# Kinetics of Loop Formation in Polymer Chains

Source: [Kinetics of Loop Formation in Polymer Chains](<https://www.sota2.com/research/paper/68fa211c7a2b558e29335347>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2025-10-28T13:52:12.017Z

Authors: Ngo Minh Toan, Greg Morrison, Changbong Hyeon, D. Thirumalai

Published: 2007-08-15T18:33:55.000Z

## Abstract

We investigate the kinetics of loop formation in flexible ideal polymer chains (Rouse model), and polymers in good and poor solvents. We show for the Rouse model, using a modification of the theory of Szabo, Schulten, and Schulten, that the time scale for cyclization is $\\tau\_c\\sim \\tau\_0 N^2$ (where $\\tau\_0$ is a microscopic time scale and $N$ is the number of monomers), provided the coupling between the relaxation dynamics of the end-to-end vector and the looping dynamics is taken into account. The resulting analytic expression fits the simulation results accurately when $a$, the capture radius for contact formation, exceeds $b$, the average distance between two connected beads. Simulations also show that, when $a \< b$, $\\tau\_c\\sim N^{\\alpha\_\\tau}$, where $1.5\<{\\alpha\_\\tau}\\le 2$ in the range $7\<N\<200$ used in the simulations. By using a diffusion coefficient that is dependent on the length scales $a$ and $b$ (with $a\<b$), which captures the two-stage mechanism by which looping occurs when $a \< b$, we obtain an analytic expression for $\\tau\_c$ that fits the simulation results well. The kinetics of contact formation between the ends of the chain are profoundly affected when interactions between monomers are taken into account. Remarkably, for $N \< 100$ the values of $\\tau\_c$ decrease by more than two orders of magnitude when the solvent quality changes from good to poor. Fits of the simulation data for $\\tau\_c$ to a power law in $N$ ($\\tau\_c\\sim N^{\\alpha\_\\tau}$) show that $\\alpha\_\\tau$ varies from about 2.4 in a good solvent to about 1.0 in poor solvents. Loop formation in poor solvents, in which the polymer adopts dense, compact globular conformations, occurs by a reptation-like mechanism of the ends of the chain.

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## Original sources and implementation

[Original paper](<https://arxiv.org/abs/0708.2077>)

[PDF](<https://arxiv.org/pdf/0708.2077.pdf>)