How to deal with large models?

Refinement-based modeling of the ErbB signaling pathway

The construction of large scale biological models is a laborious task, which is often addressed by adopting iterative routines for model augmentation, adding certain details to an initial high level abstraction of the biological phenomenon of interest. Refitting a model at every step of its development is time consuming and computationally intensive. The concept of model refinement brings about an effective alternative by providing adequate parameter values that ensure the preservation of its quantitative fit at every refinement step. We demonstrate this approach by constructing the largest-ever refinement-based biomodel, consisting of 421 species and 928 reactions. We start from an already fit, relatively small literature model whose consistency we check formally. We then construct the final model through an algorithmic step-by-step refinement procedure that ensures the preservation of the model's fit.

Automated adaptive inference of phenomenological dynamical models

Dynamics of complex systems is often driven by large and intricate networks of microscopic interactions, whose sheer size obfuscates understanding. With limited experimental data, many parameters of such dynamics are unknown, and thus detailed, mechanistic models risk overfitting and making faulty predictions. At the other extreme, simple ad hoc models often miss defining features of the underlying systems. Here we develop an approach that instead constructs phenomenological, coarse-grained models of network dynamics that automatically adapt their complexity to the available data. Such adaptive models produce accurate predictions even when microscopic details are unknown. The approach is computationally tractable, even for a relatively large number of dynamical variables. Using simulated data, it correctly infers the phase space structure for planetary motion, avoids overfitting in a biological signalling system and produces accurate predictions for yeast glycolysis with tens of data points and over half of the interacting species unobserved.

Mechanistic modelling of dynamical phenomena with many degrees of freedom runs the risk of overfitting and making faulty predictions, whereas ad hoc models may miss defining features. Here the authors develop an approach to construct dynamical models that adapt their complexity to the amount of available data.

Convergence in parameters and predictions using computational experimental design

Typically, biological models fitted to experimental data suffer from significant parameter uncertainty, which can lead to inaccurate or uncertain predictions. One school of thought holds that accurate estimation of the true parameters of a biological system is inherently problematic. Recent work, however, suggests that optimal experimental design techniques can select sets of experiments whose members probe complementary aspects of a biochemical network that together can account for its full behaviour. Here, we implemented an experimental design approach for selecting sets of experiments that constrain parameter uncertainty. We demonstrated with a model of the epidermal growth factor–nerve growth factor pathway that, after synthetically performing a handful of optimal experiments, the uncertainty in all 48 parameters converged below 10 per cent. Furthermore, the fitted parameters converged to their true values with a small error consistent with the residual uncertainty. When untested experimental conditions were simulated with the fitted models, the predicted species concentrations converged to their true values with errors that were consistent with the residual uncertainty. This paper suggests that accurate parameter estimation is achievable with complementary experiments specifically designed for the task, and that the resulting parametrized models are capable of accurate predictions.

A multiscale model of interleukin-6-mediated immune regulation in Crohn's disease and its application in drug discovery and development

In this study, we have developed a multiscale systems model of interleukin (IL)-6-mediated immune regulation in Crohn's disease, by integrating intracellular signaling with organ-level dynamics of pharmacological markers underlying the disease. This model was linked to a general pharmacokinetic model for therapeutic monoclonal antibodies and used to comparatively study various biotherapeutic strategies targeting IL-6-mediated signaling in Crohn's disease. Our work illustrates techniques to develop mechanistic models of disease biology to study drug-system interaction. Despite a sparse training data set, predictions of the model were qualitatively validated by clinical biomarker data from a pilot trial with tocilizumab. Model-based analysis suggests that strategies targeting IL-6, IL-6Rα, or the IL-6/sIL-6Rα complex are less effective at suppressing pharmacological markers of Crohn's than dual targeting the IL-6/sIL-6Rα complex in addition to IL-6 or IL-6Rα. The potential value of multiscale system pharmacology modeling in drug discovery and development is also discussed.CPT: Pharmacometrics & Systems Pharmacology (2014) 3, e89; doi:10.1038/psp.2013.64; advance online publication 8 January 2014.

