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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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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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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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Tio, Z. K., & Balakrishnan, N. P. (2022) · International Symposium on Advanced Control of Industrial Processes · doi:10.1109/adconip55568.2022.9894218
“It is not possible to estimate all parameters in a hypothesized model due to the following reasons [13]: (i) The type and number of data points in the reference mode(s) can be noisy and/or insufficient to capture all parts of the hypothesized model [14].”
BackgroundSimpson, M. J., & Maclaren, O. J. (2022) · bioRxiv (Cold Spring Harbor Laboratory) · doi:10.1101/2022.12.14.520367
FLAT LIKELIHOODS: THREE-PARAMETER WEIBULL MODEL CASE
Montoya, J. A., & Figueroa-Preciado, G. (2022) · Revista de la Facultad de Ciencias · doi:10.15446/rev.fac.cienc.v11n2.98450
Zepeda, J. E., & Montoya, J. A. (2022) · Revista de la Facultad de Ciencias · doi:10.15446/rev.fac.cienc.v11n2.97782
FLAT LIKELIHOODS: SIR-POISSON MODEL CASE
Montoya, J. A., Figueroa-Preciado, G., & Tocto-Erazo, M. R. (2022) · Revista de la Facultad de Ciencias · doi:10.15446/rev.fac.cienc.v11n2.100986
A workflow for incorporating cross-sectional data into the calibration of dynamic modelsPreprint
Fischer-Holzhausen, S., & Röblitz, S. (2023) · bioRxiv (Cold Spring Harbor Laboratory) · doi:10.1101/2023.01.17.523407
“In the second step, an identifiability analysis is performed by calculating the profile likelihoods (Murphy and Van der Vaart, 2000; Raue et al., 2009) for each model parameter numerically.”
BackgroundMethodApplications of mathematical modeling and simulation in COVID-19
Platt, G. M., Libotte, G. B., & Lobato, F. S. (2023) · Elsevier eBooks · doi:10.1016/b978-0-323-91794-0.00023-8
Activator-blocker model of transcriptional regulation by pioneer-like factorsPreprint
Onichtchouk, D., Riesle, A. J., Gao, M., et al. (2023) · Research Square · doi:10.21203/rs.3.rs-2485402/v1
Self-citationA continuation technique for maximum likelihood estimators in biological models
Cassidy, T. (2023) · White Rose Research Online (University of Leeds, The University of Sheffield, University of York) · doi:10.48550/arxiv.2303.09194
“(θi,α) = {c|PLEθi(c)−PLEθi(θ ∗ i )< ∆α} where ∆α = χ2(α,d f ) is the χ2 distribution at significance level α and d f degrees of freedom [Raue et al., 2009].”
McSNAC: A software to approximate first‐order signaling networks from mass cytometry data
Wethington, D., Mukherjee, S., & Das, J. (2023) · Quantitative Biology · doi:10.15302/j-qb-022-0308
“We estimate confidence intervals (CIs) for the elements of the M matrix following a Profile Likelihood based approach[18].”
MethodLawrence, M. J., Wilson, B. M., Reid, G., et al. (2023) · bioRxiv (Cold Spring Harbor Laboratory) · doi:10.1101/2023.05.15.540815
On near‐redundancy and identifiability of parametric hazard regression models under censoring
Rubio, F. J., Espindola, J. A., & Montoya, J. A. (2023) · Biometrical Journal · doi:10.1002/bimj.202300006
“Methods for detecting NR or PNI are classified into symbolic or numerical methods (Catchpole & Morgan, 1997; Cole, 2020).”
BackgroundMethodInfluentialImplementation and Optimisation of Time-Varying Hepatic Clearance Model Using Basis-Spline MethodsPreprint
McHugh, A. D., Chase, J. G., Zhou, T., Holder-Pearson, L. R., & Knopp, J. L. (2023) · SSRN Electronic Journal · doi:10.2139/ssrn.4570984
Modelling of postprandial glucose and insulin dynamics
van Sloun, B. (2023) · doi:10.26481/dis.20231027bs
Koopsen, T., van Osta, N., van Loon, T., et al. (2023) · Research Square · doi:10.21203/rs.3.rs-3442063/v1
Okuneye, K., Bergman, D. R., Bloodworth, J. C., et al. (2024) · doi:10.22541/au.170664414.42348543/v1
Protocol for FRAP-based estimation of nuclear import and export rates in single yeast cells
Durrieu, L., Bush, A., & Colman‐Lerner, A. (2024) · STAR Protocols · doi:10.1016/j.xpro.2024.102876
An Objective Measure of Distributional Estimability as Applied to the Phase-Type Aging Model
Nie, C., Liu, X., & Provost, S. B. (2024) · Risks · doi:10.3390/risks12020037
Modeling Intestinal Glucose Absorption from D-Xylose Data
Dursoniah, D., Folschette, M., Goutchtat, R., et al. (2024) · International Joint Conference on Biomedical Engineering Systems and Technologies (BIOSTEC) · doi:10.5220/0012358300003657
“In this paper, we consider practical identifiability based on the profile likelihood method (Raue et al., 2009).”
