Arashiro, E.

Drugowich de Felicio, J. R.

Hansmann, U. H. E.

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John von Neumann Institute for ComputingGlobal Persistence Exponent of the Helix-CoilTransition in PolypeptidesE. Arashiro, J. R. Drugowich de Felı´cio,U. H. E. Hansmannpublished inFrom Computational Biophysics to Systems Biology (CBSB07),Proceedings of the NIC Workshop 2007,Ulrich H. E. Hansmann, Jan Meinke, Sandipan Mohanty,Olav Zimmermann (Editors),John von Neumann Institute for Computing, Ju¨lich,NIC Series, Vol. 36, ISBN 978-3-9810843-2-0, pp. 79-82, 2007.c© 2007 by John von Neumann Institute for ComputingPermission to make digital or hard copies of portions of this work forpersonal or classroom use is granted provided that the copies are notmade or distributed for profit or commercial advantage and that copiesbear this notice and the full citation on the first page. To copy otherwiserequires prior specific permission by the publisher mentioned above.http://www.fz-juelich.de/nic-series/volume36Global Persistence Exponent of the Helix-Coil Transitionin PolypeptidesEveraldo Arashiro1, Jose´ Roberto Drugowich de Felı´cio2, andUlrich H. E. Hansmann11 John von Neumann Institute for Computing,Forschungszentrum Ju¨lich, 52425 Ju¨lich, GermanyE-mail: {e.arashiro, u.hansmann}@fz-juelich.de2 Departamento de Fı´sica e Matema´tica, FFCLRP, Universidade de Sa˜o Paulo,Ribeira˜o Preto, Sa˜o Paulo CEP 14040-901, BrazilE-mail: drugo@usp.brThe dynamical exponent θg that governs the behavior of the global persistence probability isobtained through short-time Monte Carlo simulations of the helix-coil transition. Our simula-tions are based on a detailed, all-atom representation of the molecules and an implicit solvationmodel to approximate the interaction with the surrounding solvent. Our results obtained for thepolyalanine and the 34-residue human parathyroid fragment PTH(1-34) are in good agreementeach other, and indicate universality of the helix-coil transition in proteinlike molecules.1 IntroductionAfter the works by Janssen, Schaub, and Schmittmann1 and Huse2, much attention hasbeen devoted to study of phase transitions and critical phenomena of dynamical systemsat the early stage of its time evolution. By using renormalization group techniques andnumerical calculations, respectively, they showed that universality and scaling behaviorare already present in this stage (henceforth named as short-time stage) after quenchingfrom high temperatures to the critical one.The values of the critical indices obtained through this technique3, 4, have shown thatshort-time Monte Carlo simulations are very efficient and can be used to estimate, withgood precision, the statical exponents and the dynamical exponent z.Far from equilibrium, another dynamic critical exponent was proposed by Majumdaret al.5 studying the behavior of the global persistence probability P (t) that the order pa-rameter has not changed its sign up to the time t. The global persistence probability P (t)can be defined asP (t) = 1−t∑t′=1ρ(t′), (1)where ρ(t′) is the fraction of the samples that have changed their state for the first time atthe instant t′.At criticality, P (t) is expected to decay algebraically asP (t)˜t−θg (2)where θg is the global persistence exponent. Since then, the study of the persistence be-havior have attracted much interest, playing an important role in the study of systems farfrom equilibrium6–8 .79Figure 1. Left: Log-log plot of the time series of the global persistency P (t) for Ala40 and various initial orderparameter values q0H. Right: Persistence exponent as a function q0H. Open symbols are the values for polyalanineand the filled symbols are the values for PTH(1-34).In this work, we have studied the dynamic critical behavior of helical proteins via short-time Monte Carlo simulations. We evaluated the dynamic critical exponent θg for (Ala)Nchains (N=10, 20, 30 and 40) to verify the effects of finite size. Our investigation is laterextended toward the 34-residue human parathyroid fragment PTH(1-34).2 ModelOur short-time MC simulations of the helix-coil transition are based on a detailed, all-atomrepresentation of proteins. The interaction between the atoms is described by a standardforce field, ECEPP/2 (Empirical Conformational Energy Program for Peptides) as imple-mented in the program package SMMP (Simple Molecular Mechanics for Proteins)9 . Theinteractions between our polypeptides and surrounding water are approximated by meansof an implicit water model, which assumes that the solvation (free) energy is proportionalto the solvent accessible surface