METABOLIC CONTROL ANALYSIS - SEPARABLE MATRICES AND INTERDEPENDENCE OF CONTROL COEFFICIENTS

A central quantity for the analysis of the interdependence of control coefficients is the Jacobian H of the pathway. For a simple metabolic chain, a is known to be tridiagonal. Its inverse H-1, which is require d to calculate control coefficients, is semi-separable. A semi-separab le n x n matrix (a(ij)) has the characteristic property that it is-dec omposable into two triangles for each of which there are vectors r = ( r(1),...,r(n)) and t = (t(1),...,t(n)) with a(ij) = r(i)t(j). The exac t definitions of semi-separability and the related separability of mat rices are given in Appendix B. Owing to the semi-separability of H-1, the determinants of all 2 x 2 sub-matrices of elements located within one of the triangles are zero. Therefore, these triangles are regions of vanishing two-miners. The flux control coefficient matrix C-J is sh own to be separable and the concentration control coefficient matrix C -S to be semi-separable. C-S has, in addition, the peculiarity that th e row vector is the same for both its upper and lower triangle. A feed back loop gives rise to a new sub-region of vanishing two-miners, ther eby disturbing the semi-separability of the upper triangle of C-S. A. recipe is given to graphically construct the regions of vanishing two- miners of concentration control coefficients. The notion of (semi-)sep arability allows assessment of all dependences of control coefficients for metabolic pathways. (C) 1998 Academic Press.