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Universally Sloppy Parameter Sensitivities in Systems Biology
Ryan N. Gutenkunst, Joshua J. Waterfall, Fergal P. Casey, Kevin S. Brown, Christopher R. Myers, James P. Sethna
TL;DR
Systems biology models often contain many difficult-to-measure parameters, raising the question of whether collective fits can support reliable predictions despite parameter uncertainty. By analyzing 17 literature models and testing implications in a signaling model, the paper finds universal sloppy sensitivity spectra, persistent parameter uncertainty after ideal collective fits, and demanding precision and completeness requirements for direct measurements.
Problem
Systems biology models contain many free biochemical parameters that are difficult to measure or constrain collectively, although useful predictions may remain possible despite large parameter uncertainties.
Method
The authors analyze Hessian-based sensitivity spectra across 17 systems biology models and test collective-fitting and direct-measurement consequences using ideal time-series data and a signaling model.
Results
Every studied model exhibits a sloppy spectrum spanning many decades, while collective fits leave parameters broadly uncertain and direct measurements require tight, complete constraints for useful predictions.
Takeaways & Limitations
Modelers should evaluate prediction uncertainty rigorously, favor collective parameter fits, and focus on prediction quality rather than obtaining tightly known individual parameters.
Takeaways & Limitations
Predictions can remain useful even when models are wrong in significant details, so successful fits and predictions do not ensure structural correctness.
Abstract
from arXiv · showhide
Quantitative computational models play an increasingly important role in modern biology. Such models typically involve many free parameters, and assigning their values is often a substantial obstacle to model development. Directly measuring \emph{in vivo} biochemical parameters is difficult, and collectively fitting them to other data often yields large parameter uncertainties. Nevertheless, in earlier work we showed in a growth-factor-signaling model that collective fitting could yield well-constrained predictions, even when it left individual parameters very poorly constrained. We also showed that the model had a `sloppy' spectrum of parameter sensitivities, with eigenvalues roughly evenly distributed over many decades. Here we use a collection of models from the literature to test whether such sloppy spectra are common in systems biology. Strikingly, we find that every model we examine has a sloppy spectrum of sensitivities. We also test several consequences of this sloppiness for building predictive models. In particular, sloppiness suggests that collective fits to even large amounts of ideal time-series data will often leave many parameters poorly constrained. Tests over our model collection are consistent with this suggestion. This difficulty with collective fits may seem to argue for direct parameter measurements, but sloppiness also implies that such measurements must be formidably precise and complete to usefully constrain many model predictions. We confirm this implication in our signaling model. Our results suggest that sloppy sensitivity spectra are universal in systems biology models. The prevalence of sloppiness highlights the power of collective fits and suggests that modelers should focus on predictions rather than on parameters.
I. RESULTS
The study analyzes sensitivity geometry across 17 systems biology models using Hessian-based ellipsoidal approximations. Every examined model has eigenvalues spanning many decades, with skewed principal axes and no clear separation between important and unimportant parameter combinations.
- Model collection and sensitivity analysis: 17 systems biology models from diverse biological systems were analyzed using a behavior-change measure based on molecular-species time courses.The collection was drawn primarily from BioModels and included models of circadian rhythm, metabolism, and signaling.
- Model collection and sensitivity analysis: Log-parameter derivatives and the Hessian of the behavior-change measure approximate constant-deviation surfaces as Np-dimensional ellipsoids.Ellipsoid widths scale as the inverse square root of Hessian eigenvalues, defining stiff and sloppy axes.
- Universal sloppy spectra: Every model’s eigenvalues span many decades, with all but one spanning more than 10^6 and sloppy axes generally over 1000 times longer than stiff axes.The spectra are normalized by their largest eigenvalue.
- Universal sloppy spectra: Eigenvalues are approximately evenly spaced in logarithm, so there is no well-defined cutoff between important and unimportant parameter combinations.This pattern is the defining spectral feature of the sloppy models examined.
