Integrin-mediated adhesion as self-sustained waves of enzymatic activation

Integrin receptors mediate interaction between the cellular actin-cytoskeleton and extracellular matrix. Based on their activation properties, we propose a reaction-diffusion model where the kinetics of the two-state receptors is modulated by their lipidic environment. This environment serves as an activator variable, while a second variable plays the role of a scaffold protein and controls the self-sustained activation of the receptors. Due to receptor diffusion which couples dynamically the activator and the inhibitor, our model connects major classes of reaction diffusion systems for excitable media. Spot and rosette solutions, characterized by receptor clustering into localized static or dynamic structures, are organized into a phase diagram. It is shown that diffusion and kinetics of receptors determines the dynamics and the stability of these structures. We discuss this model as a precursor model for cell signaling in the context of podosomes forming actoadhesive metastructures, and we study how generic signaling defects influence their organization.

Figure 1

Schematic representation of the model. Left panel: Integrin receptors have two conformational states. Unactivated integrins are represented by down triangles and activated ones by up triangles. Changing membrane composition with increasing $u$ as indicated by the color code of the bar drives integrin receptors from their unactivated to their activated state where they are immobilized by ligation to an extracellular ligand ($\perp$). Right panel: Activated integrins bind reversibly a controller protein represented as a pink circle with the symbol $v$. A covalent modification of $v$ on the integrin site leads to the inert molecule $⌀$ represented as a yellow circle. Changes in affinity between integrins and $v$ or $⌀$ are represented by right-left arrows. The complex $v·n_l$ synthesizes $u$ and thus favors the activated phase by a feedback loop.

Figure 2

Parameter space in the limit of fast adaptation of the integrin field (b case). Typical null clines for regions I, II, and II are shown as insets. Region I, with one stable fixed point, corresponds to the monostable case studied in this paper. Solutions $u\left(t\right)$ and $v\left(t\right)$ in region II are oscillatory. In region III, the null clines correspond to a bistable system with three fixed points.

Figure 3

Generic trajectories (rainbow) in the $(u,v)$ phase space with null clines (red, blue) in the case of slow adaptation of the integrin field (b case). The trajectory approaches the null cline $v=bu/n_l\left(u\right)$ (red curve) and $u$ is bounded from above when $\epsilon \to 0$ before returning to the homogeneous fixed point. The inset shows the variations of $u\left(t\right)$ as a function of time ($h=0.05,\epsilon =1,b=1,v_c=0.5$).

Figure 4

Generic trajectories (rainbow) in the $(u,v)$ phase space with null clines (red, blue) in the extreme cases of instantaneous adaptation of the integrin field (b case). The inset shows the variations of $u\left(t\right)$ as a function of time in the cases where a constant $c$ limit the production of $u$ ($c\gg 1$, dashed, brown) and in the case where there is little saturation ($c=0.1$, blue). Note that there is a strong separation of time scales between the fast increase of $u\left(t\right)$ and its slow return to the fixed point ($h=0.05,\epsilon =1,b=1,v_c=0.5$).

Figure 5

Example of a static spike obtained from a pointlike excitation. The density of free integrins (a) shown in green has almost converged to $n_{\infty }=1$. Ligated integrins (b) are shown in red. The blue dotted line represents the variations of $u$ (c) (normalized by its asymptotic value in one dimension; see (23). The dashed magenta line (d) corresponds to $v/v_h$. The distance $r$ to the symmetry axis has been normalized by the diffusion length $l_u=\sqrt{D_u\epsilon /b}$ ($\approx l_v$) of $u$ ($D_{u,v}=1,h=0.1,b=2,v_c=0.75,\epsilon =5.75$). The scale where $v$ varies is slightly larger than the characteristic scale of $u$ with $l_u\lesssim l_v$.

Figure 6

Example of a static radially symmetric rosette obtained from a pointlike excitation. Compared with the solution of Fig. 5, the excitability parameter $\epsilon$ is slightly decreased so that the the solution has the shape of a radially symmetric rosette ($D_{u,v}=1,h=0.1,b=2,v_c=0.75,\epsilon =3.65$). The color code is the same as in Fig. 5: (a) $n_f$, (b) $n_l$, (c) $u/u_{1d}$, (d) $v/v_h$.

Figure 7

Phase diagram for 2D localized structures (case b with $c=0$). D, S, and H correspond to nonstationary (expanding rosettes), static (stable), and homogeneous solutions. The insets show snapshots of $u\left(x\right)$. From left to right: Radially symmetric expanding rosettes, static rosettes near the left boundary line D-S, spike near the right boundary line S-H. Nonstationary solutions exist only on the left of the D-S line. Static solutions are obtained after convergence of the integrin field to 1 ($h=0.1,b=2,v_c=0.75$). Note that expanding rings can be unstable and fragment in the vicinity of the D-S line as shown in Fig. 12. Dashed lines have been traced by hand.

Figure 8

Phase diagram in the small $\beta h{v}_c/b$ limit ($h=0.05,v_c=0.5,b=1$). The shaded area corresponds to the domain of stationary localized solutions (inset D). This domain is bounded from below by a minimum value of the diffusion constant $D_v$ below which stationary solutions cannot be found. Only transient spikes fading away can be excited on the down right side of the shaded area (inset C). Going left from inset C to inset B, only transient expanding rings can be excited. Between the shaded area and the left dotted line traced by hand, the only solutions found are expanding rings which rapidly fragment into self-replicating spots as in Fig. 12 for a wide range of integrin diffusion constants. On the left of this dotted line, we find stable radially symmetric expanding rosettes.

