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Item Quantifying robustness of the gap gene network(Montana State University - Bozeman, College of Letters & Science, 2024) Andreas, Elizabeth Anne; Chairperson, Graduate Committee: Tomas Gedeon; Bree Cummins (co-chair)Early development of Drosophila melanogaster (fruit fly) facilitated by the gap gene network has been shown to be incredibly robust, and the same patterns emerge even when the process is seriously disrupted. We investigate this robustness using a previously developed computational framework called DSGRN (Dynamic Signatures Generated by Regulatory Networks). Our mathematical innovations include the conceptual extension of this established modeling technique to enable modeling of spatially monotone environmental effects, as well as the development of a collection of graph theoretic robustness scores for network models. This allows us to rank order the robustness of network models of cellular systems where each cell contains the same genetic network topology but operates under a parameter regime that changes continuously from cell to cell. We demonstrate the power of this method by comparing the robustness of two previously introduced network models of gap gene expression along the anterior-posterior axis of the fruit fly embryo, both to each other and to a random sample of networks with same number of nodes and edges. We observe that there is a substantial difference in robustness scores between the two models. Our biological insight is that random network topologies are in general capable of reproducing complex patterns of expression, but that using measures of robustness to rank order networks permits a large reduction in hypothesis space for highly conserved systems such as developmental networks.Item Applications and diagnostics for dimension reduction of multivariate data(Montana State University - Bozeman, College of Letters & Science, 2022) Harmon, Paul Gary; Chairperson, Graduate Committee: Mark Greenwood; This is a manuscript style paper that includes co-authored chapters.Working with high-dimensional data involves various statistical challenges. This dissertation overviews a suite of tools and methods for dimension reduction, using latent- variable models, techniques for mapping high-dimensional data, clustering, and working with multivariate responses across a variety of use cases. First, we propose and develop a method for classifying institutions of higher education is and compare with the current standard for university classification: the Carnegie Classification. We present a classification tool based on Structural Equation Models that better allows for modeling of correlated indices than the PCA-based methodology that underlies the Carnegie Classification. Additionally, we create a Shiny-based web application that allows for assessment of sensitivity to changes in the underlying characteristics of each institution. Second, we develop a novel methodology that extends the Cook's Distance diagnostic for identifying influential points in regression to a new application on high-dimensional mapping tools. We highlight a PERMANOVA-based method for calculating the difference in the shape of resulting ordinations based on inclusion/exclusion of points, similar in style to the influence diagnostic Cook's Distance for regression. We present a set simulation studies with several mapping techniques and highlight where the method works well (Classical Multidimensional Scaling) and where the methods appear to work less effectively (t-distributed Stochastic Neighbor Embedding). Additionally, we examine several real data sets and assess the efficacy of the diagnostic on thsoe data sets. Finally, we introduce a new method for feature selection in a specific type of divisive clustering, called monothetic clustering. Utilizing a penalized matrix decomposition to re- weight the input data to the monothetic clustering algorithm allows for reduction in noise features allows this clustering method to better make splits based on single features at a time, leading to better cluster results. We present a method for tuning both the number of clusters, K, and the degree of sparsity, s, as well as simulation studies that highlight the efficacy of noise reduction in monothetic clustering solutions.Item Two-year community college students' understanding of rational expressions(Montana State University - Bozeman, College of Letters & Science, 2023) Kong, Chor Wan Amy; Chairperson, Graduate Committee: Jennifer Luebeck; Megan Wickstrom (co-chair)This research study investigated the gaps in knowledge held by two-year community college students in simplifying and operating with rational expressions and how these gaps affect their learning. The study employed multiple methods, including completion of a Diagnostic Problem Set, participating in collaborative and exploratory activities, and attending task-based interviews, to elicit and assess students' understanding of rational expressions. The study analyzed and categorized participants' responses based on the participants' different perspectives and learning processes. The research also explored how collaborative and exploratory learning, as well as the use of Knowledge-Eliciting Tasks, can help identify and address students' misconceptions. Qualitative analysis of the findings identified potential causes of the learning gaps and generated recommendations for instructional strategies that can bridge these gaps and improve students' understanding of rational expressions, which is crucial to student success in algebraic subjects