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Logo of nihpaAbout Author manuscriptsSubmit a manuscriptHHS Public Access; Author Manuscript; Accepted for publication in peer reviewed journal;
 
Nature. Author manuscript; available in PMC 2012 January 28.
Published in final edited form as:
PMCID: PMC3167384
NIHMSID: NIHMS313107

Transforming binding affinities from 3D to 2D 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 proteins1. 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)2. A central problem in understanding the molecular basis of such phenomena is that equilibrium constants are generally measured in three-dimensional (3D) solution and are thus difficult to relate to the two-dimensional (2D) environment of a membrane surface. Here we present a theoretical treatment that converts 3D to 2D affinities accounting directly for the structure and dynamics of the membrane-bound molecules. Using a multi-scale simulation approach we apply the theory to explain the formation of ordered junction-like clusters by classical cadherin adhesion proteins. The approach includes atomic-scale molecular dynamics simulations to determine inter-domain flexibility, Monte-Carlo simulations of multi-domain motion, and lattice simulations of junction formation3. A finding of general relevance is that changes in inter-domain motion upon trans binding plays a crucial role in driving the lateral, cis, clustering of adhesion receptors.

It is commonplace to quantitatively characterize binding between macromolecules by measuring dissociation constants in solution, Kd(3D), which are typically defined in 3D concentration units (e.g. moles/liter). However, phenomena that take place on membrane surfaces are dependent on 2D densities and the relevant dissociation constants, Kd(2D), are defined in units such as molecules/μm2. Measurements of Kd(2D) are difficult to perform and have only been carried out in a small number of cases4-5. Thus, it would be extremely valuable to have a method available that could transform measured values for Kd(3D) into corresponding values of Kd(2D). A reasonable simplifying assumption in such a method is that the binding interface formed by any two molecules is essentially identical in 3D and in 2D. The difference in the Kds then results only from the change in dimensionality and from any other effects that arise from the constrained environment of a planar system.

Bell and coworkers6-7 transformed between two and three dimensions through the simple expression, Kd(2D)=h×Kd(3D), where h is the “confinement length”. The basic idea is that if two interacting species are confined to a region h along an axis perpendicular to the plane of a membrane, then they are effectively confined to a volume, Ah, where A is the area per molecule 6-8. This simple procedure turns a 2D system into a “quasi-3D system” since there is now a volume associated with each molecule even when it is constrained to a planar membrane. The extent of motion along the third dimension can arise from different factors such as molecular flexibility, rotations with respect to the membrane plane, membrane fluctuations and translational motion of the membranes themselves. A number of studies have used measured 3D and 2D affinities to determine h for individual systems. However, as pointed out by Dustin and coworkers 5, widely discrepant values have been obtained from the use of different methods to measure 2D affinities; for example fluorescence measurements typically yield values for h on the order of nanometers, whereas mechanical measurements have yielded values for h on the order of micrometers.5 Here we focus on cases where two flat parallel membranes are separated by a distance that allows proteins located on opposing surfaces to interact in trans and where proteins located on the same surface oligomerize in cis. The values of h that we find are on the order of nanometers as is consistent with fluorescence measurements of 2D affinities 5.

Our specific focus is on the formation of ordered structures by the type I family of classical cadherins. Cadherins have five extracellular immunoglobulin-fold EC domains but the trans binding interface is localized entirely on the membrane-distal EC1 domain 9. We have recently shown that a molecular layer seen in crystal structures of classical cadherins corresponds to the extracellular structure of adherens junctions 10. In addition to the trans interface a second, cis, interface is formed between the EC1 domain of one cadherin and a region comprising parts of the EC2 and EC3 domains of another (Figure 1). Trans cadherin binding affinities have been measured in 3D solution 11 while cis interactions are too weak to measure but have an upper limit of about 1 mM 10. We use this well-defined system as a basis for the development of general theoretical expressions that accomplish the transformation from 3D to 2D. These expressions, when used in conjunction with experimental data and our multi-scale simulations, provide a detailed description of the structural and energetic basis of junction formation and elucidate new principles that are likely to be relevant to other systems.

