Parallel tempering Monte Carlo simulation of met-enkephalin and WW-domain proteins

M. Rana; P. Singh; N. C. Verma; S. Jain; M. Rana; P. Singh; N. C. Verma; S. Jain

doi:10.3934/biophy.2026006

AIMS Biophysics

2026, Volume 13, Issue 1: 94-110. doi: 10.3934/biophy.2026006

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Research article

Parallel tempering Monte Carlo simulation of met-enkephalin and WW-domain proteins

1.
Department of Physics, Medicaps University, Indore, India
2.
Israel Institute of Technology, Haifa, Israel
3.
Department of Physics, IPS Academy, Indore, India

Received: 20 October 2025 Revised: 25 February 2026 Accepted: 02 March 2026 Published: 18 March 2026

Efficient sampling of low-energy conformational states remains a central challenge in protein folding simulations. In this study, we investigated the performance and limitations of standard Monte Carlo (MC) and parallel tempering Monte Carlo (PTMC) methods in exploring protein energy landscapes using two representative systems: the small peptide Met-enkephalin and the FBP28 WW-Domain protein. For Met-enkephalin, standard MC simulations achieved equilibrium at higher temperatures but showed insufficient sampling in low-energy regions. In contrast, PTMC significantly improved sampling efficiency across a wide temperature range, particularly in the low-energy regime, enabling robust exploration of the energy landscape. PTMC successfully identified the global minimum structure of Met-enkephalin with an energy of -11.23 kcal/mol and a root-mean-square deviation of 1.18 Å from the reference structure. The applicability of PTMC to larger proteins was further examined through extensive simulations of the WW-Domain, revealing pronounced energy fluctuations and folding–unfolding transitions at higher temperatures, while sampling at lower temperatures remained limited. The number of visits per unit energy range for two consecutive temperatures have also monitored to check the exchange rate and conformational sampling of the entire landscape. The average energy and specific heat with respect to temperature have also calculated to check the folding unfolding transition of WW-Domain. These results demonstrated the superiority of PTMC over standard MC for protein folding studies; however, they also highlighted challenges associated with larger protein systems.
- protein-folding,
- Monte-Carlo simulation (MCS),
- parallel-tempering (PT),
- energy landscape
Citation: M. Rana, P. Singh, N. C. Verma, S. Jain. Parallel tempering Monte Carlo simulation of met-enkephalin and WW-domain proteins[J]. AIMS Biophysics, 2026, 13(1): 94-110. doi: 10.3934/biophy.2026006

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Abstract

Efficient sampling of low-energy conformational states remains a central challenge in protein folding simulations. In this study, we investigated the performance and limitations of standard Monte Carlo (MC) and parallel tempering Monte Carlo (PTMC) methods in exploring protein energy landscapes using two representative systems: the small peptide Met-enkephalin and the FBP28 WW-Domain protein. For Met-enkephalin, standard MC simulations achieved equilibrium at higher temperatures but showed insufficient sampling in low-energy regions. In contrast, PTMC significantly improved sampling efficiency across a wide temperature range, particularly in the low-energy regime, enabling robust exploration of the energy landscape. PTMC successfully identified the global minimum structure of Met-enkephalin with an energy of -11.23 kcal/mol and a root-mean-square deviation of 1.18 Å from the reference structure. The applicability of PTMC to larger proteins was further examined through extensive simulations of the WW-Domain, revealing pronounced energy fluctuations and folding–unfolding transitions at higher temperatures, while sampling at lower temperatures remained limited. The number of visits per unit energy range for two consecutive temperatures have also monitored to check the exchange rate and conformational sampling of the entire landscape. The average energy and specific heat with respect to temperature have also calculated to check the folding unfolding transition of WW-Domain. These results demonstrated the superiority of PTMC over standard MC for protein folding studies; however, they also highlighted challenges associated with larger protein systems.

Acknowledgments

I sincerely thank Dr. Rajamani Raghunathan from UGC-DAE-CSR Indore, for providing the computational facilities essential for this work.

Conflict of interest

The authors declare no conflict of interest.

Author contributions

PS conceived and designed the study and established the methodology. MR performed the simulations at various stages of the work. MR, NCV, and SJ contributed to the data analysis. PS and MR prepared the manuscript and reviewed the manuscript.

