Human mammary epithelial cells exhibit a bimodal correlated random walk pattern

Abstract

Background: Organisms, at scales ranging from unicellular to mammals, have been known to exhibit foraging behavior described by random walks whose segments confirm to Lévy or exponential distributions. For the first time, we present evidence that single cells (mammary epithelial cells) that exist in multi-cellular organisms (humans) follow a bimodal correlated random walk (BCRW).

Methodology/principal findings: Cellular tracks of MCF-10A pBabe, neuN and neuT random migration on 2-D plastic substrates, analyzed using bimodal analysis, were found to reveal the BCRW pattern. We find two types of exponentially distributed correlated flights (corresponding to what we refer to as the directional and re-orientation phases) each having its own correlation between move step-lengths within flights. The exponential distribution of flight lengths was confirmed using different analysis methods (logarithmic binning with normalization, survival frequency plots and maximum likelihood estimation).

Conclusions/significance: Because of the presence of non-uniform turn angle distribution of move step-lengths within a flight and two different types of flights, we propose that the epithelial random walk is a BCRW comprising of two alternating modes with varying degree of correlations, rather than a simple persistent random walk. A BCRW model rather than a simple persistent random walk correctly matches the super-diffusivity in the cell migration paths as indicated by simulations based on the BCRW model.

Figure 1. Flagging directional and re-orientation flights using bimodal analysis .

An experimental 2-hour neuT (top) and pBabe (bottom) cell trajectory with the directional and re-orientation flights flagged using bimodal analysis. The cell track starts at origin (0,0) with the start of a directional flight indicated by an open circle and the start of re-orientation denoted by a filled triangle. A directional flight length is the summation of all the consecutive move step-lengths during the directional phase and similarly, a re-orientation flight length is the summation of a series of all the move step-lengths during the re-orientation phase. The net flight length (directional/re-orientation) refers to the net displacement from start to end during the flight.

Figure 2. Net flight length during the directional and re-orientation phases for the three cell types.

Box plots of the mean net flight lengths during the directional () and re-orientation () phases for the three cell types (pBabe ( = 15), neuN ( = 15), neuT ( = 12)). The distance traversed in directional flights is more than during re-orientation flights (statistical analysis in Table 1).

Figure 3. Survival frequency (log-linear) plots for the three cell types in different modes.

3a (left panel), the flight length survival frequency plots, filled circles (directional mode) and open circles (re-orientation mode). 3b (right panel), the net flight length survival frequency plots, filled circles (directional mode) and open circles (re-orientation mode). The straight-line behavior on the log-linear plots (survival frequency on log scale versus the lengths (flight/net flight) on linear scale) is indicative of exponential distribution of the lengths. The slopes ( for exponential distribution) along with statistical analysis are shown in Table 2.

Figure 4. Log-log frequency plots using the logarithmic binning with normalization method along with a fitted exponential function.

The logarithmically binned flight length distributions on log-log scale for the three cell types. The directional flight lengths are shown in the left panel while re-orientation flight lengths are on the right. An exponential distribution fitted to the (obtained from corresponding survival distribution) is shown in bold curve in black. The fitted exponential distribution is in good agreement with the experimental data points.

Figure 5. Probability distributions of the turn angles within the directional and re-orientation flights.

Top, pBabe, middle, neuN and bottom, neuT cells. The solid line shows the turn angle distribution during directional flights while the broken lines during re-orientation flight. The directional flights display higher persistence compared to the re-orientation flights that have a more flatter turn angle distribution.

Figure 6. Super-diffusive behavior in mean-squared displacement trends.

Simulated mean-squared displacement versus time from a simulation based on BCRW model (blue) compared to the experimental neuN data (red). The bimodal correlation contributes to prolonged super-diffusivity (high persistence) observed in epithelial cells under consideration (neuN cell type). The “*” indicates transition to the diffusive regime in the BCRW model. A fit of the experimental data using a PRW model (green) has been overlaid. Inset: Comparison of BCRW and PRW model predictions with the experimental mean-squared displacement. The squared relative difference error (difference normalized using the experimental mean-squared displacement at a given time) for predictions from BCRW and PRW model. The BCRW model predictions are in good agreement with the experiments.