Insights into Cell Membrane Microdomain Organization from Live Cell Single Particle Tracking of the IgE High Affinity Receptor FcϵRI of Mast Cells

Abstract

Current models propose that the plasma membrane of animal cells is composed of heterogeneous and dynamic microdomains known variously as cytoskeletal corrals, lipid rafts and protein islands. Much of the experimental evidence for these membrane compartments is indirect. Recently, live cell single particle tracking studies using quantum dot-labeled IgE bound to its high affinity receptor FcϵRI, provided direct evidence for the confinement of receptors within micrometer-scale cytoskeletal corrals.

In this study, we show that an innovative time-series analysis of single particle tracking data for the high affinity IgE receptor, FcϵRI, on mast cells provides substantial quantitative information about the submicrometer organization of the membrane. The analysis focuses on the probability distribution function of the lengths of the jumps in the positions of the quantum dots labeling individual IgE FcϵRI complexes between frames in movies of their motion. Our results demonstrate the presence, within the micrometer-scale cytoskeletal corrals, of smaller subdomains that provide an additional level of receptor confinement. There is no characteristic size for these subdomains; their size varies smoothly from a few tens of nanometers to a over a hundred nanometers.

In QD-IGE labeled unstimulated cells, jumps of less than 70 nm predominate over longer jumps. Addition of multivalent antigen to crosslink the QD-IgE-FcϵRI complexes causes a rapid slowing of receptor motion followed by a long tail of mostly jumps less than 70 nm. The reduced receptor mobility likely reflects both the membrane heterogeneity revealed by the confined motion of the monomeric receptor complexes and the antigen-induced cross linking of these complexes into dimers and higher oligomers. In both cases, the probability distribution of the jump lengths is well fit, from 10 nm to over 100 nm, by a novel power law. The fit for short jumps suggests that the motion of the quantum dots can be modeled as diffusion in a fractal space of dimension less than two.

Appendices

Appendix A: Examples of Long Paths and Segments

In this appendix, we extend the information about the longest paths introduced in Sect. 2. We first show the longest paths, then the longest segments, and then replot these tracks and color code the jumps by length. The main point is to show how highly variable the paths are.

In Table 12, we summarize some information about longest paths. The parameters MDX (MaxDistX) and MDY (MaxDistY) give the size in nm of the smallest rectangle that contains the path. Figures 13 and 14 show the longest path for each stimulus for the two data sets. The plot notation is the same as that used in Fig. 2. Recall that a path is made up of segments where the QD is on, separated by segments where the QD is off, and consequently these figures also illustrate the blinking of the QDs. Note that the QDs can be on or off for a long time.

Fig. 13

Data set A: tracks with the longest paths

Fig. 14

Data set B: tracks with the longest paths

Table 12 Paths, identified by the stimulus added and track number, with the largest number of time steps (nts) for the two data sets, A and B. MDX (MaxDistX) and MDY (MaxDistY) are the lengths of the sides of the smallest rectangle containing the path

Figures 15 and 16 show longest segment for each stimulus. Recall that the QD is on for all of the points in a segment. Each segment is enclosed by its convex hull. The area of the convex hull is written in the form R ², so R is the length of the side of a square with the same area.

Fig. 15

The longest segments for data set A

Fig. 16

The longest segments for data set B

In Figs. 17 and 18, we use different colors for the jump sizes L: blue for 0≤L≤70 nm, green for 70<L≤190 nm, and red for 190<L≤346 nm. From these figures, we can see how the number of jumps less than or equal to 70 nm (blue) increases as the stimulus increases, while there are very few of the longest jumps.

Fig. 17

The longest segments and their different jump lengths for data set A

Fig. 18

The longest segments and their different jump lengths for data set B

Appendix B: QD Blinking Times

It is important for our analysis to understand that, due to the blinking of the QDs, very few QDs are on at any given time step. The minimum, mean, and maximum of the dots that are on at any given time are given in Table 2. Because only a few QDs are on at a given time, statistics that are a function of time will be noisy. The lengths of the QD on times are strongly dependent on the algorithm that connects dots at successive time steps, while the off times are strongly dependent on the algorithm that connects runs of on times. It is known that the on and off times of the QDs satisfy a power law with a negative exponent of approximately 3/2 (Bachir 2006), so we fit the function that gives the number of on and off times of a given length by

$$\text{on}(i) = q_1 i^{-p_1} ,\quad\text{off}(i) = q_2 i^{-p_2}. $$

It is not expected that the on times will be affected by the stimulus, but it is possible that the off times are affected as the motion is slower and thus it may be easier to estimate where a dot that has turned off will turn on. The power p ₁ for the on times is approximately 4/3 for all conditions (see Table 13). This is somewhat slower than that given in (Bachir 2006). The power p ₂ for off times is approximately 1 (see Table 13). Note that it is common to divide the on or off times by the total number of times so that one obtains a probability distribution. However, for this to work, the power must be greater than 1. In our case, the number of off times is bounded by the number of time steps in a track, so with a power less than 1, we still get a probability distribution.

