The collection of biological data has been limited by instrumentation, the complexity of the systems themselves, and even the ability of graduate students to stay awake and record the data. However, increasing measurement capabilities and decreasing costs may soon enable the collection of reasonably sampled time course data characterizing biological systems, though in general only a subset of the system's species would be measured. This increase in data volume requires a corresponding increase in the use and interpretation of such data, specifically in the development of system identification techniques to identify parameter sets in proposed models. In this paper, we present the results of identifiability analysis on a small test system, including the identifiability of parameters with respect to different measurements (proteins and mRNA), and propose a working definition for "biologically meaningful estimation". We also analyze the correlations between parameters, and use this analysis to consider effective approaches to determining parameters with biological meaning. In addition, we look at other methods for determining relationships between parameters and their possible significance. Finally, we present potential biologically meaningful parameter groupings from the test system and present the results of our attempt to estimate the value of select groupings. / Master of Science

The identifiability effect - the human tendency to help identified victims to a greater extent than unidentified - has been proved of being an important aspect of moral judgment. However, the endowment effect - the human tendency to overestimate our properties - is unexplored within this area, such as the impact of identifiability on the endowment effect. For the purpose of examining the impact of identifiability and endowment on moral dilemmas, an experiment with 192 participants was conducted, using a charity scenario concerning African children, framed as a trolley dilemma. The results showed that a majority of the participants choose to maximize the number of children saved. No significant effects of identifiability or endowment were found. The main conclusion of the study was that the dilemma affected men and women differently. Women felt stronger feelings of sympathy, were less confident in choosing and perceived the choice more difficult than men.

The last five years have shown us the profound impact that SARS-CoV-2 pandemic has had on human kind and made us aware of the dangers that emerging pathogens can present. The goal of this dissertation is to use mathematical models in connection with data to uncover mechanistic interactions governing viral infections. To acquire a holistic understanding of the impact of viral infections, it is necessary to develop mathematical techniques and models that bridge knowledge on multiple biological scales. This dissertation explores the relationship between within-host virus dynamics, the environment and the between-host viral transmission. We will validate the models against data from SARS-CoV-2 infections, and data from infections with an emerging pathogen, the Usutu virus. Our models of SARS-CoV-2 infection looked at the relationship between infectious virus and viral RNA in the body and in the environment. Using golden hamster data and within-host mathematical models, we determined that infectious virus shedding early in infection correlates with transmission events, shedding of infectious virus diminishes late in the infection, and high viral RNA levels late in the infection are a poor indicator of transmission. We further showed that viral infectiousness increases in a density dependent manner with viral RNA and that their relative ratio is time-dependent. Such information is useful for designing interventions. Our models of Usutu virus infection looked at differences between different virus strains during bird infections. Within-host models applied to data showed heterogeneity in viral strain dynamics, and correlated high basic reproductive number with short infected cell lifespan (indicative of immune responses) and correlated low basic reproductive number with low viral peaks and longer lasting viremia (due to lower infection rates and high infected cell lifespan). We expanded the models to investigate multiscale dynamics connecting within-host scale, bird-to-vector transmission scale, and vector-borne epidemiological scale. One important direction of this dissertation is the investigation of uncertainty in parameter estimation and overall model identifiability. We conducted identifiability studies (using several theoretical tools) in the multiscale models of Usutu virus infection and in several within-host influenza models. Model identifiability is critical to the reproducibility of modeling results in any biological systems. In this dissertation, we will show how insights from such analyses inform both modeling practices and experimental design. / Doctor of Philosophy / The last five years have shown us the profound impact that SARS-CoV-2 pandemic has had on human kind and made us aware of the dangers that emerging pathogens can present. Within-host mathematical models are tools that can be used to study the dynamics of virus infections. These models help us gain an understanding of biological quantities of interest, relationships between biological processes in a quantitative and qualitative ways, and disease outcome. However, to acquire a holistic understanding of the impact of viral infections, it is necessary to develop mathematical tools and models that bridge knowledge on multiple biological scales. This dissertation explores the relationship between virus infection characteristics over time in a single host and larger biological scales including virus' release into the environment and spread of virus between hosts. Biological and public health insights about SARS-CoV-2 and Usutu virus were gained through these modeling efforts.

