 Research Article
 Open Access
Validation and selection of ODE based systems biology models: how to arrive at more reliable decisions
 Dicle Hasdemir^{1, 2},
 Huub C.J Hoefsloot^{1, 2}Email author and
 Age K. Smilde^{1, 2}
 Received: 17 December 2014
 Accepted: 16 June 2015
 Published: 8 July 2015
Abstract
Background
Most ordinary differential equation (ODE) based modeling studies in systems biology involve a holdout validation step for model validation. In this framework a predetermined part of the data is used as validation data and, therefore it is not used for estimating the parameters of the model. The model is assumed to be validated if the model predictions on the validation dataset show good agreement with the data. Model selection between alternative model structures can also be performed in the same setting, based on the predictive power of the model structures on the validation dataset. However, drawbacks associated with this approach are usually underestimated.
Results
We have carried out simulations by using a recently published High Osmolarity Glycerol (HOG) pathway from S.cerevisiae to demonstrate these drawbacks. We have shown that it is very important how the data is partitioned and which part of the data is used for validation purposes. The holdout validation strategy leads to biased conclusions, since it can lead to different validation and selection decisions when different partitioning schemes are used. Furthermore, finding sensible partitioning schemes that would lead to reliable decisions are heavily dependent on the biology and unknown model parameters which turns the problem into a paradox. This brings the need for alternative validation approaches that offer flexible partitioning of the data. For this purpose, we have introduced a stratified random crossvalidation (SRCV) approach that successfully overcomes these limitations.
Conclusions
SRCV leads to more stable decisions for both validation and selection which are not biased by underlying biological phenomena. Furthermore, it is less dependent on the specific noise realization in the data. Therefore, it proves to be a promising alternative to the standard holdout validation strategy.
Keywords
 Kinetic models
 ODE
 Differential equations
 Model validation
 Model selection
 Cross validation
 Holdout validation
Background
Ordinary differential equation (ODE) based kinetic models are able to capture all of the available kinetic information regarding a biological system. Therefore, they are used extensively in systems biology especially for the purpose of predicting time dependent profiles and steady state levels of biochemical species in conditions where experimental data is not available. Examples from the literature show that there is a common path taken by the modeling community for the construction and the analysis of ODE based systems biology models. The first step is to define the model structure and the associated kinetics. Due to serious concerns about the validity of model structures and kinetics, many studies include the parallel development and analysis of multiple alternative model structures [1–3]. The second step is the estimation of the unknown model parameters by fitting the model to the data using global and local minimization algorithms. Data here are usually in vivo time series concentration data of the observable biochemical species included in the model. At this step, uncertainty in the estimated values of the model parameters can also be quantified by constructing confidence intervals [4–6]. Last but not least, models are assessed for the quality of their fit to the data and for their predictive power on independent data. Independent data are datasets that were not used for parameter estimation. Selection between alternative model structures can also be performed at this step. A complete modeling cycle includes all these steps to achieve sufficiently good models [7, 8].
A good model has to be sufficient both in explaining the data on which it was built and in predicting independent data [9]. The first is taken into account mostly by likelihood ratio tests which can be used to reject models based on the quality of fit to the data [10–13]. The second aspect has been considered in conceptually two different ways. The first approach uses a penalized likelihood based metric such as Akaike’s (AIC) [14] or Bayesian Information Criterion (BIC) [15]. This metric is calculated using the whole dataset for parameter estimation but provides an expected value of the prediction error on an independent dataset. Therefore, it makes selecting the true complexity of a model possible because unnecessarily complex models are poor in predicting independent datasets. However, it is an ‘insample’ measure which means that the expected prediction error is valid only for the exact same experimental conditions as of the parameter estimation dataset [10]. Predicting the kinetics of the biological system under different experimental conditions is the very purpose of kinetic models, though. Therefore, modelers would like to show that the newly built model is good in qualitative or quantitative prediction of experimental data that was collected at different experimental conditions. This strategy which uses data at different experimental conditions as validation data constitutes the second approach to assess the prediction error [8, 16].

