A Bayesian approach to targeted experiment design

Abstract

Motivation: Systems biology employs mathematical modelling to further our understanding of biochemical pathways. Since the amount of experimental data on which the models are parameterized is often limited, these models exhibit large uncertainty in both parameters and predictions. Statistical methods can be used to select experiments that will reduce such uncertainty in an optimal manner. However, existing methods for optimal experiment design (OED) rely on assumptions that are inappropriate when data are scarce considering model complexity.

Results: We have developed a novel method to perform OED for models that cope with large parameter uncertainty. We employ a Bayesian approach involving importance sampling of the posterior predictive distribution to predict the efficacy of a new measurement at reducing the uncertainty of a selected prediction. We demonstrate the method by applying it to a case where we show that specific combinations of experiments result in more precise predictions.

Availability and implementation: Source code is available at: http://bmi.bmt.tue.nl/sysbio/software/pua.html.

Fig. 1.

Illustration of the effect of adding a new data point on the PPD. Shown on the top right is the PPD at one specific time point for two predictions with a subset of the samples of the chain indicated with white points. The square denotes the location of the ‘new measurement’. Prediction A refers to a prediction of which a new measurement can be performed (observable), whereas B denotes the prediction of interest. Here the grey distribution corresponds to the PPD before the new measurement, whereas the white Gaussian corresponds to the error model of the new measurement. Due to additional constraints imposed by this new measurement in combination with the old data and the model, the distribution on the hypothesis side is also updated in light of the new data point and shown in white.

Fig. 2.

Model of the JAK-STAT pathway. In this model u₁ serves as driving input, while the total concentration of STAT (x₁+x₂+2x₃) and the total concentration of phosphorylated STAT in the cytoplasm (x₂+2x₃) were measured. Note that the step from x₄ back to x₁ is associated with a delay.

Fig. 3.

Top left: one simulated time course of state 3 superimposed on the PPD. Two time points are indicated with circles. Bottom left: correlation coefficient between states 3 and 4 and SVR of state 4 based on a measurement of state 3 (SVR). The relation between the two states at the indicated time points is shown in both scatter plot and 2D histogram form. The former shows the actual samples from the PPD for one point in time. Here the dots represent simulated values belonging to different parameter sets from the MCMC chain. In the histogram the colour indicates the number of samples in a particular region which is proportional to the probability density.

Fig. 4.

Variance reduction of the peak time of dimerized STAT (x₄) with respect to two new measurements. (A) Each axis represents an experiment, where the different model outputs are numbered. Numbers 1 to 3 correspond to the first three states whereas 4 and 5 correspond to the sums of states on which the original PPD was parametrized. Note that each block on each axis corresponds to an entire time series. The block corresponding to experiments involving state 1 is shown enlarged in (B). Variance reduction is computed using the importance sampling method.

Fig. 5.

Comparison of two methods for calculating the variance reduction. Variance reduction of the peak time of dimerized STAT (x₄) with respect to two new measurements. (A) LVR. (B) Difference between the variance reduction computed by means of LVR and importance sampling (shown in Fig. 4).