Definitions of the variables of the UNRES model. The virtual-bond angle θ i is determined by the three Cα carbons at sites i, i + 1, i + 2 and is defined as the angle between the virtual-bond vector and the virtual-bond vector [Eq. (11) ]. It should be noted that, for consistency with the notation of Sec. II B , the angles θ used in this work are complements of the original angles θ, i.e., π − θ (see, e.g. Ref. 39 ). The Cα carbon atoms are represented by small open circles. The virtual-bond-dihedral angle γ i it the angle between the two planes, determined by the Cα at sites (i, i + 1, i + 2) and (i + 1, i + 2, i + 3) [Eqs. (12) and (13) ]. In addition, the UNRES energy function [Eqs. (32)–(35) ] involves the following structure, shown in the Figure: The interaction sites are peptide-bond centers (p), and side-chain ellipsoids of different sizes (SC) attached to the corresponding α-carbons with different virtual-bond lengths b SC . The UNRES energy is also a function of the coordinates of the SC and p sites which are functions of (θ, γ, α, β) and also contains terms that depend explicitly on these angles.
(a) The (N, R g ) distribution of all individual single-chain PDB proteins with resolution less than 2.0 Å and with less than 30% homology. The lower line (ν = 0.37) describes mostly α-helical proteins and the top line is a ν = 3/5 Flory line. There are practically no single-chain proteins above the Flory line. (b) The (N, R g ) distribution for all multi-chain proteins from the current PDB that are above the Flory line. The two clusters given by equation (38) , are clearly visible. The values of R 0 and ν shown in the graphs were determined by linear regression.
The spectrum of the virtual-bond and virtual-torsion angles for 1AIK (top) and 2CUO (bottom), using PDB indexing. The black lines and symbols correspond to the virtual-bond angles θ and the red lines and symbols correspond to the virtual-bond dihedral angles γ, respectively.
The distribution of individual chains on the (N, R g ) plane in the second class of phase coexistent hetero-oligomers found here. The data clearly accumulate around the top line that describes the cluster and the bottom line, the latter describing a Θ-point cluster with best-fit values R 0 = 1.234 and ν = 0.508.
Cartoon representation of the experimental structure of the AICD/Fe65 complex (PDB: 3DXC). Green: The Fe65 (longer) chain. Red: The AICD (shorter) chain. The first and the last residues of each chain are marked. Residue numbers have been taken from the 3DXC structure.
(a) The spectrum of backbone virtual-bond angles θ i (black line) and virtual-torsion angles γ i (red line) for the AICD component of 3DXC (chain B). (b) The same spectrum after application of the transformation of Eq. (22) to reveal the soliton structure. The sites are indexed with residue numbers from the PDB.
The two solitons of 3DXC. Residues are indexed with the numbers from the PDB structure. The black line denotes the residue-wise difference between the coordinates computed from the soliton and those computed from the PDB conformation. The red line denotes the Debye-Waller (one standard deviation) fluctuation distance, computed from the B-factors in the PDB. The grey area describes the estimated 0.15 Å zero-point fluctuation distance around the solitons.
The evolution of the radius of gyration for AICD in 3DXC (chain B), during the propagation of the first soliton from site 680 towards the proline at site 669; see also Figure 6(b) .
The (θ i , γ i ) profile of the 3DXC (chain B) after propagation of the first soliton onto the proline at site 669. See Figure 6 for the initial spectrum. The black lines and symbols correspond to the virtual-bond angles θ and the red lines and symbols correspond to the virtual-bond dihedral angles γ, respectively.
Plots of the time evolution of the radius of gyration of isolated AICD (top) and isolated Fe65 (bottom) during Langevin molecular dynamics simulation with UNRES.
First class of phase co-existent heterodimers. Single proteins but with multiple subchains that are in different phases.
Second class of phase co-existent heterodimers; more than one protein with subchains in different phases.
Parameter values for the two solitons implied in Figure 6(b) . For virtual-bond angles, Eq. (31) is used. For virtual-torsion angles Eq. (28) is used. It should be noted that the values of both θ and γ in these two equations are defined mod (2π). The large values of M enable us to describe the irregular structures in Figure 6(b) . These irregularities are due entirely to multi-valuedness of the angular variables.
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