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Non-equilibrium Bio-Molecular Unfolding Under Tension

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DNA Systems Under Internal and External Forcing

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Abstract

We explore how the force-mediated unfolding of simple DNA systems, despite occurring far from equilibrium, can reveal information about equilibrium free energies. The Jarzynski equality from nonequilibrium statistical physics is introduced and adapted to study the folding of DNA hairpins, validating that such techniques can be profitably applied with oxDNA coarse-grained simulations.

It is a part of probability that many improbabilities will happen.

—Agathon, as quoted in Aristotle’s Poetics 9.

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Notes

  1. 1.

    Here and throughout the chapter, G has been used to denote free energy. While Jarzynski originally proved his equality using Helmholtz free energy [23], it is valid for both Helmholtz and Gibbs free energies [60], and for this chapter, G has been used instead of F to prevent confusion with ‘force’.

  2. 2.

    Neupane et al. [71] and Manuel et al. [70] do not specify the tension used.

  3. 3.

    Arrival in equilibrium was confirmed by ensuring the system energy had reached a stable average value.

  4. 4.

    An iterative process was used to identify suitable umbrella sampling weights and thus achieve sufficient sampling, which led to lengthy simulation times.

  5. 5.

    That 360 hours yield a smaller bias than 600 hours for the harmonic protocol is a result of choice of pulling speed, discussed in the text.

  6. 6.

    This is because the fitted work distributions may fail to satisfy the Crooks fluctuation theorem [24] as \(\delta \) \(\rightarrow \) 1; see the supplementary information of Ref. [77].

References

  1. Anfinsen CB, Haber E, Sela M, White FH (1961) The kinetics of formation of native ribonuclease during oxidation of the reduced polypeptide chain. Proc Natl Acad Sci USA 47:1309–1314

    Article  ADS  Google Scholar 

  2. Anfinsen CB (1973) Principles that govern the folding of protein chains. Science 181:223–230

    Article  ADS  Google Scholar 

  3. Forloni G, Terreni L, Bertani I, Fogliarino S, Invernizzi R, Assini A, Ribizzi G, Negro A, Calabrese E, Volont MA, Mariani C, Franceschi M, Tabaton M, Bertoli A (2002) Protein misfolding in Alzheimers and Parkinsons disease: genetics and molecular mechanisms. Neurobiol Aging 23:957–976 (Brain Aging: Identifying the Brakes and Accelerators)

    Article  Google Scholar 

  4. Huang P-S, Boyken SE, Baker D (2016) The coming of age of de novo protein design. Nature 537:320

    Article  ADS  Google Scholar 

  5. Yin P, Hariadi RF, Sahu S, Choi HMT, Park SH, LaBean TH, Reif JH (2008) Programming DNA tube circumferences. Science 321:824–826

    Article  ADS  Google Scholar 

  6. Pinheiro AV, Han D, Shih WM, Yan H (2011) Challenges and opportunities for structural DNA nanotechnology. Nat Nanotechnol 6:763–772

    Article  ADS  Google Scholar 

  7. Levinthal C (1969) How to fold graciously. In DeBrunner JTP, Munck E (eds) Mossbauer spectroscopy in biological systems: proceedings of a meeting held at allerton house, Monticello, Illinois. University of Illinois Press, pp 22–24

    Google Scholar 

  8. Bryngelson JD, Wolynes PG (1987) Spin glasses and the statistical mechanics of protein folding. Proc Natl Acad Sci USA 84:7524–7528

    Article  ADS  Google Scholar 

  9. Onuchic JN, Luthey-Schulten Z, Wolynes PG (1997) Theory of protein folding: the energy landscape perspective. Annu Rev Phys Chem 48:545–600 PMID: 9348663

    Article  ADS  Google Scholar 

  10. Plotkin SS, Onuchic JN (2002) Understanding protein folding with energy landscape theory part I: basic concepts. Q Rev Biophys 35:111–167

    Article  Google Scholar 

  11. Yu H, Gupta AN, Liu X, Neupane K, Brigley AM, Sosova I, Woodside MT (2012) Energy landscape analysis of native folding of the prion protein yields the diffusion constant, transition path time, and rates. Proc Natl Acad Sci USA 109:14452–14457