- Parameter-axis alignment and model reduction: Monte Carlo principal-component analysis in the Brown et al. model indicates that Hessian-based sloppiness reflects the full nonlinear χ2 surface.The Hessian itself is only a local quadratic approximation.
- Parameter-axis alignment and model reduction: Few principal axes align with individual parameter directions, and stiff eigenvectors often contain substantial components from many parameters.The resulting eigenvectors may not provide a useful guide for model reduction, although direct parameter-correlation analyses can reveal clusters.
B. Consequences of Sloppiness
Sloppiness allows collective fits to leave individual parameters weakly constrained while preserving tight predictions, but it makes direct measurements demanding. Prediction quality depends on constraining stiff parameter combinations and on complete measurements.
- Collective fits: Collective fits can leave individual parameters poorly determined because parameter ensembles expand along insensitive sloppy directions.This prediction is expected even when fitting complete, ideal time-series data that the model fits perfectly.
- Collective fits: Large parameter-set ensembles can nevertheless yield small prediction uncertainties because fitting constrains stiff directions and model behavior varies little within the ensemble.The distinction is between uncertainty in individual parameters and uncertainty in predicted behavior.
- Direct parameter measurements: Direct measurements of all parameters to equal precision must reach the stiffest model direction to produce tight predictions.Because measurement uncertainty is generally uncorrelated with stiff and sloppy directions, the required precision can be high.
- Direct parameter measurements: A single parameter measured less precisely can produce large prediction uncertainty when its uncertainty is not aligned with a sloppy direction.The resulting parameter ensemble may cross many constant-behavior contours.
1. Parameter Values from Collective Fits
Idealized collective fits across sloppy models can leave many individual parameters poorly constrained, even when the data greatly outnumber the parameters.
- Testing collective fits: Seventeen models were tested to determine whether sloppiness prevents collective fits from precisely identifying individual parameters.The analysis used an idealized fitting procedure to test this prediction.
- Testing collective fits: 100 times as many time-series data points as parameters, each with 10% uncertainty, still left many parameters with very large confidence intervals.The relative interval size Σ compares the upper and lower 95% confidence-interval extremes minus one.
- Results: Many models retained parameters with greater than 100% uncertainty after fitting the idealized time-series data.A 95% interval spanning 1 to 100 corresponds to Σ ≈100.
- Interpretation: Correlated parameter combinations can fit data across wide ranges, reflecting the underlying sloppy eigenvalue spectrum.This pattern was also found in acceptable parameter sets from an earlier ligand-binding study.
2. Predictions from Direct Parameter Measurements
Direct measurements can constrain predictions only when they are both exceptionally precise and complete, whereas collective fitting can produce tight predictions despite loose individual parameter constraints.
- Experimental comparison: The signaling-model test compared direct measurements centered on original best-fit values with the model’s prior collective fit.The model contains 48 parameters and was originally fit to 14 time-series cell-biology experiments.
- Collective fitting: The collective fit produced a tight ERK-activity prediction despite leaving every parameter uncertain by more than a factor of 50.The prediction concerns EGF stimulation with PI3K inhibition in PC12 cells.
- Direct measurements: All 24 parameters involved in the prediction had to be measured within ±25% to match the collective fit’s tight prediction.The PI3K and C3G branches were irrelevant for this prediction, leaving 24 relevant parameters.
- Direct measurements: Leaving one of the 24 involved parameters poorly measured produced a very large, likely useless prediction uncertainty.The example left the Raf1-to-Mek activation rate constant bounded only across a total range of 1000.
- Implication: Direct measurements therefore need to be precise and complete, while collective fitting yielded tight predictions from a modest number of experiments.These results are attributed to the model’s sloppiness.
II. DISCUSSION
Across all 17 examined systems-biology models, parameter-sensitivity spectra were sloppy, with eigenvalues spanning many decades and generally not aligning with individual parameters. This sloppiness complicates parameter estimation but supports collective fitting and prediction-focused modeling with rigorous uncertainty analysis.
