The experimental data of Cloutier and Widom [2] opened the door to the possibility that WLC model might fail dramatically for tightly bent DNA configurations. What are the possible mechanisms of this failure? Within the accepted framework of DNA mechanics, there are already two possible explanations. It is already known that DNA was pre-bent [33, 34,35,32]. There are sequences, so called A tracks, that are spontaneously curved in the absence of thermal fluctuations [36]. Also the stiffness of DNA is also a function of sequence [37,9,2]. Could either of these complications lead to the three-order-of-magnitude anomaly that Cloutier and Widom had observed? The answer is no.
In the interest of brevity I will not make these extensive arguments here.
Rob Phillips, Phil Nelson, and I became convinced that the problem could only be resolved by changing the bending energy of DNA. Certainly from the perspective of macroscopic rods, it is well known that the linear-elastic model breaks down at high curvature. How can this intuitive picture be reconciled with years of experiments that showed that the WLC model described DNA statistics? One possible answer is that the cyclization experiments of Cloutier and Widom [2] probe a high-curvature regime of DNA bending that very few studies had probed before.
Our idea was to explore a model that included a rare, catastrophic breakdown of elasticity that would only appreciably change the chain statistics at high curvature. Many macroscopic systems kink, or localize curvature, in response to tight bending. Perhaps the most pedestrian example of this phenomena is the drinking straw which has a very small elastic bending regime before undergoing a kinking transition which buckles the straw. Back-of-the-envelope calculations showed that this model could reproduce exactly the behavior observed by Cloutier and Widom [2] for very rare kinking events. Not only that, but we would show that these kinks were nearly irrelevant to force- extension experiments that fit the WLC model so well. Does DNA kink?
Bibliography
[1] C. Bustamante, J. F. Marko, E. D. Siggia, and S. Smith. Entropic elasticity of lambda phage DNA. Science, 265:1599–1600, 1994. 5,96,195
[2] T. E. Cloutier and J. Widom. Spontaneous sharp bending of double-stranded DNA. Molecular Cell, 14(3):355–362, 2004. 5,6,7, 96,97,110, 114,116,117, 118,124,145,148,152,164,169, 172,173,174,179,181,185,198, 199,200,203
[3] Graeme K. Hunter. Vital forces: The discovery of the molecular basis of life. Academic Press, San Diego, 2000. 97
[4] Bruce Alberts, Dennis Bray, Julian Lewis, Martin Raff, Keith Roberts, and James D. Watson.
Molecular Biology of the Cell. Garland Publishing, New York, NY, 3rd edition, 1994. 1,11,12, 13,16,97,98,100, 101,116,167,179
[5] Bruce Alberts. The cell as a collection of protein machines: Preparing the next generation of molecular biologists. Cell, 92(3):291–294, 1998. 1,2,98
[6] Mario E. Cerritelli, Naiqian Cheng, Alan H. Rosenberg, Catherine E. McPherson, Frank P.
Booy, and Alasdair C. Steven. Encapsidated conformation of bacteriophage T7 DNA. Cell, 91(2):271–280, October 1997. 99
[7] Prashant Purohit, Jan´e Kondev, and Rob Phillips. Mechanics of DNA packaging in viruses.
Proc. Natl. Acad. Sci. USA, 100:3173–3178, 2003. 99,100
[8] M. Doi and S. F. Edwards. The Theory of Polymer Dynamics. Oxford University Press, 1986.
99,190
[9] Jonathan Widom. Role of DNA sequence in nucleosome stability and dynamics. Quarterly Reviews of Biophysics, 34(3):269–324, 2001. 100,105,116,118, 179,181,201
[10] Carmen V. Kirchhamer, Chiou-Hwa Yuh, and Eric H. Davidson. Modular cis-regulatory or- ganization of developmentally expressed genes: Two genes transcribed territorially in the sea urchin embryo, and additional examples. Proc. Natl. Acad. Sci. USA, 93:9322–9328, 1996.101, 102
[11] Lacramioara Bintu, Nicolas E Buchler, Hernan G Garcia, Ulrich Gerland, Terence Hwa, Jan´e Kondev, and Rob Phillips. Transcriptional regulation by the numbers: models.Current Opinion in Genetics & Development, 15:116–124, 2005. 101,103,114
[12] R. W. Zeller, J. D. Griffith, J. G. Moore, C. V. Kirchhamer, R. J. Britten, and E. H. Davidson.
A multimerizing transcription factor of sea urchin embryos capable of looping DNA.Proc. Natl.