    Article  ADS  Google Scholar 

  12. Woodside MT, Block SM (2014) Reconstructing folding energy landscapes by single-molecule force spectroscopy. Annu Rev Biophys 43:19–39

    Article  Google Scholar 

  13. Zhuang X, Bartley LE, Babcock HP, Russell R, Ha T, Herschlag D, Chu S (2000) A single-molecule study of RNA catalysis and folding. Science 288:2048–2051

    Article  ADS  Google Scholar 

  14. Greenleaf WJ, Woodside MT, Block SM (2007) High-resolution, single-molecule measurements of biomolecular motion. Annu Rev Biophys Biomol Struct 36:171–190

    Article  Google Scholar 

  15. Harris NC, Song Y, Kiang C-H (2007) Experimental free energy surface reconstruction from single-molecule force spectroscopy using Jarzynski’s equality. Phys Rev Lett 99:068101

    Article  ADS  Google Scholar 

  16. Gebhardt JCM, Bornschlögl T, Rief M (2010) Full distance-resolved folding energy landscape of one single protein molecule. Proc Natl Acad Sci USA 107:2013–2018

    Article  ADS  Google Scholar 

  17. Woodside MT, Behnke-Parks WM, Larizadeh K, Travers K, Herschlag D, Block SM (2006b) Nanomechanical measurements of the sequence-dependent folding landscapes of single nucleic acid hairpins. Proc Natl Acad Sci USA 103:6190–6195

    Article  ADS  Google Scholar 

  18. Engel MC, Ritchie DB, Foster DAN, Beach KSD, Woodside MT (2014) Reconstructing folding energy landscape profiles from nonequilibrium pulling curves with an inverse Weierstrass integral transform. Phys Rev Lett 113:238104

    Article  ADS  Google Scholar 

  19. Liphardt J, Dumont S, Smith SB, Tinoco I, Bustamante C (2002) Equilibrium information from nonequilibrium measurements in an experimental test of Jarzynski’s equality. Science 296:1832–1835

    Article  ADS  Google Scholar 

  20. Collin D, Ritort F, Jarzynski C, Smith SB, Tinoco I Jr, Bustamante C (2005) Verification of the Crooks fluctuation theorem and recovery of RNA folding free energies. Nature 437:231

    Article  ADS  Google Scholar 

  21. Woodside MT, Anthony PC, Behnke-Parks WM, Larizadeh K, Herschlag D, Block SM (2006a) Direct measurement of the full, sequence-dependent folding landscape of a nucleic acid. Science 314:1001–1004

    Article  ADS  Google Scholar 

  22. Harris NC, Kiang C-H (2009) Velocity convergence of free energy surfaces from single-molecule measurements using Jarzynski’s equality. Phys Rev E 79:041912

    Article  ADS  Google Scholar 

  23. Jarzynski C (1997) Nonequilibrium equality for free energy differences. Phys Rev Lett 78:2690–2693

    Article  ADS  Google Scholar 

  24. Crooks GE (1999) Entropy production fluctuation theorem and the nonequilibrium work relation for free energy differences. Phys Rev E 60:2721–2726

    Article  ADS  Google Scholar 

  25. Hummer G, Szabo A (2001) Free energy reconstruction from nonequilibrium single-molecule pulling experiments. Proc Natl Acad Sci USA 98:3658–3661

    Article  ADS  Google Scholar 

  26. Hummer G, Szabo A (2010) Free energy profiles from single-molecule pulling experiments. Proc Natl Acad Sci USA 107:21441–21446

    Article  ADS  Google Scholar 

  27. Gupta AN, Vincent A, Neupane K, Yu H, Wang F, Woodside MT (2011) Experimental validation of free-energy-landscape reconstruction from non-equilibrium single-molecule force spectroscopy measurements. Nat Phys 7:631

    Article  Google Scholar 

  28. Dellago C, Hummer G (2014) Computing equilibrium free energies using non-equilibrium molecular dynamics. Entropy 16:41–61

    Article  ADS  MathSciNet  Google Scholar 

  29. Oberhofer H, Dellago C, Geissler PL (2005) Biased sampling of nonequilibrium trajectories: can fast switching simulations outperform conventional free energy calculation methods? J Phys Chem B 109:6902–6915 PMID: 16851777