Segregation and crosstalk of D1 receptor-mediated activation of ERK in striatal medium spiny neurons upon acute administration of psychostimulants

The convergence of corticostriatal glutamate and dopamine from the midbrain in the striatal medium spiny neurons (MSN) triggers synaptic plasticity that underlies reinforcement learning and pathological conditions such as psychostimulant addiction. The increase in striatal dopamine produced by the acute administration of psychostimulants has been found to activate not only effectors of the AC5/cAMP/PKA signaling cascade such as GluR1, but also effectors of the NMDAR/Ca(2+)/RAS cascade such as ERK. The dopamine-triggered effects on both these cascades are mediated by D1R coupled to Golf but while the phosphorylation of GluR1 is affected by reductions in the available amount of Golf but not of D1R, the activation of ERK follows the opposite pattern. This segregation is puzzling considering that D1R-induced Golf activation monotonically increases with DA and that there is crosstalk from the AC5/cAMP/PKA cascade to the NMDAR/Ca(2+)/RAS cascade via a STEP (a tyrosine phosphatase). In this work, we developed a signaling model which accounts for this segregation based on the assumption that a common pool of D1R and Golf is distributed in two D1R/Golf signaling compartments. This model integrates a relatively large amount of experimental data for neurons in vivo and in vitro. We used it to explore the crosstalk topologies under which the sensitivities of the AC5/cAMP/PKA signaling cascade to reductions in D1R or Golf are transferred or not to the activation of ERK. We found that the sequestration of STEP by its substrate ERK together with the insensitivity of STEP activity on targets upstream of ERK (i.e. Fyn and NR2B) to PKA phosphorylation are able to explain the experimentally observed segregation. This model provides a quantitative framework for simulation based experiments to study signaling required for long term potentiation in MSNs.

Rule-based modeling: a computational approach for studying biomolecular site dynamics in cell signaling systems

Rule-based modeling was developed to address the limitations of traditional approaches for modeling chemical kinetics in cell signaling systems. These systems consist of multiple interacting biomolecules (e.g., proteins), which themselves consist of multiple parts (e.g., domains, linear motifs, and sites of phosphorylation). Consequently, biomolecules that mediate information processing generally have the potential to interact in multiple ways, with the number of possible complexes and posttranslational modification states tending to grow exponentially with the number of binary interactions considered. As a result, only large reaction networks capture all possible consequences of the molecular interactions that occur in a cell signaling system, which is problematic because traditional modeling approaches for chemical kinetics (e.g., ordinary differential equations) require explicit network specification. This problem is circumvented through representation of interactions in terms of local rules. With this approach, network specification is implicit and model specification is concise. Concise representation results in a coarse graining of chemical kinetics, which is introduced because all reactions implied by a rule inherit the rate law associated with that rule. Coarse graining can be appropriate if interactions are modular, and the coarseness of a model can be adjusted as needed. Rules can be specified using specialized model-specification languages, and recently developed tools designed for specification of rule-based models allow one to leverage powerful software engineering capabilities. A rule-based model comprises a set of rules, which can be processed by general-purpose simulation and analysis tools to achieve different objectives (e.g., to perform either a deterministic or stochastic simulation).

Programming biological models in Python using PySB

PySB is a framework for creating biological models as Python programs using a high-level, action-oriented vocabulary that promotes transparency, extensibility and reusability. PySB interoperates with many existing modeling tools and supports distributed model development.

PySB models are programs and leverage existing programming tools for documentation, testing, and collaborative development.

Reusable functions can encode common low-level biochemical processes as well as high-level modules, making models transparent and concise.

Modeling workflow is accelerated through close integration with Python numerical tools and interoperability with existing modeling software.

We demonstrate the use of PySB to encode 15 alternative hypotheses for the mitochondrial regulation of apoptosis, including a new ‘Embedded Together' model based on recent biochemical findings.

Mathematical equations are fundamental to modeling biological networks, but as networks get large and revisions frequent, it becomes difficult to manage equations directly or to combine previously developed models. Multiple simultaneous efforts to create graphical standards, rule-based languages, and integrated software workbenches aim to simplify biological modeling but none fully meets the need for transparent, extensible, and reusable models. In this paper we describe PySB, an approach in which models are not only created using programs, they are programs. PySB draws on programmatic modeling concepts from little b and ProMot, the rule-based languages BioNetGen and Kappa and the growing library of Python numerical tools. Central to PySB is a library of macros encoding familiar biochemical actions such as binding, catalysis, and polymerization, making it possible to use a high-level, action-oriented vocabulary to construct detailed models. As Python programs, PySB models leverage tools and practices from the open-source software community, substantially advancing our ability to distribute and manage the work of testing biochemical hypotheses. We illustrate these ideas using new and previously published models of apoptosis.

