Analysis of sloppiness in model simulations: Unveiling parameter uncertainty when mathematical models are fitted to data

Figures

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  Fig. 1.. Synthetic data generated using a measurement error of ε = 25% versus noiseless model prediction based on Eq. 1 using reference parameter values kcat = 100 min−1, [ET] = 5 μM, and KM = 146.7 μM. Dataset A is obtained at a relatively high [S], and dataset B is obtained at a relatively low [S]. Neither dataset can inform the full rate of reaction υ across the range of [S] (see also Figures’ Supplementary Legends).

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  Fig. 2.. Prior and posterior distributions for the model parameters together with the stiffest eigenparameters from scenarios 2 and 3 (Table 1) compared to their associated sets of best-fit values (MLE) and reference values (see also insets). (A) Scenario 2 parameters kcat and [ET] together with stiffest eigenparameters θ^1=KM (matrix P) and θ^1=kcat[ET] (matrices H or L and G). (B) Scenario 3 parameter [ET] together with stiffest eigenparameters θ^1=kcat (matrix G), θ^1=KM (matrix P), and θ^1=kcat[ET]/KM (matrices H or L). Parameter combinations kcat[ET] and kcat[ET]/KM are well constrained by the data in scenarios 2 and 3, respectively. Parameter KM is well constrained by the prior (posterior and prior overlapping) in both scenarios, and parameter kcat is well constrained by the data relative to the prior in scenario 3. The best-fit values for parameter KM lie far away from its reference values in both scenarios. As parameter combination kcat[ET]/KM is well constrained by the data in (B), the posterior distribution for parameter kcat is left-shifted from the reference to compensate for parameter [ET] that is right-shifted from the reference in fig. S3A (see also Figures’ Supplementary…

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  Fig. 3.. Ecosystem interaction network, consisting of pasture (V), rabbits (N), foxes (P), and threatened species (M). (A) Ecosystem network in which arrows indicate the direction of the energy transfer associated with the species interaction. (B) Synthetic time-series data for ecological abundance with measurement error of ε = 25% together with noiseless model prediction using reference parameter values (table S2), model predictions using two sets of best-fit parameter values (MLEs), and model ensemble predictions using all plausible parameter values (fig. S5). The ecosystem network model fits the synthetic time-series data, with the model ensemble propagating parameter uncertainty into species abundance predictions (see also Figures’ Supplementary Legends).

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  Fig. 4.. Prior and posterior distributions for eigenparameters θ^1, θ^2, and θ^3 (Table 2) compared to their reference values and associated sets of best-fit values (MLEs) (see also insets). Posterior distributions for all eigenparameters and their associated MLEs closely match the true values (see also Figures’ Supplementary Legends).

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  Fig. 5.. Bivariate scatterplots of the prior and posterior distributions (shaded regions) for the first stiffest eigenparameter obtained from the Bayesian methods (Table 2) compared to their reference parameter values and sets of best-fit values (MLEs). (A) θ^1 and θ^3 from matrices P and G, respectively, and (B) θ^1 from matrix G. Many samples of the posterior distribution yield similar values of the log-likelihood function (see also Figures’ Supplementary Legends).

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  Fig. 6.. BR model fit to synthetic time-series data together with the identified stiffest eigenparameter. (A) Synthetic AP data with measurement error of σ = 2 mV and time resolution of 1 ms (1 kHz) together with noiseless model prediction using reference parameter values (table S4), model predictions using the set of best-fit parameter values (MLE), and model ensemble predictions using all plausible parameter values. (B) Prior and posterior distributions for the stiffest eigenparameter θ^1=AK10.9Ax10.4/gs obtained from all matrices compared to their associated set of best-fit values and reference values (see also inset). Both MLE-based and Bayesian inference–based model predictions overlap the true AP dynamics and synthetic data. All methods lead to the same stiffest eigenparameter. The posterior distribution for the stiffest eigenparameter and their associated best-fit value (MLE) closely match the true eigenparameter value (see also Figures’ Supplementary Legends).

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  Fig. 7.. Effect of varying parameter values that are part of the stiffest eigenparameter θ^1=AK10.9Ax10.4/gs on the BR model prediction. (A) Predictions obtained with four sets of parameter values that change the value of θ^1 (sets A1 to A4; fig. S14). (B) Predictions obtained with four sets of parameter values that keep approximately constant the value of θ^1 (sets B1 to B4; fig. S14). Similar predictions are obtained with change in the parameter values when the value of θ^1 is kept approximately constant, while in the opposite case predictions strongly differ from those obtained with the initial parameter set (fig. S10) (see also Figures’ Supplementary Legends).

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  Fig. 8.. Bivariate scatterplot for the prior and posterior distributions (shaded regions) for combination of parameters AK10.9Ax10.4 and gs compared to sets of reference parameter values and best-fit values (MLE). Many samples of the posterior distribution yield similar values of the log-likelihood function with AK10.9Ax10.4∝gs (see also Figures’ Supplementary Legends).