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Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood
Open accessAbstract
MOTIVATION: Mathematical description of biological reaction networks by differential equations leads to large models whose parameters are calibrated in order to optimally explain experimental data. Often only parts of the model can be observed directly. Given a model that sufficiently describes the measured data, it is important to infer how well model parameters are determined by the amount and quality of experimental data. This knowledge is essential for further investigation of model predictions. For this reason a major topic in modeling is identifiability analysis. RESULTS: We suggest an approach that exploits the profile likelihood. It enables to detect structural non-identifiabilities, which manifest in functionally related model parameters. Furthermore, practical non-identifiabilities are detected, that might arise due to limited amount and quality of experimental data. Last but not least confidence intervals can be derived. The results are easy to interpret and can be used for experimental planning and for model reduction. AVAILABILITY: An implementation is freely available for MATLAB and the PottersWheel modeling toolbox at http://web.me.com/andreas.raue/profile/software.html. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.
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TL;DR
The paper presents a profile-likelihood method for assessing structural and practical identifiability in partially observed dynamical models and for deriving likelihood-based confidence intervals. Applied to a JAK-STAT signaling model, it identifies four structurally non-identifiable parameters and one practically non-identifiable parameter, and shows how additional measurements can resolve these ambiguities.
Key findings
- Profile likelihood distinguishes structural non-identifiability, seen as a flat profile, from practical non-identifiability, seen as a profile with a minimum that fails to cross the confidence threshold in one or both directions; identifiable parameters cross the threshold on both sides.
- In the JAK-STAT application, parameters p2, x1(0), s1, and s2 were structurally non-identifiable, with functional relations p2·x1(0)=const, s1·x1(0)=const, and s2·x1(0)=const; p3 was practically non-identifiable for increasing values.
- Table 1 reports infinite likelihood-based intervals for the structurally non-identifiable parameters, whereas Hessian-based asymptotic intervals were finite but misleadingly narrow, e.g. p2: −33 to +31, x1(0): −31 to +32, s1: −32 to +31, s2: −32 to +31; for p3, the profile-likelihood interval was −1.14 to +∞ versus −1.14 to +0.15 from the Hessian.
- Adding hypothetical measurements of x1(0)=200±20 nM and x3/(x2+x3)=0.90±0.05 at t=20 min made p2, x1(0), s1, and s2 structurally identifiable and p3 practically identifiable, while p2 remained practically non-identifiable. Profile-likelihood computation took 54±18 s per parameter on a 1 GHz CPU with 2 GB RAM.
Methods
- The approach uses an ODE model with weighted least-squares objective χ², computes the profile likelihood for each parameter by re-optimizing all other parameters at fixed values of the target parameter, and derives likelihood-based confidence intervals from χ² thresholds.
- The application is a JAK-STAT signaling model inspired by Swameye et al., calibrated to experimental data; observables are phosphorylated STAT in cytoplasm (y1) and total STAT in cytoplasm (y2) measured by quantitative western blotting, with cytoplasm and nucleus volumes of 1400 µm³ and 450 µm³.
- Parameters calibrated include p1–p4, s1, s2, and x1(0), with x2(0)–x4(0) set to zero; the model was implemented and fitted with PottersWheel, and experimental planning and model reduction were explored via profile-likelihood trajectories, Monte Carlo simulations, and likelihood-ratio thresholds.
Limitations
- The authors state that structural non-identifiability cannot be resolved by refining existing measurements or reducing noise; it requires a qualitatively new measurement that changes the observation mapping, such as observing additional species. Practical non-identifiability can be resolved by increasing the amount or quality of data and/or selecting better measurement time points.
- The authors note that recovering functional relations unambiguously from parameter changes along the profile likelihood is only possible when the corresponding manifold is one-dimensional; higher-dimensional manifolds require further analysis, such as the approach of Hengl et al.
- Evident from the text: the method is demonstrated on a single JAK-STAT model and dataset, with no broad empirical comparison across many model classes; the approach also assumes a sufficiently adequate model and normally distributed measurement noise.
