Skip to main content

A Framework for Modeling DNA Based Molecular Systems

  • Conference paper
DNA Computing (DNA 2006)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4287))

Included in the following conference series:

Abstract

In this paper, we propose a framework for a discrete event simulator for simulating the DNA based nano-robotical systems. We describe a physical model that captures the conformational changes of the solute molecules. We also present methods to simulate various chemical reactions due to the molecular collisions, including hybridization, dehybridization and strand displacement. The feasibility of such a framework is demonstrated by some preliminary results.

The work is supported by NSF EMT Grants CCF-0523555 and CCF-0432038.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Adleman, L.: Molecular computation of solutions to combinatorial problems. Science 266, 1021–1024 (1994)

    Article  Google Scholar 

  2. Alberti, P., Mergny, J.L.: DNA duplex-quadruplex exchange as the basis for a nanomolecular machine. Proc. Natl. Acad. Sci. USA 100, 1569–1573 (2003)

    Article  Google Scholar 

  3. Allison, S.A., Mazur, S.: Modeling the free solution electrophoretic mobility of short dna fragments. Biopolymers 46, 359–373 (1998)

    Article  Google Scholar 

  4. Allison, S.A., McCammon, J.A.: Multistep brownian dynamics: application to short wormlike chains. Biopolymers 23, 363–375 (1984)

    Article  Google Scholar 

  5. Aragon, S.R., Pecora, R.: Dynamics of wormlike chains. Macromolecules 18, 1868 (1985)

    Article  Google Scholar 

  6. Arridge, R.G.C.: An introduction to polymer mechanics (1985)

    Google Scholar 

  7. Arteca, G.A., Edvinsson, T., Elvingson, C.: Compaction of grafted wormlike chains under variable confinement. Phys. Chem. Chem. Phys. 3, 3737–3741 (2001)

    Article  Google Scholar 

  8. Maier, B., Bensimon, B.D., Croquette, V.: Replication by a single dna polymerase of a stretched single-stranded dna. Proc. Natl. Acad. Sci. U.S.A. 97(22), 12002–12007 (2000)

    Article  Google Scholar 

  9. Benenson, Y., Adar, R., Paz-Elizur, T., Livneh, Z., Shapiro, E.: DNA molecule provides a computing machine with both data and fuel. Proc. Natl. Acad. Sci. USA 100, 2191–2196 (2003)

    Article  Google Scholar 

  10. Benenson, Y., Gil, B., Ben-Dor, U., Adar, R., Shapiro, E.: An autonomous molecular computer for logical control of gene expression. Nature 429, 423–429 (2004)

    Article  Google Scholar 

  11. Benenson, Y., Paz-Elizur, T., Adar, R., Keinan, E., Livneh, Z., Shapiro, E.: Programmable and autonomous computing machine made of biomolecules. Nature 414, 430–434 (2001)

    Article  Google Scholar 

  12. Biswas, I., Yamamoto, A., Hsieh, P.: Branch migration through dna sequence heterology. J. Mol. Bio. (1998)

    Google Scholar 

  13. Bois, J.S., Venkataraman, S., Choi, H.M.T., Spakowitz, A.J., Wang, G., Pierce, N.A.: Topological constraints in nucleic acid hybridization kinetics. Nucleic Acids Research 33(13), 4090–4095 (2005)

    Article  Google Scholar 

  14. Bouchiat, C., Wang, M.D., Allemand, J., Strick, T., Block, S.M., Croquette, V.: Estimating the persistence length of a worm-like chain molecules from force-extension measurements. Biophys. J. 76, 409 (1999)

    Article  Google Scholar 

  15. Bustamante, C., Marko, J.F., Siggia, E.D., Smith, S.: Entropic elasticity of lambda-phage dna mechanics. Science 265, 1599 (1994)

    Article  Google Scholar 

  16. Bustamante, C., Smith, S., Liphardt, J., Smith, D.: Single-molecule studies of dna mechanics. Current Opinion in Structural Biology 10, 279 (2000)

    Article  Google Scholar 

  17. Butler, J.E., Shaqfeh, E.S.G.: Brownian dynamics simulations of a flexible polymer chain which includes continuous resistance and multi-body hydrodynamic interaction. Journal of Chemical Physics 122(014901) (2005)

