Free Energy of a Folded Polymer under Cylindrical Confinement

Monte Carlo computer simulations are used to study the conformational free energy of a folded polymer confined to a long cylindrical tube. The polymer is modeled as a hard-sphere chain. Its conformational free energy F is measured as a function of λ, the end-to-end distance of the polymer. In the case of a flexible linear polymer, F(λ) is a linear function in the folded regime with a gradient that scales as f ≡ |dF/dλ| ∼ N 0 D –1.20±0.01 for a tube of diameter D and a polymer of length N. This is close to the prediction f ∼ N 0 D –1 obtained from simple scaling arguments. The discrepancy is due in part to finite-size effects associated with the de Gennes blob model. A similar discrepancy was observed for the folding of a single arm of a three-arm star polymer. We also examine backfolding of a semiflexible polymer of persistence length P in the classic Odijk regime. In the overlap regime, the derivative scales f ∼ N 0 D –1.72±0.02 P –0.35±0.01, which is close to the prediction f ∼ N 0 D –5/3 P –1/3 obtained from a scaling argument that treats interactions between deflection segments at the second virial level. In addition, the measured free energy cost of forming a hairpin turn is quantitatively consistent with a recent theoretical calculation. Finally, we examine the scaling of F(λ) for a confined semiflexible chain in the presence of an S-loop composed of two hairpins. While the predicted scaling of the free energy gradient is the same as that for a single hairpin, we observe a scaling of f ∼ D –1.91±0.03 P –0.36±0.01. Thus, the quantitative discrepancy between this measurement and the predicted scaling is somewhat greater for S-loops than for single hairpins.