New time-scale criteria for model simplification of bio-reaction systems

A mean-field approach for modeling the propagation of perturbations in biochemical reaction networks

Often, the time evolution of a biochemical reaction network is crucial for determining the effects of combining multiple pharmaceuticals. Here we illustrate a mathematical framework for modeling the dominant temporal behaviour of a complicated molecular pathway or biochemical reaction network in response to an arbitrary perturbation, such as resulting from the administration of a therapeutic agent. The method enables the determination of the temporal evolution of a target protein as the perturbation propagates through its regulatory network. The mathematical approach is particularly useful when the experimental data that is available for characterizing or parameterizing the regulatory network is limited or incomplete. To illustrate the method, we consider the examples of the regulatory networks for the target proteins c-Myc and Chop, which play an important role in venetoclax resistance in acute myeloid leukemia. First we show how the networks that regulate each target protein can be reduced to a mean-field model by identifying the distinct effects that groups of proteins in the regulatory network have on the target protein. Then we show how limited protein-level data can be used to further simplify the mean-field model to pinpoint the dominant effects of the network perturbation on the target protein. This enables a further reduction in the number of parameters in the model. The result is an ordinary differential equation model that captures the temporal evolution of the expression of a target protein when one or more proteins in its regulatory network have been perturbed. Finally, we show how the dominant effects predicted by the mathematical model agree with RNA sequencing data for the regulatory proteins comprising the molecular network, despite the model not having a priori knowledge of this data. Thus, while the approach gives a simplified model for the expression of the target protein, it allows for the interpretation of the effects of the perturbation on the regulatory network itself. This method can be easily extended to sets of target proteins to model components of a larger systems biology model, and provides an approach for partially integrating RNA sequencing data and protein expression data. Moreover, it is a general approach that can be used to study drug effects on specific protein(s) in any disease or condition.

A new hierarchical approach to multi-level model abstraction for simplifying ODE models of biological networks and a case study: The G1/S Checkpoint/DNA damage signalling pathways of mammalian cell cycle

Model reduction is an important topic in studies of biological systems. By reducing the complexity of large models through multi-level models while keeping the essence (biological meaning) of the model, model reduction can help answer many important questions about these systems. In this paper, we present a new reduction method based on hierarchical representation and a lumping approach. We used G1/S checkpoint pathway represented in Ordinary Differential Equations (ODE) in Iwamoto et al. (2011) as a case study to present this reduction method. The approach consists of two parts; the first part represents the biological network as a hierarchy (multiple levels) based on protein binding relations, which allowed us to model the biological network at different levels of abstraction. The second part applies different levels (level 1, 2 and 3) of lumping the species together depending on the level of the hierarchy, resulting in a reduced and transformed model for each level. The model at each level is a representation of the whole system and can address questions pertinent to that level. We develop and simulate reduced models for levels-1, 2 and 3 of lumping for the G1/S checkpoint pathway and evaluate the biological plausibility of the proposed method by comparing the results with the original ODE model of Iwamoto et al. (2011). The results for continuous dynamics of the G1/S checkpoint pathway with or without DNA-damage for reduced models of level- 1, 2 and 3 of lumping are in very good agreement and consistent with the original model results and with biological findings. Therefore, the reduced models (level-1, 2 and 3) can be used to study cell cycle progression in G1 and the effects of DNA damage on it. It is suitable for reducing complex ODE biological network models while retaining accurate continuous dynamics of the system. The 3 levels of the reduced models respectively achieved 20%, 26% and 31% reduction of the base model. Moreover, the reduced model is more efficient to run (30%, 44% and 52% time reduction for the three levels) and generate solutions than the original ODE model. Simplification of complex mathematical models is possible and the proposed reduction method has the potential to make an impact across many fields of biomedical research.

BioUML: an integrated environment for systems biology and collaborative analysis of biomedical data

BioUML (homepage: http://www.biouml.org, main public server: https://ict.biouml.org) is a web-based integrated environment (platform) for systems biology and the analysis of biomedical data generated by omics technologies. The BioUML vision is to provide a computational platform to build virtual cell, virtual physiological human and virtual patient. BioUML spans a comprehensive range of capabilities, including access to biological databases, powerful tools for systems biology (visual modelling, simulation, parameters fitting and analyses), a genome browser, scripting (R, JavaScript) and a workflow engine. Due to integration with the Galaxy platform and R/Bioconductor, BioUML provides powerful possibilities for the analyses of omics data. The plug-in-based architecture allows the user to add new functionalities using plug-ins. To facilitate a user focus on a particular task or database, we have developed several predefined perspectives that display only those web interface elements that are needed for a specific task. To support collaborative work on scientific projects, there is a central authentication and authorization system (https://bio-store.org). The diagram editor enables several remote users to simultaneously edit diagrams.

Towards abstraction of computational modelling of mammalian cell cycle: Model reduction pipeline incorporating multi-level hybrid petri nets

