Skip to main content
SpringerLink
Log in

Genetic code, hamming distance and stochastic matrices

  • Published:
Bulletin of Mathematical Biology Aims and scope Submit manuscript

Abstract

In this paper we use the Gray code representation of the genetic code C = 00, U = 10, G = 11 and A = 01 (C pairs with G, A pairs with U) to generate a sequence of genetic code-based matrices. In connection with these code-based matrices, we use the Hamming distance to generate a sequence of numerical matrices. We then further investigate the properties of the numerical matrices and show that they are doubly stochastic and symmetric. We determine the frequency distributions of the Hamming distances, building blocks of the matrices, decomposition and iterations of matrices. We present an explicit decomposition formula for the genetic code-based matrix in terms of permutation matrices, which provides a hypercube representation of the genetic code. It is also observed that there is a Hamiltonian cycle in a genetic code-based hypercube.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • He, M. (2003a). Genetic code, attributive mappings and stochastic matrices. Bull. Math. Biol. (in press) (doi: 10.1016/j.bulm.2003.10.002).

  • He, M. (2003b). Double helical sequences and doubly stochastic matrices. Symmetry: Culture and Science: Symmetries in Genetic Information (in press).

  • He, M. (2003c). Symmetry in structure of genetic code. Proceedings of the Third All-Russian Interdisciplinary Scientific Conference “Ethics and the Science of Future. Unity in Diversity”, February 12–14, Moscow.

  • Jimenéz-Montaño, M. A., C. R. Mora-Basáñez and T. Pöschel (1994). On the hypercube structure of the genetic code, in Proc. 3. Int. Conf. on Bioinformatics and Genome Research, A. H. Lim and A. C. Cantor (Eds), World Scientific, p. 445.

  • Knight, R. D., S. J. Freeland and L. F. Landweber (1999). Selection, history and chemistry: the three faces of the genetic code. TIBS 24, 241–247.

    Google Scholar 

  • Petoukhov, S. V. (1999). Genetic code and the ancient Chinese book of changes. Symmetry: Culture Sci. 10, 211–226.

    MathSciNet  Google Scholar 

  • Petoukhov, S. V. (2001). The Bi-periodic Table of Genetic Code and Number of Protons, Foreword of K. V. Frolov, Moscow, 258 (in Russian).

  • Petoukhov, S. V. (2002). Binary sub-alphabets of genetic language and problem of unification bases of biological languages, IX International Conference “Mathematics, Computer, Education”, Russia, Dubna, January 28–31, 191 (in Russian).

  • Romanovsky, V. (1931). Sur les zeros des matrices stocastiques. C. R. Acad. Sci. Paris 192, 266–269 [Zbl. 1 (1932) 055].

    MATH  Google Scholar 

  • Štambuk, N. (2000). Universal metric properties of the genetic code. Croatica Chemica ACTA 73, 1123–1139.

    Google Scholar 

  • Swanson, R. (1984). A unifying concept for the amino acid code. Bull. Math. Biol. 46, 187–203.

    Article  MATH  MathSciNet  Google Scholar 

  • Yang, C. M. (2003). The naturally designed spherical symmetry in the genetic code (manuscript).

Download references

Author information

Authors and Affiliations

  1. Division of Math, Science and Technology, Nova Southeastern University, 3301 College Avenue, Fort Lauderdale, FL, 33314, USA

    Matthew X. He

  2. Department of Biomechanics, Mechanical Engineering Research Institute, Russian Academy of Sciences, Moscow, 101830, Russia

    Sergei V. Petoukhov

  3. Dipartimento di Matematica “Guido Castelnuovo”, Università degli Studi di Roma, “La Sapienza”, Rome, Italy

    Paolo E. Ricci

Authors
  1. Matthew X. He
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sergei V. Petoukhov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Paolo E. Ricci
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Matthew X. He.

Rights and permissions

Reprints and permissions

About this article

Cite this article

He, M.X., Petoukhov, S.V. & Ricci, P.E. Genetic code, hamming distance and stochastic matrices. Bull. Math. Biol. 66, 1405–1421 (2004). https://doi.org/10.1016/j.bulm.2004.01.002

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1016/j.bulm.2004.01.002

Keywords

Search

Navigation