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Protein Folding Kinetics and Thermodynamics from Atomistic Simulation

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Protein Folding Kinetics and Thermodynamics from Atomistic Simulation Protein folding kinetics and thermodynamics SPECIAL FEATURE from atomistic simulation Stefano Pianaa,1,2, Kresten Lindorff-Larsena,1,2, and David E. Shawa,b,2 aD. E. Shaw Research, New York, NY 10036; and bCenter for Computational Biology and Bioinformatics, Columbia University, New York, NY 10032 Edited by Peter G. Wolynes, Rice University, Houston, TX, and approved June 13, 2012 (received for review March 2, 2012) Advances in simulation techniques and computing hardware have folding mechanism, it has been difficult to produce consistent created a substantial overlap between the timescales accessible to predictions of the thermodynamics of villin folding (10–13), and atomic-level simulations and those on which the fastest-folding where comparison has been possible between kinetic models built proteins fold. Here we demonstrate, using simulations of four from multiple short simulations and independent, long-equili- variants of the human villin headpiece, how simulations of spon- brium MD simulations (14), substantial differences have been taneous folding and unfolding can provide direct access to thermo- observed in the folding free energies. dynamic and kinetic quantities such as folding rates, free energies, Recent advances in computer hardware have, however, ex- folding enthalpies, heat capacities, Φ-values, and temperature- tended the timescale accessible to simulation up to the millise- jump relaxation profiles. The quantitative comparison of simula- cond (15), thus creating a broad overlap with the microsecond tion results with various forms of experimental data probing timescale characteristic of fast-folding proteins such as villin, different aspects of the folding process can facilitate robust assess- and allowing for the direct calculation of equilibrium thermody- ment of the accuracy of the calculations while providing a detailed namic and kinetic properties from the simulation data (16, 17). structural interpretation for the experimental observations. In the Substantial improvements have also been made in the molecular example studied here, the analysis of folding rates, Φ-values, and mechanics force fields used in MD simulations (14, 18–20). folding pathways provides support for the notion that a norleucine Taking advantage of improvements in both simulation speeds double mutant of villin folds five times faster than the wild-type and force fields, we have employed equilibrium MD simulations sequence, but following a slightly different pathway. This work to study the folding kinetics and thermodynamics of several BIOPHYSICS AND showcases how computer simulation has now developed into a variants of villin. Some researchers have suggested that tempera- COMPUTATIONAL BIOLOGY mature tool for the quantitative computational study of protein ture-jump experiments may underestimate the folding time of folding and dynamics that can provide a valuable complement to villin (4, 5) and may be more sensitive to processes other than experimental techniques. protein folding. It should be noted, however, that some of these theoretical estimates were based on models that combine the re- Amber ff99SB*-ILDN ∣ enthalpy ∣ heat capacity ∣ pre-exponential factor ∣ sults of non-equilibrium, short, individual trajectories, thereby transition path time introducing additional sources of uncertainty in the quantitative comparisons between the simulation results and the experimental roteins are synthesized in the cell or in vitro as unstructured observations. The equilibrium simulations presented here over- Ppolypeptide chains that, in most cases, self-assemble into their come this difficulty as in each trajectory at least 30 folding and functionally active three-dimensional shapes. This process, called unfolding events are observed, making it possible to directly com- protein folding, occurs on a broad range of timescales ranging pute thermodynamic and kinetic quantities without the need to from microseconds to seconds and higher. From a purely physi- build approximate models to describe the system. Further, we cal-chemical perspective, it should be possible in principle to compared simulations of different variants allowing the direct characterize the folding mechanism of a given protein at atomis- calculation of Φ-values in a manner analogous to experiments tic resolution and to reconstruct its free-energy landscape, given (17, 21). only its primary sequence, through molecular dynamics (MD) si- Most of the results presented here are in good agreement with mulations based on elementary physical principles. This direct previous experimental findings, with the notable exception of the approach has been rarely pursued because even the simplest sys- heat capacity for