Bayesian and frequentist analyses of two-state single-molecule diffusion trajectories

Abstract The motion and interactions of proteins are key components in processes in living cells. Such interactions are generally initiated by the binding of two bio-molecular species to form temporary complexes. Though in most single molecule localisation microscopy experiments, only one species is tracked using fluorescent tags; it is therefore unknown whether a tracked molecule is unbound or bound (i.e. two-state kinetics). That difficulty is passed on to the trajectory analysis, the aim of which is to infer kinetic binding/unbinding rates and diffusion constants for the unbound molecule and the complex. Currently, two main such analysis methods are employed in the field. One is the Rayleigh mixture model (RMM), which has been used extensively by experimental researchers for several decades. The other method is a hidden Markov model (HMM), which is now generally considered as state-of-the-art. Yet it is unknown how and why the HMM is superior, and if the RMM retains some advantage, i.e. the methodological transition lacks a thorough methods comparison. We here provide such a comparison. To this end, we extend the RMM such that it estimates the same parameters as the HMM, which means estimating the individual binding/unbinding rates, rather than solely their fraction (as previously). Furthermore, we incorporate the nested sampling algorithm into the HMM approach, thus removing the need for manual adjustment of tunable hyper-parameters found in previous Markov–Chain Monte–Carlo-based approaches. With these adaptations, we then do an in-depth comparison of the frequentist RMM versus the Bayesian HMM on simulated data. We find the HMM to display the strongest inference with reliable uncertainty estimates. Our extended RMM has the advantage of being computationally very fast. For both methods, we publicly share our computer implementations.