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35 Own interpretation · 14 Own result · 6 No full text of the sources · 6 No basis given · 1 Partly supported · 1 Sources not found · 1 Not checked yet · accuracy not yet measured
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“The largest discrepancy occurs for practically non-identifiable parameters, where asymptotic confidence intervals are significantly smaller than likelihood-based confidence intervals.”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > Structural non-identifiability
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“The variability of the predicted model observables y along the profile likelihood of p 3 remains consistent with the measurement errors, as shown in Figure 6.”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > Structural non-identifiability
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“The resulting plots of profile likelihood versus parameter reveal four structurally non-identifiable parameters p 2 , x 1 (0), s 1 , s 2 by their flat profile likelihood, see Figure 3, upper panel.”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > 5 APPLICATION
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“For identifiable parameters the discrepancies are small.”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > Structural non-identifiability
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“The functional relations h connecting these structurally non-identifiable parameters can be recovered from the changes of the remaining parameters, while calculating the profile likelihood of the structurally non-identifiable parameters, as shown in Figure 4.”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > Structural non-identifiability
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“The recalculated profile likelihood reveals, that parameters p 2 , x 1 (0), s 1 , s 2 become structurally identifiable (Fig. 3, lower panel), by measuring only one of them.”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > Experimental planning
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“The profile likelihood reveals that parameter p 3 is practically non-identifiable for increasing parameter values.”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > Structural non-identifiability
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“Nevertheless, a lower confidence bound can be derived.”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > Structural non-identifiability
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“In this case, asymptotic approximation leads to large but nevertheless finite confidence intervals for the structurally non-identifiable parameters.”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > Structural non-identifiability
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“All internal model states can only be identified up to a common factor.”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > Structural non-identifiability
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In the paper
“An approach to detect structural non-identifiability by the corresponding functional relations was introduced by Hengl et al. (2007).It is able to detect flatness of likelihood for arbitrary models, but it is not intended to detect practical non-identifiability.”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > 3 EXISTING METHODS
What the sources say
- Partly supportedcites Hengl et al., 2007 sourceExperimental — accuracy not yet measured
The Approach and Conclusion describe detecting structural non-identifiability through functionally related parameters in nonlinear models. But the Introduction explicitly lists practical identifiability analysis, accounting for data quality, as an aim of the method, contrary to the claim that it is not intended for that purpose. The passages do not specifically describe the method as detecting likelihood flatness. (LLM, conf=0.90)
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“Spots of largest variability suggest where and when a measurement of a species most efficiently determines theses parameters: x 1 at times t = 0 or t > 50; x 2 to x 4 at times 5 < t < 30.”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > Experimental planning
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“Calculating the profile likelihood takes 54 ± 18 s per parameter, using an implementation of the approach embedded in the Potters-Wheel fitting toolbox (1 GHz CPU, 2 GB RAM), which is described in the Supplementary Material.”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > 5 APPLICATION
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“Parameter p 3 becomes practically identifiable, only previously structurally non-identifiable parameter p 2 remains practically non-identifiable.”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > Experimental planning
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“The trajectories of the internal model states x along the profile likelihood of the practically non-identifiable parameter p 3 shown in Figure 6 comprises large antipodal variability of species x 2 and x 3 , revealing that the experimental setup is inappropriate to estimate this parameter with confidence.”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > Experimental planning
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“h 1 (θ sub ) = p 2 •x 1 (0)-const = 0 h 2 (θ sub ) = s 1 •x 1 (0)-const = 0 ( 1 1 ) h 3 (θ sub ) = s 2 •x 1 (0)-const = 0”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > Structural non-identifiability
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In the paper
“The approach was applied to a model of the JAK-STAT signaling pathway.Non-identifiable parameters were detected, revealing limitations in the experimental setup.Additional measurements that efficiently improve parameter identification were suggested and validated.”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > 6 DISCUSSION
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“They are exact if the observables y depend linearly on θ.”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > 2.2.1 Asymptotic confidence intervals
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“Unfortunately, these methods become rapidly infeasible with increasing model size (Margaria et al., 2001;White et al., 2001).Practical non-identifiability cannot be detected, since experimental data are disregarded.”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > 3 EXISTING METHODS
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“Even identifiable parameters can only be determined within confidence intervals, which contain the true value of the parameter with a desired probability (Lehmann and Leo, 1983).”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > 1 INTRODUCTION
What the sources say
- Not matchedcites Lehmann and Leo, 1983
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In the paper
“These methods are appropriate if functional relations h between the parameters emerging from structural non-identifiability are linear.This is often not the case for reaction networks modeled by ODE, because observables depend non-linearly on the parameters.Practical non-identifiability cannot be detected, because a quadratic approximation is not able to explain increasing but limited behavior of”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > 3 EXISTING METHODS
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“However, even for the simplest reaction network the observables y depend nonlinearly on θ and the amount and quality of experimental data is often insufficient.Therefore, asymptotic confidence intervals might not be appropriate (Joshi et al., 2006).”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > 2.2.1 Asymptotic confidence intervals
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- Not checked — no full text of the sourcecites Joshi et al., 2006 source
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“Likelihood-based confidence intervals are considered superior to asymptotic confidence intervals for finite samples (Neale and Miller, 1997).”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > 2.2.2 Finite sample confidence intervals
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- Not checked — no full text of the sourcecites Neale and Miller, 1997 source
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In the paper
“Along a practical non-identifiability, the observable y change only negligibly remaining compliant with the given measurement accuracy.Nevertheless, model behavior in terms of internal states x might vary strongly.”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > )]:
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“This arises frequently if amount and quality of experimental data is insufficient and manifests in a confidence interval that is infinite.”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > 2.3.2 Practical non-identifiability
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“The points of passover represent likelihoodbased confidence intervals as defined in Equation ( 8) (Royston, 2007).”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > 4 APPROACH
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- Not checked — no full text of the sourcecites Royston, 2007 source
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In the paper
“The results are easy to interpret”
- Also stated in: § 6 DISCUSSION · plain
“The approach results in easily interpretable plots of profile likelihood versus parameter.”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > Abstract
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- Also stated in: § 6 DISCUSSION · plain
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In the paper
“The profile likelihood of a practically non-identifiable parameter has a minimum, but is not excessing a threshold α for increasing and/or decreasing values of θ i (see Definition 1).”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > 4 APPROACH
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“This gives insight into a complex and high-dimensional parameter space.”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > 6 DISCUSSION
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“It can be automated, but an explicit advantage is that the output might be evaluated visually.”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > 6 DISCUSSION
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“The approach can be applied to any parameter estimation problem, where a likelihood or a similar objective criterion is available, e.g.partial differential equations (PDE) or stochastic differential equations (SDE).”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > 6 DISCUSSION
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In the paper
“The profile likelihood can be calculated for each parameter separately.Thereby it is possible to restrict the analysis to the parameters relevant for the biological question.Moreover, this allows to perfectly parallelize the approach, which is a major benefit for its scalability.”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > 6 DISCUSSION
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In the paper
“Working in logarithmic parameter space linearizes the functional relations given in Equation (11).”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > Structural non-identifiability
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In the paper
“The impact of new measurements can be evaluated by Monte Carlo simulations.”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > Experimental planning
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In the paper
“Thus, identifiability analysis ensures that the model complexity is tailored to the information content given by the experimental data.”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > 6 DISCUSSION
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The authors state this as an assumption.
