Curvature-induced clustering of cell adhesion proteins

Cell adhesion proteins typically form stable clusters that anchor the cell membrane to its environment. Several works have suggested that cell membrane protein clusters can emerge from a local feedback between the membrane curvature and the density of proteins. Here, we investigate the effect of such a curvature-sensing mechanism in the context of cell adhesion proteins. We show how clustering emerges in an intermediate range of adhesion and curvature-sensing strengths. We identify key differences with the tilt-induced gradient sensing mechanism we previously proposed (Lin et al., arXiv:2307.03670).

Figure 1

(a) Model sketch of the membrane-protein-substrate system. Proteins (green) pin the membrane (blue) to the substrate (grey) by adhesion to ligands (brown). Triangles (orange) indicate that the adhesion protein induces a local spontaneous curvature into the cell membrane. (b) The spontaneous curvature $c_0\left(\varphi \right)$ of a cell membrane is assumed to be a linear function of the fraction of bound adhesion proteins $\varphi$.

Figure 2

The chemical potential $h_{\varphi }$ and the spontaneous curvature $c_{0,max}$ dictate the cluster formation of curvature-inducing cell adhesion proteins. (a) Stability of the homogeneous state ($\overline{e}, \overline{\varphi }$), regulated by the rescaled chemical potential ${\widehat{h}}_{\varphi }=h_{\varphi }/\left(k_0{e}_0^2\right)$ and the rescaled spontaneous curvature ${\widehat{c}}_{0,max}=c_{0,max}\sqrt{\kappa }/\left(e_0\sqrt{k_0}\right)$. This phase diagram is obtained by linear stability analysis, where the yellow region indicates cluster formation $[{\beta }_{min}<0$, see Eq. (22)]. The red solid line refers to a first-order transition between the dense phase ($\varphi \approx 1$) and the dilute phase ($\varphi \approx 0$), see Eq. (14); while the red dashed line refers to the $\varphi =1/2$ line. See the inset for these two different behaviors. The green solid line (respectively, the blue solid line) refers to the lower (respectively, upper) bound (with respect to $c_{0,max}$) of the cluster formation regime, obtained by solving the ${\beta }_{min}=0$ condition numerically. Purple symbols represent the numerical simulation results of Eqs. (8) and (9) [see (b)–(d) for the corresponding patterns]: purple squares refer to the dense phase; purple triangles refer to the cluster phase; purple circles refer to the dilute phase. The two red circles indicate the triple point (bottom-left one) and the critical point (top-right one). The square box patterns refer to some typical patterns of these three different phases, obtained by numerical simulations. The gray dashed lines indicate critical values of $h_{\varphi }^{\mathrm{cr},0}, h_{\varphi }^{\mathrm{cr},1}, h_{\varphi }^{\mathrm{cr},2}, h_{\varphi }^{\mathrm{cr},3}, c_{0,max}^{\mathrm{cr},1}$, and $c_{0,max}^{\mathrm{cr},2}$. Inset: The homogeneous state $\overline{\varphi }$ as a function of the rescaled curvature ${\widehat{c}}_{0,max}$. (b), (c) Typical patterns of bound cell adhesion proteins at different values of $c_{0,max}$. Parameters: (b) ${\widehat{h}}_{\varphi }=h_{\varphi }/\left(k_0{e}_0^2\right)=1$; (c) ${\widehat{h}}_{\varphi }=h_{\varphi }/\left(k_0{e}_0^2\right)=5$. (d) Typical patterns of bound cell adhesion proteins at different values of $h_{\varphi }$. Here ${\widehat{c}}_{0,max}=c_{0,max}\sqrt{\kappa }/\left(e_0\sqrt{k_0}\right)=1.2$. In (b)–(d), the color codes correspond to $\varphi \left(\mathit{x}\right)$; simulation box size $L\times L$ with $L=16e_0$. (e) The free-energy density $f$ as a function of the spontaneous curvature $c_{0,max}$, where ${\widehat{h}}_{\varphi }=h_{\varphi }/\left(k_0{e}_0^2\right)=1$. Solid line: Theoretically predicted homogeneous steady state, obtained by numerical resolution of Eq. (10). Symbols: Numerical simulations. See Sec. 2c and Table 1 for other parameter values.

