Past exam of the biological physics course of the University of Oxford / 2015 / 6 / d / Solution

Single-molecule fluorescence microscopy can localize isolated fluorescent proteins and link successive positions into trajectories. For ideal free Brownian motion projected into two dimensions, mean squared displacement satisfies $⟨|\mathbf{r}(t+\Delta t)-\mathbf{r}(t){|}^2⟩=4D\Delta t$. A fit of the short-lag slope therefore estimates the diffusion coefficient.

If each measured coordinate has independent localization error of variance $s^2$, two measured positions give a constant offset $4s^2$ in the two-dimensional mean squared displacement. Finite exposure averages the motion and biases that relation too. Estimates should therefore account for localization error, motion blur, missed links, and confinement. Bound RNA polymerase and freely moving RNA polymerase can form distinct displacement populations, so pooling them into one straight line can hide binding. Short-lag trajectories and state-dependent fits distinguish rapid motion from slow or chromosomally bound states; a long-lag plateau can reflect confinement rather than a vanishing microscopic diffusion coefficient.