5Simultaneous measurements of single-molecule positions and orientations provide critical insight 6 into a variety of biological and chemical processes. Various engineered point spread functions (PSFs) 7 have been introduced for measuring the orientation and rotational diffusion of dipole-like emitters, 8 but the widely used Cramér-Rao bound only evaluates performance for one specific orientation at 9 a time. Here, we report a performance metric, termed variance upper bound (VUB), that yields 10 a global maximum CRB for all possible molecular orientations, thereby enabling the measurement 11 performance of any PSF to be computed efficiently (~1000 times faster than calculating average 12 CRB). Our VUB reveals that the simple polarized standard PSF provides robust and precise ori-13 entation measurements if emitters are near a refractive index interface. Using the polarized PSF, 14 we measure the orientations and positions of Nile red (NR) molecules transiently bound to amy-15 loid aggregates. Our super-resolved images reveal the main binding mode of NR to amyloid fibers, 16 as well as structural heterogeneities within amyloid fibrillar networks, that cannot be resolved by 17 single-molecule localization alone.
19A key strength of single-molecule localization mi-20 croscopy (SMLM) is its ability to measure the full distri-21 bution of the phenomena under study and avoid ensemble 22 averaging. Going beyond standard SMLM to measure 23 SM position and orientation simultaneously is critical 24 for understanding a variety of nanoscale biological and 25 chemical processes, such as the motions of molecular mo-26 tors [1-3], the complex higher-order structures of DNA 27 [4, 5], and reaction kinetics within catalytic nanoparti-28 cles [6]. To perform these measurements, molecular po-29 sition and orientation must be encoded within the shape 30 of the image produced by a microscope, i.e., its point-31 spread function (PSF). However, balancing the need to 32 resolve various orientations unambiguously with the need 33 to detect SMs efficiently with high signal-to-background 34 ratio (SBR) remains a challenge that limits the adoption 35 of Single-Molecule Orientation Localization Microscopy 36 (SMOLM) for biological and chemical applications; exist-37 ing techniques either cannot discriminate between similar 38 types of molecular motions or cannot detect weak fluo-39 rescent emitters. 40 In order to design a PSF for measuring molecular ori-41 entation, one must have a figure of merit for compari-42 son. The Cramér-Rao bound (CRB), which is the best-43 possible precision achievable by an unbiased estimator, 44 has been used extensively for evaluating [7] and opti-45 mizing [8, 9] SMLM techniques. However, since CRB 46 quantifies measurement performance locally, in the sense 47 65 tection channels, exhibiting a polarized (standard) PSF, 66 provides superior measurement precision when molecules 67 are near a refractive index interface, especially when they 68 lie perpendicular to the optical axis, even under low SBR. 69 ...