Inhibition of enzymes.

Reduction of protein levels by RNAi mediated suppression.

Gene deletions.

Overexpression of genes in gene networks.

Doseresponse experiments in which different doses of triggering chemicals are used to stimulate the system.
These validation scenarios are popularly applied since the common goal of the modelers is to demonstrate the models’ competency under challenging conditions. Estimating the parameters of a model in certain experimental conditions and showing their competency in other conditions within these scenarios requires multiple datasets under different conditions and, therefore, it is an example to the holdout validation strategy. That is, a predetermined set of conditions are held out of the training data and used as validation data instead. However, rules about the application of this strategy are not straightforward.
There are potential pitfalls associated with the application of holdout validation strategies in the validation and selection of kinetic systems biology models. These arise due to the lack of a satisfying answer to the question: which part of the dataset should be held out of the parameter estimation and instead should be used as the validation dataset? We carried out simulations to demonstrate the phenomena that hinder us from giving satisfactory answers to this question which can also be referred to as the problem of selecting an appropriate holdout partitioning scheme for the data. The problem arises due to incomplete biological knowledge of the system and unknown true values of the model parameters. This makes the problem a paradoxical one since our knowledge about the system will never be complete and the true values of the model parameters are themselves what we are looking for. However, statistics literature offers an established method which is independent of this knowledge, namely crossvalidation.
Crossvalidation (CV) is a resampling method traditionally used for model selection, determining the optimal complexity of a model or assessment of its generalizability in statistics [17, 18]. It is based on the partitioning of the data in training and test sets. The training set is used to build the model and the predictions of the model on the test set are used for model assessment. Since the test set is completely independent of the parameter estimation process, selection will not be biased towards more complicated models. The efficiency of crossvalidation and its difference from holdout validation strategy lies in the fact that the partitioning is made not in a predetermined but in a random way and the procedure is repeated multiple times so that each partition can be used as test set at least once. Different variants of CV exist such as leaveoneout, kfold and stratified kfold crossvalidation. A comprehensive evaluation and comparison of these methods can be found in [19–21] for classifier selection and in [22] for the selection of regression models.
CV has been applied in different ways in the ODE based modeling framework. Partitions can consist of different experiments (such as different cell types, experimental conditions or cultures), data belonging to different biochemical species in the same experiment or different data points within the time profile of the same biochemical species. In [23], the authors present an example for the latter. In this work, prediction errors on test sets obtained by using an ODE based model are compared to the residuals from an unsupervised data analysis method which does not make any use of biochemical knowledge. Better predictions found by using the unsupervised principal components analysis (PCA) method give hints on the low informative level of the ODE model leading to a rejection of the proposed ODE model structure. CV by using different species from the same or different cultures with different experimental conditions was considered by [24]. In that study, prediction errors were used to select not between two single models but between two families of models each constituted of models with slightly different topologies. Both approaches use a kfold stratified partitioning scheme in which time points or species were approximately equally distributed between k different partitions. The prediction errors from different test sets are averaged for the final measure of the predictive power.
Existence of only very few examples like we mentioned above show that CV has been highly neglected in the field. Also, the risks associated with the holdout validation strategy have been underestimated. The conceptual differences between the two methods and the difference between their outcomes have not been presented in detail. Therefore, with this paper we aim to present a detailed comparison of the holdout and crossvalidation methods by using simulations and emphasize the advantages of CV over holdout partitioning schemes. More details on our implementation of CV are given in the Methods section.
The reason for choosing simulations for our demonstrative purposes is that simulations and synthetic data allows us to know the ground truth, in this case the true model parameters and the true model structure. Therefore, we can analyze the results we achieved in different partitioning schemes in a comparative manner. We mainly look at the effect of different partitioning schemes in the outcome of model validation and selection. However, we report also results related to its effect on parameter estimation which is very influential on validation and selection in order to present a complete explanation.
