Bulletin of Mathematical Biology

, Volume 76, Issue 3, pp 515–540 | Cite as

Parametric Resonance in DNA

  • Deborah LacitignolaEmail author
  • Giuseppe Saccomandi
Original Article


We consider a simple mesoscopic model of DNA in which the binding of the RNA polymerase enzyme molecule to the promoter sequence of the DNA is included through a substrate energy term modeling the enzymatic interaction with the DNA strands. We focus on the differential system for solitary waves and derive conditions—in terms of the model parameters—for the occurrence of the parametric resonance phenomenon. We find that what truly matters for parametric resonance is not the ratio between the strength of the stacking and the inter-strand forces but the ratio between the substrate and the inter-strands. On the basis of these results, the standard objection that longitudinal motion is negligible because of the second order seems to fail, suggesting that all the studies involving the longitudinal degree of freedom in DNA should be reconsidered when the interaction of the RNA polymerase with the DNA macromolecule is not neglected.


DNA mesoscopic models RNA polymerase Solitary waves Hill’s equation Parametric resonance 



The present work has been performed under the auspices of the italian National Group for Mathematical Physics (GNFM-Indam). The research is supported by PRIN-2009 project Matematica e meccanica dei sistemi biologici e dei tessuti molli. The authors warmly thank Ivonne Sgura for her precious contribution in achieving numerical simulations provided in Sect. 6. We also thank the anonymous referees and the handling Editor for their helpful comments and remarks.


