Risky Population. Spatial Dynamics. Law of Mass Action. Catalyzed Product. Two-Stage Nutrient Uptake Model. Iodine Compartment Model. The Brusselator.

Fitzhugh-Nagumo Neuron Model. Solar Radiation to Nonobstructed Inclined Surfaces. Mating and Mutation of Alleles. Natural Selection, Mutation, and Fitness. Odor-Sensing Model. Stochastic Resonance. Heartbeat Model. Bat Thermoregulation. The Optimum Plant. Infectious Diseases. Adaptive Population Control. Roan Herds. Population Dynamics of Voles. Lemming Model. Multistage Insect Models. For a given noise level, we generated perturbed data sets by adding Gaussian random numbers with mean zero and standard deviation scaled by a multiple of the empirical standard deviation see the error bars in Fig.

Thus, the noise level is defined as a multiple of the empirical standard deviation. The dynamic elastic-net was then fitted to each output sample and the corresponding area under the curve for each component of the estimated model error was computed. The plots for these AUC values versus the noise level are shown in Fig. The median values of the AUC for the components are largely independent of the noise level, but the variability of the AUC estimates increases with measurement noise. Nevertheless, the AUC values for and are always much larger than zero, whereas the AUC of and is close or even equal to zero for many samples with higher noise level.

This increases the confidence that the nodes and Fig. To ease visualisation, box plots at a given noise level are slightly offset.

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The impact of parameter uncertainty in the nominal model was assessed in a similar way. Parameter estimation algorithms 4 , 10 , 26 applied to the nominal model using the experimental data Fig. These confidence intervals were again scaled by the noise level, yielding an interval for each parameter from which uniform random samples were drawn.

Again, we generated modified parameter vectors per noise level. For each parameter sample, the system 4 was taken as the nominal model and the AUC of the resulting estimates was recorded Fig. Again, there is no systematic trend for the AUC of the different components of the estimated error. However, the variation of the AUC increases much faster than in Fig. Apart from the different sampling distributions used, this effect is related to the definition of the model error w , which is always defined with respect to the nominal model confer eqution 2a.

Hence, the estimated model error contains contributions from both structural and parameter misspecifications in the nominal model. Nevertheless, it is still possible to infer the dominant components and with high confidence. Similar results were found for the sensitivity against the number of measurement time points Supplementary text, Fig. As a test case for a larger system, we used a recent model for the coordination of photomorphogenic UV-B signalling in plants The model consists of 11 ODEs describing the dynamics of protein concentrations coupled by 10 chemical reactions Fig.

We considered this model as the nominal model in order to test the dynamic elastic-net method for a situation, where the ground truth is known. The model error was simulated by adding the hidden inputs to the nodes and. The output function is a linear combination of 7 different state variables see Supplementary text for all equations. Synthetic data were sampled at discrete time points from the outputs of the true model and Gaussian random perturbations were added to simulate measurement noise Fig.

The dynamic elastic-net with the nominal model was used to reconstruct the model error and the true state from these simulated data. The absolute area under the curve for each component of the model error estimate clearly indicates that the states and are targeted by hidden inputs Fig.

### Introduction

This illustrates the sparsity of the dynamic elastic-net estimate, which is a clear advantage over pure L 2 regularisation. Most importantly, the discrepancy between the true and the estimated state trajectory is almost zero Fig. The target points of the simulated model errors are indicated by the red arrows. As for any inverse method, there are limitations of the dynamic elastic-net method.

Some model errors are unobservable, because there exists a different hidden input function which generates an output which is identical to the output obtained for , see the Supplementary text for a simple example. Other model errors might be practically unobservable, because the output for another hidden input function might not be distinguishable within the measurement errors.

A special case are model errors which have no or almost no effect on the output at all. These will not be noticed during modelling and the nominal model will be accepted.

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## Learning (from) the errors of a systems biology model

To further test the ability of the dynamic elastic-net to infer the states targeted by the model error, i. First, we simulated model errors targeting a single node k in the same way as before. For the nodes and there was no effect on the output see again Fig. In addition, we simulated hidden inputs for all remaining two node combinations.

For each of these 36 simulated true models we tested the ability of the dynamic elastic-net to recover the correct target nodes from the AUC of the estimated. By this stringent criterion, we found that two single node errors targeting or were not correctly detected and another single node was predicted to be the target of the model error Fig. This indicates, that these model errors are unobservable and the observed output data can be explained by different inputs to different nodes.

With two exceptions 8, 3 and 7, 6 , the mistakes made by the algorithm for simulated pairwise model errors involve these two state nodes 1 and 4. However, with exception of the combination 1, 4 , at least one node is correctly predicted. Nodes and are omitted, since the simulated error signal had no effect on the output. The rows and the columns correspond to the true target nodes of the model error and the numbers in the cells are the nodes found by the dynamic elastic-net NA means that no second node was assigned.

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Gray cells indicate errors made by the dynamic elastic-net for unobservable model errors. The true target nodes of the model error are , but the dynamic elastic-net predicts the target nodes. The other two combinations and of the nodes did not fit the output data. These results demonstrate the inherent limitations of any attempt to recover the model error from observed outputs. For an unobservable model error, the true model error might correspond to a slightly larger value of the error functional 3c than the minimum obtained by the dynamic elastic-net.

