Cyclization and Relaxation Dynamics of Finite-Length Collapsed Self-Avoiding Polymers

Julian Kappler, Frank Noé, and Roland R. Netz

Phys. Rev. Lett. 122, 067801 – Published 12 February 2019

Abstract

We study the cyclization and relaxation dynamics of ideal as well as interacting polymers as a function of chain length $N$ . For the cyclization time $τ_{cyc}$ of ideal chains we recover the known scaling $τ_{cyc} \sim N^{2}$ for different backbone models, for a self-avoiding slightly collapsed chain we obtain from Langevin simulations and scaling theory a modified scaling $τ_{cyc} \sim N^{5 / 3}$ . The cyclization and relaxation dynamics of a finite-length collapsed chain scale differently; this unexpected dynamic multiscale behavior is rationalized by the crossover between swollen and collapsed chain behavior.

Received 21 June 2018
Revised 23 November 2018

DOI:https://doi.org/10.1103/PhysRevLett.122.067801

Physics Subject Headings (PhySH)

Polymer behavior Polymer conformation & topology Polymer conformation changes

Peptides Polymers

Langevin equation Non-Markovian processes Polymer theory

Polymers & Soft MatterPhysics of Living SystemsStatistical Physics & Thermodynamics

Authors & Affiliations

Julian Kappler¹, Frank Noé², and Roland R. Netz¹

¹Department of Physics, Freie Universität Berlin, 14195 Berlin, Germany
²Department of Mathematics and Computer Science, Freie Universität Berlin, 14195 Berlin, Germany

Article Text (Subscription Required)

Click to Expand

Supplemental Material (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 122, Iss. 6 — 15 February 2019

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Illustration of different backbone models considered. In the Gaussian model, neighboring monomers are bound by harmonic potentials. In the freely jointed model, the bond length $b$ between neighboring monomers is constrained. In the freely rotating chain model both bond length $b$ and bond angle $θ$ are constrained.
Reuse & Permissions
Figure 2
Comparison of analytically and numerically calculated memory kernels for the end-to-end distance vector ${\vec{R}}_{ete}$ of a Gaussian chain.
Reuse & Permissions
Figure 3
(a) Simulation snapshots of Gaussian and GLJ chains. (b) Effective potential $U$ for the end-to-end distance for $N = 1000$ . (c) Mean squared end-to-end distance as a function of chain length $N$ . Dashed and dash-dotted lines show predictions for FR and Gaussian/FJ chain models, respectively. The dotted line indicates the power law $⟨ R_{ete}^{2} ⟩ \sim N^{2 / 3}$ . (d) Mean-squared displacement of the end-to-end distance. For better visibility, the FR chain MSD is multiplied by 5. (e) Mean squared distance $⟨ R_{0, i}^{2} ⟩ = ⟨ ({\vec{R}}_{0} - {\vec{R}}_{i})^{2} ⟩$ between the terminal monomer and the $i$ th monomer of a GLJ chain. The red dashed line represents the mean end-to-end distance $⟨ R_{ete}^{2} ⟩$ . (f) Memory kernels for chain length $N = 1000$ . The power laws indicated by black bars in (d) and (f) are justified by fits in the Supplemental Material [35].
Reuse & Permissions
Figure 4
(a) Effective potential $U$ for the end-to-end distance of a GLJ chain as a function of $R_{ete}$ (upper plot) and corresponding cyclization time $τ_{cyc}$ (lower plot) as a function of the starting position $R_{s}$ . The cyclization radius $R_{cyc} = 3 b \approx 0.46 nm$ is denoted by a black vertical dashed line; colored vertical dashed lines denote the respective positions of the minima of $U$ . The inset illustrates the end-to-end distance $R_{ete}$ and the cyclization radius $R_{cyc}$ . (b) Cyclization time as a function of chain length $N$ , together with scaling predictions indicated by lines. Except for the blue empty circles, where $R_{cyc} = 0.03 nm$ , we use $R_{cyc} = 3 b \approx 0.46 nm$ . For $R_{s}$ the minimum of $U$ is used. (c) Chain relaxation time $τ_{rel}$ (as defined in Fig. 5) as function of $N$ .
Reuse & Permissions
Figure 5
The chain relaxation time $τ_{rel}$ is defined by the intersect of the intermediate-time power-law behavior of $⟨ Δ R_{ete}^{2} (t) ⟩$ and the long-time limiting value $\lim_{t \to \infty} ⟨ Δ R_{ete}^{2} (t) ⟩ = 2 Var (R_{ete})$ calculated directly from simulations. The dashed black lines illustrate the definition of $τ_{rel}$ for $N = 2000$ . The inset shows the MSD of a terminal monomer, $⟨ Δ R_{0}^{2} (t) ⟩ = ⟨ [{\vec{R}}_{0} (t) - {\vec{R}}_{0} (0)]^{2} ⟩$ , for $N = 2000$ . The different scaling regimes are denoted by black bars.
Reuse & Permissions

Physical Review Letters