monte carlo simulations are carried out in which cadherins domains , each treated as a rigid body described at the level of c atoms , are allowed to move with respect to the membrane surface via random changes in the three euler angles ,  ,  and ,  , of the ec5 domain and via motions around the dihedral angles in the hinge regions as indicated in figure 3  ranges over 360 , while  and  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   which defines normal modes along which inter - domain motion is allowed . the block normal mode approach   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 simulations 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 - cadherin 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  , a value determined from md simulations as preserving the dimer interface . values of hm , ht , m , t , m and t were obtained directly from the conformations generated in the mc simulations .
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 .