Human Mammary Epithelial Cells Exhibit a Bimodal Correlated Random Walk Pattern Alka A. Potdar 1,2 , Junhwan Jeon 1,2 , Alissa M. Weaver 2,3 , Vito Quaranta 2,3 , Peter T. Cummings 1,2,4 * 1 Department of Chemical and Biomolecular Engineering, Vanderbilt University, Nashville, Tennessee, United States of America, 2 Vanderbilt Integrative Cancer Biology Center, Nashville, Tennessee, United States of America, 3 Department of Cancer Biology, Vanderbilt University Medical Center, Nashville, Tennessee, United States of America, 4 Center for Nanophase Materials Sciences, Oak Ridge National Laboratory, Oak Ridge, Tennessee, United States of America 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 Le ´vy 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. Citation: Potdar AA, Jeon J, Weaver AM, Quaranta V, Cummings PT (2010) Human Mammary Epithelial Cells Exhibit a Bimodal Correlated Random Walk Pattern. PLoS ONE 5(3): e9636. doi:10.1371/journal.pone.0009636 Editor: Jo ¨ rg Langowski, German Cancer Research Center, Germany Received September 18, 2009; Accepted February 14, 2010; Published March 10, 2010 Copyright: ß 2010 Potdar et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Funding: This study was funded by National Cancer Institute grant: Multiscale Mathematical Modeling of Cancer Invasion (grant number: 5U54CA113007-02). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. Competing Interests: The authors have declared that no competing interests exist. * E-mail: peter.cummings@vanderbilt.edu Introduction Cell migration is an important process for a wide range of domains from bacteria to mammals. For prokaryotes (e.g., bacteria), migration is important to locate food sources [1]. Similar goals may apply to unicellular eukaryotes (e.g., Dictyoste- lium). In contrast, in ‘‘higher’’ multi-cellular eukaryotes (e.g., mammals) cell migration is involved in physiological processes as well as pathogenic conditions such as cancer metastasis [2]. Mammalian cell migration is generally thought of as having a ‘‘purpose’’ (such as embryogenesis [3] and immune response [4]) other than locating nutrients and it is believed that these cells follow orders from a higher ‘‘programming center’’. When these orders are misinterpreted or disregarded, cancer may occur. This paper, instead, is informed by a different premise: namely, that individual cell migration in random motility conditions can be interpreted as a problem of how to efficiently perform a search to locate randomly distributed ‘‘target items’’ (such as nutrients and growth factors) which could only be detected in limited vicinity. This is analogous to animal foraging problem where animals come to adopt an optimal search strategy to locate food. Random walk theories have been used to model animal displacements to explain optimal foraging, predator-prey relation- ships, etc. For long-times, animal movements can be modeled as uncorrelated random walks with normal diffusion (mean-squared displacement (MSD), v ~ r(t){ ~ r(0) ð Þ 2 w scaling as t a where, a~1) [5,6], where ~ r(t) is the position of the animal at time t and the average (,.) is over all the members of the population. Anomalous diffusion arises when a=1, with av1 corresponding to sub-diffusive motion, aw1 to super-diffusive motion and ‘ballistic motion’ for the case of a~2. The directional persistence in animal movements has been modeled using correlated random walks or Le ´vy motion [7]. Correlated random walks have an exponentially decreasing distribution of move step-lengths (dis- tance traveled in one sampling time) [8] and the shape of the turn angle distribution between these move step-lengths controls the directional memory. Le ´vy motion (Le ´vy walks or Le ´vy flights [9,10,11] where Le ´vy walk has a finite mean-squared displacement while a Le ´vy flight does not) comprises of random walks wherein long flights or steps are separated by shorter jumps. These walks are described by the power-law probability distribution function for the flight or step-length l , given by P(l )&l {m , 1vmv3 where PLoS ONE | www.plosone.org 1 March 2010 | Volume 5 | Issue 3 | e9636