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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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In the paper
“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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In the paper
“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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In the paper
“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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In the paper
“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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In the paper
“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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In the paper
“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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In the paper
“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
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- The authors state this as an assumption: An appropriate model M that sufficiently describes the available experimental data is assumed.
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In the paper
“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
Strength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes? Load-bearing: unclear✓ Based: passesThe paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“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
Strength:plain· not yet calibratedCausal:direct· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes? Load-bearing: unclear✓ Based: passesThe paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“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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In the paper
“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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In the paper
“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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In the paper
“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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In the paper
“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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In the paper
“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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A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
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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In the paper
“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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In the paper
“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
What the sources say
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In the paper
“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
What the sources say
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In the paper
“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
What the sources say
- Not checked — no full text of the sourcecites Joshi et al., 2006 source
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In the paper
“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
What the sources say
- 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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In the paper
“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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In the paper
“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
What the sources say
- 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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In the paper
“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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In the paper
“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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In the paper
“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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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
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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The source backs part of the claim, not all of it — or backs it for a different population, condition or comparison.
In the citing paper
“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]
Experimental — accuracy not yet measuredThe 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
“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]
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]
Self-check
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.
Concentration unit conflicts with figure labels
MinorThe text says concentrations are modeled in nanometers, a unit of length, while the figure labels use nM, a unit of concentration.
“Species concentrations are modeled in nanometer.”
“time / min x 1 / nM”
Citations
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McNamara, R., Monsalve-Bravo, G. M., Stein, S. R., Francis, G. D., & Allenby, M. C. (2026) · bioRxiv (Cold Spring Harbor Laboratory) · doi:10.64898/2026.04.08.717345
Jurj, S. L. (2026) · AI in Medicine · doi:10.3390/aimed1020011
“The authors in [44] showed that even moderately complex models may exhibit substantial non-identifiability under typical experimental conditions.”
Non-linear colloidal flow and capture derived from Margules activity model
Khazali, N., Russell, T., Xu, C., Malgaresi, G., & Bedrikovetsky, P. (2026) · Advances in Water Resources · doi:10.1016/j.advwatres.2026.105309
Mapping immune imprinting zones enables predictive vaccination optimization
Yang, W., Jin, K., Li, W., Ding, Z., Yu, H., Zhou, Z., Su, W., Zhu, X. L., Ma, X., Wan, N., et al. (2026) · Cell Reports · doi:10.1016/j.celrep.2026.117271 · Cited by 1
Guyon, J. R. (2026) · Springer eBooks · doi:10.1007/978-3-032-18230-2_2
Wiens, A., Bohner, T., Udell, B. J., & Thogmartin, W. E. (2026) · Journal of Agricultural Biological and Environmental Statistics · doi:10.1007/s13253-025-00709-2
Chahet, Y., Mimouni, A., Amraoui, M. E., Amrani, A. E., Bouaichi, A., & Bejjit, L. (2026) · Sustainability · doi:10.3390/su18094486
Scattering and sputtering on the lunar surface
Canu-Blot, R., Wieser, M., Rollero, U., Maynadié, T., Barabash, S., Wieser, G. S., Zhang, A., Wang, W., & Wang, C. (2026) · Astronomy and Astrophysics · doi:10.1051/0004-6361/202659499
Gevertz, J. L., & Kareva, I. (2026) · Mathematical Biosciences · doi:10.1016/j.mbs.2026.109713
Cogan, N. G., Nooranidoost, M., Peltier, M., & Romero, S. (2026) · Journal of The Royal Society Interface · doi:10.1098/rsif.2025.0938
Ohnemus, S., Dasí, A., Ramadan, A. F., Greiner, J., Wülfers, E. M., Stoyek, M. R., Tillert, L., Vierock, J., Quinn, T. A., Kohl, P. K., et al. (2026) · The Journal of Physiology · doi:10.64898/2026.05.11.724228 · Cited by 1
“In addition to the here analysed defibrillation scenarios, we recommend future studies comparing defibrillation efficacy as a function of ChR conductance, expression density, and spatial expression patterns, for example through profile-likelihood-based approaches [77, 78].”
CFD analysis of surface saturation effects on wafer-scale uniformity in Si3N4 LPCVD
Park, J., & Kim, T. (2026) · Thermal Science and Engineering Progress · doi:10.1016/j.tsep.2026.104745
Pant, B., Levine, M. E., Nande, A., Garcia, R., Dewey, G., Link, N., & Santillana, M. (2026) · Infectious Disease Modelling · doi:10.1016/j.idm.2026.05.004
“…of parameter unidentifiability, while highlighted here in an epidemiological context, represents a broader class of inverse problems prevalent across computational science (Bellman & Åstr € om, 1970; Cobelli & Distefano, 1980; Raue et al., 2009; Villaverde et al., 2016; Walter & Pronzato, 1997).”
