title: How long does it take to equilibrate the unfolded state of a protein?
abstract: How long does it take to equilibrate the unfolded state of a protein? The answer to this question has important implications for our understanding of why many small proteins fold with two state kinetics. When the equilibration within the unfolded state U is much faster than the folding, the folding kinetics will be two state even if there are many folding pathways with different barriers. Yet the mean first passage times (MFPTs) between different regions of the unfolded state can be much longer than the folding time. This seems to imply that the equilibration within U is much slower than the folding. In this communication we resolve this paradox. We present a formula for estimating the time to equilibrate the unfolded state of a protein. We also present a formula for the MFPT to any state within U, which is proportional to the average lifetime of that state divided by the state population. This relation is valid when the equilibration within U is very fast as compared with folding as it often is for small proteins. To illustrate the concepts, we apply the formulas to estimate the time to equilibrate the unfolded state of Trp-cage and MFPTs within the unfolded state based on a Markov State Model using an ultra-long 208 microsecond trajectory of the miniprotein to parameterize the model. The time to equilibrate the unfolded state of Trp-cage is ∼100 ns while the typical MFPTs within U are tens of microseconds or longer.
authors: Department of Chemistry and Chemical Biology, Rutgers, the State University of New Jersey, Piscataway, New Jersey, 08854.
authors: Ronald M
authors: RM
authors: Levy
authors: Wei
authors: W
authors: Dai
authors: Nan-Jie
authors: NJ
authors: Deng
authors: Dmitrii E
authors: DE
authors: Makarov
mesh_terms: D007700:Kinetics; D056004:Molecular Dynamics Simulation; D011487:Protein Conformation; D017510:Protein Folding; D055550:Protein Stability; D011506:Proteins / Q000737:chemistry*; D013816:Thermodynamics; D013997:Time Factors
chemical_list: D011506:Proteins
keywords: Markov state model; mean first passage time; protein folding; protein unfolded state
references: Protein Sci. 2004 Mar;13(3):822-9
references: 14978313
references: Nat Struct Biol. 1997 Jan;4(1):10-9
references: 8989315
references: J Chem Theory Comput. 2011 Oct 11;7(10):3412-3419
references: 22125474
references: Nature. 2003 May 8;423(6936):193-7
references: 12736690
references: Protein Sci. 2000 Mar;9(3):452-65
references: 10752607
references: Biochemistry. 1991 Oct 29;30(43):10428-35
references: 1931967
references: Phys Biol. 2009 Feb 10;6(1):015001
references: 19208936
references: Proc Natl Acad Sci U S A. 1988 Jul;85(14):5082-6
references: 2455892
references: Structure. 1996 Oct 15;4(10):1133-9
references: 8939749
references: Proc Natl Acad Sci U S A. 1997 Jan 7;94(1):148-50
references: 8990176
references: J Am Chem Soc. 2010 Feb 10;132(5):1526-8
references: 20070076
references: Proc Natl Acad Sci U S A. 2010 Jun 15;107(24):10890-5
references: 20534497
references: Annu Rev Biophys Biomol Struct. 2001;30:361-96
references: 11340064
references: J Chem Theory Comput. 2013 Mar 12;9(3):1743-53
references: 26587632
references: J Chem Theory Comput. 2012;8(9):3044-3052
references: 24027492
references: J Phys Chem B. 2009 Aug 27;113(34):11702-9
references: 19655770
references: J Am Chem Soc. 2004 Nov 17;126(45):14686-7
references: 15535670
references: Proc Natl Acad Sci U S A. 2010 Jan 19;107(3):1088-93
references: 20080558
references: J Chem Phys. 2013 Oct 14;139(14):145104
references: 24116650
references: Proteins. 2003 Sep 1;52(4):510-23
references: 12910451
references: Nat Struct Biol. 1998 Mar;5(3):229-35
references: 9501917
references: J Phys Chem B. 2013 Oct 24;117(42):12787-99
references: 23705683
references: Fold Des. 1998;3(4):R81-91
references: 9710577
references: Trends Biochem Sci. 2000 Jul;25(7):331-9
references: 10871884
references: Proteins. 1995 Mar;21(3):167-95
references: 7784423
references: Annu Rev Phys Chem. 1997;48:545-600
references: 9348663
references: Proc Natl Acad Sci U S A. 2002 Mar 19;99(6):3535-9
references: 11904417
references: Nat Chem Biol. 2011 Jun 17;7(7):401-4
references: 21685880
references: Proc Natl Acad Sci U S A. 2012 Oct 30;109(44):17800-6
references: 22492978
references: Science. 2011 Oct 28;334(6055):517-20
references: 22034434
references: Protein Sci. 2006 Mar;15(3):564-82
references: 16501227
references: J Am Chem Soc. 2013 Mar 27;135(12):4729-34
references: 23458553
references: Curr Opin Struct Biol. 2011 Feb;21(1):4-11
references: 21081274
references: Proc Natl Acad Sci U S A. 2012 Dec 18;109(51):20919-24
references: 23213246