Trp-Cage 是一個由20個胺基酸組成的小蛋白質，我們使用分子動力學軟體--- GROMACS 在電腦上進行摺疊模擬，在參數設定上，我們使用真實水溶液的模型以及Gromos 43a1的力場參數，而平行溫度法也在此研究中被使用來增加尋找可能結構的速度，平行溫度法的設定中，我們使用26個溫度( 250K~478K)當作交換的依據，在GROMACS進行一段短時間的模擬之後依據公式來計算機率並決定是否交換彼此溫度，在本文的研究中，我們利用平行溫度法收集到104000個不同結構，總模擬時間為4.16 μs，除了蛋白質折疊的研究之外，另一個主題是關於分布計算環境的建構，我們取名為Protein@CBL，所有分子動力學計算皆在這個平台上進行，相關的架構及流程說明在本文中皆有敘述。
此蛋白質模擬的初始結構為extended structure，針對摺疊過程中的一些參數加以討論，例如總能量，位能，機率分布，首次摺疊時間，radius of gyration，鹽橋作用，親水厭水的影響以及定壓熱容的變化，大部分的參數皆是對應於RMSD的變化，其中需要特別指出的是：當RMSD為0.7 nm、0.5 nm及0.35 nm時，其對應的首次折疊時間分別為0.6 ns、8 ns及20 ns，其顯示出這個蛋白質的折疊過程可以依據這三個RMSD值分成四個階段：extended state、molten globule、free-energy barrier 及native like，在摺疊過程的初期(extended state)，構型主要是受到牛頓力學的影響，當進入第二階段(molten globule)，力學的影響力逐漸減少而Entropy的影響逐漸增加，當RMSD ~ 0.5nm時，親水厭水的作用使得結構趨近穩定，不會再有太大變化，進入第三階段時，Entropy的作用變得相對重要，最後進入Native like，最佳結構 (RMSD = 0.23 nm) 也在經過152 ns的模擬之後出現，但是最佳結構卻不相對於最低能量，從本文的分析中可以估計，如果想得到更佳的結構 (RMSD < 0.2 nm)，總模擬時間需要約10 μs，Extended state的位能大約比平均值大540 KJ/mol，Molten globule的位能大約比平均值低 450 KJ/mol

Trp-cage is a mini-protein composed of 20 residues. Here its folding is simulated in computer using the method of molecular dynamics with the aid of the software package GROMACS. Simulation was conducted in explicit solvation model and the GROMOS 96 43a1 force field was used. The parallel-tempering, or replica-exchange, algorithm was employed to enhance the efficiency of the exploration of conformational space. Folding of replicas of the peptide were simulated in twenty-six temperatures ranging from 250K to 478K are temperatures of pairs of replicas were swapped at certain time intervals according to a fixed rule. A total of 104,000 conformations were collected for analysis, representing an accumulated simulation time of 4.16 μs. A part of the present project is the construction of a distributed computing facility, called Protein@CBL, on which simulations reported here were conducted. Details on both the hardware and software aspects of Protein@CBL are described here.
All simulations began with the Trp-cage replica being in an extended state. A number of parameters measuring the folding state of the replica, including total energy and potential energy, state probability, first arrival time, radius of gyration, separation of hydrophobic and hydrophilic residues, salt bridge bond length, and heat capacity were computed. In many cases these were expressed as functions of RMSD, the rms positional deviation relative to the native conformation. The three values, RMSD ~0.70, 0.50, and 0.35 nm, with first-arrival times of 0.6, 8 and 20 ns, respectively, demarcated the folding into four regimes: extended state, molten globule, free-energy barrier, and native-like. Folding was essentially kinetic in the extended state regime, dominated by kinetics but impeded by rapid loss of entropy in the molten globule regime and largely entropic thereafter. At RMSD ~0.50 nm the hydrophilic and hydrophobic residues became cleanly separated into an outer shell and an inner core. The most native-like state had a first-arrival time of 152 ns and an RMSD of 0.23 nm. The lowest energy state given by the GROMOS force-field is probably not native-like and the average time needed to fold to a state with RMSD<0.20 nm is estimated to be 10 μs. An extended state was about 540 kJ/mol above the average potential of a molten globule and a lowest energy state was about 450 kJ/mol below.

[ 1 ] Berman HM, Westbrook J, Feng Z, Gilliland G, Bhat TN, Weissig T, Shindyalov
IN, Bourne PE. The protein data bank. Nucleic Acids Res. 2000;28: 235-242.
[ 2 ] Neidigh JW, Fesinmeyer RM, Andersen NH. Designing a 20-residue protein ,
Nat. Struct. Bio 2002;9: 425-430.
[ 3 ] Simmerling C, Strockbine B, Roitberg AE. All-atom structure prediction and
folding simulations of a stable protein. J. Am. Chem. Soc. 2002;38:11258-11259.
[ 4] van Gunsteren, W. F., Berendsen, H. J. C. Computer simulation of molecular
dynamics: Methodology, applications, and perspectives in chemistry. Angew.
Chem. Int. Ed. Engl. 29:992–1023, 1990.
