On Modelling Metabolism-Repair by Convolution and Partial Realization

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.