Abstract
We study folding of Trp-cage miniprotein in the conditions when the native state of the protein is stable and unfolding events are improbable, which corresponds to physiological conditions. Using molecular dynamics simulations with an implicit solvent model, an ensemble of folding trajectories from unfolded (practically extended) states of the protein to the native state was generated. To get insight into the folding kinetics, the free energy surface and kinetic network projected on this surface were constructed. This, “conventional” analysis of the folding reaction was followed by a recently proposed hydrodynamic description of protein folding (Chekmarev et al. in Phys Rev Lett 100(1):018107, 2008), in which the process of the first-passage folding is viewed as a stationary flow of a folding “fluid” from the unfolded to native state. This approach is conceptually different from the previously used approaches and thus allows an alternative view of the folding dynamics and kinetics of Trp-cage, the conclusions about which are very diverse. In agreement with most previous studies, we observed two characteristic folding pathways: in one pathway (I), the collapse of the hydrophobic core precedes the formation of the \(\alpha\)-helix, and in the other pathway (II), these events occur in the reverse order. We found that although pathway II is complicated by a repeated partial protein unfolding, it contributes to the total folding flow as little as ≈10 %, so that the folding kinetics remain essentially single-exponential.
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References
Abaskharon RM, Culik RM, Woolley GA, Gai F (2015) Tuning the attempt frequency of protein folding dynamics via transition-state rigidification: application to Trp-cage. J Phys Chem Lett 6(3):521–526
Ahmed Z, Beta IA, Mikhonin AV, Asher SA (2005) UV-resonance Raman thermal unfolding study of Trp-cage shows that it is not a simple two-state miniprotein. J Am Chem Soc 127(31):10943–10950
Brooks BR, Brooks CL, MacKerell AD, Nilsson L, Petrella RJ, Roux B, Won Y, Archontis G, Bartels C, Boresch S et al (2009) CHARMM: the biomolecular simulation program. J Comput Chem 30(10):1545–1614
Byrne A, Williams DV, Barua B, Hagen SJ, Kier BL, Andersen NH (2014) Folding dynamics and pathways of the Trp-cage miniproteins. Biochemistry 53(38):6011–6021
Chekmarev SF (2013) Protein folding: complex potential for the driving force in a two-dimensional space of collective variables. J Chem Phys 139(14):145103
Chekmarev SF (2015a) Equilibration of protein states: a time dependent free-energy disconnectivity graph. J Phys Chem B 119(26):8340–8348
Chekmarev SF (2015b) Protein folding as a complex reaction: a two-component potential for the driving force of folding and its variation with folding scenario. PLoS One 10(4):0121640
Chekmarev SF, Krivov SV, Karplus M (2005) Folding time distributions as an approach to protein folding kinetics. J Phys Chem B 109(11):5312–5330
Chekmarev SF, Palyanov AY, Karplus M (2008) Hydrodynamic description of protein folding. Phys Rev Lett 100(1):018107
Chowdhury S, Lee MC, Duan Y (2004) Characterizing the rate-limiting step of Trp-cage folding by all-atom molecular dynamics simulations. J Phys Chem B 108(36):13855–13865
Culik RM, Serrano AL, Bunagan MR, Gai F (2011) Achieving secondary structural resolution in kinetic measurements of protein folding: a case study of the folding mechanism of Trp-cage. Angew Chem Int Ed 123(46):11076–11079
Day R, Paschek D, Garcia AE (2010) Microsecond simulations of the folding/unfolding thermodynamics of the Trp-cage miniprotein. Proteins: Struct Funct Bioinform 78(8):1889–1899
Deng NJ, Dai W, Levy RM (2013) How kinetics within the unfolded state affects protein folding: an analysis based on Markov state models and an ultra-long MD trajectory. J Phys Chem B 117(42):12787–12799