Table 13 The power-law decay for the on and off times of the QDs for data sets A and B

In the off times data, the off times are zero after 32 time steps. This is caused by the path construction algorithm, which connects segments of on times, having a limit of 32 off times in between the on times. Because of this, the power-law fit was made using the first 32 data points.

Appendix C: Problem with Large Jumps

During our analysis, we noticed a problem with the large jump lengths. We used the material in Sect. 3.6 with 500 bins to estimate the PDF of the jump lengths and display these in Fig. 19. We see that the data analysis algorithms that construct the paths introduce a dramatic reduction in the number of jumps at 346 nm. Therefore, in our analysis we discarded all jumps bigger than 346 nm. This is a very small percentage of the total data: 2,069 jumps or less than 0.5 % of the data for data set A; and 1,668 jumps or less than 0.5 % of the data for data set B.

Fig. 19

PDFs of all the jump lengths for the data sets A and B

Appendix D: Analysis Details for the Tail Data

In this appendix, we provide some of the details about the tail data that was not discussed in Sect. 5. We begin by showing that the tail jump components can be modeled as mean zero. We next compute the autocorrelations coefficients to support modeling the jumps as independent. Thus, as with the unstimulated data, we will model the tail jump data as IID and mean zero and estimate the statistics for the data in the tails using the formulas in Sect. 3.4. Next, we give more details concerning the PDFs of the tail jump components and the jump angles.

In Table 14, we expand on information about the statistics of the tail jump components first presented in Table 9. Note that the dimensionless means μ/σ are small, which justifies modeling the tail data as mean zero. Also, note that the standard deviations σ _x and σ _y decrease with increasing stimulus except for the slight increase at stimulus 10 μg/ml for both data sets.

Table 14 Tail data sets A and B: stimulus s, mean, standard deviation and mean zero test for the x and y jump components of the PDFs shown in Figs. 20 and 21

Table 15 gives the first six autocorrelation coefficients for the tail data. We include the data for the unstimulated case for comparison. The second column is the stimulus s. The third column is the time in seconds when the tail data starts. The remaining columns are the autocorrelation coefficients. C ₀=1 because the coefficients are normalized by the square of the standard deviation (variance) and thus are dimensionless. For both the A and B datasets, C ₁ goes from positive to negative as the stimulus increases. For each k>1 the coefficients are essentially independent of the stimulus with the exception of one value in data set B with s=1.000 and C ₄=−0.009. With three exceptions, the coefficients slowly decrease in size with increasing k≥2. The behavior of C ₁ is quite different from the coefficients with k≥2.

Table 15 The autocorrelation coefficients, C _k (k=0,…,5), for the tails. Recall that C ₀=1.0 because the coefficients are normalized by the square of the standard deviation

In Figs. 20 and 21, we display the PDFs of the jump components in the tail data along with their fits by mean zero normal distributions. The PDFs of the jump components in the tails were found by dividing the interval −346≤Δx,Δy≤346 into 500 equal subintervals that were used to bin the components. The mean and standard deviation of the jumps were estimated using (18). The normal fit is given by a mean zero normal PDF with the same standard deviation as the data that are given in Table 14. As with the unstimulated data, the plots in these figures indicate that the PDFs are not normally distributed which is confirmed by the two-sample Kolmogorov–Smirnov test. In all cases, we observe that for approximately |x|,|y|<50 nm, there is an excess of short jumps compared to the normal distribution and that this excess is bigger than in the unstimulated data. For approximately 50<|x|,|y|<190 nm, there are fewer jumps than in a normal distribution. Note that a bound B on |x| and |y| corresponds to a bound of $\sqrt{2} B$ on the jump size r, so 50 nm in the components corresponds to approximately 70 nm for the total jump lengths.

Fig. 20

Distributions and their normal fits of the jump components in the tails of data set A

Fig. 21

Distributions and their normal fits of the jump components in the tails of data set B

To analyze the jump angles, as before, we divide [−π,π] into 500 bins and then bin the angles and compute their PDFs, which are displayed in Figs. 22 and 23. As with the unstimulated data the mean angle is 0.1592 as is true for the uniform distribution which is equal to π/2. The two-sample Kolmogorov–Smirnov test gives H of zero under all stimuli for a confidence value of 0.0001, so we cannot reject the null hypothesis that the angles are uniformly distributed. The asymptotic p-values shown in Table 16 confirm this. This is very strong support for modeling the angles as uniformly distributed.

Fig. 22

Data angles and generated random angles in the tails of data set A

Fig. 23

Data angles and generated random angles in the tails of data set B

Table 16 The p-values for the two-sample Kolmogorov–Smirnov test for the jump angles