Many problems in Biology can be modelled by gene regulatory networks involving systems of ordinary differential equations. Typically, such systems depend on many parameters that can not be directly measured and should thus be indirectly inferred from experimental data.

Such inverse parameter identification problems are typically ill-posed since the solution (in the least square sense) may not depend continuously on the measurement data. That is, small measurement errors might lead to large errors in the solution and hence to poor parameter estimates. This effect becomes more pronounced with increasing number of unknown parameters. Furthermore, unregularized parameter estimation methods fail to determine the parameters of the model due to strong correlations between them.

Parameter correlation is, however, common in inverse and ill-posed problems.

To address this issue we employ a numerical method in the framework of Bayesian inversion, combining a priori sensitivity analysis, model reduction using a sparsity promoting maximum a posteriori (MAP) estimate and subsequent uncertainty quantification using Markov chain Monte Carlo (MCMC) sampling of the obtained posterior density. This method has three stages:

1. In the first stage it identifies a subset of parameters which may not be reliably identifiable from the available measurement data.
2. In the second stage it selects among several solutions compatible with the measured data, one with the maximum number of parameters from the subset obtained in a) equal to zero. This yields a reduced model, nevertheless capable of reproducing the measurement data.
3. In the third stage it assesses the reliability of the estimated parameter values.

We apply the method to a problem from the paper by Ashyraliev, Jaeger and Blom (2008), who thoroughly analysed the quality of parameter estimates for the gap gene system of the vinegar fly Drosophila melanogaster. They concluded that none of the parameters can be determined due to extreme parameter correlations.

Based on synthetic data, we exhibit that the proposed method is able to eliminate non-identifiable parameters, as well as to produce parameter estimates for the reduced system. We also found that adaptive MCMC sampling of the posterior distribution is an efficient strategy to assess the uncertainty resulting from measurement data, the mathematical models and their computational implementation.

References

M. Ashyraliyev, J. Jaeger, and J.G. Blom, Parameter estimation and determinability analysis applied to drosophila gap gene circuits, BMC Systems Biology 2:83 (2008).

N. Sfakianakis and M. Simon, Inverse modeling of the Drosophila gap gene system: Sparsity promoting bayesian paremeter estimation and uncertainty quantication, Proceedings of the 10th international workshop on computational Systems Biology 86:91 (2013).