Contact order revisited: influence of protein size on the folding rate

Abstract

Guided by the recent success of empirical model predicting the folding rates of small two-state folding proteins from the relative contact order (CO) of their native structures, by a theoretical model of protein folding that predicts that logarithm of the folding rate decreases with the protein chain length L as L(2/3), and by the finding that the folding rates of multistate folding proteins strongly correlate with their sizes and have very bad correlation with CO, we reexamined the dependence of folding rate on CO and L in attempt to find a structural parameter that determines folding rates for the totality of proteins. We show that the Abs_CO = CO x L, is able to predict rather accurately folding rates for both two-state and multistate folding proteins, as well as short peptides, and that this Abs_CO scales with the protein chain length as L(0.70 +/- 0.07) for the totality of studied single-domain proteins and peptides.

Figure 1.

Natural logarithm of observed folding rate in water, ln(k_f), versus relative contact order (CO) for various proteins and peptides: proteins having two-state folding kinetics at all the denaturant concentrations (circles ), proteins having multistate folding kinetics in water (and at low denaturant concentrations; triangles ), and short peptides (crosses ). The figure includes peptides and proteins listed in Table 1; CO is computed after Equation 1 from the PDB coordinates (Bernstein et al. 1977). If several folding rates are observed for some protein (see Table 1), ln(k_f) is the mean value of their natural logarithms. The dashed line represents the best linear fit for two-state folders only (the negative correlation coefficient is as significant as −0.75; the fitted dependence is y = 16.94 − 0.76x); the dotted line represents the best linear fit for multistate folders only (the correlation coefficient is +0.26; namely, it has the opposite sign compared with that for the two-state folders; y = −1.55 + 0.26x); the solid line represents the best linear fit for the totality of all peptides and proteins presented (the correlation coefficient is insignificant, +0.10 only; y = 2.37 + 0.10x).

Figure 2.

Logarithm of relative contact order versus logarithm of chain length. See legend to Figure 1 ▶ for specification of the symbols and other details. The dashed line represents the best linear fit for two-state folders only (the correlation coefficient is 0.02; y = 2.68 + 0.01x); the dotted line represents the best linear fit for multistate folders only (the correlation coefficient is −0.54; y = 4.41 − 0.40x); the solid line represents the best linear fit for the totality of all peptides and proteins (the correlation coefficient is −0.50; y = 3.95 − 0.30x). The linear regression coefficients 0.01, −0.40, and −0.30 are determined with errors ±0.16, ±0.13, and ±0.07, respectively.

Figure 3.

Logarithm of observed folding rate in water ln(k_f) versus Abs_CO = CO × L. See legend to Figure 1 ▶ for specification of the symbols and other details. The dashed line represents the best linear fit for two-state folders only (the fitted dependence is y = 9.44 − 0.36x; the correlation coefficient is −0.51); the dotted line represents the best linear fit for multistate folders only (the fitted dependence is y = 8.56 − 0.44x; the correlation coefficient is −0.78); the solid line represents the best linear fit for the totality of all peptides and proteins presented (the fitted dependence is y = 11.15 − 0.54x; the correlation coefficient is −0.74). (Inset) Correlation coefficients between the logarithm of experimental folding rate in water and the value of CO × L^P depending on the value of power P. Error bars, standard errors in correlation coefficients. The curve "2-STATE" concerns the two-state folding proteins only, and the curve "ALL" concerns the totality of all studied peptides and proteins; the curve for the multistate folding proteins is not shown because it is close to the curve "ALL" up to the standard error in correlation coefficients.