Optimization and model reduction in the high dimensional parameter space of a budding yeast cell cycle model

Background

Parameter estimation from experimental data is critical for mathematical modeling of protein regulatory networks. For realistic networks with dozens of species and reactions, parameter estimation is an especially challenging task. In this study, we present an approach for parameter estimation that is effective in fitting a model of the budding yeast cell cycle (comprising 26 nonlinear ordinary differential equations containing 126 rate constants) to the experimentally observed phenotypes (viable or inviable) of 119 genetic strains carrying mutations of cell cycle genes.

Results

Starting from an initial guess of the parameter values, which correctly captures the phenotypes of only 72 genetic strains, our parameter estimation algorithm quickly improves the success rate of the model to 105–111 of the 119 strains. This success rate is comparable to the best values achieved by a skilled modeler manually choosing parameters over many weeks. The algorithm combines two search and optimization strategies. First, we use Latin hypercube sampling to explore a region surrounding the initial guess. From these samples, we choose ∼20 different sets of parameter values that correctly capture wild type viability. These sets form the starting generation of differential evolution that selects new parameter values that perform better in terms of their success rate in capturing phenotypes. In addition to producing highly successful combinations of parameter values, we analyze the results to determine the parameters that are most critical for matching experimental outcomes and the most competitive strains whose correct outcome with a given parameter vector forces numerous other strains to have incorrect outcomes. These “most critical parameters” and “most competitive strains” provide biological insights into the model. Conversely, the “least critical parameters” and “least competitive strains” suggest ways to reduce the computational complexity of the optimization.

Conclusions

Our approach proves to be a useful tool to help systems biologists fit complex dynamical models to large experimental datasets. In the process of fitting the model to the data, the tool identifies suggestive correlations among aspects of the model and the data.

Multi-level model for the investigation of oncoantigen-driven vaccination effect

Background

Cancer stem cell theory suggests that cancers are derived by a population of cells named Cancer Stem Cells (CSCs) that are involved in the growth and in the progression of tumors, and lead to a hierarchical structure characterized by differentiated cell population. This cell heterogeneity affects the choice of cancer therapies, since many current cancer treatments have limited or no impact at all on CSC population, while they reveal a positive effect on the differentiated cell populations.

Results

In this paper we investigated the effect of vaccination on a cancer hierarchical structure through a multi-level model representing both population and molecular aspects. The population level is modeled by a system of Ordinary Differential Equations (ODEs) describing the cancer population's dynamics. The molecular level is modeled using the Petri Net (PN) formalism to detail part of the proliferation pathway. Moreover, we propose a new methodology which exploits the temporal behavior derived from the molecular level to parameterize the ODE system modeling populations. Using this multi-level model we studied the ErbB2-driven vaccination effect in breast cancer.

Conclusions

We propose a multi-level model that describes the inter-dependencies between population and genetic levels, and that can be efficiently used to estimate the efficacy of drug and vaccine therapies in cancer models, given the availability of molecular data on the cancer driving force.

Model composition through model reduction: a combined model of CD95 and NF-κB signaling pathways

Background

Many mathematical models characterizing mechanisms of cell fate decisions have been constructed recently. Their further study may be impossible without development of methods of model composition, which is complicated by the fact that several models describing the same processes could use different reaction chains or incomparable sets of parameters. Detailed models not supported by sufficient volume of experimental data suffer from non-unique choice of parameter values, non-reproducible results, and difficulty of analysis. Thus, it is necessary to reduce existing models to identify key elements determining their dynamics, and it is also required to design the methods allowing us to combine them.

Results

Here we propose a new approach to model composition, based on reducing several models to the same level of complexity and subsequent combining them together. Firstly, we suggest a set of model reduction tools that can be systematically applied to a given model. Secondly, we suggest a notion of a minimal complexity model. This model is the simplest one that can be obtained from the original model using these tools and still able to approximate experimental data. Thirdly, we propose a strategy for composing the reduced models together. Connection with the detailed model is preserved, which can be advantageous in some applications. A toolbox for model reduction and composition has been implemented as part of the BioUML software and tested on the example of integrating two previously published models of the CD95 (APO-1/Fas) signaling pathways. We show that the reduced models lead to the same dynamical behavior of observable species and the same predictions as in the precursor models. The composite model is able to recapitulate several experimental datasets which were used by the authors of the original models to calibrate them separately, but also has new dynamical properties.

Conclusion

Model complexity should be comparable to the complexity of the data used to train the model. Systematic application of model reduction methods allows implementing this modeling principle and finding models of minimal complexity compatible with the data. Combining such models is much easier than of precursor models and leads to new model properties and predictions.