Methods
Simulated data
The pathway can be triggered by using an NaCl shock and is activated via two parallel upstream signaling routes. The activity of the upstream routes is encoded by a binary input parameter which indicates that the route is either active or not. The level of the NaCl shock is also an input parameter which can be manipulated. Therefore, the model can be used also for deletion mutants where only one of the signaling routes is active, following different doses of NaCl shock, by changing only those two input parameters. It includes additional 20 free parameters which can be estimated from data.
We generated 100 different realizations of synthetic data by adding error to the time profiles obtained by the model. We added heterogeneous noise where the noise term for each concentration value was drawn from a normal distribution with a standard deviation equal to 10 % of the concentration value itself which reflects realistic noise levels and structure for these type of experiments. The time series data contained 15 time points during a course of 160 minutes.
Data partitioning schemes
Holdout partitioning schemes
Stratified random crossvalidation scheme
Stratified crossvalidation is based on the idea of providing training sets in which all classes of data are represented approximately equally. In our case, different classes of data are different cell types and different doses. This makes it a suitable scheme for our purposes, because we anticipate biased parameter estimates and hence, biased model validation and selection when training sets are dominated by certain classes of data. A stratified scheme would typically avoid such a problem and would give more robust results across different runs of crossvalidation compared to other crossvalidation approaches.
Parameter estimation
For each partitioning scheme, we estimated the parameters of both the true model (the model that we accepted as the ground truth and generated the data upon that) and the simplified model (the model that lacked one of the important regulatory interactions in the true model which can be seen in Fig. 1). We repeated the parameter estimation process by using 100 different realizations of the data. The estimation of the parameters required the minimization of the difference between the data and model predictions. We carried out the minimization using the local minimizer ‘lsqnonlin’ function of Matlab [27, 28]. We considered a local optimizer to be sufficient since we work with generated data and could use the true values of parameters as starting points.
In the case of real data, true values of the parameters are not known. However, usually there is prior information on the ranges of the values that the parameters can take. In such a situation, uniformly distributed random starting points can be generated in these ranges and optimization with lower and upper bounds can be performed starting from the different initial points, aiming to find the same minimum in a sufficient number of runs. We took this approach for a single example noise realization. We assumed that the parameter ranges span intervals that are twice as big as the true values of the parameters. We started the optimization from 80 different starting points and could achieve the same minimum with the one achieved when the true values of the parameters were used as the starting point, in 20 % of the runs. The correlation between the parameter estimate vectors of these runs was above 0.99. This finding confirmed that performing parameter estimation in a more realistic setting does not affect the minimum achieved if there is good prior information on the parameter ranges. Therefore, we use fixed starting points (true values of the parameters) throughout the study due to its substantial advantage in computational power.
Measures used for the analysis of the simulations
We analyzed five main features from the simulations, namely the amount of bias in the parameter estimates, the predictive power of the models on the validation datasets, the number of wrong decisions in which the simplified model structure was selected over the true model structure and the distance between the predicted profiles by the true and the simplified model structures (model separation).
We use normalized standard deviation of parameter estimates as a measure of identifiability levels of parameters. We obtain the standard deviation of the estimates by calculating the Fisher Information Matrix.
Results and discussion
Scenario 1: partitioning of data from different cell types
Existence of realizations with very high prediction errors in box plots with low medians shows that extremely bad predictions can occur even when the median prediction error is not very high. Examples of this can be observed also in the Sho1 scheme shown in Fig. 9 c and d. Indeed, the models trained by using only the Sho1 data can lead to extremely high prediction errors both on Sln1 and WT data. This can be seen from the existence of realizations with a percentage prediction error above 30 % and 15 %, respectively. On the other hand, models trained by using only the WT data perform well in predicting the Sln1 data but not the Sho1 data (maximum of medians = 1.11 % vs 4.20 % in Fig. 9 e and f). However, they are still better than those obtained by the models trained by the Sln1 data (maximum of medians = 4.20 % vs 8.85 % in Fig. 9 f and a).