  1. Cadoni, M., De Leo, R., Demelio, S., & Gaeta, G. (2008). Twist solitons in complex macromolecules: from DNA to polyethylene. Int. J. Non-Linear Mech., 43, 1094–1107. CrossRefzbMATHGoogle Scholar
  2. Chou, K. C., Maggiora, G. M., & Mao, B. (1989). Quasi-continuum models of twist-like and accordion-like low-frequency motions in DNA. Biophys. J., 56, 295–305. CrossRefGoogle Scholar
  3. Crick, F. H. C., & Watson, J. D. (1954). The complementary structure of deoxyribonucleic acid. Proc R. Soc. Lond. A, 223, 80–96. CrossRefGoogle Scholar
  4. Dauxois, T., Peyrard, M., & Bishop, A. R. (1993). Entropy driven DNA denaturation. Phys. Rev. E, 47, R44–R47. CrossRefzbMATHGoogle Scholar
  5. Derks, G., & Gaeta, G. (2011). A minimal model of DNA dynamics in interaction with RNA-polymerase. Physica D, 240, 1805–1817. MathSciNetCrossRefzbMATHGoogle Scholar
  6. Duc, L. H., Ilchmann, A., & Taraba, P. (2006). On stability of linear time-varying second-order differential equations. Q. Appl. Math., 64, 137–151. MathSciNetCrossRefzbMATHGoogle Scholar
  7. Edwards, G. S., Davis, C. C., Saffer, J. D., & Swicord, M. L. (1984). Resonant microwave absorption of selected DNA molecules. Phys. Rev. Lett., 53, 1284–1287. CrossRefGoogle Scholar
  8. Englander, S. W., Kallenbach, N. R., Heeger, A. J., Krumhansl, J. A., & Litwin, S. (1980). Nature of the open state in long polynucleotide double helices: possibility of soliton excitations. Proc. Natl. Acad. Sci. USA, 77, 7222–7226. CrossRefGoogle Scholar
  9. Goel, A., Frank-Kamenetskii, M. D. F., Ellenberger, T., & Herschbach, D. (2001). Tuning DNA ‘strings’. Proc. Natl. Acad. Sci. USA, 98, 8485–8489. CrossRefGoogle Scholar
  10. Gore, J., Bryant, Z., Nellmann, M., Le, M., Cozzarelli, N., & Bustamante, C. (2006). DNA overwinds when stretched. Nature, 442, 836–839. CrossRefGoogle Scholar
  11. Hairer, E., Lubich, C., & Wanner, G. (2006). Geometric numerical integration. In Structure-preserving algorithms for ordinary differential equations (2nd ed.). Berlin: Springer. Google Scholar
  12. Homma, S., & Takeno, S. (1984). A coupled base-rotator model for structure and dynamics of DNA. Prog. Theor. Phys., 72, 679–693. MathSciNetCrossRefzbMATHGoogle Scholar
  13. Koch, S. J., & Wang, M. D. (2003). Dynamic force spectroscopy of protein-DNA interactions by unzipping DNA double helix. Phys. Rev. Lett., 91, 1–4. Google Scholar
  14. Kocsis, A., & Swigon, D. (2012). DNA stretching modeled at the base pair level: overtwisting and shear instability in elastic linkages. Int. J. Non-Linear Mech., 47, 639–654. CrossRefGoogle Scholar
  15. Komarova, N. L., & Soffer, A. (2005). Nonlinear waves in double-stranded DNA. Bull. Math. Biol., 67, 701–718. MathSciNetCrossRefGoogle Scholar
  16. Lacitignola, D., Saccomandi, G., & Sgura, I. Parametric resonance in a mesoscopic discrete DNA model (2014, submitted). Google Scholar
  17. Lamba, O. P., Wang, A. H. J., & Thomas, G. J. Jr. (1995). Low frequency dynamics and Raman scattering of crystals of B, A, and Z-DNA and fibers of C-DNA. Biopolymers, 28, 667–678. CrossRefGoogle Scholar
  18. Lankas, F., Sponer, J., Hobza, P., & Langowski, J. (2000). Sequence-dependent elastic properties of DNA. J. Mol. Biol., 299, 695–709. CrossRefGoogle Scholar
  19. Lindsay, S. M., & Powell, J. (1983). Light scattering of lattice vibrations of DNA. In E. Clementi & R. H. Sarma (Eds.), Structure and dynamics: nucleic acids and proteins, New York: Adenine Press. Google Scholar
  20. Lindsay, S. M., Powell, J., Prohofsky, E. W., & Devi-Prasad, K. V. (1983). Lattice modes, soft modes and local modes in double helical DNA. In E. Clementi & R. H. Sarma (Eds.), Structure and motion: nucleic acids, proteins, New York: Adenine Press. Google Scholar
  21. Magnus, W., & Winkler, S. (1966). Hill’s equation. Interscience tracts in pure and applied mathematics: Vol. 20. New York: Interscience Publishers. zbMATHGoogle Scholar
  22. Marko, J. F., & Siggia, E. D. (1995). Stretching DNA. Macromolecules, 28, 8759–8770. CrossRefGoogle Scholar
  23. Murray, P. J., Edwards, C. M., Tindall, M. J., & Maini, P. K. (2009). From a discrete to a continuum model of cell dynamics in one dimension. Phys. Rev. E, 80, 1–10. CrossRefGoogle Scholar
  24. Muto, V. (2011). Solitons oscillations for DNA dynamics. Acta Appl. Math., 115, 5–15. MathSciNetCrossRefzbMATHGoogle Scholar
  25. Peyrard, M. (2004). Nonlinear dynamics and statistical physics of DNA. Nonlinearity, 17, R1–R40. MathSciNetCrossRefzbMATHGoogle Scholar
  26. Revyakin, A., Liu, C., Ebright, R. H., & Strick, T. R. (2006). Abortive initiation and productive initiation by RNA polymerase involve DNA scrunching. Science, 314, 1139–1143. CrossRefGoogle Scholar
  27. Saccomandi, G., & Sgura, I. (2006). The relevance of nonlinear stacking interactions in simple models of double-stranded DNA. J. R. Soc. Interface, 3, 655–667. CrossRefGoogle Scholar
  28. Saenger, W. (1984). Principles of nucleic acid structure. Springer advanced texts in chemistry. Berlin: Springer. CrossRefGoogle Scholar
  29. Scott, A. C. (1985). Soliton oscillations in DNA. Phys. Rev. A, 31, 3518–3519. MathSciNetCrossRefGoogle Scholar
  30. Shahinpoor, M. (1978). The role of parametric self-excitation in DNA self-replication. J. Theor. Biol., 70, 17–22. CrossRefGoogle Scholar
  31. Sinden, R. R. (1994). DNA structure and function. San Diego: Academic Press. Google Scholar
  32. Slutsky, M., & Mirny, L. A. (2004). Kinetics of protein-DNA interaction: facilitated target location in sequence-dependent potential. Biophys. J., 87, 4021–4035. CrossRefGoogle Scholar
  33. Smale, S. T., & Kadonaga, J. T. (2003). The RNA polymerase II core promoter. Annu. Rev. Biochem., 72, 449–479. CrossRefGoogle Scholar
  34. Stoker, J. J. (1950). Nonlinear vibrations in mechanical and electrical systems. New York: Wiley-Interscience. zbMATHGoogle Scholar
  35. Tributsch, H. (1975). Parametric energy conversion—a possible, universal approach to bioenergetics in biological structures. J. Theor. Biol., 52, 17–56. CrossRefGoogle Scholar
  36. Verhulst, F. (1996). Nonlinear differential equations and dynamical systems. Berlin: Springer. CrossRefzbMATHGoogle Scholar
  37. Vitt, A., & Gorelik, G. (1933). Oscillations of an elastic pendulum as an example of the oscillations of two parametrically coupled linear systems. J. Tech. Phys., 3, 294–307. Google Scholar
  38. Weidlich, T., Lindsay, S. M., Lee, S. A., Tao, N. J., Lewen, G. D., Peticolas, W. L., Thomas, G. A., & Rupprecht, A. (1988). Low-frequency Raman spectra of DNA: a comparison between two oligonucleotide crystals and highly crystalline films of calf thymus DNA. J. Phys. Chem., 92, 3315–3317. CrossRefGoogle Scholar
  39. Xiao, J., Lin, J., & Zhang, G. (1987). The influence of longitudinal vibration on soliton excitation in DNA double helices. J. Phys. A, Math. Gen., 20, 2425–2432. MathSciNetCrossRefGoogle Scholar
  40. Yakushevich, L. V. (2004). Nonlinear physics of DNA. Chichester: Wiley. CrossRefzbMATHGoogle Scholar
  41. Yin, H., Wang, M. D., Svoboda, K., Landick, R., Gelles, J., & Block, S. M. (1995). Transcription against an applied force. Science, 270, 1653–1657. CrossRefGoogle Scholar
  42. Zhang, C. T. (1989). Harmonic and subharmonic resonances of microwave absorption in DNA. Phys. Rev. A, 40, 2148–2153. CrossRefGoogle Scholar

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© Society for Mathematical Biology 2013

Authors and Affiliations

  1. 1.Dipartimento di Ingegneria Elettrica e dell’InformazioneUniversità degli Studi di Cassino e del Lazio MeridionaleCassinoItaly
  2. 2.Dipartimento di Ingegneria IndustrialeUniversità degli Studi di PerugiaPerugiaItaly

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