Excitable reaction-diffusion waves of curvature-inducing proteins on deformable membrane tubes

Living cells employ excitable reaction-diffusion waves for internal cellular functions, in which curvature-inducing proteins are often involved. However, the role of their mechanochemical coupling is not well understood. Here, we report the membrane deformation induced by the excitable reaction-diffusion waves of curvature-inducing proteins and the alternation in the waves due to the deformation, using a coarse-grained simulation of tubular membranes with a modified FitzHugh-Nagumo model. Protein-propagating waves deform tubular membranes and large deformations induce budding and erase waves. The wave speed and shape are determined by a combination of membrane deformation and spatial distribution of the curvature-inducing protein. Waves are also undulated in the azimuthal direction depending on the condition. Rotationally symmetric waves locally deform the tubes into a symmetric shape but maintain a straight shape on average. Our simulation method can be applied to other chemical reaction models and used to investigate various biomembrane phenomena.

Figure 1

Phase plane of the FitzHugh-Nagumo model with the modified protein unbinding process for $A=0.4, B=0.001, C=0.1$, and $G=0$. Purple and green lines are nullclines for $\partial u/\partial t=0$ and $\partial v/\partial t=0$, respectively. Black lines indicate trajectories after the equilibrium state is stimulated.

Figure 2

Initiation of propagating waves. (a) The profiles of the stimuli $Q_u$ and $Q_v$ to initiate the propagating wave. (b) Propagating waves induced by the stimuli shown in (a). The color variation indicates different concentration of the curvature-inducing protein, $u$.

Figure 3

Typical sequential snapshots of propagating waves on nondeformable tubular membranes for (a) $I=0.002$, (b) $I=0$, (c) $I=-0.002$, and (d) $I=-0.01$. The color variation indicates different concentration of the curvature-inducing protein, $u$.

Figure 4

Effect of constant inputs $I$ on the wave properties with nondeformable membranes. In the gray shaded area, propagating waves disappear.

Figure 5

Typical sequential snapshots of propagating waves on deformable tubular membranes with a small radius ($R_{\mathrm{t}}=9.79a$) and small spontaneous curvatures $C_0$. (a) $C_0{R}_{\mathrm{t}}=1$ and ${\kappa }_1/{\kappa }_0=2$. (b) $C_0{R}_{\mathrm{t}}=3$ and ${\kappa }_1/{\kappa }_0=3$. The color variation indicates different concentration of the curvature-inducing protein, $u$. The corresponding movies are provided in Supplemental Movies S1 and S2 [47].

Figure 6

Typical sequential snapshots of propagating waves on deformable tubular membranes with the small radius ($R_{\mathrm{t}}=9.79a$) with large spontaneous curvatures $C_0$. $C_0{R}_{\mathrm{t}}=4$ and ${\kappa }_1/{\kappa }_0=4$. The corresponding movie is provided in Supplemental Movie S3 [47].

Figure 7

Time evolution of $x$ coordinate of the center of mass of the wave ($g_{\mathrm{wave},x}$), on deformable tubular membranes with the small radius ($R_{\mathrm{t}}=9.79a$). The simulation condition is the same as that of Figs. 5 and 6.

Figure 8

Time evolution of the wave width on deformable tubular membranes with a small radius ($R_{\mathrm{t}}=9.79a$). (a) ${\kappa }_1/{\kappa }_0$ varied at $C_0{R}_{\mathrm{t}}=3$. (b) $C_0{R}_{\mathrm{t}}$ varied at ${\kappa }_1/{\kappa }_0=6$.

Figure 9

Typical sequential snapshots of propagating waves on deformable wide tubes ($R_{\mathrm{t}}=19.6a$) at $C_0{R}_{\mathrm{t}}=4$ and ${\kappa }_1/{\kappa }_0=4$. The color variation indicates different concentration of the curvature-inducing protein, $u$. The corresponding movie is provided in Supplemental Movie S4 [47].

Figure 10

Typical sequential snapshots of propagating waves on deformable wide tubes ($R_{\mathrm{t}}=19.6a$) at $C_0{R}_{\mathrm{t}}=8$ and ${\kappa }_1/{\kappa }_0=8$. The color variation indicates different concentration of the curvature-inducing protein, $u$. The corresponding movie is provided in Supplemental Movie S6 [47].

Figure 11

Time evolution of the $x$ coordinate of vertices in the wave ($r_{\mathrm{wave},x}$) on a deformable wide tube ($R_{\mathrm{t}}=19.6a$) at $C_0{R}_{\mathrm{t}}=8$ and ${\kappa }_1/{\kappa }_0=8$. The simulation condition is the same as that of Fig. 10.

Figure 12

Pattern diagrams with deformable tubular membranes for $R_{\mathrm{t}}=9.79a$ and $19.6a$ (narrow tube and wide tube, respectively). The symbols represent the simulation results. Overlapped symbols indicate the coexistence of multiple patterns. The dashed lines are given by Eq. (12) with $u=0.2$.

Figure 13

Typical snapshots of undulated waves and cross-section views of the waves at front and rear edges. (a) $C_{0}R=4, {\kappa }_1/{\kappa }_0=4$, and $R_{\mathrm{t}}=19.6a$ (wide tube). (b) $C_0{R}_{\mathrm{t}}=0$ and ${\kappa }_1/{\kappa }_0=32$, and $R_{\mathrm{t}}=19.6a$ (wide tube). (c) $C_0{R}_{\mathrm{t}}=0$ and ${\kappa }_1/{\kappa }_0=12$, and $R_{\mathrm{t}}=9.8a$ (narrow tube). The color variation indicates different concentration of the curvature-inducing proteins, $u$.

Figure 14

Time evolution of the magnitude of undulation of wave fronts, $\langle d_{\mathrm{RMSD}}\rangle$ for $C_0{R}_{\mathrm{t}}=4, {\kappa }_1/{\kappa }_0=4$, and $R_{\mathrm{t}}=19.6a$ (wide tube), with $q\mathrm{th}$ amplitudes of DFT for (a) $u$ and (b) $\tilde{H}$ in an azimuthal direction, $A_{u_q}$ and $A_{H_q}$, respectively ($q=1$ and 2). The simulation condition is the same as that of Figs. 9 and 13.

Figure 15

Calculation of local geodesic lines and $\langle d_{\mathrm{RMSD}}\rangle$ for a narrow tube ($R_{\mathrm{t}}=9.79a, C_0{R}_{\mathrm{t}}=2$, and ${\kappa }_1/{\kappa }_0=6$). (a) Lines and points obtained in the calculation process of the local geodesic line. (b) Snapshots of waves and local geodesic lines at $t/\tau =2000$ and $t/\tau =3000$. The color in snapshots indicates the concentration of curvature-inducing proteins, $u$. (c) Time development of $\langle d_{\mathrm{RMSD}}\rangle$.