Power-Law Formalism

It should be emphasized that Eq. (22) is already based on a number of preconditions. In particular, the intracellular medium may significantly deviate from a well-stirred ideal solution [141 143], While the use of Eq. (22) is often justified, several authors have suggested to allow noninteger exponents in the expression of elementary rate equations [96,142,144] corresponding to a more general form of mass-action kinetics. A related concept, the power-law formalism, developed by M. Savageau and others [145 147], is addressed in Section VII.C. 

A core constituent of BST is to represent all metabolic rate equations as power law functions. Using the power-law formalism, each reaction rate is written as a product... 

To simulate the overall network behavior, the power-law formalism is applied in two different ways. Within a generalized mass-action model (GMA), each biochemical interconversion is modeled with a power-law term, resulting in a differential equation analogous to Eq. (5)... 

The approximation of biochemical rate equations by linear-logarithmic (lin-log) equations [318] seeks to avoid several drawbacks of the power-law formalism. Using the lin-log framework, all reaction rates are described by their dependencies on logarithmic concentrations, based on deviations from a... 

Similar to generalized mass-action models, lin-log kinetics provide a concise description of biochemical networks and are amenable to an analytic solution, albeit without sacrificing the interpretability of parameters. Note that lin-log kinetics are already written in term of a reference state v° and S°. To obtain an approximate kinetic model, it is thus sometimes suggested to choose the reference elasticities according to simple heuristic principles [85, 89]. For example, Visser et al. [85] report acceptable result also for the power-law formalism when setting the elasticities (kinetic orders) equal to the stoichiometric coefficients and fitting the values for allosteric effectors to experimental data. 

Despite the obvious correspondence between scaled elasticities and saturation parameters, significant differences arise in the interpretation of these quantities. Within MCA, the elasticities are derived from specific rate functions and measure the local sensitivity with respect to substrate concentrations [96], Within the approach considered here, the saturation parameters, hence the scaled elasticities, are bona fide parameters of the system without recourse to any specific functional form of the rate equations. Likewise, SKM makes no distinction between scaled elasticities and the kinetic exponents within the power-law formalism. In fact, the power-law formalism can be regarded as the simplest possible way to specify a set of explicit nonlinear functions that is consistent with a given Jacobian. Nonetheless, SKM seeks to provide an evaluation of parametric representation directly, without going the loop way via auxiliary ad hoc functions. 

The power-law formalism was used by Savageau [27] to examine the implications of fractal kinetics in a simple pathway of reversible reactions. Starting with elementary chemical kinetics, that author proceeded to characterize the equilibrium behavior of a simple bimolecular reaction, then derived a generalized set of conditions for microscopic reversibility, and finally developed the fractal kinetic rate law for a reversible Michaelis-Menten mechanism. By means of this fractal kinetic framework, the results showed that the equilibrium ratio is a function of the amount of material in a closed system, and that the principle of microscopic reversibility has a more general manifestation that imposes new constraints on the set of fractal kinetic orders. So, Savageau concluded that fractal kinetics provide a novel means to achieve important features of pathway design. 

The power-law formalism is a mathematical language or representation with a structure consisting of ordinary nonlinear differential equations whose elements are products of power-law functions. The power-law formalism meets two of the most important criteria for judging the appropriateness of a kinetic representation for complex biological systems the degree to which the formalism is systematically structured, which is related to the issue of mathematical tractability, and the degree to which actual systems in nature conform to the formalism, which is related to the issue of accuracy. 

The Power-Law Formalism A Representation for Intact Biochemical Systems... 

TCA Cycle in Dictyostelium A Case Study Generality of the Power-Law Formalism Conclusions Summary... 

The relatively simple form of the rate law in the Power-Law Formalism has several important implications for the characterization of the molecular elements of the system. In particular, there are known methods for estimating the kinetic parameters and the amount of data required for the estimation is minimal. 

As is clear from Eqn. (28), there are two kinds of parameters in the Power-Law Formalism multiplicative and exponential. These are familiar from chemical and biochemical kinetics and are referred to as rate constants and kinetic orders, respectively. 

Kinetic analysis of rate laws in the Power-Law Formalism is particularly simple and straightforward. By taking the logarithms of both sides of Eqn. (28) one can convert the product of power laws into a sum of terms involving logarithms of the variables. 

A plot of In Vi as a function of In Xj, while all other X s are held constant at their nominal values in situ, yields a straight line whose slope determines the exponential parameter g,[. Given that there will always be a certain amount of experimental error associated with each assay, it will in general require about 10 data points to obtain a reasonably good estimate of the slope. Thus, we can conclude that 0n assays will be needed to estimate the kinetic parameters of the rate law in the Power-Law Formalism when there are n variables that influence the rate law under consideration. 