    Article  Google Scholar 

  30. Lechner W, Dellago C (2007) On the efficiency of path sampling methods for the calculation of free energies from non-equilibrium simulations. J Stat Mech Theory Exp 2007:P04001

    Article  Google Scholar 

  31. Oberhofer H, Dellago C (2008) Optimum bias for fast-switching free energy calculations. Comput Phys Commun 179:41–45 (Special issue based on the Conference on Computational Physics 2007)

    Article  ADS  Google Scholar 

  32. Vaikuntanathan S, Jarzynski C (2011) Escorted free energy simulations. J Chem Phys 134:054107

    Article  ADS  Google Scholar 

  33. Chelli R (2012) Local sampling in steered Monte Carlo simulations decreases dissipation and enhances free energy estimates via nonequilibrium work theorems. J Chem Theory Comput 8:4040–4052

    Article  Google Scholar 

  34. Athènes M, Marinica M-C (2010) Free energy reconstruction from steered dynamics without post-processing. J Comput Phys 229:7129–7146

    Article  ADS  MathSciNet  MATH  Google Scholar 

  35. Jensen MØ, Park S, Tajkhorshid E, Schulten K (2002) Energetics of glycerol conduction through aquaglyceroporin GlpF. Proc Natl Acad Sci USA 99:6731–6736

    Article  ADS  Google Scholar 

  36. Grubmüller H, Heymann B, Tavan P (1996) Ligand binding: molecular mechanics calculation of the streptavidin-biotin rupture force. Science 271:997–999

    Article  ADS  Google Scholar 

  37. Paci E, Karplus M (1999) Forced unfolding of fibronectin type 3 modules: an analysis by biased molecular dynamics simulations. J Mol Bio 288:441–459

    Article  Google Scholar 

  38. Chen AA, García AE (2012) Mechanism of enhanced mechanical stability of a minimal RNA kissing complex elucidated by nonequilibrium molecular dynamics simulations. Proc Natl Acad Sci USA 109:E1530–E1539

    Article  ADS  Google Scholar 

  39. Sotomayor M, Schulten K (2007) Single-molecule experiments in vitro and in silico. Science 316:1144–1148

    Article  ADS  Google Scholar 

  40. Izrailev S, Stepaniants S, Balsera M, Oono Y, Schulten K (1997) Molecular dynamics study of unbinding of the avidin-biotin complex. Biophys J 72:1568–1581

    Article  Google Scholar 

  41. Isralewitz B, Gao M, Schulten K (2001) Steered molecular dynamics and mechanical functions of proteins. Curr Opin Struct Biol 11:224–230

    Article  Google Scholar 

  42. Sancho DD, Best RB (2017) Complex dynamics in single molecule force spectroscopy from simple simulation models. Biophys J 112:196a

    Article  Google Scholar 

  43. He C, Genchev GZ, Lu H, Li H (2012) Mechanically untying a protein slipknot: multiple pathways revealed by force spectroscopy and steered molecular dynamics simulations. J Am Chem Soc 134:10428–10435

    Article  Google Scholar 

  44. Braun O, Hanke A, Seifert U (2004) Probing molecular free energy landscapes by periodic loading. Phys Rev Lett 93:158105

    Article  ADS  Google Scholar 

  45. Imparato A, Pelizzola A, Zamparo M (2007b) Ising-like model for protein mechanical unfolding. Phys Rev Lett 98:148102

    Article  ADS  Google Scholar 

  46. Imparato A, Luccioli S, Torcini A (2007a) Reconstructing the free-energy landscape of a mechanically unfolded model protein. Phys Rev Lett 99:168101

    Article  ADS  Google Scholar 

  47. Dudko OK, Graham TGW, Best RB (2011) Locating the barrier for folding of single molecules under an external force. Phys Rev Lett 107:208301

    Article  ADS  Google Scholar 

  48. Mitternacht S, Luccioli S, Torcini A, Imparato A, Irbäck A (2009) Changing the mechanical unfolding pathway of FnIII10 by tuning the pulling strength. Biophys J 96:429–441

    Article  Google Scholar 

  49. Colizzi F, Bussi G (2012) RNA unwinding from reweighted pulling simulations. J Am Chem Soc 134:5173–5179 PMID: 22313354