    Google Scholar 

  18. Carri, G.A., Marucho, M.: Statistical mechanics of worm-like polymers from a new generating function. J. Chem. Phys. 121(12), 6064–6077 (2004)

    Article  Google Scholar 

  19. Chelyapov, N., Brun, Y., Gopalkrishnan, M., Reishus, D., Shaw, B., Adleman, L.: DNA triangles and self-assembled hexagonal tilings. J. Am. Chem. Soc. 126, 13924–13925 (2004)

    Article  Google Scholar 

  20. Chen, Y., Mao, C.: Putting a brake on an autonomous DNA nanomotor. J. Am. Chem. Soc. 126, 8626–8627 (2004)

    Article  Google Scholar 

  21. Chen, Y., Wang, M., Mao, C.: An autonomous DNA nanomotor powered by a DNA enzyme. Angew. Chem. Int. Ed. 43, 3554–3557 (2004)

    Article  Google Scholar 

  22. Cocco, S., Marko, J.F., Monasson, R.: Theoretical models for single-molucule dna and rna experiments: from elasticity to unzipping. In: CRAS, special issue dedicated to Single Molecule Experiments (to appear, 2002)

    Google Scholar 

  23. Desruisseaux, C., Long, D., Drouin, G., Slater, G.W.: Electrophoresis of composite molecular objects. 1. relation between friction, charge and ionic strength in free solution. Macromolecules 34, 44–59 (2001)

    Article  Google Scholar 

  24. Dessinges, M.N., Maier, B., Zhang, Y., Peliti, M., Bensimon, D., Croquette, V.: Stretching single stranded dna, a model polyelectrolyte. Phys. Rev. Lett. 89, 248102 (2002)

    Article  Google Scholar 

  25. Dimitrakopoulos, P.: Stress and configuration relaxation of an initially straight flexible polymer. J. Fluid Mech. 513, 265–286 (2004)

    Article  MATH  Google Scholar 

  26. Doyle, P.S., Underhill, P.T.: Brownian dynamics simulations of polymers and soft matter. In: Yip, S. (ed.) Handbook of Materials Modeling, pp. 2619–2630 (2005)

    Google Scholar 

  27. Dirks, R.M., Bois, J.S., Schaeffer, J.M., Winfree, E., Pierce, N.A.: Thermodynamic analysis of interacting nucleic acid strands. In: SIAM Rev. (in press)

    Google Scholar 

  28. Feng, L., Park, S.H., Reif, J.H., Yan, H.: A two-state DNA lattice switched by DNA nanoactuator. Angew. Chem. Int. Ed. 42, 4342–4346 (2003)

    Article  Google Scholar 

  29. Fixman, M., Kovac, J.: Polymer conformation statistics iii: Modified gaussian models of the stiff chains. J. Chem. Phys. 58, 1564–1568 (1973)

    Article  Google Scholar 

  30. Flamm, C., Fontana, W., Hofacker, I.L., Schuster, P.: RNA folding at elementary step resolution. RNA 6(3), 325–338 (2000)

    Article  Google Scholar 

  31. Fournier, J.B.: Wormlike chain or tense string? a question of resolution. Continuum Mechanical Thermodynamics 14, 241 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  32. Frank-Kamenetskii, M.D.: Biophysics of dna molecule. Phys. Rep. 288, 13–60 (1997)

    Article  Google Scholar 

  33. Gillespie, D.T.: Exact stochastic simulation of coupled chemical reactions. J. Phys. Chem. 81, 2340–2361 (1977)

    Article  Google Scholar 

  34. Gillespie, D.T.: Approximate accelerated stochastic simulation of chemically reacting systems. J. Chem. Phys. 115, 1716–1733 (2001)

    Article  Google Scholar 

  35. Hartemink, A.J., Gifford, D.K.: Thermodynamics simulation of deoxyoligonucleotide hybridization for dna computation (1997)

    Google Scholar 

  36. Heath, P.J., Gebe, J.A., Allison, S.A., Schurr, J.M.: Comparison of analytical theory with brownian dynamics simulations for small linear and circular dnas. Macromolecules 29, 3583 (1996)