In the Power-Law Formalism each rate law is represented as a product of power-law functions [e.g., Eqn. (28)]. The fundamental equations governing the behavior of the intact biochemical system are Kirchhoff s flux equations, which are obtained by combining the rate laws for synthesis and degradation of each molecular constituent. There are a number of general strategies for combining the individual rate laws to obtain Kirchhoff s flux equations (see below) the simplest of these strategies allows one to write a local representation as... 

There are two types of fundamental parameters in the Power-Law Formalism, rate constants and kinetic orders the definitions of sensitivity with respect to changes in these parameters are summarized below. A full discussion of the relationships among these sensitivities is given in Savageau and Sorribas (1989). 

A recent investigation of the current model for the tricarboxylic acid cycle in Dictyostelium discoideum may be viewed as a case study that illustrates the uses of local representation within the Power-Law Formalism (Shiraishi and Savageau, 1992a,b,c,d 1993). First, it demonstrates that systemic analysis is a powerful tool to evaluate the quality of biochemical models especially those representing the function of a complex system in vivo. Second, it demonstrates that systemic analysis allows one to diagnose deficiencies and to predict modifications that are likely to improve the model. 

To what degree do actual systems in nature conform to the Power-Law Formalism We can consider this issue from three different perspectives local representation, fundamental representation, and recast representation (Savageau, 1995). 

Each of the formalisms considered in this subsection—Mass-Action and Michaelis-Menten—is able to serve as the foundation for representing diverse phenomena, but each also has known limitations. The Power-Law Formalism may be considered more fundamental than either of these because it includes them as special cases and is not subject to their limitations. 

Although the Power-Law Formalism was originally developed from the notion of a local representation in logarithmic space (Savageau, 1969b), it was subsequently discovered that nearly any nonlinear function or set of differential equations can be transformed exactly into the Power-Law Formalism (Savageau and Voit, 1987). Thus, this formalism provides a canonical nonlinear representation for most nonlinear functions. An example will make this clear. 

As a local representation, the S-system variant within the Power-Law Formalism is more accurate than the GMA variant, which is generally more accurate than the conventional Linear Formalism. As a fundamental representation, the Power-Law Formalism includes as special cases the Mass-Action, Michaelis-Menten, and Linear Formalisms that are considered to accurately represent natural phenomena... 

The Power-Law Formalism described above provides an attractive alternative. In this formalism, the rates of formation and removal of each elemental component of the system are described by a product of power-law functions, one power function for each variable affecting the rate process in question. These elemental rate laws can be combined according to several general strategies to yield a description of the intact system. The strategy leading to the local S-system repre-... 

The Power-Law Formalism possesses a number of advantages that recommend it for the analysis of integrated biochemical systems. As discussed above, we saw that estimation of the kinetic parameters that characterize the molecular elements of a system in this representation reduces to the straightforward task of linear regression. Furthermore, the experimental data necessary for this estimation increase only as the number of interactions, not as an exponential function of the number of interactions, as is the case in other formalisms. The mathematical tractability of the local S-system representation is evident in the characterization of the intact system and in the ease with which the systemic behavior can be related to the underlying molecular determinants of the system (see above). Indeed, the mathematical tractability of this representation is the very feature that allowed proof of its consistency with experimentally observed growth laws and allometric relationships. It also allowed the diagnoses of deficiencies in the current model of the TCA cycle in Dictostelium and the prediction of modifications that led to an improved model (see above). 

Savageau, M. A. (1995). Power-law formalism A canonical nonlinear approach to modeling and analysis. Proc. First World Congress of Nonlinear Analysts, Walter de Gruyter, Berlin. 

Exact Representation. Althou the power-law formalism was originally derived as an approximation, one can often obtain an exact representation of rational-function nonllnearltles by Introducing additional variables (8). This obviously extends the generality and utility of the power-law formalism. Any disadvantage caused by the Increase In number of variables Is more than offset by the decrease In complexity of the nonllnearltles. The number of parameter values plus Initial conditions necessary to characterize a system remains the same In either representation. 

The power-law formalism Is Justified by four First, Its validity rests on theoretical... 

The power-law formalism provides a Judicious compromise that has many of the advantages of linear analysis without Its severe limitations. (a) The functional form Is known and the Interactions and parameters are readily specified, (b) From experimental data one can extract parameter values for the Individual rate laws by linear regression In a logarithmic space,... 

The development of the power-law formalism, reviewed briefly In the first portion of this paper, began with a consideration of the fundamental nonllnearltles that characterize the kinetic behavior of biological systems at the molecular level (2,5). From this molecular-biological context a "dynamical systems" approach to biochemical and genetic networks was developed (3-7,11,42). 

---