As a summary of the observations on the predictive power, we can say two things. Firstly, models trained by using only the data from one of the deletion mutants is poor in predicting the data from the other. Secondly, models trained by using the data from the WT cell can predict the data from one of the deletion mutants better than the other one. The poor predictions might easily lead to misleading decisions on model validation. True model structures might fail to be validated due to weak predictive power of some partitioning schemes. To study the reasons leading to weak predictive power we investigated the parameter estimation quality.
The asymmetrical behaviour of the predictive power (that is, the WT and the Sln1 schemes were good in predicting Sln1 and WT validation data, respectively but none of them could achieve good predictions on the Sho1 validation dataset) stems from an underlying biological property which is the inequality of the two phosphorylation branches in the model. Although the two branches (Fig. 1) act redundantly for the ultimate goal of Hog1 protein phosphorylation, the fluxes in each branch are not equal. As also mentioned in [25], the Sho1 branch active deletion mutant produces less output in terms of phosphorylated Hog1 protein. This biological fact manifests itself also in the data. The WT data is characterized more by the activity in the Sln1 activation branch rather than the Sho1 branch. In other words, the Hog1PP levels in the WT cell are affected more by the changes in the Sln1 branch parameters than by the changes in the Sho1 branch parameters. Therefore, the WT data can substitute for the Sln1 data for training the models. However, the cost of excluding the Sho1 data from the training set is higher due to the asymmetry we mentioned above. The Sho1 branch parameters are weakly identifiable when the Sho1 data is not used for parameter estimation. This asymmetry in the information content of the data is clearly the output of the pathway machinery. This machinery is summarized into a model with a model structure and parameter values. Therefore, the decisions of model validation using a holdout strategy is dependent on the underlying biological properties (the asymmetrical branch structure in this particular example) and reflections of these properties in the model parameters (parameter values that allow less flux in one of the branches). The data partitioning task, hence, proves to be a difficult one since the prior knowledge about the underlying biology would never be complete.
Another important observation is the NaCl dose dependency of the predictive power using the Sho1 data. The predictive power using Sho1 data was especially lower in the lowest two doses compared to the higher doses as can be seen in Fig. 9 a and f (maximum of the medians 8.85 % vs. 4.65 % in the Sln1 scheme and 4.20 % vs. 1.52 % in the WT scheme).
In this section, we focused on partitioning schemes which use the data from only one cell type for parameter estimation. Our results show the importance of having a variety of different validation sets. This is because decisions of model validation and selection vary considerably depending on the experimental conditions of different validation sets due to unknown values of the underlying parameters.
Scenario 2: partitioning of data in different doses
These results showed us that predictive power becomes lower with increasing distances between the training and the validation sets, where the distance is measured in terms of the dose of the triggering chemical. This means that the risk of invalidating the true model structure increases when the validation set is too distant from the training set. However, the limits between which the model parameters stay applicable depend very much also on the cell type as we have observed. The predictive power on the Sho1 data deteriorated more rapidly compared to the other cell types. These observations helped us to identify a serious pitfall of doseresponse strategy: as long as we do not have realistic prior information on the limits for which we expect the estimated values of the model parameters to be applicable, we face the risk of invalidating a true model structure by overchallenging the model. Unfortunately, determination of the limits is not possible beforehand since it depends on the underlying biological properties which will never be completely known.
