= 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 $K_d=k_{\rm off}/k_{\rm on}$. 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 $T=300\,\mathrm K$,
$$
\boxed{D=\frac{k_BT}{6\pi\eta a}=3.66\,\mathrm{\mu m^2\,s^{-1}}.}
$$
A $2\,\mathrm{nm}$ free probe would have $D\simeq110\,\mathrm{\mu m^2\,s^{-1}}$, 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 are
$$
\boxed{\sqrt{\langle|\Delta\mathbf r|^2\rangle}=\sqrt{6Dt}=1.48\,\mathrm{\mu m},\qquad
\langle|\Delta\mathbf r|\rangle=4\sqrt{Dt/\pi}=1.37\,\mathrm{\mu m}.}
$$
A two-dimensional image instead gives projected root-mean-square displacement $\sqrt{4Dt}=1.21\,\mathrm{\mu m}$ and mean projected distance $\sqrt{\pi Dt}=1.07\,\mathrm{\mu m}$. 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.
Back to article page