[ 5] Fraaije, J. G. E. M. Dynamic density functional theory for microphase separation
kinetics of block copolymer melts. J. Chem. Phys. 99:9202–9212, 1993.
[ 6] van Gunsteren, W. F., Berendsen, H. J. C. Algorithms for macromolecular
dynamics and constraint dynamics. Mol. Phys. 34:1311–1327, 1977.
[ 7] Geman, Geman. IEEE Trans. Patt. Anal. Mach. Int. 6:721, 1984.
[ 8] Nilges, M., Clore, G. M., Gronenborn, A. M. Determination of three-dimensional
structures of proteins from interproton distance data by dynamical simulated
annealing from a random array of atoms. FEBS Lett. 239:129–136, 1988.
[ 9] van Schaik, R. C., Berendsen, H. J. C., Torda, A. E., van Gunsteren, W. F. A
structure refinement method based on molecular dynamics in 4 spatial
dimensions. J. Mol. Biol. 234:751–762, 1993.
[ 10] Zimmerman, K. All purpose molecular mechanics simulator and energy
minimizer. J. Comp. Chem. 12:310–319, 1991.
[ 11] Ryckaert, J. P., Ciccotti, G., Berendsen, H. J. C. Numerical integration of the
Cartesian equations of motion of a system with constraints; molecular dynamics
of n-alkanes. J. Comp. Phys. 23:327–341, 1977.
[ 12] Miyamoto, S., Kollman, P. A. SETTLE: An analytical version of the SHAKE
and RATTLE algorithms for rigid water models. J. Comp. Chem. 13:952–962,
1992.
[ 13] Hess, B., Bekker, H., Berendsen, H. J. C., Fraaije, J. G. E. M. LINCS: A linear
constraint solver for molecular simulations. J. Comp. Chem. 18:1463–1472,
1997.
[ 14] Berendsen, H. J. C., Postma, J. P. M., DiNola, A., Haak, J. R. Molecular
dynamics with coupling to an external bath. J. Chem. Phys. 81:3684–3690, 1984.
[ 15] Nos´e, S. A molecular dynamics method for simulations in the canonical
ensemble. Mol. Phys. 52:255–268, 1984
[ 16 ] Hoover, W. G. Canonical dynamics: equilibrium phase-space distributions.
Phys. Rev. A 31: 1695–1697, 1985.
[ 17] Berendsen, H. J. C. Transport properties computed by linear response through
weak coupling to a bath. In: Computer Simulations in Material Science. Meyer,
M., Pontikis, V. eds. Kluwer 1991 139–155.
[ 18] Nos´e, S. A molecular dynamics method for simulations in the canonical
ensemble. Mol. Phys. 52:255–268, 1984.
[ 19] ] Hoover, W. G. Canonical dynamics: equilibrium phase-space distributions.
Phys. Rev. A 31:1695–1697, 1985.
[ 20] Recently the news has reported that over 50% of US households now have a
personal computer.
[ 21] Moore''s Law states that for a given cost, the amount of computing power
available for that cost doubles approximately every eighteen months.
[ 22] The Internet Domain Survey at http://www.nw.com claims over 43,000,000
connected computers ("hosts") as of January 1999. The data collected by this
survey clearly shows an exponential growth trend.
[ 23] Zero-knowledge proofs are one way to establish that assigned work has been
done to an arbitrarily high confidence level. Full double checking is one type of
zero-knowledge proof.
[ 24] Geyer GJ, Thompson EA. Annealing markov chain Monte Carlo with
applications to ancestral inference. J. Am. Stat. Assn. 1995;90(431):909-920.
[ 25] Falcioni, M.; Deem, M.W. J. Chem. Phys. 1999, 110, 1754-1766.
[ 26] Wu, M.G.; Deem, M.W. Mol. Phys. 1999, 97, 559-580.
[ 27] Yan, Q.L.; Pablo, J.J.de J. Chem. Phys. 1999, 111, 9509-9516.
[ 28] N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.N. Teller, E.Teller,
Equations of state calculations by fast computing machines. J. Chem. Phys.,
21:10871092,1953
[ 29] D. Frenkel, B. Smit, Understanding Molecular Simulation, Academic
Press ,2002
[ 30] P.J. in''t Veld, G.C. Rutledge, TemperatureDependent Elasticity of a
Semicrystalline Interphase Composed of Freely Rotating Chains,
Macromolecules, 36:73587365,2003
[ 31] MP92 to 3em, Simulated tempering: A new monte carlo scheme, Europhys. Lett.
19 (1992), 451
[ 32] P. Lyubartsev, A.A. Martinovski andS. V. Shevkunov, and P.N. Vorontsov-
Velyaminov, JCP 96 (1992), 1776
[ 33] Hukushima K, Nemoto K. Exchange Monte Carlo method and application to
spin glass simulations. J. Phys. Soc. Jpn. 1996; 65: 1604-1608.