    Article  Google Scholar 

  37. Hur, J.S., Shaqfeh, E.S.G.: Brownian dynamics simulations of single dna molecule in shear flow. J. Rheol. 44(4), 713–742 (2000)

    Article  Google Scholar 

  38. Isambert, H., Siggia, E.D.: Modeling RNA folding paths with pseudoknots: application to hepatitis delta virus ribozyme. Proc. Natl. Acad. Sci. USA 97(12), 6515–6520 (2000)

    Article  Google Scholar 

  39. James, H.M., Guth, E.: Theory of the elastic properties of rubber. Journal of Chemical Physics 10, 455–481 (1943)

    Article  Google Scholar 

  40. Jendrejack, R.M., Pablo, J.J., Graham, M.D.: Stochastic simulations of dna in flow: Dynamics and the effects of hydrodynamic interactions. Journal of Chemical Physics 116(17), 7752 (2002)

    Article  Google Scholar 

  41. Santalucia Jr., J.: A unified view of polymer, dumbbell and oligonucleotide dna nearest-neighbor thermodynamics. PNAS 95, 1460–1465 (1998)

    Article  Google Scholar 

  42. Kierzek, A.M.: Stocks: Stochastic kinetic simulations of biochemical systems with gillespie algorithm. Bioinformatics 18, 470–481 (2002)

    Article  Google Scholar 

  43. Klenin, K., Merlitz, H., Langowski, J.: A brownian dynamics program for the simulation of linear and circular dna and other wormlike chain polyelectrolytes. Biophys. J 74(2), 780–788 (1998)

    Article  Google Scholar 

  44. Kovac, J., Crabb, C.: Modified gaussian model for rubber elasticity. 2. the wormlike chain. Macromolecules 15(2), 537 (1982)

    Article  Google Scholar 

  45. Kuhn, M., Grun, F.: Relationships between elastic constants and stretching double refraction of highly elastic substances. Kolloid-Z 101, 294 (1942)

    Article  Google Scholar 

  46. Kutter, S.: Elasticity of polymers with internal topological constraints. PhD Thesis (August 2002)

    Google Scholar 

  47. LaBean, T.H., Yan, H., Kopatsch, J., Liu, F., Winfree, E., Reif, J.H., Seeman, N.C.: The construction, analysis, ligation and self-assembly of DNA triple crossover complexes. J. Am. Chem. Soc. 122, 1848–1860 (2000)

    Article  Google Scholar 

  48. Ladoux, B., Quivy, J.P., Doyle, P.S., Almouzni, G., Viovy, J.L.: Direct imaging of single-molecules: from dynamics of a single dna chain to the study of complex dna-protein interactions. Sci. Prog. 84, 267 (2001)

    Article  Google Scholar 

  49. Langowski, J.: Polymer chain models of dna and chromatin (manuscript, 2006)

    Google Scholar 

  50. Larson, R.G., Hu, H., Smith, D.E., Chu, S.: Brownian dynamics simulation of a dna molecule in an extensional flow field. J. Rheol. 43(2), 267–304 (1999)

    Article  Google Scholar 

  51. Larson, R.G., Perkins, T., Smith, D., Chu, S.: Hydrodynamics of a dna molecule in a flow field. Phys. Rev. E. 55, 1794–1797 (1997)

    Article  Google Scholar 

  52. Larson, R.G., Perkins, T.T., Smith, D.E., Chu, S.: Brownian dynamics simulations of a dna molecule in an extensional flow field. J. Rheol. 43, 267 (1999)

    Article  Google Scholar 

  53. Li, J., Tan, W.: A single DNA molecule nanomotor. Nano Lett. 2, 315–318 (2002)

    Article  Google Scholar 

  54. Liu, D., Wang, M., Deng, Z., Walulu, R., Mao, C.: Tensegrity: Construction of rigid DNA triangles with flexible four-arm dna junctions. J. Am. Chem. Soc. 126, 2324–2325 (2004)

