Temperature dependence and counter effect of the correlations of folding rate with chain length and with native topology

There is a controversy about the major determinants of the folding rate of small single-domain proteins. To shed light on this issue, we examined a possibility that the major determinants may change depending on temperature by conducting molecular dynamics simulations for 17 small single-domain proteins using an off-lattice Go-like model over a wide range of temperature. It was shown that the rank order of the folding rates is temperature dependent, which indicates that the major determinants are dependent on temperature. It was also found that as temperature is decreased, the correlation of the folding rate with the chain length becomes weakened, whereas that with the native topology becomes enhanced. Our simulation results, therefore, may provide a clue to reconcile the apparent controversy between the study by Plaxco et al. based on experimental data and the previous theoretical and subsequent simulation studies: the former showed that the folding rate of two-state folders does not correlate with the chain length but correlates well with the native topology, whereas the latter showed that the folding rate does correlate with the chain length. We propose a possible scenario reconciling the controversy, explaining the reason why the correlation of the folding rate with the chain length became weakened and that with the native topology became enhanced with decreasing temperature.

Figure 1

Temperature dependence of the folding rates for 17 proteins. To make it easy to identify the data set for each protein, the results are divided into four panels (a)–(d). Lines are drawn for guides for eyes and labeled by PDB codes (see the text).

Figure 2

Scatter plots of the folding rate at $T_{\mathrm{f}}$ against $L$, ACO, $\left(\mathrm{RCO}\right)\times L^{0.6}$, RCO, TCD, and LRO, respectively. See text for terminologies. Lines represent the least-squares linear fits.

Figure 3

The same results as shown in Fig. 2 except that the folding rates are those obtained at lower temperature $\left(0.4T_{\mathrm{f}}\right)$.

Figure 4

Temperature dependences of the correlation coefficients between the folding rate and the quantities examined in Figs. 2, 3. Error bars were estimated by transforming the correlation coefficient (Pearson’s $r$ value) into the Fisher’s $z^{\prime }$ value, which is assumed to obey a normal distribution with the standard error of $1∕\sqrt{N-3}$, where $N$ is the number of the data.

Figure 5

Temperature dependence of the folding rate for the $\alpha$-helix protein group (a) and for the $\beta$-sheet protein group (b) (data are the same as shown in Fig. 1). Plots between the folding rate and the chain length for $\alpha$-helix and $\beta$-sheet protein groups observed at $T_{\mathrm{f}}$ (c) and $0.4T_{\mathrm{f}}$ (d). The lines represent least-squares linear fits obtained, respectively, for the $\alpha$-helix protein group and for the $\beta$-sheet protein group.

Figure 6

Schematic illustration of the temperature dependences of the folding rate for the four typical protein groups: (i) with smaller RCO and smaller $L$, (ii) smaller RCO and larger $L$, (iii) larger RCO and smaller $L$, (iv) and larger RCO and larger $L$. The groups with smaller RCO are colored in light gray, and those with larger RCO are colored in dark gray. The groups with smaller $L$ are indicated by thin lines, and those with larger $L$ are indicated by thick lines.