< Back to Article

Bayesian parameter estimation for dynamical models in systems biology

Fig 3

Parameter estimation for a simple two-state model.

(A) Top row: Network diagram of the two-state model with states, x₁(t), x₂(t), input function u(t), and four unknown parameters, . Bottom row: Trajectories of the input function u(t) and corresponding state trajectories. The input has at least one non-zero derivative to ensure that all model parameters are globally structurally identifiable following [55]. (B) Marginal posterior distributions of the model parameters show increasing uncertainty in the parameter estimates (e.g. widening and flattening) with increasing levels of additive normally distributed measurement noise with mean zero. We control the noise level by setting the noise covariances to the specified percentage of the standard deviation of each state variable. The dashed black vertical lines indicate each parameter’s nominal (true) value. Marginal posteriors are visualized by fitting a kernel density estimator to 20,000 MCMC samples obtained using CIUKF-MCMC with the delayed rejection adaptive Metropolis (DRAM) MCMC algorithm after discarding the first 10,000 samples as burn-in. (C) Posterior distributions of the trajectory of x₁(t) reflect increasing parameter estimation uncertainty in panel B. The true trajectory (solid black line) shows the dynamics with the nominal parameters, dashed black lines show that trajectory with the most probable set of parameters (MAP point), and the empty circles show the noisy data at the specified noise level. The 95% credible interval shows the region between the 2.5th and 97.5th percentiles that contains 95% of the 5, 000 trajectories. (D) Marginal posterior distributions of the model parameters show increasing uncertainty (widening and flattening) with increasing data sparsity (fewer samples). We simulate data sparsity by sampling the simulation from 0 ≤ t ≤ 2 with three time steps, Δt = 0.05 (40 experimental samples), Δt = 0.1 (20 experimental samples) and Δt = 0.2 (10 experimental samples). Marginal posteriors are fit to 20,000 MCMC samples obtained as in panel B. (E) Posterior distributions of the trajectory of x₁(t) reflect increasing parameter estimation uncertainty seen in panel D.

doi: https://doi.org/10.1371/journal.pcbi.1010651.g003