6.1 Cytoskeleton as a Continuum
115
Fig. 6.2 Diagram of stability regions of spirals, asters, and vortices,
as indicated by their outlines. Ordinate: Scaled difference between the
bend and splay orientational elastic moduli. Abscissa: Scaled activity
(Kruse et al, 2005)
sufficient either if chemistry remains behind the scene. Of course, Prost et al are well
aware of this:
Some aspects of biological systems do escape this generic description. The active gel
theory has to be complemented by what is commonly called signaling. Note that signaling
pathways can depend in subtle ways on mechanical stresses, which brings a further twist to
the nonlinear physics aspect of active gel behavior.
The debut application of the active gel theory (Kruse et al, 2004, 2005) was the
description of common structures observed most clearly in in vitro assays (recall
Fig. 5.8): asters, vortices, and spirals. All these are topologically equivalent forms
of unit-charge defects of either sign, characteristic to polar media. Activity causes
spontaneous pairwise generation of defects of opposite sign, leading to a persistent
turbulent pattern. We have already discussed this in Sects. 2.5–2.7 for the case of
active fluids; topology is insensitive to physics and is the same in viscous and elastic
media. However, the question of which particular structure of a defect is preferred
depends on quantitative details. The diagram in Fig. 6.2 shows that spirals are stable
when activity is contractile and the absolute value of the difference between splay
and bend orientational elasticity is not too great, whereas asters and vortices are
preferred in the case of tensile or not too great contractile activity, with asters being
stable when the bend elastic modulus is larger, and vortices, the other way around.
The location of curved boundaries in Fig. 6.2 depends on other parameters of the
problem, and would also change if the anisotropy of the viscoelastic coefficients was
taken into account.
Some caution is required here. We can speak about the structure of the inner
core of a defect only when its size is much smaller than the “macroscopic” scale
of the system but much larger than the “microscopic” scale. For the cytoskeleton,
the former is the cell size measured in tens of microns. However, the latter is not a
molecular scale of actin monomers but a characteristic distance between nodes of
the network, which may be in the range of tens of nanometers or more. There may
not be much leeway here, but it is more important that the alignment of the filaments
is constrained by the structure of the network: recall that even angles at branching
points are fixed (Sect. 5.3). In in vitro assays, there are no such constraints, and the
scale contrast is better: the macroscopic scale is that of the laboratory device, but
the microscopic scale is not molecular either, as it is set by the persistence length of
115
Fig. 6.2 Diagram of stability regions of spirals, asters, and vortices,
as indicated by their outlines. Ordinate: Scaled difference between the
bend and splay orientational elastic moduli. Abscissa: Scaled activity
(Kruse et al, 2005)
sufficient either if chemistry remains behind the scene. Of course, Prost et al are well
aware of this:
Some aspects of biological systems do escape this generic description. The active gel
theory has to be complemented by what is commonly called signaling. Note that signaling
pathways can depend in subtle ways on mechanical stresses, which brings a further twist to
the nonlinear physics aspect of active gel behavior.
The debut application of the active gel theory (Kruse et al, 2004, 2005) was the
description of common structures observed most clearly in in vitro assays (recall
Fig. 5.8): asters, vortices, and spirals. All these are topologically equivalent forms
of unit-charge defects of either sign, characteristic to polar media. Activity causes
spontaneous pairwise generation of defects of opposite sign, leading to a persistent
turbulent pattern. We have already discussed this in Sects. 2.5–2.7 for the case of
active fluids; topology is insensitive to physics and is the same in viscous and elastic
media. However, the question of which particular structure of a defect is preferred
depends on quantitative details. The diagram in Fig. 6.2 shows that spirals are stable
when activity is contractile and the absolute value of the difference between splay
and bend orientational elasticity is not too great, whereas asters and vortices are
preferred in the case of tensile or not too great contractile activity, with asters being
stable when the bend elastic modulus is larger, and vortices, the other way around.
The location of curved boundaries in Fig. 6.2 depends on other parameters of the
problem, and would also change if the anisotropy of the viscoelastic coefficients was
taken into account.
Some caution is required here. We can speak about the structure of the inner
core of a defect only when its size is much smaller than the “macroscopic” scale
of the system but much larger than the “microscopic” scale. For the cytoskeleton,
the former is the cell size measured in tens of microns. However, the latter is not a
molecular scale of actin monomers but a characteristic distance between nodes of
the network, which may be in the range of tens of nanometers or more. There may
not be much leeway here, but it is more important that the alignment of the filaments
is constrained by the structure of the network: recall that even angles at branching
points are fixed (Sect. 5.3). In in vitro assays, there are no such constraints, and the
scale contrast is better: the macroscopic scale is that of the laboratory device, but
the microscopic scale is not molecular either, as it is set by the persistence length of