Suárez, J. I., Roa, S., Valdés, V., et al. (2026) · Biotechnology and Bioengineering · doi:10.1002/bit.70246
“• Profile likelihood analysis (Raue et al. 2009) yielded non‐linear 95% confidence regions, forming closed, bounded contours for all calibrated parameters (Table 3; Supporting Information Figures S2–S3 suggesting that the calibrated parameter set captures the dominant system behavior.”
InfluentialIzza, R. N., Vanheusden, F. J., Nursyafi, F. S., et al. (2026) · Biomedical Engineering Letters · doi:10.1007/s13534-026-00584-x
“Because circulation system personalization used only SBP and DBP from a single beat, strict unique identifiabil - ity of E , R scale , and C scale is not expected, consistent with established limitations in parameter identifiability under sparse observations[ 24].”
Synthetic-data augmented calibration for expert-informed rare disease modelsPreprint
Yang, H., Rachel, T., Litwin, T., et al. (2026) · bioRxiv (Cold Spring Harbor Laboratory) · doi:10.64898/2026.05.18.725833
“Such an approach is vulnerable to noisy and sparse data, because the parameter calibration can not properly distinguish the relevant from the irrelevant model variables, causing model and parameter identifiability issues [15, 16].”
Self-citationCvetković, S., Ostojić, M., Milošević, M., & Platiša, M. M. (2026) · Physiological Measurement · doi:10.1088/1361-6579/ae7175
Gupta, M. K., Zoabi, R., Yado, S., Starruß, J., Geiger, B., & Hatzikirou, H. (2026) · BMC Cancer · doi:10.64898/2026.06.03.729495
“Profile likelihood analysis 20 of a single synthetic ROI confirmed that the parameters were not practically identifiable under an individual curve fitting strategy (SI Fig.”
Gonçalves, R. F., & Rebouças, T. C. (2026) · Journal of Water Sanitation and Hygiene for Development · doi:10.2166/washdev.2026.183
Damiano, M. D., & Shertzer, K. W. (2026) · Fisheries Research · doi:10.1016/j.fishres.2026.107789
Röblitz, S. (2026) · Journal of Molecular Endocrinology · doi:10.1530/jme-25-0211
Dinsa, D. W., Keno, T. D., & Deressa, C. T. (2026) · International Journal of Mathematics and Mathematical Sciences · doi:10.1155/ijmm/3690384
Zhou, Q., Ouyang, Q., & Wang, H. (2026) · Biomedicines · doi:10.3390/biomedicines14071454
A minimal mathematical model to study B-cell receptor signalling mediated calcium dynamics
Chatterjee, P., Arora, G., Kundu, J., Halder, S., & Chatterjee, S. (2026) · Applied Mathematical Modelling · doi:10.1016/j.apm.2026.117174
Kinetic model analysis of solvent free etherification of glycerol over Ca1+xAl1-xLaxO3 catalyst
Chowdhury, S., Siang, O. K., Gholami, Z., et al. (2026) · Chemical Engineering Journal Advances · doi:10.1016/j.ceja.2026.101353
Jenner, A. L., Araujo, R. P., Levi, N. L., et al. (2026) · medRxiv · doi:10.64898/2026.07.02.26357107
Celis, A. I. S., Agosin, E., Pérez‐Correa, J. R., & Saa, P. A. (2026) · Bioprocess and Biosystems Engineering · doi:10.1007/s00449-026-03384-w
“These structures reflect intrinsic compensatory mechanisms where multiple parameters can jointly compensate to produce similar trajectories [13, 57 ].”
Probabilistic prediction of pavement icing onset using a DeepONet-Accelerated Bayesian framework
Yu, S., Chen, J., Yang, Y., et al. (2026) · Cold Regions Science and Technology · doi:10.1016/j.coldregions.2026.105074
Diagnosing Identifiability in Two-Tower Models for Unbiased Learning to Rank
Fris, S., & Hager, P. (2026) · International ACM SIGIR Conference on Research and Development in Information Retrieval (SIGIR) · doi:10.1145/3805712.3809640
“Despite its long history, profile likelihood remains widely used to assess model identifiability in applied domains such as systems biology and bioinformatics, which inspired our approach [29].”
