Linearization in Various Forms

The kinetic equations governing gene expression can be expected to be highly nonlinear and therefore difficult to investigate. Linearization of the kinetic equations is tempting and certainly facilitates matters. A number of investigations [3-5] have studied such an approach and we discuss briefly one of them. 

Young et al. [4] propose a reverse engineering method in which they start with general chemical kinetic equations (see eq. (11.1)) and then linearize these equations (see eq. (11.3)) for the concentrations of each mRNA in the system the time variation of each mRNA is a linear function of its own concentration a linear function of all other mRNA concentrations with coefficients that give the type (positive, negative, or zero) and strength of interaction between two mRNAs a constant perturbation term (external stimulus) and a noise term dependent on time only. The coefficients form the connectivity matrix. The system of these N mRNAs is originally presumed to be in a stationary 

The concept of sparse networks is crucial to the success of this approach, but the authors were unable to quantify this notion of sparseness and to pinpoint the critical size of a network as a function of the average number of connections for which the method suffices. The method is not applicable for obtaining the connectivity of small, and likely highly connected, networks that govern specific biological functions. 

An important consideration has been omitted in [3-5], which are devoted to this approach, and that is the usually rather large experimental errors associated with microarray measurements. It is important to know how such errors propagate in the calculations and their effects on the proper identification of the connectivity matrix. A simple example worked out in detail in section 12.5 shows the possible multiplicative effects of such errors in nonlinear kinetic equations. Until this problem is addressed, the approach of linearization must be viewed with caution. 

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