MethodQuantitative modelling of signaling in aggressive B cell lymphoma unveils conserved core networkPreprint
Klinger, B., Rausch, I., Sieber, A., Kutz, H., Kruse, V., Kirchner, M., Mertins, P., Kieser, A., Blüthgen, N., & Kube, D. (2024) · bioRxiv (Cold Spring Harbor Laboratory) · doi:10.1101/2024.03.25.586526
“We performed a profile likelihood analysis on that unveiled that about half of the pathway coefficients (11 of 21, asterisks FIGURE 2E ) are significantly different, i.e., the 95% pointwise confidence intervals (Raue et al., 2009) do not overlap.”
MethodUsing Bayesian priors to overcome non-identifiablility issues in Hidden Markov modelsPreprint
Münch, J. L., Schmauder, R., Paul, F., & Habeck, M. (2024) · bioRxiv (Cold Spring Harbor Laboratory) · doi:10.1101/2024.04.20.590387
“In contrast, strategies such as profile likelihood [33] In chemical physics, it is a well-known phenomenon that often the overall rate of a process in a CRN relies heavily on the slowest chemical reaction, called the rate-limiting step.”
InfluentialSMT based parameter identifiable combination detection for non-linear continuous and hybrid dynamics
Ghosh, D., & Mandal, C. (2024) · Formal Aspects of Computing · doi:10.1145/3665920
“A model can be practically non-identifiable even though it is structurally identifiable if the amount and quality of experimental data (sparse and noisy) are insufficient [Ashyraliyev et al. 2009 ; Raue et al. 2009 ].”
Practical and Numerical Considerations
Cunniffe, N. J., Hamelin, F., Iggidr, A., Rapaport, A., & Sallet, G. (2024) · SpringerBriefs on PDEs and Data Science · doi:10.1007/978-981-97-2539-7_5
Modeling relaxation experiments with a mechanistic model of gene expression
Estavoyer, M., Dufeu, M., Ranson, G., Lefort, S., Voeltzel, T., Maguer-Satta, V., Gandrillon, O., & Lepoutre, T. (2024) · BMC Bioinformatics · doi:10.1186/s12859-024-05816-4
“In the case of CD34 + cell relaxation, the linear regression line is given by the slope + ≈ 0.418 , and for CD34 − cells, the slope is − ≈ 0.422 .”
BackgroundInfluentialQuantitative modeling of signaling in aggressive B cell lymphoma unveils conserved core network
Klinger, B., Rausch, I., Sieber, A., Kutz, H., Kruse, V., Kirchner, M., Mertins, P., Kieser, A., Blüthgen, N., & Kube, D. (2024) · PLoS Computational Biology · doi:10.1371/journal.pcbi.1012488
Parra-Rojas, C. A., Fanelli, D., & McKane, A. J. (2026) · Ecological Modelling · doi:10.1016/j.ecolmodel.2026.111477
Model Predictive Control of an Underground Mine Cooling Water Operation
Anastasiou, J., Louw, T. M., & Shipman, W. J. (2024) · IFAC-PapersOnLine · doi:10.1016/j.ifacol.2024.10.228
Modeling Fibrous Dysplasia Progression and its Therapeutic Intervention
Beltran-Vargas, J. C., de Castro, L. F., Calvo, G. F., & Pérez‐García, V. M. (2024) · Mathematical Modelling of Natural Phenomena · doi:10.1051/mmnp/2024022
“According to the results obtained in [18] and summarized in Table 2, two sets of parameters, θ 1 and θ 2 , defined as FD: are proposed for adjusting the models using the profile-likelihood method [30].”
Identifiability and Kalman Filter Parameter Estimation Applied to Biomolecular Controller Motifs
Haus, E. S., Overland, M. H., Rotondo, D., et al. (2025) · Linköping electronic conference proceedings · doi:10.3384/ecp212.053
“Biological systems are typically only partially observable due to experimental limitations, and the available sys-tem outputs may be a function of several states and/or parameters (Raue et al., 2009).”
