Transforming binding affinities from three dimensions to two with application to cadherin clustering

Abstract

Membrane-bound receptors often form large assemblies resulting from binding to soluble ligands, cell-surface molecules on other cells and extracellular matrix proteins. For example, the association of membrane proteins with proteins on different cells (trans-interactions) can drive the oligomerization of proteins on the same cell (cis-interactions). A central problem in understanding the molecular basis of such phenomena is that equilibrium constants are generally measured in three-dimensional solution and are thus difficult to relate to the two-dimensional environment of a membrane surface. Here we present a theoretical treatment that converts three-dimensional affinities to two dimensions, accounting directly for the structure and dynamics of the membrane-bound molecules. Using a multiscale simulation approach, we apply the theory to explain the formation of ordered, junction-like clusters by classical cadherin adhesion proteins. The approach features atomic-scale molecular dynamics simulations to determine interdomain flexibility, Monte Carlo simulations of multidomain motion and lattice simulations of junction formation. A finding of general relevance is that changes in interdomain motion on trans-binding have a crucial role in driving the lateral, cis-, clustering of adhesion receptors.

Figure 1. Structures of cis dimers formed from cadherin monomers (designated MM) and from trans dimers (designated TT)

All coordinates are taken from the crystal structure of C-cadherin ectodomains. Note that each TT structure has only a single cis interface because the binding regions of the two monomers in a trans dimer face in different directions. This property enables the formation of a 2D lattice in which each pair of trans dimers makes only a single cis interaction ^,,.

Figure 2. Essential coordinates that characterize the dimerization processes of classical cadherins in a 2D membrane environment

The five domains of cadherin's extracellular regions are represented by ellipsoids. Trans dimers (shown in blue) can be formed from two cadherin monomers from two apposing cell surfaces. The molecules are only free to diffuse in two dimensions and rotational motion is constrained. A third dimension is introduced through variations in the perpendicular displacement with respect to the membrane surface, defined by the variable h. h is different for the monomer and trans dimer. In general, h_M will be larger than h_T since trans binding will limit molecular motion. The rotational degrees of freedom for EC1 domains are characterized by the three Euler angles, ϕ, θ and ψ, as depicted in the right panel.

Figure 3. Monte-Carlo simulations of the flexibility of the cadherin ectodomain

The rotations of the EC5 domain with respect to the membrane plane depend on the three Euler angles, Φ, Θ and Ψ of that domain, as shown in the upper left panel. The inter-domain hinge motion indicated by a red arrow is shown in the upper right panel. The lower part of the figure gives the superposition of different conformations in monomer and trans dimer generated by the simulations. The range of values for h, Δψ and Δθ can be obtained from the statistical distribution of simulation results. The decreased flexibility of the trans dimer with respect to the monomer is evident in the figure. Movies describing molecular motion of the monomers and dimers are included in Supplemental Material.

Figure 4. Simulation of junction formation

The lattice in the left panel is a snapshot from a MC simulation where cadherin monomers on apposing cells are colored in red and green, respectively, and trans dimers are colored blue. A diffusion trap mechanism in which the trap region comprises 20×20 lattice sites (in yellow), in the center of a 2D lattice of 100×100 sites, with periodic boundary conditions, was used in the simulations. Trans dimer formation can only take place in the trap region, as the distance between membranes in the surrounding region is too large to allow trans dimer formation. The cadherins form ordered clusters in the trap region, as indicated. Details of the structure appear in reference 10. A movie describing the formation of the ordered junction is included in Supplemental Material. The simulations are carried out using the calculated $K_d^{(2D)}(\mathit{\text{trans}})$ for the trans dimerization of E-cadherin (Table S1) that is derived from experimental measurements. The total concentration of monomers in each of the two adhering surfaces (either free or trans dimerized) is 1%, while the local concentration in the trap region is much higher (18.5%). The corresponding molecular structures of monomers on both cell surfaces, and part of the cluster formed by eight trans dimers are reconstructed in the right panel from the crystal structure of C-cadherin using the same color code. The figure displays the Cα backbone with spheres placed on each carbon atom to improve clarity.