Dynamics of the minimally frustrated helices determine the hierarchical folding of small helical proteins

In this paper we aim at determining the key residues of small helical proteins in order to build up reduced models of the folding dynamics. We start by arguing that the folding process can be dissected into concurrent fast and slow dynamics. The fast events are the quasiautonomous coil-to-helix transitions occurring in the minimally frustrated initiation sites of folding in the early stages of the process. The slow processes consist in the docking of the fluctuating helices formed in these critical sites. We show that a neural network devised to predict native secondary structures from sequence can be used to estimate the probabilities of formation of these helical traits as they are embedded in the protein. The resulting probabilities are shown to correlate well with the experimental helicities measured in the same isolated peptides. The relevance of this finding to the hierarchical character of folding is confirmed within the framework of a diffusion-collision-like mechanism. We demonstrate that thermodynamic and topological features of these critical helices allow accurate estimation of the folding times of five proteins that have been kinetically studied. This suggests that these critical helices determine the fundamental events of the whole folding process. A remarkable feature of our model is that not all of the native helices are eligible as critical helices, whereas the whole set of the native helices has been used so far in other reconstructions of the folding mechanism. This stresses that the minimally frustrated helices of these helical proteins comprise the minimal set of determinants of the folding process.

Figure 1

Information entropy profile vs residue number of the crystallized fragment (255–316) of thermolysine (PDB code, 1TRL) calculated according to the procedure of Ref. 20 (see text). The step function superimposed to the curve indicates the location of the $\alpha$-helices predicted by the neural network. The ISs of folding are defined as the segments classified as helical regions by the neural network that correspond to the minima of the entropy plot that lie below a threshold entropy $S=0.416$ [20, 21]. The ISs comprise those residues that deviate from the entropy minimum by less than 0.05 [20]. The predicted helices 282–294 and 300–311 contain two minima that fulfill this criterion. The ISs span the 287–291 and 303–306 regions (black bars). The first helix does not include any IS.

Figure 2

Derivation of thermodynamic parameters of the IS helices from the information entropy profile, through a graphical estimation of Eq. (6). The curve reproduces a stretch of the entropy plot corresponding to the last NIS helix of protein 1HRC (see Table and Fig. 4). The entropy minimum $S\left(x_0\right)$ is found in $x_0=96$. The horizontal dashed line signals the average entropy $⟨S⟩$ of the NIS helix under study. The residues are divided in two subsets $\mathcal{A}=\left\{\text{residues}\text{with}S<⟨S⟩\right\}$ and $\mathcal{B}=\left\{\text{residues}\text{with}S⩾⟨S⟩\right\}={\mathcal{B}}_L\cup {\mathcal{B}}_R$. The horizontal dotted line represents the average entropy ${⟨S⟩}_A$ calculated over the set $\mathcal{A}$. The vertical arrow represents $1-{⟨S⟩}_A$ that estimates $K_{nucl}$. This amounts to setting $S_{\mathit{min}}={⟨S⟩}_A$ in the graphical estimate of $K_{nucl}\approx 1-S_{\mathit{min}}$ referred to in the discussion to Eq. (6). This choice has given optimal results in implementing the DC model. As explained in the text, the alternative choice $S_{\mathit{min}}=S\left(x_0\right)=S\left(96\right)$ made on comparing CD data and calculated helicities (see Fig. 3), implies that the set $\mathcal{A}$ has shrunk to $x_0$. To compute the average slopes ${\mathbf{\nabla }}^{L}S$ and ${\mathbf{\nabla }}^{R}S$ of the entropy profile (dashed lines in ${\mathcal{B}}_L$ and ${\mathcal{B}}_R$), a least squares procedure has been applied to the data in ${\mathcal{B}}_L$ and ${\mathcal{B}}_R$.

Figure 3

Comparison of the helicities $H\left(NN\right)$ calculated through the neural network (NN) and experimental helicities $H\left(CD\right)$. Circular dichroism values (CD) were measured on the isolated peptides indicated in the panel. The labels for ARA, COMA, CIII, 3LZM, and Sigma are the same used in the original paper reporting the CD values (see Table 3 of Ref. 34). In addition we have considered the data for two helical peptides of the thermolysine segment (shown in Fig. 1) (spanning the 281–295 and the 301–310 regions), that were drawn from Ref. 35, and the H and G helices of myoglobin (PDB code, 1MBD), that were studied in Ref. 36. The NN values were estimated by processing the full sequence of the above mentioned proteins with our neural network-based procedure. The experimental helicities $H\left(CD\right)$ are usually expressed as mean helicity of each of the $N$ residues of the peptide, where $N$ is the number of the helical residues as determined from crystallographic data. To remedy the discrepancies between $N$ and the number of helical residues predicted by the neural network, $n$, we set $H\left(CD\right)N=n\beta$. The scatter plot compares $H\left(CD\right)$ with $H\left(NN\right)=\beta n∕N$ ($H\left(CD\right)$ and $H\left(NN\right)$ have been expressed in percent). The equation in the left corner refers to the least square line.

Figure 4

Entropy profiles of 2ABD and 1HRC, based on the secondary structure prediction of the neural network. The average entropy of the ISs of 1HRC, estimated at the minima of the 60s and 90s NIS helices (see Table ) is clearly higher than the average level of the entropy minima of the NIS helices 1,2, and 4 of 2ABD. The effects of noise in the entropy signal are visible in that two very short helices are not predicted by the neural network.