Fig 1.
Objective functions and the role of a measurement model.
Mechanistic models of biological processes are typically encoded as systems of (ordinary) differential equations (Eq 1). Model calibration relies on an objective function (Eq 6)—or in a Bayesian setting, a likelihood function (Eq 7)—quantifies the degree of dissimilarity or similarity between model variables and corresponding measurements. Note, the objective or likelihood function uses an implied measurement model (Eq 6) which converts modeled variables x(t) to a quantity y(ti, θ) that can be compared to data . In physics and engineering, where measurements are typically quantitative, this implicit measurement model suffices. For nonquantitative measurements and observations, the measurement model must be defined explicitly in consideration of the nonquantitative measurements’ properties.
Fig 2.
Considerations for integrating different measurements and observations into mechanistic models.
Defining a measurement as (nominal (A), ordinal (B) or semi-quantitative (C)), and applying that measurement to a mechanistic model, requires consideration of the measurement process. Properties of the measurement such as the measurand (the property targeted by experimental measurement), its marker(s), the measurement technology and the goals of the experiment or model, dictate how we can apply the measurement within our mechanistic model. We discuss three scenarios (A., B., and C.) and their impact on how we model the resulting measurement.
Fig 3.
Predicted Bid truncation dynamics of aEARM trained to different sized ordinal datasets.
Multiple Bayesian optimizations were run on the A.) abridged Extrinsic Apoptosis Reaction Model (aEARM) using different sized ordinal dataset to probe how dataset size influenced certainty of aEARM predictions. B.) Initiator caspase reporter (IC-RP) fluorescence time-course measurements (at 180s intervals) were measured (top left) as a proxy for truncated tBid (data from Albeck et al21). The plot shows the mean (dotted orange line) ± 1 standard deviation (shaded region) for each time point. The 95% credible region (top right) of posterior predictions (shaded region) for tBID concentration in aEARM, calibrated to fluorescence measurements of IC-RP and EC-RP (See also Fig C in S1 Text). The median prediction (solid-line) and ground truth (dotted line) tBID concentration trajectories are shown. In the next four rows (from top to bottom), Ordinal measurements of tBID (left) at every 1500, 300, 180 and 60s interval, respectively. The 95% credible region of predictions (shaded region), median prediction (solid line) and true (dotted line) tBID dynamics for aEARM calibrated to ordinal measurements of tBID and cPARP occurring at every 1500, 300, 180 and 60s timepoint are plotted in plots on the right. The plots for cPARP ordinal measurements and predictions are found in Fig A in S1 Text.
Fig 4.
Predicted Bid truncation dynamics of aEARM trained to nominal and ordinal datasets.
A.) Nominal cell death (x) vs survival (o) outcomes data for cells treated with 10ng/mL (orange) and 50ng/mL (grey) of TRAIL and with known relative values of DISC formation (x-axis). The 95% credible region (shaded region) of posterior predictions of tBID dynamics of aEARM calibrated to nominal data (right plot). The median prediction (solid-line) and true (dotted line) are also plotted. B.) Ordinal measurements for initiator caspase-DISC colocalization (IC-DISC) at 300s intervals (left plot). The 95% credible region (shaded region) of posterior predictions of tBID dynamics of aEARM calibrated to ordinal IC-DISC data (right plot), and C.) of aEARM calibrated to nominal and ordinal IC-DISC data. The median prediction (solid-line) and true (dotted line) were also plotted. The fit to IC-DISC data is shown in Fig G S1 Text.
Fig 5.
Predicted Bid truncation dynamics of aEARM trained to ordinal data using different measurement model parameterizations.
A.) and B.) The 95% credible region of posterior predictions (shaded region) of tBID dynamics for aEARM calibrated to ordinal measurements two fixed parameterizations for the measurement model (see Table C in S1 Text). The adjacent panels plot the measurement models predicted probability of class membership (x-axis) as a function of normalized tBID concentration (y-axis). C.) D.) and E.) The 95% credible region of posterior predictions (shaded region) of tBID dynamics of aEARM calibrated to ordinal measurements uniform, Cauchy (scale = 0.05) and Cauchy (scale = 0.005) prior distributions for the parameterizations of θj (the distance between offset βj and the preceding offset βj−1) for the measurement model, respectively. In each, the median prediction (solid line) and true (dotted line) tBID dynamics are also shown. The adjacent panels give the 95% credible region of posterior predictions of the probability of class membership (x-axis) as a function of normalized tBID concentration (y-axis). The left and their adjacent panels share the y-axis (normalized tBID concentration) Four accompanying plots show the prior (blue), posterior (orange) and true (dashed line) values of measurement model parameters.
Fig 6.
Measurement model predicts features of cell death vs. survival using aEARM calibrated to cell death datasets.
Normalized predicted values of the features used in the cell death vs. survival measurement model–the x-axis is the maximum Bid truncation rate, and the y-axis is the time at maximum Bid truncation rate (top row) or an unrelated non-apoptotic signal (middle row)–for corresponding to observed cell death (x) and survival (o) outcomes. These feature values are modeled by aEARM parameterized by 100 parameter vectors randomly drawn from the posterior; for each parameterization, 5 out of the total simulated population of 400 cells were plotted. The grey and orange curves, in these plots, are 0.05 contours for the estimated density of simulated cell populations produced for each of the 100 parameter vectors–grey and orange correspond to 50 and 10ng/ml TRAIL treatments, respectively. The measurement model predicts a probability of cell death vs survival based on simulated values of the above features. The lower right region of the plots in the top row. (i.e., early maximization of Bid truncation and higher maximal Bid truncation rates) is associated with higher probability of cell death. The shaded region is the 95% credible region of the posterior prediction of the line marking 50% probability of cell death or survival. The black and blue lines are the median predicted and true 50% probability lines, respectively. The bottom row plots the posterior distributions of the weight for each feature (i.e., the product of the slope term and feature coefficient encoded in the measurement model): maximum Bid truncation rate (green), time at maximum Bid truncation (orange) and unrelated non-apoptotic signal (blue). Plots in the left column are predictions of aEARM calibrated to the cell death vs. survival dataset. Plots right column were those of aEARM calibrated to the cell death vs survival + ordinal IC-DISC combined dataset.
Fig 7.
Posterior predictions of aEARM trained to published fractional cell death data.
A. Fractional cell death in WT (blue) and high- and low- expression of dominant negative FADD (green and orange respectively) in HeLa cells treated with 0 to 200ng/mL TRAIL. These data come from Wajant et al. 1998 [29]. The aEARM and accompanying measurement model were calibrated to these data. B. The posterior predictions of the Gaussian process modeled mean fractional cell death values for WT and high- and low- expression of dominant negative FADD in HeLa cell treated with 0 to 200ng/mL TRAIL. C. Posterior predictions of the Gaussian process modeled mean fractional cell death values for WT and BID overexpressed (TAT-Bid) HeLa cells treated with and without 100ng/mL TRAIL. Fractional cell death predictions for these experimental conditions, which were excluded from our training dataset, correspond to fractional cell death measurements by Orzechowska et al [30]. The 95% credible region of the posterior prediction (D.) of tBID dynamics in cells treated with 25g/mL TRAIL. (E.) Posterior distributions of the weight for each feature extracted from tBID dynamics.