InfluentialOn the structural identifiability of nonlinear models of ligand binding dynamics
White, C., Rottschäfer, V., & Bridge, L. (2026) · Journal of Pharmacokinetics and Pharmacodynamics · doi:10.1007/s10928-026-10040-z · Cited by 1
“Existing approaches include profile likelihood methods [ 20, 26, 27, 48 ].”
Pesce, F., Strizzi, C. T., Granata, A., Bracaccia, M. E., Muzi, L., Liakopoulos, V., & Grandaliano, G. (2026) · Journal of Nephrology · doi:10.1093/joneph/aajag074
Xie, M., & Ma, T. (2026) · Journal of Process Control · doi:10.1016/j.jprocont.2026.103795
Complexity of coupled behaviour-disease models and their relative performance against empirical dataPreprint
Frimpong, S., & Bauch, C. T. (2026) · medRxiv · doi:10.64898/2026.07.23.26358796
Quantifying target antigen-dependent CAR T-cell performance against AML
Shah, S., Mueller, J., Vogel, E., Raatz, M., Boettcher, S., Traulsen, A., Manz, M. G., & Altrock, P. M. (2026) · Cancer Cell International · doi:10.1186/s12935-026-04419-8 · Cited by 1
“Instead of the profile likelihood approach [40], we used pairwise joint posterior distribution analysis [30].”
Beyond the fit: assessing bias-variance trade-off in johnson-cook constitutive modeling for AA5182
Knecht, A., Panchal, A., & Russcher, S. (2026) · Mechanical Systems and Signal Processing · doi:10.1016/j.ymssp.2026.114764
Assessing the Role of Model Complexity in Virtual Clinical Trial Outcomes
Gevertz, J. L., & Wares, J. R. (2026) · Bulletin of Mathematical Biology · doi:10.1007/s11538-026-01715-2
Jenner, A. L., Araujo, R. P., Levi, N. L., Ungerechts, G., Engeland, C. E., & Heidbuechel, J. P. W. (2026) · PLoS Computational Biology · doi:10.1371/journal.pcbi.1014533
Uncertainty quantification, sensitivity analysis, and model comparison
Lazebnik, T. (2026) · Elsevier eBooks · doi:10.1016/b978-0-44-355525-1.00017-5
Saptono, S., Vergiagara, V., Riyadi, F. A., et al. (2026) · Journal of Engineering and Applied Science · doi:10.1186/s44147-026-01153-w
“The result is weak practical identifiability [2, 27, 33].”
(2026) · Elsevier eBooks · doi:10.1016/b978-0-44-355525-1.00020-5
Nies, T., Ebeling, J., & Matuszyńska, A. (2026) · PLANT PHYSIOLOGY · doi:10.1093/plphys/kiag535
“Optimal experimental design and Fisher information analysis identify which perturbations and measurements will most efficiently resolve parameters that matter for a given model structure (Raue et al. 2009).”
Sorption and diffusion of vegetable oil into high-density polyethylene packaging
Touffet, M., & Wallecan, J. (2026) · Food Packaging and Shelf Life · doi:10.1016/j.fpsl.2026.101825
Kheyfets, V. O., Ichimura, K., Heerdt, P. M., et al. (2026) · Annals of Biomedical Engineering · doi:10.1007/s10439-026-04107-8
Hu, L., Wang, Z., Sun, J., et al. (2026) · Energy & Fuels · doi:10.1021/acs.energyfuels.6c02724
Golovko, V. V. (2026) · Sensors · doi:10.3390/s26165056
“Therefore, this profile is used to assess uncertainty in the baseline offset, not as the primary half-life estimate.”
Goryanin, I., Damms, B., & Goryanin, I. (2026) · CPT Pharmacometrics & Systems Pharmacology · doi:10.1002/psp4.70322 · Cited by 1
Picó, J., Arboleda-García, A., Penas, D. R., et al. (2026) · Nature Communications · doi:10.1038/s41467-026-76332-7
Zhong, H. (2026) · SSRN Electronic Journal · doi:10.2139/ssrn.7291018
Hidirova, M. B., Satin, M., & Turg'unov, A. (2026) · IEEE International Conference of Young Professionals in Electron Devices and Materials (EDM) · doi:10.1109/edm69524.2026.11632149
Jung, J., Park, T., Park, J., et al. (2026) · Water Research · doi:10.1016/j.watres.2026.126716
Finite-thickness corrections to the wellbore skin-factor condition for aquifer testsPreprint
Lin, Y., Lin, J., & Liu, C. (2026) · SSRN Electronic Journal · doi:10.2139/ssrn.7292182
Enver, A., Tepe, N., Sara, M. Y., Dileköz, E., & Bolay, H. (2026) · Biomedical Signal Processing and Control · doi:10.1016/j.bspc.2026.111244
Symbolic regression enables coarse-grained model discovery of intracellular signalling dynamicsPreprint
Pomereu, T. D., & Fröhlich, F. (2026) · bioRxiv (Cold Spring Harbor Laboratory) · doi:10.64898/2026.08.20.745973
Sun, T.-C., DarAssi, M. H., Alfwzan, W. F., et al. (2026) · Fractal and Fractional · doi:10.3390/fractalfract10090595
Zhang, S. (2026) · Advances in Differential Equations and Control Processes · doi:10.59400/adecp4304
Kottegoda, C., Codeço, C. T., Struchiner, C. J., et al. (2026) · Infectious Disease Modelling · doi:10.1016/j.idm.2026.08.001
“Parameter identifiability, which is typically assessed with profile likelihood analysis (Raue et al., 2009), Fisher Information Matrix (Miao et al., 2011; Roy Frieden, 2004), among other methods, was not formally evaluated in this study.”
