  Single-molecule localization microscopy allows practitioners to locate and
track labeled molecules in biological systems. When extracting diffusion
coefficients from the resulting trajectories, it is common practice to perform
a linear fit on mean-square-displacement curves. However, this strategy is
suboptimal and prone to errors. Recently, it was shown that the increments
between observed positions provide a good estimate for the diffusion
coefficient, and their statistics are well-suited for likelihood-based analysis
methods. Here, we revisit the problem of extracting diffusion coefficients from
single-particle tracking experiments subject to static and dynamic noise using
the principle of maximum likelihood. Taking advantage of an efficient
real-space formulation, we extend the model to mixtures of subpopulations
differing in their diffusion coefficients, which we estimate with the help of
the expectation-maximization algorithm. This formulation naturally leads to a
probabilistic assignment of trajectories to subpopulations. We employ the
theory to analyze experimental tracking data that cannot be explained with a
single diffusion coefficient. We test how well a dataset conforms to the
assumptions of a diffusion model and determine the optimal number of
subpopulations with the help of a quality factor of known analytical
distribution. To facilitate use by practitioners, we provide a fast open-source
implementation of the theory for the efficient analysis of multiple
trajectories in arbitrary dimensions simultaneously.
