### On multistability and constitutive relations of cell motion on fibronectin lanes

Behnam Amiri, Johannes C.J. Heyn, Christoph Schreiber, Joachim O. Rädler, Martin Falcke

Biophysical Journal (2023)

Cell motility on flat substrates exhibits coexisting steady and oscillatory morphodynamics, the biphasic adhesion-velocity relation, and the universal correlation between speed and persistence (UCSP) as simultaneous observations common to many cell types. Their universality and concurrency suggest a unifying mechanism causing all three of them. Stick-slip models for cells on one-dimensional lanes suggest multistability to arise from the nonlinear friction of retrograde flow. This study suggests a mechanical mechanism controlled by integrin signaling on the basis of a biophysical model and analysis of trajectories of MDA-MB-231 cells on fibronectin lanes, which additionally explains the constitutive relations. The experiments exhibit cells with steady or oscillatory morphodynamics and either spread or moving with spontaneous transitions between the dynamic regimes, spread and moving, and spontaneous direction reversals. Our biophysical model is based on the force balance at the protrusion edge, the noisy clutch of retrograde flow, and a response function of friction and membrane drag to integrin signaling. The theory reproduces the experimentally observed cell states, characteristics of oscillations, and state probabilities. Analysis of experiments with the biophysical model establishes a stick-slip oscillation mechanism, and explains multistability of cell states and the statistics of state transitions. It suggests protrusion competition to cause direction reversal events, the statistics of which explain the UCSP. The effect of integrin signaling on drag and friction explains the adhesion-velocity relation and cell behavior at fibronectin density steps. The dynamics of our mechanism are nonlinear flow mechanics driven by F-actin polymerization and shaped by the noisy clutch of retrograde flow friction, protrusion competition via membrane tension, and drag forces. Integrin signaling controls the parameters of the mechanical system.

### On the adhesion–velocity relation and length adaptation of motile cells on stepped fibronectin lanes

C. Schreiber, B. Amiri, J. C. J. Heyn, J. O. Rädler and M. Falcke

PNAS (2021)

Cells exert forces on their environment by contracting actin networks, friction of intracellular F-actin flow, and polymerization when they move, e.g., during tumor metastasis or development. In this context, the relation between adhesion and cell velocity is a general cell-type-independent observation, the investigation of which bears the chance of understanding basic mechanisms. Restricting cell motion to one-dimensional lanes simplifies the problem and allows for comparison to mathematical models. Polymerization at the cell’s leading edge drives F-actin network flow and pushes the membrane. The drag of detaching the cell, the membrane, and the cell body resist motion. Since only velocity-controlled forces shape motion, cells can move even across highly adhesive areas without getting stuck.

### Area and Geometry Dependence of Cell Migration in Asymmetric Two-State Micropatterns

A. Fink, D. B. Brückner, C. Schreiber, P. J. F. Röttgermann, C. P. Broedersz, J. O. Rädler

Biophysical Journal (2020)

Microstructured surfaces provide a unique framework to probe cell migration and cytoskeletal dynamics in a standardized manner. Here, we report on the steady-state occupancy probability of cells in asymmetric two-state microstructures that consist of two fibronectin-coated adhesion sites connected by a thin guidance cue. We study the dynamics of human breast carcinoma cells (MDA-MB-231) in microstructures as a function of area, shape, and orientation of the adhesion sites. On square adhesive sites with different areas, we find that the occupancy probability ratio is directly proportional to the ratio of corresponding adhesion site areas. Sites of equal area but different shape lead to equal occupancy if shapes are isotropic (e.g., squared or circular). In contrast, an asymmetry in the occupancy is induced by anisotropic shapes like rhombi, triangles, or rectangles that enable motion in the direction perpendicular to the transition axis.

### Stochastic Nonlinear Dynamics of Confined Cell Migration in Two-State Systems

D.B. Brückner, A. Fink, C. Schreiber, P. J. F. Röttgermann, J. O. Rädler and C. P. Broedersz

Nature Physics (2019)

Migrating cells in physiological processes, including development, homeostasis and cancer, encounter structured environments and are forced to overcome physical obstacles. Yet, the dynamics of confined cell migration remains poorly understood, and thus there is a need to study the complex motility of cells in controlled confining microenvironments. Here, we develop two-state micropatterns, consisting of two adhesive sites connected by a thin constriction, in which migrating cells perform repeated stochastic transitions. This minimal system enables us to obtain a large ensemble of single-cell trajectories. From these trajectories, we infer an equation of cell motion, which decomposes the dynamics into deterministic and stochastic contributions in position–velocity phase space. Our results reveal that cells in two-state micropatterns exhibit intricate nonlinear migratory dynamics, with qualitatively similar features for a cancerous (MDA-MB-231) and a non-cancerous (MCF10A) cell line. In both cases, the cells drive themselves deterministically into the thin constriction; a process that is sped up by noise. Interestingly, however, these two cell lines have distinct deterministic dynamics: MDA-MB-231 cells exhibit a limit cycle, while MCF10A cells show excitable bistable dynamics. Our approach yields a conceptual framework that may be extended to understand cell migra-tion in more complex confining environments.

