6.2 Crawling Cells
117
Gordian knot. A formal variable φ, called the phase field, is introduced, governed by
a diffusion equation with a cubic nonlinearity that has two stable uniform solutions,
say, φ = 1 and φ = 0 which, if they coexist, are separated by a narrow border region.
The two values are associated first with the interior, and second with the exterior of
the cell; thus, the boundary, rather than being sharp, becomes diffuse. This formal
variable can be associated with the thickness or mass per unit area of the cell.
The phase field should be coupled to physical variables. In simulations by Shao et
al (2012), it was part of a full-fledged model based on the viscous flow equation of the
actin network and including contributions from the membrane surface tension and
friction due to adhesion to the substrate. The hydrodynamic part was complemented
by convection–diffusion equations for actin and myosin, taking into account the actin
polymerization rate. The phase variable enters these equations in such a way that
all terms in the transport equations, as well as the velocity, vanish at φ = 0, and the
flow feeds back by advecting the phase variable. A plethora of parameters in this
system should in principle be sufficient to describe all experimental situations, and
the publication contains many pictures of the various shapes of migrating cells.
Conversely, Ziebert and Aranson (2013) adopted a radical approach. The only
physical field in their model is the averaged vector field p representing the actin
filament network, which defines both its magnitude and its polar orientation. Its
scalar product with the gradient of the phase field imitates the advection of the
cell’s interface due to polymerization or breakdown of actin filaments; in a later
publication (Ziebert and Aranson, 2014), it is also made dependent on the density
of adhesions. In its turn, the phase field affects the physical vector variable in
such a way that its magnitude can be finite only when φ approaches unity and its
orientation is correlated with the gradient of the phase field in a way encouraging
polarization of the cell; moreover, the strength of this correlation is associated with
the adhesion strength, and the model is extended to include substrate compliance. A
global constraint involving both variables keeps both the cell area and the magnitude
of p (imitating actin density) finite. This is a kind of stripped-down active gel model,
Fig. 6.4 (a), (b) Shapes of crawling cells and traction patterns (Ziebert et al, 2016). (c), (d) A cell
reacts upon encountering a softer substrate, colored black (Ziebert and Aranson, 2013)
117
Gordian knot. A formal variable φ, called the phase field, is introduced, governed by
a diffusion equation with a cubic nonlinearity that has two stable uniform solutions,
say, φ = 1 and φ = 0 which, if they coexist, are separated by a narrow border region.
The two values are associated first with the interior, and second with the exterior of
the cell; thus, the boundary, rather than being sharp, becomes diffuse. This formal
variable can be associated with the thickness or mass per unit area of the cell.
The phase field should be coupled to physical variables. In simulations by Shao et
al (2012), it was part of a full-fledged model based on the viscous flow equation of the
actin network and including contributions from the membrane surface tension and
friction due to adhesion to the substrate. The hydrodynamic part was complemented
by convection–diffusion equations for actin and myosin, taking into account the actin
polymerization rate. The phase variable enters these equations in such a way that
all terms in the transport equations, as well as the velocity, vanish at φ = 0, and the
flow feeds back by advecting the phase variable. A plethora of parameters in this
system should in principle be sufficient to describe all experimental situations, and
the publication contains many pictures of the various shapes of migrating cells.
Conversely, Ziebert and Aranson (2013) adopted a radical approach. The only
physical field in their model is the averaged vector field p representing the actin
filament network, which defines both its magnitude and its polar orientation. Its
scalar product with the gradient of the phase field imitates the advection of the
cell’s interface due to polymerization or breakdown of actin filaments; in a later
publication (Ziebert and Aranson, 2014), it is also made dependent on the density
of adhesions. In its turn, the phase field affects the physical vector variable in
such a way that its magnitude can be finite only when φ approaches unity and its
orientation is correlated with the gradient of the phase field in a way encouraging
polarization of the cell; moreover, the strength of this correlation is associated with
the adhesion strength, and the model is extended to include substrate compliance. A
global constraint involving both variables keeps both the cell area and the magnitude
of p (imitating actin density) finite. This is a kind of stripped-down active gel model,
Fig. 6.4 (a), (b) Shapes of crawling cells and traction patterns (Ziebert et al, 2016). (c), (d) A cell
reacts upon encountering a softer substrate, colored black (Ziebert and Aranson, 2013)