Anomalous dynamics of cell migration

Abstract

Cell movement-for example, during embryogenesis or tumor metastasis-is a complex dynamical process resulting from an intricate interplay of multiple components of the cellular migration machinery. At first sight, the paths of migrating cells resemble those of thermally driven Brownian particles. However, cell migration is an active biological process putting a characterization in terms of normal Brownian motion into question. By analyzing the trajectories of wild-type and mutated epithelial (transformed Madin-Darby canine kidney) cells, we show experimentally that anomalous dynamics characterizes cell migration. A superdiffusive increase of the mean squared displacement, non-Gaussian spatial probability distributions, and power-law decays of the velocity autocorrelations is the basis for this interpretation. Almost all results can be explained with a fractional Klein-Kramers equation allowing the quantitative classification of cell migration by a few parameters. Thereby, it discloses the influence and relative importance of individual components of the cellular migration apparatus to the behavior of the cell as a whole. data analysis ͉ fractional dynamics ͉ non-Brownian motion N early all cells in the human body are mobile at a given time during their life cycle. Embryogenesis, wound-healing, immune defense, and the formation of tumor metastases are well known phenomena that rely on cell migration. Extensive experimental work revealed a precise spatial and temporal coordination of multiple components of the cellular migration machinery such as the actin cytoskeleton, cell-substrate and cell-cell interactions, and the activity of ion channels and transporters (1-4). These findings are the basis for detailed molecular models representing different microscopic aspects of the process of cell migration such as the protrusion of the leading edge of the lamellipodium, or actin dynamics (5). Mathematical continuum models, in contrast, focus on collective properties of the entire cell to explain requirements for the onset of motion and some typical features of cell motility (6). These models are usually limited to small spatiotemporal scales. Therefore, they provide little information about how the integration of protrusion of the lamellipodium, retraction of the rear part, and force transduction onto the extracellular matrix lead to the sustained long-term movement of the entire cell. This process is characterized by alternating phases of directed migration, changes of direction, and polarization. The coordinated interaction of these phases suggests the existence of intermittency (7) and of strong spatiotemporal correlations. It is therefore an important question whether the long-term movement of the entire cell can still be understood as a simple diffusive behavior like usual Brownian motion (8, 9) or whether more advanced concepts of dynamic modeling have to be applied (10, 11). Results and DiscussionWe performed migration experiments and analyzed the trajectories of two migrating transformed renal epithelial MadinDar...

Anomalous Fluctuation Relations

“…1.6; see Ref. [43] for full details of the experiments. According to the Langevin description of Brownian motion outlined in Section 1.2.3, Brownian motion is characterized by a MSD σ 2 x,0 (t) ∼ t (t → ∞) designating normal diffusion.…”

“…[47,48,49,50]. Examples of applications for this type of stochastic modeling are given by generalized elastic models [66], polymer dynamics [67] and biological cell migration [43]. We now split this class into two specific cases.…”

“…Fig. 1.6 depicts the path of a single biological cell crawling on a substrate measured in an in vitro experiment [43]. At first sight, the path looks like the trajectory of a Brownian particle generated, e.g., by the ordinary Langevin dynamics of Eq.…”

“…Paradigmatic examples are diffusion processes where the long-time mean square displacement does not grow linearly in time: That is, x 2 ∼ t α , where the angular brackets denote an ensemble average, does not increase with α = 1 as expected for Brownian motion but either subdiffusively with α < 1 or superdiffusively with α > 1 [36,37,38]. After pioneering work on amorphous semiconductors [39], anomalous transport phenomena have more recently been observed in a wide variety of complex systems, such as plasmas [40], nanopores [41], epidemic spreading [42], biological cell migration [43] and glassy materials [44], to mention a few [45,46]. This raises the question to which extent conventional FRs are valid for anomalous dynamics.…”

“…The cell frequently changes its shape and direction during migration, as is shown by several cell contours extracted during the migration process. The inset displays phase contrast images of the cell at the beginning and to the end of its migration process [43].…”