Junctional angle of a bihanded helix

Helical filaments having sections of reversed chirality are common phenomena in the biological realm. The apparent angle between the two sections of opposite handedness provides information about the geometry and elasticity of the junctional region. In this paper, the governing differential equations for the local helical axis are developed, and asymptotic solutions of the governing equations are solved by perturbation theory. The asymptotic solutions are compared with the corresponding numerical solutions, and the relative error at second order is found to be less than 1.5% over a range of biologically relevant curvature and torsion values from 0 to 1/2 in dimensionless units.

Figure 1

Schematic of two concatenated helices of opposite handedness, a left-handed helix (L) on the left, and a right-handed helix (R) on the right. (a) The tangent $\widehat{\mathbf{t}}$, normal $\widehat{\mathbf{n}}$, and binormal $\widehat{\mathbf{b}}$ at the junction can be made identical; (b) connecting the two helices smoothly. Local helical axis $\widehat{\mathbf{z}}$ and “block angle” $\alpha$ are indicated.

Figure 2

The variation of torsion $\tau$ along the backbone of the helical filament. (a) Torsion curves at different front widths $\xi$ (0.1 for dashed red curve, 0.4 for green solid curve, 0.8 for black solid curve) if there is no shift in torsion. (b) In the asymmetric case $\left(\Delta \ne 0\right)$, the torsion on the two sides of the origin is no longer antisymmetric.

Figure 3

The geometrical explanation of the angle ${\alpha }_0$ (adapted from Ref. [15]). $R_{\pm }$ are the radii of the helices, and $P_{\pm }$ are the pitches of the helices.

Figure 4

The variation of block angle before and after shifting as a function of shift ratio $d$ and torsion-curvature ratio $\lambda$. The isolines are the variation ${\tilde{\alpha }}_0-{\alpha }_0$ in degrees.

Figure 5

The evolution of the helical axis $\widehat{\mathbf{z}}$ on a sphere at different front widths.

Figure 6

Block-angle correction as a function of K and T, including the quadratic term in the Taylor series for $cos\alpha$. (a) Contour plot of block-angle correction labeled in degrees. (b) Relative errors between numerical and perturbation results.

Figure 7

Block-angle correction as a function of K and T (without quadratic analysis). (a) Contour plot of block-angle correction labeled in degrees. (b) Relative errors between numerical and perturbation results.

Figure 8

Truncated form of block-angle correction [see Eq. (28)] as a function of K and T. (a) Contour plot of block-angle correction labeled in degrees. (b) Relative errors between the numerical and perturbation results.

Figure 9

The $\Delta \alpha$ isolines represented in degrees as a function of dimensionless curvature and torsion for different percentages (a) 1%, (b) 30%, (c) 60%, (d) 90% of torsion shift $\Delta /\Omega$.