Maximum likelihood-based estimation of diffusion coefficient is quick and reliable method for analyzing estradiol actions on surface receptor movements

Abstract

The rapid effects of estradiol on membrane receptors are in the focus of the estradiol research field, however, the molecular mechanisms of these non-classical estradiol actions are poorly understood. Since the lateral diffusion of membrane receptors is an important indicator of their function, a deeper understanding of the underlying mechanisms of non-classical estradiol actions can be achieved by investigating receptor dynamics. Diffusion coefficient is a crucial and widely used parameter to characterize the movement of receptors in the cell membrane. The aim of this study was to investigate the differences between maximum likelihood-based estimation (MLE) and mean square displacement (MSD) based calculation of diffusion coefficients. In this work we applied both MSD and MLE to calculate diffusion coefficients. Single particle trajectories were extracted from simulation as well as from α-amino-3-hydroxy-5-methyl-4-isoxazolepropionic acid (AMPA) receptor tracking in live estradiol-treated differentiated PC12 (dPC12) cells. The comparison of the obtained diffusion coefficients revealed the superiority of MLE over the generally used MSD analysis. Our results suggest the use of the MLE of diffusion coefficients because as it has a better performance, especially for large localization errors or slow receptor movements.

Keywords: MLE; diffusion coefficient; maximum likelihood; mean square displacement; receptor movements.

FIGURE 1

The parameters extracted by mean square displacement (MSD) (A) and maximum likelihood-based estimation (MLE) (B) based parameter estimation on three set of simulated trajectories. Each point on the graphs represents a set of parameters calculated from a trajectory. The value of diffusion coefficients is shown on the x-axis of both graphs. The y-axis represents another parameters provided by the diffusion coefficient’s estimation, namely they are the y-intercept of the linear fitting and the extracted localization error for the MSD and MLE graph, respectively. The number of trajectories is 1,000 in each group.

FIGURE 2

Mean and standard deviation (SD) values of diffusion coefficients extracted from a set of trajectories (N = 1,000) simulated with the following diffusion coefficients: (A) 0.01μm²/s, (B) 0.02μm²/s, (C) 0.05μm²/s, (D) 0.1μm²/s, (E) 0.2μm²/s, (F) 0.5μm²/s as a function of the length of trajectories. The diffusion coefficients were extracted by both the mean square displacement (MSD) (black) and maximum likelihood-based estimation (MLE) (red) method.

FIGURE 3

The coefficients of variation (the ratio of the SD and the mean from Figure 2) as a function of the length of trajectories. The diffusion coefficients for the simulation were: (A) 0.01μm²/s, (B) 0.02μm²/s, (C) 0.05μm²/s, (D) 0.1μm²/s, (E) 0.2μm²/s, (F) 0.5μm²/s.

FIGURE 4

Distribution of diffusion coefficients derived from trajectories recorded on immobile particles. The measurement was carried out on different temperatures and the extracted trajectories were analyzed by the mean square displacement (MSD) (A) and the maximum likelihood-based estimation (MLE) (B) method. The inserted table shows the mean and SD values for each group, respectively.

FIGURE 5

The effect of E2 treatment on the diffusion coefficient of GluR2-AMPAR molecules in the soma’s plasma membranes of dPC12, live-cell. The E2 treatments were carried out by the concentration of 100 pM (A,B) and 100 nM (C,D). Both the mean square displacement (MSD) (A,C) and the maximum likelihood-based estimation (MLE) (B,D) methods were used for further analysis to obtain the diffusion coefficients from the recorded trajectories. The graphs represent the groups as mean and SD values. The probability values of significant differences calculated by Kolmogorov–Smirnov test (*p < 0.05) and the number of trajectories in each group are also shown.

FIGURE 6

The trajectories length distribution from GluR2-AMPAR molecules in the soma’s plasma membranes of dPC12, live-cell in control state and after administration of 100 nM E2.