A five-steps plan to obtain a power-law model

In the next lines we illustrate the main steps necessary to obtain a power-law model of a biochemical system. We will use as example a very simple model, which describes the MAPK cascade Raf1/MEK/ERK.

Step1. Analysing and organising prior knowledge about the pathway: the map of the pathway

The first step is to obtain from the literature all the biological information available about the structure of the pathway: essential interactions and involved molecules (in their different states) must be listed and stated. In our case, proteins involved in the pathway are the kinases Raf1, MEK and ERK either in their inactivated (MEK, ERK) or activated (pMEK and pERK) states. With respect to the essential biochemical interactions which integrate this pathway, we have detected:

• Activated Raf1 (Raf1*) can activate MEK; this process is inhibited by activated ERK (pERK).
• Activated pMEK is inactivated to MEK after a certain time.
• Activated MEK (pMEK) activates ERK.
• Activated pERK decays to its inactivated from after a certain time.

With this information we can set up a preliminary graph of the pathway, which is showed in the next figure:

Step2. Identifying the variables involved in each interaction

Once the processes to be modelled are listed, variables involved (or potentially involved) must be identify and included in the rate equations. In our case, variables involved in each signal rate are indicated in the next figure:

Step3. Setting up the differential equations that integrate the model

Once the list of dependent variables is know, we set up the differential equations describing the dynamics of such variables. In each differential equation, any biochemical rate contributing to the value of such variable is included. The rate is included with positive sign if it increases the value of the variable and with negative sign if it provokes a decrease in such value. The information from the previous steps is use to formulate the equations. In our case study, these are the ordinary differential equations describing the pathway:

Step4. Conversion of rate equations into power-law terms

At this point, the rate equations are written as power-law terms. The characteristic structure of power-law equations is derived for each term, in which parameters will be not only rate constants but also kinetic orders. e.g. in our example for the signal rate V₁ we have:

Prior knowledge about the system can be used to constrain the possible values of kinetic orders. In our case study, all the kinetic orders must be positive real numbers excepting the kinetic order which describes the inhibition of V₁ by pERK. This kinetic order must be negative because inhibition is modelled with negative kinetic orders in simplified power-law models like our case study. When we expand all the rates as power-law terms and apply these definitions in the differential equations we obtain:

Step5. Parameter estimation and quality assesment.

Once the structure of the model is obtained, parameters of the models must be estimated from data. In metabolic systems in which a previous Michaelis-Menten type model is available, one possibility is to obtain a first-order Taylor expansion of the model in the logarithmic space around the preferential steady-state of the system (see Section "Parameter Estimation" or Voit 2000 for further explanations). Another possibility when a previous model is available is to obtain a least-square approach for the power-law rates in the manner Hernandez-Bermejo et al. (2000) describes. In a general case in which real quantitative data is available from perturbatory time courses, quantitative data together with algorithms of optimisation for dynamical data can be used to estimate the values of the parameters (Gonzalez et al. 2006) .

Finally, the quality of the model must be tested. This means that models must prove robustness with respect to small perturbations in the values of parameters (using for instance sensitivities). The model must be as well validated using independent experiments or by qualitative analysis. In addition, the identifiability of the parameters computed must be analysed (Burham et al 1998). As a results of these different analyses a refinement in structure of the model could be necessary, which implies to repeat steps from one to five.