A Molecular Ruler for Measuring Quantitative Distance Distributions
Figure 3
Transformation of scattering interference profiles into distance distributions.
(A) Model coordinates of a 12 base-pair DNA duplex bearing a gold nanocrystal at each end. Thioglucose ligands are not shown. Various types of scattering interference between gold nanocrystals and DNA are illustrated with labeled arrows. The probe-probe interference pattern (vi) is obtained by subtracting the scattering profiles for the single-labeled A sample (i+ii+iii), and B sample (iii+iv+v) from the sum of the double-labeled sample (i+ii+iii+iv+v+vi) and the unlabeled sample (iii). (B) The scattering profiles for the 10 base-pair double-labeled (Blue), single-labeled (Purple, Magenta; indistinguishable), and unlabeled (Green) DNA duplexes. The probe-probe scattering interference pattern (Black) is obtained by adding the double-labeled and unlabeled profiles and subtracting off the single-labeled profiles. The residual difference between this interference pattern and the transform of the probability distribution in panel D is plotted in Red, and offset downward. See Fig. S6 for a log-log plot of the scattering profiles. (C) Scattering interference basis profiles corresponding to pairs of nanocrystals with center-to-center separation distances between 1 and 200 Å. The profiles are plotted at 25 Å increments for clarity. (D) Distance distributions obtained by decomposing the scattering interference pattern in panel B into a linear combination of the basis profiles shown in panel C. Three different transformation methods are illustrated. They are offset vertically from one another. The bottom profile (Green, 56.3 ű3.1 Å) was obtained using 1 Å resolution basis functions and a non-negative least-squares optimizer. The middle profile (Red, 56.8 ű3.2 Å) was obtained using 5 Å resolution basis functions and a non-negative least-squares optimizer. The top profile (Blue, 56.8 ű3.0 Å) was obtained using 1Å resolution basis functions and a non-negative least-squares optimizer with a maximum entropy regularization term. The least-squares transformation of a scattering interference profile into a distance distribution is unique for coarsely sampled distance basis functions. The highest basis set resolution that retains this property is 1/(2*Smax), where Smax is the scattering angle at which the signal-to-noise ratio reaches ∼2. For the data presented here, Smax≈0.08 Å−1, giving a natural basis function resolution of 6 Å. Transforms at basis set resolutions higher than this value generally require some form of regularization to break the degeneracy between multiple possible solutions.