Journal of Mathematical Biology

, Volume 64, Issue 6, pp 1087–1108 | Cite as

The effects of density on the topological structure of the mitochondrial DNA from trypanosomes

  • Y. Diao
  • K. Hinson
  • R. Kaplan
  • M. Vazquez
  • J. Arsuaga


Trypanosomatida parasites, such as trypanosoma and lishmania, are the cause of deadly diseases in many third world countries. A distinctive feature of these organisms is the three dimensional organization of their mitochondrial DNA into maxi and minicircles. In some of these organisms minicircles are confined into a small disk volume and are topologically linked, forming a gigantic linked network. The origins of such a network as well as of its topological properties are mostly unknown. In this paper we quantify the effects of the confinement on the topology of such a minicircle network. We introduce a simple mathematical model in which a collection of randomly oriented minicircles are spread over a rectangular grid. We present analytical and computational results showing that a finite positive critical percolation density exists, that the probability of formation of a highly linked network increases exponentially fast when minicircles are confined, and that the mean minicircle valence (the number of minicircles that a particular minicircle is linked to) increases linearly with density. When these results are interpreted in the context of the mitochondrial DNA of the trypanosome they suggest that confinement plays a key role on the formation of the linked network. This hypothesis is supported by the agreement of our simulations with experimental results that show that the valence grows linearly with density. Our model predicts the existence of a percolation density and that the distribution of minicircle valences is more heterogeneous than initially thought.


Kinetoplast DNA–DNA minicircle networks Topological structures Trypanosome Unsplittable links Confinement Percolation density 

Mathematics Subject Classification (2000)