Efficiently Establishing Practical Identifiability Via Active LearningPreprint
Liu, X., Wanika, L., Chappell, M. J., et al. (2025) · SSRN Electronic Journal · doi:10.2139/ssrn.5125554
Malcata, F. X. (2024) · Springer eBooks · doi:10.1007/978-3-031-41155-7_6
Shioda, K., Watanabe, M., Freeman, M. C., Lévy, K., & Brouwer, A. F. (2025) · medRxiv · doi:10.1101/2025.02.20.25322587
Magnusson, R. (2025) · bioRxiv (Cold Spring Harbor Laboratory) · doi:10.1101/2025.03.24.644951
“From the perspective of parametric uncertainty in mechanistic modeling, the most relevant analysis is arguably the profile likelihood [9, 23].”
Digitale Seed-Train-Zwillinge und statistische Methoden
Rodríguez, T. H., & Frahm, B. (2025) · Springer eBooks · doi:10.1007/978-3-031-75698-6_5
Investigating the Impact of Measurement Variance on Gene Circuit Model ParameterizationPreprint
Spartalis, T. R., Tang, W., & Tang, X. (2025) · bioRxiv (Cold Spring Harbor Laboratory) · doi:10.1101/2025.04.02.646175
“While weighted cost function has attracted wide attentions in various fields, [27–29] the design of objective function, especially on simple weighted objective functions for synthetic gene circuits model parameterization is still under-investigated.”
Global Dynamics of a Within‐Host Model for Immune Response With a Generic Distributed Delay
Al‐Darabsah, I. (2025) · Mathematical Methods in the Applied Sciences · doi:10.1002/mma.11021
Model selection and parameter identification analysis in epidemiological models
Zhao, Y., Wu, H., Li, M. Y., et al. (2025) · Mathematics and Computers in Simulation · doi:10.1016/j.matcom.2025.05.012
Neubauer, P. B., Anane, E., Junne, S. G., & Bournazou, M. N. C. (2025) · Springer eBooks · doi:10.1007/978-3-031-75700-6_1
Analysis of commonly used meal models for type 1 diabetes from an identification perspective
Clara, F.-N., Sala-Mira, I., José-Luis, D., & Jorge, B. (2025) · IFAC-PapersOnLine · doi:10.1016/j.ifacol.2025.06.017
Babaev, D. S., Kutumova, E. O., & Kolpakov, F. (2025) · Pharmaceutics · doi:10.3390/pharmaceutics17070935
“This understanding is necessary for further investigation of model predictions and can be provided by analyzing the parameters for identifiability [49,50].”
Debanath, K., Aich, S., & Srizon, A. Y. (2025) · SSRN Electronic Journal · doi:10.2139/ssrn.5366003
Optimal experiment design for practical parameter identifiability and model discrimination
Liu, Y., Maini, P. K., & Baker, R. E. (2026) · Mathematical Biosciences · doi:10.1016/j.mbs.2026.109710 · Cited by 5
“…is one of the main techniques for practical identifiability analysis, and has been used in many studies [ liu2024ParameterIdentifiabilityModel , raue2009StructuralPracticalIdentifiability , simpson2022ParameterIdentifiabilityModel , simpson2020PracticalParameterIdentifiabilitya ,…”
Shaikh, R., & Reeves, G. T. (2025) · bioRxiv (Cold Spring Harbor Laboratory) · doi:10.1101/2025.08.04.668461
Liu, Y., Pathirana, D., & Hasenauer, J. (2025) · IFAC-PapersOnLine · doi:10.1016/j.ifacol.2025.07.160
Model reduction and analysis: A case study of a malaria control model
Korsah, M. A., Johnston, S. T., Tiedje, K. E., et al. (2026) · Journal of Theoretical Biology · doi:10.1016/j.jtbi.2026.112478
Parameter Identification for PDEs via Qualitative Constraints
Blanco‐Rodríguez, R., Humphrey, A., Anderson, A., & Hernandez-Vargas, E. A. (2025) · IFAC-PapersOnLine · doi:10.1016/j.ifacol.2025.08.069
Reliable machine learning initialization methods for the calibration of Dynamic Energy Budget modelsPreprint
Oliveira, D. F., Marques, G. M., Santos, F., et al. (2025) · SSRN Electronic Journal · doi:10.2139/ssrn.5434102
Making the Case for Geometric Identifiability in Electrochemical Impedance Spectroscopy
Masoudi, Y., Alavi, S. M. M., & Azad, N. L. (2025) · IEEE Transactions on Control Systems Technology · doi:10.1109/tcst.2025.3603205
On the structural and practical identifiability of multi‐echo BBB‐ASL tracer kinetic models
Manson, T. J., Thomas, D. L., Günther, M., Tippett, L. J., Dragunow, M., Morgan, C., & Suresh, V. (2025) · Magnetic Resonance in Medicine · doi:10.1002/mrm.70075
“16 One widely used method is the profile-likelihood method, 24,25 used recently to analyze DCE MRI tracer kinetic models.”