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

Figure 2(a) describes the trans dimerization reaction when cadherins are restricted to the membrane surface. As mentioned above, we assume that the binding interfaces are the same in solution and on a membrane surface so that the energetic contributions to binding are identical. ΔE(3D) = ΔE(2D) Hence, the difference in the binding affinities is entirely entropic. Since the trans dimerization interface is located on EC1, the difference between 3D and 2D affinities is related to the probability that two EC1 domains will encounter one another in an orientation that allows binding. This in turn depends on the local concentration of EC1 domains and on their freedom of rotational motion. As indicated in the figure, we use hM and hT to denote the range of EC1 motion normal to the membrane plane corresponding to monomeric and trans dimeric cadherins, respectively. Thus, as opposed to Bell's expression6-7 we allow for different values of the confinement length between monomer and dimer, and hence their local concentrations, a factor that will prove crucial in the discussion below. To calculate hM and hT we make the simplifying assumption that the two adhering membranes are flat and parallel to each other, as illustrated schematically in Figure 2. Assuming a cadherin density of 80 molecules per square micrometer11 the lateral intermolecular distance is about 100 nm (which becomes much smaller once clustering begins). Estimates based on bending rigidity suggest that, over this lateral distance range, spontaneous fluctuations in membrane height are typically only a fraction of a nanometer12-13, significantly less than the variations in h due to molecular flexibility considered in this work. Of course cells in-vivo can extend membranous protrusions such as filopodia, which at some scale are not flat. Consideration of such issues goes beyond the scope of the current work, however the treatment given here should provide a good starting point for these more complex instances.

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

The factors that enter into our treatment of rotational motion are shown in Figure 2(b), where the orientation of the EC1 binding site is described in terms of the three Euler angles, ϕ, θ and ψ. In 3D, all three rotational angles are unrestricted. In contrast, there are restrictions on the rotational freedom of the membrane bound molecules, except for the angle ϕ which corresponds to motion around the z axis. The rotational entropy is related to the integral over the three Euler angles14, ϕ, θ and ψ, which yields 8π2 in 3D and a value, Ω < 8π2, for membrane bound molecules (see Supplemental material). Here, Ω = (ΔωM)2ωT where ΔωM = 2πΔψM(1 − cosΔθM) and ΔωT = 2πΔψT(1 − cosΔθT) are the “rotational phase space volumes” of monomer and trans dimer, respectively (see Supplemental Information for details). In parallel to the confinement lengths hM and hT, ΔωM and ΔωT describe the “confinement” in rotational motion in the constrained environment of the membrane.

In Supplemental Information we derive the expression:

Kd(2D)(trans)Kd(3D)(trans)=Ω8π2×hM2hT=18π2×(ΔωMhM)2ΔωThT
(1)

Equation 1 is quite general although, as presented here, the variables refer specifically to the EC1 domains of cadherins. Note that it is straightforward to transform from 3D to 2D if hM, hT, ΔθM, ΔψM, ΔθT and ΔψT are known. These geometric variables will depend on the structures and flexibility of the proteins involved and on the constraints imposed by the membrane environment.

It is instructive to consider the special, hypothetical, case where the reactive EC1s of monomers and dimers can freely diffuse within the same (“reaction”) volume, so that hM = hT [equivalent] h and, in addition, monomer and dimer rotations in 2D are totally unrestricted, as in 3D (Ω/8π2 = 1). In this case Equation 1 simply reduces to Bell's expression6-7 which, however, does not account for real differences in binding free energies in 2D and 3D. Real differences are due to two effects: (i) Because hM > hT and ΔωM > ΔωT, the volume available to monomers in 2D is larger than that available to trans dimers, implying a smaller binding affinity as compared to the 3D case. (ii). The rotational entropy loss upon binding in 2D is smaller than that in 3D, as quantitatively represented by Ω/8π2 < 1, resulting in enhancement of the binding affinity in 2D as compared to 3D. These two effects will thus partly compensate each other, as demonstrated below in quantitative terms based on molecular level simulations.

