On Modelling Metabolism-Repair by Convolution and Partial Realization

  • Franz Pichler
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6927)


John Casti introduced by several papers [1], [2], [3] a mathematical modelling method for metabolism-repair in biological cells following the approach of Robert Rosen [4]. As a result Casti was able to determine algebraic criteria which describe for the linear time-invariant case functional conditions for repair and replication. Furthermore Casti was interested to compute for the metabolism map f, for the repair map P and for the replication map β by means of the realization algorithm of Kalman-Ho [5], which originally was developed for applications in control engineering, a state space representation given by the associated minimal dynamical system. Different authors, coming mainly from the field of mathematical biology, have made use of the results of Casti and have tried to extend the results [6], [7]. In this lecture and in the paper we repeat partly the results of John Casti but take the narrower point of view in describing the relevant I/O operations by discrete time convolution. Furthermore Casti computes on the basis of the impulse response h by the method of Kalman-Ho the associated state space representations (F,G,H). By this approach he gets for the metabolism map f, the repair map P and the replication map β a algorithmic representations with the expectation to get so additional modelling means for the solution of associated biological problems. The application of the Kalman-Ho algorithm for realization requires, however, that the Hankel matrix of the associated impulse responses is finite dimensional. In the case of biological cells, the validity of this assumption seems to be difficult to prove. Any biological cell is in its inners a highly complex system which is reflected by its metabolism map and the associated impulse response. John Casti stated on this point, that it would be desirable to be able to compute partial realizations, which does not require finite dimensionality of the impulse responses. Our presentation follows this direction. We make use of the partial realization method as introduced for the scalar case by Rissanen [8] and for the multi-variable case by Rissanen-Kailath [9]. In an earlier paper we used the partial realization method of Rissanen to generalize the Massey-Berlekamp algorithm of linear cryptanalysis for the case of multi-variable pseudo random sequences [10] The implementation of the partial realization method of Rissanen-Kailath was reported by Jochinger [11]. It is our hope that our work can stimulate biological research in metabolism-repair by the use of the modelling approach of Rosen-Casti.


Impulse Response Biological Cell State Space Representation Hankel Matrix Convolution Operation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Casti, J.L.: Linear Metabolism Repair Systems. Int. Journal General Systems 14, 143–167 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Casti, J.L.: The Theory of Metabolism Repair Systems. Applied Mathematics and Computation 28, 113–154 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Casti, J.L.: Newton, Aristotle, an the Modeling of Living Systems. In: Casti, J., Karlqvist, A. (eds.) Newton to Aristotle, pp. 47–89. Birkhäuser, Boston (1989)CrossRefGoogle Scholar
  4. 4.
    Rosen, R.: Some Relational Models: The Metabolism-Repair Systems. In: Rosen, R. (ed.) Foundations of Mathematical Biology, vol. 2. Academic Press, New York (1972)Google Scholar
  5. 5.
    Ho, B.L., Kalman, R.E.: Effective Construction of Linear State-Variable Models from Input/Output Functions. Regelungstechnik, Oldenbourg, 545–548 (1966)Google Scholar
  6. 6.
    Zhang, Y., Sugisaka, M., Xu, C.: A New Look at Metabolism-Repair Systems - A Living System on Screen. Artificial Life and Robotics 3, 225–229 (1999)CrossRefGoogle Scholar
  7. 7.
    Nomura, T.: An Attempt for Description of Quasi-Autopoietic Systems Using Metabolism-Repair Systems. Evolutionary Systems Department. ATR Human Information Processing Research Laboratories. 2-2, Hikaridai, Soraku-gun, Kyoto 619-02, Japan (9 pages)Google Scholar
  8. 8.
    Rissanen, J.: Recursive Identification of Linear Systems. SIAM Journal on Control, 9–3, 420–430 (1971)Google Scholar
  9. 9.
    Rissanen, J., Kailath, T.: Partial Realizations of Random Systems. Automatica 8, 389–396 (1972)CrossRefzbMATHGoogle Scholar
  10. 10.
    Pichler, F.: Linear Complexity Measures for Multi-valued CryptographicData Streams by Application of the Rissanen Partial Realization Method. In: Moreno-Díaz, R., Pichler, F., Quesada-Arencibia, A. (eds.) EUROCAST 2009. LNCS, vol. 5717, pp. 41–46. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  11. 11.
    Jochinger, D.: A Software Implementation of the Rissanen Method for Partial Linear Systems Realization. In: Moreno-Díaz, R., Pichler, F., Quesada-Arencibia, A. (eds.) EUROCAST 2009. LNCS, vol. 5717, pp. 47–52. Springer, Heidelberg (2009)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Franz Pichler
    • 1
  1. 1.Johannes Kepler University LinzLinzAustria

Personalised recommendations