instantaneous center. This blurring effect reduces the apparent bead excursion. Suppose for example that the bead has a momentary excursion to a large value of $x$. Subsequently, its stretched tether will pull it inward, so that the average position during the video frame has a smaller value of $x$. We quantified this effect using a 901bp DNA and a 490nm diameter bead at 1, 5, 10, 30, 50, 100, 200, 300 and 500$\, \msunit$ exposures (). Longer exposures indeed reduce the apparent RMS motion of the bead. The effect is minimal for exposure times smaller than 30ms, but decreases sharply above this value.

These effects can be considered from a theoretical perspective (see the Supplement to [@Towles2008]). The effect of the tether on the bead may be approximated as a harmonic restoring force. If the bead starts at a distance $\rho_0$ from the center, then its average position drifts inward under the influence of this force. Averaging that trajectory over the video frame gives a blurred trajectory with center at $S(\rho_0)\rho_0$, where the blur factor is $$S(\rho_0)=\frac{T_{s}}{\delta t}[1-{{\mathrm e}^{-\delta t/T_{s}}}]\label{e:1a}$$ The time constant $T_s$ can in principle be estimated from first principles, but in practice we fit it to data 