The measurement noise is normally distributed.
“The measurement noise (t) is assumed to be normally distributed.”
The authors state this as an assumption.
An appropriate model M that sufficiently describes the available experimental data is assumed.
“Furthermore, an appropriate model M that sufficiently describes the available experimental data is assumed.”
The paper relies on this without support.
No approach for explicit testing of practical non-identifiability is available.
“Similar to a clear formal definition, an approach for explicit testing of practical non-identifiability is not available to our knowledge.”
The authors state this as an assumption.
The initial values of species x2 to x4 are equal to zero.
“The initial values of species x 2 to x 4 are assumed to be equal to zero.”
The authors state this as an assumption.
For the model-reduction argument, a parameter θi is practically non-identifiable for decreasing parameter value.
“Assume that a parameter θ i is practically non-indentifiable for decreasing parameter value.”
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“Such parameter estimations usually yield statistical information related to parameter precision (importantly, not accuracy) such as standard errors or correlation matrices, and sometimes information derived from more sophisticated approaches such as profile likelihood.39”
From Best Practices to Maximize the Use and Reuse of Quantitative and Systems Pharmacology Models: Recommendations From the United Kingdom Quantitative and Systems Pharmacology Network (2019) · cites this paper as [39]
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“Fortunately, recent literature is correcting this (e.g. [129–140]).”
From Reverse engineering and identification in systems biology: strategies, perspectives and challenges (2014) · cites this paper as [133]
The cited source says what the claim attributes to it.
In the citing paper
“Raue et al. (2009) referred to likelihood ratio C.I.s as finite sample C.I.s, to underscore the superiority of likelihood ratio C.I.s over Wald-type C.I.s.”
From rSalvador: An R Package for the Fluctuation Experiment (2017) · cites this paper as Raue et al. 2009
In the citing paper
“When two models are nested (Mð hPL Þ and Mð hopt Þ) their likelihood ratio is approximately distributed according to a v 2 p distribution.Here p refers to the number of degrees of freedom which is defined as the difference in the number of parameters (which in this case is 1).”
From Parameter uncertainty in biochemical models described by ordinary differential equations (2013) · cites this paper as [36]
In the citing paper
“In such cases, more reliable confidence intervals can be obtained using the Profile Likelihood method [36].”
From Parameter uncertainty in biochemical models described by ordinary differential equations (2013) · cites this paper as [36]
In the citing paper
“Techniques for this exist in both a frequentist [13,36,91] as well as a Bayesian setting [87,90].”
From Parameter uncertainty in biochemical models described by ordinary differential equations (2013) · cites this paper as [36]
In the citing paper
“Given the data, not all parameters are identifiable (Raue et al., 2009).”
From A bayesian approach to targeted experiment design (2012) · cites this paper as Raue et al., 2009
In the citing paper
“Methods for an approximate sampling of the parameter space, e.g. the Markov Chain Monte Carlo (MCMC) methods [11,12], and bootstrap based approaches [4,13] are numerically demanding and provide only rough approximations for ODE models. Therefore, it is difficult to control the coverage of the prediction confidence intervals for such approaches.”
From Likelihood based observability analysis and confidence intervals for predictions of dynamic models (2012) · cites this paper as [4]
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Percentages, sums and reported differences re-computed from the paper’s own numbers.
Each contradiction between two passages listed here was confirmed by an independent check.
Rated minor: it would not touch a result — a value no result rests on, a p-value that stays on the same side of the significance level, or a conflict that is clerical, resolved elsewhere in the paper, or outside its own findings.
In the paper
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > 5 APPLICATION
“Species concentrations are modeled in nanometer.”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > Structural non-identifiability
“time / min x 1 / nM”
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Self-citationMathematical modeling improves EC 50estimations from classical dose–response curves
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“We performed profile likelihood calculations [15] for a few of the model parameters that were of extra interest for the analyses.”
MethodInfluentialO’Donovan, S. D., Lenz, M., Vink, R., Roumans, N. J. T., de Kok, T. M. C. M., Mariman, E. C. M., Peeters, R., van Riel, N. A. W., van Baak, M. A., & Arts, I. C. W. (2019) · PLoS Computational Biology · doi:10.1371/journal.pcbi.1007400 · Cited by 22
“For an identifiable parameter the error measure C(p) would be expected to increase as the parameter value deviated from its optimum [31, 32].”
BackgroundStructural and practical identifiability analysis in bioengineering: a beginner’s guide
Wanika, L., Egan, J. R., Swaminathan, N., Duran‐Villalobos, C. A., Branke, J., Goldrick, S., & Chappell, M. J. (2024) · Journal of Biological Engineering · doi:10.1186/s13036-024-00410-x · Cited by 20
“In parameter estimation, the data are assessed to evaluate how probable it is that the data can be generated based on the estimated parameter value, this statistic is known as the likelihood (usually, the goal is to minimise the negative log likelihood [26]).”
