de Gennes Narrowing Describes the Relative Motion of Protein Domains

The relative motion of structural domains is essential for the biological function of many proteins. Here, by analyzing neutron scattering data and performing molecular dynamics simulations, we find that interdomain motion in several proteins obeys the principle of de Gennes narrowing, in which the wave vector dependence of the interdomain diffusion coefficient is inversely proportional to the interdomain structure factor. Thus, the rate of interdomain motion is inversely proportional to the probability distribution of the spatial configurations of domains.

Figure 1

$D_{\mathrm{inter}}(q)$ versus $S(q)$ for PGK derived from neutron scattering experiments. (a) Structure of PGK consisting of two domains (left and right) of almost equal mass connected by a helix (middle). (b) $D_{\text{inter}}(q)$ of PGK in aqueous solution (see the Supplemental Material [11] and Fig. S1). The value at the lowest $q$ was measured by dynamic light scattering while all the other data were collected using NSE ([4]). (c) Interdomain structure factor. The form factor of the whole protein molecule was measured by SANS [4] and that for each single domain was calculated based on the crystal structure (PDB entry 3PGK [20]).

Figure 2

$D_{\text{inter}}(q)$ versus $S(q)$ for PGK derived from MD simulations. (a) Effective diffusion coefficient, $D_{\text{inter}}(q)$. The empty squares denote the protein internal dynamics derived from the trajectory obtained by superimposing the MD trajectories of the protein molecule onto a reference structure. $D_{\text{inter}}(q)$ with the intradomain fluctuations subtracted, i.e., representing the interdomain motion only, is plotted as solid squares and was obtained by further fitting the structure of the two domains [blue and red in Fig. 1] in the reference state independently to the corresponding domains in each frame of the superimposed trajectory. (b) Interdomain structure factor.

Figure 3

Schematic illustration of (a) model A and (b) model B. (c) $D_{\text{eff}}(q)$ calculated from models A and B. (d) $D_{\text{eff}}(q)$ versus $1/S(q)$ derived from model A. $S(q)$ calculated from the two models are identical as global rotation does not alter the relative position of the two spheres and has, thus, no effect on the structure factor.

Figure 4

Test of DGN using analysis of MD trajectories of PGK. (a) Distribution of interdomain distances and the corresponding potential of mean force. (b) Time-averaged velocity of the interdomain motion and the curvature of the potential of mean force. The position of $r^{*}$ as noted in the text is marked by a dashed line. Figure 4 displays the absolute values.