\\
\end{pmatrix},\label{eq:cov}$$ where $$\sigma_{x_{i}x_{j}} = \frac{1}{N}\sum_{k=1}^{N} x_{i}^kx_{j}^k - \bar x_i\bar x_j 
$$ are the second moments of the bead’s position. Here $N$ is the number of video frames and $x_{1}^{k}$, $x_{2}^{k}$ are the in-plane coordinates (i.e. the position $x,y$) of the microsphere for frame $k$ as obtained from the data. The eigenvalues ($\lambda_1, \lambda_2$) of the covariance matrix indicate the squares of the major and minor axes corresponding to the in-plane displacement of the bead and are equal for a perfectly symmetric motion. We took $s=\sqrt{{\lambda_{\rm max}}/{\lambda_{\rm min}}} \leq 1.1$ as our acceptable threshold. (d,e) displays scatter plots for the in-plane motion of two beads to illustrate the distinction between symmetric and asymmetric tethers. The first plot passes the symmetry test and would serve as a qualified tether; the second would be rejected. Asymmetric bead trajectories may be caused by multiple DNA tethers [@Pouget2004].

“Uniformity” qualifies tethers on the basis of the consistency of their motion over time and eliminates beads showing non-specific binding events, such as binding of DNA to the bead or glass surface for short periods. To detect these events automatically, we refine a procedure used in . We first divide the entire time series into 10 