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 required to calculate control coefficients, is semi-separable. A semi-separable n x n matrix (a(ij)) has the characteristic property that it is-decomposable 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 exact definitions of semi-separability and the related separability of matrices 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 shown to be separable and the concentration control coefficient matrix C-S to be semi-separable. C-S has, in addition, the peculiarity that the row vector is the same for both its upper and lower triangle. A feedback loop gives rise to a new sub-region of vanishing two-miners, thereby 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-)separability allows assessment of all dependences of control coefficients for metabolic pathways. (C) 1998 Academic Press.
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