# The Matrix Methods

### Introduction

In previous sections it was described how to obtain the control coefficients of a metabolic pathway knowing the values of the enzyme elasticities, using the summation thoerems, the connectivity relations and several additional relations. For several reasons, it is useful if those operations can be tied up together and applied in one uniform way. Several matrix methods have been developed that fulfil this task.

### The Matrix Method of Fell & Sauro

Fell & Sauro (1985) were the first to describe a matrix method. Originally, their method was only for flux-control coefficients and could be applied to pathways with branches, substrate cycles and moiety-conserved cycles. Later extensions to this method were published making the method capable of dealing with concentration-control coefficients (Westerhoff & Kell 1987, Sauro et al. 1987) and even more complex structures (Small & Fell 1989).

### The Matrix Method of Reder

Reder (1988) developed a matrix method which is general and complete. In this method the matrices are reduced to become invertible, the operations that lead to such invertible matrices are equivalent to the special theorems developed for complex structures (see above). Reder's method is based on finding the structural properties of the pathway, that is the properties which are not dependent on the values of the elasticities but only on the stoichiometric relationships. This method, albeit requiring some advanced knowledge of linear algebra, is the most straightforward to incorporate in computational applications.

### The Matrix Method of Cascante et al.

Cascante et al. (1989a, b) focused on the similarities between MCA and Savageau's Biochemical Systems Theory (BST) and developed a matrix method in which one constructs a matrix of global properties and a matrix of local properties. This is, of course, similar to the two other methods cited above, in which one obtains the matrix of global properties (control coefficients) by inversion of the matrix of local properties (elasticity coefficients).

### Recent Developments in Matrix Methods

More recently, Westerhoff et al. (1994) developed a matrix method that accomplishes the reverse of the previous ones. With it one can determine the values of the enzyme elasticities from know values of the control coefficients. This is a potentially very useful method as enzyme elasticities are very difficult to measure in vivo or in situ while the control coefficients are rather more easy (but still have many difficulties, see next section).

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