Simultaneous orientation and 3D localization microscopy with a Vortex point spread function.

Fitting model

We use standard Maximum Likelihood Estimation (MLE) using an image formation model that describes the expected photon count across the image as a function of the molecule position r 0 = ( x 0 , y 0 , z 0 ), signal photon count N of the entire PSF on the camera, background photons per pixel b , and dipole orientation Ω 0 = ( ϕ 0 , θ 0 ) with the degree of orientational constraint g 2 , giving a total of 8 parameters. The underlying PSF model is the fully vectorial PSF model, as in earlier work 15 , 25 , 53 – 55 , but now extended to estimate the dipole orientation along with the degree of orientational constraint. The image formation model is based on a weighted sum of the freely rotating dipole PSF and the fixed dipole PSF corresponding to the equilibrium dipole orientation 36 , plus a constant background: 1 documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$H({{{{{{{bf{r}}}}}}}},{{Omega }})=Nleft[frac{(1-{g}_{2})}{3}{H}_{{{{{{{{rm{free}}}}}}}}}({{{{{{{bf{r}}}}}}}})+frac{{g}_{2}}{3}{H}_{{{{{{{{rm{fixed}}}}}}}}}({{{{{{{bf{r}}}}}}}},{{Omega }})right]+frac{b}{{a}^{2}}$$end{document} H ( r , Ω ) = N ( 1 − g 2 ) 3 H free ( r ) + g 2 3 H fixed ( r , Ω ) + b a 2 where a is the pixel size and 0 ≤ g 2 ≤ 1 represents the degree of orientational constraint, with the limiting cases of a fully free dipole g 2 = 0 and a fully fixed dipole g 2 = 1. This seems simplistic compared to more complex rotational diffusion models 50 , 56 – 58 . The appropriateness of the model for rotational diffusion faster than the fluorescence lifetime, however, is demonstrated in ref. 36 . An asymmetric rotational diffusion model would require 2 additional fitting parameters, raising the total amount of parameters to 10 which we expect is not realistic to fit with