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Collapse Kinetics and Chevron Plots from Simulations of Denaturant-Dependent Folding of Globular Proteins

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Collapse Kinetics and Chevron Plots from Simulations of Denaturant-Dependent Folding of Globular Proteins Collapse kinetics and chevron plots from simulations of denaturant-dependent folding of globular proteins Zhenxing Liua, Govardhan Reddya, Edward P. O’Briena, and D. Thirumalaia,b,1 aBiophysics Program, Institute for Physical Science and Technology, University of Maryland, College Park, MD 20742; and bDepartment of Chemistry, University of Maryland, College Park, MD 20742 Edited* by William A. Eaton, National Institutes of Health-National Institute of Diabetes and Digestive and Kidney Diseases, Bethesda, MD, and approved March 8, 2011 (received for review December 24, 2010) Quantitative description of how proteins fold under experimental (folding arm) followed by an increase (unfolding arm) at high conditions remains a challenging problem. Experiments often use ½C. In previous computational studies, chevron plots were gen- urea and guanidinium chloride to study folding whereas the nat- erated by mapping temperature or certain interaction parameters ural variable in simulations is temperature. To bridge the gap, we in the force-field to ½C (16). Although these studies have pro- use the molecular transfer model that combines measured denatur- vided insights into the role of nonnative interactions they fall ant-dependent transfer free energies for the peptide group and short of directly probing folding of proteins in the presence of amino acid residues, and a coarse-grained Cα-side chain model denaturants. Here, we solve the long-standing problem of how for polypeptide chains to simulate the folding of src SH3 domain. to calculate chevron plots, and in the process describe the entire Stability of the native state decreases linearly as ½C (the concentra- folding reaction of a small protein in near quantitative detail. tion of guanidinium chloride) increases with the slope, m, that is in Molecular simulations of coarse-grained off-lattice models (14, excellent agreement with experiments. Remarkably, the calculated 24, 25), are ideally suited to make (semi)quantitative predictions folding rate at ½C¼0 is only 16-fold larger than the measured of the folding reaction. A combination of statistical mechanical k k value. Most importantly ln obs ( obs is the sum of folding and approach and simulations using simple off-lattice models has unfolding rates) as a function of ½C has the characteristic V (chev- provided insights into protein folding mechanisms. In particular, ron) shape. In every folding trajectory, the times for reaching the these approaches have shown how linear free energy relations for native state, interactions stabilizing all the substructures, and the seemingly complex folding reactions emerge naturally (26). BIOPHYSICS AND mf COMPUTATIONAL BIOLOGY global collapse coincide. The value of m (mf is the slope of the fold- More importantly, the theoretical ideas rationalize the robustness ing arm of the chevron plot) is identical to the fraction of buried of folding in terms of topology alone as illustrated in the simula- solvent accessible surface area in the structures of the transition tions of structure-based models of small globular proteins (27). state ensemble. In the dominant transition state, which does not Of particular relevance to the present study is the demonstration vary significantly at low ½C, the core of the protein and certain that solvation forces determine the origin of barriers in SH3 loops are structured. Besides solving the long-standing problem domains, which helped rationalize the effect of specific mutations of computing the chevron plot, our work lays the foundation for on the folding rates (28). Here, we extend the molecular transfer CHEMISTRY incorporating denaturant effects in a physically transparent man- model (MTM) (29–31) to map in quantitative detail the thermo- ner either in all-atom or coarse-grained simulations. dynamics and to a lesser extent kinetics of folding of src SH3 do- Δ main as a function of GdmCl and urea. We show that GNUð½CÞ kinetic cooperativity ∣ self-organized polymer model ∣ pathway diversity ∣ decreases linearly as ½C, with the slopes (m) being in quantitative protein denaturation agreement with experiments. Simulations of the ½C-dependent ln kobs has the characteristic chevron shape, and the values of nderstanding how proteins fold in quantitative molecular the slopes of the folding (mf ) and unfolding (mu) arms are in ex- Udetail can provide insights into protein aggregation and dy- cellent agreement with experiments. There is a single dominant namics of formation of multisubunit complexes. As a result there transition state whose structure shows that the core β-sheets are have been intense efforts in deciphering the folding mechanisms well packed and the stiff distal loop is ordered. Contacts