- Parameter estimation: Sloppiness helps explain why collective fits can leave individual parameter estimates poorly constrained even when the data are comprehensive.The authors relate this difficulty to the broad sensitivity spectrum and its geometry in parameter space.
- Predictive modeling: Collective parameter fits remain useful because uncertainty analysis can distinguish reliable predictions from those that depend strongly on uncertain parameters.Predictions based solely on a best-fit parameter set are considered of little value without uncertainty bounds.
- Experimental strategy: Experiments should test predictions with tight uncertainty estimates or use optimal experimental design to reduce uncertainty in crucial model predictions.The appropriate choice depends on whether the goal is testing model assumptions or improving prediction certainty.
- Model exchange: Because models and fitted parameters depend on their informing data, exchanging models requires standards such as SBML to support automated data exchange as well.The authors identify data exchange as a practical requirement for their modeling approach.
- Universality of sloppiness: Every one of the 17 studied models exhibited a sloppy spectrum, with sensitivity eigenvalues spanning many decades roughly evenly and not aligning with single parameters.The same pattern was observed across the model collection and described as apparently universal.
- Predictive modeling: Models should be judged primarily by the quality of their predictions rather than by the precision of individual parameter values.Rigorous sensitivity analysis can identify predictions that remain trustworthy despite poorly known parameters.
A. Hessian Computations
The Hessian is evaluated at θ∗, where second-derivative terms vanish, and its first derivatives can be obtained by integrating sensitivity equations rather than finite differences.
- At θ∗, the second-derivative terms in the Hessian vanish.
- Sensitivity equations provide the first derivatives needed for Equation 3.This avoids finite-difference derivatives, which are troublesome in sloppy systems.
- The projection of the uncertainty ellipsoid onto parameter axis i is proportional to q_i,i.
- The intersection of the uncertainty ellipsoid with parameter axis i is proportional to 1/Hχ2_i,i.The projection and intersection share the same proportionality constant.
B. Parameter Uncertainties
The Hessian is rescaled to represent a collective fit with a specified number and precision of data points, then used to estimate parameter uncertainties and confidence intervals.
- Multiplying Hχ2 by Nd/f^2 rescales it to Nd data points with uncertainty fraction f of each species’ maximal value.
- In the quadratic approximation, the rescaled Hessian determines each parameter’s one-standard-deviation uncertainty in log θi.
- The relative size of parameter θi’s 95% confidence interval is Σi = exp(4σlog θi) −1.
C. Prediction Uncertainties
Prediction uncertainties are estimated by propagating randomly sampled parameter sets through the Erk time-course calculation.
- 1000 parameter sets consistent with stated parameter uncertainties were randomly generated for prediction-uncertainty calculations.
- Each sampled parameter set produced an Erk time course, and shaded regions contain the central 95% of trajectories at each timepoint.
D. Software
The computations used SloppyCell, with reproducibility materials supplied as SBML files and scripts, alongside supplementary texts and a dataset containing χ2-Hessians.
- All computations were performed in the open-source modeling environment SloppyCell version 0.81.
- Dataset S1 contains SBML files and SloppyCell scripts for reproducing the presented calculations.
- Supplementary texts cover stiffest eigenvectors, poorly determined parameters, fragile predictions, model rescaling, and binding-study eigenvalue analysis.
- Dataset S1 includes SBML files, SloppyCell scripts, and χ2-Hessians.
A. Accession Numbers
The paper identifies specific BioModels database accession numbers and acknowledges funding support.
- BioModels database: The discussed BioModels database entries include BIOMD0000000005, BIOMD0000000003, BIOMD0000000035, and other listed accession numbers.The passage lists thirteen BioModels accessions labeled across panels (a), (c)–(e), and (h)–(q).
- Funding: RNG was supported by an NIH Molecular Biophysics Training Grant, T32-GM-08267.