MethodA tutorial on Bayesian inference for dynamical modeling of eye-movement control during reading
Engbert, R., & Rabe, M. M. (2024) · Journal of Mathematical Psychology · doi:10.1016/j.jmp.2024.102843 · Cited by 19
Pegu, R., Prakash, A., Borah, P., et al. (2024) · Chemosphere · doi:10.1016/j.chemosphere.2024.142782 · Cited by 19
Parametric sensitivity analysis for stochastic molecular systems using information theoretic metrics
Tsourtis, A., Pantazis, Y., Katsoulakis, M. A., & Harmandaris, V. (2015) · The Journal of Chemical Physics · doi:10.1063/1.4922924 · Cited by 20
Performance comparison of parameter estimation techniques for unidentifiable models
Graciano, J. E. A., Mendoza, D. F., & Roux, G. A. C. L. (2014) · Computers & Chemical Engineering · doi:10.1016/j.compchemeng.2014.01.009 · Cited by 19
Quantitative Measurement of Protein Relocalization in Live Cells
Bush, A., & Colman‐Lerner, A. (2013) · Biophysical Journal · doi:10.1016/j.bpj.2012.12.030 · Cited by 19
Bayesian model selection validates a biokinetic model for zirconium processing in humans
Schmidl, D., Hug, S., Li, W. B., Greiter, M. B., & Theis, F. J. (2012) · BMC Systems Biology · doi:10.1186/1752-0509-6-95 · Cited by 19
“Model likelihood Technical limitations, such as missing in vivo measurement techniques for all involved compartments as well as noisy data introduce model uncertainties to biological systems [27].”
BackgroundSBML-PET-MPI: a parallel parameter estimation tool for Systems Biology Markup Language based models
Zi, Z. (2011) · Bioinformatics · doi:10.1093/bioinformatics/btr038 · Cited by 19
Dobre, S., Bastogne, T., Profeta, C., Barberi‐Heyob, M., & Richard, A. (2012) · Automatica · doi:10.1016/j.automatica.2012.05.004 · Cited by 20
“This raises particular challenges such as the identification of highdimensional nonlinear dynamical systems and more precisely the analysis of their parameter identifiability [13,19,27].”
BackgroundcupSODA: A CUDA-Powered Simulator of Mass-Action Kinetics
Nobile, M. S., Besozzi, D., Cazzaniga, P., Mauri, G., & Pescini, D. (2013) · Lecture notes in computer science · doi:10.1007/978-3-642-39958-9_32 · Cited by 19
“These issues play a fundamental role in standard computational investigations of biological systems [1, 2], which usually rely on methods such as parameter sweep analysis [3], sensitivity analysis [4], structure and parameter identifiability [5], parameter estimation [6–8] and reverse engineering of model topologies […”
MethodHepatoDyn: A Dynamic Model of Hepatocyte Metabolism That Integrates 13C Isotopomer Data
Foguet, C., Marín, S., Selivanov, V. A., Fanchon, É., Lee, W. P., Guinovart, J. J., de Atauri, P., & Cascante, M. (2016) · PLoS Computational Biology · doi:10.1371/journal.pcbi.1004899 · Cited by 19
“A parameter is defined as identifiable if the confidence interval for its estimated value at a given significance level is finite [48, 49].”
BackgroundShedding light on grey noise in diabetes modelling
Mansell, E. J., Docherty, P. D., & Chase, J. G. (2016) · Biomedical Signal Processing and Control · doi:10.1016/j.bspc.2016.06.007 · Cited by 21
Accurate estimates of age at maturity from the growth trajectories of fishes and other ectotherms
Honsey, A. E., Staples, D. F., & Venturelli, P. (2016) · Ecological Applications · doi:10.1002/eap.1421 · Cited by 20
“We used profile likelihoods to generate likelihood intervals (Kalbfleisch and Sprott 1970, Taper and Lele 2011) for evaluating parameter estimability in application (Raue et al. 2009).”
Scheiblauer, J., Scheiner, S., Joksch, M., et al. (2018) · Journal of Theoretical Biology · doi:10.1016/j.jtbi.2018.05.016 · Cited by 19
Lee, D., Jayaraman, A., & Kwon, J. S. (2019) · IET Systems Biology · doi:10.1049/iet-syb.2018.5079 · Cited by 19
“For many intracellular signalling pathways, only a small subset of θ is identifiable from the experimental measurements [22, 25].”
BackgroundBlasi, T., Buettner, F., Strasser, M. K., Marr, C., & Theis, F. J. (2017) · Physical Biology · doi:10.1088/1478-3975/aa609a · Cited by 18
“We evaluate if the parameters of both steady-state distributions are identifiable and therefore capable of describing the data by calculating their profile likelihoods (Raue et al 2009).”
MethodMiddendorf, T. R., & Aldrich, R. W. (2016) · The Journal of General Physiology · doi:10.1085/jgp.201611703 · Cited by 19
Hoffmann, H., Thiede, C., Glauche, I., Bornhaeuser, M., & Roeder, I. (2020) · Journal of The Royal Society Interface · doi:10.1098/rsif.2020.0091 · Cited by 17
HCV Spread Kinetics Reveal Varying Contributions of Transmission Modes to Infection Dynamics
Durso-Cain, K., Kumberger, P., Schälte, Y., Fink, T., Dahari, H., Hasenauer, J., Uprichard, S. L., & Graw, F. (2021) · Viruses · doi:10.3390/v13071308 · Cited by 18
“The 95% confidence intervals of estimates were determined using the profile likelihood approach [24].”
MethodSimpson, M. J., Baker, R. E., Buenzli, P. R., Nicholson, R., & Maclaren, O. J. (2022) · Journal of Theoretical Biology · doi:10.1016/j.jtbi.2022.111201 · Cited by 17
From Fitting the Average to Fitting the Individual: A Cautionary Tale for Mathematical Modelers
Luo, M. C., Nikolopoulou, E., & Gevertz, J. L. (2022) · Frontiers in Oncology · doi:10.3389/fonc.2022.793908 · Cited by 18
“The profile likelihood PL for the parameter q is defined in (47) as:”
BackgroundParameter estimation in behavioral epidemic models with endogenous societal risk-response
Osi, A., & Ghaffarzadegan, N. (2024) · PLoS Computational Biology · doi:10.1371/journal.pcbi.1011992 · Cited by 19
“The uncertainties and biases accompanying parameter identifiability escalate as both the number of jointly estimated parameters and the model’s complexity increase [42,45].”