Figure 3

The membrane surface tension $\sigma$ regulates cluster formation. (a) The phase diagram of cluster formation is regulated by the membrane surface tension $\tilde{\sigma }=\sigma e_0^2/\left(k_{B}T\right)$. (b), (c) The critical spontaneous curvature values $c_{0,max}^{\mathrm{cr},1}$ and $c_{0,max}^{\mathrm{cr},2}$ as a function of the membrane surface tension $\sigma$: (b) $D_{\varphi }=1k_{B}T$; (c) $D_{\varphi }=0$. These data points are obtained from the numerical resolution of the condition ${\beta }_{min}=0$, defined in Eq. (22). (d) The membrane surface tension $\sigma$ regulates the cluster patterns of cell adhesion proteins; snapshots of the steady state reached in simulations for several values of the surface tension (parameter values: ${\widehat{h}}_{\varphi }=h_{\varphi }/\left(k_0{e}_0^2\right)=1, {\widehat{c}}_{0,max}=c_{0,max}\sqrt{\kappa }/\left(e_0\sqrt{k_0}\right)=1.2$; other parameter values in Sec. 2c and Table 1).

Figure 4

Phase diagram of the rescaled wave number ${\widehat{q}}_{\mathrm{cluster}}=q_{\mathrm{cluster}}{e}_0$, with $q_{\mathrm{cluster}}$ defined in Eq. (29), as the wave number maximizing the instability growth rate ${\lambda }_{-}\left(q\right)$ as a function of the rescaled chemical potential ${\widehat{h}}_{\varphi }=h_{\varphi }/\left(k_0{e}_0^2\right)$ and the rescaled spontaneous curvature ${\widehat{c}}_{0,max}=c_{0,max}\sqrt{\kappa }/\left(e_0\sqrt{k_0}\right)$. Inset: The cluster diameter $d_{\mathrm{cluster}}$ measured in simulation (magenta crosses) and predicted cluster size (red line) as a function of the spontaneous curvature $c_{0,max}$, where ${\widehat{h}}_{\varphi }=h_{\varphi }/\left(k_0{e}_0^2\right)=1$. See Sec. 2c and Table 1 for other parameter values.

Figure 5

Swift-Hohenberg theory interpretation of the clustering pattern transition of cell adhesion proteins upon varying the spontaneous curvature $c_{0,max}$: line structures appear when the bifurcation parameter $\mu$ [see Eq. (34)] is maximal, see point C. The magenta and blue arrows indicate the two pattern transition routes discussed in the main text. Inset: Typical patterns of bound adhesion proteins. Parameters: ${\widehat{h}}_{\varphi }=h_{\varphi }/\left(k_0{e}_0^2\right)=1$; see Sec. 2c and Table 1 for other parameter values.

Figure 6

The membrane–substrate rest-length distance $e_0$ affects the cluster formation of cell adhesion proteins. Stability of the homogeneous state ($\overline{e}, \overline{\varphi }$), regulated by the rescaled membrane–substrate rest-length distance ${\check{e}}_0=e_0/\sqrt[4]{\kappa /k_0}$ and the rescaled spontaneous curvature ${\check{c}}_{0,max}=c_{0,max}\sqrt[4]{\kappa /k_0}$. This phase diagram is obtained by linear stability analysis, where the yellow region corresponds to ${\beta }_{min}<0$. Purple symbols represent the location of the numerical simulation data represented in the inset: squares refer to the homogeneous dense state; triangles refer to the cluster state; circles refer to the homogeneous dilute state. The two red circles indicate the location of the triple and critical points. Inset: Typical patterns of bound cell adhesion proteins at different values of $e_0$ where ${\check{c}}_{0,max}=c_{0,max}\sqrt[4]{\kappa /k_0}=3.8$. See Sec. 2c and Table 1 for other parameter values.

Figure 7

The protein interaction $\chi$ regulates the cluster formation of cell adhesion proteins. The phase diagram of cluster formation is regulated by the protein interaction $\widehat{\chi }=\chi /\left(k_0{e}_0^2\right)$. Inset: Typical patterns of clusters regulated by the protein interaction $\chi$, where ${\widehat{h}}_{\varphi }=h_{\varphi }/\left(k_0{e}_0^2\right)=1$ and ${\widehat{c}}_{0,max}=c_{0,max}\sqrt{\kappa }/\left(e_0\sqrt{k_0}\right)=1.2$, see the purple triangle on the phase diagram. Other parameter values are listed in Sec. 2c and Table 1.

Figure 8

Phase diagram of the number of homogeneous states, regulated by the rescaled chemical potential ${\widehat{h}}_{\varphi }=h_{\varphi }/\left(k_0{e}_0^2\right)$ and the rescaled spontaneous curvature ${\widehat{c}}_{0,max}=c_{0,max}\sqrt{\kappa }/\left(e_0\sqrt{k_0}\right)$. This phase diagram is obtained by the numerical solution of Eq. (10). The region surrounded by the yellow line refers to the cluster phase (${\beta }_{min}<0$), obtained by linear stability analysis. See Sec. 2c and Table 1 for other parameter values.