Force-Clamp Experiments Reveal the Free-Energy Profile and Diffusion Coefficient of the Collapse of Protein Molecules

We present force-clamp data on the collapse of ubiquitin polyproteins from a highly extended state to the folded length, in response to a quench in the force from 110 pN to 5 or 10 pN. Using a recent method for free-energy reconstruction from the observed nonequilibrium trajectories, we find that their statistics is captured by simple diffusion along the end-to-end length. The estimated diffusion coefficient of $\sim 100{\mathrm{nm}}^2{\mathrm{s}}^{-1}$ is significantly slower than expected from viscous effects alone, possibly because of the internal degrees of freedom of the protein. The free-energy profiles give validity to a physical model in which the multiple protein domains collapse all at once and the role of the force is approximately captured by the Bell model.

Figure 1

A typical force-clamp trajectory of the unfolding and refolding of a polyubiquitin chain with $N_d=3$ domains. The circled numbers indicate the various stages of the pulling experiment, with a schematic representation of the protein conformation in each stage shown in the inset. Here we are interested in the collapse from stage 3 to 5: for details on the other stages, see Sec. 1 of the Supplemental Material [19].

Figure 2

Collapsing trajectories are grouped by their total length ($N_d=3$) and recorded from the time of the force quench from 110 to 10 pN in (a) and 5 pN in (b). The experimental trajectories are compared with those generated by simulations of diffusive dynamics on the reconstructed free-energy profiles.

Figure 3

The nonequilibrium distribution $\rho (x)$ for each $N_d$ collected at a force quench of 10 pN in (a) and 5 pN in (b). The linear rescaling by $N_d$ is shown in the inset, which indicates a cooperative mechanism for the collapse.

Figure 4

(a) Experimental free-energy profiles as a function of the end-to-end length, rescaled by $N_d$. (b) Diffusion coefficients $D(x)$ derived from Eq. (5) at 10 pN (solid lines) compare well with those estimated from the free-energy reconstruction in Eq. (2) (dashed lines). The inset shows ${\tau }_c$ dependence on $N_d$ (squares), consistent with simulated data (circles).