Nucleation of Ligand-Receptor Domains in Membrane Adhesion

We present a comprehensive model for the nucleation of domains in membrane adhesion. We determine the critical number of bonds in a nucleus and calculate the probability distribution of nucleation time from a discrete master equation. The latter is characterized by only four effective rates, which account for cooperative effects between bonds. We validate our model by finding excellent agreement with extensive Langevin simulations. In the range of parameters typical for cell adhesion, we find the critical number of bonds to be small. Furthermore, we find a characteristic separation between the bonds at which nucleation is particularly fast, pointing to potential regulatory mechanisms that could be used to control the cell recognition processes.

Figure 1

Left: Snapshot of a nucleation site from a simulation. Right: Explicitly calculated membrane profiles for a seed with zero (black), one (green), and five (blue) bonds, the shapes of the latter two being nearly identical. Effective rates for the association ($K_n^{\mathrm{on}}$) and the dissociation ($K_n^{\mathrm{off}}$) of bonds in a growing seed are displayed.

Figure 2

Phase diagrams showing regions of unstable adhesion ($N_c=\infty$) and regions with a particular number of bonds forming a stable seed ($N_c=1,2,3,\dots$) for $\kappa =10k_{B}T$ ($k_{\mathrm{B}}$ being the Boltzmann constant and $T$ temperature), $\gamma =3.125\times {10}^{-7}{k}_{B}T{\mathrm{nm}}^{-4}$, $h_0=80\mathrm{nm}$, and $l_0=40\mathrm{nm}$, ${ϵ}_b=6.97k_{B}T$, ${\rho }_l=1.5625\times {10}^{-4}{\mathrm{nm}}^{-2}$, and $\lambda =\infty$. These parameters are used throughout unless otherwise indicated.

Figure 3

The effective binding rate $K_1^{\mathrm{on}}$ for a formation of a bond at the distance $r$ from a preexisting bond for $\lambda =\infty$, 0.05, and $0.02k_{B}T{\mathrm{nm}}^{-2}$, shown with dashed black, full blue, and the dotted red lines, respectively. Furthermore, we set $\alpha =10\mathrm{nm}$ and $k_0={10}^6{\mathrm{s}}^{-1}$. Inset: Average membrane profile in the envelop of local fluctuation amplitude $\pm \sigma$ around a stiff bond placed at $r=0$.

Figure 4

Characteristic nucleation time as a function of the number density of ligands for several binding affinities (left), and as a function of the distance between bonds for different stiffness of the receptors (right). The numerical solution of Eq. (6) is shown with the symbols while the approximate solutions given by Eq. (7) and Eq. (SI-30) in the Supplemental Material [36] are displayed with the dashed black and full red lines. This short-dashed lines are guidance for the eye.

Figure 5

Comparison of modeling (lines) and simulations (symbols). Probability distribution of nucleation time $\mathcal{P}(\tau )$ for a single radially growing seed (up). The probability distribution of nucleation time ${\mathcal{P}}_{N_R}(\tau )$ generated from a segment of a membrane nucleating on $N_R=1024$ receptors simultaneously (down). For curves with ${ϵ}_b=6.97k_{B}T$, we take $k_0\simeq {10}^6{\mathrm{s}}^{-1}$, $D=5\times {10}^6\mathrm{nm}{\mathrm{s}}^{-2}$, whereas for ${ϵ}_b=8.12k_{B}T$, we use $k_0\simeq 7\times {10}^5{\mathrm{s}}^{-1}$, $D=5\times {10}^7\mathrm{nm}{\mathrm{s}}^{-2}$.