Du W, Bolhuis PG (2014) Sampling the equilibrium kinetic network of Trp-cage in explicit solvent. J Chem Phys 140(19):195102
Eaton WA, Muñoz V, Hagen SJ, Jas GS, Lapidus LJ, Henry ER, Hofrichter J (2000) Fast kinetics and mechanisms in protein folding. Annu Rev Biophys Biomol Struct 29(1):327–359
Ferrara P, Apostolakis J, Caflisch A (2000) Thermodynamics and kinetics of folding of two model peptides investigated by molecular dynamics simulations. J Phys Chem B 104(20):5000–5010
Ferrara P, Apostolakis J, Caflisch A (2002) Evaluation of a fast implicit solvent model for molecular dynamics simulations. Proteins: Struct Funct Bioinform 46(1):24–33
Fraley C, Raftery AE (2002) Model-based clustering, discriminant analysis, and density estimation. J Am Stat Assoc 97(458):611–631
Halabis A, Zmudzinska W, Liwo A, Oldziej S (2012) Conformational dynamics of the Trp-cage miniprotein at its folding temperature. J Phys Chem B 116(23):6898–6907
Han W, Schulten K (2013) Characterization of folding mechanisms of Trp-cage and WW-domain by network analysis of simulations with a hybrid-resolution model. J Phys Chem B 117(42):13367–13377
Hanwell MD, Curtis DE, Lonie DC, Vandermeersch T, Zurek E, Hutchison GR (2012) Avogadro: an advanced semantic chemical editor, visualization, and analysis platform. J Cheminform 4(1):17
Jolliffe I (2002) Principal component analysis. Springer, New York
Juraszek J, Bolhuis P (2006) Sampling the multiple folding mechanisms of Trp-cage in explicit solvent. Proc Natl Acad Sci USA 103(43):15859–15864
Juraszek J, Saladino G, van Erp T, Gervasio F (2013) Efficient numerical reconstruction of protein folding kinetics with partial path sampling and pathlike variables. Phys Rev Lett 110(10):108106
Kalgin IV, Chekmarev SF (2011) Turbulent phenomena in protein folding. Phys Rev E 83(1):011920
Kalgin IV, Chekmarev SF (2015) Folding of a \(\beta\)-sheet miniprotein: probability fluxes, streamlines, and the potential for the driving force. J Phys Chem B 119(4):1380–1387
Kalgin IV, Karplus M, Chekmarev SF (2009) Folding of a SH3 domain: standard and “hydrodynamic” analyses. J Phys Chem B 113(38):12759–12772
Kalgin IV, Chekmarev SF, Karplus M (2014) First passage analysis of the folding of a \(\beta\)-sheet miniprotein: Is it more realistic than the standard equilibrium approach? J Phys Chem B 118(16):4287–4299
Kannan S, Zacharias M (2014) Role of tryptophan side chain dynamics on the Trp-cage mini-protein folding studied by molecular dynamics simulations. PloS One 9(2):88383
Kim SB, Dsilva CJ, Kevrekidis IG, Debenedetti PG (2015) Systematic characterization of protein folding pathways using diffusion maps: application to Trp-cage miniprotein. J Chem Phys 142(8):085101
Lai Z, Preketes NK, Mukamel S, Wang J (2013) Monitoring the folding of Trp-cage peptide by two-dimensional infrared (2dir) spectroscopy. J Phys Chem B 117(16):4661–4669
Landau L, Lifshitz E (1987) Fluid mechanics. Pergamon, New York
Lindorff-Larsen K, Piana S, Dror RO, Shaw DE (2011) How fast-folding proteins fold. Science 334(6055):517–520
Linhananta A, Boer J, MacKay I (2005) The equilibrium properties and folding kinetics of an all-atom go model of the Trp-cage. J Chem Phys 122(11):114901
Marinelli F (2013) Following easy slope paths on a free energy landscape: the case study of the Trp-cage folding mechanism. Biophys J 105(5):1236–1247
Marinelli F, Pietrucci F, Laio A, Piana S (2009) A kinetic model of Trp-cage folding from multiple biased molecular dynamics simulations. PLoS Comput Biol 5(8):1000452
Marino KA, Bolhuis PG (2012) Confinement-induced states in the folding landscape of the Trp-cage miniprotein. J Phys Chem B 116(39):11872–11880
