Conformational analysis

This protocol is extracted from research article:

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

**
Sci Adv**,
Jul 29, 2020;
DOI:
10.1126/sciadv.aaw8331

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

Procedure

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$${\mathbf{r}}_{\perp}(s)=\mathbf{r}(s)-{r}_{u}(s)\mathbf{u}$$(1)with **r**(*s*) the position of the discretized backbone segment with curvilinear abscissa *s* and *r _{u}*(

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 ${\widehat{c}}_{\mathit{vw}}$$${\widehat{c}}_{\mathit{vw}}(k)={\widehat{r}}_{\perp v}(k)\times {\widehat{r}}_{\perp w}^{*}(k)$$(3)where ${\widehat{r}}_{\perp x}={\widehat{\mathbf{r}}}_{\perp}\xb7\mathbf{x}$ for **x** ∈ {**v**, **w**} and ${\widehat{r}}_{\perp w}^{*}$ is the complex conjugate of ${\widehat{r}}_{\perp w}$. 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$$\mathrm{H}(k)=\frac{2\times \mathfrak{I}\{{\stackrel{\u02c6}{c}}_{\mathrm{vw}}(k)\}}{{\stackrel{\u02c6}{c}}_{\mathrm{vv}}(k)+{\stackrel{\u02c6}{c}}_{\mathrm{ww}}(k)}$$(4)with $\mathfrak{I}\{{\widehat{c}}_{\mathit{vw}}\}$ the imaginary part of ${\widehat{c}}_{\mathit{vw}}$. 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, ${\widehat{\mathbf{r}}}_{\perp}(k)$ describes an ideal circular helical deformation mode with pitch 1/*k* and handedness determined by the sign of H.

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