BackgroundA model-based strategy to investigate the role of microRNA regulation in cancer signalling networks
Nikolov, S. G., Vera, J., Schmitz, U., & Wolkenhauer, O. (2010) · Theory in Biosciences · doi:10.1007/s12064-010-0109-5 · Cited by 18
“However, the reduced ability to produce enough reliable quantitative data, together with some intrinsic mathematical problems of this approach (like identifiability, Raue et al. 2009; or sloppiness, Gutenkunst et al. 2007) turns parameter estimation in signalling networks into complex task.”
BackgroundModel calibration and uncertainty analysis in signaling networks
Heinemann, T., & Raue, A. (2016) · Current Opinion in Biotechnology · doi:10.1016/j.copbio.2016.04.004 · Cited by 17
“For signaling models and non-linear models in general, asymptotic confidence intervals based on the Fischer information matrix are not reliable [29].”
BackgroundMethodInfluentialSelf-citationKurzhunov, D., Borowiak, R., Reisert, M., et al. (2017) · NeuroImage · doi:10.1016/j.neuroimage.2017.05.029 · Cited by 18
“11 To calculate CMRO2 values, the method of PL analysis (Kurzhunov et al., 2016; Raue et al., 2009) was used which was implemented in a MATLAB-based Data2Dynamics framework (Raue et al.”
BackgroundMethodInfluentialPyCoTools: a Python toolbox for COPASI
Welsh, C., Fullard, N., Proctor, C. J., Guimera, A. M., Isfort, R. J., Bascom, C. C., Tasseff, R. A., Przyborski, S. A., & Shanley, D. P. (2018) · Bioinformatics · doi:10.1093/bioinformatics/bty409 · Cited by 17
“Finally, if the profile exceeds this threshold in both directions the parameter is identifiable and the parameter values at which the profile exceeds the threshold are the upper and lower confidence boundaries for the parameter (Raue et al., 2009).”
BackgroundMethodInfluentialRemli, M. A., Mohamad, M. S., Deris, S., Samah, A. A., Omatu, S., & Corchado, J. M. (2018) · Expert Systems with Applications · doi:10.1016/j.eswa.2018.09.020 · Cited by 18
A Comparison of Methods for Quantifying Prediction Uncertainty in Systems Biology
Villaverde, A. F., Raimúndez, E., Hasenauer, J., & Banga, J. R. (2019) · IFAC-PapersOnLine · doi:10.1016/j.ifacol.2019.12.234 · Cited by 16
“To facilitate a statistical interpretability, we consider the same threshold as used in profile calculation (Raue et al., 2009).”
MethodLizarralde-Bejarano, D. P., Rojas-Díaz, D., Arboleda-Sánchez, S., & Puerta-Yepes, M. E. (2020) · PLoS ONE · doi:10.1371/journal.pone.0229668 · Cited by 18
A kinetic analysis of mouse rod and cone photoreceptor responses
Reingruber, J., Ingram, N. T., Griffis, K. G., & Fain, G. (2020) · The Journal of Physiology · doi:10.1113/jp279524 · Cited by 18
“We verified that the model and the fitting parameters are identifiable (Raue et al. 2009).”
MethodDifferential equations in data analysis
Dattner, I. (2020) · Wiley Interdisciplinary Reviews Computational Statistics · doi:10.1002/wics.1534 · Cited by 20
“Another review is that of Kreutz and Timmer (2009), who focused on experimental designs aimed at improving parameter estimation of dynamic models.”
BackgroundMethodParameter estimation and uncertainty quantification using information geometry
Sharp, J. A., Browning, A. P., Burrage, K., & Simpson, M. J. (2022) · Journal of The Royal Society Interface · doi:10.1098/rsif.2021.0940 · Cited by 18
“In this work we focus on the most commonly employed additive noise model [5,11,18,19,27].”
BackgroundMethodMurphy, R. J., Maclaren, O. J., Calabrese, A. R., Thomas, P. B., Warne, D. J., Williams, E. D., & Simpson, M. J. (2022) · Journal of The Royal Society Interface · doi:10.1098/rsif.2022.0560 · Cited by 16
Ryser, M. D., Gulati, R., Eisenberg, M. C., et al. (2018) · American Journal of Epidemiology · doi:10.1093/aje/kwy214 · Cited by 18
Polansky, L., Newman, K. B., & Mitchell, L. (2020) · Biometrics · doi:10.1111/biom.13267 · Cited by 13
“…under which parameter identifiability is ensured theoretically, data-specific features can result in practically nonidentifiable, or nearly redundant, parameters, sometimes identified by flat profile likelihoods and infinitely large confidence intervals in a frequentist context (Raue et al., 2009).”
