End-to-end distance, persistence length, and entropic elasticity
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L/Lp (ratio)
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⟨R²⟩ theory
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⟨R²⟩ simulated
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Regime
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Worm-like chain (WLC): a continuum polymer model where the chain resists bending with stiffness EI, characterized by persistence length Lp = EI/kBT. The tangent-tangent correlation decays as ⟨t(s)·t(0)⟩ = exp(−s/Lp). Mean-squared end-to-end distance: ⟨R²⟩ = 2Lp·L·[1 − (Lp/L)(1 − e^(−L/Lp))]. When L ≪ Lp (stiff rod): ⟨R²⟩ ≈ L². When L ≫ Lp (flexible coil): ⟨R²⟩ ≈ 2Lp·L (Gaussian). DNA has Lp ≈ 50 nm; actin ≈ 15 μm; microtubules ≈ 1 mm. The right panel shows ⟨R²⟩/L² as a function of Lp, with the orange dot marking simulated result. The WLC force-extension curve (Marko-Siggia 1995) fits single-molecule DNA stretching experiments with remarkable precision.