A minimal physical model for a crawling cell

Theriot et. al., Plos Biology, 1001059 (2011)

E. Tjhung, A.T., D. Marenduzzo and M. Cates, submitted to Nat. Comm.

E. Tjhung, D. Marenduzzo, M. E. Cates

A. Tiribocchi

Bari Xmas Theory Workshop, 23rd December 2013

Motivations: the study of motile cells

✗ Crawling is one of the mechanism used by cells to acquire motility (further ways are swimming and grabbing)

✗ Crawling is important for many processes as wound healing, tissue development, immunological responses, cancer.

✗ The physics behind this phenomenon is still unclear

GrabbingSwimmingCrawling

E.L. Barnhart et al, Plos. Biol. 9(5)e1001059, (2011).

Poincloux et al, Plos. PNAS108, 1943 (2011).

J.C. Adams, Cell. Mol. Life Sc. 58, 371(2001).

Structure of an eukaryotic cell✔ Cytoskeleton: a dense network of protein filaments (actin) cross

linked with motor proteins (myosin)

How does a cell crawl?

E. L. Barnhart et al., PLoSBiology, 9(5):e1001059, 2011

✔ a quasi-2d ‘lamellipodium’ (packed with actin fibers) is pushed forward

✔ Motion triggered by actin polymerisation at the front of the cell and actomyosincontraction at the rear.

✔ Focal adhesions provide friction with the substrate and prevent the actinfilaments from slipping (‘no-slip’ BC)

Acto-myosin pairs generates a contractile stress due to the activity of the motor protein. At a coarse-grained level the process can be described as a force dipole [1].

[1] T. B. Liverpool, M. C. Marchetti (2008), Cell Motility, 7:177206.

Continuum model: Dynamic equations

( )[ ]

∂∂+∂+∂

c

FM=cwPvct δ

δααααα

treadmillingadvectiondiffusive flux

σαβactive=ζ c Pα Pβ

active stress

|P|2

E. Tjhung et al., PNAS 109, 12381 (2012).

Crawling cell in 2d

✔ A nonzero w is localized close to the substrate on which the droplet sits

small w: motion without protrusion

large w: motion with lamellipodium

(activity is not needed)

The formation of the protrusion can also be understood

as a non-equilibrium transition

W0

Crawling (shapes and dynamics) in 3d

Fried-egg shape

Pseudopodium

Phagocytic cup

Lamellipodium

P.T. Yam et al., J. Cell. Biol. 178, 1207-1221(2007).

Barnhart E. L. et al., Plos. Biol. 178, e1001059(2011).

Mercanti V. et al., J. Cell. Biol., 119, 4079 (2006).

Zhang H. et al., J. Cell. Sci., 115, 1733 (2002).

Crawling (shapes and dynamics) in 3d (2)

Polarization field Velocity field

Further dynamics in 3d

Temporal sequence of an oscillatory motion

B.A. Camley et al., PRL 111, 158102 (2013); E. Barnhart et al., Plos. Biol., 1001059 (2011).

Motion of a contractile cell on a substrate in absence of treadmilling and “soft” anchoring.

Further dynamics in 3d (2)

E. Tjhung et al., PNAS 109, 12381 (2012).

Conclusions

● Variation of the treadmilling rate, contractility and anchoringparameters leads in 3d to a several series of cell morphologies,resembling structures seen in eukaryotic cells including bothlamellipodia and pseudopods

● A minimum model with simple physical ingredients captures many of the observed features of motile and spreading cells.

● Some aspects of cellular motility, although controlled by the cell's complex biochemical feedback network, do not directly depend upon these for their basic operational principles.

OutlookWhat if we have several non coalescing motile droplets?

● Experiments show that active droplets tend to accumulate near the surface (WCK Poon, J.Phys. Cond. Mat. 14, R859 (2002))

● Theory describes this behavior assuming a propulsion velocity decaying with the density ofthe droplets (or particles) (J. Tailleur, M. Cates, PRL 100, 218103 (2008), Cates et al., PNAS 107, 11715 (2010)).