Selecting fast-folding proteins by their rate of convergence

We propose a general method for predicting potentially good folders from a given number of amino acid sequences. Our approach is based on the calculation of the rate of convergence of each amino acid chain towards the native structure using only the very initial parts of the dynamical trajectories. It does not require any preliminary knowledge of the native state and can be applied to different kinds of models, including atomistic descriptions. We tested the method within both the lattice and off-lattice model frameworks and obtained several so far unknown good folders. The unbiased algorithm also allows one to determine the optimal folding temperature and takes at least 3–4 orders of magnitude fewer time steps than those needed to compute folding times.

Figure 1

Thick solid line: The normalized rate of convergence versus temperature for the designed sequence S${}_0$ for the time period $t_0=500$. Dash-dotted and thin solid line: The same for the sequences S${}_1$ and S${}_3$, respectively. Note that the folding temperature of S${}_3$ is approximately $1.2T_f\simeq 30$ as can be seen from Figs. 9(a) and 9(b) in [25]. Dashed line: The normalized rate of convergence $S_{\mathrm{bad}}$ of a typical bad folder (in this case a homopolymer). The vertical dotted line corresponds to the folding temperature of S${}_0$. The temperature is given in dimensionless Miyazawa-Jernigan units multiplied by 100

Figure 2

Normalized rate of convergence $R_N(t_0,T)$ vs time step $t_0$ for fixed temperature $T=0.001987k_B^{-1}$ of the 6 analyzed sequences in the off-lattice model (see the text). For each point, $R_N(t_0,T)$ was calculated averaging over 100 conformation pairs. Inset: First stages of the time development of $R_N(t_0,T)$. The different behavior of good and bad folders is already evident.

Figure 3

Upper panel: Temperature dependence of the normalized rate of convergence $R_N(t_0,T)$ for the 6 sequences considered within the off-lattice model. $t_0={10}^7$ time steps. Lower panel: Specific-heat curves of the sequences SEQ1, SEQ2, and SEQ3, characterized as good folders by our method.

Figure 4

Time evolution of the root-mean-square deviation $\theta$ of sequences SEQ1, SEQ2, and SEQ3 (with respect to their global minimum structures). The value of $\theta$ was computed according to (5). Time axis is plotted in logarithmic scale. $\theta$ decays exponentially in time for the three sequences. Threshold value of 3.9 Å is denoted by the dotted line. SEQ2 shows the fastest folding Monte Carlo dynamics followed by SEQ1 and SEQ3 (folding times are given in text).

Figure 5

Native-state conformations for some of the sequences designed using the rate-of-convergence method developed in this work. Lower panel: S${}_2$ (left) and S${}_3$ (right) obtained in the framework of the lattice model. Dotted lines connect those monomers that are in contact. Upper panel: SEQ2 and SEQ3, designed within the off-lattice model.