Driving the Model to Its Limit: Profile Likelihood Based Model Reduction

Abstract

In systems biology, one of the major tasks is to tailor model complexity to information content of the data. A useful model should describe the data and produce well-determined parameter estimates and predictions. Too small of a model will not be able to describe the data whereas a model which is too large tends to overfit measurement errors and does not provide precise predictions. Typically, the model is modified and tuned to fit the data, which often results in an oversized model. To restore the balance between model complexity and available measurements, either new data has to be gathered or the model has to be reduced. In this manuscript, we present a data-based method for reducing non-linear models. The profile likelihood is utilised to assess parameter identifiability and designate likely candidates for reduction. Parameter dependencies are analysed along profiles, providing context-dependent suggestions for the type of reduction. We discriminate four distinct scenarios, each associated with a specific model reduction strategy. Iterating the presented procedure eventually results in an identifiable model, which is capable of generating precise and testable predictions. Source code for all toy examples is provided within the freely available, open-source modelling environment Data2Dynamics based on MATLAB available at http://www.data2dynamics.org/, as well as the R packages dMod/cOde available at https://github.com/dkaschek/. Moreover, the concept is generally applicable and can readily be used with any software capable of calculating the profile likelihood.

Fig 1. Emergence of the Michaelis-Menten approximation.

A: Fast complex formation and decay (blue trajectories) result in the Michaelis-Menten approximation, slow formation and decay (orange trajectories) in a significant discrepancy to the data. B: The path in parameter space leading to the Michaelis-Menten approximation runs parallel to the contour lines of the log-likelihood function. C: The log-likelihood defines a significance threshold, which is not exceeded in the limit of fast formation/decay rates. Slower rates quickly lead to significant deviations from the data.

Fig 2. Reduction of the cascade toy model 1 (scenario 1).

A: Model scheme before (upper) and after (lower) reduction with lumped states X and pX. B: X(t) and pX(t) for fitted parameters. C: Profile likelihood for parameter k₁. D: Relationship of k₁ to k₂ obtained by re-optimisation along the profile likelihood (panel C). E: Profile likelihood for the identifiable parameter k in the reduced model. F: ppX(t) with simulated data after fitting, with comparison between a one and two step conversion.

Fig 3. Reduction of the cascade toy model 2 (scenario 2).

A: Model scheme before (upper) and after (lower) reduction. The difference is the basal deactivation from pX to X with rate constant k₄. B: X(t) and pX(t) with data for fitted parameters. C: Profile likelihood for parameter k₄. D: Relationship of the remaining parameters to the profile shown in C. E: The reaction associated with the parameter k₄ is removed by the reduction, while k₅ is identifiable. F: pX(t) and X(t) with comparison before and after reduction.

Fig 4. Reduction through functional relation (scenario 3).

A: Model scheme before (upper) and after (lower) reduction. The difference is the removal of state Z, and the algebraic replacement pZ(t) = α ⋅ pY(t). B: pY(t) and pZ(t) with simulated data after fitting. C: Profile likelihood for parameter k_d,Z. D: Re-optimisation of remaining parameters along the profile likelihood for k_d,Z. E: The profile likelihood for the identifiable parameter α in the reduced model. F: Comparison of trajectories for pY(t) and pZ(t).

Fig 5. Model reduction of weakly activated signalling pathway (scenario 4).

A: Model scheme before (upper) and after (lower) reduction. The difference is the omission of state X. B: X(t) for parameter sets along the profile likelihood of k_on (high to low values of k_on from bottom to top, i.e. cyan to blue). C: Parameter profile likelihood of k_on, depicting a practical non-identifiability to small values. D: Relation of scale to k_on. E: After model reduction, the parameter k_on is structurally non-identifiable and can be set to an arbitrary value. F: Comparison of model fits before and after reduction. Both curves overlap and cannot be statistically distinguished based on the data.

Fig 6. Typical flow-chart for model reduction based on the profile likelihood.

The depicted steps are to be applied for each parameter individually, starting with the calculation of the respective profile likelihood. After detection, the procedure resolves non-identifiabilities by fixing parameters, removing reactions, performing algebraic substitutions, or context-specific reductions. The method terminates when all parameters of interest are identifiable.

Fig 7. Reelin-induced signalling pathway.

A: Scheme of the Reelin-induced signalling pathway. Sub modules that underwent model reduction are framed. B: Experimental data and model trajectories of the Reelin model. Data points and their measurement errors are shown on a log-scale for the measurements of total Dab1, phosphorylated Dab1, Akt and SFKs for time points between zero and 240 minutes. Dotted and dashed lines indicate the model response for the parameter set before and after model reduction, respectively.

Fig 8. Summary of model reduction steps.

A: The profile likelihood of the inhibitor release from SFKs. B: The re-optimised paths of the remaining parameters with respect to the profile of the inhibitor release. C: Parameter profile likelihood of the Akt deactivation. D: Coupling of the Akt activation, log(Akt_act), to the Akt deactivation shown in panel C. E: Identifiable partitioning of SFKs with and without bound inhibitor. F: The scaling factor scale_pAkt, which links the observations of pAkt to pDab1, is identifiable.