Ere identified as having ring structures. We did 
not include oligomers containing homodimers as the unit of the symmetry 
to examine effects solely due to circular symmetry.Elastic Network Model (ENM) and Normal Mode Analysis of TRAPThe potential energy of the ENM was defined as the sum of Hookean pairwise energy functions [20,21], 2 P 0 V =2 ?Dr0 DRc Dri,j D{Dri,j D , where ri,j ri {rj denotesi,jthe vector connecting atoms i and j, r0 is the vector of the i,j reference Acid Yellow 23 custom synthesis structure (see below), and Rc is the cut-off distance. The strength of the potential, K, is an arbitrary constant assumed to be 22948146 independent of the atom type. The normal modes were obtained by the diagonalization of the (mass-weighted) Hessian matrix H under the harmonic approximation of the potential energy, V =2 t Hq, where q is the (mass-weighted) Cartesian coordinates of the atoms, and the superscript t denotes transposition. The covariance matrix C of the fluctuations of the Cartesian coordinates was obtained by C kB TH{1 , where kB is the Boltzmann constant and T is temperature. In the normal mode analysis, to eliminate the influences of crystal packing and to obtain a structure with perfect rotational symmetry, we calculated a reference structure r0 by energy i,j minimization using the symmetry operator in the IMAGE facility of CHARMM (version c35b1) [36]. The PDB structures used in the analysis were the 11-mer wild-type TRAP from B. stearothermophilus (PDB code: 1C9S chain A [37]) and the engineered 12-mer TRAP (PDB code: 2EXS chain B [18]). These chains were used as the subunits of the two TRAP models. To make the chain length the same for both TRAPs, we used only the coordinates of residues 7?2, and ignored residues 1?, 73?6, and the linker peptides in the 12-mer. In the minimization, the CHARMM 22 force field [38] with CMAP corrections [39] was used. A distance-dependent dielectric constant was applied to account for solvent screening. After the 100 steps of steepest descent minimization, the coordinates of Ca atoms in the purchase ITI007 minimized structures were used as the reference structures of the ENM. The Ca RMSD between ?the subunits of the 11-mer and 12-mer structures was 0.741 A.Where uk comprises an orthonormal basis for the conformation space of a single subunit fuk ; k 1, . . . ,3Nsubunit g (Nsubunit is the number of Ca atoms in a subunit), R represents the rotation of 2p=n around the symmetry axis, v exp?pi=n ? and the asterisk denotes the complex conjugate. Since the irreducible representation Tp is complex, the complex subspaces ep and its complex conjugate e?must be combined to p give a physically meaningful symmetry subspace of double the dimension [26]. In the case of C11, since q1? q1 ,q2? q11 ,q3? q10 , . . . ,q6? q7 , the real physically meank k k k k k k k ??are ingful irreducible representations T’ p fT’ T1 ,T’ T2 zT11 , T’ T3 zT10 , . . . ,T’ 15755315 T6 zT7 g. The 1 2 3 6 first subspace, T’1 , contains 3Nsubunit degrees of freedom, including the global translation and rotation, while the other subspaces, fT’ , . . . ,T’ g, contains 6Nsubunit degrees of freedom (T’2 includes 2 6 the translations and rotations) and doubly degenerate normal modes. For C12, they are fT’ T1 ,T’ T2 zT12 , 1 2 T’ = T3 zT11 , . . . ,T’ T6 zT8 ,T’ T7 g. Simonson and Perahia 3 6 7 [26] showed that a normal mode with frequency f in the subspace T’ of the Cn group produces a displacement of the subunit m of the p form: cos?pft m{1 Xkk cos {1 p{1 ??zBk sin {1 p{1 uk , where.Ere identified as having ring structures. We did not include oligomers containing homodimers as the unit of the symmetry to examine effects solely due to circular symmetry.Elastic Network Model (ENM) and Normal Mode Analysis of TRAPThe potential energy of the ENM was defined as the sum of Hookean pairwise energy functions [20,21], 2 P 0 V =2 ?Dr0 DRc Dri,j D{Dri,j D , where ri,j ri {rj denotesi,jthe vector connecting atoms i and j, r0 is the vector of the i,j reference structure (see below), and Rc is the cut-off distance. The strength of the potential, K, is an arbitrary constant assumed to be 22948146 independent of the atom type. The normal modes were obtained by the diagonalization of the (mass-weighted) Hessian matrix H under the harmonic approximation of the potential energy, V =2 t Hq, where q is the (mass-weighted) Cartesian coordinates of the atoms, and the superscript t denotes transposition. The covariance matrix C of the fluctuations of the Cartesian coordinates was obtained by C kB TH{1 , where kB is the Boltzmann constant and T is temperature. In the normal mode analysis, to eliminate the influences of crystal packing and to obtain a structure with perfect rotational symmetry, we calculated a reference structure r0 by energy i,j minimization using the symmetry operator in the IMAGE facility of CHARMM (version c35b1) [36]. The PDB structures used in the analysis were the 11-mer wild-type TRAP from B. stearothermophilus (PDB code: 1C9S chain A [37]) and the engineered 12-mer TRAP (PDB code: 2EXS chain B [18]). These chains were used as the subunits of the two TRAP models. To make the chain length the same for both TRAPs, we used only the coordinates of residues 7?2, and ignored residues 1?, 73?6, and the linker peptides in the 12-mer. In the minimization, the CHARMM 22 force field [38] with CMAP corrections [39] was used. A distance-dependent dielectric constant was applied to account for solvent screening. After the 100 steps of steepest descent minimization, the coordinates of Ca atoms in the minimized structures were used as the reference structures of the ENM. The Ca RMSD between ?the subunits of the 11-mer and 12-mer structures was 0.741 A.Where uk comprises an orthonormal basis for the conformation space of a single subunit fuk ; k 1, . . . ,3Nsubunit g (Nsubunit is the number of Ca atoms in a subunit), R represents the rotation of 2p=n around the symmetry axis, v exp?pi=n ? and the asterisk denotes the complex conjugate. Since the irreducible representation Tp is complex, the complex subspaces ep and its complex conjugate e?must be combined to p give a physically meaningful symmetry subspace of double the dimension [26]. In the case of C11, since q1? q1 ,q2? q11 ,q3? q10 , . . . ,q6? q7 , the real physically meank k k k k k k k ??are ingful irreducible representations T’ p fT’ T1 ,T’ T2 zT11 , T’ T3 zT10 , . . . ,T’ 15755315 T6 zT7 g. The 1 2 3 6 first subspace, T’1 , contains 3Nsubunit degrees of freedom, including the global translation and rotation, while the other subspaces, fT’ , . . . ,T’ g, contains 6Nsubunit degrees of freedom (T’2 includes 2 6 the translations and rotations) and doubly degenerate normal modes. For C12, they are fT’ T1 ,T’ T2 zT12 , 1 2 T’ = T3 zT11 , . . . ,T’ T6 zT8 ,T’ T7 g. Simonson and Perahia 3 6 7 [26] showed that a normal mode with frequency f in the subspace T’ of the Cn group produces a displacement of the subunit m of the p form: cos?pft m{1 Xkk cos {1 p{1 ??zBk sin {1 p{1 uk , where.
