Determination of duplex twist and backbone writhe

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 asti(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)Twi=12πli0lidsi ti(si){ni(si)×dni(si)dsi}(5)

For stiff origamis, whose centerline curve ri=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=12πlc0lcdsu·{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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