Figure 1.
The soft (visco-)elastic adhesive layer of thickness h is sandwiched between the much less deformable substrate (e.g. a glass slide or PDMS) and the cell, here assumed to be moving in positive x-direction with speed V. The cell exerts a dipolar force in the layer, illustrated schematically by the propulsion force Fp balanced by the friction force Ff. Correspondingly, a pair of forces of opposite polarity is applied to the cell. The induced deformation in the soft layer is modeled by the extension of an effective spring, with effective spring constant G.
Figure 2.
The main figure shows a typical stick-slip motion. The inset displays steady-state motion for high substrate stiffness and otherwise same parameters and initial conditions. The integrated adhesion density is shown in red and the velocity
of the cell in blue. The absolute value of the extension (in the direction of motion) of the effective spring modeling the substrate,
, is shown in green. The dashed box is the time window where Figure 2 displays the spatially resolved dynamics. Parameters:
,
,
,
,
,
,
; initial area of the cell
(initial radius
). Other parameter as given in Table 1.
Figure 3.
Spatially resolved dynamics during a stick-slip cycle.
The cell’s boundary is given by the green curves. The arrows display the local averaged actin filament orientation. The density of adhesion sites is color coded (with white color corresponding to , blue to
and red to
). The direction of motion is to the right. The time window is the one marked by the dashed box in Figure 2. Between
and the next attachment event, the cell will relax to an almost round state, similar to the one displayed for
. Parameters as in Fig. 2.
Figure 4.
a) The different modes of motility in the plane of propulsion parameter vs. substrate stiffness
for
and
. b) The different modes of motility in the plane of rate of adhesion formation
vs. substrate stiffness
for
,
. For both parts, parameters where a cell stops after an initial perturbation are marked by blue circles. If stick-slip motion is persistent this is marked by red diamonds and if the cell acquires a steadily moving state (continuous gliding), by a green square. Other parameters as in Fig. 2.
Figure 5.
Stick-slip cycle in the reduced ODE model.
The main plot shows the nullclines (dashed line) and
(solid line) and the limit cycle (with each quarter period marked in a different color) obtained by numerical integration of Eqs. (10). Starting at the upper part of the blue trajectory, the cell increasingly adheres and
becomes more negative as the cell exerts more and more force on the substrate. When
, adhesive contacts break and
rapidly decreases (lower part of the blue trajectory) until the dynamics reaches the
-nullcline (red branch). There
relaxes while
almost stays zero, but effectively slowly grows as at small values of
the cell slows down and adhesion can restart (green trajectory) followed again by rapid attachment (blue trajectory). Parameters:
,
,
,
,
,
,
,
,
. The two panels on the right show
and
for one period of the stick-slip cycle.
Figure 6.
Response to step-like changes in substrate properties.
A) Motion of a cell on a substrate where the adhesive strength is modulated by a step in the rate of adhesion formation , corresponding to a varying density of adhesive ligands (e.g. different surface coverages of fibronectin). The blue region has
, the black one
, substrate stiffness is
. B,C) Motion of a cell on a substrate where the substrate stiffness exhibits a step (the blue region is a rigid substrate with stiffness
, the black one is much softer,
). B) The cell bounces off the step for
,
within the entire cell and
. C) The cell overcomes the step and continues in the same direction for
,
. All other parameters as in Fig. 4.
Figure 7.
Motion of cells on substrates with alternating stripes of high/low adhesiveness.
a) Motion of a cell on a rigid substrate () with alternating stripes of high adhesiveness parameter
(blue) and no adhesiveness (
for black stripes). The cell positions itself symmetrically and moves parallel to the stripes in a steady fashion. b) Motion of a cell on a substrate with
, and with alternating stripes of low adhesiveness parameter
(blue) and
(black). After moving initially along the stripes, the cell turns and moves perpendicular to the stripes in a stick-slip fashion. Parameters are as in Fig. 4 except
. c) Select trajectories of the center of mass of cells moving on stripe-patterned substrates with different values of the adhesion formation rate
and substrate stiffness
. Gray stripes correspond to high adhesiveness regions, white stripes correspond to zero adhesiveness regions (
). For high
and
, the cell displays persistent and steady motion along the stripes (black curve). For intermediate values the predominant motion is along the stripes with excursions into the perpendicular directions (blue curve). Finally, for low adhesiveness the motion is perpendicular to the stripes with reversals.
Figure 8.
Trajectories of cells on substrates with three different width ratios of adhesive to non-adhesive stripes.
In the corresponding snapshots, the adhesive stripes are shown by the blue color and the green arrows indicate the direction of motion. All parameters are as for Fig. 7, except for (all parameters are the same in the three cases, only the width ratio changed).
Table 1.
Parameters.
Figure 9.
Subcritical onset of motion and stationary cell shapes.
Panel a) displays the normalized velocity vs. the normalized driving force
. The main plot shows the solutions of Eq. (16), the solid line corresponding to the stable moving branch and the dashed line to the unstable branch. The inset shows results obtained by numerical solution of the full model for
,
,
,
and for different values of
and
as indicated. b) A typical stable moving shape corresponding to
and
. c) A typical stable moving shape corresponding to
and
.
Figure 10.
Effect of adhesion strength and motor activity on the velocity and shape.
The upper panel shows the cell’s velocity as a function of the propulsion parameter for fixed
, mimicking an increase of the substrate adhesiveness as explained in the text. Two different levels of the contractile motor activity have been investigated, modeled by different values of the parameter
. The lower panel displays the corresponding aspect ratios (see Ref. [8] for its definition). Parameters as in Fig. 2, except
,
.