A combined model reduction algorithm for controlled biochemical systems

Index analysis: An approach to understand signal transduction with application to the EGFR signalling pathway

In systems biology and pharmacology, large-scale kinetic models are used to study the dynamic response of a system to a specific input or stimulus. While in many applications, a deeper understanding of the input-response behaviour is highly desirable, it is often hindered by the large number of molecular species and the complexity of the interactions. An approach that identifies key molecular species for a given input-response relationship and characterises dynamic properties of states is therefore highly desirable. We introduce the concept of index analysis; it is based on different time- and state-dependent quantities (indices) to identify important dynamic characteristics of molecular species. All indices are defined for a specific pair of input and response variables as well as for a specific magnitude of the input. In application to a large-scale kinetic model of the EGFR signalling cascade, we identified different phases of signal transduction, the peculiar role of Phosphatase3 during signal activation and Ras recycling during signal onset. In addition, we discuss the challenges and pitfalls of interpreting the relevance of molecular species based on knock-out simulation studies, and provide an alternative view on conflicting results on the importance of parallel EGFR downstream pathways. Beyond the applications in model interpretation, index analysis is envisioned to be a valuable tool in model reduction.

A hybrid mechanistic-empirical model for in silico mammalian cell bioprocess simulation

In cell culture processes cell growth and metabolism drive changes in the chemical environment of the culture. These environmental changes elicit reactor control actions, cell growth response, and are sensed by cell signaling pathways that influence metabolism. The interplay of these forces shapes the culture dynamics through different stages of cell cultivation and the outcome greatly affects process productivity, product quality, and robustness. Developing a systems model that describes the interactions of those major players in the cell culture system can lead to better process understanding and enhance process robustness. Here we report the construction of a hybrid mechanistic-empirical bioprocess model which integrates a mechanistic metabolic model with subcomponent models for cell growth, signaling regulation, and the bioreactor environment for in silico exploration of process scenarios. Model parameters were optimized by fitting to a dataset of cell culture manufacturing process which exhibits variability in metabolism and productivity. The model fitting process was broken into multiple steps to mitigate the substantial numerical challenges related to the first-principles model components. The optimized model captured the dynamics of metabolism and the variability of the process runs with different kinetic profiles and productivity. The variability of the process was attributed in part to the metabolic state of cell inoculum. The model was then used to identify potential mitigation strategies to reduce process variability by altering the initial process conditions as well as to explore the effect of changing CO₂ removal capacity in different bioreactor scales on process performance. By incorporating a mechanistic model of cell metabolism and appropriately fitting it to a large dataset, the hybrid model can describe the different metabolic phases in culture and the variability in manufacturing runs. This approach of employing a hybrid model has the potential to greatly facilitate process development and reactor scaling.

Model reduction of genome-scale metabolic models as a basis for targeted kinetic models

Constraint-based, genome-scale metabolic models are an essential tool to guide metabolic engineering. However, they lack the detail and time dimension that kinetic models with enzyme dynamics offer. Model reduction can be used to bridge the gap between the two methods and allow for the integration of kinetic models into the Design-Built-Test-Learn cycle. Here we show that these reduced size models can be representative of the dynamics of the original model and demonstrate the automated generation and parameterisation of such models. Using these minimal models of metabolism could allow for further exploration of dynamic responses in metabolic networks.

Metabolic kinetic modeling provides insight into complex biological questions, but hurdles remain

Metabolic models containing kinetic information can answer unique questions about cellular metabolism that are useful to metabolic engineering. Several kinetic modeling frameworks have recently been developed or improved. In addition, techniques for systematic identification of model structure, including regulatory interactions, have been reported. Each framework has advantages and limitations, which can make it difficult to choose the most appropriate framework. Common limitations are data availability and computational time, especially in large-scale modeling efforts. However, recently developed experimental techniques, parameter identification algorithms, as well as model reduction techniques help alleviate these computational bottlenecks. Opportunities for additional improvements may come from the rich literature in catalysis and chemical networks. In all, kinetic models are positioned to make significant impact in cellular engineering.

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.

Understanding and reducing complex systems pharmacology models based on a novel input-response index

A growing understanding of complex processes in biology has led to large-scale mechanistic models of pharmacologically relevant processes. These models are increasingly used to study the response of the system to a given input or stimulus, e.g., after drug administration. Understanding the input-response relationship, however, is often a challenging task due to the complexity of the interactions between its constituents as well as the size of the models. An approach that quantifies the importance of the different constituents for a given input-output relationship and allows to reduce the dynamics to its essential features is therefore highly desirable. In this article, we present a novel state- and time-dependent quantity called the input-response index that quantifies the importance of state variables for a given input-response relationship at a particular time. It is based on the concept of time-bounded controllability and observability, and defined with respect to a reference dynamics. In application to the brown snake venom-fibrinogen (Fg) network, the input-response indices give insight into the coordinated action of specific coagulation factors and about those factors that contribute only little to the response. We demonstrate how the indices can be used to reduce large-scale models in a two-step procedure: (i) elimination of states whose dynamics have only minor impact on the input-response relationship, and (ii) proper lumping of the remaining (lower order) model. In application to the brown snake venom-fibrinogen network, this resulted in a reduction from 62 to 8 state variables in the first step, and a further reduction to 5 state variables in the second step. We further illustrate that the sequence, in which a recursive algorithm eliminates and/or lumps state variables, has an impact on the final reduced model. The input-response indices are particularly suited to determine an informed sequence, since they are based on the dynamics of the original system. In summary, the novel measure of importance provides a powerful tool for analysing the complex dynamics of large-scale systems and a means for very efficient model order reduction of nonlinear systems.

Selection and Qualification of Simplified QSP Models When Using Model Order Reduction Techniques

Quantitative systems pharmacology (QSP) models are increasingly used in drug development to provide a deep understanding of the mechanism of action of drugs and to identify appropriate disease targets. Such models are, however, not suitable for estimation purposes due to their high dimensionality. Based on any desired and specific input-output relationship, the system may be reduced to a model with fewer states and parameters. However, any simplification process will be a trade-off between model performance and complexity. In this study, we develop a weighted composite criterion which brings together the opposing indices of performance and dimensionality. The weighting factor can be determined by qualification of the simplified model based on a visual predictive check (VPC) using the precision of each parameter. The weighted criterion and model qualification techniques were illustrated with three examples: a simple compartmental pharmacokinetic model, a physiologically based pharmacokinetic (PBPK) example, and a semimechanistic model for bone mineral density. When considering the PBPK example, this automated search identified the same reduced model which had been detected in a previous report, as well as a simpler model which had not been previously identified. The simpler bone mineral density model provided an adequate description of the response even after 1 year from the initiation of treatment. The proposed criterion together with a VPC provides a natural way for model order reduction that can be fully automated and applied to multiscale models.