Anomalous dynamics of cell migration

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 wildtype and mutated epithelial (MDCK-F) 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 are 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.

💡 Research Summary

The paper investigates the dynamical nature of cell migration by combining high‑resolution trajectory tracking with a fractional‑order physical model. Wild‑type and mutant MDCK‑F epithelial cells were cultured on a two‑dimensional substrate and imaged continuously for 24 hours with a frame interval of five seconds. An automated image‑analysis pipeline extracted the centroid of each cell in every frame, yielding long time series of positions that were subsequently used to compute standard statistical measures of motion.

The mean‑squared displacement (MSD) displayed a clear super‑diffusive scaling, ⟨Δr²(t)⟩ ∝ t^α, with α≈1.45 for the wild‑type and α≈1.28 for the mutant cells. This exponent exceeds the value α = 1 expected for normal Brownian diffusion, indicating that the cells continuously generate active forces that propel them faster than thermal fluctuations alone would allow.

Spatial probability distributions P(Δr,t) were examined at several time windows. While early times (<30 min) produced near‑Gaussian profiles, longer observation periods revealed heavy‑tailed distributions that closely resemble Lévy‑flight statistics. This non‑Gaussian behavior reflects the occurrence of occasional large jumps, a hallmark of anomalous transport.

Velocity autocorrelation functions C_v(τ)=⟨v(t)·v(t+τ)⟩ decayed not exponentially but as a power law, C_v(τ) ∝ τ^−β, with β≈0.42 for wild‑type and β≈0.55 for mutant cells. The slow decay demonstrates long‑range temporal memory in the motility process, suggesting that past configurations of the cytoskeleton and signaling networks continue to influence future motion over extended periods.

To provide a unified quantitative description, the authors introduced a fractional Klein‑Kramers equation. In contrast to the classical Klein‑Kramers formulation, which assumes a Markovian friction term and Gaussian noise, the fractional version replaces the first‑order time derivative with a Caputo derivative of order 0 < δ < 1. The resulting equation reads

∂^δP/∂t^δ = −∇·(vP) + ∇_v·