 Solution

The small binding RNA can serve conceptually as an RNA aptamer recognition probe. Coupling recognition to an optical label would make a specifically bound virus appear as a moving fluorescent spot. Single-particle tracking estimates successive spot positions and fits their displacement statistics. Specificity is assessed by comparison with a nonbinding recognition probe or a sample without the target, while the much faster diffusion of unbound probes distinguishes freely moving label from target-associated signal. Subnanomolar dissociation constant supports appreciable occupancy at low probe abundance, but affinity alone does not determine the dissociation lifetime because $K_d=k_{\mathrm{off}}/k_{\mathrm{on}}$. This describes detection principles, not an operational virus preparation or labelling protocol.

For a dilute spherical particle of radius $a=60\mathrm{nm}$ in bulk water, the Stokes–Einstein relation gives, at an explicitly assumed $T=300\mathrm{K}$,A $2\mathrm{nm}$ free probe would have $D\simeq 110\mu {\mathrm{m}}^2{\mathrm{s}}^{-1}$, roughly thirty times larger. The recognition probe changes the target's hydrodynamic size only modestly in this estimate.

For isotropic Brownian motion, the mean vector displacement is zero. In $t=0.1\mathrm{s}$, distinct measures of how far it moves areA two-dimensional image instead gives projected root-mean-square displacement $\sqrt{4Dt}=1.21\mu \mathrm{m}$ and mean projected distance $\sqrt{\pi Dt}=1.07\mu \mathrm{m}$. The projected mean squared displacement is $4D\Delta t$; independent localization errors of variance ${\sigma }^2$ in each coordinate add $4{\sigma }^2$. Finite exposure blurs motion, and nearby walls reduce diffusion; those effects must be separated from particle size when interpreting a trajectory.