Universality in conformations and transverse fluctuations of a semi-flexible polymer in a crowded environment

We study the universal aspects of polymer conformations and transverse fluctuations for a single swollen chain characterized by a contour length L and a persistence length ℓp in two dimensions (2D) and three dimensions (3D) in the bulk, as well as in the presence of excluded volume (EV) particles of different sizes occupying different area/volume fractions. In the absence of the EV particles, we extend the previously established universal scaling relations in 2D [Huang et al., J. Chem. 140, 214902 (2014)] to include 3D and demonstrate that the scaled end-to-end distance ⟨RN2⟩/(2Lℓp) and the scaled transverse fluctuation ⟨l⊥2⟩/L as a function of L/ℓp collapse onto the same master curve, where ⟨RN2⟩ and ⟨l⊥2⟩ are the mean-square end-to-end distance and transverse fluctuations. However, unlike in 2D, where the Gaussian regime is absent due to the extreme dominance of the EV interaction, we find that the Gaussian regime is present, albeit very narrow in 3D. The scaled transverse fluctuation in the limit L/ℓp ≪ 1 is independent of the physical dimension and scales as ⟨l⊥2⟩/L∼(L/ℓp)ζ-1, where ζ = 1.5 is the roughening exponent. For L/ℓp ≫ 1, the scaled fluctuation scales as ⟨l⊥2⟩/L∼(L/ℓp)ν-1, where ν is the Flory exponent for the corresponding spatial dimension (ν2D = 0.75 and ν3D = 0.58). When EV particles of different sizes for different area or volume fractions are added into 2D and 3D systems, our results indicate that the crowding density either does not or does only weakly affect the universal scaling relations. We discuss the implications of these results in living matter by showing the experimental result for a dsDNA on the master plot.