THE MACHINERY OF CELL CRAWLING -Amoeboidal cell crawling on solid substrates is characterized by protrusions that seemingly appear randomly along the cell periphery and drive the cell forward. For many cell types, it is known that the protrusions result from polymerization of the actin cytoskeleton. However, little is known about how the formation of protrusions is triggered and whether the appearance of subsequent protrusions is coordinated. Recently, the spontaneous formation of actin-polymerization waves was observed. These waves have been proposed to orchestrate the cytoskeletal dynamics during cell crawling. Here, we study the impact of cytoskeletal polymerization waves on cell migration using a phase-field approach. In addition to directionally moving cells, we find states reminiscent of amoeboidal cell crawling. In this framework, new protrusions are seen to emerge from a nucleation process, generating spiral actin waves in the cell interior. Nucleation of new spirals does not require noise, but occurs in a state that is apparently displaying spatio-temporal chaos....... When a cell crawls, part of its fluid cytoplasm briefly turns rigid. This transformation depends on the orderly assembly and disassembly of a protein scaffold.The pool of actin filaments is constantly turning over and disassembled filaments are replaced. Under physiological conditions, actin monomers do not assemble spontaneously to form the nucleus of a nascent filament. To this end, NPFs like the Arp2/3 complex or members of the formin family are used in cells. The Arp2/3 complex binds to an existing actin filament and then stays attached to the pointed end of the newly created actin filament. In contrast, formins stay attached for some time to the barbed end of the newly generated actin filament and assist elongation. If there is a negative feedback from existing filaments on active NPFs, for which some experimental evidence exists [14], then polymerization waves can emerge spontaneously

Many eukaryotic cells can crawl on solid substrates, for example, to move towards nutrients or
other cells, to avoid toxins, or in response to other external stimuli [1]. Depending on the cell
type and environmental conditions, several distinct types of cell crawling can be identified. Two
extreme examples are provided by fish keratocytes that essentially maintain their shape and
move directionally [2], and by the slime mold Dictyostelium discoideum or human neutrophils
that display an irregular crawling via the formation and retraction of protrusions (pseudopods).
The irregular pattern of motion is called amoeboidal movement and has been studied notably in
the context of chemotaxis [3].
These different kinds of cell crawling rely on the same molecular machinery, namely, the
actin cytoskeleton, which is a network of structurally polar, filamentous protein aggregates. For
some cell types, it is known that the polymerization of actin filaments pushes the leading edge
forward and thus provides the machinery of cell crawling [4]. In other cell types, the motorinduced
contraction of the actin cytoskeleton squeezes the intracellular fluid, which in turn leads
to the extension of a protrusion [5] and has been recently analyzed from a theoretical point of
view [6–8]. However, it is largely unknown how the assembly of the actin network and its
contractility is regulated on a cellular scale to achieve the crawling patterns observed in living
cells, in particular, amoeboidal crawling.
Actin assembly can be regulated through external cues. For example, the direction of
motion of a crawling cell can be controlled by an external gradient. Notably, chemical gradients
are read out by cells able to perform chemotaxis. Remarkably, though, many eukaryotic cells
and even cell fragments can also spontaneously polarize and choose a direction of motion
[9–12]. Furthermore, some cells actively search for other cells by performing a random walk
type of movement that is largely independent of external stimuli. Whereas mechanisms for this
random cell motion have not been studied systematically so far, some ideas exist about how the
cells can spontaneously polarize [13]. In particular, it has been suggested that cytoskeletal
polymerization waves could play an important role in this context [14, 15].
Polymerization waves can emerge spontaneously from the interaction of actin filaments
and nucleation promoting factors (NPFs) [14, 16–19]. Actin assembly is a process depending on
the release of chemical energy through the hydrolysis of adenosine triphosphate (ATP).
Opposite ends of actin filaments differ in their assembly kinetics, with the so-called barbed end
usually growing on average, whereas the so-called pointed end typically shrinks on average, a
phenomenon called treadmilling 
The average propulsion velocity is largely independent of the polymerization
velocity va. Instead, it depends sensitively on the rates of actin nucleation and of nucleator
binding and unbinding. This behavior can be rationalized by considering a dimensionless form
of equations (1)–(7). As we show in appendix B, for the chosen parameters, the actin density T
and polarization p are enslaved by the density na of active nucleators. This property also
suggests that details of the actin polymerization process are not important for the overall system
behavior. Indeed, other polymerization dynamics in combination with the NPF dynamics
similar to the one used here have been found to spontaneously produce waves [14, 16, 18, 19].
Even though the polymerization velocity does not have a major impact on the waveʼs
propagation velocity or the cellʼs migration speed, which are set by the rates of nucleator
activation and inactivation as well as the actin nucleation rate, changes in va can lead to the
qualitative changes in the type of motion, as we will discuss now.
3.2. Irregular motion
Decreasing the growth velocity from v˜ = 0.39 a to v˜ = 0.27 a , the state changes qualitatively: the
fragment is still moving, however, its trajectory is no longer straight; see figure 4(a). Instead it
consists of curved segments that are interspersed by points at which the cell discontinuously
changes direction. We would like to emphasize that this apparently random motion results from

deterministic dynamic equations (1)–(7) without addition of noise