As mentioned above, many membrane receptors form lateral clusters on the cell surface driven by the formation of a distinct inter-protein cis interface2, which for the specific case of cadherins has been characterized crystallographically10,15. Asymmetric cis interfaces can form between two monomers, as well as between two trans dimers, as shown in Figure 1. In Supplemental Information we derive equations for the 2D dissociation constants appropriate to the cis dimerization of cadherin monomers, Kd(2D)(cis)MM and trans dimers, Kd(2D)(cis)TT. We show there that

Kd(2D)(cis)MMKd(2D)(cis)TT=(ΔωMhMΔωThT)2
(2)

Equation 2, which accounts for differences in the strength of cis interactions between monomers and trans dimers, provides physical insights as to the coupling between trans and cis interactions. Even if cis dimers formed from trans dimers have an identical interface to that formed between monomers, the affinities will be different due to differences in their respective rotational and vibrational flexibilities, as reflected by the factors ΔωMωT and hM/hT, respectively. Qualitatively, since both factors are larger than unity, it follows that the lateral attraction between trans dimers is stronger than that between monomers.

In Methods we describe a multi-scale simulation approach that yields estimates of the six variables hM, hT, ΔθM, ΔψM, ΔθT and ΔψT that define the transformation between 3D and 2D. It is evident from the simulations (see Figure 3) that trans and/or cis dimer formation places significant constraints on the molecular system. Values of h, Δθ and Δψ are reduced by approximately a factor of 2-3 in going from a monomer to a trans or a cis dimer (i.e., hT < hM; ΔθT < ΔθM; ΔψT < ΔψM), an effect that will tend to weaken binding affinities (Supplemental Table S1). Table S1 also reports 3D and 2D Kd's for the dimerization reactions occurring in solution and on the membrane. Notably, the values of Kd(2D) for trans interactions reported in Table S1 (ranging from 15 to 250 μm-2) for N-cadherin are in the range obtained from measurements on molecules associated with the T cell system 4-5,16, while those for E-cadherin are about an order of magnitude weaker due largely to the greater values of Kd(3D).

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

The most dramatic effect seen in the simulations is the difference in Kd(2D) of 3-5 orders of magnitude for lateral, cis, dimerization affinities between monomers vs. trans dimers. The increased binding affinity for trans dimers has a clear physical explanation. The association of two cadherin monomers into a cis dimer places severe constraints on the inter-domain mobility of both ectodomains such that the spread of allowable values of h, Δθ, and Δψ is significantly reduced, thus resulting in a large entropic penalty for dimerization. In contrast, inter-domain mobility is already reduced in trans dimers so that the additional entropic penalty associated with the cis dimerization of two trans dimers is small compared to that between monomers.

We have previously described the process of adherens junction formation as a phase transition between a dilute phase of monomers and trans dimers that diffuse over the surface of a cell, and a condensed lattice composed of trans dimers, interacting laterally via a well-defined cis interface3. Using lattice simulations we showed that the formation of a condensed ordered phase requires trans and cis interactions of sufficient magnitude. The results of such simulations, using the 2D binding affinities reported in Table S1, illustrate the formation of well-defined lateral clusters (Figure 4). Thus, converting the measured 3D cadherin binding affinities into 2D free energies yields interactions of sufficient strength to drive trans dimer formation, and cis interactions between trans dimers of sufficient strength to drive the formation of ordered clusters of these dimers. That is, the values of Kd(2D) derived here from a combination of experiment, theory and simulations predict that cadherin ectodomains will form junctions, as is observed. In contrast, owing to the one-dimensional nature of cis interaction between monomers (see Figure 1), and because of their small magnitude, monomer oligomerization is negligible3.