BackgroundModel selection for parameter identifiability problem in Bayesian inference of building energy model
Yi, D. H., & Park, C. S. (2021) · Energy and Buildings · doi:10.1016/j.enbuild.2021.111059 · Cited by 17
Data collection design for calibration of crop models using practical identifiability analysis
Coudron, W., Gobin, A., Boeckaert, C., Cuypere, T. D., Lootens, P., Pollet, S., Verheyen, K., Frenne, P. D., & Swaef, T. D. (2021) · Computers and Electronics in Agriculture · doi:10.1016/j.compag.2021.106457 · Cited by 17
Browning, A. P., & Simpson, M. J. (2023) · PLoS Computational Biology · doi:10.1371/journal.pcbi.1010844 · Cited by 17
Müller, D. F., Wibbing, D., Herwig, C., & Kager, J. (2023) · Computers & Chemical Engineering · doi:10.1016/j.compchemeng.2023.108203 · Cited by 17
Systems Biology of Monovalent Cation Homeostasis in Yeast
Ariño, J., Aydar, E., Drulhe, S., Ganser, D., Jorrín-Novo, J. V., Kahm, M., Krause, F., Petrezsélyová, S., Yenush, L., Zimmermannová, O., et al. (2014) · Advances in microbial physiology/Advances in Microbial Physiology · doi:10.1016/b978-0-12-800143-1.00001-4 · Cited by 17
Argyri, K. D., Dionysiou, D. D., Misichroni, F. D., & STAMATAKOS, G. S. (2016) · Biology Direct · doi:10.1186/s13062-016-0114-9 · Cited by 18
Cenci, S., & Saavedra, S. (2018) · Journal of The Royal Society Interface · doi:10.1098/rsif.2018.0695 · Cited by 17
Identifiability of Control-Oriented Glucose-Insulin Linear Models: Review and Analysis
Hoyos, J. D., Villa-Tamayo, M. F., Builes-Montaño, C., Ramírez-Rincón, A., Godoy, J. L., García-Tirado, J., & Rivadeneira, P. (2021) · IEEE Access · doi:10.1109/access.2021.3076405 · Cited by 13
“Structural identifiability refers to the theoretical possibility to find a unique solution (set of parameters θ ) in the identification problem, where confidence intervals are not infinite [85].”
BackgroundOkuneye, K., Bergman, D. R., Bloodworth, J. C., Pearson, A. T., Sweis, R. F., & Jackson, T. L. (2021) · Computational and Systems Oncology · doi:10.1002/cso2.1019 · Cited by 17
“The structural identifiability of themodel is analyzed using theMATLABpackageGenSSI (see [36, 39, 40] for complete details).”
BackgroundMethodKernel identification in continuous fluidized bed spray agglomeration from steady state data
Otto, E., Dürr, R., Strenzke, G., et al. (2021) · Advanced Powder Technology · doi:10.1016/j.apt.2021.05.028 · Cited by 17
Moleriu, R. D., Zaharie, D., Moatar-Moleriu, L., Lukinich-Gruia, A. T., Mic, A. A., & Mic, F. A. (2014) · Journal of Theoretical Biology · doi:10.1016/j.jtbi.2014.01.020 · Cited by 17
Vereecken, E., Roels, S., & Janßen, H. (2017) · Building and Environment · doi:10.1016/j.buildenv.2017.12.030 · Cited by 19
Guidelines for mechanistic modeling and analysis in cardiovascular research
Colebank, M. J., Oomen, P. J. A., Witzenburg, C. M., Grosberg, A., Beard, D. A., Husmeier, D., Olufsen, M. S., & Chesler, N. C. (2024) · American Journal of Physiology-Heart and Circulatory Physiology · doi:10.1152/ajpheart.00766.2023 · Cited by 18
Tivay, A., Jin, X., Lo, A. K.-Y., Scully, C. G., & Hahn, J. (2020) · Frontiers in Physiology · doi:10.3389/fphys.2020.00452 · Cited by 16
Automated model calibration with parallel MCMC: Applications for a cardiovascular system model
Argus, F. J., Zhao, D., Gamage, T. P. B., Nash, M. P., & Talou, G. D. M. (2022) · Frontiers in Physiology · doi:10.3389/fphys.2022.1018134 · Cited by 16
“Structural and practical identifiability (Raue et al., 2009; Miao et al., 2011) are focused on ensuring the identified parameter set is unique.”
BackgroundMethodResultInfluentialA meta‐analysis of mental rotation in the first years of life
Enge, A., Kapoor, S., Kieslinger, A.-S., & Skeide, M. A. (2023) · Developmental Science · doi:10.1111/desc.13381 · Cited by 18
High-resolution kinetic characterization of the RIG-I-signaling pathway and the antiviral response
Burkart, S. S., Schweinoch, D., Frankish, J., Sparn, C., Wüst, S., Urban, C., Merlo, M., Magalhães, V. G., Piras, A., Pichlmair, A., et al. (2023) · Life Science Alliance · doi:10.26508/lsa.202302059 · Cited by 18
“Structural and practical identi fi ability of the core and coupled model were assessed by calculating the pro fi le likelihood estimates for every optimized parameter as implemented in the data2-dynamics framework (Raue et al, 2009, 2015).”
Activator-blocker model of transcriptional regulation by pioneer-like factors
Riesle, A. J., Gao, M., Rosenblatt, M., et al. (2023) · Nature Communications · doi:10.1038/s41467-023-41507-z · Cited by 16
Self-citationDistilling identifiable and interpretable dynamic models from biological data
Massonis, G., Villaverde, A. F., & Banga, J. R. (2023) · PLoS Computational Biology · doi:10.1371/journal.pcbi.1011014 · Cited by 19
Lawrence, M. J., Wilson, B. M., Reid, G. K., Hawthorn, C., English, G., Black, M., Leadbeater, S., McKindsey, C. W., & Trudel, M. J. (2023) · Animal Biotelemetry · doi:10.1186/s40317-023-00351-0 · Cited by 15
“Models were visually inspected using profile likelihood plots [44, 48].”