Bayesian Validation of Dynamic Systems for Biological Networks
Son, D., & Kim, J. (2025) · Journal of Computational Biology · doi:10.1177/15578666251382251
Cortés‐Ríos, J., Rodriguez-Fernández, M., Sorger, P. K., & Fröhlich, F. (2025) · PLoS Computational Biology · doi:10.1371/journal.pcbi.1012890
“…the model consistently predicted convergence to a single steady state, both from the primary mode of CMD angles (black line, used for model training) and other points in the cell cycle (grey lines, not used for model To assess parameter uncertainties, we used the profile likelihood approach [58].”
González‐Parra, G., Luebben, G., Villanueva, R. J., Navarro-González, F. J., & Bhakta, B. (2025) · Mathematics and Computers in Simulation · doi:10.1016/j.matcom.2025.10.015
Statistical analysis for fractional differential equation models
Caraveo-Balderas, L. A., Laos, J. A. M., & Ramírez-Ramírez, L. L. (2025) · International Journal of Dynamics and Control · doi:10.1007/s40435-025-01927-9
Moor, D., Zetterdahl, E., van Vliet, P., et al. (2025) · ACM International Conference on Information and Knowledge Management (CIKM) · doi:10.1145/3746252.3761070
Wang, Y., Ricciuto, D. M., Mao, J., et al. (2026) · Geoscientific model development · doi:10.5194/gmd-19-5827-2026
“Increase in AGNPP shrub , decrease in NPP moss , and increase in HR from the coldest to the warmest treatments are visually discernible in both the observations (Fig.”
Regression modelling in traffic crash reconstruction
Sit, H. Y. (2025) · Forensic Science International · doi:10.1016/j.forsciint.2025.112766
Quantitative analysis of ion transport in human airway epithelial cells
O'Donoghue, D. (2014) · UCL Discovery (University College London)
Duchesne, R., Guillemin, A., Gandrillon, O., et al. (2021) · Figshare · doi:10.6084/m9.figshare.16735746
Koopsen, T., van Osta, N., van Loon, T., et al. (2024) · Figshare · doi:10.6084/m9.figshare.26719698
Koopsen, T., van Osta, N., van Loon, T., et al. (2024) · Figshare · doi:10.6084/m9.figshare.25815072
Koopsen, T., van Osta, N., van Loon, T., et al. (2024) · Figshare · doi:10.6084/m9.figshare.26719701
de Cellès, M. D., Campbell, H., Borrow, R., Taha, M., & Opatowski, L. (2020) · Figshare · doi:10.6084/m9.figshare.12172065
Lawrence, M. J., Wilson, B. M., Reid, G. K., et al. (2024) · Figshare · doi:10.6084/m9.figshare.26643991
Koopsen, T., van Osta, N., van Loon, T., et al. (2024) · Figshare · doi:10.6084/m9.figshare.25815615
Parameter-wise predictions and sensitivity analysis for random walk models in the life sciences
Liu, Y., Warne, D. J., & Simpson, M. J. (2025) · Journal of Theoretical Biology · doi:10.1016/j.jtbi.2025.112347 · Cited by 1
Ueda, R., & Shoji, H. (2025) · Forma · doi:10.55653/forma.2025.001.004
Rodrigues, J. A. (2025) · Springer proceedings in mathematics & statistics · doi:10.1007/978-3-031-97950-7_3
Patrascu, A. T. (2026) · SSRN Electronic Journal · doi:10.2139/ssrn.6017434
Bayesian inference of gene regulatory networks at stochastic steady state
Gupta, A., Yoon, R., & Josić́, K. (2026) · Journal of The Royal Society Interface · doi:10.1098/rsif.2026.0040
The Role of Host Immunity and the Environment in Seasonal Disease Dynamics
Kosmacher, G., Max, D. S., Rapti, Z., et al. (2026) · Bulletin of Mathematical Biology · doi:10.1007/s11538-025-01582-3
“In reality, fitting models to time series is even more challenging due to practical identifiability issues (Raue et al. 2009).”