    Article  Google Scholar 

  55. Liu, Q., Wang, L., Frutos, A.G., Condon, A.E., Corn, R.M., Smith, L.M.: DNA computing on surfaces. Nature 403, 175–179 (2000)

    Article  Google Scholar 

  56. Malevanets, A., Yoemans, J.M.: Dynamics of short polymer chains in solution. Europhysics Letters 52(2), 231 (2000)

    Article  Google Scholar 

  57. Mao, C., LaBean, T.H., Reif, J.H., Seeman, N.C.: Logical computation using algorithmic self-assembly of DNA triple-crossover molecules. Nature 407, 493–496 (2000)

    Article  Google Scholar 

  58. Mao, C., Sun, W., Seeman, N.C.: Designed two-dimensional DNA holliday junction arrays visualized by atomic force microscopy. J. Am. Chem. Soc. 121, 5437–5443 (1999)

    Article  Google Scholar 

  59. Mao, C., Sun, W., Shen, Z., Seeman, N.C.: A DNA nanomechanical device based on the B-Z transition. Nature 397, 144–146 (1999)

    Article  Google Scholar 

  60. Marko, J., Siggia, E.D.: Bending and twisting elasticity of dna. Macromolecules 27, 981 (1994)

    Article  Google Scholar 

  61. Marko, J.F., Siggia, E.D.: Stretching dna. Macromolecules 28, 8759 (1995)

    Article  Google Scholar 

  62. Meagher, R.J., Won, J., McCormick, L.C., Nedelcu, S., Bertrand, M.M., Bertarm, J.L., Drouin, G., Barron, A.E., Slaters, G.W.: End-labeled free-solution electrophoresis of dna. Electrophoresis 26, 331–350 (2005)

    Article  Google Scholar 

  63. Mercier, J., Slater, G.W.: Solid phase dna amplification: a brownian dynamics study of crowding effects. Biophysical Journal 89, 32–42 (2005)

    Article  Google Scholar 

  64. Murphy, M.C., Rasnik, I., Cheng, W., Lohman, T.M., Ha, T.: Probing single-stranded dna conformation flexibility using fluorescence spectroscopy. Biophysical Journal 86, 2530–2537 (2004)

    Article  Google Scholar 

  65. Odijk, T.: Stiff chains and filaments under tension. Macromolecule 28, 7016–7018 (1995)

    Article  Google Scholar 

  66. Panyutin, I.G., Hsieh, P.: The kinetics of spontaneous dna branch migration. Proc. Natl. Acad. Sci. USA 91(6), 2021–2025 (1994)

    Article  Google Scholar 

  67. Pedersen, J.S., Laso, M., Schurtenberger, P.: Monte carlo study of excluded volume effects in wormlike micelles and semiflexible polymers. Phys. Rev. E 54(6), 5917–5920 (1996)

    Article  Google Scholar 

  68. Peyret, N., Seneviratne, P.A., Allawi, H.T., Santalucia, J.: Nearest-neighbor thermodynamics and nmr of dna sequences with internal aa,cc,gg and tt mismatches. Biochemistry 38, 3468 (1999)

    Article  Google Scholar 

  69. Rao, C., Arkin, A.: Stochastic chemical kinetics and the quasi-steady-state assumption: application to the gillespie algorithm. J. of Chem. Phys. 118, 4999–5010 (2003)

    Article  Google Scholar 

  70. Reif, J.H.: The design of autonomous DNA nanomechanical devices: Walking and rolling DNA. In: The 8th International Meeting on DNA Based Computers (DNA 8) (2002)

    Google Scholar 

  71. Rief, M., Clausen-Schaumann, H., Gaub, H.E.: Sequence-dependent mechanics of single dna molecules. Nature Structural Biology 6, 346–349 (1999)

    Article  Google Scholar 

  72. Sales-Pardo, M., Guimera, R., Moreira, A.A., Widom, J., Amaral, L.A.: Mesoscopic modeling for nucleic acid chain dynamics. Phys. Rev. E Stat. Nonlin. Soft. Matter Phys. 71, 051902 (2005)

    Article  Google Scholar 

  73. Santalucia, J., Hicks, D.: The thermodynamics of dna structural motifs. Annu. Rev. Biophys. Biomol. Struct. 33, 415 (2004)