Acad. Sci. USA, 92:2989–2993, 1995. 102
[13] Karsten Rippe, Peter R. von Hippel, and J org Langowski. Action at a distance: DNA-looping and initiation of transcription. Trends Biochem. Sci., 20(12):500–506, 1995. 2, 102, 110,114, 118,124,152,179
[14] J. Muller, S. Oehler, and B Muller-Hill. Repression of lac promoter as a function of distance, phase and quality of an auxiliary lac operator. J. Mol. Biol., 257:21–29, 1996. 2,102,103,118, 124,152,202
[15] J. Muller, A. Barker, S. Oehler, and B. Muller-Hill. Dimeric lac repressors exhibit phasedepen- dent co-operativity. J. Mol. Biol., 284:851–857, 1998. 2, 102,103, 118,124,152
[16] Karsten Rippe. Making contacts on a nucleic acid polymer. Trends Biochem. Sci., 26(12):733–
740, 2001. 2, 102,110,114,118,124,152
[17] O. Kratky and G. Porod. Rotgenuntersuchung geloster fadenmolekule. Rec. Trav. Chim., 68(12):1106–1122, 1949. 104,109, 123,125
[18] H. Yamakawa. Helical Wormlike Chains in Polymer Solutions. Springer, Berlin, 1997. 5,104, 108,109,113,123,125,137,146, 160,164,192,199,202
[19] Rob Phillips and Jan’e Kondev. Physical Biology of the Cell. Garland Press, 2006. To be published. 106
[20] R. P. Feynman and A. R. Hibbs. Quantum Mechanics and Path Integrals. McGraw-Hill, New York, 1965. 106,130
[21] J. J. Sakurai. Modern Quantum Mechanics. Addison-Wesley, Reading, Massachusetts, 2nd edition, 1994. 107,131,186,187, 188,207,208
[22] A. J. Spakowitz and Z.-G. Wang. Exact results for a semiflexible polymer chain in an aligning field.Macromolecules, 37:5814–5823, 2004.108,133,138,140,144,159,169,180,193,196,211 [23] A. J. Spakowitz and Z.-G. Wang. End-to-end distance vector distribution with fixed end orien- tations for the wormlike chain model. Phys. Rev. E, 2005. In preparation. 108, 169, 180,192, 193,211,212
[24] A. Y. Grosberg and A. R. Khokhlov. Statistical physics of macromolecules. AIP Press, New York, 1994. 109,130,137
[25] H. Jacobson and W. H. Stockmayer. Intramolecular reaction in polycondensations 1. The theory of linear systems. J. Chem. Phys., 18(12):1600–1606, 1950. 110,111,146,198,199
[26] D. Shore, J Langowski, and R. L. Baldwin. DNA flexibility studied by covalent closure of short fragments into circles. Proc. Natl. Acad. Sci. USA, 170:4833–4837, 1981. 110, 112, 113, 115, 117,148,173,179,198,199
[27] D. Shore and R. L. Baldwin. Energetics of DNA twisting 1. Relation between twist and cy- clization probability. Journal of Molecular Biology, 170(4):957–981, 1983. 110, 113, 117,148, 173,198,199
[28] P. J. Hagerman. Investigation of the flexibility of DNA using transient electric birefringence.
Bioploymers, 20:1503–1535, 1981. 110,111
[29] J. Shimada and H. Yamakawa. Ring-closure probabilities for twisted wormlike chains – appli- cations to DNA. Macromolecules, 17:689–698, 1984. 113,123,149,198,199
[30] Lacramioara Bintu, Nicolas E Buchler, Hernan G Garcia, Ulrich Gerland, Terence Hwa, Jan´e Kondev, Thomas Kuhlman, and Rob Phillips. Transcriptional regulation by the numbers:
applications. Current Opinion in Genetics & Development, 15:125–135, 2005. 114
[31] T. E. Cloutier and Jonathan Widom. DNA twisting flexibility and the formation of sharply looped protein–DNA complexes. Proc. Natl. Acad. Sci. USA, 102:3634–3650, 2005. 117 [32] M. Vologodskaia and A. Vologodskii. Contribution of the intrinsic curvature to measured DNA
persistence length. J. Mol. Biol., 317(2):205–213, 2002. 117,118,148, 199
[33] E. N. Trifonov, R. K.-Z. Tan, and S. C. Harvey. Static persistence length of DNA. In W. K.
Olson, M. H. Sarma, and M. Sundaralingam, editors,DNA bending and curvature, pages 243–
254. Adenine Press, Schenectady NY, 1987. 118,137
[34] Jan Bednar, Patrick Furrer, Vsevolod Katritch, Alicja Z Stasiak, Jacques Dubochet, and An- drzej Stasiak. Determination of DNA persistence length by cryo-electron microscopy. Separa- tion of static and dynamic contributions to the apparent peristence length of DNA. Journal of Molecular Biology, 254:579–594, 1995. 118, 167,184
[35] P. Nelson. Sequence-disorder effects on DNA entropic elasticity.Phys. Rev. Lett., 80:5810–5812, 1998. 118,137
[36] J. C. Marini, S. D. Levene, D. M. Crothers, and P. T. Englund. Bent helical structure in kinetoplast. Proc Natl Acad Sci USA, 79:7664–7668, 1982. 118
[37] J. D. Kahn, E. Yun, and D. M. Crothers. Detection of localized DNA flexibility. Nature, 368(6467):163–166, 1994. 118,173
Chapter 6
Exact theory of kinkable elastic polymers
This chapter is a reproduction of Ref. [1].
The importance of nonlinearities in material constitutive relations has long been appreciated in the continuum mechanics of macroscopic rods. Although the moment (torque) response to bending is almost universally linear for small deflection angles, many rod systems exhibit a high-curvature softening. The signature behavior of these rod systems is a kinking transition in which the bending is localized. Recent DNA cyclization experiments by Cloutier and Widom have offered evidence that the linear-elastic bending theory fails to describe the high-curvature mechanics of DNA. Motivated by this recent experimental work, we develop a simple and exact theory of the statistical mechanics of linear-elastic polymer chains that can undergo a kinking transition. We characterize the kinking behavior with a single parameter and show that the resulting theory reproduces both the low- curvature linear-elastic behavior which is already well described by the Wormlike Chain model, as well as the high-curvature softening observed in recent cyclization experiments.