Estimation of diffusion constants from single molecular measurement without explicit tracking

Abstract

Background: Time course measurement of single molecules on a cell surface provides detailed information about the dynamics of the molecules that would otherwise be 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.

Methods: To circumvent this problem, we develop an algorithm to estimate diffusion constants without relying on SPT. The proposed algorithm is based on a probabilistic model of the distance to the nearest point in subsequent frames. This probabilistic model generalizes the model of single particle Brownian motion under an isolated environment into the one surrounded by indistinguishable multiple particles, with a mean field approximation.

Results: We demonstrate that the proposed algorithm provides reasonable estimation of diffusion constants, even when other methods suffer 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.

Conclusions: The proposed algorithm based on the probabilistic model of indistinguishable Brownian particles provide accurate estimation of diffusion constants even in the regime where the traditional SPT methods underestimate them due to linking errors.

Keywords: Brownian motion; Diffusion constants; Expectation maximization algorithm; Probabilistic model; Single molecular measurement.

Fig. 1

Schematic of the probabilistic model. a a typical distribution of particles at t + Δt (thick circles) with an indication of the position of a representative particle at t (dashed circle). b the case where the nearest particle is the original particle. c the case where the nearest particle is a surrounding particle. Gray color indicates the identification of the original particle. The large dotted circles indicate the distance to the nearest particle. The distance to the nearest neighbor of the origin at the subsequent time frame is modeled by the probabilistic model with respect to the diffusion constant of the original particle and the particle density at the origin

Fig. 2

Mean square displacement to the nearest particle. A comparison of MSDN and MSD. The black straight line corresponds to the expected MSD, while the black curve is the expected MSDN, with D=1 μm²/s and ρ=1 particles/μm². The points are the mean MSDN directly calculated from corresponding simulated data. The error bars indicate the standard deviation from one thousand independent simulations. The red line indicates the asymptotic value of the expected MSDN at Δt → ∞

Fig. 3

Comparison of the performance of different algorithms in a uniform distribution. Box plots summarizing a comparison of the algorithms. The x axis is the particle density and the y axis is the estimated diffusion constant. The red line indicates the true diffusion constant. a local SPT. b global SPT. c PICS and d PNN

Fig. 4

Comparison of the performance of PICS and PNN in a uniform distribution with false detections. Box plots summarizing the comparison of PICS (a and b) and PNN (c and d). The top row is for PICS and the bottom row is for PNN. The first column is the result before introducing the state corresponding to the false detections. The second column is the result after introducing the state for false detection compensation. The x axis is the particle density and the y axis is the estimated diffusion constant. The red line indicates the true diffusion constant

Fig. 5

Comparison of the performance of PICS and PNN in a Gaussian distribution. a a representative snapshot of the particle distribution. b, c, and d box plots summarizing the comparison between PICS and PNN under a Gaussian distribution. b PICS. c PNN, where the known particle density distribution for the simulation is used for the diffusion constant estimation. d PNN where the particle density distribution is estimated from the data using a k nearest neighbor algorithm. The x axis is the mean particle density over the area of interest, and the y axis is the estimated diffusion constant. The red line indicates the true diffusion constant

Fig. 6

The performance of different algorithms in image based simulations. Scatter plots summarizing the performance of the algorithms. The x axis is the true diffusion constant used for the simulation and the y axis is the estimated diffusion constant. The red line indicates the diagonal line corresponding to the successful estimation. a PNN. b PICS. c local SPT and d PICS applied to the corresponding ground truth data

Fig. 7

3D visualization of particle positions and states. 3D representation of the time course simulated data of diffusing particles. The z axis corresponds to time while the other two axes correspond to the x- and y-axes of the original data. a the original data. b, the same data depicted in color (red: slower particle (0.2 μm²/s), cyan: faster particle (2 μm²/s)) after removing the false detection. c the same data depicted in colors based on the particle states inferred by PNN