    Article  Google Scholar 

  74. Sahu, S., Wang, B., Reif, J.H.: A Framework for Modeling DNA Based Molecular Systems. Technical Report, Duke University (2006)

    Google Scholar 

  75. Sha, R., Liu, R., Millar, D.P., Seeman, N.C.: Atomic force microscopy of parallel DNA branched junction arrays. Chemistry and Biology 7, 743–751 (2000)

    Article  Google Scholar 

  76. Sherman, W.B., Seeman, N.C.: A precisely controlled DNA biped walking device. Nano Lett. 4, 1203–1207 (2004)

    Article  Google Scholar 

  77. Shin, J.S., Pierce, N.A.: A synthetic DNA walker for molecular transport. J. Am. Chem. Soc. 126, 10834–10835 (2004)

    Article  Google Scholar 

  78. Simmel, F.C., Yurke, B.: Using DNA to construct and power a nanoactuator. Phys. Rev. E 63, 041913 (2001)

    Article  Google Scholar 

  79. Simmel, F.C., Yurke, B.: A DNA-based molecular device switchable between three distinct mechanical states. Appl. Phys. Lett. 80, 883–885 (2002)

    Article  Google Scholar 

  80. Smith, S.B., Finzi, L., Bustamante, B.: Direct mechanical measurements of the elasticity of single dna molecules by using magnetic beads. Science 258, 1122 (1992)

    Article  Google Scholar 

  81. Smith, S.B., Cui, Y., Bustamante, C.: Overstretching b-dna: the elastic response of individual double-stranded and single-stranded dna molecules. Science 271, 795–799 (1996)

    Article  Google Scholar 

  82. Somasi, M., Khomami, B., Woo, N.J., Hur, J.S., Shaqfeh, E.S.G.: Brownian dynamics simulations of bead-rod and bead-spring chains: numerical algorithms and coarse-graining issues. J. Non-Newtonian Fluid Mech. 108, 227–255 (2002)

    Article  MATH  Google Scholar 

  83. Stellwagen, E., Stellwagen, N.C.: Determining the electrophoretic mobility and translational diffusion coefficients of dna molecules in free solution. Electrophoresis 23(16), 2794–2803 (2002)

    Article  Google Scholar 

  84. Storm, C., Nelson, P.C.: Theory of high-force dna stretching and overstretching. Physical Review E 67, 051906 (2003)

    Article  MathSciNet  Google Scholar 

  85. Thompson, B.J., Camien, M.N., Warner, R.C.: Kinetics of branch migration in double-stranded dna. Proc. Natl. Acad. Sci. USA 73(7), 2299–2303 (1976)

    Article  Google Scholar 

  86. Tian, Y., He, Y., Chen, Y., Yin, P., Mao, C.: Molecular devices - a DNAzyme that walks processively and autonomously along a one-dimensional track. Angew. Chem. Intl. Ed. 44, 4355–4358 (2005)

    Article  Google Scholar 

  87. Tinoco, I., Bustamante, C.: The effect of force on thermodynamics and kinetics of single molecule reactions. Biophys Chem 513, 101–102 (2002)

    Google Scholar 

  88. Turberfield, A.J., Mitchell, J.C., Yurke, B., Mills Jr., A.P., Blakey, M.I., Simmel, F.C.: DNA fuel for free-running nanomachines. Phys. Rev. Lett. 90, 118102 (2003)

    Article  Google Scholar 

  89. Turberfield, A.J., Yurke, B., Mills Jr., A.P.: DNA hybridization catalysts and molecular tweezers. In: DNA5 (2000)

    Google Scholar 

  90. Turner, T.E., Schnell, S., Burrage, K.: Stochastic approaches for modelling in vivo reactions. Computational Biology and Chemistry (2004)