57M25 92B99 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Avliyakulov NK, Lukes J, Ray DS (2004) Mitochondrial histone-like DNA-binding proteins are essential for normal cell growth and mitochondrial function in Crithidia fasciculata. Eukaryot Cell 3: 518–526CrossRefGoogle Scholar
  2. Benne R, van den Burg J, Brakenhoff JPJ, Sloof P, Van Boom JH, Tromp MC (1986) Major transcript of the frameshifted coxll gene from trypanosome mitochondria contains four nucleotides that are not encoded in the DNA. Cell 46: 819–826CrossRefGoogle Scholar
  3. Chen J, Rauch CA, White JH, Englund PT, Cozzarelli NR (1995) The topology of the kinetoplast DNA network. Cell 80(1): 61–69CrossRefGoogle Scholar
  4. Chen J, Englund PT, Cozzarelli NR (1995) Changes in network topology during the replication of kinetoplast DNA. EMBO J 14(24): 6339–6347Google Scholar
  5. De Gennes PG (1979) Scaling concepts in polymer physics. Cornel University Press, New YorkGoogle Scholar
  6. Deng JS, Rubin RL, Lipscomb MF, Sontheimer RD, Gilliam JN (1984) Reappraisal of the specificity of the Crithidia luciliae assay for nDNA antibodies: evidence for histone antibody kinetoplast binding. Am J Clin Pathol 82(4): 448–452Google Scholar
  7. Desjeux P (1992) Human Leishmaniasis: epidemiology and public health aspects. World Health Stat Q 45: 267–275Google Scholar
  8. Diao Y, van Rensburg EJ (1998) Percolation of linked circles. Topology and Geometry in Polymer Science. In: Whittington SG et al (eds) IMA volumes in mathematics and its applications, vol 103, pp 79–88Google Scholar
  9. Diao Y (1994) Unsplittability of random links. J Knot Theory Ramif 3(3): 379–389MathSciNetzbMATHCrossRefGoogle Scholar
  10. Duda R (1998) Protein chainmail: catenated protein in viral capsids. Cell 94: 55–60CrossRefGoogle Scholar
  11. Englund PT (1979) Free minicircles of kinetoplast DNA in Crithia fasciculata. J Biol Chem 254: 4895–4900Google Scholar
  12. Ferguson M, Torri AF, Ward DC, Englund PT (1992) In situ hybridization to the Crithidia fasciculata kinetoplast reveals two antipodal structures involved in kinetoplast DNA replication. Cell 70: 621–629CrossRefGoogle Scholar
  13. Hines JC, Ray DS (1998) The Crithidia fasciculata KAP1 gene encodes a highly basic protein associated with kinetoplast DNA. Mol Biochem Parasitol 94: 41–52CrossRefGoogle Scholar
  14. Kesten H (1982) Percolation theory for mathematicians. Birkhauser, BaselzbMATHGoogle Scholar
  15. Kreuzer KN, Cozzarelli NR (1980) Formation and resolution of DNA catenanes by DNA gyrase. Cell 20(1): 245–254CrossRefGoogle Scholar
  16. Liu B et al (2005) Fellowship of the rings, the replication of kinetoplast DNA. Trends Parasitol 21(8): 363–369CrossRefGoogle Scholar
  17. Lukes J et al (2002) Kinetoplast DNA network: evolution of an improbable structure. Eukaryot Cell 1(4): 495–502CrossRefGoogle Scholar
  18. Marini JC, Miller KG, Englund PT (1980) Decatenation of kinetoplast DNA by topoisomerases. J Biol Chem 255: 4976–4979Google Scholar
  19. Melendy T, Sheline C, Ray DS (1988) Localization of a type II DNA topoisomerase to two sites at the periphery of the kinetoplast DNA of Crithidia fasciculata. Cell 23: 1083–1088CrossRefGoogle Scholar
  20. Müller-Nedebock KK, Edwards SF (1998) Entanglement in polymers: I. Annealed probability for loops. J Phys A Math Gen 32: 3283–3300CrossRefGoogle Scholar
  21. Pasion SG, Hines JC, Aebersold R, Ray DS (1992) Molecular cloning and expression of the gene encoding the kinetoplast-associated type II DNA topoisomerase of Crithidia fasciculata. Mol Biochem Parasitol 50: 57–67CrossRefGoogle Scholar
  22. Perez-Morga D, Englund PT (1993) The structure of replicating kinetoplast DNA networks. J Cell Biol 123: 1069–1079CrossRefGoogle Scholar
  23. Pickett GT (2006) DNA origami technique for olympic gels. Europhys Lett 76: 616–622CrossRefGoogle Scholar
  24. Raphaël E, Gay C, de Gennes PG (1997) Progressive construction of an olympic gel. J Stat Phys 89: 111–118CrossRefGoogle Scholar
  25. Rassi A Jr, Rassi A, Marin-Neto JA (2010) Chagas disease. Lancet 375(9723): 1388–1402CrossRefGoogle Scholar
  26. Rauch CA et al (1993) The absence of supercoiling in kinetoplast DNA minicircles. EMBO J 12: 403–411Google Scholar
  27. Ray DS, Hines JC, Anderson M (1992) Kinetoplast-associated DNA topoisomerase in Crithidia fasciculata: crosslinking of mitochondrial topoisomerase II to both minicircles and maxicircles in cells treated with the topoisomerase inhibitor VP16. Nucleic Acids Res 20: 3353–3356CrossRefGoogle Scholar
  28. Savill NJ, Higgs PG (1999) A theoretical study of random segregation of minicircles in trypanosomatids. Proc R Soc Lond B 266: 611–620CrossRefGoogle Scholar
  29. Shafi KVPM et al (1999) Olympic ring formation from newly prepared Barium Hexaferrite nanoparticle suspension. J Phys Chem B 103: 3358–3360CrossRefGoogle Scholar
  30. Shapiro TA, Klein VA, Englund PT (1989) Drug-promoted cleavage of kinetoplast DNA minicircles: Evidence for type II topoisomerase activity in trypanosome mitochondria. J Biol Chem 264: 4173–4178Google Scholar
  31. Shapiro T, Englund P (1995) The structure and replication of kinetoplast DNA. Annu Rev Microbiol 49: 117–143CrossRefGoogle Scholar
  32. Simarro PP et al (2010) The Atlas of human African trypanosomiasis: a contribution to global mapping of neglected tropical diseases. Int J Health Geogr 9(1): 57–75CrossRefGoogle Scholar
  33. Simpson L, Sbicego S, Aphasizhev R (2003) Uridine insertion/deletion RNA editing in trypanosome mitochondria: A complex business. RNA 9: 265–276CrossRefGoogle Scholar
  34. Stauffer D, Aharony A (1994) Introduction to percolation theory. CRC Press, New YorkGoogle Scholar
  35. Welburn SC, Fevre EM, Coleman PG, Odiit M, Maudlin I (2001) Sleeping sickness: a tale of two diseases. Trends Parasitol 17(1): 19–24CrossRefGoogle Scholar
  36. World health organisation (2009) Neglected tropical diseases. Hidden successes, Emerging opportunities. WHO Library Cataloguing-in-Publication Data, pp 38–39Google Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  • Y. Diao
    • 1
  • K. Hinson
    • 1
  • R. Kaplan
    • 2
  • M. Vazquez
    • 2
  • J. Arsuaga
    • 2
  1. 1.Department of Mathematics and StatisticsUniversity of North Carolina CharlotteCharlotteUSA
  2. 2.Department of MathematicsSan Francisco State UniversitySan FranciscoUSA

Personalised recommendations