Diffusive Dynamics of Contact Formation in Disordered Polypeptides

Experiments measuring contact formation between probes in disordered chains provide information on the fundamental time scales relevant to protein folding. However, their interpretation usually relies on one-dimensional (1D) diffusion models, as do many experiments probing a single distance. Here, we use all-atom molecular simulations to capture both the time scales of contact formation, as well as the scaling with peptide length for tryptophan triplet quenching experiments, revealing the sensitivity of the experimental quenching times to the configurational space explored by the chain. We find a remarkable consistency between the results of the full calculation and from Szabo-Schulten-Schulten theory applied to a 1D diffusion model, supporting the validity of such models. The significant reduction in diffusion coefficient at the small probe separations which most influence quenching rate, suggests that contact formation and Förster resonance energy transfer correlation experiments provide complementary information on diffusivity.

Figure 1

Decays of the tryptophan triplet state. Overall decays calculated from simulations with the ff03* (black), ff03w (red), and ff03ws (green) force fields are shown for peptides ${\mathrm{AGQ}}_n$ for $n=1–6$, together with the corresponding experimental decays (broken purple lines). Symbols: simulation data; lines: single exponential fits to data.

Figure 2

Dependence of quenching times (inverse of quenching rates) on chain length. Top, middle, and bottom rows show the overall, reaction-limited, and diffusion-limited quenching times, respectively. Simulation data for Amber ff03*, ff03w, and ff03ws are shown by black, red, and green symbols, respectively. Filled and empty symbols are for step-function and exponential distance dependence, respectively. Experimental data are shown by purple symbols, and $n_b^{3/2}$ scaling expected for a Gaussian chain by the broken line. Left panels show the rates calculated directly from simulation and right panels show those calculated from the 1D diffusion model using SSS theory.

Figure 3

Distribution of Trp-Cys distance for number of peptide bonds $n_b$ for ${\mathrm{AGQ}}_n$ peptides, for three force fields (cyan shaded curves). Symbols show the mean distance at each number of peptide bonds and purple curves are power law fits.

Figure 4

Position-dependent diffusion coefficients $D(r_{\mathrm{C}\mathrm{W}})$, for each peptide ${\mathrm{AGQ}}_n$, $n=1–6$. Error bars calculated from the division of data into 5 nonoverlapping blocks. Vertical lines indicate the backbone extension for a fully extended chain. Empty symbols for $n=3$ are results from a 5 nm simulation box (vs 4 nm for solid symbols). Horizontal line is constant diffusion coefficient $D_{\text{const}}$ needed to fit the data for $n=3–6$.