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.
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Here we adopt for such quantities the usual definitions introduced in linear elasticity.
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.
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.
Crick, F. H. C., & Watson, J. D. (1954). The complementary structure of deoxyribonucleic acid. Proc R. Soc. Lond. A, 223, 80–96.
Dauxois, T., Peyrard, M., & Bishop, A. R. (1993). Entropy driven DNA denaturation. Phys. Rev. E, 47, R44–R47.
Derks, G., & Gaeta, G. (2011). A minimal model of DNA dynamics in interaction with RNA-polymerase. Physica D, 240, 1805–1817.
Duc, L. H., Ilchmann, A., & Taraba, P. (2006). On stability of linear time-varying second-order differential equations. Q. Appl. Math., 64, 137–151.
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.
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.
Goel, A., Frank-Kamenetskii, M. D. F., Ellenberger, T., & Herschbach, D. (2001). Tuning DNA ‘strings’. Proc. Natl. Acad. Sci. USA, 98, 8485–8489.
Gore, J., Bryant, Z., Nellmann, M., Le, M., Cozzarelli, N., & Bustamante, C. (2006). DNA overwinds when stretched. Nature, 442, 836–839.
Hairer, E., Lubich, C., & Wanner, G. (2006). Geometric numerical integration. In Structure-preserving algorithms for ordinary differential equations (2nd ed.). Berlin: Springer.
Homma, S., & Takeno, S. (1984). A coupled base-rotator model for structure and dynamics of DNA. Prog. Theor. Phys., 72, 679–693.
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.
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.
Komarova, N. L., & Soffer, A. (2005). Nonlinear waves in double-stranded DNA. Bull. Math. Biol., 67, 701–718.
Lacitignola, D., Saccomandi, G., & Sgura, I. Parametric resonance in a mesoscopic discrete DNA model (2014, submitted).
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.
Lankas, F., Sponer, J., Hobza, P., & Langowski, J. (2000). Sequence-dependent elastic properties of DNA. J. Mol. Biol., 299, 695–709.
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.
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.
Magnus, W., & Winkler, S. (1966). Hill’s equation. Interscience tracts in pure and applied mathematics: Vol. 20. New York: Interscience Publishers.
Marko, J. F., & Siggia, E. D. (1995). Stretching DNA. Macromolecules, 28, 8759–8770.
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.
Muto, V. (2011). Solitons oscillations for DNA dynamics. Acta Appl. Math., 115, 5–15.
Peyrard, M. (2004). Nonlinear dynamics and statistical physics of DNA. Nonlinearity, 17, R1–R40.
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.
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.
Saenger, W. (1984). Principles of nucleic acid structure. Springer advanced texts in chemistry. Berlin: Springer.
Scott, A. C. (1985). Soliton oscillations in DNA. Phys. Rev. A, 31, 3518–3519.
Shahinpoor, M. (1978). The role of parametric self-excitation in DNA self-replication. J. Theor. Biol., 70, 17–22.
Sinden, R. R. (1994). DNA structure and function. San Diego: Academic Press.
Slutsky, M., & Mirny, L. A. (2004). Kinetics of protein-DNA interaction: facilitated target location in sequence-dependent potential. Biophys. J., 87, 4021–4035.
Smale, S. T., & Kadonaga, J. T. (2003). The RNA polymerase II core promoter. Annu. Rev. Biochem., 72, 449–479.
Stoker, J. J. (1950). Nonlinear vibrations in mechanical and electrical systems. New York: Wiley-Interscience.
Tributsch, H. (1975). Parametric energy conversion—a possible, universal approach to bioenergetics in biological structures. J. Theor. Biol., 52, 17–56.
Verhulst, F. (1996). Nonlinear differential equations and dynamical systems. Berlin: Springer.
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.
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.
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.
Yakushevich, L. V. (2004). Nonlinear physics of DNA. Chichester: Wiley.
Yin, H., Wang, M. D., Svoboda, K., Landick, R., Gelles, J., & Block, S. M. (1995). Transcription against an applied force. Science, 270, 1653–1657.
Zhang, C. T. (1989). Harmonic and subharmonic resonances of microwave absorption in DNA. Phys. Rev. A, 40, 2148–2153.
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.
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Lacitignola, D., Saccomandi, G. Parametric Resonance in DNA. Bull Math Biol 76, 515–540 (2014). https://doi.org/10.1007/s11538-013-9930-6
- DNA mesoscopic models
- RNA polymerase
- Solitary waves
- Hill’s equation
- Parametric resonance