Bayesian parameter estimation for dynamical models in systems biology
Fig 6
Parameter estimation results for the MAPK model in the limit cycle regime with data only sampled from the oscillations.
(A) Marginal posterior distributions of the model parameters. Distributions are visualized by fitting a kernel density estimator to 1,305,720 (30 walkers with 43,524 steps each) MCMC samples obtained using CIUKF-MCMC with the affine invariant ensemble sampler (AIES) after discarding the first 7,447 samples per walker as burn-in. (B) Posterior distribution of the trajectory of x3(t) in the limit cycle regime. The true trajectory (dashed black line) shows the dynamics with the nominal parameters, the dotted blue line shows the trajectory evaluated at the MAP point, and the points show the noisy data (covariance is 1% of the variance of the true trajectory). The 95% credible interval shows the region between the 2.5th and 97.5th percentiles that contains 95% of 30,000 posterior trajectories. (C) Sample posterior trajectories (50 out of 30,000 total) reveal that most trajectories closely match the true limit cycles. Additionally, several trajectories that reach a fixed point are shown. (D) Quantification of the fraction of the 30,000 sample trajectories that produce limit cycles oscillations, 90.6% (27,182 samples), or reach a fixed point, 9.4% (2,818 samples). (E) Histograms quantify the variability in limit cycle amplitude and period for the 27,182 trajectories that show limit cycle oscillations. We define the limit cycle amplitude as the peak-to-peak difference for one oscillation, and the period is the time to complete an oscillation. The vertical black lines show these quantities for the true trajectory.