(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 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 Å±3.1 Å) was obtained using 1 Å resolution basis functions and a non-negative least-squares optimizer. The middle profile (Red, 56.8 Å±3.2 Å) was obtained using 5 Å resolution basis functions and a non-negative least-squares optimizer. The top profile (Blue, 56.8 Å±3.0 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*S_{max}), where S_{max} is the scattering angle at which the signal-to-noise ratio reaches ∼2. For the data presented here, S_{max}≈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.

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