DNA Mechanics: Worm-Like Chain

Semiflexible Polymers

DNA is neither a rigid rod nor a freely flexible string — it is a semiflexible polymer. Its mechanical properties are captured by the worm-like chain (WLC) model, which characterizes flexibility through the persistence length L_p.

Persistence Length

The persistence length is the distance over which thermal fluctuations bend the polymer significantly:

• dsDNA: L_p ≈ 50 nm (~150 bp)
• ssDNA: L_p ≈ 1–3 nm
• Actin: L_p ≈ 17 μm
• Microtubule: L_p ≈ 5 mm

DNA Contour Length

Each base pair contributes 0.34 nm to the contour length:

$L = N_{bp} \times 0.34 \text{ nm/bp}$

A 1 kbp DNA fragment has L = 340 nm, but fits in a cell because it is highly coiled.

End-to-End Distance (Gaussian Regime)

For polymers much longer than L_p (L >> L_p), the mean squared end-to-end distance is:

$\langle r^2 \rangle = 2 L_p L$

$r_{\text{rms}} = \sqrt{2 L_p L}$

For 1 kbp DNA: r_rms = √(2 × 50 × 340) ≈ 184 nm (much smaller than L = 340 nm).

Marko-Siggia Force-Extension

When DNA is stretched by an external force F, the force-extension relationship is (Marko & Siggia, 1995):

$F = \frac{k_B T}{L_p} \left[ \frac{1}{4\left(1 - x/L\right)^2} - \frac{1}{4} + \frac{x}{L} \right]$

• x = end-to-end extension (0 ≤ x < L)
• L = contour length
• L_p = persistence length

This formula diverges as x → L (infinite force required for full extension) and reduces to Hookean behavior for small extensions.

Your Task

Implement three functions:

1. dna_contour_length_nm(N_bp) — contour length from number of base pairs
2. end_to_end_rms_nm(L_nm, L_p_nm=50) — RMS end-to-end distance (Gaussian regime)
3. wlc_force_pN(x_nm, L_nm, L_p_nm=50, T_K=310) — stretching force in picoNewtons

For wlc_force_pN: convert lengths to meters, compute force in Newtons, then return in pN (1 pN = 10⁻¹² N). Use k_B = 1.381 × 10⁻²³ J/K.