Secant varieties are among the main protagonists in tensor decomposition, whose study involves both pure and applied mathematic areas. Despite they have been studied for decades, several aspects of their geometry are still mysterious, among which identifiability and singularity of their points. In this thesis we study the secant varieties of lines of Grassmannians and of Spinor varieties. As first result, we completely determine their posets of orbits under the action of the groups SL and Spin, respectively. Then we solve the problems of identifiability and tangential-identifiability of points in the secant varieties of lines: as a consequence, we also determine the second Terracini locus to a Grassmannian and to a Spinor variety. Our main result concerns the singular locus of the secant variety of lines: we completely determine it for Grassmannians, and we give lower and upper bounds for Spinor varieties. Finally, we partially describe the poset of orbits in the secant variety of lines of any cominuscule variety.

<p>In the system identification community a popular framework for the problem of estimating a parametrized model structure given a sequence of input and output pairs is given by the prediction-error method. This method tries to find the parameters which maximize the prediction capability of the corresponding model via the minimization of some chosen cost function that depends on the prediction error. This optimization problem is often quite complex with several local minima and is commonly solved using a local search algorithm. Thus, it is important to find a good initial estimate for the local search algorithm. This is the main topic of this thesis.</p><p>The first problem considered is the regressor selection problem for estimating the order of dynamical systems. The general problem formulation is difficult to solve and the worst case complexity equals the complexity of the exhaustive search of all possible combinations of regressors. To circumvent this complexity, we propose a relaxation of the general formulation as an extension of the nonnegative garrote regularization method. The proposed method provides means to order the regressors via their time lag and a novel algorithmic approach for the \textsc{arx} and \textsc{lpv-arx} case is given.</p><p> </p><p>Thereafter, the initialization of linear time-invariant polynomial models is considered. Usually, this problem is solved via some multi-step instrumental variables method. For the estimation of state-space models, which are closely related to the polynomial models via canonical forms, the state of the art estimation method is given by the subspace identification method. It turns out that this method can be easily extended to handle the estimation of polynomial models. The modifications are minor and only involve some intermediate calculations where already available tools can be used. Furthermore, with the proposed method other a priori information about the structure can be readily handled, including a certain class of linear gray-box structures. The proposed extension is not restricted to the discrete-time case and can be used to estimate continuous-time models.</p><p> </p><p>The final topic in this thesis is the initialization of discrete-time systems containing polynomial nonlinearities. In the continuous-time case, the tools of differential algebra, especially Ritt's algorithm, have been used to prove that such a model structure is globally identifiable if and only if it can be written as a linear regression model. In particular, this implies that once Ritt's algorithm has been used to rewrite the nonlinear model structure into a linear regression model, the parameter estimation problem becomes trivial. Motivated by the above and the fact that most system identification problems involve sampled data, a version of Ritt's algorithm for the discrete-time case is provided. This algorithm is closely related to the continuous-time version and enables the handling of noise signals without differentiations.</p>

<p>Active insulin receptors (IR) phosphorylate insulin receptor substrate (IRS), but it is not clear whether IRS is phosphorylated mainly by IR at the plasma membrane or by internalized IR in the cytosol. In this thesis, structural identifiability analysis and parameter sensitivity analysis is performed for models of the first steps in the metabolic insulin signaling pathway. In particular, the identifiability of the kinetic parameters governing IRS phosphorylation are investigated.</p><p>Given measurements of the relative increase in phosphorylation degree of IR and IRS, the structural identifiability analysis revealed that the parameters governing IRS phosphorylation are non-identifiable, but their ratio is identifiable. This is sufficient to study whether phosphorylation of IRS proceeds more rapidly by IR at the plasma membrane or by internalized IR in the cytosol. In the examined model structure, internalization of insulin receptors is shown to be necessary to reproduce the experimental data.</p><p>Sensitivity analysis of the parameters governing IRS phosphorylation showed that many parameters need to be known in order to obtain ``practical identifiability'' of the interesting parameters.</p>

Active insulin receptors (IR) phosphorylate insulin receptor substrate (IRS), but it is not clear whether IRS is phosphorylated mainly by IR at the plasma membrane or by internalized IR in the cytosol. In this thesis, structural identifiability analysis and parameter sensitivity analysis is performed for models of the first steps in the metabolic insulin signaling pathway. In particular, the identifiability of the kinetic parameters governing IRS phosphorylation are investigated. Given measurements of the relative increase in phosphorylation degree of IR and IRS, the structural identifiability analysis revealed that the parameters governing IRS phosphorylation are non-identifiable, but their ratio is identifiable. This is sufficient to study whether phosphorylation of IRS proceeds more rapidly by IR at the plasma membrane or by internalized IR in the cytosol. In the examined model structure, internalization of insulin receptors is shown to be necessary to reproduce the experimental data. Sensitivity analysis of the parameters governing IRS phosphorylation showed that many parameters need to be known in order to obtain ``practical identifiability'' of the interesting parameters.

