivalence class of S.

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

Kullback-Leibler Divergence without equivalence classes: dimers, in YR-alphabet

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

KL Divergence with Equivalence Classes: Dimer

If S = X1X2, then

• If

Xi ∈ {A, T, C, G}, there are 6 [S].

• If Xi ∈ {Y , R},

there are 2 [S].

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

Kullback-Leibler Divergence without equivalence classes: trimers, in YR-alphabet

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

KL Divergence with Equivalence Classes: Trimer

If S = X1X2X3, then

• If

Xi ∈ {A, T, C, G}, there are 12 [S].

• If Xi ∈ {Y , R},

there are 2 [S].

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

Kullback-Leibler Divergence without equivalence classes: tetramers, in YR-alphabet

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

KL Divergence with Equivalence Classes: Tetramer

If S = X1X2X3X4, then

• If

Xi ∈ {A, T, C, G}, there are 39 [S].

• If Xi ∈ {Y , R},

there are 4 [S].

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

Codons

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

Summary of the results

• The Kullback-Leibler divergence between equilibrium

distributions discerns purines and pyrimidines in dimers, trimers and tetramers.

• The clusterings obtained when adding the equivalence classes
• f reading strand and translations also distinguishes the

purines and pyrimidines, and they are more consistent with

• ur physical knowledge.
• The redundancy of sequences in the codons is not

immediately explained by the Kullback-Leibler divergence between equilibrium distributions.

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

Spectral Analysis of the Periodic Stiffness Matrix

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

Spectra of sequence-dependent stiffness matrices computed from Molecular Dynamic5 and cgDNA

MD: cgDNA:

5Julien Delafontaine. “Spectral analysis of a coarse-grain model of DNA”. .

MA thesis. Ecole Polytechnique F´ ed´ erale de Lausanne, 2011.

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

Reminder: Periodic Parameters6

6Jaros

law G

• lowacki. “Computation and Visualization for Multiscale

Modelling of DNA Mechanics”. PhD thesis. Ecole Polytechnique F´ ed´ erale de Lausanne, 2016.

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

Decomposition of the Periodic Stiffness Matrix Kp(XY )

+

Σ(XY )

+

Q(XY ) Q⊺(XY )

=

Kp(XY )

∈ R24×24

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

Decomposition of the Periodic Stiffness Matrix Kp(S2(XY ))

Kp(S2(XY )) =

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

Decomposition of the Periodic Stiffness Matrix Kp(S2(XY ))

= Σ(XY ) Q(XY )+ Q(XY )⊺ Q(XY )+ Σ(XY ) Q(XY )⊺ Q(XY ) =

Σ = Σ⊺ ∈ R24×24; Q ∈ R24×24.

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

Eigenvalues of Kp(AGAG)

σ(Kp(AGAG)) = {λi, i = 1, . . . , 48}

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

Eigenvalues of Kp(AGAG)

σ(Kp(AGAG)) = σ(Kp(AG)) ∪ σ(Σ − (Q + Q⊺)) Denote by

• ui 24 linearly independent

eigenvectors of Kp(AG),

• vi 24 linearly independent

eigenvectors of Σ − (Q + Q⊺). Then (ui, ui) and (vi, −vi) are 48 linearly independent eigenvectors

• f Kp(AGAG).

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

Eigenvalues of Kp(MnMn)

This result can be generalized to polymers that are composed of an n-mer repeated twice. Σ(Mn), Q(Mn) ∈ R12n×12n = Σ(Mn) Q(Mn)+ Q(Mn)⊺ Q(Mn)+ Σ(Mn) Q(Mn)⊺ ⇒ σ(Kp(MnMn)) = σ(Σ + (Q + Q⊺)) ∪ σ(Σ − (Q + Q⊺))

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

Eigenvectors of Kp(S4(XY ))

Let

• ui ∈ R24 the eigenvectors of Kp(XY ) = Σ + (Q + Q⊺),
• vi ∈ R24 the eigenvectors of Σ − (Q + Q⊺),
• wi ∈ R48 the eigenvectors of
• Σ

Q − Q⊺ Q⊺ − Q Σ

• .

Then linear independent eigenvectors of Kp(S4(XY )) are of the form     ui ui ui ui     ,     vi −vi vi −vi     ,

• wi

−wi

• .

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

Spectrum of Kp(S4(AG))

σ(S4(AG)) = {µ+

i , i=1,...,24} ∪ {µ− i , i=1,...,24} ∪ {λi, i=1,...,48}

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

The 48 remaining eigenvalues of Kp(S4(XY )) are the eigenvalues of this matrix ˜ M

= Σ Q − Q⊺ Q⊺ − Q Σ

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

The 48 remaining eigenvalues of Kp(S4(XY )) are the eigenvalues of this matrix ˜ M

The eigenspace associated with any eigenvalue of this matrix has the form x y

• ,

−y x

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

Summary of the results

For k = 1, the eigenvalues of the periodic stiffness matrix Kp(S2k(Mn)) = Kp(MnMn) ∈ R2(12n)×2(12n) are the union of the eigenvalues of

• Σ(Mn) + Q(Mn) + Q⊺(Mn) = Kp(Mn) ∈ R12n×12n, and
• Σ(Mn) − (Q(Mn) + Q⊺(Mn)) ∈ R12n×12n.

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

Summary of the results

For k > 1, the eigenvalues and eigenvectors of the periodic stiffness matrix Kp(S2k(Mn)) ∈ RN×N are the union of the eigenvalues of

Kp(S2k−1(Mn)) ∈ RN/2×N/2

Σ(S2k−2

(Mn))

Q(S2k−2

(Mn))+

Q⊺(S2k−2

(Mn))

Q⊺(S2k−2

(Mn))

Σ(S2k−2

(Mn)) +Q(S2k−2 (Mn))

And ˜ M ∈ RN/2×N/2

Σ(S2k−2

(Mn))

Q(S2k−2

(Mn))−

Q⊺(S2k−2

(Mn))

Q⊺(S2k−2

(Mn))

Σ(S2k−2

(Mn)) −Q(S2k−2 (Mn))

Moreover, the multiplicity of any eigenvalue of ˜ M is even.

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

Thank you for your attention!

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