Basic analytical tools for power-law models

Logarithmic Gains

The logarithmic gains (log gains) predict and measure the influence that changes in the value of independent variables (X_d) have in the value of dependent variables (X_i). The formal definition is:

Since it is a logarithmic coefficient, the interpretation of a log gain could be the answer to the following question: How much does the value of X_d change when X_i varies its value a 1%?

It is also possible to define in a similar way a logarithmic gain describing the effects that changes in an independent variable provoke in the steady-state value of the fluxes in the model:

Log gains are used to test the quality of models. In metabolic systems the absolute value of log gains is very often smaller than ten:

This is an intrinsic feature of metabolic systems and indicates that drastic responses to changes in independent variables are not common in such processes. Therefore, metabolic models with very high log gains must be revised and refined. In other kinds of biochemical systems, high log gains could be a constitutive characteristic of the system, and thus, the property of low log gains does not translate directly to other systems (see this example).

The values of logarithmic gains are a general property of any S-system models. This means that all the possible steady-states of the system have the same values for log gains, which only depend on the structure of the system and the values of kinetic orders. Then, this coefficients are a valuable information about the global behaviour of a particular s-system models (and the system they represent).

In case of GMA models and S-systems with moiety conservation (additional algebraic equations that complement differential equations to describe the system), the log gains have the same definition, but they are not global properties of the models, and they must be calculated for any steady-state studied.

Sensitivities

Sensitivites are other coefficients similar to logarithmic gains. They measure the influence that changes in the value of parameters (rate constants or kinetic orders) have in the steady-state value of dependent variables and fluxes:

They are useful for analysing the robustness of the models with respect changes in parameters. Very often they are used as a test of quality of the model. High sensitivies (higher than ten) means that the model is not robust enough and requires further refinement.