Keshyagol, K. (2026) · International Journal of Electrochemical Science · doi:10.1016/j.ijoes.2026.101485
Li, C., Zhang, Y., Liu, X., Qu, Y., Lai, X., & Lei, J. (2026) · PLoS Computational Biology · doi:10.1371/journal.pcbi.1014690
“…analytical protocols—such as structural profile likelihoods, rigorous Fisher information matrix diagnostics, and local coordinate identifiability frameworks [46–50]—will systematically map the uncertainty boundaries, aggressively elevating the robustness of in silico translational modeling.”
Wanika, L., Rusten, I. S., Kermode, J. R., & Chappell, M. J. (2026) · CPT Pharmacometrics & Systems Pharmacology · doi:10.1002/psp4.70326
Pandelani, T., & Nemavhola, F. J. (2026) · Bioengineering · doi:10.3390/bioengineering13090991
“If not, parameters should be fixed to plausible priors or the model should be simplified [52,56,57,137,143–155].”
InfluentialKebbab, Y., Louar, M. A., Rebhi, L., et al. (2026) · Computer Methods in Biomechanics & Biomedical Engineering · doi:10.1080/10255842.2026.2724360
Ali, R., & Kubra, K. T. (2026) · Results in Control and Optimization · doi:10.1016/j.rico.2026.100799
Yun, D., Park, T., & Shin, H. (2026) · SSRN Electronic Journal · doi:10.2139/ssrn.7367218
Spoto, S. E. (2026) · Fractal and Fractional · doi:10.3390/fractalfract10090603
Sun, R., Cecilia, A., Costa‐Castelló, R., et al. (2026) · Computers & Chemical Engineering · doi:10.1016/j.compchemeng.2026.109867
Mining Stored-Specimen Studies for Information about Cancer Natural History
Lange, J. M., & Etzioni, R. (2026) · Cancer Epidemiology Biomarkers & Prevention · doi:10.1158/1055-9965.epi-26-0731
Ferreira, A. N. S., Félix, J. H. D. S., de Oliveira, L. A. S., et al. (2026) · Computational Biology and Chemistry · doi:10.1016/j.compbiolchem.2026.109353
Lamo, P., Lopez, J., & Tigira, S. (2026) · SSRN Electronic Journal · doi:10.2139/ssrn.7397323
How AI Can Advance Mathematical Biology: Opportunities, Challenges, and Future Directions
Rabiei, K., Rempała, G. A., Laubenbacher, R., & Hao, W. (2026) · Bulletin of Mathematical Biology · doi:10.1007/s11538-026-01744-x · Cited by 1
“…identifiability concerns whether they can be estimated reliably from finite and noisy data, and observability determines whether hidden states or unknown inputs can be reconstructed from the available outputs (Raue et al. 2009; Villaverde et al. 2019; Simpson et al. 2020; Wang and Hao 2025).”