In cases where the differences between the number of wrong decisions is too low for a meaningful comparison, the model separation, ΔTS is more informative. Figure 14 shows the percentage of the realizations in which one specific scheme resulted in better model separation than the other scheme. The percentages are based on the number of realizations in which a correct decision was made by using both the lowest and the highest dose schemes. For example, we know that both schemes result in a correct decision in 77 realizations of the Sln1 data at 0.1 M. NaCl shock (data not shown). The first pie chart in Fig. 14 shows that in 99 % of these 77 realizations, the highest dose scheme resulted in better separation between the two model structures than the lowest dose scheme. As can be seen from this figure, model separation obtained by the highest dose scheme is higher than that obtained by the lowest dose scheme in almost all realizations of 0.1 M.  0.2 M. Sln1 and WT data. At 0.6 M. dose, the situation is reverse and the lowest scheme provides a better separation of the two model structures, in most of the realizations of all three cell types. These findings support the observation we made from the number of wrong decisions: model selection becomes problematic with too close training and validation sets. This is mainly because the simplified model might also predict well in the close proximity of the training dose (See Additional file 1: Figure S1). However, it will perform worse than the true model structure as the training and validation sets become more distant from each other. However, too much distance can also pose a problem for model selection due to increased uncertainty in the predictive power. Uncertainty in the predictions shows that different noise realizations can either give very good or very bad predictions. High levels of uncertainty reveals itself in the wide box plots of especially Sho1 validation data in Fig. 12 b, 12 e and Additional file 1: Figure S1, showing a wide dispersion of predictive power across different noise realizations. In these regions where the uncertainty is high, it becomes more difficult to anticipate the predictive powers of the true and the simplified model structures on a single noise realization. This hampers also model selection. Using Sho1 validation data results in such a situation where uncertainty is very high at certain doses. This is why the trends in model selection that we have presented in this section cannot be observed on the Sho1 validation data as sharply as on the other cell types.
We can understand the risks associated with high uncertainty in a holdout strategy, if we remember that in a single real experiment we have only one realization of noise. The outcomes of both model validation and selection depend highly on the specific noise realization in the data but we have only one realization available. This means that it is highly probable that we end up in wrong decisions just due to experimental noise. Therefore, we need partitioning schemes that minimize the effect of idiosyncratic noise realisations and lead to similar decisions for all of them. The stratified random cross validation scheme is promising in this sense as we will explain in the following section.
Introducing variation in the training and the test data
In the previous sections, we showed the pitfalls that we might come across if we use single doses or single cell types as validation data. Therefore, we stress the importance of consensus results obtained from a collection of different validation sets. In this section, we take it one step further and introduce variation of experimental conditions also in the training data. We do it in three different ways as described by the adapted scenarios and the stratified cross validation scheme in the Methods section. First, we include two different cell types in the training data, namely in the Sln1/Sho1, Sln1/WT and Sho1/WT schemes. Second, we include four different doses from each cell type in the training data, namely in the low doses and the high doses schemes. These are examples of holdout validation strategies just like the previous two scenarios. However, unlike those, the training and the validation sets include a variety of different cell types or doses. The third way is not an example of a holdout strategy. It is the stratified random crossvalidation (SRCV) scheme about which we have given the details in the Methods section. With this approach we can introduce variation in the training and validation sets in terms of both cell types and doses at the same time.
In addition to the risks associated with model validation, the Sln1/WT scheme performes poorly also in model selection with 16 wrong decisions. Therefore, we conclude that the Sln1/WT scheme is not a good scheme for model validation and selection whereas the Sln1/Sho1 and the Sho1/WT are sensible partitioning schemes. The SRCV scheme results in prediction errors that are comparable with the sensible partitioning schemes (Fig. 15 a). Furthermore, it results in no wrong decisions and it gives the highest model separation compared to the Sln1/Sho1 and Sho1/WT schemes which also result in all correct decisions (Fig. 15 b). In addition to this, the predictive power of such a scheme is less dependent on the noise realization compared to the other schemes as can be seen from the smaller box plots in Fig. 15 a. This indicates the low amount of uncertainty in the predictions.