    Article  Google Scholar 

  50. Dietz H, Berkemeier F, Bertz M, Rief M (2006) Anisotropic deformation response of single protein molecules. Proc Natl Acad Sci USA 103:12724–12728

    Article  ADS  Google Scholar 

  51. Srinivas N, Ouldridge TE, Šulc P, Schaeffer JM, Yurke B, Louis AA, Doye JPK, Winfree E (2013) On the biophysics and kinetics of toehold-mediated DNA strand displacement. Nucleic Acids Res 41:10641–10658

    Article  Google Scholar 

  52. ten Wolde PR, Frenkel D (1997) Enhancement of protein crystal nucleation by critical density fluctuations. Science 277:1975–1978

    Article  Google Scholar 

  53. Ouldridge TE, Louis AA, Doye JPK (2011) Structural, mechanical, and thermodynamic properties of a coarse-grained DNA model. J Comput Phys 134:085101

    Google Scholar 

  54. Šulc P, Romano F, Ouldridge TE, Rovigatti L, Doye JPK, Louis AA (2012) Sequence-dependent thermodynamics of a coarse-grained DNA model. J Comput Phys 137:135101

    Google Scholar 

  55. Snodin BEK, Randisi F, Mosayebi M, Šulc P, Schreck JS, Romano F, Ouldridge TE, Tsukanov R, Nir E, Louis AA, Doye JPK (2015) Introducing improved structural properties and salt dependence into a coarse-grained model of DNA. J Chem Phys 142:234901

    Article  ADS  Google Scholar 

  56. Schreck JS, Romano F, Zimmer MH, Louis AA, Doye JPK (2016) Characterizing DNA star-tile-based nanostructures using a coarse-grained model. ACS Nano 10:4236–4247

    Article  Google Scholar 

  57. Romano F, Chakraborty D, Doye JPK, Ouldridge TE, Louis AA (2013) Coarse-grained simulations of DNA overstretching. J Chem Phys 138:085101

    Article  ADS  Google Scholar 

  58. Mosayebi M, Louis AA, Doye JPK, Ouldridge TE (2015) Force-induced rupture of a DNA duplex: from fundamentals to force sensors. ACS Nano 9:11993–12003

    Article  Google Scholar 

  59. Schuss Z (1980) Theory and applications of stochastic processes, vol 170, 1st edn. Springer, New York. ISBN 978-1-4419-1604-4

    Google Scholar 

  60. Cuendet MA (2006) The Jarzynski identity derived from general Hamiltonian or non-Hamiltonian dynamics reproducing NVT or NPT ensembles. J Chem Phys 125:144109

    Article  ADS  Google Scholar 

  61. Gore J, Ritort F, Bustamante C (2003) Bias and error in estimates of equilibrium free-energy differences from nonequilibrium measurements. Proc Natl Acad Sci USA 100:12564–12569

    Article  ADS  MathSciNet  MATH  Google Scholar 

  62. Jarzynski C (2006) Rare events and the convergence of exponentially averaged work values. Phys Rev E 73:046105

    Article  ADS  Google Scholar 

  63. Jarzynski C (2002) What is the microscopic response of a system driven far from equilibrium?. Springer, Berlin, pp 63–82

    MATH  Google Scholar 

  64. Jarzynski C (2008) Nonequilibrium work relations: foundations and applications. Eur Phys J B 64(3):331–340

    Article  ADS  MathSciNet  Google Scholar 

  65. Jarzynski C (2003) Thermal fluctuations in systems driven away from equilibrium. In: Proceedings of SPIE–The international society for optical engineering, vol 5110

    Google Scholar 

  66. Jarzynski C (2009) Nonequilibrium statistical mechanics: lectures for the Boulder School for Condensed Matter and Materials Physics. https://www.youtube.com/watch?v=LXcQx6Bu3OQ&list=PLdV51-1AzT3NRgkJUQYErIzC6wqW7n-tn&index=16. Accessed 11 Feb 2016

  67. Hummer G, Szabo A (2005) Free energy surfaces from single-molecule force spectroscopy. Acc Chem Res 38:504–513

    Article  Google Scholar 

  68. Torrie G, Valleau J (1977) Nonphysical sampling distributions in Monte Carlo free-energy estimation: umbrella sampling. J Comput Phys 23:187–199

