Ten misundertandings about power-law models

1. The power-law models were formulated for a metabolic systems and they are not valid for other biochemical systems. Power-law models were initially formulated for metabolic networks, but the assumptions that support its use are fully satisfied in other biochemical systems (Schnell and Turner 2004): The biochemical processes governed by an anomalous diffusion-reaction process due to:

• 1: Non-homogeneity of the cytosol
• 2: Crowding with several kinds of macromolecules from isolated proteins to ribosomes and filaments
• 3: Volume exclusion effect

2. Power-law models are steady-state approach not valid in transient processes. As any other model, Power-law models are a steady-state approach when only steady-state data is used to estimate the parameters describing the system. If dynamical data from perturbatory experiments and transient response experiments are used, the models are not steady-state approach (see Voit 2000).

3. The power-law models are not valid because they suppose flux aggregation. At the same level of detail in the equations, a power-law model is not more aggregated than a conventional kinetic model. The problems of aggregation appear in the specific case of S-system models, which is a sub-type of power-law models. Flux aggregation is not used in power-law models with elementary reactions (Voit 2000).

4. The non-integer kinetic orders have no biological meaning. The non-integer kinetic orders relates to the effects on the system dynamics of the anomalous difussion-reaction in crowded non-homogeneous medium. The more crowded, non-homogeneous and non-isotropic the medium is, the higher the values of the associated kinetic orders (Savageau 1998).

5. The power-law models cannot reproduce saturation. When a model in elementary steps is considered, a power-law model can reproduce any dynamical property of a conventional kinetic model (saturation, sigmoidicity, thresholds, bifurcations…) and some more related to the higher non-linearity of its equations. Only when you compare an aggregated power-law model with a Michelis-Menten like kinetics the saturation is not reproduced. Moreover, the saturation described by this equation is possibly not representing the conditions inside the cell -where concentration in substrate is never so high- but a secondary undesirable effect of the use of test-tube experiments ([[SavTestTube][Savageau 1992]).

6. The estimation of kinetic orders makes these models not identifiable. When experimental data are accessible in the necessary quality and quantity, the kinetic orders can be estimated. The parameter estimation requires advanced algorithms and sometimes has higher computational requirements, but is possible (see Almeida 2003, Tucker 2006, Gonzalez 2006).

7. The increased problems of identifiability in power-law models are not compensated by any advantage; therefore simpler models should be used. This is a case-dependent question. If the system considered is simple and the data available not very rich from a dynamical point of view, a conventional kinetic model or even a Michaelis-Menten like model could be the best choice. In a more general case, the use of power-law model allows an improved flexibility to reproduce high non-linearity in the system (see section "Pros and Cons of different ODE modeling frameworks").

8. If a power law models turns out to be necessary, it means that a model more detailed kinetic model could be formulated. This is not always a feasible possibility for modeling a system. There are two possible scenarios:

• In some cases, the complete structure of the network of chemical reactions is not considered and these non-integer kinetic orders take into account this lack of knowledge about the system (similarly to the case of chemical kinetics). This is a very common situation for example in the case of the metabolic systems, or when “nuclear interactions” are considered. In that case, the expansion as a conventional kinetic model of elementary steps could be enough to reproduce the complexity of the system if the effect of anomalous reaction-diffusion processes is not significant.
• But in a general case where the structure of the model considers all the "elemental reactions" occurring in the system, and effects associated to crowded and non-homogeneity are important, this strategy could be not enough (see document below). In the perfect case of having this "complete structural information", the non-integer kinetic orders help to model the effect of the anomalous diffusion of massive molecules in a crowded medium -the case of the cytosol.

9. Power-law models have been historically used in engineering as working models (empirical models); they can be used for control analysis purposes but not for understanding a process. Power-law models or conventional kinetic models are not the right modeling framework for understanding the underlying microscopic mechanism of the biochemical processes. Biochemical processes involve very complex molecules (for example, proteins), which mechanistics behaviour can only be studied with enough accuracy with Quantum Chemistry or similar approaches. The purpose of power-law and kinetic models is to study systemic-global properties of biochemical networks (See Vera et al 2006).