Hydrodynamic Description of Protein Folding

A hydrodynamic description of protein folding is proposed and illustrated with a lattice protein model, which has a free energy surface (FES) typical of proteins with two-state folding kinetics. The flows from the unfolded to the native state are concentrated in a limited region of the FES. The rest is occupied by a flow “vortex”, which does not lead to the native state. In contrast with intermediates that are associated with local minima, the vortex is not visible on the FES. The hydrodynamic interpretation thus provides new insights into the mechanism of protein folding and can be a useful complement to standard analyses.

Figure 1

Native structure of the model lattice protein.

Figure 2

Free energy surface for the model protein, $T=0.575$. The solid curves show the streamlines $\Psi (q,r)=\mathrm{const}$ [Eq. (2)] calculated on the basis of the $q$ component of the flow vector $\mathbf{j}$ [see Fig. 3 and Eq. (1)]. The lower and upper black thick lines correspond to $\Psi (q,r)=0$ and $\Psi (q,r)=1$, respectively, and the thin lines between them to $\Psi (q,r)=0.1,0.2,\dots ,0.9$. The white line shows the flow vortex and corresponds to $\Psi (q,r)=1.01$.

Figure 3

The map of flow vectors $\mathbf{j}=(j_q,j_r)$, $T=0.575$. To make small flows visible, the magnitude of each vector is increased by a factor of 10; the very small vectors are not represented consistently because of size of the arrow heads.

Figure 4

An example of folding trajectories, $T=0.575$. Straight lines bridge the points separated by the time interval of $4\times {10}^3$ MC steps.

Figure 5

Time dependence of the fractions of native helical and interhelical contacts corresponding to the folding trajectory of Fig. 4. In the native state, they are equal approximately to 0.76 and 0.21, respectively, (32 helical and 9 interhelical contacts, Fig. 1). Structures (a)–(d) depict the conformations occurred at 0 (initial structure), $5.8\times {10}^5$, $1.04\times {10}^6$, and $1.16\times {10}^6$ (native structure) MC steps, respectively.