### Ring-Shaped Microlanes and Chemical Barriers as a Platform for Probing Single-Cell Migration

C. Schreiber, F. J. Segerer, E. Wagner, A. Roidl & J. O. Rädler

Scientific Reports (2016)

Quantification and discrimination of pharmaceutical and disease-related effects on cell migration requires detailed characterization of single-cell motility. In this context, micropatterned substrates that constrain cells within defined geometries facilitate quantitative readout of locomotion. Here, we study quasi-one-dimensional cell migration in ring-shaped microlanes. We observe bimodal behavior in form of alternating states of directional migration (run state) and reorientation (rest state). Both states show exponential lifetime distributions with characteristic persistence times, which, together with the cell velocity in the run state, provide a set of parameters that succinctly describe cell motion. By introducing PEGylated barriers of different widths into the lane, we extend this description by quantifying the effects of abrupt changes in substrate chemistry on migrating cells. The transit probability decreases exponentially as a function of barrier width, thus specifying a characteristic penetration depth of the leading lamellipodia. Applying this fingerprint-like characterization of cell motion, we compare different cell lines, and demonstrate that the cancer drug candidate salinomycin affects transit probability and resting time, but not run time or run velocity. Hence, the presented assay allows to assess multiple migration-related parameters, permits detailed characterization of cell motility, and has potential applications in cell biology and advanced drug screening.

### Emergence and Persistence of Collective Cell Migration on Small Circular Micropatterns

F. J. Segerer, F. Thüroff, A. Piera Alberola, E. Frey, J.O. Rädler

Physical Review Letters (2015)

The spontaneous formation of vortices is a hallmark of collective cellular activity. Here, we study the onset and persistence of coherent angular motion as a function of the number of cells N confined in circular micropatterns. We find that the persistence of coherent angular motion increases with N but exhibits a pronounced discontinuity accompanied by a geometric rearrangement of cells to a configuration containing a central cell. Computer simulations based on a generalized Potts model reproduce the emergence of vortex states and show in agreement with experiment that their stability depends on the interplay of the spatial arrangement and internal polarization of neighboring cells. Hence, the distinct migrational states in finite size ensembles reveal significant insight into the local interaction rules guiding collective migration.

### Flow and diffusion in channel-guided cell migration

Marel, A.-K., Zorn, M., Klingner, C., Wedlich-Söldner, R., Frey, E., and Rädler, J. O.

Biophysical Journal (2014)

In confluent epithelial sheets, the dynamics have been found to be highly heterogeneous, exhibiting spontaneous formation of swirls, long-range correlations, and glass-like dynamic arrest as a function of cell density. In contrast, the flow-like properties of one-sided cell-sheet expansion in confining geometries are not well understood. Here, we studied the short- and long-term flow of Madin-Darby canine kidney (MDCK) cells as they moved through microchannels. Using single-cell tracking and particle image velocimetry (PIV), we found that a defined averaged stationary cell current emerged that exhibited a velocity gradient in the direction of migration and a plug-flow-like profile across the advancing sheet. The observed flow velocity can be decomposed into a constant term of directed cell migration and a diffusion-like contribution that increases with density gradient. The diffusive component is consistent with the cell-density profile and front propagation speed predicted by the Fisher-Kolmogorov equation. Our work thus suggests that active cell migration manifests itself in an underlying, spatially uniform drift as well as in randomized bursts of short-range correlated motion that lead to a diffusion-mediated transport.

- M. Dietrich, H. Le Roy, D. B. Brückner, H. Engelke, R. Zantl, J. O. Rädler, and C. P. Broedersz

Guiding 3D Cell Migration in Deformed Synthetic Hydrogel Microstructures, Soft Matter (2018), 15 - Zorn, M. L., Marel, A.-K., Segerer, F. J., Rädler, J. O.

Phenomenological approaches to collective behavior in epithelial cell migration

Biochimica et Biophysica Acta (BBA) - Molecular Cell Research, (2015) - A.-K. Marel, A. Piera Alberola, J. O. Rädler

Proliferation and Collective Migration of Small Cell Groups Released from Circular Patches

Biophysical Reviews and Letters Vol. 7, Nos. 1 & 2, 15–28, (2012)