folding, which appears to be smaller than the tems representing a protein immersed in water consist of several value extracted from calorimetric data. We find that the double thousand atoms, and simulating their behavior on the timescales norleucine (Nle/Nle) mutant (3) folds approximately five times typical of protein folding is computationally extremely demand- faster than the wild-type protein, thus supporting the original in- ing. The discovery and design of fast-folding proteins (1) signifi- terpretation of the experimental data (3). In agreement with our cantly narrowed the timescale gap between simulations and ex- previous observations (14), the results reported here also indicate periments, making such simulations feasible, at least for the that both the number of helical residues and the Trp side-chain fastest-folding proteins. environment are sensitive to the folding/unfolding process, sup- The C-terminal fragment of the villin headpiece [referred to in porting the notion that experiments that probe these quantities, the remainder of this paper simply as “villin” (2)], one of the fast- like infrared (IR) and fluorescence-detected temperature-jump, est-folding protein domains known (3), has proven to be an ex- may be used to determine folding and unfolding rates (22). cellent target for folding simulations with physics-based force fields and an atomistically detailed representation of both the so- Author contributions: S.P., K.L.-L., and D.E.S. designed research; S.P. and K.L.-L. performed lute and the surrounding solvent (4–7). Until recently, the length research; S.P. and K.L.-L. analyzed data; and S.P., K.L.-L., and D.E.S. wrote the paper. — of such simulations was limited to a few microseconds a time- The authors declare no conflict of interest. scale sufficient to capture, at best, a single folding event (8, 9). This article is a PNAS Direct Submission. With this limitation, it has been difficult to directly connect the 1S.P. and K.L.-L. contributed equally to this work. data produced by short, non-equilibrium simulations to experi- 2To whom correspondence may be addressed. E-mail: Stefano.Piana-Agostinetti@ mental observations, unless sufficient statistics were generated to DEShawResearch.com or [email protected] or David.Shaw@ allow the construction of coarse-grained kinetic models that ap- DEShawResearch.com. proximate the underlying folding dynamics (10, 11). While these This article contains supporting information online at www.pnas.org/lookup/suppl/ models have proven useful for obtaining certain insights into the doi:10.1073/pnas.1201811109/-/DCSupplemental. www.pnas.org/cgi/doi/10.1073/pnas.1201811109 PNAS ∣ October 30, 2012 ∣ vol. 109 ∣ no. 44 ∣ 17845–17850 Downloaded by guest on October 1, 2021 Table 1. Folding kinetics and thermodynamics from equilibrium MD simulations of wild-type (WT) and two variants of villin headpiece C-terminal fragment T L μ n ΔG ΔH ΔC τ μ hτ i μ k μ −1 Variant sim (K) ( s) f f v f ( s) TP ( s) 0 ( s ) MD VH MD ΔΔH WT (HP-35) 345 398 30 0.8(2) −15.1(4) 0.2(2) 19(5) 0.5(1) 0.65 WT (HP-35) 360 319 31 1.6(2) −18(3) −19(7) 0.1(2) 0.2(2) 16(4) 0.24(5) 1.49 WT-F10L 345 371 30 1.6(3) −10(2) 0.4(2) 24(4) 0.39(7) 0.90 Nle/Nle 360 305 61 −0.6(2) −16(3) 0.1(1) 3.2(6) 0.19(2) 1.05 Nle/Nle 370 395 150 0.0(1) −18.2(8) −22(8) 0.07(2) 0.1(2) 2.3(2) 0.15(1) 1.05 Nle/Nle 380 301 140 0.7(1) −21.2(9) −26(5) 0.0(4) 0.1(1) 3.0(4) 0.12(1) 2.33 Nle/Nle-F10L 360 301 110 0.4(1) −14.7(8) 0.2(1) 3.5(5) 0.21(2) 1.00 Nle/Nle-F10L 370 300 130 0.8(1) −16(1) −15(5) 0.3(1) 0.1(1) 2.9(3) 0.18(1) 1.10 The temperature of each MD simulation is reported together with the total length (L) and the total number of observed folding and unfolding events (n). The trajectories have been partitioned into folded and unfolded segments using a transition-based assignment (14, 26). The folding free energy ΔG −1 ΔH −1 ( f,kcalmol ) is calculated from the ratio of the folded and unfolded fractions. Folding enthalpies ( f, kcal mol ) were calculated either from the folding free energy at different temperature using the van’t Hoff equation (VH)orasΔH ¼ ΔU þ VΔP where ΔU is the difference in average ΔC −1 −1 force-field energy in the folded and unfolded states (MD). The heat capacities ( v, kcal mol K ) were calculated either from difference in the fluctuations of the force-field energy between the folded and unfolded states (MD) or from the temperature dependency of the folding enthalpy ΔΔH τ k ( ). The folding time ( f) is calculated as the average waiting time in the unfolded state. The pre-exponential factor for folding ( 0) is estimated hτ i ’ from the folding time and the mean transition path time ( TP ) using Kramers theory (40) Results and Discussion and the folding heat capacities calculated from the temperature Equilibrium Reversible Folding Simulation of the Villin Headpiece dependency of the folding free energy and enthalpy, respectively.
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