involving of proteins using a variety of experiments (1–6), theories (7–13), the hydrophobic core and the other secondary structural ele- and simulations (14–16). Despite these advances, a number of ments, overall collapse, and formation of the entire native struc- issues such as the denaturant-dependent characteristics of the ture form nearly simultaneously at zero and low ½C. Our work unfolded states and the link between protein collapse and fold- solves the long-standing problem of describing the entire dena- ing remain unclear (17–21). Single molecule experiments have turant-dependent folding of a protein, including the creation of provided clear evidence for folding pathway diversity, and have the chevron plot, albeit at a coarse-grained level. shown that polypeptide chains undergo almost continuous col- lapse as the denaturant concentration (½C) is decreased (1, Results 19–21). These and other experiments (5, 22) raise the need for Structure of src SH3. To establish the utility of MTM we computed computational models whose predictions can be directly com- several experimentally measurable thermodynamic and kinetic pared to experiments, which often use denaturants [guanidinium properties of the 56-residue src SH3 domain (Fig. 1A) (29) (see chloride (GdmCl) and urea] to initiate folding and unfolding. Methods for details). The core of SH3 domains, whose folding has Global thermodynamic and kinetic behavior of small proteins been extensively investigated using experiments (32–35) and si- often exhibit two-state behavior (23), which implies that at all values of ½C only the population of native (N) and the unfolded Author contributions: Z.L., G.R., E.P.O, and D.T. designed research; Z.L., G.R., and D.T. (U) states are detectable. Experimentally two-state behavior is performed research; D.T. contributed new reagents/analytic tools; Z.L., G.R., and D.T. Δ characterized by (i) the free energy of stability, GNUð½CÞ analyzed data; and Z.L., G.R., and D.T. wrote the paper. − ð¼ GN ð½CÞ GU ð½CÞÞ,ofN with respect to U, varies linearly The authors declare no conflict of interest. with ½C, and (ii) the logarithm of the relaxation rate kobs, which *This Direct Submission article had a prearranged editor. is the sum of folding and unfolding rates, is V-shaped (“chevron 1To whom correspondence should be addressed. E-mail: [email protected]. ” plot ) when plotted as a function of ½C. For two-state systems This article contains supporting information online at www.pnas.org/lookup/suppl/ chevron plots show a linear decrease in ln kobs as ½C increases doi:10.1073/pnas.1019500108/-/DCSupplemental. www.pnas.org/cgi/doi/10.1073/pnas.1019500108 PNAS Early Edition ∣ 1of6 Downloaded by guest on September 29, 2021 mulations (28, 36–38), consists of two well-packed three stranded Δ − f NBA ð½CÞ f NBAð½CÞ leading to GNUð½CÞ ¼ kBTs lnð1− Þ. The line- β f NBA ð½CÞ -sheets that are connected by RT, n-src, and short distal loops Δ Δ 0 ar fit (Fig. 1C), GNUð½CÞ ¼ GNUð½ Þ þ m½C, yields (Fig. 1A). Δ 0 −3 4 ∕ 1 34 ∕ GNUð½ Þ ¼ . kcal mol and m ¼ . kcal mol·M. We also computed ΔG C using free energy profiles as a function of χ Denaturant-Dependent Folding Thermodynamics. Given the approx- NUð½ Þ (Eq. S16), and obtained m ¼ 1.47 kcal∕mol·M. The experimen- imations in our model (see Methods and SI Text), it is difficult 1 5 ∕ to get absolute free energy differences between the folded and tally inferred m value using circular dichroism is . kcal mol·M 1 6 ∕ unfolded states from simulations. To make comparisons with whereas m ¼ . kcal mol·M using fluorescence measurements. The values from our simulation for m ð1.34–1.47Þ kcal∕mol·M experiments, we choose a simulation temperature, Ts, at which the calculated free energy of stability of the native state (N) with are in excellent agreement with experimental estimates, which Δ − further shows that MTM can predict the global thermodynamic respect to the unfolded state (U), GNUðTsÞ (GN ðTsÞ GU ðTsÞ), 295 Δ properties accurately. and the measured free energy at TEð¼ KÞ GNUðTEÞ coin- Δ Δ 0 Although the calculated value of ΔG 0 is in good agree- cide. The use of GNUðTsÞ¼ GNUðTEÞ (in water with ½C¼ ) NUð½ Þ to fix Ts is equivalent to choosing a overall reference energy ment with the experimental result, it is not as accurate as m-value Δ 295 scale in the simulations. For src SH3, GNUðTE ¼ KÞ¼ because of the known difficulties in obtaining accurate results at −4 1 ∕ 0 339 . kcal mol at ½C¼ (34), which results in Ts ¼ K. low GdmCl concentrations (40). To ensure that MTM can predict Δ 0 Besides the choice of Ts no other parameter is adjusted to fit GNUð½ Þ accurately we calculated f NBAð½CÞ for urea (green 339 Δ any other aspect of src SH3 folding. With Ts ¼ K fixed, we squares in Fig. 1B) from which we obtained GNUð½CÞ (Fig. 1C). Δ 0 −4 2 ∕ calculated the dependence of fraction of molecules in the native The linear fit yields GNUð½ Þ ¼ . kcal mol, which is in S18 basin of attraction (39), f NBAð½CÞ,on½C using Eq.
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