Likelihood based observability analysis and confidence intervals for predictions of dynamic models

BACKGROUND

Predicting a system's behavior based on a mathematical model is a primary task in Systems Biology. If the model parameters are estimated from experimental data, the parameter uncertainty has to be translated into confidence intervals for model predictions. For dynamic models of biochemical networks, the nonlinearity in combination with the large number of parameters hampers the calculation of prediction confidence intervals and renders classical approaches as hardly feasible.

RESULTS

In this article reliable confidence intervals are calculated based on the prediction profile likelihood. Such prediction confidence intervals of the dynamic states can be utilized for a data-based observability analysis. The method is also applicable if there are non-identifiable parameters yielding to some insufficiently specified model predictions that can be interpreted as non-observability. Moreover, a validation profile likelihood is introduced that should be applied when noisy validation experiments are to be interpreted.

CONCLUSIONS

The presented methodology allows the propagation of uncertainty from experimental to model predictions. Although presented in the context of ordinary differential equations, the concept is general and also applicable to other types of models. Matlab code which can be used as a template to implement the method is provided at http://www.fdmold.uni-freiburg.de/∼ckreutz/PPL.

Decoding complex biological networks - tracing essential and modulatory parameters in complex and simplified models of the cell cycle

Background

One of the most well described cellular processes is the cell cycle, governing cell division. Mathematical models of this gene-protein network are therefore a good test case for assessing to what extent we can dissect the relationship between model parameters and system dynamics. Here we combine two strategies to enable an exploration of parameter space in relation to model output. A simplified, piecewise linear approximation of the original model is combined with a sensitivity analysis of the same system, to obtain and validate analytical expressions describing the dynamical role of different model parameters.

Results

We considered two different output responses to parameter perturbations. One was qualitative and described whether the system was still working, i.e. whether there were oscillations. We call parameters that correspond to such qualitative change in system response essential. The other response pattern was quantitative and measured changes in cell size, corresponding to perturbations of modulatory parameters. Analytical predictions from the simplified model concerning the impact of different parameters were compared to a sensitivity analysis of the original model, thus evaluating the predictions from the simplified model. The comparison showed that the predictions on essential and modulatory parameters were satisfactory for small perturbations, but more discrepancies were seen for larger perturbations. Furthermore, for this particular cell cycle model, we found that most parameters were either essential or modulatory. Essential parameters required large perturbations for identification, whereas modulatory parameters were more easily identified with small perturbations. Finally, we used the simplified model to make predictions on critical combinations of parameter perturbations.

Conclusions

The parameter characterizations of the simplified model are in large consistent with the original model and the simplified model can give predictions on critical combinations of parameter perturbations. We believe that the distinction between essential and modulatory perturbation responses will be of use for sensitivity analysis, and in discussions of robustness and during the model simplification process.

A Boolean probabilistic model of metabolic adaptation to oxygen in relation to iron homeostasis and oxidative stress

Background

In aerobically grown cells, iron homeostasis and oxidative stress are tightly linked processes implicated in a growing number of diseases. The deregulation of iron homeostasis due to gene defects or environmental stresses leads to a wide range of diseases with consequences for cellular metabolism that remain poorly understood. The modelling of iron homeostasis in relation to the main features of metabolism, energy production and oxidative stress may provide new clues to the ways in which changes in biological processes in a normal cell lead to disease.

Results

Using a methodology based on probabilistic Boolean modelling, we constructed the first model of yeast iron homeostasis including oxygen-related reactions in the frame of central metabolism. The resulting model of 642 elements and 1007 reactions was validated by comparing simulations with a large body of experimental results (147 phenotypes and 11 metabolic flux experiments). We removed every gene, thus generating in silico mutants. The simulations of the different mutants gave rise to a remarkably accurate qualitative description of most of the experimental phenotype (overall consistency > 91.5%). A second validation involved analysing the anaerobiosis to aerobiosis transition. Therefore, we compared the simulations of our model with different levels of oxygen to experimental metabolic flux data. The simulations reproducted accurately ten out of the eleven metabolic fluxes. We show here that our probabilistic Boolean modelling strategy provides a useful description of the dynamics of a complex biological system. A clustering analysis of the simulations of all in silico mutations led to the identification of clear phenotypic profiles, thus providing new insights into some metabolic response to stress conditions. Finally, the model was also used to explore several new hypothesis in order to better understand some unexpected phenotypes in given mutants.

Conclusions

All these results show that this model, and the underlying modelling strategy, are powerful tools for improving our understanding of complex biological problems.

