Topological and kinetic determinants of the modal matrices of dynamic models of metabolism

Large-scale kinetic models of metabolism are becoming increasingly comprehensive and accurate. A key challenge is to understand the biochemical basis of the dynamic properties of these models. Linear analysis methods are well-established as useful tools for characterizing the dynamic response of metabolic networks. Central to linear analysis methods are two key matrices: the Jacobian matrix (J) and the modal matrix (M-1) arising from its eigendecomposition. The modal matrix M-1 contains dynamically independent motions of the kinetic model near a reference state, and it is sparse in practice for metabolic networks. However, connecting the structure of M-1 to the kinetic properties of the underlying reactions is non-trivial. In this study, we analyze the relationship between J, M-1, and the kinetic properties of the underlying network for kinetic models of metabolism. Specifically, we describe the origin of mode sparsity structure based on features of the network stoichiometric matrix S and the reaction kinetic gradient matrix G. First, we show that due to the scaling of kinetic parameters in real networks, diagonal dominance occurs in a substantial fraction of the rows of J, resulting in simple modal structures with clear biological interpretations. Then, we show that more complicated modes originate from topologically-connected reactions that have similar reaction elasticities in G. These elasticities represent dynamic equilibrium balances within reactions and are key determinants of modal structure. The work presented should prove useful towards obtaining an understanding of the dynamics of kinetic models of metabolism, which are rooted in the network structure and the kinetic properties of reactions.

! ( ) = !! • ! ! ! , where ! is the corresponding eigenvalue and !! is the inner product of left eigenvector ! and steady state concentration vector x (v in the case of flux mode) at time 0. The coefficient in left eigenvector ! represents the contribution of element response to the mode response. As the effect of perturbation propagates through time, slower modes get excited and faster modes relax. When a mode relaxes, we have ! = 0. The direction at which the mode moves depends on the elements in the mode and the elements in the perturbation vector. For example, when the perturbation is the change of ATP concentration, the first concentration mode to respond must have a non-zero ATP coefficient in that left eigenvector.

Diagonal dominance and the Gershgorin circle theorem
Gershgorin circle theorem can help explain eigenvalues in cases where diagonal dominance occurs in the Jacobian matrix. In this theorem, each row of the Jacobian matrix forms a Gershgorin circle, with the diagonal element !! being the center and the absolute sum of offdiagonal elements | !" | !!! being the radius. The Gershgorin circle theorem constrains the eigenvalues based on the relative magnitudes of diagonal and off-diagonal elements in a matrix. As illustrated in Fig 2B, the theorem is helpful in associating eigenvalues with certain Gershgorin circles, particularly when the row is strongly diagonally dominant, where the diagonal element is much greater than the sum of off-diagonal elements in absolute value.
However, as the diagonal dominance gets weaker, the radius of the circle becomes larger and it becomes difficult to associate a unique eigenvalue to each circle. In the case where the row is not diagonally dominant, the circle can reach the positive part of the x-axis and possibly contain positive eigenvalues when the system is unstable. In our system, we did not find any positive eigenvalues. It is also worth mentioning that eigenvalues will fall in any Gershgorin circles but not all Gershgorin circles will contain an eigenvalue, especially in cases where the circles overlap with each other. However, even though overlapping occurs frequently in our model, we found that 99.0% of the circles had an eigenvalue inside, which may not be guaranteed for all models.

Negative diagonal elements in the Jacobian matrix encourage system stability
During the course of analysis of diagonal dominance and eigenvalues of the Jacobian matrix, we noticed a critical relationship between diagonal dominance in metabolic networks and system stability. The local stability of a fixed point (steady-state) of a dynamic system is determined by the presence of only negative eigenvalues of the Jacobian matrix. We note that the diagonal elements are always negative in the Jacobian matrix of a biochemical network, due to the formulation of mass action rate law. Because of this, the eigenvalues that fall in the Gershgorin circles formed from diagonally dominant rows are guaranteed to be negative, ensuring the stability of the system. On the other hand, when the row is not diagonally dominant, the Gershgorin circle crosses 0 on the x-axis, enabling positive eigenvalues to appear.
Examining the impact of the negative diagonal elements on system stability, we replaced the diagonal elements of J with zero value one at a time and recalculated the eigenvalues of J. This effectively moves center of the related Gershgorin circle to the origin, while the radius is unchanged as it is set by the off-diagonal elements. We found that there are always positive eigenvalues after modifying the diagonal element and the largest positive eigenvalue tends to be close to the replaced diagonal element in absolute value (S2 Fig). Since the Gershgorin circle surrounds the origin, it would seem that the resulting eigenvalue could be either positive or negative; however, we did not observe this to be the case. The mathematical origin of this phenomenon is thus not entirely clear to us. Still, we concluded that the negative diagonal elements of the Jacobian matrix are clearly critical for system stability, with the effect magnified by diagonal dominance. Enzyme forms are in italic. '&' means the enzyme form is bound with metabolite(s). '_T' means that the enzyme is at the tense state. G6P, Glucose 6-phosphate; NADP, Nicotinamide adenine dinucleotide phosphate (oxidized); NADPH, Nicotinamide adenine dinucleotide phosphate (reduced); OAA, Oxaloacetate; PPi, Diphosphate; 5MDRU1P, 5-Methylthio-5-deoxy-D-ribulose 1-phosphate; 6PGC, 6-Phospho-D-gluconate; 6PGL, 6-phospho-D-glucono-1,5-lactone; GND, phosphogluconate dehydrogenase; G6PDH, glucose 6-phosphate dehydrogenase; GAPDH, glyceraldehyde 3-phosphate dehydrogenase; PFK, phosphofructokinase; PGI, glucose 6phosphate isomerase.