Figure 9

Values of $\epsilon$ at the S-D (${\epsilon }_{\mathrm{sd}}$, lower curve) and at the H-S boundary line (${\epsilon }_{\mathrm{hs}}$, upper curve) at fixed $d=D_v/D_u=1$. Static solutions exist only in the colored region. The width of this region, ${\epsilon }_{\mathrm{sd}}-{\epsilon }_{\mathrm{hs}}$, decreases with $v_c$ when $\beta v_{c}h/b$ approaches a critical value below which no localized solution exists ($h=0.1,b=2$).

Figure 10

Example of a phase diagram for $c>0$ in (11). Taking $c>0$ limits the relative fraction of activated receptors which saturates for large $u$. Comparing with the phase diagram of Fig. 7, all phases are shifted to lower excitability parameter $\epsilon$, but the general topology is unchanged ($h=0.44,b=2,c=0.05$).

Figure 11

Density plot for $u$ when an radially symmetric rosette experiences breathing oscillations near the D-S bifurcation line. The shape oscillates with time between the two shapes shown in the insets. For the values of the parameters used in this simulation ($D_v=D_u=0.5D_n=1,c=25$), the D-S and the S-H bifurcation points almost coincide and the shape finally collapses.

Figure 12

Density plots of the activator $u$ at successive times for an expanding rosette when parameters are chosen on the left-hand side vicinity of the D-S boundary line of Fig. 7 ($\epsilon =3.50,D_v/D_u=1.0,D_n=2$). The rosette starts expanding with radial symmetry in 1 and thereafter fragments into self-replicating spots in panels 2, 3, and 4. For larger integrin diffusion constant, $D_n=10D_u$, the radially symmetric rosette is stable. In this case, it stops expanding due to boundary conditions and adopts a square shape.

Figure 13

Density plots of the activator $u$. A collision between two expanding rosettes induces four characteristic different shape instabilities depending on receptor kinetics and diffusion constant. Case (a) is an annihilation of rosettes which initially expanded in the kinetic dominant regime where the recruitment of integrin receptors is small ($D_u=D_v=D_n,{\epsilon }_{\text{int}}=0.5,\epsilon =0.5$). Scaling down both $k_{\text{on}}$ and $k_{\text{off}}$ induces a zig-zag instability in case (b) followed by budding of self-replicating small spots granulating the space, ${\epsilon }_{\text{int}}=0.7$. For different excitation, $\epsilon =1$, and faster receptor kinetics, ${\epsilon }_{\text{int}}=0.15$, the rosettes exhibit in case (c) a pearling instability which propagates from the point of contact between the two rosettes. For much larger diffusion length of the receptors, $D_n=10D_{u,v},\epsilon =1,{\epsilon }_{\text{int}}=1$, the two rosettes repel each other without fusing or splitting into spots ($h=0.05,b=1,v_c=0.5$).

Figure 14

Expanding rosette in the case of a small controller diffusion length. The left panel represents the density map for the activator $u$. The right panel is for ligated integrin receptors $n_l$. Starting with a noncircular initial condition the rosette recovers asymptotically a radially symmetrical shape. As shown by comparison between the figures, this regime is characterized by strong gradients in signaling components across the rosette, $n_l$ being maximal at the intrados.

Figure 15

Density map of the activator $u$ when the activator and the controller variable have comparable diffusion lengths. The two panels correspond to successive snapshots of spiral waves. Starting from a noncircular initial condition, the rosette started expanding to recover a circular shape. At later time, a dynamic ablation defect was numerically simulated by stetting the concentration fields of integrin receptors to zero in a small spot. Subsequent diffusion of integrin receptors towards this spot immediately destabilized the rosette, and the tip started to trace an involute shape (left panel). As shown in Fig. 16, the tips of the spirals are highly excited regions. They recover by fusion and expel receptors in excess to an inside double spiral (right panel). The outside rosette started again to expand while the inside spirals circle around their core ($h=0.1,b=1,D_u=D_n,D_v=0.1D_u,\epsilon =0.1$).

Figure 16

Distribution of ligated integrin receptors in the spiral wave of Fig. 15. The tips of the spirals having small curvature radii are the most excited regions where receptors concentrate. Note that ligated integrins are preferentially concentrated in the intrados of the rosette in the outside circular regions.

Figure 17

Density map of the controller $v$, left panel, and of the density of ligated integrins, $n_l$, for an expanding rosette colliding with a defect. The defect is simulated by taking $v_c=0$ inside the left blue circle, so that $v$ cannot be activated within the circle. While expanding, the rosette encircles the defect at right angle as shown in the right panel for the distributions of ligated integrins. The rosette will resume asymptotically its circular shape after passing over the defect and fusing the two arms together. Parameters are those of Fig. 15.

Figure 18

Cut of $u(x,y)$ and $v(x,y)$ along the the $y=0$ axis for a $2-d$ fluctuating spike for $R_n=R_u=R_v=1$ in Eq. (24). Despite the strong fluctuations, the shape of the spike fluctuates around its mean position without breaking into multiple spots, $\epsilon =2,h=0.1,v_c=0.35,1/{\tau }_n=0.1$).