Torsional Network Model: Normal Modes in Torsion Angle Space Better Correlate with Conformation Changes in Proteins

We introduce the torsional network model (TNM), an elastic network model whose degrees of freedom are the torsion angles of the protein backbone. Normal modes of the TNM displace backbone atoms including ${\mathrm{C}}_{\beta }$ maintaining their covalent geometry. For many proteins, low frequency TNM modes are localized in torsion space yet collective in Cartesian space, reminiscent of hinge motions. A smaller number of TNM modes than anisotropic network model modes are enough to represent experimentally observed conformation changes. We observed significant correlation between the contribution of each normal mode to equilibrium fluctuations and to conformation changes, and defined the excess correlation with respect to a simple neutral model. The stronger this excess correlation, the lower the predicted free energy barrier of the conformation change and the fewer modes contribute to the change.

Figure 1

Torsional versus Cartesian collectivity of the four lowest frequency normal modes for 21 proteins performing large conformation changes, using TNM and ANM with ${\mathrm{C}}_{\alpha }$ interactions. The diagonal $y=x$ is plotted for comparison.

Figure 2

Top: For each protein pair, we plot $\kappa [(c^{\alpha }{)}^2]/N_{\mathrm{res}}$, estimating the effective number of modes contributing to the conformation change divided by $N_{\mathrm{res}}$, versus $\rho =r[(c_{\alpha }{\omega }_{\alpha }{)}^2,{\omega }_{\alpha }^{-2}]$. The larger $\rho$, the smaller the predicted energy barrier. Conformation changes that are likely not functional or are regulatory are indicated with concentric symbols. Bottom: Comparison of predicted thermal fluctuations and observed angular conformation change of adenylate kinase (see text). Central figure: open and close conformation (PDB: 4ake, 1ank). $\Delta q_a^2$ of the conformation change is shown in color code, from dark (more rigid) to light (more dynamic). Bottom: thermal fluctuations $⟨q_a^2⟩$ predicted with the ANM (left) and TNM (right). Functional residues are drawn as spheres [31].