Cell cycle is a large biochemical network and it is crucial to simplify it to gain a clearer understanding and insights into the cell cycle. This is also true for other biochemical networks. In this study, we present a model abstraction scheme/pipeline to create a minimal abstract model of the whole mammalian cell cycle system from a large Ordinary Differential Equation model of cell cycle we published previously (Abroudi et al., 2017). The abstract model is developed in a way that it captures the main characteristics (dynamics of key controllers), responses (G1-S and G2-M transitions and DNA damage) and the signalling subsystems (Growth Factor, G1-S and G2-M checkpoints, and DNA damage) of the original model (benchmark). Further, our model exploits: (i) separation of time scales (slow and fast reactions), (ii) separation of levels of complexity (high-level and low-level interactions), (iii) cell-cycle stages (temporality), (iv) functional subsystems (as mentioned above), and (v) represents the whole cell cycle - within a Multi-Level Hybrid Petri Net (MLHPN) framework. Although hybrid Petri Nets is not new, the abstraction of interactions and timing we introduced here is new to cell cycle and Petri Nets. Importantly, our models builds on the significant elements, representing the core cell cycle system, found through a novel Global Sensitivity Analysis on the benchmark model, using Self Organising Maps and Correlation Analysis that we introduced in (Abroudi et al., 2017). Taken the two aspects together, our study proposes a 2-stage model reduction pipeline for large systems and the main focus of this paper is on stage 2, Petri Net model, put in the context of the pipeline. With the MLHPN model, the benchmark model with 61 continuous variables (ODEs) and 148 parameters were reduced to 14 variables (4 continuous (Cyc_Cdks - the main controllers of cell cycle) and 10 discrete (regulators of Cyc_Cdks)) and 31 parameters. Additional 9 discrete elements represented the temporal progression of cell cycle. Systems dynamics simulation results of the MLHPN model were in close agreement with the benchmark model with respect to the crucial metrics selected for comparison: order and pattern of Cyc_Cdk activation, timing of G1-S and G2-M transitions with or without DNA damage, efficiency of the two cell cycle checkpoints in arresting damaged cells and passing healthy cells, and response to two types of global parameter perturbations. The results show that the MLHPN provides a close approximation to the comprehensive benchmark model in robustly representing systems dynamics and emergent properties while presenting the core cell cycle controller in an intuitive, transparent and subsystems format.

Model reduction in mathematical pharmacology : Integration, reduction and linking of PBPK and systems biology models

In this paper we present a framework for the reduction and linking of physiologically based pharmacokinetic (PBPK) models with models of systems biology to describe the effects of drug administration across multiple scales. To address the issue of model complexity, we propose the reduction of each type of model separately prior to being linked. We highlight the use of balanced truncation in reducing the linear components of PBPK models, whilst proper lumping is shown to be efficient in reducing typically nonlinear systems biology type models. The overall methodology is demonstrated via two example systems; a model of bacterial chemotactic signalling in Escherichia coli and a model of extracellular regulatory kinase activation mediated via the extracellular growth factor and nerve growth factor receptor pathways. Each system is tested under the simulated administration of three hypothetical compounds; a strong base, a weak base, and an acid, mirroring the parameterisation of pindolol, midazolam, and thiopental, respectively. Our method can produce up to an 80% decrease in simulation time, allowing substantial speed-up for computationally intensive applications including parameter fitting or agent based modelling. The approach provides a straightforward means to construct simplified Quantitative Systems Pharmacology models that still provide significant insight into the mechanisms of drug action. Such a framework can potentially bridge pre-clinical and clinical modelling - providing an intermediate level of model granularity between classical, empirical approaches and mechanistic systems describing the molecular scale.

Methods of Model Reduction for Large-Scale Biological Systems: A Survey of Current Methods and Trends

Complex models of biochemical reaction systems have become increasingly common in the systems biology literature. The complexity of such models can present a number of obstacles for their practical use, often making problems difficult to intuit or computationally intractable. Methods of model reduction can be employed to alleviate the issue of complexity by seeking to eliminate those portions of a reaction network that have little or no effect upon the outcomes of interest, hence yielding simplified systems that retain an accurate predictive capacity. This review paper seeks to provide a brief overview of a range of such methods and their application in the context of biochemical reaction network models. To achieve this, we provide a brief mathematical account of the main methods including timescale exploitation approaches, reduction via sensitivity analysis, optimisation methods, lumping, and singular value decomposition-based approaches. Methods are reviewed in the context of large-scale systems biology type models, and future areas of research are briefly discussed.

A combined model reduction algorithm for controlled biochemical systems

BACKGROUND

Systems Biology continues to produce increasingly large models of complex biochemical reaction networks. In applications requiring, for example, parameter estimation, the use of agent-based modelling approaches, or real-time simulation, this growing model complexity can present a significant hurdle. Often, however, not all portions of a model are of equal interest in a given setting. In such situations methods of model reduction offer one possible approach for addressing the issue of complexity by seeking to eliminate those portions of a pathway that can be shown to have the least effect upon the properties of interest.

METHODS

In this paper a model reduction algorithm bringing together the complementary aspects of proper lumping and empirical balanced truncation is presented. Additional contributions include the development of a criterion for the selection of state-variable elimination via conservation analysis and use of an 'averaged' lumping inverse. This combined algorithm is highly automatable and of particular applicability in the context of 'controlled' biochemical networks.

RESULTS

The algorithm is demonstrated here via application to two examples; an 11 dimensional model of bacterial chemotaxis in Escherichia coli and a 99 dimensional model of extracellular regulatory kinase activation (ERK) mediated via the epidermal growth factor (EGF) and nerve growth factor (NGF) receptor pathways. In the case of the chemotaxis model the algorithm was able to reduce the model to 2 state-variables producing a maximal relative error between the dynamics of the original and reduced models of only 2.8% whilst yielding a 26 fold speed up in simulation time. For the ERK activation model the algorithm was able to reduce the system to 7 state-variables, incurring a maximal relative error of 4.8%, and producing an approximately 10 fold speed up in the rate of simulation. Indices of controllability and observability are additionally developed and demonstrated throughout the paper. These provide insight into the relative importance of individual reactants in mediating a biochemical system's input-output response even for highly complex networks.

CONCLUSIONS

Through application, this paper demonstrates that combined model reduction methods can produce a significant simplification of complex Systems Biology models whilst retaining a high degree of predictive accuracy. In particular, it is shown that by combining the methods of proper lumping and empirical balanced truncation it is often possible to produce more accurate reductions than can be obtained by the use of either method in isolation.

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.