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Conformational analysis

Chiral shape fluctuations and the origin of chirality in cholesteric phases of DNA origamis

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The discretized origami backbones are obtained by averaging the center-of-mass locations of their bonded nucleotides over the six constituent duplexes within each transverse plane along the origami contour (22). We define the molecular frame R = [u v w] of each conformation as the principal frame of its backbone gyration tensor, such that u and v correspond to the respective direction of maximum and minimum dispersion of the origami backbone (33). Shape fluctuations are described by the contour variations of the transverse position vector$r⊥(s)=r(s)−ru(s)u$(1)with r(s) the position of the discretized backbone segment with curvilinear abscissa s and ru(s) ≡ r(s) · u, assuming the backbone center of mass to be set to the origin of the frame. Denoting by Δs the curvilinear length of each segment, the Fourier components of r read as(2)

Using the convolution theorem, the spectral coherence between the two transverse components of an arbitrary backbone deformation mode may be quantified by their Fourier-transformed cross-correlation function $ĉvw$$ĉvw(k)=r̂⊥v(k)×r̂⊥w*(k)$(3)where $r̂⊥x=r̂⊥·x$ for x ∈ {v, w} and $r̂⊥w*$ is the complex conjugate of $r̂⊥w$. It is shown in section S4 that a helicity order parameter H(k) for a deformation mode with arbitrary wave number k about the filament long axis u may be derived in the form$H(k)=2×I{cˆvw(k)}cˆvv(k)+cˆww(k)$(4)with $I{ĉvw}$ the imaginary part of $ĉvw$. One may check that −1 ≤ H(k) ≤ 1, with H(k) = ± 1 if and only if the two transverse Fourier components bear equal amplitudes and lie in perfect phase quadrature. In this case, $r̂⊥(k)$ describes an ideal circular helical deformation mode with pitch 1/k and handedness determined by the sign of H.

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