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 vectorr(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 asr̂(k)=sΔs r(s)×e2iπks(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 ĉvwĉvw(k)=r̂v(k)×r̂w*(k)(3)where r̂x=r̂·x for x ∈ {v, w} and r̂w* is the complex conjugate of r̂w. 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 formH(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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