Delineating Parameter Unidentifiabilities in Complex Models

Scientists use mathematical modelling to understand and predict the properties of complex physical systems. In highly parameterised models there often exist relationships between parameters over which model predictions are identical, or nearly so. These are known as structural or practical unidentifiabilities, respectively. They are hard to diagnose and make reliable parameter estimation from data impossible. They furthermore imply the existence of an underlying model simplification. We describe a scalable method for detecting unidentifiabilities, and the functional relations defining them, for generic models. This allows for model simplification, and appreciation of which parameters (or functions thereof) cannot be estimated from data. Our algorithm can identify features such as redundant mechanisms and fast timescale subsystems, as well as the regimes in which such approximations are valid. We base our algorithm on a novel quantification of regional parametric sensitivity: multiscale sloppiness. Traditionally, the link between parametric sensitivity and the conditioning of the parameter estimation problem is made locally, through the Fisher Information Matrix. This is valid in the regime of infinitesimal measurement uncertainty. We demonstrate the duality between multiscale sloppiness and the geometry of confidence regions surrounding parameter estimates made where measurement uncertainty is non-negligible. Further theoretical relationships are provided linking multiscale sloppiness to the Likelihood-ratio test. From this, we show that a local sensitivity analysis (as typically done) is insufficient for determining the reliability of parameter estimation, even with simple (non)linear systems. Our algorithm provides a tractable alternative. We finally apply our methods to a large-scale, benchmark Systems Biology model of NF-$\kappa$B, uncovering previously unknown unidentifiabilities

On the existence of identifiable reparametrizations for linear compartment models

The parameters of a linear compartment model are usually estimated from experimental input-output data. A problem arises when infinitely many parameter values can yield the same result; such a model is called unidentifiable. In this case, one can search for an identifiable reparametrization of the model: a map which reduces the number of parameters, such that the reduced model is identifiable. We study a specific class of models which are known to be unidentifiable. Using algebraic geometry and graph theory, we translate a criterion given by Meshkat and Sullivant for the existence of an identifiable scaling reparametrization to a new criterion based on the rank of a weighted adjacency matrix of a certain bipartite graph. This allows us to derive several new constructions to obtain graphs with an identifiable scaling reparametrization. Using these constructions, a large subclass of such graphs is obtained. Finally, we present a procedure of subdividing or deleting edges to ensure that a model has an identifiable scaling reparametrization

Comparative flux control through the cytoplasmic phase of cell wall biosynthesis

The introduction of antibacterial drugs in the middle of the last century heralded a new era in the treatment of infectious disease. However the parallel emergence of antibiotic resistance and decline in new drug discovery threatens these advances. The development of new antibacterials must therefore be a high priority. The biosynthesis of the bacterial cell wall is the target for several clinically important antibacterials. This extracellular structure is essential for bacterial viability due to its role in the prevention of cell lysis under osmotic pressure. Its principal structural component, peptidoglycan, is a polymer of alternating N-acetyl-glucosamine (GlcNAc) and N-acetyl muramic acid (MurNAc) residues crosslinked by peptide bridges anchored by pentapeptide stems attached to the MurNAc moieties. The biosynthesis of peptidoglycan proceeds in three phases. The first, cytoplasmic, phase is catalysed by six enzymes. It forms a uridine diphosphate (UDP) bound MurNAc residue from UDP-GlcNAc and attaches the pentapeptide stem. This phase is a relatively unexploited target for antibacterials, being targeted by a single clinically relevant antibacterial, and is the subject of this thesis. The Streptococcus pneumoniae enzymes were kinetically characterised and in silico models of this pathway were developed for this species and Escherichia coli. These models were used to identify potential drug targets within each species. In addition the potentially clinically relevant interaction between an inhibitor of and feedback loops within this pathway was investigated. The use of direct parameter estimation instead of more traditional approaches to kinetic characterisation of enzymes was found to have significant advantages where it could be successfully applied. This approach required the theoretical analysis of the models used to determine whether unique parameter vectors could be determined. Such an analysis has been completed for a broad range of biologically relevant enzymes. In addition a relatively new approach to such analysis has been developed

Identifiability of Points and Rigidity of Hypergraphs under Algebraic Constraints

Identifiability of data is one of the fundamental problems in data science. Mathematically it is often formulated as the identifiability of points satisfying a given set of algebraic relations. A key question then is to identify sufficient conditions for observations to guarantee the identifiability of the points. This paper proposes a new general framework for capturing the identifiability problem when a set of algebraic relations has a combinatorial structure and develops tools to analyze the impact of the underlying combinatorics on the local or global identifiability of points. Our framework is built on the language of graph rigidity, where the measurements are Euclidean distances between two points, but applicable in the generality of hypergraphs with arbitrary algebraic measurements. We establish necessary and sufficient (hyper)graph theoretical conditions for identifiability by exploiting techniques from graph rigidity theory and algebraic geometry of secant varieties

Jeffreys priors for mixture estimation: properties and alternatives

While Jeffreys priors usually are well-defined for the parameters of mixtures of distributions, they are not available in closed form. Furthermore, they often are improper priors. Hence, they have never been used to draw inference on the mixture parameters. The implementation and the properties of Jeffreys priors in several mixture settings are studied. It is shown that the associated posterior distributions most often are improper. Nevertheless, the Jeffreys prior for the mixture weights conditionally on the parameters of the mixture components will be shown to have the property of conservativeness with respect to the number of components, in case of overfitted mixture and it can be therefore used as a default priors in this context.Comment: arXiv admin note: substantial text overlap with arXiv:1511.0314

Identifiability of linear compartmental tree models

A foundational question in the theory of linear compartmental models is how to assess whether a model is identifiable -- that is, whether parameter values can be inferred from noiseless data -- directly from the combinatorics of the model. We completely answer this question for those models (with one input and one output) in which the underlying graph is a bidirectional tree. Such models include two families of models appearing often in biological applications: catenary and mammillary models. Our proofs are enabled by two supporting results, which are interesting in their own right. First, we give the first general formula for the coefficients of input-output equations (certain equations that can be used to determine identifiability). Second, we prove that identifiability is preserved when a model is enlarged in specific ways involving adding a new compartment with a bidirected edge to an existing compartment.Comment: 32 page