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Determination of duplex twist and backbone writhe
This protocol is extracted from research article:
Chiral shape fluctuations and the origin of chirality in cholesteric phases of DNA origamis

Procedure

Let ri,1(si) and ri,2(si) be a set of continuous curves interpolating the positions of the nucleotide centers of mass of the ith constituent duplex of an arbitrary origami conformation. The unit tangent and normal vectors ti and ni at a given curvilinear abscissa si respectively read as$ti(si)≡dri(si)dsini(si)≡ri,1(si)−ri,2(si)∥ri,1(si)−ri,2(si)∥$where ri ≡ (ri,1 + ri,2)/2 is the continuous duplex centerline with contour length li. The average twist density Tw of each duplex may then be obtained from the sum of the local stacking angles between consecutive base pairs (40)(5)

For stiff origamis, whose centerline curve $r≡∑i=16ri/6$ does not display any turning points in u (dru/ds > 0), the so-called polar writhe Wr of the filament backbone simply reduces to the local contribution (41)$Wr=12πlc∫0lcdsu·{t(s)×dt(s)ds}1+u·t(s)$(6)where tdr/ds is the unit backbone tangent vector. It may then be shown that Wr > 0 (resp. Wr < 0) if t winds about u in a right-handed (resp. left-handed) fashion (41). Equations 5 and 6 are evaluated numerically through standard quadrature methods, using cubic spline interpolations for all discrete curves (40). A Savitzky-Golay filter of order 9 (42) was preliminarily applied to the backbone curve to weed out irrelevant short-wavelength contour fluctuations arising from our geometric definition of the origami centerline (40). The last 𝒪(10) base pair planes at each of the filament extremities were excluded from the calculations to limit the influence of end effects.

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