area and utilizes the parameter set of Ref. 10 that is oftenused in conjunction with the ECEPP force field.Crucial for our analysis is the definition of an order parameter. Our analog to themagnetization in spin systems is the number of helical residues qH = 2 ∗ 〈nH(T )〉 /(N −2) − 1. Here we define a residue as helical if its backbone dihedral angles (φ, ψ) , takevalues in the range (−70◦±30◦,−37◦±30◦) (their common values in an α helix), and theresidue exhibits the hydrogen bonding pattern observed in α helices. The normalizationfactorN−2 (N the number of residues) is chosen because the flexible terminal residues areusually not part of an α helix. Our definition ensures that −1 ≤ qH ≤ 1 and qH(Tc) = 0.3 ResultsWe start our investigation of dynamic critical exponent θg in the helix-coil transition bysimulating polyalanine chains of lengths N=10, 20, 30, and 40 in your respectively criticaltemperatura Tc = 315, 415, 450 and 470 which were found in our earlier works11, 12. Ourresults are averaged over 5000 independent runs . Errors are estimated by dividing these800.00 0.05 0.100.550.600.650.700.750.800qH gg = 0.729(6)   Figure 2. The scaling obtained for the 5 smaller q0H. The persistence exponent found is θg = 0.729(6).5000 runs in bins of 100 runs and calculating the fluctuation of the averages calculated foreach bin.In order to estimate the exponent θg one needs to prepare the initial states carefullyto obtain precise values of the initial q0H = qH(t = 0). The exponent q0H is obtainedfrom Eq. 2. Figure 1 (Left) shows the corresponding log-log plots of P (t) for Ala40 andvarious initial values of q0H . This value, and the ones corresponding to polyalanine chainsof lenght N= 10, 20 and 30 are listed in Tab. 1.q0H Ala10 q0H Ala20 q0H Ala30 q0H Ala400.25 0.402(6) 0.11 0.687(7) 0.0714 0.708(7) 0.0526 0.710(8)0.50 0.248(7) 0.22 0.469(7) 0.1429 0.626(6) 0.1053 0.696(7)0.33 0.317(8) 0.2143 0.488(7) 0.1579 0.605(7)0.2857 0.374(7) 0.2105 0.503(7)Table 1. Global persistence exponent θg for polyalanine chains of length N = 10, 20, 30 and 40.q0H PTH(1− 34)0.0625 0.705(6)0.125 0.658(7)0.1875 0.562(6)0.25 0.395(7)Table 2. Global persistence exponent θg for the 34-residue human parathyroid fragment PTH(1-34).The general behavior of the global persistence for each polyalanina indicate that whenq0H decreases, the value for the persistence exponent θg increases. In Fig 1 (Right) weshow the θg of all polyalanine as a function of the q0H . We can observe that the value of81θg increase quickly, with the decrease of the q0H , but tends to stabilize by θg = 0.7. Wealso estimate the persistence exponent for the PTH(1-34), Tc = 545 (Tab. 2). In Fig 2we show that the values de θg for PTH(1-34) (closed simbols) have the same behaviorthose the polyalanine (open simbols). From the 5 smaller q0H the persistence exponentsθg = 0.729(6) was estimated for this helical protein.4 ConclusionThe behavior of exponent θg = 0.729(6) for polyalanine and the 34-residue humanparathyroid fragment PTH(1-34) are in good agreement each other, and indicate univer-sality of the helix-coil transition in proteinlike molecules.AcknowledgmentsTwo of the authors (E.A. and J.R.D.F.) acknowledge support by Capes and CNPq (Brazil),and another author (U.H.E.H.) was supported by a research grant from the National ScienceFoundation (CHE-0313618).References1. H. K. Janssen, B. Schaub, and B. Schmittmann, Z. Phys. B: Condens. Matter 73, 539(1989).2. D.A. Huse, Phys. Rev. B 40, 304 (1989).3. B. Zheng, Int. J. Mod. Phys. B 12, 1419 (1998).4. E. Arashiro, J.R. Drugowich de Felicio, Phys. Rev. E 67 046123 (2003).5. S. N. Majumdar, A. J. Bray, S. J. Cornell and C. Sire, Phys. Rev. Lett. 77 3704(1996).6. S. N. Majumdar and A. J. Bray, Phys. Rev. Lett. 91, 030602 (2003).7. H. A. Fernandes, E. Arashiro, J. R. Drugowich de Felicio, A. A. Caparica, Physica A366, 255 (2006).8. R. da Silva, N. A. Alves, J. R. Drugowich de Felicio, Phys. Rev. E 67, 057102 (2003).9. F. Eisenmenger, U. H. E. Hansmann, Sh. Hayryan, and C.-K. Hu, Comput.Phys.Commun. 138, 192 (2001).10. T. Ooi, M. Obatake, G. Nemethy, and H. A. Scheraga, Proc. Natl. Acad., Sci. U.S.A.8, 3086 (1987).11. E. Arashiro, J. R. Drugowich de Felı´cio, and U. H. E. Hansmann, Phys. Rev. E 73,040902 (2006).12. E. Arashiro, J. R. Drugowich de Felı´cio, and U. H. E. Hansmann, J. Chem. Phys. 126,045107 (2007).82

Global Persistence Exponent of the Helix-Coil Transition in Polypeptides

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Global Persistence Exponent of the Helix-Coil Transition in Polypeptides

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