# Dynamic Looping of a Free-Draining Polymer

## Abstract

Here, we revisit the celebrated Wilemski--Fixman (WF) treatment for the looping time of a free-draining polymer. The WF theory introduces a sink term into the Fokker--Planck equation for the $3(N+1)$-dimensional Ornstein--Uhlenbeck process of polymer dynamics, which accounts for the appropriate boundary condition due to the formation of a loop. The assumption for WF theory is considerably relaxed. A perturbation method approach is developed that justifies and generalizes the previous results using either a delta sink or a Heaviside sink. For both types of sinks, we show that under the condition of a small dimensionless $$\epsilon$$, the ratio of capture radius to the Kuhn length, we are able to systematically produce all known analytical and asymptotic results obtained by other methods. This includes most notably the transition regime between the $N^2$ scaling of Doi, and $$N\sqrt{N}/\epsilon$$ scaling of Szabo, Schulten, and Schulten. The mathematical issue at play is the nonuniform convergence of $$\epsilon\to 0$$ and $$N\to\infty$$, the latter being an inherent part of the theory of a Gaussian polymer. Our analysis yields a novel term in the analytical expression for the looping time with small $$\epsilon$$, which was previously unknown. Monte Carlo numerical simulations corroborate the analytical findings. The systematic method developed here can be applied to other systems modeled by multidimensional Smoluchowski equations.

- Authors:

- Publication Date:

- Research Org.:
- Pacific Northwest National Lab. (PNNL), Richland, WA (United States)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1427921

- Report Number(s):
- PNNL-SA-125393

Journal ID: ISSN 0036-1399; KJ0401000

- DOE Contract Number:
- AC05-76RL01830

- Resource Type:
- Journal Article

- Journal Name:
- SIAM Journal of Applied Mathematics

- Additional Journal Information:
- Journal Volume: 78; Journal Issue: 1; Journal ID: ISSN 0036-1399

- Country of Publication:
- United States

- Language:
- English

### Citation Formats

```
Ye, Felix X. -F., Stinis, Panos, and Qian, Hong.
```*Dynamic Looping of a Free-Draining Polymer*. United States: N. p., 2018.
Web. doi:10.1137/17M1127260.

```
Ye, Felix X. -F., Stinis, Panos, & Qian, Hong.
```*Dynamic Looping of a Free-Draining Polymer*. United States. doi:10.1137/17M1127260.

```
Ye, Felix X. -F., Stinis, Panos, and Qian, Hong. Thu .
"Dynamic Looping of a Free-Draining Polymer". United States. doi:10.1137/17M1127260.
```

```
@article{osti_1427921,
```

title = {Dynamic Looping of a Free-Draining Polymer},

author = {Ye, Felix X. -F. and Stinis, Panos and Qian, Hong},

abstractNote = {Here, we revisit the celebrated Wilemski--Fixman (WF) treatment for the looping time of a free-draining polymer. The WF theory introduces a sink term into the Fokker--Planck equation for the $3(N+1)$-dimensional Ornstein--Uhlenbeck process of polymer dynamics, which accounts for the appropriate boundary condition due to the formation of a loop. The assumption for WF theory is considerably relaxed. A perturbation method approach is developed that justifies and generalizes the previous results using either a delta sink or a Heaviside sink. For both types of sinks, we show that under the condition of a small dimensionless $\epsilon$, the ratio of capture radius to the Kuhn length, we are able to systematically produce all known analytical and asymptotic results obtained by other methods. This includes most notably the transition regime between the $N^2$ scaling of Doi, and $N\sqrt{N}/\epsilon$ scaling of Szabo, Schulten, and Schulten. The mathematical issue at play is the nonuniform convergence of $\epsilon\to 0$ and $N\to\infty$, the latter being an inherent part of the theory of a Gaussian polymer. Our analysis yields a novel term in the analytical expression for the looping time with small $\epsilon$, which was previously unknown. Monte Carlo numerical simulations corroborate the analytical findings. The systematic method developed here can be applied to other systems modeled by multidimensional Smoluchowski equations.},

doi = {10.1137/17M1127260},

journal = {SIAM Journal of Applied Mathematics},

issn = {0036-1399},

number = 1,

volume = 78,

place = {United States},

year = {2018},

month = {1}

}