    Google Scholar 

  91. Vologodskii, A.V.: Monte carlo simulation of dna topological properties (preprint, 2004)

    Google Scholar 

  92. Voter, A.F.: Introduction to kinetic monte carlo method. Springer, NATO publishing unit (2005)

    Google Scholar 

  93. Wetmur, J.G., Davidson, N.: Kinetics of renaturation of dna. J. Mol. Biol. 31, 349–370 (1968)

    Article  Google Scholar 

  94. Winfree, E.: Complexity of restricted and unrestricted models of molecular computation. In: Lipton, R.J., Baum, E.B. (eds.) DNA Based Computers 1. DIMACS, vol. 27, pp. 187–198. American Mathematical Society (1996)

    Google Scholar 

  95. Winfree, E.: Simulation of computing by self-assembly. Technical Report 1998.22, Caltech (1998)

    Google Scholar 

  96. Winfree, E., Liu, F., Wenzler, L.A., Seeman, N.C.: Design and self-assembly of two-dimensional DNA crystals. Nature 394(6693), 539–544 (1998)

    Article  Google Scholar 

  97. Winfree, E., Yang, X., Seeman, N.C.: Universal computation via self-assembly of DNA: Some theory and experiments. In: Landweber, L.F., Baum, E.B. (eds.) DNA Based Computers II. DIMACS, vol. 44, pp. 191–213. American Mathematical Society (1999)

    Google Scholar 

  98. Wolfinger, M.T., Svrcek-Seiler, W.A., Flamm, C., Hofacker, I.L., Stadler, P.F.: Exact Folding Dynamics of RNA Secondary Structures. J. Phys. A: Math. Gen. 37, 4731–4741 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  99. Yamakawa, H., Yoshizaki, T.: Dynamics of helical wormlike chains. Journal of Chemical Physics 75(2), 1016 (1981)

    Article  MathSciNet  Google Scholar 

  100. Yan, H., LaBean, T.H., Feng, L., Reif, J.H.: Directed nucleation assembly of DNA tile complexes for barcode patterned DNA lattices. Proc. Natl. Acad. Sci. USA 100(14), 8103–8108 (2003)

    Article  Google Scholar 

  101. Yan, H., Park, S.H., Finkelstein, G., Reif, J.H., LaBean, T.H.: DNA-templated self-assembly of protein arrays and highly conductive nanowires. Science 301(5641), 1882–1884 (2003)

    Article  Google Scholar 

  102. Yan, H., Zhang, X., Shen, Z., Seeman, N.C.: A robust DNA mechanical device controlled by hybridization topology. Nature 415, 62–65 (2002)

    Article  Google Scholar 

  103. Yan, J., Marko, J.F.: Localized single-stranded bubble mechanism for cyclization of short double helix dna. Phys. Rev. Lett. 93(10), 108108 (2004)

    Article  Google Scholar 

  104. Yin, P., Sahu, S., Turberfield, A.J., Reif, J.H.: Design of autonomous DNA cellular automata. In: Proc. 11th International Meeting on DNA Computing, pp. 376–387 (2005)

    Google Scholar 

  105. Yin, P., Turberfield, A.J., Sahu, S., Reif, J.H.: Design of an autonomous DNA nanomechanical device capable of universal computation and universal translational motion. In: Proc. 10th International Meeting on DNA Computing, pp. 344–356 (2004)

    Google Scholar 

  106. Yurke, B., Mills, A.P., Turberfield, A.J.: A molecular machine made of and powdered by DNA. Biophysics 78, 2629 (2000)

    Google Scholar 

  107. Yurke, B., Turberfield, A.J., Mills Jr., A.P., Simmel, F.C., Neumann, J.L.: A DNA-fuelled molecular machine made of DNA. Nature 406, 605–608 (2000)

    Article  Google Scholar 

  108. Zhang, Y., Zhou, H., Ou-Yang, Z.: Stretching single-stranded dna: Interplay of electrostatic, base-pairing, and base-pair stacking interactions. Biophys J 81(2), 1133–1143 (2001)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2006 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Sahu, S., Wang, B., Reif, J.H. (2006). A Framework for Modeling DNA Based Molecular Systems. In: Mao, C., Yokomori, T. (eds) DNA Computing. DNA 2006. Lecture Notes in Computer Science, vol 4287. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11925903_19

Download citation

  • DOI: https://doi.org/10.1007/11925903_19

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-49024-1

  • Online ISBN: 978-3-540-68423-7

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics