Kinetic Jacobian matrix

Adesina has shown that it is superfluous to carry out the inversion required by Equation 5-255 at every iteration of the tri-diagonal matrix J. The vector y"is readily computed from simple operations between the tri-diagonal elements of the Jacobian matrix and the vector. The methodology can be employed for any reaction kinetics. The only requirement is that the rate expression be twice differentiable with respect to the conversion. The following reviews a second order reaction and determines the intermediate conversions for a series of CFSTRs. 

The basic idea is very simple In many scenarios the construction of an explicit kinetic model of a metabolic pathway is not necessary. For example, as detailed in Section IX, to determine under which conditions a steady state loses its stability, only a local linear approximation of the system at this respective state is needed, that is, we only need to know the eigenvalues of the associated Jacobian matrix. Similar, a large number of other dynamic properties, including control coefficients or time-scale analysis, are accessible solely based on a local linear description of the system. 

Figure 26. The proposed workflow of structural kinetic modeling Rather than constructing a single kinetic model, an ensemble of possible models is evaluated, such that the ensemble is consistent with available biological information and additional constraints of interest. The analysis is based upon a (thermodynamically consistent) metabolic state, characterized by a vector S° and the associated flux v° v(S°). Since based only on the an evaluation of the eigenvalues of the Jacobian matrix are evaluated, the approach is (computationally) applicable to large scale system. Redrawn and adapted from Ref. 296.

For any arbitrary metabolic network, the Jacobian matrix can be decomposed into a sum of three fundamental contributions A term M eg that relates to allosteric regulation. A term M in that relates to the kinetic properties of the network, as specified by the dissociation and Michaelis Menten parameters. And, finally, a term that relates to the displacement from thermodynamic equilibrium. We briefly evaluate each contribution separately. 

As already discussed in Section VII.B.2, reactions close to equilibrium are dominated by thermodynamics and the kinetic properties have no, or only little, influence on the elements of the Jacobian matrix. Furthermore, thermodynamic properties are, at least in principle, accessible on a large-scale level [329,330]. In some cases, thermodynamic properties, in conjunction with the measurements of metabolite concentrations described in Section IV, are thus already sufficient to specify some elements of the Jacobian in a quantitative way. 

The Jacobian matrix for the mass action law kinetic equation (77) is ... 

When the principal linear law of conservation is of the form LrriiZi = const., elementary reactions entering into the mechanism without interactions are (d/m A - (dlmj)AJ and the corresponding kinetic equations and Jacobian matrix will be... 

A different dependence of the parameters in kinetic equations was reported by Horiuti [11] who suggested a method for determining the number of independent parameters. The method consists of the numerical estimation of a rank for some Jacobian matrix. (It is known that this procedure can result in a considerable error.) Later, these problems were analyzed in detail by Spivak and Gorskii [52, 53] but they did not aim at the elucidation of the physico-chemical reasons for the appearance of dependent and undeterminable parameters. It is this aspect that we will discuss below. 

We consider only the case that the kinetic terms Fi(p , Py, 4>) and F2ip , Py, 4>) depend linearly on the light intensity 4>- This covers both the photosensitive BZ reaction and the CDIMA reaction, see Sect. 13.7.1. The influence of the projected light is additive if /j and /2 are constants otherwise it is multiplicative. We assume that the system (13.157) has a unique steady state (Pn 4>), Py 4>)) which is stable. This requires that conditions (1.27) are satisfied, i.e., the trace of the Jacobian matrix... 

Simplihcations of chemical kinetics generally become an alternative. Small mechanisms of a low number of species are often reduced using the assumptions of steady-state and partial equilibrium. Farge mechanisms are reduced using a combination of techniques such as direct relation graph (DRG), to obtain a skeleton mechanism and techniques based on the sensitivity analysis of the eigenvalues and eigenvectors of the Jacobian matrix of the chemical system to obtain a reduced mechanism. 

For the two radial coordinates, we use the radial sinc-DVR given by Colbert and Miller [42]. Considering the scattering coordinate first, a grid of R values is defined by R = iAR where i = 1,2,3,. The point at zero is automatically deleted because of the Jacobian weight at the origin. The radial kinetic energy matrix element is... 

In terms of the generalized parameter matrices, the Jacobian is given as product of a simple matrix multiplication. Using explicit kinetic parameters, the estimation of the Jacobian can be tedious and computationally demanding, prohibiting the analysis of large ensembles of models. 

We emphasize that all quantities that specify the elements of the matrix of partial derivatives are local quantities. The properties of the network only enter in terms of the (net)flux distribution v° that obeys the flux balance equation AT0 = 0. That is, reactions see other reactions only via the flux distribution. The locality of kinetic properties also allows the straightforward specification of an explicit kinetic model that corresponds to a given Jacobian. 

Let us determine the matrices J and F belonging to the kinetic system of ODEs above. These two types of matrices will be used several dozen times in the following chapters. For example, the Jacobian is used within the solution of stiff differential equations (Sect. 6.7), the calculation of local sensitivities (Sect. 5.2) and in timescale analysis (Sect. 6.2), whilst matrix F is used for the calculation of local sensitivities (Sect. 5.2). Carrying out the appropriate derivations, the following matrices are obtained ... 

It is clear that, based on small perturbations of concentrations, lifetimes can be related to chemical kinetic systems. These lifetimes do not belong to species, however, but to combinations of species concentrations defined by the left eigenvectors of the Jacobian, called modes. A matrix Jacobian of size x has eigenvalues, and therefore, the number of modes is identical to the number of variables. In the case of a linear system (in reaction kinetics, this means that the mechanism consists of first-order and zeroth-order reactions only), the Jacobian is constant and does not depend on the values of variables (concentrations). If the system is nonlinear, which is the case for most reaction kinetic systems, the Jacobian depends on the values of variables, i.e. the timescales depend on the concentrations. In other words, the set of timescales belong to a given point in the space of concentrations (phase space) and are different from location to location (or from time point to time point if the concentrations change in time). 

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