Impact of Hydrodynamic Interactions on the Kinetic Pathway of Protein Folding

Protein folding is a fundamental process critical to cellular function and human health, but it remains a grand challenge in biophysics. Hydrodynamic interaction (HI) plays a vital role in the self-organization of soft and biological materials, yet its role in protein folding is not fully understood despite folding occurring in a fluid environment. Here, we use the fluid particle dynamics method to investigate many-body hydrodynamic couplings between amino acid residues and fluid motion in the folding kinetics of a coarse-grained four-$\alpha$-helices bundle protein. Our results reveal that HI helps select fast folding pathways to the native state without being kinetically trapped, significantly speeding up the folding kinetics compared to its absence. First, the directional flow along the protein backbone expedites protein collapse. Then, the incompressibility-induced squeezing flow effects retard the accumulation of non-native hydrophobic contacts, thus preventing the protein from being trapped in local energy minima during the conformational search of the native structure. We also find that the significance of HI in folding kinetics depends on temperature, with a pronounced effect under biologically relevant conditions. Our findings suggest that HI, particularly the short-range squeezing effect, may be crucial in avoiding protein misfolding.

Figure 1

Folding kinetics of four-$\alpha$-helices bundle protein in the absence of HI. Temporal change of order parameter $\chi$ and protein volume for (a) folded and (b) misfolded trajectories in BD simulations (at $k_{\mathrm{B}}T/ϵ=0.6$). The inset of (a) shows a typical initial configuration of a swollen amino acid coil at $k_{\mathrm{B}}{T}_{\mathrm{init}}/ϵ=1$. The protein volume is determined by calculating the convex hull [51] of amino acid residues.

Figure 2

Folding kinetics of four-$\alpha$-helices bundle protein in the presence of HI. Temporal change of order parameter $\chi$ and protein volume for (a),(b) folded and (c) misfolded trajectories in FPD simulations (at $k_{\mathrm{B}}T/ϵ=0.6$). The presence of HI promotes the selection of type I and type II folding pathways and significantly reduces the folding time. We classify folding pathways into two types based on the chronological order of collapse and folding. See text for details.

Figure 3

Characterization of type I and type II folding pathways. We show the results for the folded trajectories $f1$ [Fig. 2; type I folding pathway] and $f4$ [Fig. 2; type II folding pathway]. (a) Temporal change of the number of native $N_{\mathrm{c}}$ (red line), non-native $N_{\mathrm{n}}$ (blue line), interhelix non-native (black line), and intrahelix non-native (magenta line) hydrophobic contact (we use cutoff $3\sigma$) in the trajectory $f1$. (b) Top: temporal change of correct folded helices in the trajectory $f1$. Here, we calculate the order parameter $\chi (t)$ for each of the four helices and regard a helix as folded if $\chi (t)<0.3$. Only when helix $i$ is folded properly (${\chi }_i<0.3$), do we indicate a point on the axis of $\mathrm{ID}i$. To avoid the overlap between different lines, folded helices with ID 1–4 are labeled as 1–4, respectively. Bottom: temporal change of the relative orientation between helix pair 1-2, pair 2-3, and pair 3-4 in the trajectory $f1$. Note that, in the native state, the neighboring helices are almost antiparallel. (c) Simulation snapshots of protein folding at various times in the trajectory $f1$. (d)–(f) The same characterization of the trajectory $f4$.

Figure 4

Roles of hydrodynamics in protein folding. (a) Temporal change of ensemble-averaged protein volume in BD and FPD simulations. The inset shows the growing native hydrophobic contact. (b) The local pressure $P$ of the solvent upon volume shrinking of protein. The solvent is squeezed out from the protein domain to the surroundings to satisfy the incompressibility condition $\nabla ·\mathit{v}=0$. Temporal evolution of ensemble-averaged non-native hydrophobic contact along with the interhelix and intrahelix parts in (c) FPD and (d) BD simulations. The arrows indicate the typical collapsing time determined from (a). (e) Temporal change of interhelix non-native hydrophobic contact between nearest (1–2, 2–3, and 3–4), second-nearest (1–3 and 2–4), and distant (1–4) neighboring helices. (f) Temporal change of order parameter $\chi$ of individual $\alpha$ helix in BD and FPD simulations. (g) Folding kinetics of an isolated $\alpha$ helix. Temporal change of order parameter $\chi$ for a single $\alpha$ helix in FPD and BD simulations. We utilize an initial configuration under $k_{\mathrm{B}}{T}_{\mathrm{init}}/ϵ=1$. The inset shows the native state of a single $\alpha$ helix.

Figure 5

The native conformation of four-$\alpha$-helices bundle protein. The amino acid residues contain hydrophobic (red beads), hydrophilic (blue beads), and neutral ones (yellow beads). The linking section between the four $\alpha$ helices is known as the turning region.

Figure 6

Characterization of misfolded pathways. We show the results for misfolded trajectories $u1$ and $u2$ [Fig. 2]. (a) Temporal change of the number of native $N_{\mathrm{c}}$ (red line), non-native $N_{\mathrm{n}}$ (blue line), interhelix non-native (black line), and intrahelix non-native (magenta line) hydrophobic contacts (we use the cutoff distance of $3\sigma$) in trajectory $u1$. (b) Temporal change of correct folded helices in trajectory $u1$, with only helix 2 folded correctly in the late stage. (c) Simulation snapshots of protein folding at various times in trajectory $u1$. (d)–(f) Similar characterization of trajectory $u2$ [Fig. 2]. In the bottom panel of (e), we show the temporal change in the relative orientation between two helices. While the $\alpha$ helices form in the late stage [(e), top], the relative orientation between helices 2–3 and 3–4 is incorrect.

Figure 7

Folding kinetics of four-$\alpha$-helices bundle protein at the temperature $k_{\mathrm{B}}T/ϵ=0.65$. Temporal change of the order parameter $\chi$ and protein volume for type I, type II, and misfolded pathways in (a)–(c) FPD and (d)–(f) BD simulations. The protein volume is determined by calculating the convex hull [51] of amino acid residues.