Probing the origins of two-state folding.

Many protein systems fold in a two-state manner. Random models, however, rarely display two-state kinetics and thus such behavior should not be accepted as a default. While theories for the prevalence of two-state kinetics have been presented, none sufficiently explain the breadth of experimental observations. A model, making minimal assumptions, is introduced that suggests two-state behavior is likely for any system with an overwhelmingly populated native state. We show two-state folding is a natural consequence of such two-state thermodynamics, and is strengthened by increasing the population of the native state. Further, the model exhibits hub-like behavior, with slow interconversions between unfolded states. Despite this, the unfolded state equilibrates quickly relative to the folding time. This apparent paradox is readily understood through this model. Finally, our results compare favorable with measurements of folding rates as a function of chain length and Keq, providing new insight into these relations.

[1]  S. N. Dorogovtsev,et al.  Laplacian spectra of, and random walks on, complex networks: are scale-free architectures really important? , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[2]  Natalya S. Bogatyreva,et al.  KineticDB: a database of protein folding kinetics , 2008, Nucleic Acids Res..

[3]  A. Fersht,et al.  Folding of chymotrypsin inhibitor 2. 1. Evidence for a two-state transition. , 1991, Biochemistry.

[4]  Maxim B. Prigozhin,et al.  Microsecond folding experiments and simulations: a match is made. , 2013, Physical chemistry chemical physics : PCCP.

[5]  V. Muñoz,et al.  Experimental Identification of Downhill Protein Folding , 2002, Science.

[6]  F. Rao,et al.  The protein folding network. , 2004, Journal of molecular biology.

[7]  G. Hong,et al.  Nucleic Acids Research , 2015, Nucleic Acids Research.

[8]  M. Gruebele,et al.  Observation of strange kinetics in protein folding. , 1999, Proceedings of the National Academy of Sciences of the United States of America.

[9]  Amelia A. Fuller,et al.  An experimental survey of the transition between two-state and downhill protein folding scenarios , 2008, Proceedings of the National Academy of Sciences.

[10]  Sheena E Radford,et al.  An expanding arsenal of experimental methods yields an explosion of insights into protein folding mechanisms , 2009, Nature Structural &Molecular Biology.

[11]  Kyle A. Beauchamp,et al.  Markov state model reveals folding and functional dynamics in ultra-long MD trajectories. , 2011, Journal of the American Chemical Society.

[12]  Athi N. Naganathan,et al.  Dynamics of one-state downhill protein folding , 2009, Proceedings of the National Academy of Sciences.

[13]  G. Illies,et al.  Communications in Mathematical Physics , 2004 .

[14]  Martin Gruebele,et al.  Folding at the speed limit , 2003, Nature.

[15]  B. Schuler,et al.  Two-state folding observed in individual protein molecules. , 2004, Journal of the American Chemical Society.

[16]  Hue Sun Chan,et al.  Simple two‐state protein folding kinetics requires near‐levinthal thermodynamic cooperativity , 2003, Proteins.

[17]  P. Henklein,et al.  An unlocking/relocking barrier in conformational fluctuations of villin headpiece subdomain , 2010, Proceedings of the National Academy of Sciences.

[18]  Kevin W Plaxco,et al.  Cooperativity and the origins of rapid, single‐exponential kinetics in protein folding , 2006, Protein science : a publication of the Protein Society.

[19]  J. Hofrichter,et al.  Sub-microsecond protein folding. , 2006, Journal of molecular biology.

[20]  M. Karplus,et al.  How does a protein fold? , 1994, Nature.

[21]  Vincent A. Voelz,et al.  Atomistic folding simulations of the five-helix bundle protein λ(6−85). , 2011, Journal of the American Chemical Society.

[22]  Martin Gruebele,et al.  Tuning λ6- 85 towards downhill folding at its melting temperature , 2007 .

[23]  E. Jaynes.  The Minimum Entropy Production Principle , 1980 .

[24]  F. Chung,et al.  Spectra of random graphs with given expected degrees , 2003, Proceedings of the National Academy of Sciences of the United States of America.