MethodIterative design of training data to control intricate enzymatic reaction networks
van Sluijs, B., Zhou, T., Helwig, B., Baltussen, M. G., Nelissen, F. H. T., Heus, H. A., & Huck, W. T. S. (2024) · Nature Communications · doi:10.1038/s41467-024-45886-9 · Cited by 16
Browning, A. P., Taşcă, M., Falcó, C., & Baker, R. E. (2024) · Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences · doi:10.1098/rspa.2023.0911 · Cited by 15
Automated piecewise power-law modeling of biological systems
Machina, A., Ponosov, A., & Voit, E. O. (2010) · Journal of Biotechnology · doi:10.1016/j.jbiotec.2009.12.016 · Cited by 16
In silico model‐based inference: A contemporary approach for hypothesis testing in network biology
Klinke, D. J. (2014) · Biotechnology Progress · doi:10.1002/btpr.1982 · Cited by 16
Predicting pathogen-specific CD8 T cell immune responses from a modeling approach
Crauste, F., Terry, E., Mercier, I. L., et al. (2015) · Journal of Theoretical Biology · doi:10.1016/j.jtbi.2015.03.033 · Cited by 16
Mathematical modelling of arsenic transport, distribution and detoxification processes in yeast
Talemi, S. R., Jacobson, T., Garla, V. N., Navarrete, C., Wagner, A., Tamás, M. J., & Schaber, J. (2014) · Molecular Microbiology · doi:10.1111/mmi.12631 · Cited by 17
Županič, A., Méplan, C., Huguenin, G. V. B., et al. (2016) · RNA · doi:10.1261/rna.055749.115 · Cited by 16
“We ran the identifiability analysis for each model/experimental data combination in PottersWheel us-ing the profile likelihood method, as described by Raue et al. (2009).”
Pant, S. (2018) · Journal of The Royal Society Interface · doi:10.1098/rsif.2017.0871 · Cited by 19
Vidal, T., & Andrieu, B. (2019) · Annals of Botany · doi:10.1093/aob/mcz168 · Cited by 16
Zhou, W., Huang, R., Liu, K., & Zhang, W. (2020) · International Journal of Energy Research · doi:10.1002/er.5118 · Cited by 16
Zhan, C., Tse, C. K., Lai, Z., Chen, X., & Mo, M. (2020) · JMIR Public Health and Surveillance · doi:10.2196/18880 · Cited by 16
“Finally, the model has a large set of parameters, and the relative importance of each parameter is not identical [25,26].”
BackgroundImplications of dimensional analysis in bioreactor models: Parameter estimation and identifiability
Lira-Parada, P. A., Pettersen, E., Biegler, L. T., & Bar, N. (2021) · Chemical Engineering Journal · doi:10.1016/j.cej.2021.129220 · Cited by 16
Identifiability and physical interpretability of hybrid, gray-box models - a case study
Hotvedt, M., Grimstad, B., & Imsland, L. (2021) · IFAC-PapersOnLine · doi:10.1016/j.ifacol.2021.08.273 · Cited by 16
“(Brun et al., 2001; Raue et al., 2009; Deconinck and Roels, 2017; Brastein et al., 2019), but few have studied this for hybrid, graybox models.”
BackgroundCompactness regularization in the analysis of dipolar EPR spectroscopy data
Ibáñez, L. F., Jeschke, G., & Stoll, S. (2022) · Journal of Magnetic Resonance · doi:10.1016/j.jmr.2022.107218 · Cited by 16
Sparse-grids uncertainty quantification of part-scale additive manufacturing processes
Chiappetta, M., Piazzola, C., Carraturo, M., et al. (2023) · International Journal of Mechanical Sciences · doi:10.1016/j.ijmecsci.2023.108476 · Cited by 16
“This kind of graphical analysis of the negative log-likelihood function is widely employed in the context of parameter estimation for dynamical systems, where is known as “profile likelihood inspection”, see e.g. [71].”
MethodPant, S., & Lombardi, D. (2015) · Mathematical Biosciences · doi:10.1016/j.mbs.2015.08.005 · Cited by 17
Sips, F. L. P., Tiemann, C. A., Oosterveer, M. H., Groen, A. K., Hilbers, P. A. J., & van Riel, N. A. W. (2014) · PLoS Computational Biology · doi:10.1371/journal.pcbi.1003579 · Cited by 16
Analyzing and Constraining Signaling Networks: Parameter Estimation for the User
Florian, G., Fengos, G., Felizzi, F., & Iber, D. (2012) · Methods in molecular biology · doi:10.1007/978-1-61779-833-7_2 · Cited by 16
“13 gives misleading results whenever the above-stated assumptions are heavily violated, a fact which can be difficult to evaluate beforehand (7).”
BackgroundParameter redundancy and identifiability in hidden Markov models
Cole, D. J. (2019) · METRON · doi:10.1007/s40300-019-00156-3 · Cited by 16
Schug, H., Maner, J., Begnaud, F., et al. (2019) · Environmental Science & Technology · doi:10.1021/acs.est.9b04281 · Cited by 16
Ceballos‐Núñez, V., Müller, M. M., & Sierra, C. A. (2020) · Theoretical Ecology · doi:10.1007/s12080-020-00455-w · Cited by 16
Simulation and Analysis Methods for Stochastic Compartmental Epidemic Models
Ganyani, T., Faes, C., & Hens, N. (2020) · Annual Review of Statistics and Its Application · doi:10.1146/annurev-statistics-061120-034438 · Cited by 17
Framework for enhancing the estimation of model parameters for data with a high level of uncertainty
Libotte, G. B., dos Anjos, L., Almeida, R. C., Malta, S. M. C., & Silva, R. S. (2022) · Nonlinear Dynamics · doi:10.1007/s11071-021-07069-9 · Cited by 17
Marucci, L., Santini, S., di Bernardo, M., & di Bernardo, D. (2010) · Journal of Mathematical Biology · doi:10.1007/s00285-010-0350-z · Cited by 18
Bayesian Uncertainty Quantification for Differential EquationsPreprint
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Prague, M., Wittkop, L., Clairon, Q., Dutartre, D., Thiébaut, R., & Hejblum, B. P. (2020) · HAL · Cited by 16
A guide to state–space modeling of ecological time series
Auger‐Méthé, M., Newman, K. B., Cole, D. J., Empacher, F., Gryba, R., King, A. A., Barajas, V. L., Flemming, J. M., Nielsen, A., Petris, G., et al. (2021) · Ecological Monographs · doi:10.1002/ecm.1470 · Cited by 335
Santesarti, G., Marino, M., Viola, F., et al. (2025) · Journal of Non-Newtonian Fluid Mechanics · doi:10.1016/j.jnnfm.2025.105438 · Cited by 15
“Specifically, they found that fitting experimental data using standard non-linear optimization procedures can yield multiple sets of model parameters that provide nearly identical fits to the rheological data.”
