A mathematical analysis for the Brownian dynamics of a DNA tether

In the single-particle tracking experiment, the internal motion of a single DNA or polymer molecule whose one end is attached to a microsphere (optica l marker) and the other end is anchored to a substratum is studied (Finzi a nd Gelles, 1995). The stochastic Brownian dynamics of the sphere reflect th e spontaneous fluctuations, thus the physical characteristics, of the DNA o r polymer molecule (Qian and Elson, 1999, Qian, 2000). Tn this paper, two c ontinuous models of polymer molecules, a flexible elastic string and a weak ly bentable elastic rod, are analyzed. Both models are cast mathematically in terms of linear stochastic differential equations. Based on Fourier anal yses, we calculate the mean square displacement (MSD) of the particle motio n, the key observable in the experiment. We obtain for both models the shor t-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 a re controlled by the tethered particle, with linear dependence on t. (iii) The two models show characteristic difference for long-time behavior: The l ongest relaxation time is proportional to L-2 for long elastic string and t o L for short elastic string, but is proportional to L-4 for both long and short weakly bentable rod.