Determining minimal output sets that ensure structural identifiability

Structural identifiability of biomolecular controller motifs with and without flow measurements as model output

Controller motifs are simple biomolecular reaction networks with negative feedback. They can explain how regulatory function is achieved and are often used as building blocks in mathematical models of biological systems. In this paper we perform an extensive investigation into structural identifiability of controller motifs, specifically the so-called basic and antithetic controller motifs. Structural identifiability analysis is a useful tool in the creation and evaluation of mathematical models: it can be used to ensure that model parameters can be determined uniquely and to examine which measurements are necessary for this purpose. This is especially useful for biological models where parameter estimation can be difficult due to limited availability of measureable outputs. Our aim with this work is to investigate how structural identifiability is affected by controller motif complexity and choice of measurements. To increase the number of potential outputs we propose two methods for including flow measurements and show how this affects structural identifiability in combination with, or in the absence of, concentration measurements. In our investigation, we analyze 128 different controller motif structures using a combination of flow and/or concentration measurements, giving a total of 3648 instances. Among all instances, 34% of the measurement combinations provided structural identifiability. Our main findings for the controller motifs include: i) a single measurement is insufficient for structural identifiability, ii) measurements related to different chemical species are necessary for structural identifiability. Applying these findings result in a reduced subset of 1568 instances, where 80% are structurally identifiable, and more complex/interconnected motifs appear easier to structurally identify. The model structures we have investigated are commonly used in models of biological systems, and our results demonstrate how different model structures and measurement combinations affect structural identifiability of controller motifs.

A plausible identifiable model of the canonical NF-κB signaling pathway

An overwhelming majority of mathematical models of regulatory pathways, including the intensively studied NF-κB pathway, remains non-identifiable, meaning that their parameters may not be determined by existing data. The existing NF-κB models that are capable of reproducing experimental data contain non-identifiable parameters, whereas simplified models with a smaller number of parameters exhibit dynamics that differs from that observed in experiments. Here, we reduced an existing model of the canonical NF-κB pathway by decreasing the number of equations from 15 to 6. The reduced model retains two negative feedback loops mediated by IκBα and A20, and in response to both tonic and pulsatile TNF stimulation exhibits dynamics that closely follow that of the original model. We carried out the sensitivity-based linear analysis and Monte Carlo-based analysis to demonstrate that the resulting model is both structurally and practically identifiable given measurements of 5 model variables from a simple TNF stimulation protocol. The reduced model is capable of reproducing different types of responses that are characteristic to regulatory motifs controlled by negative feedback loops: nearly-perfect adaptation as well as damped and sustained oscillations. It can serve as a building block of more comprehensive models of the immune response and cancer, where NF-κB plays a decisive role. Our approach, although may not be automatically generalized, suggests that models of other regulatory pathways can be transformed to identifiable, while retaining their dynamical features.

Mathematical Modelling in Plant Synthetic Biology

Mathematical modelling techniques are integral to current research in plant synthetic biology. Modelling approaches can provide mechanistic understanding of a system, allowing predictions of behaviour and thus providing a tool to help design and analyse biological circuits. In this chapter, we provide an overview of mathematical modelling methods and their significance for plant synthetic biology. Starting with the basics of dynamics, we describe the process of constructing a model over both temporal and spatial scales and highlight crucial approaches, such as stochastic modelling and model-based design. Next, we focus on the model parameters and the techniques required in parameter analysis. We then describe the process of selecting a model based on tests and criteria and proceed to methods that allow closer analysis of the system's behaviour. Finally, we highlight the importance of uncertainty in modelling approaches and how to deal with a lack of knowledge, noisy data, and biological variability; all aspects that play a crucial role in the cooperation between the experimental and modelling components. Overall, this chapter aims to illustrate the importance of mathematical modelling in plant synthetic biology, providing an introduction for those researchers who are working with or working on modelling techniques.

Assessing the role of initial conditions in the local structural identifiability of large dynamic models

Structural identifiability is a binary property that determines whether or not unique parameter values can, in principle, be estimated from error-free input-output data. The many papers that have been written on this topic collectively stress the importance of this a priori analysis in the model development process. The story however, often ends with a structurally unidentifiable model. This may leave a model developer with no plan of action on how to address this potential issue. We continue this model exploration journey by identifying one of the possible sources of a model's unidentifiability: problematic initial conditions. It is well-known that certain initial values may result in the loss of local structural identifiability. Nevertheless, literature on this topic has been limited to the analysis of small toy models. Here, we present a systematic approach to detect problematic initial conditions of real-world systems biology models, that are usually not small. A model's identifiability can often be reinstated by changing the value of such problematic initial conditions. This provides modellers an option to resolve the "unidentifiable model" problem. Additionally, a good understanding of which initial values should rather be avoided can be very useful during experimental design. We show how our approach works in practice by applying it to five models. First, two small benchmark models are studied to get the reader acquainted with the method. The first one shows the effect of a zero-valued problematic initial condition. The second one illustrates that the approach also yields correct results in the presence of input signals and that problematic initial conditions need not be zero-values. For the remaining three examples, we set out to identify key initial values which may result in the structural unidentifiability. The third and fourth examples involve a systems biology Epo receptor model and a JAK/STAT model, respectively. In the final Pharmacokinetics model, of which its global structural identifiability has only recently been confirmed, we indicate that there are still sets of initial values for which this property does not hold.

An efficient procedure to assist in the re-parametrization of structurally unidentifiable models

An efficient method that assists in the re-parametrization of structurally unidentifiable models is introduced. It significantly reduces computational demand by combining numerical and symbolic identifiability calculations. This hybrid approach facilitates the re-parametrization of large unidentifiable ordinary differential equation models, including models where state transformations are required. A model is first assessed numerically, to discover potential structurally unidentifiable parameters. We then use symbolic calculations to confirm the numerical results, after which we describe the algebraic relationships between the unidentifiable parameters. Finally, the unidentifiable parameters are substituted with new parameters and simplification ensures that all the unidentifiable parameters are eliminated from the original model structure. The novelty of this method is its utilisation of numerical results, which notably reduces the number of symbolic calculations required. We illustrate our procedure and the detailed re-parametrization process in 5 examples: (1) an immunological model, (2) a microbial growth model, (3) a lung cancer model, (4) a JAK/STAT model, and (5) a small linear model with a non-scalable re-parametrization.