Dynamics of essential collective motions in proteins: Theory

A general theoretical background is introduced for characterization of conformational motions in protein molecules, and for building reduced coarse-grained models of proteins, based on the statistical analysis of their phase trajectories. Using the projection operator technique, a system of coupled generalized Langevin equations is derived for essential collective coordinates, which are generated by principal component analysis of molecular dynamic trajectories. The number of essential degrees of freedom is not limited in the theory. An explicit analytic relation is established between the generalized Langevin equation for essential collective coordinates and that for the all-atom phase trajectory projected onto the subspace of essential collective degrees of freedom. The theory introduced is applied to identify correlated dynamic domains in a macromolecule and to construct coarse-grained models representing the conformational motions in a protein through a few interacting domains embedded in a dissipative medium. A rigorous theoretical background is provided for identification of dynamic correlated domains in a macromolecule. Examples of domain identification in protein G are given and employed to interpret NMR experiments. Challenges and potential outcomes of the theory are discussed.

Figure 1

The number of domains in fragment B1 of protein G containing more than two atoms, as a function of the interdomain distance $d$ (a); the number of atoms in all domains, $N_{\mathrm{tot}}$ (solid line), and the number of atoms in the largest domain, $N_1$ (dashed line), as functions of $d$ (b); and the difference $N_{\mathrm{tot}}-N_1$ (c) [42].

Figure 2

(Color) Three largest domains identified in protein G for $k_{\max }=10$: general view (a) and schematic (b). The domains are shown with green, red, and blue.

Figure 3

A sketch illustrating identification of domains with different numbers of essential coordinates $k_{\max }$. The plane $\left\{E_1\text{−}{E}_2\right\}$ represents a “high-dimensional” essential space, and the axis $E_1$ represents a “low-dimensional” subspace. The dashed lines indicate the hypothetic domains identified with the different dimensionalities.

Figure 4

(Color) Comparison of three largest domains identified in protein G for $k_{\max }=1$, 2, 5, 10, 20, and 40 [45].

Figure 5

Experimental NMR-derived order parameters $S^2$ for streptococcal protein G [57]: (a) adapted from Fig. 7 of Ref. 55 and reprinted in part with permission of The National Academy of Sciences of the USA; (b) adapted from Fig. 1(a) of Ref. 52 and reprinted in part with permission of the American Chemical Society. In (a), the bold line shows the relaxation-derived order parameter in the subnanosecond time scale [54] and thin line shows the order parameter derived from RDC data in the submillisecond scale [55]. In (b), relaxation data were employed representing the subnanosecond regimes [52]. The circles, squares, and triangles correspond to different mutants introduced to minimize cross-strand interactions [52].

Figure 6

The domain order parameters $S_D$ computed for $k_{\max }=10$ and progressively increasing interdomain distance $d$. See Fig. 7 for the largest domains identified in each case.

Figure 7

(Color) Comparison of the largest domains in protein G for various interdomain distances $d$ and $k_{\max }=10$.