Figure 1.
The GC is represented in the mathematical model as a disk subdivided in angular sectors. We consider the settings of the chemotactic assay for neural cells, where a pipette (here on the right) establishes a steady graded field of a chemotropic molecule.
Table 1.
Nomenclature of the species entering the mathematical model.
Figure 2.
Proposed path for GC chemotaxis induced by netrin binding with DCC receptors.
Solid arrows indicate the prevalent direction of chemical reactions, the dashed arrow indicates physical displacement (recruitment) of receptors induced by open calcium channels.
Figure 3.
Schematization of the antagonist pathways induced by netrin binding with DCC (left column) and DCC-UNC5 receptor complex (right column).
The dashed arrows indicate the feedback effect in the closed loop model.
Figure 4.
DCC model: concentration profiles as a function of time (in minutes).
Curves are relative to the four most representative angular sectors, perceiving a cue concentration ranging from minimal to maximal value, respectively. Legend: green [Netrin] = 9.52 nM (max value), cyan [Netrin] = 9.36 nM, blue [Netrin] = 9.16 nM, magenta [Netrin] = 9.08 nM, red [Netrin] = 9.07 nM (min value).
Figure 5.
DCC model: abscissa (in []) of the barycenter of the molecules as a function of time (in minutes).
Figure 6.
DCC model: concentration profiles of selected species after 2 h as a function of the angle .
Figure 7.
DCC model: concentration profiles as a function of time (in minutes), integration till T = 12 h.
Curves are relative to the four most representative angular sectors, perceiving a cue concentration ranging from minimal to maximal value, respectively. Legend: green [Netrin] = 9.52 nM (max value), cyan [Netrin] = 9.36 nM, blue [Netrin] = 9.16 nM, magenta [Netrin] = 9.08 nM, red [Netrin] = 9.07 nM (min value).
Figure 8.
Effect of the variation of the feedback coefficient .
Top: abscissa (in
) of the barycenter of bound DCC receptors as a function of the feedback coefficient
. Bottom: abscissa
(in
) of the barycenter of open calcium channels as a function of the abscissa of the barycenter of bound receptors (in
). Each marker corresponds to a simulation carried out with a different value of
.
Figure 9.
Effect of the variation of the parameter .
Abscissa (in ) of the barycenter of the bound receptors as a function of
, index of the enzymatic amplification. Below a non–zero threshold, no polarization occurs.
Figure 10.
Diffusive vs. convective phenomena.
When diffusion overwhelms drift effects, the species are homogenized and polarization is not created. The abscissa of the bounded DCC receptors tends to zero, as if under an uniform chemotropic field.
Table 2.
Diffusion and feedback coefficients.
Figure 11.
Spectral analysis of the model.
Eigenvalues (in absolute value) of the Jacobian matrix of the DCC simplified model at steady state, as a function of the angular coordinate (for symmetry represented here only in the range
).
Figure 12.
Concentration contours as a function of time (in minutes, axis) and angular coordinate
(
axis).
Figure 13.
Concentration contours as a function of time (in minutes, axis) and angular coordinate
(
axis).
Figure 14.
Study of the effect of the [cAMP]/[cGMP] ratio.
Top: [cAMP]/[cGMP] ratio as a function of time (in minutes) for different values of (reported in the legend). Bottom. Position
of the abscissa of the barycenter of the open calcium channels (in
) as a function of the ratio [cAMP]/[cGMP] after
. Each marker corresponds to a simulation carried out with a different value of
.
Table 3.
Total values of species concentration.
Table 4.
Initial value of species concentration.
Table 5.
Kinetic constants.
Figure 15.
Scatter plot of the position of the barycenter of bound DCC receptors (in ) as a function of the total parameter variation.
Perturbation in the barycenter displacement is lower than in more than
of the tests.