Limits of accuracy for parameter estimation and localisation in Single-Molecule Microscopy via sequential Monte Carlo methods

Assessing the quality of parameter estimates for models describing the motion of single molecules in cellular environments is an important problem in fluorescence microscopy. We consider the fundamental data model, where molecules emit photons at random times and the photons arrive at random locations on the detector according to complex point spread functions (PSFs). The random, non-Gaussian PSF of the detection process and random trajectory of the molecule make inference challenging. Moreover, the presence of other nearby molecules causes further uncertainty in the origin of the measurements, which impacts the statistical precision of estimates. We quantify the limits of accuracy of model parameter estimates and separation distance between closely spaced molecules (known as the resolution problem) by computing the Cramer-Rao lower bound (CRLB), or equivalently the inverse of the Fisher information matrix (FIM), for the variance of estimates. This fundamental CRLB is crucial, as it provides a lower bound for more practical scenarios. While analytic expressions for the FIM can be derived for static molecules, the analytical tools to evaluate it for molecules whose trajectories follow SDEs are still mostly missing. We address this by presenting a general SMC based methodology for both parameter inference and computing the desired accuracy limits for non-static molecules and a non-Gaussian fundamental detection model. For the first time, we are able to estimate the FIM for stochastically moving molecules observed through the Airy and Born & Wolf PSF. This is achieved by estimating the score and observed information matrix via SMC. We sum up the outcome of our numerical work by summarising the qualitative behaviours for the accuracy limits as functions of e.g. collected photon count, molecule diffusion, etc. We also verify that we can recover known results from the static molecule case.