Table 1.
Parameters in the models.
Figure 1.
The spherical coordinates of a point will be denoted by r, and
. The shaded region corresponds to the cell cytoplasm and the unshaded to the nucleus.
Figure 2.
Visualization of the travelling wave solution.
Panel A: two stable stationary solutions for ,
,
and
. Panel B: propagation of the traveling wave connecting the two stable solutions shown at Panel A, initiated by a formation of a cluster of receptors at the upper cell pole. The aggregated fraction of receptors is F = 0.01 and the cluster size is
. For t = 250 s the whole cell is activated. The minimum and maximum value of the active kinase concentration are given in each panel. In Panel A the active kinase distribution
is spherically symmetric. In Panel B the active kinase distribution
is not reaching the spherical symmetry even for t = 250 s the small cluster of receptors at the upper pole causes that locally the active kinase concentration is elevated with respect to spherically symmetric case (0.77 versus 0.88).
Figure 3.
The dependence of the fraction of the aggregated receptors F on the fraction of cell surface 1/i occupied by the cluster for rn = 0, 0.8, 0.9, 0.95 and 0.98, for ,
,
.
Figure 4.
The dependence of the minimal fraction of the aggregated receptors sufficient for the cell activation , on thickness of the cytoplasmic layer (
) for
,
,
.
Figure 5.
Activation of the cell by the displacement of the nucleus.
The shift of the nucleus by 0.08 causes that locally thickness of the cytoplasm drops to 0.02. This leads to the local cell activation, thereafter the wave of the kinase activity spreads over the rest of the cytoplasm. The concentration of the active kinase is the highest at the pole in which the cytoplasm is the thinnest. The remaining parameters for this simulation are ,
,
.
Figure 6.
Time snapshots of the active kinase concentration .
The initial condition is ,
. The traveling wave of activation is induced by a local aggregation of the receptors;
,
,
. Panel A:
,
, time interval is
, activation time
; Panel B:
,
,
,
; Panel C:
,
,
,
; Panel D:
,
,
,
.
Figure 7.
The dependence of the critical fraction of the aggregated receptors on the fraction of cell surface occupied by the cluster for 2D (red) and 3D (blue) models for and
.
The minimal fraction of aggregated receptors for 2D model equals 0.0032 at i = 5×105. Right Panel: the zoomed region around point
for 2D model.