Schematic of membrane-inserted receptors. (a) Trimer (orange shaded area) of dimers (red bars) with geometric parameters. C_T is the intrinsic curvature of a trimer. (b) Cluster of three trimers at the cell pole. Trimer-trimer coupling strength J₀ is indicated by green arrows. Also shown are the cell wall and the inner membrane (gray) with curvature C_m,p. The height profile h(r) describes the cluster-membrane deformation as measured relative to the preferred height due to cell wall and turgor pressure. (c) Same cluster at a lateral position with membrane curvature C_m,l. Red arrows indicate energetically unfavorable membrane deformations due to the cluster-membrane curvature mismatch.

Cluster-membrane height profiles. (a) Profile h(r) as a function of distance r from the center for a small cluster of radius R = 10 nm. (Inset) Profile for a large cluster of radius R = 100 nm. (b) Three-dimensional profile for a small cluster. (Left inset) Three-dimensional profile for a large cluster. For parameters κ_c, κ_m, and λ, see Table 1. (Right inset) Large cluster for a very small pinning modulus (λ = 10⁻⁵k_BT/nm⁴).

Predicted receptor clustering in cocci cells. Average cluster radius, 〈R〉, as a function of coccus radius. Solid lines of varying thickness correspond to the three densities from Fig. 5b.

Electron micrographs of chemoreceptor clusters. (a) Extended clusters in membrane preparations (8–11). Clusters of similar size are observed at the poles of living cells (12,13). Image is courtesy of Michael Manson. (b) Self-assembled round micelle at receptor-dimer resolution (23). Image printed with written permission from publisher. All experiments are based on Tsr-receptor overexpression, as well as membrane extraction, negative staining, and freezing for imaging.

Average interaction energy J(N) per trimer (in units of coupling strength J₀) as a function of cluster size (number of trimers, N). Trimers are shown by orange disks, and trimer-trimer coupling of strength J₀ by green bars. Clusters are assumed to have a triangular lattice structure. Symbols represent small compact clusters of minimal circumference. Solid and dashed lines represent the “ring model” and “fit-to-symbols model”, respectively (see Model section). The horizontal dotted line shows the average interaction energy for an infinitely large cluster. (Inset) Possible mechanisms of trimer-trimer coupling including that mediated by receptor-bound CheA-CheW and membrane deformations based on large hydrophobic transmembrane domains.

Quantifying polar receptor clustering. (a) Cluster-membrane energy density, i.e., energy per trimer as a function of cluster size N. Total energy density ɛ (black) is the sum of repulsive elastic energy density ɛ_el (blue) and attractive cluster energy density ɛ_a (red). Solid lines correspond to cell poles and dashed lines to lateral positions. Thin solid and dashed blue lines indicate asymptotic limits of the elastic energy density for at the poles and lateral positions, respectively. Polar clusters are stabilized by energy density Δɛ. (b) Distribution of cluster sizes for three values of the trimer density (occupancy fraction), ρ = 0.4, 0.1, and 0.025 at the poles (solid lines) and lateral positions (dashed lines). (c) Average cluster radius 〈R〉 at the poles (black solid line) and lateral area (black dashed line) as a function of trimer density, ρ, based on standard parameters, including trimer-trimer coupling, J₀, trimer curvature, C_T, and cluster bending stiffness, κ_c from Table 1. Error bars indicate robustness to parameter changes: upper error bars for poles and lateral area are 1.1 for J₀ (red), 0.98 for C_T (dimer-dimer distance reduced by 1 nm (green)), and 0.92 for κ_c (blue); lower error bars for poles and lateral area are 0.9 for J₀ (red), 1.03 for C_T (dimer-dimer distance increased by 1 nm (green)), and 1.09 for κ_c (blue). (Inset) Average cluster size (number of trimers) 〈N〉 as a function of trimer density for standard parameters.