References

[1]	Skolnick J, Kolinski A (1989) Computer simulations of globular protein folding and tertiary structure. Annu Rev Phys Chem 40: 207-235. https://doi.org/10.1146/annurev.pc.40.100189.001231
[2]	Markwick PRL, McCammon JA (2011) Studying functional dynamics in biomolecules using accelerated molecular dynamics. Phys Chem Chem Phys 213: 20053-20065. https://doi.org/10.1039/C1CP22100K
[3]	Corey RB, Pauling LC (1953) Fundamental dimensions of polypeptide chains. Proc R Soc Lond B Biol Sci 141: 10-20. https://doi.org/10.1098/rspb.1953.0011
[4]	Deng C, Wu J, Cheng R, et al. (2014) Functional polypeptide and hybrid materials: precision synthesis via alpha-amino acid N-carboxyanhydride polymerization and emerging biomedical applications. Prog Polym Sci 39: 330-364. https://doi.org/10.1016/j.progpolymsci.2013.10.008
[5]	Alberts B, Johnson A, Lewis J, et al. (2002) Molecular Biology of the Cell. New York: Garland Science.
[6]	Faure G, Ogurtsov AY, Shabalina SA, et al. (2016) Role of mRNA structure in the control of protein folding. Nucleic Acids Res 44: 10898-10911. https://doi.org/10.1093/nar/gkw671
[7]	Lipovsek D, Pluckthun A (2004) In vitro protein evolution by ribosome display and mRNA display. J Immunol Methods 290: 51-67. https://doi.org/10.1016/j.jim.2004.04.008
[8]	Szilagyi A, Kardos J, Osvath S, et al. (2007) Handbook of Neurochemistry and Molecular Neurobiology. New York: Springer.
[9]	Onuchic JN, Wolynes PG (2004) Theory of protein folding. Curr Opin Struct Biol 14: 70-75. https://doi.org/10.1016/j.sbi.2004.01.009
[10]	Daggett V (2006) Protein folding–simulation. Chem Rev 106: 1898-1916. https://doi.org/10.1021/cr0404242
[11]	Lumry R, Eyring H (1954) Conformation changes of proteins. J Phys Chem 58: 110-120. https://doi.org/10.1021/j150512a005
[12]	Straub JE, Thirumalai D (1993) Exploring the energy landscape in proteins. Proc Natl Acad Sci USA 90: 809-813. https://doi.org/10.1073/pnas.90.3.809
[13]	Whitfield TW, Bu L, Straub JE (2002) Generalized parallel sampling. Phys A: Stat Mech Appl 305: 157-171. https://doi.org/10.1016/S0378-4371(01)00656-2
[14]	Kirkpatrick TR, Thirumalai D (1987) Dynamics of the structural glass transition and the p-spin-interaction spin-glass model. Phys Rev Lett 58: 2091. https://doi.org/10.1103/PhysRevLett.58.2091
[15]	Onuchic JN, Luthey-Schulten Z, Wolynes PG (1997) Theory of protein folding: the energy landscape perspective. Annu Rev Phys Chem 48: 545-600. https://doi.org/10.1146/annurev.physchem.48.1.545
[16]	Bernetti M, Bertazzo M, Masetti M (2020) Data-driven molecular dynamics: a multifaceted challenge. Pharmaceuticals (Basel) 13: 253. https://doi.org/10.3390/ph13090253
[17]	Perilla JR, Goh BC, Cassidy CK, et al. (2015) Molecular dynamics simulations of large macromolecular complexes. Curr Opin Struct Biol 31: 64-74. https://doi.org/10.1016/j.sbi.2015.03.007
[18]	Wales DJ (2018) Exploring energy landscapes. Annu Rev Phys Chem 69: 401-425. https://doi.org/10.1146/annurev-physchem-050317-021219
[19]	Singh P, Sarkar SK, Bandyopadhyay P (2014) Wang-Landau density of states based study of the folding-unfolding transition in the mini-protein Trp-cage (TC5b). J Chem Phys 141: 015103. https://doi.org/10.1063/1.4885726
[20]	Frantz DD, Freeman DL, Doll JD (1990) Reducing quasi-ergodic behavior in Monte Carlo simulations by J-walking: applications to atomic clusters. J Chem Phys 93: 2769-2784. https://doi.org/10.1063/1.458863
[21]	Marinari E, Parisi G (1992) Simulated tempering: a new Monte Carlo scheme. Europhys Lett 19: 451-458. https://doi.org/10.1209/0295-5075/19/6/002
[22]	Earl DJ, Deem MW (2005) Parallel tempering: theory, applications, and new perspectives. Phys Chem Chem Phys 7: 3910-3916. https://doi.org/10.1039/b509983h
[23]	Okamoto Y (2004) Generalized-ensemble algorithms: enhanced sampling techniques for Monte Carlo and molecular dynamics simulations. J Mol Graph Model 22: 425-439. https://doi.org/10.1016/j.jmgm.2003.12.009