In systems biology, models often contain a large number of unknown or only roughly known parameters that must be estimated through the fitting of data. This work examines the question of whether or not these parameters can in fact be estimated from available measurements. Structural or a priori identifiability of unknown parameters in biochemical reaction networks is considered. Such systems consist of continuous time, nonlinear differential equations. Several methods for analyzing identifiability of such systems exist, most of which restate the question as one of observability by expanding the state space to include parameters. However, these existing methods were not developed with biological systems in mind, so do not necessarily address the specific challenges posed by this type of problem. In this work, such methods are considered for the analysis of a representative biological system, the NF-kappaB signal transduction pathway. It is shown that existing observability-based strategies, which rely on finding an analytical solution, require significant simplifications to be applicable to systems biology problems that are seldom feasible. The analytical nature of the solution imposes restrictions on the size and complexity of systems that these methods can handle. This conflicts with the fact that most currently studied systems biology models are rather large networks containing many states and parameters. In this thesis, a new simulation based method using an empirical observability Gramian for determining identifiability is proposed. Computational and numerical sensitivity issues for this method are considered. An algorithm, based on this method, is developed and demonstrated on a simple biological example of microbial growth with Michaelis-Menten kinetics. The new method is applied to the motivating NF-kappaB example to show its suitability for use in systems biology. / Thesis (Master, Chemical Engineering) -- Queen's University, 2008-08-05 22:20:32.561

Capture-recapture (CRC) models use two or more samples, or lists, to estimate the size of a population. In the canonical example, a researcher captures, marks, and releases several samples of fish in a lake. When the fish that are captured more than once are few compared to the total number that are captured, one suspects that the lake contains many more uncaptured fish. This basic intuition motivates CRC models in fields as diverse as epidemiology, entomology, and computer science. We use simulations to study the performance of conventional log-linear models for CRC. Specifically we evaluate model selection criteria, model averaging, an asymptotic variance formula, and several small-sample data adjustments. Next, we argue that interpretable models are essential for credible inference, since sets of models that fit the data equally well can imply vastly different estimates of the population size. A secondary analysis of data on survivors of the World Trade Center attacks illustrates this issue. Our main chapter develops local log-linear models. Heterogeneous populations tend to bias conventional log-linear models. Post-stratification can reduce the effects of heterogeneity by using covariates, such as the age or size of each observed unit, to partition the data into relatively homogeneous post-strata. One can fit a model to each post-stratum and aggregate the resulting estimates across post-strata. We extend post-stratification to its logical extreme by selecting a local log-linear model for each observed point in the covariate space, while smoothing to achieve stability. Local log-linear models serve a dual purpose. Besides estimating the population size, they estimate the rate of missingness as a function of covariates. Simulations demonstrate the superiority of local log-linear models for estimating local rates of missingness for special cases in which the generating model varies over the covariate space. We apply the method to estimate bird species richness in continental North America and to estimate the prevalence of multiple sclerosis in a region of France.

The three structural possibilities marking a noun with an English article are a, the, and Ø (the absence of an article). Although these structural possibilities are simple, they encode a multitude of semantic and pragmatic functions, and it is these complex form-function interactions that this study explores and explains using a semantic map model. The semantic map that is proposed contains three dimensions, which I refer to as Grammatical Number, Referentiality, and Discourse Mode. Each of these dimensions contains a number of further semantic values or pragmatic functions - which I will label "attributes" - that are implicated in English article choice. Various semantic map versions are tested and compared with a methodological approach that uses data collected in a controlled protocol from an elicited conversational discourse. The version that performed best is used as a basis for proposing a comprehensive semantic map that includes the following dimensions and dimensional attributes: a Number dimension with 3 attributes (singular, plural, and uncountable); a Referentiality dimension with 11 attributes, including 7 referential attributes that describe kinds of identifiability (proper names, shared lexis, shared speech situation, frame, current discourse, identifiable to speaker only ["new reference"], and identifiable to neither speaker nor listener [non-specific]) as well as 4 non-referential attributes (categorization, general non-referential expressions, finite verb [verb-object] "noun incorporation", and idioms); and a Discourse Mode dimension with 4 attributes (headline, immediacy, normal, and reintroducing). This model of English articles contributes to the field of research on articles as well as to the field of English language instruction and learning. In addition, it is suggested that the methodological paradigm used to test the semantic map model may be useful as an experimental paradigm for testing semantic maps of other constructions and languages.