Biomedical engineering strategies in system design space

Modern systems biology and synthetic bioengineering face two major challenges in relating properties of the genetic components of a natural or engineered system to its integrated behavior. The first is the fundamental unsolved problem of relating the digital representation of the genotype to the analog representation of the parameters for the molecular components. For example, knowing the DNA sequence does not allow one to determine the kinetic parameters of an enzyme. The second is the fundamental unsolved problem of relating the parameters of the components and the environment to the phenotype of the global system. For example, knowing the parameters does not tell one how many qualitatively distinct phenotypes are in the organism's repertoire or the relative fitness of the phenotypes in different environments. These also are challenges for biomedical engineers as they attempt to develop therapeutic strategies to treat pathology or to redirect normal cellular functions for biotechnological purposes. In this article, the second of these fundamental challenges will be addressed, and the notion of a "system design space" for relating the parameter space of components to the phenotype space of bioengineering systems will be focused upon. First, the concept of a system design space will be motivated by introducing one of its key components from an intuitive perspective. Second, a simple linear example will be used to illustrate a generic method for constructing the design space in which qualitatively distinct phenotypes can be identified and counted, their fitness analyzed and compared, and their tolerance to change measured. Third, two examples of nonlinear systems from different areas of biomedical engineering will be presented. Finally, after giving reference to a few other applications that have made use of the system design space approach to reveal important design principles, some concluding remarks concerning challenges and opportunities for further development will be made.

Integrated computational model of cell cycle and checkpoint reveals different essential roles of Aurora-A and Plk1 in mitotic entry

Understanding the regulation of mitotic entry is one of the most important goals of modern cell biology, and computational modeling of mitotic entry has been a subject of several recent studies. However, there are still many regulation mechanisms that remain poorly characterized. Two crucial aspects are how mitotic entry is controlled by its upstream regulators Aurora-A and Plk1, and how mitotic entry is coordinated with other biological events, especially G2/M checkpoint. In this context, we reconstructed a comprehensive computational model that integrates the mitotic entry network and the G2/M checkpoint system. Computational simulation of this model and subsequent experimental verification revealed that Aurora-A and Plk1 are redundant to the activation of cyclin B/Cdk1 during normal mitotic entry, but become especially important for cyclin B/Cdk1 activation during G2/M checkpoint recovery. Further analysis indicated that, in response to DNA damage, Chk1-mediated network rewiring makes cyclin B/Cdk1 more sensitive to the down-regulation of Aurora-A and Plk1. In addition, we demonstrated that concurrently targeting Aurora-A and Plk1 during G2/M checkpoint recovery achieves a synergistic effect, which suggests the combinational use of Aurora-A and Plk1 inhibitors after chemotherapy or radiotherapy. Thus, the results presented here provide novel insights into the regulation mechanism of mitotic entry and have potential value in cancer therapy.

Modelling the response of FOXO transcription factors to multiple post-translational modifications made by ageing-related signalling pathways

FOXO transcription factors are an important, conserved family of regulators of cellular processes including metabolism, cell-cycle progression, apoptosis and stress resistance. They are required for the efficacy of several of the genetic interventions that modulate lifespan. FOXO activity is regulated by multiple post-translational modifications (PTMs) that affect its subcellular localization, half-life, DNA binding and transcriptional activity. Here, we show how a mathematical modelling approach can be used to simulate the effects, singly and in combination, of these PTMs. Our model is implemented using the Systems Biology Markup Language (SBML), generated by an ancillary program and simulated in a stochastic framework. The use of the ancillary program to generate the SBML is necessary because the possibility that many regulatory PTMs may be added, each independently of the others, means that a large number of chemically distinct forms of the FOXO molecule must be taken into account, and the program is used to generate them. Although the model does not yet include detailed representations of events upstream and downstream of FOXO, we show how it can qualitatively, and in some cases quantitatively, reproduce the known effects of certain treatments that induce various single and multiple PTMs, and allows for a complex spatiotemporal interplay of effects due to the activation of multiple PTM-inducing treatments. Thus, it provides an important framework to integrate current knowledge about the behaviour of FOXO. The approach should be generally applicable to other proteins experiencing multiple regulations.

Qualitatively distinct phenotypes in the design space of biochemical systems

Although characterization of the genotype has undergone revolutionary advances as a result of the successful genome projects, the chasm between our understanding of a fully characterized gene sequence and the phenotypic repertoire of the organism is as broad and deep as it was in the pre-genomic era. There are two fundamental unsolved problems that provide the context for the challenges in relating genotype to phenotype. We address one of these and describe a generic method for constructing a system design space in which qualitatively distinct phenotypes can be identified and counted, their relative fitness analyzed and compared, and their tolerance to change measured.