A heuristic approach to explore some of these slightly suboptimal solutions is to rerun the dynamic elastic-net with some of the estimated target nodes from the first run excluded and to check, whether the output data can satisfactory be fitted with the same level of sparsity. This is illustrated in Fig. Refitting the dynamic elastic under the constraint identifies the correct nodes 9, 1 , see Fig. The two other combinations and do not provide a satisfactory fit to the data Supplemental Fig.

For the UVB-signaling network we find, that the slightly suboptimal solutions identified by this heuristics always contain the correct target node configuration. The combinatorial explosion of this strategy should typically not be a problem, thanks to the sparsity of the dynamic elastic-net predictions. The decision, which of the predicted target node sets, or , is the correct one can in practice only be made when additional states are measured. However, this example shows, how the dynamic elastic-net provides useful information to select further states for experimental observation 20 , Efficient computational methods to learn from incomplete model drafts and to direct model improvement are urgently needed.

Our proposed dynamic elastic-net approach provides suggestions for the location of these model errors in the network and estimates their dynamic time courses from measured output data. The sparsity of the proposed target points for the model error promotes model improvements in the most parsimonious way.

Even for an incomplete nominal model the algorithm can provide estimates for the system states which are not experimentally accessible. This is in stark contrast to many other state estimators including the Kalman Filter 29 for linear systems and its various extensions for nonlinear systems 30 , 31 , which usually require a correctly specified model.

Not all model errors can uniquely be determined from the output. For such unobservable model errors, our strategy to explore alternative, slightly suboptimal solutions might indicate alternative explanations for observed discrepancies between the data and the nominal model. In addition, this approach can also be informative for selecting additional nodes required for observing the state from output measurements 20 , Further research is needed to establish the relationship between the network topology and the observability of a model error.

Model errors arising in kinetic reaction systems can originate from erroneous rate equations or lacking reactions. The dynamic elastic-net can detect both types of errors as hidden inputs to the corresponding nodes of the network, but it can not discriminate between these errors. However, knowing the nodes affected by a model error might already be very informative for systematic model improvement.

As our method is designed for generic ODE models, it can also be applied to challenging modelling tasks in engineering, robotics and in the earth sciences. Our work also raises fundamental questions regarding successful modelling strategies. The approach to manually include more and more details into the model to compensate the initial model errors is often not practical or at least very time consuming.

The dynamic elastic-net hence paves the way towards a more principled and systematic way, in which models could be adapted based on experimental data. For parameter values and mathematical details see the Supplementary text. How to cite this article : Engelhardt, B. Learning from the errors of a systems biology model. Gunawardena, J. BMC Biol 12 , 29 Cvijovic, M. Bridging the gaps in systems biology. Mol Genet Genomics , — Li, C. BioModels Database: An enhanced, curated and annotated resource for published quantitative kinetic models. BMC Syst. Swameye, I. Identification of nucleocytoplasmic cycling as a remote sensor in cellular signaling by databased modeling.

USA , — Sunnaker, M. Sci Signal 6 , ra41 Topological augmentation to infer hidden processes in biological systems. Bioinformatics 30 , — Babtie, A. Topological sensitivity analysis for systems biology. Kahm, M. PLoS Comput Biol 8 , e Von Bertalanffy, L. The theory of open systems in physics and biology. Science , 23—29 Balsa-Canto, E. An iterative identification procedure for dynamic modeling of biochemical networks. BMC Syst Biol 4 , 11 Bachman, J. New approaches to modeling complex biochemistry.

Nat Methods 8 , — Melas, I.

PLoS Comput Biol 9 , e Rodriguez-Fernandez, M. Simultaneous model discrimination and parameter estimation in dynamic models of cellular systems. Global dynamic optimization approach to predict activation in metabolic pathways.

Mook, D. Minimum model error estimation for poorly modeled dynamic systems. Kolodziej, J. A novel approach to model determination using the minimum model error estimation. In Proceedings of the American Control Conference, Schelker, M. Comprehensive estimation of input signals and dynamics in biochemical reaction networks. Bioinformatics 28 , i—i Zou, H. Regularization and variable selection via the Elastic Net. B 67 , — Ouyang, X. Coordinated photomorphogenic UV-B signaling network captured by mathematical modeling.

Liu, Y. Observability of complex systems. Pontryagin, L. The mathematical theory of optimal processes. Fleming, W. Deterministic and stochastic optimal control. Gerdts, M. Tibshirani, R. Regression shrinkage and selection via the lasso: a retrospective: Regression Shrinkage and Selection via the Lasso.

## [] Modeling Global Dynamics from Local Snapshots with Deep Generative Neural Networks

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Identifiability and observability analysis for experimental design in nonlinear dynamical models. Chaos 20 , Kalman, R. Julier, S. New extension of the Kalman filter to nonlinear systems. Crassidis, J. Optimal estimation of dynamic systems. Download references. All authors reviewed the manuscript.