Soliton concepts and protein structure

Structural classification shows that the number of different protein folds is surprisingly small. It also appears that proteins are built in a modular fashion from a relatively small number of components. Here we propose that the modular building blocks are made of the dark soliton solution of a generalized discrete nonlinear Schrödinger equation. We find that practically all protein loops can be obtained simply by scaling the size and by joining together a number of copies of the soliton, one after another. The soliton has only two loop-specific parameters, and we compute their statistical distribution in the Protein Data Bank (PDB). We explicitly construct a collection of 200 sets of parameters, each determining a soliton profile that describes a different short loop. The ensuing profiles cover practically all those proteins in PDB that have a resolution which is better than 2.0 Å, with a precision such that the average root-mean-square distance between the loop and its soliton is less than the experimental $B$-factor fluctuation distance. We also present two examples that describe how the loop library can be employed both to model and to analyze folded proteins.

Figure 1

The distribution of the parameter values $c_1$ and $c_2$ in Eq. (15) in the 200 solitons we have constructed. As expected, these two distributions are practically identical.

Figure 2

The distribution of of the parameter values $m_1$ and $m_2$ in Eq. (15) for the 200 solitons. As in Fig. 1 the distributions are highly symmetric; the difference is not statistically meaningful. The $\alpha$ helices and $\beta$ strands (8) are also clearly identifiable in the parameter values.

Figure 3

The distribution of of the parameter values for the torsion angle (12) in our 200 solitons. Observe that the parameter $e$ clusters in two regions, around $-1$ and below $-{10}^{-4}$. Furthermore, the parameter $b$ has very large values, in excess of $\pm$10${}^6$, and the spread is very large. The fundamental region of the torsion angle ${\tau }_i$ is $[-\pi ,\pi )$, and the large values reveal that, as solitons, the loops cover the spheres $(\kappa ,\tau )\sim {\mathbb{S}}^2$ several times, i.e., each of the loops is a multiple-soliton configuration. This explains why a very regular soliton such as (15),(12) can model the apparently highly irregular ${\kappa }_i$ and ${\tau }_i$ profiles such as those we commonly find in the PDB.

Figure 4

The distribution of the RMSD between the original 200 loops and their soliton approximations.

Figure 5

An example showing how our 200 explicit solitons cover the 289-amino-acid protein with PDB code 1KZQ in our data set, in terms of the bond angles ${\kappa }_i$. We use cutoff values 0.3 Å (top two panels), 0.5 Å (middle two panels), and 0.7 Å (bottom two panels). Gray (green) dots and lines correspond to sites and structures that are described by the solitons with the cutoff accuracy or better, while black dots and lines correspond to sites where the local distance exceeds the cutoff value; isolated black dots indicate local fluctuations in $B$ factors. Three or more consecutive black dots indicate the presence of a loop that is not covered by our 200 solitons. Note that at a resolution of 0.7 Å (bottom two panels) there is only one such loop.

Figure 6

(a) shows the PDB data for the bond and torsion angles for the monomer A in 3DLK, for sites 398–414. (b) displays the ${\kappa }_i$ profile, after we introduced the gauge transformation (9). As explained in the text, the two solitons are located between sites 405 and 412. In (c) we compare the $B$ factor of the 3DLK (upper line in red) with the distance between its backbone and the two-soliton configuration (lower line in black). The shaded area around the two-soliton configuration is the 0.15 Å fluctuation regime around the soliton. In (d) we compare the (${\kappa }_i,{\tau }_i$) distributions for the PDB data (black and red for sites 398–416) and the two-soliton configuration (blue and green for sites 405–412). There is practically no difference.