Ideal, best packing, and energy minimizing double helices

We study optimal double helices with straight axes (or the fattest tubes around them) computationally using three kinds of functionals; ideal ones using ropelength, best volume packing ones, and energy minimizers using two one-parameter families of interaction energies between two strands of types $r^{-\alpha}$ and $\frac1r\exp(-kr)$. We compare the numerical results with experimental data of DNA.

💡 Research Summary

The paper investigates the geometry of infinitely long double helices with straight axes, focusing on three distinct optimality criteria: (1) ideality measured by average ropelength, (2) maximal volume packing, and (3) minimization of two families of pairwise interaction energies. The helices are defined by two strands Γ₁(a) and Γ₂(a) of unit radius, slope a>0, and pitch P=2πa. The thickness ρ(a) of the double helix is the minimum of the curvature radius √(1+a²) and half the doubly‑critical self‑distance; the latter equals 2 for a≥1 and is obtained from the transcendental equation sinθ₀ = aθ₀ for a<1.

For the ideal case, the average ropelength ARL(a)=4π√(1+a²)/ρ(a) is minimized. Numerical computation shows the minimum at a≈0.822074, giving a pitch‑to‑thickness ratio P/ρ≈5.93. This agrees with earlier work on ideal knots and confirms that when P/ρ exceeds 2π the two tubes touch along the axis, while for smaller ratios the contact curve is helical.

The second functional, the packing proportion PR(a)=2√(1+a²)·ρ(a)²·a/(1+ρ(a))², measures the volume of the fattest tubular neighbourhood relative to the volume of the circumscribing cylinder. Its maximum occurs at a≈0.635805, yielding P/ρ≈4.80, which represents the most efficient cylindrical packing of the two tubes.

The third line of inquiry introduces two one‑parameter families of interaction potentials. The first is a modified Coulomb potential V(r)=r^{‑α} with α>1. The mutual energy per twist is
AE(α)(a)=2π(1+a²)∫_{‑∞}^{∞}