Figure 3.

Internal rotation and wobbling of the proximal and distal domains of Ub₂ on the surface of lipid-based nanoparticles. (A) K63- and (B) K48-linked Ub₂. The distal and proximal domains of Ub₂ are shown as red and blue ribbons, respectively, with Ub₁ in gray for comparison. Each domain rotates about an internal axis of rotation, approximately perpendicular to the lipid bilayer, and simultaneously wobbles in a cone centered around this axis. (C) Relationship between the axes (−x, −y, and z) of the inertia tensor (black) in the molecular frame with the internal rotation axes of Ub₁ (green) and the distal (red) and proximal (blue) domains of K48-linked Ub₂. θ is the angle subtended by each internal rotation axis and the z axis of the intertia tensor; φ (not shown) is the angle subtended by the x axis of the inertia tensor and the projection of the internal rotation axis on the x–y plane. The angle formed between the internal rotation axis of Ub₁ (green) and those of the distal (red) and proximal (blue) domains of Ub₂ are denoted by ω′ and ω″, respectively. These angles are calculated from the polar angles (θ_i, φ_i) defining the position of axis i (i = 1,2) in the inertia frame using the relationship cos(ω) = cos(θ₁) cos(θ₂) + sin(θ₁) sin(θ₂)cos(φ₁ − φ₂). The U-[¹⁵N/²H]-labeled domains of Ub₂ are shown as filled-in circles in the cartoons in panels A and B.