Figure 4
Simulation of junction formation

It is important to note that the treatment we present is based entirely on forces localized to the extracellular region. This is justified for cadherins since junction-like structures form when cytoplasmic regions are deleted 10,17. However, as has recently been demonstrated for TCR-MHC interactions, cytoskeletal forces can affect the kinetic and thermodynamic properties of extracellular domains 18. Thus, although we expect cadherin junction formation in vivo to be affected significantly by cytoplasmic involvement, the process is almost certain to depend on the principles of ordered ectodomain assembly uncovered here.

Finally, the concepts and methods introduced in this work should facilitate the analysis of both trans and cis binding interactions between other flexible membrane-bound molecules. For example, Dustin and coworkers have shown that chimeras of CD48 with two or three additional Ig-like domains are 10 times less efficient in adhesion than wild type, despite sharing the same binding interface with CD2 19. The entropic penalty associated with restricting inter-domain motion as a consequence of trans binding provides a simple explanation of these observations and, more generally, offers a mechanism to control binding affinities of membrane-bound receptors that is not available to molecules that are free in solution.

Methods Summary

Monte Carlo simulations are carried out in which cadherins domains, each treated as a rigid body described at the level of Cα atoms, are allowed to move with respect to the membrane surface via random changes in the three Euler angles, Φ, Θ and, Ψ, of the EC5 domain and via motions around the dihedral angles in the hinge regions as indicated in Figure 3 Φ ranges over 360°, while Θ and Ψ are restricted to a limited range (0° in one set of simulations 30° in the other). Motions around the flexible linker regions are described with the Elastic Network Model 20-21 which defines normal modes along which inter-domain motion is allowed. The Block Normal Mode approach 21-22 was applied to partition the structure of cadherin ectodomain into five rigid blocks, each corresponding to one EC domain. The six lowest-frequency modes, each of which describes a collective motion of the entire ectodomain, were used to generate alternate conformations. Fluctuations of the distance between the centers of mass were obtained from MD simulations23 and were used to calibrate the size of the MC steps along the normal modes.

In each MC step, the EC5 domain was allowed to randomly rotate and then the conformation of the whole ectodomain was changed along one of the normal modes starting with the C-cadherin monomer conformation. For trans and cis dimers, two ectodomains were first placed in conformations generated from the crystal structure of C-cadherin15 after which MC steps were taken. Two monomers were defined as forming a dimer if the RMSD obtained from a structural superposition was lower than 6 Å, a value determined from MD simulations23 as preserving the dimer interface. Values of hM, hT, ΔθM, ΔθT, ΔψM and ΔψT were obtained directly from the conformations generated in the MC simulations.

Supplementary Material

Acknowledgments

This work was supported by National Science Foundation Grant MCB-0918535 (to B.H.) and National Institutes of Health Grant R01 GM062270-07 (to L.S.). The Financial support of the US-Israel Binational Science Foundation (Grant No. 2006-401, to A.B.-S., B.H., and L.S.), and the Archie and Marjorie Sherman chair (to A.B.-S.), is gratefully acknowledged. We thank Professor Erich Sackmann for a very helpful email exchange concerning membrane fluctuations.

Footnotes

Full methods are available in the online version of the paper at www.nature.com/nature

Supplementary Information is linked to the online version of the paper at www.nature.com/nature.

Author Contributions: Y.W., J.V., L.S., B.H., and A.B.-S. designed research; Y.W. performed research; J.V. carried out the all atom MD simulations; Y.W., B.H., and A.B.-S. analyzed data; Y.W., A.B.-S. and B.H. contributed analytic tools; and Y.W., L.S., B.H., and A.B.-S. wrote the paper.

Reprints and permissions information is available at www.nature.com/reprints.

The authors declare no competing financial interests.

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