[24]	Sugita Y, Okamoto Y (1999) Replica-exchange molecular dynamics method for protein folding. Chem Phys Lett 314: 141-151. https://doi.org/10.1016/S0009-2614(99)01123-9
[25]	Cheng X, Cui G, Hornak V, et al. (2005) Modified replica exchange simulation methods for local structure refinement. J Phys Chem B 109: 8220-8230. https://doi.org/10.1021/jp045437y
[26]	Srinivasaraghavan K, Zacharias M (2009) Folding of Trp-cage mini protein using temperature and biasing potential replica-exchange molecular dynamics simulations. Int J Mol Sci 10: 1121-1137. https://doi.org/10.3390/ijms10031121
[27]	Lyman E, Ytreberg FM, Zuckerman DM (2006) Resolution exchange simulation. Phys Rev Lett 96: 028105. https://doi.org/10.1103/PhysRevLett.96.028105
[28]	Liu P, Kim B, Friesner RA, et al. (2005) Replica exchange with solute tempering: a method for sampling biological systems in explicit water. Phys Rev Lett 102: 018101. Proc Natl Acad Sci USA 102 13749–13754. https://doi.org/10.1073/pnas.0506346102
[29]	Singh P, Sarkar SK, Bandyopadhyay P (2011) Understanding the applicability and limitations of Wang–Landau method for biomolecules: Met-enkephalin and Trp-cage. Chem Phys Lett 514: 357-361. https://doi.org/10.1016/j.cplett.2011.08.053
[30]	Takahashi A (2021) Handbook of Hormones. Amsterdam: Elsevier. https://doi.org/10.1016/C2019-1-01019-4
[31]	Macias MJ, Gervais V, Civera C, et al. (2000) Structural analysis of WW domains and design of a WW prototype. Nat Struct Biol 7: 375-379. https://doi.org/10.1038/75144
[32]	Crane JC, Koepf EK, Kelly JW, et al. (2000) Mapping the transition state of the WW domain beta-sheet. J Mol Biol 298: 283-292. https://doi.org/10.1006/jmbi.2000.3665
[33]	Hansmann UH (1997) Parallel tempering algorithm for conformational studies of biological molecules. Chem Phys Lett 281: 140-150. https://doi.org/10.1016/S0009-2614(97)01198-6
[34]	Zhan L, Chen JZY, Liu WK (2006) Conformational study of met-enkephalin based on the ECEPP force fields. Biophys J 91: 2399-2404. https://doi.org/10.1529/biophysj.106.083899
[35]	Sindhikara D, Spronk SA, Day T, et al. (2017) Improving accuracy, diversity, and speed with prime macrocycle conformational sampling. J Chem Inf Model 57: 1881-1894. https://doi.org/10.1021/acs.jcim.7b00052
[36]	Nguyen H, Jager M, Moretto A, et al. (2003) Tuning the free-energy landscape of a WW domain by temperature, mutation, and truncation. Proc Natl Acad Sci USA 100: 3948-3953. https://doi.org/10.1073/pnas.0538054100
[37]	Maisuradze GG, Liwo A, Scheraga HA (2010) Relation between free energy landscapes of proteins and dynamics. J Chem Theory Comput 6: 583-595. https://doi.org/10.1021/ct9005745
[38]	Beccara SA, Skrbic T, Covino R, et al. (2012) Dominant folding pathways of a WW domain. Proc Natl Acad Sci USA 109: 2330-2335. https://doi.org/10.1073/pnas.1111796109
[39]	Weinzierl S Introduction to Monte Carlo methods (2000). https://doi.org/10.48550/arXiv.hep-ph/0006269
[40]	Santana R, Larranaga P, Lozano JA (2008) Protein folding in simplified models with estimation of distribution algorithms. IEEE Trans Evol Comput 12: 418-438. https://doi.org/10.1109/TEVC.2007.906095
[41]	Suruzhon M, Bodnarchuk MS, Ciancetta A, et al. (2022) Enhancing ligand and protein sampling using sequential Monte Carlo. J Chem Theory Comput 18: 3894-3910. https://doi.org/10.1021/acs.jctc.1c01198
[42]	Berg BA, Neuhaus T (1991) Multicanonical algorithms for first order phase transitions. Phys Lett B 267: 249-253. https://doi.org/10.1016/0370-2693(91)91256-U
[43]	Bhanot G (1988) The Metropolis algorithm. Rep Prog Phys 51: 429. https://doi.org/10.1088/0034-4885/51/3/003
[44]	Rahman A, Saikia B, Gogoi CR, et al. (2022) Advances in the understanding of protein misfolding and aggregation through molecular dynamics simulation. Prog Biophys Mol Biol 175: 31-48. https://doi.org/10.1016/j.pbiomolbio.2022.08.007
[45]	Noe F (2008) Probability distributions of molecular observables computed from Markov models. J Chem Phys 128: 244103. https://doi.org/10.1063/1.2916718