and college academic achievement.Item Coding Code: Qualitative Methods for Investigating Data Science Skills(Informa UK Limited, 2023-11) Theobold, Allison S.; Wickstrom, Megan H.; Hancock, Stacey A.Despite the elevated importance of Data Science in Statistics, there exists limited research investigating how students learn the computing concepts and skills necessary for carrying out data science tasks. Computer Science educators have investigated how students debug their own code and how students reason through foreign code. While these studies illuminate different aspects of students’ programming behavior or conceptual understanding, a method has yet to be employed that can shed light on students’ learning processes. This type of inquiry necessitates qualitative methods, which allow for a holistic description of the skills a student uses throughout the computing code they produce, the organization of these descriptions into themes, and a comparison of the emergent themes across students or across time. In this article we share how to conceptualize and carry out the qualitative coding process with students’ computing code. Drawing on the Block Model to frame our analysis, we explore two types of research questions which could be posed about students’ learning. Supplementary materials for this article are available online.Item Modeling saline fluid flow in subglacial ice-walled channels(Montana State University - Bozeman, College of Letters & Science, 2022) Jenson, Amy Jo; Chairperson, Graduate Committee: Scott McCallaSubglacial hydrological systems have impacts on ice dynamics as well as nutrient and sediment transport. There has been an extensive effort to understand the dynamics of subglacial drainage through numerical modeling, however these models have focused on freshwater, neglecting the consideration of brine. Saline fluid can exist in cold-based glacier systems where freshwater cannot. Therefore, there exist subglacial hydrological systems where the only fluid is brine. Understanding the routing of saline fluid is important for understanding geochemical and microbiological processes in these saline cryospheric habitats. In this thesis, I present a model of channelized drainage from a hypersaline subglacial lake and highlight the impact of saline fluid on melt rates in an ice-walled channel. The model results show that channel walls grow more quickly when fluid contains higher salt concentrations, which results in greater peak discharge and faster drainage for a fixed lake volume. This model provides a framework to assess the relative impact of brine on discharge and drainage duration.Item Analysis of dynamic biological systems imagery(Montana State University - Bozeman, College of Letters & Science, 2022) Dudiak, Cameron Drew; Chairperson, Graduate Committee: Scott McCallaBiological systems pose considerable challenges when attempting to isolate experimental variables of interest and obtain viable data. Developments in image analysis algorithms and techniques allow for further mathematical interpretation, model integration, and even model optimization ('training'). We formulate two distinct methods for obtaining robust quantitative data from time-series imagery of two biological systems: Paenibacillus dendritiformis bacterial colonies, and human gastric organoids. Boundary parameterizations of P. dendritiformis are extracted from timelapse image sequences displaying colony repulsion, and are subsequently used to 'train' a previously developed nonlocal PDE model through the means of error minimization between observation and simulation. Particle tracking is conducted for small colloidal beads embedded within human gastric organoids, and then used to perform particle tracking analysis. This information is analyzed to quantify the local complex viscoelastic properties of organoids' interior mucosal environment.Item Ribosome Abundance Control in Prokaryotes(Springer Science and Business Media LLC, 2023-10) Shea, Jacob; Davis, Lisa; Quaye, Bright; Gedeon, TomasCell growth is an essential phenotype of any unicellular organism and it crucially depends on precise control of protein synthesis. We construct a model of the feedback mechanisms that regulate abundance of ribosomes in E. coli, a prototypical prokaryotic organism. Since ribosomes are needed to produce more ribosomes, the model includes a positive feedback loop central to the control of cell growth. Our analysis of the model shows that there can be only two coexisting equilibrium states across all 23 parameters. This precludes the existence of hysteresis, suggesting that the ribosome abundance changes continuously with parameters. These states are related by a transcritical bifurcation, and we provide an analytic formula for parameters that admit either state.Item Leveraging social networks for identification of people living with HIV who are virally unsuppressed(Wolters Kluwer Health, Inc., 2023-10) Cummins, Breschine; Johnson, Kara; Schneider, John A.; Del Vicchio, Natasha; Moshiri, Niema; Wertheim, Joel O.; Goyal, Ravi; Skaathun, BrittObjectives: This study investigates primary peer-referral engagement (PRE) strategies to assess which strategy results in engaging higher numbers of people living with HIV (PLWH) who are virally unsuppressed. Design: We develop a modeling study that simulates an HIV epidemic (transmission, disease progression, and viral evolution) over 6 years using an agent-based model followed by simulating PRE strategies. We investigate two PRE strategies where referrals are based on social network strategies (SNS) or sexual partner contact tracing (SPCT). Methods: We parameterize, calibrate, and validate our study using data from Chicago on Black sexual minority men to assess these strategies for a population with high incidence and prevalence of HIV. For each strategy we calculate the number of PLWH recruited who are undiagnosed or out-of-care and the number of direct or indirect transmissions. Results: SNS and SPCT identified 256.5 (95% C.I.: [234,279]) and 15 (95% C.I.: [7,27]) PLWH, respectively. Of these, SNS identified 159 (95% C.I.: [142,177]) PLWH out-of-care and 32 (95% C.I.: [21, 43]]) PLWH undiagnosed compared to 9 (95% C.I.: [3,18]) and 2 (95% C.I.: [0,5]) for SPCT. SNS identified 15.5 (95% C.I.: [6,25]) and 7.5 (95% C.I.: [2, 11]]) indirect and direct transmission pairs, while SPCT identified 6 (95% C.I.: [0,8]) and 5 (95% C.I.: [0,8]), respectively. Conclusions: With no testing constraints, SNS is the more effective strategy to identify undiagnosed and out-of-care PLWH. Neither strategy is successful at identifying sufficient indirect or direct transmission pairs to investigate transmission networks.Item Detecting punctuated evolution in SARS-CoV-2 over the first year of the pandemic(Frontiers Media SA, 2023-02) Surya, Kevin; Gardner, Jacob D.; Organ, Chris L.The Severe Acute Respiratory Syndrome Coronavirus 2 (SARS-CoV-2) evolved slowly over the first year of the Coronavirus Disease 19 (COVID-19) pandemic with differential mutation rates across lineages. Here, we explore how this variation arose. Whether evolutionary change accumulated gradually within lineages or during viral lineage branching is unclear. Using phylogenetic regression models, we show that ~13% of SARS-CoV-2 genomic divergence up to May 2020 is attributable to lineage branching events (punctuated evolution). The net number of branching events along lineages predicts ~5% of the deviation from the strict molecular clock. We did not detect punctuated evolution in SARS-CoV-1, possibly due to the small sample size, and in sarbecovirus broadly, likely due to a different evolutionary process altogether. Punctuation in SARS-CoV-2 is probably neutral because most mutations were not positively selected and because the strength of the punctuational effect remained constant over time, at least until May 2020, and across continents. However, the small punctuational contribution to SARS-CoV-2 diversity is consistent with the founder effect arising from narrow transmission bottlenecks. Therefore, punctuation in SARS-CoV-2 may represent the macroevolutionary consequence (rate variation) of a microevolutionary process (transmission bottleneck).Item Faithful sets of topological descriptors and the algebraic K-theory of multi-parameter zig-zag grid persistence modules(Montana State University - Bozeman, College of Letters & Science, 2023) Schenfisch, Anna Katherine; Chairperson, Graduate Committee: Tomas Gedeon and Brittany Fasy (co-chair)Given a geometric simplicial complex, the uncountable set of (augmented) persistence diagrams corresponding to lower-star filtrations taken with respect to all possible directions uniquely correspond to the simplicial complex, i.e., the set is faithful. While this hints towards interesting applications in shape comparison, the set of all possible directions is uncountably infinite, and so has no hope of computability. In practice, one might use a finite approximation, but faithfulness of this approximation is not guaranteed. Motivated by the need for both computability and provable faithfulness, we provide an explicit description of a finite faithful set of augmented persistence diagrams. We then show this construction applies to augmented Euler characteristic curves and augmented Betti curves, and is stable under particular perturbations. In the specific case where the underlying complex is a graph, we provide an improved construction that utilizes a radial binary search. We then shift focus to comparing the cardinalities of minimal faithful sets of descriptors as a way to define and order equivalence classes of topological descriptor types. Focusing on six topological descriptor types commonly used in practice, we give a partial order on their corresponding equivalence classes, as well as give bounds on the sizes of minimum faithful sets for each descriptor type. Next, we broaden our view to zig-zag grid persistence modules, functors whose domain categories are posets with grid-like structure. We begin by explicitly defining such persistence modules in terms of constructible cosheaves over stratified Euclidean space, including a careful treatment of augmented persistence modules, which are analogous to the aforementioned augmented descriptors who played a central role in discussion of faithful sets. Exodromy gives us an equivalence between persistence modules as a functor category and as constructible cosheaves; we furthermore show the equivalence of these categories with a category of constructible functors out of Rd with a fixed stratification and localized at weak equivalences, essentially "standardizing" modules so that the category has a clear monoid structure. We compute the algebraic K-theory of zig-zag grid persistence modules, using a double inductive argument to show the K-theory is additive over strata. Finally, we identify connections to related topics, such as the virtual diagrams of Bubenik and Elchensen, as well as Euler characteristic and Betti curves/surfaces/manifolds. We hope a study of K-groups will provide interesting insights into the nature of persistence modules, and we indicate ways in which the zeroth and first K-groups may be interpreted