Meuzelaar H, Marino KA, Huerta-Viga A, Panman MR, Smeenk LE, Kettelarij AJ, van Maarseveen JH, Timmerman P, Bolhuis PG, Woutersen S (2013) Folding dynamics of the Trp-cage miniprotein: evidence for a native-like intermediate from combined time-resolved vibrational spectroscopy and molecular dynamics simulations. J Phys Chem B 117(39):11490–11501
Mok KH, Kuhn LT, Goez M, Day IJ, Lin JC, Andersen NH, Hore P (2007) A pre-existing hydrophobic collapse in the unfolded state of an ultrafast folding protein. Nature 447(7140):106–109
Mou L, Jia X, Gao Y, Li Y, Zhang JZ, Mei Y (2014) Folding simulation of Trp-cage utilizing a new AMBER compatible force field with coupled main chain torsions. J Chem Theory Comput 13(04):1450026
Neidigh JW, Fesinmeyer RM, Andersen NH (2002) Designing a 20-residue protein. Nat Struct Mol Biol 9(6):425–430
Neria E, Fischer S, Karplus M (1996) Simulation of activation free energies in molecular systems. J Chem Phys 105(5):1902–1921
Neuweiler H, Doose S, Sauer M (2005) A microscopic view of miniprotein folding: Enhanced folding efficiency through formation of an intermediate. Proc Natl Acad Sci USA 102(46):16650–16655
Orsi M, Ding W, Palaiokostas M (2014) Direct mixing of atomistic solutes and coarse-grained water. J Chem Theor Comput 10(10):4684–4693
Paschek D, Hempel S, García AE (2008) Computing the stability diagram of the Trp-cage miniprotein. Proc Natl Acad Sci USA 105(46):17754–17759
Qiu L, Pabit SA, Roitberg AE, Hagen SJ (2002) Smaller and faster: The 20-residue Trp-cage protein folds in 4 μs. J Am Chem Soc 124(44):12952–12953
Rovó P, Farkas V, Hegyi O, Szolomájer-Csikós O, Tóth GK, Perczel A (2011) Cooperativity network of Trp-cage miniproteins: probing salt-bridges. J Pept Sci 17(9):610–619
Rovó P, Stráner P, Láng A, Bartha I, Huszár K, Nyitray L, Perczel A (2013) Structural insights into the Trp-cage folding intermediate formation. Chem Eur J 19(8):2628–2640
Sabelko J, Ervin J, Gruebele M (1999) Observation of strange kinetics in protein folding. Proc Natl Acad Sci USA 96(11):6031–6036
Seeber M, Cecchini M, Rao F, Settanni G, Caflisch A (2007) WORDOM: a program for efficient analysis of molecular dynamics simulations. Bioinformatics 23(19):2625–2627
Shao Q, Shi J, Zhu W (2012) Enhanced sampling molecular dynamics simulation captures experimentally suggested intermediate and unfolded states in the folding pathway of Trp-cage miniprotein. J Chem Phys 137(12):125103
Snow CD, Zagrovic B, Pande VS (2002) The Trp-cage: folding kinetics and unfolded state topology via molecular dynamics simulations. J Am Chem Soc 124(49):14548–14549
Sobolev S (1964) Partial differential equations of mathematical physics. Pergamon Press, Oxford
Streicher WW, Makhatadze GI (2007) Unfolding thermodynamics of Trp-cage, a 20 residue miniprotein, studied by differential scanning calorimetry and circular dichroism spectroscopy. Biochemistry 46(10):2876–2880
Van Kampen NG (1992) Stochastic processes in physics and chemistry, vol 1. North-Holland, Amsterdam
Weber HJ, Arfken GB (2004) Essential mathematical methods for physicists. Elsevier, San Diego
Zheng W, Gallicchio E, Deng N, Andrec M, Levy RM (2011) Kinetic network study of the diversity and temperature dependence of Trp-cage folding pathways: combining transition path theory with stochastic simulations. J Phys Chem B 115(6):1512–1523
Zhou R (2003) Trp-cage: folding free energy landscape in explicit water. Proc Natl Acad Sci USA 100(23):13280–13285
Acknowledgments
This work was performed under a grant from the Russian Science Foundation (No. 14-14-00325).
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Andryushchenko, V.A., Chekmarev, S.F. A hydrodynamic view of the first-passage folding of Trp-cage miniprotein. Eur Biophys J 45, 229–243 (2016). https://doi.org/10.1007/s00249-015-1089-7
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DOI: https://doi.org/10.1007/s00249-015-1089-7