Polar patterns of driven filaments

Abstract

The emergence of collective motion exhibited by systems ranging from flocks of animals to self-propelled microorganisms to the cytoskeleton is a ubiquitous and fascinating self-organization phenomenon1,2,3,4,5,6,7,8,9,10,11,12. Similarities between these systems, such as the inherent polarity of the constituents, a density-dependent transition to ordered phases or the existence of very large density fluctuations13,14,15,16, suggest universal principles underlying pattern formation. This idea is followed by theoretical models at all levels of description: micro- or mesoscopic models directly map local forces and interactions using only a few, preferably simple, interaction rules12,17,18,19,20,21, and more macroscopic approaches in the hydrodynamic limit rely on the systems’ generic symmetries8,22,23. All these models characteristically have a broad parameter space with a manifold of possible patterns, most of which have not yet been experimentally verified. The complexity of interactions and the limited parameter control of existing experimental systems are major obstacles to our understanding of the underlying ordering principles13. Here we demonstrate the emergence of collective motion in a high-density motility assay that consists of highly concentrated actin filaments propelled by immobilized molecular motors in a planar geometry. Above a critical density, the filaments self-organize to form coherently moving structures with persistent density modulations, such as clusters, swirls and interconnected bands. These polar nematic structures are long lived and can span length scales orders of magnitudes larger than their constituents. Our experimental approach, which offers control of all relevant system parameters, complemented by agent-based simulations, allows backtracking of the assembly and disassembly pathways to the underlying local interactions. We identify weak and local alignment interactions to be essential for the observed formation of patterns and their dynamics. The presented minimal polar-pattern-forming system may thus provide new insight into emerging order in the broad class of active fluids8,23,24 and self-propelled particles17,25.

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Figure 1: Schematic of the high-density motility assay.
Figure 2: Phase behaviour as a function of the filament density.
Figure 3: Swirling pattern of coherently moving filaments.
Figure 4: Directional persistence and instability mechanisms.
Figure 5: Cellular automaton simulations.

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Acknowledgements

We thank A. Baskaran and C. Marchetti for discussions. Financial support from the DFG in the framework of the SFB 863 and the German Excellence Initiatives via the ‘Nano-Initiative Munich (NIM)’ and the Technische Universität München - Institute for Advanced Study is gratefully acknowledged. V.S. and C.W. acknowledge support from the Elite Network of Bavaria by the graduate programmes CompInt and NanoBioTechnology.

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Authors

Contributions

A.R.B., C.S. and V.S. conceived and performed the experiments. C.W. and E.F. conceived and designed the simulations. C.W. performed and analysed the simulations. A.R.B., C.S., V.S., C.W. and E.F. interpreted the data. A.R.B., V.S., C.W. and E.F. wrote the paper.

Corresponding author

Correspondence to Andreas R. Bausch.

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The authors declare no competing financial interests.

Supplementary information

Supplementary Information

This file contains Supplementary Materials and Methods, Supplementary Figures 1-3 with legends and additional References. (PDF 472 kb)

Supplementary Movie 1

This movie shows a homogeneous cluster moving in a dilute and disordered background right above the critical filament density pc (p = 5.5 uµ-2, labeling ratio 1:200, filament length ~10 µm, 40x objective). (MOV 4191 kb)

Supplementary Movie 2

This movie shows density waves observed at a filament density of pm-2 for two different magnifications (first part: 40x objective, second part: 100x objective). (labeling ratio 1:320, filament length ~10 µm). (MOV 4062 kb)

Supplementary Movie 3

This movie shows swirling pattern and movement of the central region of the swirl (pm-2, labeling ratio 1:320, filament length ~10 µm, 40x objective). (MOV 4356 kb)

Supplementary Movie 4

This movie shows a close up of the central region of a typical swirling pattern (pm-2, labeling ratio 1:320, filament length ~10 µm, 100x objective). (MOV 4864 kb)

Supplementary Movie 5

This movie, which is in 2 parts, shows oscillatory orientational bending of coherently moving filaments (first part) and the gradually developing splay pattern (second part) (pm-2, labeling ratio 1:320, filament length ~10 µm, 100x objective). (MOV 3858 kb)

Supplementary Movie 6

This movie shows the emergence of patterns in the cellular automaton simulations as observed in the entire simulation box with reflective boundary conditions (r = 83%, ς= 10, α = 10, simulation time 0 – 5000, ‘labeling ratio’ 1:40). (MOV 4000 kb)

Supplementary Movie 7

This movie shows a close up on an emergent polar nematic pattern in a region of observation far away from the boundaries (r = 83%, ς= 10, α= 10, simulation time 0 – 5000, ‘labeling ratio’ 1:40). (MOV 3835 kb)

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Schaller, V., Weber, C., Semmrich, C. et al. Polar patterns of driven filaments. Nature 467, 73–77 (2010). https://doi.org/10.1038/nature09312

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