= 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 $\langle|\mathbf r(t+\Delta t)-\mathbf r(t)|^2\rangle=4D\Delta t$. 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>.
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