[25]  C. Brooks,et al.  Native states of fast-folding proteins are kinetic traps. , 2013, Journal of the American Chemical Society.

[26]  J. Herskowitz,et al.  Proceedings of the National Academy of Sciences, USA , 1996, Current Biology.

[27]  M. Stephanov,et al.  Random Matrices , 2005, hep-ph/0509286.

[28]  M. Marahiel,et al.  Extremely rapid protein folding in the absence of intermediates , 1995, Nature Structural Biology.

[29]  F. Young.  Biochemistry , 1955, The Indian Medical Gazette.

[30]  M. Gruebele.  Downhill protein folding: evolution meets physics. , 2005, Comptes rendus biologies.

[31]  T. Schindler,et al.  Diffusion control in an elementary protein folding reaction. , 1997, Proceedings of the National Academy of Sciences of the United States of America.

[32]  R. Rosenfeld.  Nature , 2009, Otolaryngology--head and neck surgery : official journal of American Academy of Otolaryngology-Head and Neck Surgery.

[33]  Victor Muñoz,et al.  Atom-by-atom analysis of global downhill protein folding , 2006, Nature.

[34]  D. Barrick.  What have we learned from the studies of two-state folders, and what are the unanswered questions about two-state protein folding? , 2009, Physical biology.

[35]  Mark Newman,et al.  The structure and function of networks , 2002 .

[36]  O. Bagasra,et al.  Proceedings of the National Academy of Sciences , 1914, Science.

[37]  Journal of Molecular Biology , 1959, Nature.

[38]  C. Tracy,et al.  Mathematical Physics © Springer-Verlag 1996 On Orthogonal and Symplectic Matrix Ensembles , 1995 .

[39]  Hue Sun Chan,et al.  Towards a consistent modeling of protein thermodynamic and kinetic cooperativity: how applicable is the transition state picture to folding and unfolding? , 2002, Journal of molecular biology.

[40]  M. Gruebele,et al.  Folding λ-repressor at its speed limit , 2004 .

[41]  Martin Gruebele,et al.  Engineering a β-sheet protein toward the folding speed limit , 2005 .

[42]  Kingshuk Ghosh,et al.  Maximum Caliber: a variational approach applied to two-state dynamics. , 2008, The Journal of chemical physics.

[43]  W. Eaton,et al.  Searching for "downhill scenarios" in protein folding. , 1999, Proceedings of the National Academy of Sciences of the United States of America.

[44]  J. Onuchic,et al.  Funnels, pathways, and the energy landscape of protein folding: A synthesis , 1994, Proteins.

[45]  Frank Noé,et al.  Markov models of molecular kinetics: generation and validation. , 2011, The Journal of chemical physics.

[46]  Alex Dickson,et al.  Quantifying hub-like behavior in protein folding networks. , 2012, Journal of chemical theory and computation.

[47]  Martin Gruebele,et al.  Rate−Temperature Relationships in λ-Repressor Fragment λ6-85 Folding† , 2004 .

[48]  Albert-László Barabási,et al.  Statistical mechanics of complex networks , 2001, ArXiv.

[49]  Yuefeng Tang,et al.  Dynamic NMR line-shape analysis demonstrates that the villin headpiece subdomain folds on the microsecond time scale. , 2003, Journal of the American Chemical Society.

[50]  V. Pande,et al.  A simple model predicts experimental folding rates and a hub-like topology. , 2012, The journal of physical chemistry. B.

[51]  Lisa J Lapidus,et al.  Extremely slow intramolecular diffusion in unfolded protein L , 2010, Proceedings of the National Academy of Sciences.

[52]  Vijay S Pande,et al.  Protein folded states are kinetic hubs , 2010, Proceedings of the National Academy of Sciences.

[53]  Mohamed A. Marahiel,et al.  Conservation of rapid two-state folding in mesophilic, thermophilic and hyperthermophilic cold shock proteins , 1998, Nature Structural Biology.

[54]  Vijay S Pande,et al.  Simple theory of protein folding kinetics. , 2010, Physical review letters.