[46]	Koneru JK, Reid KM, Robustelli P Performing all-atom molecular dynamics simulations of intrinsically disordered proteins with replica exchange solute tempering (2025). https://doi.org/10.48550/arXiv.2505.01860
[47]	Predescu C, Predescu M, Ciobanu CV (2005) On the efficiency of exchange in parallel tempering Monte Carlo simulations. J Phys Chem B 109: 4189-4196. https://doi.org/10.1021/jp045073+
[48]	Ozboyaci M, Kokh DB, Corni S, et al. (2016) Modeling and simulation of protein–surface interactions: achievements and challenges. Q Rev Biophys 49: e4. https://doi.org/10.1017/s0033583515000256
[49]	Wei G, Xi W, Nussinov R, et al. (2016) Protein ensembles: how does nature harness thermodynamic fluctuations for life? The diverse functional roles of conformational ensembles in the cell. Chem Rev 116: 6516-6551. https://doi.org/10.1021/acs.chemrev.5b00562
[50]	Kumar S, Rosenberg JM, Bouzida D, et al. (1992) The weighted histogram analysis method for free-energy calculations on biomolecules. I. The method. J Comput Chem 13: 1011-1021. https://doi.org/10.1002/jcc.540130812
[51]	Chodera JD, Swope WC, Pitera JW, et al. (2007) Use of the weighted histogram analysis method for the analysis of simulated and parallel tempering simulations. J Chem Theory Comput 3: 26-41. https://doi.org/10.1021/ct0502864
[52]	Nemethy G, Gibson KD, Palmer KA, et al. (1992) Energy parameters in polypeptides. 10. Improved geometrical parameters and nonbonded interactions for use in the ECEPP/3 algorithm, with application to proline-containing peptides. J Phys Chem 96: 6472-6484. https://doi.org/10.1021/j100194a068
[53]	Bashford D, Case DA (2000) Generalized Born models of macromolecular solvation effects. J Comput Aided Mol Des 14: 129-152. https://doi.org/10.1146/annurev.physchem.51.1.129
[54]	Zhou R (2003) Free energy landscape of protein folding in water: explicit vs. implicit solvent. Proteins 53: 148-161. https://doi.org/10.1002/prot.10483
[55]	Garcia AE, Onuchic JN (2003) Folding a protein in a computer: an atomic description of the folding/unfolding of protein A. Proc Natl Acad Sci USA 100: 13898-13903. https://doi.org/10.1073/pnas.2335541100
[56]	Okur A, Wickstrom L, Layten M, et al. (2006) Improved efficiency of replica exchange simulations through use of a hybrid explicit/implicit solvation model. J Chem Theory Comput 2: 420-433. https://doi.org/10.1021/ct050196z
[57]	Walter JC, Barkema GT (2015) An introduction to Monte Carlo methods. Physica A 418: 78-87. https://doi.org/10.1016/j.physa.2014.06.014
[58]	Evans DA, Wales DJ (2003) The free energy landscape and dynamics of met-enkephalin. J Chem Phys 119: 9947-9955. https://doi.org/10.1063/1.1616515
[59]	Perez JJ, Villar HO, Loew GH (1992) Characterization of low-energy conformational domains for Met-enkephalin. J Comput Aided Mol Des 6: 175-190. https://doi.org/10.1007/BF00129427
[60]	Dias AMGC, Teixeira GDG, Barbosa AJM, et al. (2025) Design and evolution of a synthetic small protein scaffold based on the WW domain. Protein Sci 34: e70164. https://doi.org/10.1002/pro.70164
[61]	Ferrenberg AM, Swendsen RH (1989) Optimized Monte Carlo data analysis. Phys Rev Lett 63: 1195-1198. https://doi.org/10.1103/PhysRevLett.63.1195
[62]	Gallicchio E, Andrec M, Felts AK, et al. (2005) Temperature weighted histogram analysis method, replica exchange, and transition paths. J Phys Chem B 109: 6722-6731. https://doi.org/10.1021/jp045294f
[63]	Cobos ES, Iglesias M, Ruiz-Sanz J, et al. (2009) Thermodynamic characterization of the folding equilibrium of the human nedd4-WW4 domain: at the frontiers of cooperative folding. Biochemistry 48: 1404-1412. https://doi.org/10.1021/bi9007758
[64]	Beccara S, Skrbic T, Covino R, et al. (2012) Dominant folding pathways of a WW domain. Proc Natl Acad Sci USA 109: 2330-2335. https://doi.org/10.1073/pnas.1111796109
[65]	Chai YL, Chin KH, Hansmann UHE, et al. (2003) Parallel tempering simulations of HP-36. Proteins 52: 436-445. https://doi.org/10.1002/prot.10351

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