A mathematical analysis for the Brownian dynamics of a DNA tether |
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Authors: | Hong Qian |
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Institution: | (1) Department of Applied Mathematics, University of Washington, Seattle, WA 98195, USA. e-mail: qian@amath.washington.edu, US |
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Abstract: | In the single-particle tracking experiment, the internal motion of a single DNA or polymer molecule whose one end is attached
to a microsphere (optical marker) and the other end is anchored to a substratum is studied (Finzi and Gelles, 1995). The stochastic
Brownian dynamics of the sphere reflect the spontaneous fluctuations, thus the physical characteristics, of the DNA or polymer
molecule (Qian and Elson, 1999, Qian, 2000). In this paper, two continuous models of polymer molecules, a flexible elastic string and a weakly bentable elastic rod, are analyzed. Both models are cast
mathematically in terms of linear stochastic differential equations. Based on Fourier analyses, we calculate the mean square
displacement (MSD) of the particle motion, the key observable in the experiment. We obtain for both models the short-time
asymptotics for the MSD, as well as the long-time behavior in terms of the smallest non-zero eigenvalues. It is shown that:
(i) the long-time dynamics of continuous elastic string model quantitatively agree with that of the discrete bead-spring model.
(ii) The short-time MSD of both models are controlled by the tethered particle, with linear dependence on t. (iii) The two models show characteristic difference for long-time behavior: The longest relaxation time is proportional to L
2
for long elastic string and to L for short elastic string, but is proportional to L
4
for both long and short weakly bentable rod.
Received: 26 March 1998 / Revised version: 9 June 2000 / Published online: 14 September 2000 |
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