InfluentialOtarola, L. L., & Domarchi, A. P. (2026) · Fluid Phase Equilibria · doi:10.1016/j.fluid.2026.114848
Liu, Z., Li, B., Yang, P., He, J., Wang, Y., Zou, G., Zhang, X., Li, X., & Gao, H. (2026) · Science China Physics Mechanics and Astronomy · doi:10.1007/s11433-026-3107-2
Predicting Pancreatogenic Disease Progression Via Hybrid Physiological and Neural ODE Modeling
Adamic, M., & Lu, W. (2026) · Lecture notes on data engineering and communications technologies · doi:10.1007/978-3-032-29430-2_18
Hallinger, L., Kirst, C., Singer, J. P., & Bohlen, O. S. (2026) · SSRN Electronic Journal · doi:10.2139/ssrn.7417281
A Nonlinear Dynamical Model of Leukemia Progression with CAR T-cell Therapeutic Intervention
Farveen, R., & Dhanalakshmi, P. (2026) · The interdisciplinary journal of Discontinuity Nonlinearity and Complexity · doi:10.5890/dnc.2027.03.006
Adeniran, I., & Degens, H. (2026) · bioRxiv (Cold Spring Harbor Laboratory) · doi:10.64898/2026.09.04.749086
“…deconvolute these nanoscale forces stems from a fundamental limitation in the inverse modelling of macroscopic fluorescence data: parameter equifinality (30), closely related to the structural and practical non-identifiability widely documented in kinetic models of biochemical systems (31 – 35).”
Let there be multifunctionality: Uncovering the dynamical zoo of the AC-DC genetic circuit
Maretvadakethope, S., Aor, I., Fischer, M. M., Pedrosa, À. P., Schaerli, Y., & Perez‐Carrasco, R. (2026) · Cell Systems · doi:10.1016/j.cels.2026.101715
A Real‐Time Digital Twin for Human Cardiovascular Applications
Snijders, S. C., Rutten, M. C. M., Sels, J.-W. E. M., Tonino, W. A. L., & Huberts, W. (2026) · International Journal for Numerical Methods in Biomedical Engineering · doi:10.1002/cnm.70209
PARAMETRIC SYNTHESIS IN SPECTROSCOPY BASED ON SPECTRAL SENSITIVITY ANALYSIS
Bilak, Y. Y., & Reblian, A. M. (2026) · Kibernetyka ta Systemnyi Analiz · doi:10.34229/kca2522-9664.26.5.11
Karkadakattil, A. (2026) · ISSS Journal of Micro and Smart Systems · doi:10.1007/s41683-026-00160-6
Vásquez, C. L. T., Hernández, R. A. L., Saa, P. A., & Pérez‐Correa, J. R. (2026) · Journal of Food Engineering · doi:10.1016/j.jfoodeng.2026.113321
Tran, S. T., Saur, G., Abel, R., et al. (2026) · SSRN Electronic Journal · doi:10.2139/ssrn.7450641
Temporal Coverage and Forcing Completeness Govern Inference From Dynamic Surface‐Tension Curves
Al‐Sou'od, K. A., & Ghanem, R. A. (2026) · Journal of Surfactants and Detergents · doi:10.1002/jsde.70100
Working-state descriptors for theory-guided CO 2 hydrogenation catalysis
Ou, Q., Yang, Z., & 韩一帆 (2026) · Chemical Communications · doi:10.1039/d6cc04679g
Sharan Adhithya, S., Shivakumar, S. P., Manavendra, G., & Bharath, P. B. (2026) · SSRN Electronic Journal · doi:10.2139/ssrn.7459678
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Mavris, I., & Valyrakis, M. (2026) · Water · doi:10.3390/w18182311
Mathematical Sociology: Models, Structures, and Open Problems
Volchenkov, D. (2026) · Mathematics · doi:10.3390/math14183363
Secchi, M., Mangifesta, M., & Sciarra, N. (2026) · Geosciences · doi:10.3390/geosciences16090378
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Boassen, G. H. B., Li, C., Luo, H., et al. (2026) · Environmental Pollution · doi:10.1016/j.envpol.2026.129154
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Self-citationStańczak, M., & Niazi, S. K. (2026) · Cells · doi:10.3390/cells15181711
“The remainder of this Perspective treats that gap as the central risk of hybrid modeling [22–24].”
InfluentialZhao, L., Xu, H., Wang, P., et al. (2026) · Sensors · doi:10.3390/s26185933
“This is a local identifiability analysis: F reflects the local geometric sensitivity of the reconstructed leads to the equivalent parameters at the operating point, not unique recovery of true anatomical or ionic parameters [39].”
Unified dynamics of capillary-driven drainage and spontaneous imbibition in porous media
Nourani, M. (2026) · Physics of Fluids · doi:10.1063/5.0347450
Shaale, M. M., Mushanyu, J., Nyabadza, F., et al. (2026) · Mathematical Biosciences & Engineering · doi:10.3934/mbe.2026102
Daryakenari, N. A., & Kohandel, M. (2026) · CPT Pharmacometrics & Systems Pharmacology · doi:10.1002/psp4.70343
Almquist, J., Chia, Y. L., Tummala, R., et al. (2026) · Clinical Pharmacology & Therapeutics · doi:10.1002/cpt.70495
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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