When only doses were allowed to vary in the training set as in the case of the low doses and the high doses scheme, there was no significant difference in the predictive powers of the two schemes (Fig. 15 c). This revealed that none of the schemes posed more risk of invalidating the true model structure compared to the other scheme. However, there was a large difference in the model separation achieved by the two schemes (median = 16.9 vs. 5.06 in the low and high doses schemes respectively, shown in Fig. 15 d). This shows that the high doses scheme is unsuitable for model selection. A simulation showing weak model separation according to the highest dose scheme can be seen in Additional file 1: Figure S2. The SRCV scheme performed better than the unsuitable holdout partitioning scheme for model selection (median ΔTS = 8.18) and the predictive power was in the range of the two holdout partitioning schemes (Fig. 15 c).
The observations explained above can also be anticipated from the identifiability levels. The standard deviations of parameters at all three runs of the SRCV scheme were comparable to those obtained in the sensible holdout partitioning schemes for model validation (see Additional file 1: Figure S3) and were never higher than those obtained in the unsuitable schemes.
These results indicated that a stratified CV scheme is favorable for both model validation and selection. In most of the comparisons, it achieves predictive power and model separation as high as the optimal holdout partitioning scheme. In addition, it leads to lower uncertainty which means that the outcomes of model validation and selection depend less on the specific noise realization. More importantly, it never performs worse than unsuitable holdout partitioning schemes. The importance of this last statement lies in the fact that finding a sensible holdout partitioning scheme can never be guaranteed. It depends highly on the biology and therefore, on the model structure and the model parameters most of which are typically unknown prior to modeling. Therefore, there are no rules that can be set beforehand to make the finding of sensible partitioning schemes certain. Those factors might hinder us from opting for a sensible scheme. However, SRCV offers a judicious and reliable partitioning scheme for which no biological knowledge is required. Its good performance relies on two properties. Firstly, it is iterative which means that it allows each piece of data to contribute as both training and validation datasets in an iterative manner and summarizes the results as the average of different iterations. Secondly it offers random stratified partitioning, so it allows fair partitioning of the data while it prevents from certain cell types or doses dominating the training data. Therefore, issues like parameter estimation and model validation/selection are not biased in a certain direction as an artifact of an underlying biological property of the system, in contrast to the holdout validation schemes we have extensively investigated with this study. In addition, we achieve this by using a CV scheme with 3 folds and no repeats and hence, the computational time increases only three times compared to the holdout schemes.
On the other hand, our additional simulations with two more complex models revealed that a prerequisite for model selection based on predictive power has to be mentioned. The first more complex model included one additional parameter (Hog1 dependent Fps1 degradation) whereas the second model included three additional parameters (Hog1 dependent Fps1 degradation, Fps1 production and protein dependent Fps1 degradation). We have found out that the additional parameters were estimated very close to 0. Median of the parameters changed between 0.8×10^{−5} and 0.4×10^{−7} in all of the schemes. This means that both of the complex models boiled down to the true model structure. Therefore, the differences in the prediction errors obtained with the complex and the true model structures were very small. For example, the difference in the prediction errors of the true and the complex model structure was, in average 2.03 % of the prediction error of the true model structure obtained with the Sln1/Sho1 scheme. However, this value was 146 % in our simulations with the simplified model structure. At this level of extreme similarity between the model structures, model selection based on tiny differences between the predictive powers of the models leads to random conclusions that are heavily dependent on the specific noise realization in the data. Instead in such situations, investigating the estimated values of the additional parameters gives clue if a more complex model is needed or not. From this we derive the following important conclusion regarding the scalability of our approach. The guidelines we present in our manuscript are aimed for more reliable decision making in model selection when the selection is made based on the predictive powers of the models. In cases where such model selection is not applicable, our guidelines are obviously not applicable either.
