## 1 Introduction  
Single molecule localization microscopy (SMLM) is one of the most widely used imaging tools in molecular biology. Benefitting from the advantage of sub-diffraction-limit spatial resolution by localization of individual molecular blinking events from sequential detections, SMLM has been popularly used to characterize the spatial organization of membrane proteins [1; 2; 3]. Growing evidence suggests that the protein clustering (aggregation) at the cell membrane, one of the most important organization states, is associated with many cell functional changes and diseases: overexpression and clustering of epidermal growth factor receptor (EGFR) are observed in many cancers [4; 5; 6]; clustering of nicotinic acetylcholine receptors (nAChRs) in high concentrations is critical for muscle function [7; 8]; clustering of ion channels (Na\({}^{+}\), Ka\({}^{+}\), Ca\({}^{2+}\) channels) is essential for signal transduction [9; 10; 11; 12]; etc. Thus, quantitative analysis of protein clusters, such as cluster number, cluster size, protein density of individual cluster, etc., is extremely important because it leads to a better understanding of the function-related clustering process of proteins. So far, the clusters detected by SMLM are commonly analyzed by Ripley's K function (see in Materials and Methods) to give detailed information. In this approach, by analyzing the localization of proteins in a two dimensional super-resolution SMLM image, clustering is firstly identified if the density of proteins within a distance r of another protein is greater than that expected for a spatial point pattern distributed randomly within the same distance [13].  
Then, other interesting cluster parameters, such as the cluster area and the molecule density inside or outside the cluster, can be extracted [14; 15]. However, because of the inherent properties of mathematical methods and the extreme small size of protein clusters in molecular biology, this method has limitations (i.e., the average radius of clusters estimated by analyzing [PERSON]'s K function has dependence not only on the distance of cluster separation, but also on the density of proteins inside/outside the cluster domain, and the measured cluster properties by analyzing [PERSON]'s K function relate to the localization uncertainty of SMLM [15; 16]). In summary, so far, the accuracy study of the quantitative analysis of clusters by analyzing [PERSON]'s K function remains elusive, but it is of the utmost need to contextualize the increasing use of Ripley's K function in molecular biology.  
In our work, to study the accuracy of cluster analysis by using the peak analysis method as described in [PERSON] et al. [16], in which disk-shaped domains are randomly drawn so that protein clustering occurs on a modeled SMLM image. The density of proteins inside the domain was set much higher than that outside domain. Varying the parameters of simulated clusters, we evaluated their effects on the accuracy of quantitative analysis of clusters by the peak analysis method. Together with experimental membrane protein cluster analysis, our results present a better understanding and characterization of cluster properties.