Metabolic Control Analysis: Separable Matrices and Interdependence of Control Coefficients

A central quantity for the analysis of the interdependence of control coefficients is the JacobianHof the pathway. For a simple metabolic chain,His known to be tridiagonal. Its inverseH−1, which is required to calculate control coefficients, is semi-separable. A semi-separablen×nmatrix (aij) has the characteristic property that it is decomposable into two triangles for each of which there are vectorsr=(r1, . . . ,rn) andt=(t1, . . . ,tn) withaij=ritj. The exact definitions of semi-separability and the related separability of matrices are given in Appendix B. Owing to the semi-separability ofH−1, the determinants of all 2×2 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 matrixCJis hown to be separable and the concentration control coefficient matrixCsto be semi separable.Cshas, 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 ofCs. 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.