Differential equations, parameter fitting, and identifiability analysis
When a system is composed of many identical and well intermixed entities, differential equations are a good choice. For example, the dynamics of specific factors in blood samples are well described by ordinary differential equations. The big advantage of this methodology is that they are easily solved and solutions may even be found analytically with pencil and paper. We also prefer differential equations for modelling more complex molecular networks over alternative approaches, like boolean networks, which involve less parameters but often lead to artificial solutions because natura not facit saltum (Gottfried Leibniz).
Given a model and a set of experimental data points, the model parameters are not always known. We take care that all our models are formulated in a language in which every parameter has a physiological meaning and is, thus, in principle measurable. In some cases, the measurements are difficult. Then the remaining unknown parameters have to be determined by reverse engineering: The read-out of the model has to fit the data. It is important to realize that not every data set contains information on the unknown model parameters. This can be determined by an identifiability analysis. Correlations between parameters are determined by bootstrapping based on refitting with artificially modified data sets.
In order to fit model parameters, a measure for the quality of the model read-out is defined based on deviations of the model output from the data. This measure is minimized by scanning the parameter space. In most cases, the absolute minimum of the measure is not found by standard or stochastic difference methods. Therefore, we prefer to use differential evolution algorithms. When the model gets too complex, finding an acceptable best fit might still be impossible.
We have invented an iterative method with which the complex problem is decomposed into simpler subsystems. Each of these subsystems is solvable by differential evolution algorithms. The, thus, generated approximate fit of the total system is then used as a starting point for a global fit. This method outperformed the unmodified global fit in all network topologies tested so far.
Sebastian C. Binder, Jaber Dehghany, Michael Meyer-Hermann
Binder SC, Hernandez-Vargas EA, Meyer-Hermann M. Reducing complexity: an iterative strategy applied to parameter selection for regulatory networks. Computer Physics Communications 190 (2015) 15-22.
- Systems Immunology- Prof. Dr. Michael Meyer-Hermann