Probabilistic Nearest Neighbor Estimation of Diffusion Constants from Single Molecular Measurement without Explicit Tracking

Time course measurement of single molecules on a cell surface provides detailed information on the dynamics of the molecules, which is otherwise inaccessible. To extract the quantitative information, single particle tracking (SPT) is typically performed. However, trajectories extracted by SPT inevitably have linking errors when the diffusion speed of single molecules is high compared to the scale of the particle density. To circumvent this problem we developed an algorithm to estimate diffusion constants without relying on SPT. We demonstrated that the proposed algorithm provides reasonable estimation of diffusion constants even when other methods fail due to high particle density or inhomogeneous particle distribution. In addition, our algorithm can be used for visualization of time course data from single molecular measurements. Introduction Sensing the extracellular environment is crucial for cells to properly respond and function. The information from the environment is typically encoded in microscopic molecular signals, and they are recognized by cell surface receptors. The signaling of cell surface receptors involves several physical processes, such as ligation to their ligands, oligomerization, and subsequent binding to the downstream signaling components in cytosol. Although many details on these processes have been inferred from biochemical, genetic, and molecular or cell biological studies, its physical and dynamical aspects at microscopic level are still largely unknown (1). Recent development of single molecular measurement, such as total internal reflection fluorescence (TIRF) microscopy (2), provides a chance to directly observe the dynamics of these processes from time course images of single molecules on cell surfaces (3, 4). A typical workflow for such data is single particle tracking (SPT) (5). In SPT, the positions of particles in each time frame are first detected. With the help of the sophisticated detection algorithms, the spatial resolution of detected position could be sub-pixel order (6). The next step is linking, where the trajectory of each molecule is inferred by connecting seemingly identical particles in subsequent frames. Usually, the nearest particles in the subsequent frames with global consistency are identified as the same particles (7, 8) The identified trajectories of particles will be further analyzed qualitatively to find biologically relevant physical parameters. Diffusion constant, which characterizes the diffusion speed of the particles, is one of such important physical parameters, and have been the target for subsequent analyses (9, 10). It has been shown that diffusion constants of membrane proteins such as cell surface receptors can change along biophysical events like binding to their ligand or cytosolic adaptor molecules. For example, the diffusion constants of epidermal growth factor receptor (EGFR) that belongs a family of receptor tyrosine kinase, are found to decrease after binding to EGF, and transduce signals via subsequent binding with its adaptor Grb2 protein (11, 12). It was also shown that intracellular signaling proteins functioning on the membrane have multiple states each of which have different diffusion constants (13, 14). Though SPT methods are widely used, they encounter difficulties when the density of particles is higher. One of the difficulties is the diffraction limit of microscope. If the particle density becomes comparable to the scale defined by the diffraction limit, the chance of having two different particles within the diffraction limit cannot be ignored. Then, we may not be able to resolve the positions of the two particles, which lead to errors in the particle detection. The other difficulty occurs when the particle density becomes comparable to the scale of diffusion in the time resolution of the measurement. In this situation, the expected area of diffusion of a particle tends to contain several irrelevant particles purely by chance. Since, in typical experiments, visualized molecules are indistinguishable just from the fluorescent signals, linking errors are inevitable. Then, trajectories from such an erroneous SPT might lead to a biased estimation of diffusion constants, and different biological interpretation. Note that, this problem of linking error may occur if the diffusion speed is high enough, even in the regime that the detection error coming from the diffraction limit is negligible. In this paper, we address this problem of linking error in diffusion constant estimation. As we have seen, the problem arises from the impossibility of the perfect hard linking of the identical particles in SPT. Here, instead of hardly linking the nearest particles in subsequent frames, we only assign a probability of such possible identification with respect to the particle density around the position, and directly estimate the diffusion constant without specifying concrete trajectories. The resultant algorithm, which successfully estimates diffusion constants even under high particle density condition, shows some resemblance to another SPT free diffusion constant estimation method, particle image correlation spectroscopy (PICS) (15), which was inspired by image correlation microscopy (16–19). The main advantage of our algorithm towards PICS is that our algorithm can be applied to the cases with inhomogeneous distribution of single molecules, while PICS assumes homogeneous distribution. In this paper, first, we introduce a probabilistic model of the position of nearest particles of a diffusing particle surrounded by indistinguishable particles and formulate the inference of diffusion constants in terms of maximum likelihood estimation based on this probabilistic model. In a simple setting with a homogeneous particle distribution, our algorithm can be considered as a natural generalization of the canonical diffusion constant estimation from the mean square displacement (MSD) to the case of finite density of surrounding particles. Our algorithm is further generalized to allow multiple states with different diffusion constant with a help of the expectation maximization (EM) algorithm (20). Comparison of the performance of our proposed method based on simulated artificial data of diffusion indicated the advantage of the proposed method over other diffusion constant estimation methods. In addition to the estimation of diffusion constants, we also demonstrate that the algorithm could infer the state of each molecule and visualize the single molecular data with such information.