Transient Storage Model Parameter Optimization Using the Simulated Annealing Method
Tsai, C. H., Rucker, D. F., Brooks, S. C., et al. (2022) · Water Resources Research · doi:10.1029/2022wr032018 · Cited by 15
“The local optima problem (non-identifiability) refers to the occurrence of a valley shape of similar values in the distribution surface of an objective function, where one can vary the parameters without changing the value of objective function as long as one stays in the valley (Raue et al., 2009).”
BackgroundBaker, E. A., Manenti, S., Reali, A., et al. (2023) · GEM - International Journal on Geomathematics · doi:10.1007/s13137-023-00219-8 · Cited by 15
“To assess the validity of this assumption, we carry out a so-called identifiability analysis (Piazzola et al, 2021; Guillaume et al., 2019; Raue et al., 2009), where we repeatedly perform the Bayesian inversion using different sets of synthetic data, i.”
MethodLuebben, G., González‐Parra, G., & Cervantes, B. (2023) · Mathematical Biosciences & Engineering · doi:10.3934/mbe.2023481 · Cited by 15
“However, an issue arises since the parameters would not be identifiable [141, 142, 164].”
BackgroundStatistical inference of the rates of cell proliferation and phenotypic switching in cancer
Gunnarsson, E. B., Foo, J., & Leder, K. (2022) · Journal of Theoretical Biology · doi:10.1016/j.jtbi.2023.111497 · Cited by 15
Identifiability investigation of within-host models of acute virus infection
Liyanage, Y. R., Heitzman-Breen, N., Tuncer, N., et al. (2024) · Mathematical Biosciences & Engineering · doi:10.3934/mbe.2024325 · Cited by 15
Ilan, Y. (2024) · Journal of Personalized Medicine · doi:10.3390/jpm15010010 · Cited by 18
“This method helps quantify the identifiability of individual parameters and model predictions but does not provide information about parameter relationships [90–92]. ii.”
How modulation of the tumor microenvironment drives cancer immune escape dynamics
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Integration based profile likelihood calculation for PDE constrained parameter estimation problems
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Quaglio, M., Fraga, E. S., & Galvanin, F. (2018) · Process Safety and Environmental Protection · doi:10.1016/j.cherd.2018.04.041 · Cited by 15
“It is known that the assumption of normality of the parameter estimate distributions is typically not satisfied if the model is nonlinear in the parameters, even in the case of exact model structure [1].”
BackgroundIdentifiability and Reconstruction of Biochemical Reaction Networks from Population Snapshot Data
Eugenio Cinquemani, B. (2018) · Processes · doi:10.3390/pr6090136 · Cited by 14
“However a formal definition of practical identifiability is not obvious and still open (see [1,4,6,20], among others).”
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Guidelines for benchmarking of optimization-based approaches for fitting mathematical models
Kreutz, C. (2019) · Genome biology · doi:10.1186/s13059-019-1887-9 · Cited by 15
Self-citationSensitivity Analysis and Practical Identifiability of Some Mathematical Models in Biology
Игоревна, К. О., Andornaya, D. V., & Kabanikhin, S. I. (2020) · Journal of Applied and Industrial Mathematics · doi:10.1134/s1990478920010123 · Cited by 15
Metadata
- Authors
- Raue, A., Kreutz, C., Maiwald, T., Bachmann, J., Schilling, M., Klingmüller, U., & Timmer, J.
- Affiliations
- University of Freiburg (DE)
- Corresponding author
- Raue, A.
- Venue
- Bioinformatics
- Publisher
- Oxford University Press (OUP)
- Published
- 2009-06-08
- Volume
- 25
- Issue
- 15
- Pages
- 1923-1929
- ISSN
- 1367-4803 (ISSN-L)
- 1367-4811
- OpenAlex
- W2168648059
- Cited by
- 1,442
- References
- 29
- Citations by year
- 2026: 209 · 2025: 107 · 2024: 103 · 2023: 82 · 2022: 98 · 2021: 118 · 2020: 97 · 2019: 116 · 2018: 94 · 2017: 78 · 2016: 87 · 2015: 64 · 2014: 52 · 2013: 54 · 2012: 39
- FWCI
- 10.99
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- Bioinformatics (publishedVersion)
- PubMed (publishedVersion)
- OPUS (Augsburg University) (submittedVersion)
- Funders
- Deutsches Krebsforschungszentrum
- European Commission (223188, EU-FP7, FP7 HEALTH)
- Bundesministerium für Bildung und Forschung (FRISYS 0313921, 0313074D, 0313921)
- FP7 Health
- Topics
- Life Sciences › Biochemistry, Genetics and Molecular Biology › Molecular Biology › Gene Regulatory Network Analysis
- Physical Sciences › Engineering › Control and Systems Engineering › Control Systems and Identification
- Life Sciences › Biochemistry, Genetics and Molecular Biology › Molecular Biology › Microbial Metabolic Engineering and Bioproduction
- Keywords
- identifiability analysis, profile likelihood, parameter estimation, confidence intervals, experimental design, model reduction
- MeSH terms
- Algorithms, Models, Biological, Probability, Computational Biology
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