Elucidating distinct ion channel populations on the surface of hippocampal neurons via single-particle tracking recurrence analysis

Protein and lipid nanodomains are prevalent on the surface of mammalian cells. In particular, it has been recently recognized that ion channels assemble into surface nanoclusters in the soma of cultured neurons. However, the interactions of these molecules with surface nanodomains display a considerable degree of heterogeneity. Here, we investigate this heterogeneity and develop statistical tools based on the recurrence of individual trajectories to identify subpopulations within ion channels in the neuronal surface. We specifically study the dynamics of the ${\mathrm{K}}^{+}$ channel Kv1.4 and the ${\mathrm{Na}}^{+}$ channel Nav1.6 on the surface of cultured hippocampal neurons at the single-molecule level. We find that both these molecules are expressed in two different forms with distinct kinetics with regards to surface interactions, emphasizing the complex proteomic landscape of the neuronal surface. Further, the tools presented in this work provide new methods for the analysis of membrane nanodomains, transient confinement, and identification of populations within single-particle trajectories.

Figure 1

Method for identification of confined motion based on number of visits to current site. The method is validated using numerical simulations. (a) FBM with $\alpha =0.1$, i.e., Hurst exponent $H=0.05$. At each successive points pair, such as those marked in red, a circle is drawn as indicated in the figure. In this case there are 17 points (visits) found within the circular area. (b) FBM with $\alpha =0.9$. Here the tracer visits the region within the selected circle only once. (c) $y$ coordinate of two-dimensional intermittent FBM with $\alpha =0.1$ (first, third, and fifth segment) and $\alpha =0.9$ (second and fourth segment). The vertical dashed lines show the true switching points. The parts of the trajectories identified as confined (low $\alpha$) are marked in red. The two-dimensional trajectories are presented on the right. (d) $V_j$ statistic, i.e., number of recurrent visits, as a function of time. Again, the vertical dashed lines show the true switching points. The threshold for finding confined regions in this example is $V_{\mathrm{th}}=6$.

Figure 2

Segmentation of Kv1.4 trajectories. (a) 92 individual Kv1.4 trajectories in one hippocampal neuron. (b) Zoom of the trajectory indicated with a thin black arrow. The periods of confinement within this trajectory are shown in red. (c) Histogram of the number of visits to current site, $V_j$ (117,195 sites; 649 trajectories). As explained in the text, the sites are determined by the exploration in 150 ms. This metric is used to evaluate how compact the random walk is and to identify regions of confinement. (d) Time series of number of visits $V_j$ for the trajectory in panel (b). When the number of visits is above the threshold $V_{\mathrm{th}}=11$, the particle is considered to be in the confined state. At 4.6 s, $V_j$ drops below the threshold during three points in the time series. However, because the dwell time is not longer than three points, i.e., a single time window, this region is considered to remain in the confined state. This region is indicated by a thick yellow arrow in panel (b). (e, f) time series of localization along $x$ and $y$ for the trajectory shown in panel (b). The regions that are detected to be confined are shown in red and the identified switching point marked by dashed vertical line.

Figure 3

Determination of number of distinct classes among Kv1.4 trajectories (649 trajectories, 9 cells). (a) Histogram of fraction of the observation time that a particle spends in the confined state. (b) Regime variance test statistic $C_k={\sum }_{i=1}^k{\varphi }_i^2$. The regime variance test rejects the hypothesis of a single regime with $p={10}^{-3}$. The inset shows the regime variance test applied to 500 realizations of numerical simulations of the type shown in Fig. 1. In these simulations, the test does not reject the hypothesis of a single class of trajectories ($p=0.29$). (c, d) Examples of Kv1.4 trajectories in the two distinct classes. The fraction $\varphi$ is given for each example. (e) Complementary cumulative distribution function of the residence times in confined states for each of the two classes (Class 1: 1 659 sojourn times, 575 trajectories; Class 2: 575 sojourn times, 74 trajectories).

Figure 4

Confining regions are the same for the Kv1.4 channels in both classes. (a) Cumulative distribution function and (b) box plots for the confining regions area for particles in both classes. Box plots show 5–95% quantiles as whiskers together with quartiles and median (Class 1: 1 165 domains; Class 2: 113 domains). The inset in panel (a) shows the characterization of the confining domain in a sample trajectory using the radii of gyration along the principal axes. The first 20 points of the confining domain are employed to find the radii of gyration. The obtained confining ellipse is shown together with the principal axes. (c) Cumulative distribution function and (d) box plots for the confining major semiaxes.

Figure 5

Nav1.6 channels in the soma of hippocampal neurons (386 trajectories, 4 cells). (a) Trajectories of Nav1.6 in one example cell (79 trajectories). (b) Residence times in confining regions for the two classes found using the silhouette method (Class 1: 915 times, 307 trajectories; Class 2: 706 times, 79 trajectories). (c) Cumulative distribution function (CDF) and (d) box plots of the confining regions area for particles in both classes. Box plots show 5–95% quantiles as whiskers together with quartiles and median (Class 1: 605 domains; Class 2: 171 domains). (e) CDF and (f) box plots of the confining major semiaxes.