Subdiffusive motion of a polymer composed of subdiffusive monomers

Abstract

We use Brownian dynamics simulations and analytical theory to investigate the physical principles underlying subdiffusive motion of a polymer. Specifically, we examine the consequences of confinement, self-interaction, viscoelasticity, and random waiting on monomer motion, as these physical phenomena may be relevant to the behavior of biological macromolecules in vivo. We find that neither confinement nor self-interaction alter the fundamental Rouse mode relaxations of a polymer. However, viscoelasticity, modeled by fractional Langevin motion, and random waiting, modeled with a continuous time random walk, lead to significant and distinct deviations from the classic polymer-dynamics model. Our results provide diagnostic tools--the monomer mean square displacement scaling and the velocity autocorrelation function--that can be applied to experimental data to determine the underlying mechanism for subdiffusive motion of a polymer.

FIG. 1

Ensemble-averaged MSD of a single circular polymer under spherical confinement. Series of simulations for decreasing radius of confinement: r → ∞ (black), r = 6 (pink), r = 5 (orange), r = 4 (green), r = 3 (red), and r = 1 (blue). The inset shows a typical snapshot from the r = 3 simulations.

FIG. 2

Mean-square displacement of the midpoint monomer 〈(R⃗_mid(t) − R⃗_mid(0))²〉 / (b²N) versus the dimensionless time τ = t/[N²b²ξ/(k_BT)]^1/α for α = 1.0 (red), α = 0.7 (purple), and α = 0.4 (blue). The dotted curves correspond the the long-time asymptotic behavior for these three α values, and the dashed curves give the short-time scaling of τ^α/2.

FIG. 3

The midpoint monomer MSD versus time t from an ensemble average over 1000 numerical simulations with 1 bead, 2 beads, 3 beads, 4 beads, and 100 beads (see text for parameter values). For the curves from 1-bead, 2-beads, and 3-beads simulations, we include the free-diffusion scaling MSD_mid ~ t^αFD. The short-time scaling of MSD_mid ~ t¹ is common to all simulations.

FIG. 4

The midpoint-monomer velocity autocorrelation function $C_{\upsilon }^{(\text{mid},\delta )}$ from an ensemble average over 2000 simulations of a free polymer chain (black) and a CTRW polymer (blue). The red dashed curve is represents analytical results for a free polymer chain.