On characterizing membrane protein clusters with model-free spatial correlation approaches

Abstract

Spatial aggregation or clustering of membrane proteins could be important for their functionality, e.g., in signaling, and nanoscale imaging can be used to study its origins, structure and function. Such studies require accurate characterization of clusters, both for absolute quantification and hypothesis testing. A set of model-free quantification approaches -- free of specific cluster models-- have been proposed for this purpose. They include the radius of maximal aggregation ra obtained from the maxima of the empirical Besag L(r) - r function as an estimator of cluster size, and the estimation of various cluster parameters based on an exponential approximation for the Pair Correlation Function(PCF). However, the parameter identifiability and bias and scaling due to their model-free nature are not clear. In practice, the clusters might exhibit specific patterns, and the behavior of these estimators in such cases must be studied. Here, we theoretically analyze these approaches for a set of cluster models, and obtain information about their identifiability and bias. We find that the ratio between ra and true cluster size depends on both the true size as well as the number of clusters per unit area, or other corresponding parameters, in a model-dependent manner. In particular, ra scales with respect to the true size by a factor that can be arbitrarily large, depending on models and parameter values. For the method based on PCF approximation, for most models we analyzed, the ratios between approximate and true model parameters were found to be constants that depend only on models and independent of other parameters. For the models analyzed, this ratio was within {+/-}100%. Our theoretical approach was validated by means of simulations. We also discuss some general issues in inference using second-order spatial properties. While precision could also be key, such information on identifiability and accuracy provides clarity on estimation, can lead to better inference, and can also fuel more accurate method development.