Conclusions
Our results showed that the final decisions on model validation and selection can differ significantly when different holdout partitioning schemes are employed. The selection of a sensible holdout partitioning scheme that will help us to make reliable decisions depends on the biology. A good biological knowledge on the system and, hence, prior information on the structure and the true parameter values of the model are essential. Unfortunately, this is not possible in many instances. This turns the problem of finding a sensible partitioning scheme for model validation and selection into a Catch 22 problem. When the determination of a sensible partitioning scheme fails, we face the risk of invalidating true model structures or of failing to select the true model structure over the other alternatives. Examples of the first situation are very difficult to find in the literature, though, because, only successful validation examples are usually presented, leading to a ‘verification bias’. Furthermore, partitioning schemes that are sensible for model selection are not necessarily suitable for model validation. Datasets from very similar experimental conditions have only weak model selection capability whereas datasets from very diverse experimental conditions are not appropriate for model selection either due to high uncertainty in the predictions. However, using a proper crossvalidation approach such as stratified random crossvalidation can help us to overcome these problems while being independent of any prior biological knowledge.
With the SRCV approach, we can partition the data randomly into training and validation sets iteratively and arrive at consensus decisions by averaging over all different validation datasets. SRCV performs at least as well as sensible holdout partitioning schemes for both model validation and selection. On top of that, this comes without the risk of opting for an incorrect partitioning scheme which would lead us to biased conclusions. Furthermore, the decisions given within a SRCV scheme are less affected by the specific realization of the experimental noise. Due to all these reasons that we mention, SRCV proves to be a judicious, unbiased and promising alternative to the holdout validation strategy for the validation and selection of ODE based models.
Declarations
Acknowledgements
This project was financed by the Netherlands Metabolomics Centre (NMC) which is a part of the Netherlands Genomics Initiative/Netherlands Organisation for Scientific Research. The authors thank Gooitzen Zwanenburg for reading the manuscript and Jacky L. Snoep for fruitful discussions.
Authors’ Affiliations
References
 Link H, Kochanowski K, Sauer U. Systematic identification of allosteric proteinmetabolite interactions that control enzyme activity in vivo. Nat Biotechnol. 2013; 31(4):357–61.PubMedView ArticleGoogle Scholar
 Marucci L, Santini S, di Bernardo M, di Bernardo D. Derivation, identification and validation of a computational model of a novel synthetic regulatory network in yeast. J Math Biol. 2011; 62(5):685–706.PubMedView ArticleGoogle Scholar
 Maiwald T, Timmer J. Dynamical modeling and multiexperiment fitting with potterswheel. Bioinformatics. 2008; 24(18):2037–043.PubMed CentralPubMedView ArticleGoogle Scholar
 Schaber J, Klipp E. Modelbased inference of biochemical parameters and dynamic properties of microbial signal transduction networks. Curr Opin Biotechnol. 2011; 22(1):109–16.PubMedView ArticleGoogle Scholar
 Kirk PDW, Stumpf MPH. Gaussian process regression bootstrapping: exploring the effects of uncertainty in time course data. Bioinformatics (Oxford, England). 2009; 25(10):1300–6. doi:http://dx.doi.org/10.1093/bioinformatics/btp139.
 Joshi M, SeidelMorgenstern A, Kremling A. Exploiting the bootstrap method for quantifying parameter confidence intervals in dynamical systems. Metab Eng. 2006; 8(5):447–55. doi:http://dx.doi.org/10.1016/j.ymben.2006.04.003.