A minimal activator-inhibitor-repressor model of the hepatic circadian clock
Delpierre, P., & Lefranc, M. (2026) · Biophysical Journal · doi:10.1016/j.bpj.2026.01.024
Pant, B. P., Saucedo, O., & Pogudin, G. (2026) · medRxiv · doi:10.64898/2026.01.16.26344284
On Guaranteed Error Bounds for Parameter Identification of Linear Difference Equations
Lomov, A. (2026) · Computational Mathematics and Modeling · doi:10.1007/s10598-025-09660-w
“With a probabilistic approach, an asymptotic distribution of O (cid:2).y/ with respect to the growing number of observations y allows to get the confidence intervals based on the asymptotic O (cid:2).y/ variance matrix and the Fisher matrix [3].”
InfluentialPicó, J., Arboleda-García, A., Penas, D. R., et al. (2026) · bioRxiv (Cold Spring Harbor Laboratory) · doi:10.64898/2026.01.21.700808
“This criterion provides a robust estimate of the number of independent parameter combinations supported by the experimental data [7, 46, 47].”
Kumaki, R. (2026) · SSRN Electronic Journal · doi:10.2139/ssrn.6054334
Ciptaningtyas, D., Fitriani, N., Thoriq, A., Pratopo, L. H., & Shiina, T. (2026) · Computers and Electronics in Agriculture · doi:10.1016/j.compag.2026.111519
Patrascu, A. T. (2026) · SSRN Electronic Journal · doi:10.2139/ssrn.6219001
Raman-guided sample subset selection for cost-efficient offline calibration in bioprocesses
Wilms, T., Schwenke, F., King, R., & Knorn, S. (2026) · Bioprocess and Biosystems Engineering · doi:10.1007/s00449-026-03402-x
“Practical identifiability has multiple methodological interpretations in the literature, which are sensitivity-matrix-based classifications (as used here) versus confidence-interval-based or profile-likelihood-based approaches [64].”
Rasool, F., Akhoon, R. H., & Gani, K. M. (2026) · SSRN Electronic Journal · doi:10.2139/ssrn.6460112
Integrating multiple observation data sources to facilitate movement modelling of killer whale pods
Thompson, P. R., Dowd, M. G., Yurk, H., et al. (2026) · Movement Ecology · doi:10.1186/s40462-026-00645-x
“We used likelihood profiling techniques to estimate exact 95% confidence intervals for each of our model parameters [109].”
Mohammed, M. A. Y., Karami, H., & Chowell, G. (2026) · Mathematical Biosciences & Engineering · doi:10.3934/mbe.2026043
“However, the ability to recover meaningful parameters from data critically depends on identifiability, that is, whether model parameters can be uniquely determined from available observations [27, 28, 29].”
Delannée, V. (2017) · theses.fr (ABES) · doi:10.70675/010f60d5z3d06z4eadz831az2edcdae592fa
“Puis, l’algorithme "trust-region" implémenté dans Matlab modifie de manière itérative ce jeu de paramètres afin de minimiser la distance entre les données expérimentales et les données théoriques en utilisant la méthode des moindres carrées [183], où σk,l,i représentent l’erreur interne liée à l’estimation des paramèt…”
Mathematical modeling and statistical inference to better understand arbovirus dynamics
Champagne, C. (2018) · theses.fr (ABES) · doi:10.70675/90ac8387z6a19z487dz96f0z3b460f81238c
Arenas, A. J., Cogollo, M. R., & González‐Parra, G. (2026) · Nonlinear Dynamics · doi:10.1007/s11071-026-12363-5
Erythroid differentiation in vitro under the lens of mathematical modelling
Duchesne, R. (2019) · theses.fr (ABES) · doi:10.70675/ff05850fzb60bz4e5fza98cz6059b1f59210
Frédérique, R. (2019) · theses.fr (ABES) · doi:10.70675/04ae36b0zbde1z4458z8c5bz91a9b55ac585
Identifiability of the thermal performance of a building envelope from poorly informative data
Juricic, S. (2020) · theses.fr (ABES) · doi:10.70675/c57b4282z77acz4c74z9649ze23a3f953e85 · Cited by 2
Martinelli, J. (2022) · theses.fr (ABES) · doi:10.70675/5488e904z57a3z4fd6z83d2z795ca0557b09
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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