    Article  ADS  Google Scholar 

  69. Kumar S, Rosenberg JM, Bouzida D, Swendsen RH, Kollman PA (1992) The weighted histogram analysis method for free-energy calculations on biomolecules. I. The method. J Comput Chem 13:1011–1021

    Article  Google Scholar 

  70. Manuel AP, Lambert J, Woodside MT (2015) Reconstructing folding energy landscapes from splitting probability analysis of single-molecule trajectories. Proc Natl Acad Sci USA 112:7183–7188

    Article  ADS  Google Scholar 

  71. Neupane K, Wang F, Woodside MT (2017) Direct measurement of sequence-dependent transition path times and conformational diffusion in DNA duplex formation. Proc Natl Acad Sci USA 114:1329–1334

    Article  Google Scholar 

  72. Pfitzner E, Wachauf C, Kilchherr F, Pelz B, Shih WM, Rief M, Dietz H (2013) Rigid DNA beams for high-resolution single-molecule mechanics. Angew Chem Int Ed 52:7766–7771

    Article  Google Scholar 

  73. Hinczewski M, Gebhardt JCM, Rief M, Thirumalai D (2013) From mechanical folding trajectories to intrinsic energy landscapes of biopolymers. Proc Natl Acad Sci USA 110:4500–4505

    Article  ADS  Google Scholar 

  74. Zhang Q, Brujić J, Vanden-Eijnden E (2011) Reconstructing free energy profiles from nonequilibrium relaxation trajectories. Curr Opin Chem Biol 144:344–366

    MathSciNet  MATH  Google Scholar 

  75. Yunger Halpern N, Jarzynski C (2016) Number of trials required to estimate a free-energy difference, using fluctuation relations. Phys Rev E 93:052144

    Article  ADS  Google Scholar 

  76. Rohwer CM, Angeletti F, Touchette H (2015) Convergence of large-deviation estimators. Phys Rev E 92:052104

    Article  ADS  Google Scholar 

  77. Palassini M, Ritort F (2011) Improving free-energy estimates from unidirectional work measurements: theory and experiment. Phys Rev Lett 107:060601

    Article  ADS  Google Scholar 

  78. Then H, Engel A (2008) Computing the optimal protocol for finite-time processes in stochastic thermodynamics. Phys Rev E 77:041105

    Article  ADS  Google Scholar 

  79. Geiger P, Dellago C (2010) Optimum protocol for fast-switching free-energy calculations. Phys Rev E 81:021127

    Article  ADS  Google Scholar 

  80. Schmiedl T, Seifert U (2007) Optimal finite-time processes in stochastic thermodynamics. Phys Rev Lett 98:108301

    Article  ADS  Google Scholar 

  81. Romeo M, Da Costa V, Bardou F (2003) Broad distribution effects in sums of lognormal random variables. Euro Phys J B 32:513–525

    Article  ADS  Google Scholar 

  82. Wood RH, Muhlbauer WCF, Thompson PT (1991) Systematic errors in free energy perturbation calculations due to a finite sample of configuration space: sample-size hysteresis. J Phys Chem 95:6670–6675

    Article  Google Scholar 

  83. Zuckerman DM, Woolf TB (2002) Theory of a systematic computational error in free energy differences. Phys Rev Lett 89:180602

    Article  ADS  Google Scholar 

  84. Hummer G (2001) Fast-growth thermodynamic integration: error and efficiency analysis. J Chem Phys 114:7330–7337

    Article  ADS  Google Scholar 

  85. Rothemund PWK (2006) Folding DNA to create nanoscale shapes and patterns. Nature 440:297–302

    Article  ADS  Google Scholar 

  86. Douglas SM, Marblestone AH, Teerapittayanon S, Vazquez A, Church GM, Shih WM (2009) Rapid prototyping of 3D DNA-origami shapes with caDNAno. Nucleic Acids Res 37:5001–5006

    Article  Google Scholar 

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  1. John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA, USA

    Dr. Megan Clare Engel

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Engel, M.C. (2019). Non-equilibrium Bio-Molecular Unfolding Under Tension. In: DNA Systems Under Internal and External Forcing. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-030-25413-1_3

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