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, H is known to be tridiagonal. Its inverse H-1, which is required to calculate control coefficients, is semi-separable. A semi-separable nxn matrix (aij) has the characteristic property that it is decomposable into two triangles for each of which there are vectors r=(r1, . . . ,rn) and t=(t1, . . . ,tn) with aij=ritj. 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 2x2 sub-matrices of elements located within one of the triangles are zero. Therefore, these triangles are regions of vanishing two-minors. The flux control coefficient matrix CJ is hown to be separable and the concentration control coefficient matrix Cs to be semi separable. Cs 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-minors, thereby disturbing the semi-separability of the upper triangle of Cs. A recipe is given to graphically construct the regions of vanishing two-minors of concentration control coefficients. The notion of (semi-)separability allows assessment of all dependences of control coefficients for metabolic pathways.Copyright 1998 Academic Press