 du Preez FB, van Niekerk DD, Kooi B, Rohwer JM, Snoep JL. From steadystate to synchronized yeast glycolytic oscillations i: model construction. FEBS J. 2012; 279(16):2810–822.PubMedView ArticleGoogle Scholar
 du Preez FB, van Niekerk DD, Snoep JL. From steadystate to synchronized yeast glycolytic oscillations ii: model validation. FEBS J. 2012; 279(16):2823–836.PubMedView ArticleGoogle Scholar
 Klipp E, Liebermeister W, Wierling C, Kowald A, Lehrach H, Herwig R. Systems Biology: A Textbook. Weinheim: WILEYVCH Verlag GmbH & Co. KGaA; 2009.Google Scholar
 Cedersund G, Roll J. Systems biology: model based evaluation and comparison of potential explanations for given biological data. FEBS J. 2009; 276(4):903–22.PubMedView ArticleGoogle Scholar
 Müller T, Faller D, Timmer J, Swameye I, Sandra O, Klingmüller U. Tests for cycling in a signalling pathway. J R Stat Soc: Ser C: Appl Stat. 2004; 53(4):557–68.View ArticleGoogle Scholar
 Williams DA. Discrimination between regression models to determine the pattern of enzyme synthesis in synchronous cell cultures. Biometrics. 1970; 26:23–32.PubMedView ArticleGoogle Scholar
 Johansson R, Strålfors P, Cedersund G. Combining test statistics and models in bootstrapped model rejection: it is a balancing act. BMC Syst Biol. 2014; 8(1):46.PubMed CentralPubMedView ArticleGoogle Scholar
 Akaike H. A new look at the statistical model identification. Automatic Control IEEE Trans. 1974; 19(6):716–23.View ArticleGoogle Scholar
 Schwarz G, et al. Estimating the dimension of a model. Ann Stat. 1978; 6(2):461–4.View ArticleGoogle Scholar
 Kadam KL, Rydholm EC, McMillan JD. Development and validation of a kinetic model for enzymatic saccharification of lignocellulosic biomass. Biotechnol Prog. 2004; 20(3):698–705.PubMedView ArticleGoogle Scholar
 Efron B, Tibshirani RJ. An Introduction to the Bootstrap. Florida: CRC press LLC; 1994.Google Scholar
 Stone M. Crossvalidatory choice and assessment of statistical predictions. J R Stat Soc Ser B Methodol. 1974; 36:111–47.Google Scholar
 Kohavi R, et al. A study of crossvalidation and bootstrap for accuracy estimation and model selection. In: Ijcai: 1995. p. 1137–45.Google Scholar
 Weiss SM. Small sample error rate estimation for knn classifiers. IEEE Trans Pattern Anal Mach Intell. 1991; 13(3):285–9.View ArticleGoogle Scholar
 BragaNeto UM, Dougherty ER. Is crossvalidation valid for smallsample microarray classification?Bioinformatics. 2004; 20(3):374–80.PubMedView ArticleGoogle Scholar
 Breiman L, Spector P. Submodel selection and evaluation in regression. the xrandom case. International statistical review/revue internationale de Statistique. 1992; 60:291–319.Google Scholar
 Hasdemir D, Hoefsloot HC, Westerhuis JA, Smilde AK. How informative is your kinetic model?: using resampling methods for model invalidation. BMC Syst Biol. 2014; 8(1):61. doi:http://dx.doi.org/10.1186/17520509861.
 Kuepfer L, Peter M, Sauer U, Stelling J. Ensemble modeling for analysis of cell signaling dynamics. Nat Biotechnol. 2007; 25(9):1001–6. doi:http://dx.doi.org/10.1038/nbt1330.
 Schaber J, Baltanas R, Bush A, Klipp E, ColmanLerner A. Modelling reveals novel roles of two parallel signalling pathways and homeostatic feedbacks in yeast. Mol Syst Biol. 2012; 8(622):622. doi:http://dx.doi.org/10.1038/msb.2012.53.
 Le Novere N, Bornstein B, Broicher A, Courtot M, Donizelli M, Dharuri H, et al. Biomodels database: a free, centralized database of curated, published, quantitative kinetic models of biochemical and cellular systems. Nucleic acids research. 2006; 34(suppl 1):689–91.View ArticleGoogle Scholar
 Coleman TF, Li Y. On the convergence of interiorreflective newton methods for nonlinear minimization subject to bounds. Math Prog. 1994; 67(13):189–224.View ArticleGoogle Scholar
 Coleman TF, Li Y. An interior trust region approach for nonlinear minimization subject to bounds. SIAM J Optim. 1996; 6(2):418–45.View ArticleGoogle Scholar
Copyright
This is an Open Access article distributed under the terms of the Creative Commons Attribution License(http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.