[ 34] Ulrich H.E. Hansmann and Yuko Okamoto, Numerical comparison of three
recentlyproposedalgorithms in the protein folding problem, J. Comput. Chem 18
(1997), 920
[ 35 ] Hansmann UHE. Parallel tempering algorithm for conformational studies of
biological molecules. Chem. Phys. Lett. 1997;281: 140-150.
[ 36] Wanga F, Jordan KD. Parallel-tempering Monte Carlo simulations of the finite
temperature behavior of (H2O)6- , J. of Chem. Phys., 2003;119:11645-11653.
[ 37] Neirotti JP, Calvo F, Freeman DL, Doll JD. Phase changes in 38-atom
Lennard-Jones clusters I. A parallel tempering study in the canonical ensemble. J.
of Chem. Phys, 2000; 112:10340-10349.
[ 38] Kihara D, Lu H, Kolinski A, Skolnick JP. An ab initio protein structure
prediction method that uses threading-based tertiary restraints. Natl. Acad. Sci.
USA. 2001; 98:10125-10130.
[ 39 ] Hansmann UHE, Okamoto Y. New Monte Carlo algorithms for protein folding.
Curr. Opin. Struct. Biol. 1999;9: 177-184.
[ 40 ] Hansmann UHE, Okamoto Y. The generalized-ensemble approach for protein
folding simulations. In: Stauffer D, editor. Annual reviews in computational
physics. 1998; Singapore, World Scientific.
[ 41 ] Shirakura T, Matsubara F. Reexamination of the SG transition in the
two-dimensional +/-J Ising model. J. Phys. Soc. Jpn. 1996;65: 3138-3141.
[ 42 ] Sugita Y, Okamoto Y. Replica-exchange molecular dynamic method for protein
folding. Chem. Phys. Lett. 1999;314: 141-151.
[ 43 ] Neidigh JW, Fesinmeyer RM, Pricket KS ,Andersen NH. Exendin-4 and
glucagon-like-peptide-1: NMR structural comparisons in the solution and
micelle-associated states. Biochemistry 2001;40, 13188-13200.
[ 44 ] Wang JM, Cieplak P, Kollman PA. How well does a restrained electrostatic
potential (RESP) model perform in calculating conformational energies of
organic and biological molecules? J. Comput. Chem. 2000;21: 1049-1074.
[ 45 ] Qiu LL, Pabit SA, Roitberg AE, Hagen SJ. Smaller and faster: the 20-residue
trp-cage protein folds in 4 us J. Am. Chem. Soc. 2002;124: 12952-12953.
[ 46 ] Snow CD, Zagrovic B, Pande VS. The trp cage: folding kinetics and unfolded
state topology via molecular dynamics simulations. J. Am. Chem. Soc. 2002;
124: 14548-14549.
[ 47 ] Berendsen HJC, Van der Spoel D, Van Drunen R. GROMACS: A
message-passing parallel molecular dynamics implementation. Comp. Phys.
Comm. 1995;91: 43-56.
[ 48 ] Lindahl E, Hess B, Van der Spoel D. GROMACS 3.0: A package for molecular
simulation and trajectory analysis. J. Mol. Mod. 2002;7: 306-317.
[ 49 ] Nose S. A molecular dynamics method for simulations in the canonical
ensemble. Mol. Phys. 1984;52: 255-268.
[ 50 ] Hoover W. Canonical dynamics: equilibrium phase-space distributions. Phys.
Rev. 1985;A31: 1695-1697.
[ 51 ] Hess B, Bekker H, Berendsen HJC, Fraaije JGEM. LINCS: A linear constraint
solver for molecular simulations. J. Comp. Chem. 1997;18: 1463-1472.
[ 52 ] LO JL, YU CP, LEE HC. TBA
[ 53 ] Hukushima K, Nemoto K. Exchange Monte Carlo method and applications to
spin glass simulations. J. Phys. Soc. (Japan) 1996;65: 1604-1608.
[ 54] Pitera JW, Swope W. Understanding folding and design: replica-exchange
simulations of "trp-cage" miniproteins. PNAS. 2003; 100: 7587-7592.
[ 55] Zhou R. Trp-cage:Folding free energy landscape in explicit water. PNAS. 2003;
100(23):13280-13285.
[ 56] Lin CY, Hu CK, Hansmann UHE. Parallel Tempering Simulations of HP-36.
PROTEINS:Structure, Function and Genetics. 2003;52: 436-445.
[ 57] F. Calvo et al., J. Chem. Phys. 112, 10350 (2000).
[ 58] P. Lyubartsev, A.A. Martinovski andS. V. Shevkunov, and P.N. Vorontsov-
Velyaminov, JCP 96 (1992), 1776
[ 59] U. H. E. Hansmann, Eur. Phys. J. B 12 (1999), 607
[ 60] Zhang Y, Kihara D, Skolnick J. Local energy landscape flattening: Parallel hyperbolic Monte Carlo sampling of protein folding. PROTEINS: Struct. Func. and Gen. 20002;48:192201.
[61] Chowdhury S, Lee MC, Xiong G, Duan Y. Ab initio Folding Simulation of the Trpcage Mini-protein Approaches NMR Resolution J. Mol. Biol. 2003;327:711-717.