Simulated demonstration of apparent clustering arising from over-counting individual labeled molecules with a finite effective PSF.

<p>(A) Labeled molecules centered at black stars are convolved by a Gaussian PSF with half-width σ = 2 in arbitrary units (AU) (red areas). In this example, the red areas represent the finite resolution of the measurement that could arise from multiple factors, including finite localization precision in a super-resolution fluorescence localization measurement or the finite size of labeling antibodies in an SEM measurement. Blue points are examples of signals detected with probability given by the intensity of the red area. Here the over counting ratio (OCR) is 3, meaning each labeled molecule is counted on average 3 times. (B) Red labeled molecules are confined within gray circular domains with an average radius of 25 AU, while green labeled molecules are distributed at random. Both labeled molecules have an average surface density AU⁻² and AU. (C) Correlation functions calculated from B for structures as indicated. Red (green) signals are sampled at random from red (green) PSF areas with OCR = 1, as described in A. g(r) for red centers and gray domains are equivalent within error, but g(r) for red signals shows additional clustering at short r, in agreement with Eqn 1. Green signals are also clustered at short r as described by Eqn 2, while g(r) for green centers is random within error. (D) Simulated g(r) for labeled red molecules partitioned into the gray domains as in B but with different average surface densities (ρ). Apparent clustering at short r decreases as ρ is increased, but long range correlations are unchanged, consistent with Eqn 1. (E) Modified Ripley's functions, (L(r)−r)/r, calculated from clustered red centers is slightly lower than but resembles functions calculated for red signals at large r. As expected, modified Ripley's functions for randomly distributed green centers do not show significant clustering over any radius. In contrast, functions calculated from green signals show significant apparent clustering over large distances.