The Knot Spectrum of Confined Random Equilateral Polygons
It is well known that genomic materials (long DNA chains) of living organisms are often packed compactly under extreme confining conditions using macromolecular self-assembly processes but the general DNA packing mechanism remains an unsolved problem. It has been proposed that the topology of the packed DNA may be used to study the DNA packing mechanism. For example, in the case of (mutant) bacteriophage P4, DNA molecules packed inside the bacteriophage head are considered to be circular since the two sticky ends of the DNA are close to each other. The DNAs extracted from the capsid without separating the two ends can thus preserve the topology of the (circular) DNAs. It turns out that the circular DNAs extracted from bacteriophage P4 are non-trivially knotted with very high probability and with a bias toward chiral knots. In order to study this problem using a systematic approach based on mathematical modeling, one needs to introduce a DNA packing model under extreme volume confinement condition and test whether such a model can produce the kind of knot spectrum observed in the experiments. In this paper we introduce and study a model of equilateral random polygons con_ned in a sphere. This model is not meant to generate polygons that model DNA packed in a virus head directly. Instead, the average topological characteristics of this model may serve as benchmark data for totally randomly packed circular DNAs. The difference between the biologically observed topological characteristics and our benchmark data might reveal the bias of DNA packed in the viral capsids and possibly lead to a better understanding of the DNA packing mechanism, at least for the bacteriophage DNA. The purpose of this paper is to provide information about the knot spectrum of equilateral random polygons under such a spherical confinement with length and confinement ratios in a range comparable to circular DNAs packed inside bacteriophage heads.
- Department of Mathematics and Statistics, University of North Carolina at Charlotte, Charlotte, NC 28223, USA, firstname.lastname@example.org
- Department of Mathematics, Western Kentucky University, Bowling Green, KY 42101, USA,, email@example.com
- Department of Mathematics, Western Kentucky University, Bowling Green, KY 42101, USA
- Department of Mathematics, University of St. Thomas, St. Paul, MN 55105, USA, firstname.lastname@example.org
- Department of Computer Science, Western Kentucky University, Bowling Green, KY 42101, USA, email@example.com
-  J. Arsuaga, B. Borgo, Y. Diao, and R. Scharein. The growth of the mean average crossing number of equilateral polygons in con_nement. J. Phys. A: Math. Theor., 42:465202, 2009.[Crossref][WoS]
-  Javier Arsuaga, Mariel Vazquez, Paul McGuirk, Sonia Trigueros, De Witt Sumners, and Joaquim Roca. DNA knots reveal a chiral organization of DNA in phage capsids. Proc. Natl. Acad. Sci. USA, 102:9165-9169, 2005.[Crossref]
-  J. Cantarella, T. Deguchi, and C. Shonkwiler. Probability theory of random polygons from the quaternionic viewpoint. Communications on Pure and Applied Mathematics, to appear.
-  P. Cromwell. Knots and Links. Cambridge University Press, 2004.
-  Tetsuo Deguchi and Kyoichi Tsurusaki. A statistical study of random knotting using the Vassiliev invariants. J. Knot Theory Rami_cations, 3(3):321-353, 1994. Random knotting and linking (Vancouver, BC, 1993).
-  Y. Diao, C. Ernst, A. Montemayor, and U. Ziegler. Generating equilateral random polygons in con_nement ii. J. Phys. A: Math. Theor., 45:275203, 2012.[Crossref]
-  Y. Diao, C. Ernst, A. Montemayor, and U. Ziegler. Generating equilateral random polygons in con_nement iii. J. Phys. A: Math. Theor., 45:465003, 2012.[Crossref]
-  Yuanan Diao. The knotting of equilateral polygons in R3. J. Knot Theory Rami_cations, 4(2):189-196, 1995.
-  Yuanan Diao, Akos Dobay, Robert B. Kusner, Kenneth C. Millett, and Andrzej Stasiak. The average crossing number of equilateral random polygons. J. Phys. A: Math. Gen., 36(46):11561-11574, 2003.[Crossref]
-  Yuanan Diao, Claus Ernst, Anthony Montemayor, and Uta Ziegler. Generating equilateral random polygons in con_nement. J. Phys. A: Math. Theor., 44:405202, 2011.[Crossref]
-  Akos Dobay, Pierre-Edouard Sottas, Jacques Dubochet, and Andrzej Stasiak. Predicting optimal lengths of random knots. Lett. Math. Phys., 55(3):239-247, 2001. Topological and geometrical methods (Dijon, 2000).[Crossref]
-  Bruce Ewing and Kenneth C. Millett. Computational algorithms and the complexity of link polynomials. In Progress in knot theory and related topics, pages 51-68. Hermann, Paris, 1997.
-  Jim Hoste and Morwen Thistlethwaite. Knotscape. http://www.math.utk.edu/_morwen/knotscape.html. Program for computing topological information about knots.
-  S. V. Jablan, L. M. Radovi, and R. Sazdanovi. Basic polyhedra in knot theory. Kragujevac J. Math., 28:155-164, 2005.
-  P. J. Jardine and D. L. Anderson. DNA packaging in double-stranded DNA phages. In Richard Calendar, editor, The bacteriophages, pages 49-65. Oxford University Press, 2006.
-  K. V. Klenin, A. V. Vologodskii, V. V. Anshelevich, A. M. Dykhne, and M. D. Frank-Kamenetskii. E_ect of excluded volume on topological properties of circular DNA. J. Biomol. Struct. Dyn., 5:1173-1185, 1988.[Crossref]
-  D. Marenduzzo, E. Orlandini, A. Stasiak, D.W. Sumners, L. Tubiana, and C. Micheletti. DNA-DNA interactions in bacteriophage capsids are responsible for the observed DNA knotting. Proc Natl Acad Sci U S A., 106(52):22269-74, 2009.[Crossref]
-  K. C. Millett. Monte carlo explorations of polygonal knot spaces. In C. McA. Gordon, V. F. R. Jones, L. H. Kau_man, S. Lambropoulou, and J. H. Przytycki, editors, Proceedings of the International Conference on Knot Theory and its Rami_cations held in Delphi, August 7-15, 1998, volume 24 of Ser. Knots Everything, pages 306-334, Singapore, 2000. World Sci. Publishing.
-  Kenneth C. Millett. Physical knot theory: an introduction to the study of the influence of knotting on the spatial characteristics of polymers. In Ser. Knots Everything, volume 46, pages 346-378. World Sci. Publ., Hackensack, NJ, 2012.
-  Patrick Plunkett, Michael Piatek, Akos Dobay, John C. Kern, Kenneth C. Millett, Andrzej Stasiak, and Eric J. Rawdon. Total curvature and total torsion of knotted polymers. Macromolecules, 40(10):3860-3867, 2007.[WoS]
-  M. Thistlethwaite. unraveller. Program for simplifying crossing codes, program provided through private communication.
-  M. Thistlethwaite. On the structure and scarcity of alternating links and tangles. Journal of Knot Theory and Its Rami_cations, 07(07):981-1004, 1998.
-  R. Varela, K. Hinson, J. Arsuaga, and Y. Diao. A fast ergodic algorithm for generating ensembles of equilateral random polygons. J. Phys. A: Math. Theor., 42(9):1-13, 2009.[WoS]
-  L. Zirbel and K. Millett. Characteristics of shape and knotting in ideal rings. J. Phys. A: Math. Theor., 45:225001, 2012. [WoS][Crossref]