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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.
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 McCalla
Subglacial 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.
Analysis of dynamic biological systems imagery
(Montana State University - Bozeman, College of Letters & Science, 2022) Dudiak, Cameron Drew; Chairperson, Graduate Committee: Scott McCalla
Biological 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.
Preservice teachers' construction of computational thinking practices through mathematical modeling activities
(Montana State University - Bozeman, College of Letters & Science, 2022) Adeolu, Adewale Samson; Chairperson, Graduate Committee: Mary Alice Carlson and Elizabeth Burroughs (co-chair)
The importance of learning computational thinking practices in K-12 settings is gaining momentum in the United States and worldwide. As a result, studies have been conducted on integrating these practices in mathematics teaching and learning. However, there is little study that focuses on how to prepare pre-service teachers who will teach the practices in K-12 settings. I investigated how pre-service teachers collaborated to develop computational thinking practices when working on modeling activities with computational tools. To carry out this research, I studied nine pre-service teachers working on modeling tasks for a semester. Five participants recorded their screens and were invited to participate in a stimulated recall interview. Using the interactional analysis procedures, findings showed that the presence of computational tools influenced the positioning (leadership and distributed authority) and collaborative processes (dividing and offloading labor, giving and receiving feedback, accommodation, and refining ideas) pre-service teachers used during modeling. This study found that pre-service teachers used ten computational thinking practices, which are sub-grouped into four broader practices -- data practices, mimicking and mathematizing, model exploration and extension, and model communication. This dissertation also found that pre-service teachers' mathematical knowledge and their ability to code were interdependent. From a research point of view, this study extends our knowledge of the social constructivist theory of doing research in the context of pre-service teachers engaging in modeling activities with computational tools. From the teacher education perspective, this study emphasizes the need to consider the impact of computational tools on the interactions of pre-service teachers during modeling. The study also reveals the need to structure the mathematical modeling curriculum to lead to a better learning experience for pre-service teachers.
Bayesian hierarchical latent variable models for ecological data types
(Montana State University - Bozeman, College of Letters & Science, 2022) Stratton, Christian Alexander; Chairperson, Graduate Committee: Jennifer Green and Andrew Hoegh (co-chair); This is a manuscript style paper that includes co-authored chapters.
Ecologists and environmental scientists employ increasingly complicated sampling designs to address research questions that can help explain the impacts of climate change, disease, and other emerging threats. To understand these impacts, statistical methodology must be developed to address the nuance of the sampling design and provide inferences about the quantities of interest; this methodology must also be accessible and easily implemented by scientists. Recently, hierarchical latent variable modeling has emerged as a comprehensive framework for modeling a variety of ecological data types. In this dissertation, we discuss hierarchical modeling of multi-scale occupancy data and multi-species abundance data. Within the multi-scale occupancy framework, we propose new methodology to improve computational performance of existing modeling approaches, resulting in a 98% decrease in computation time. This methodology is implemented in an R package developed to encourage community uptake of our method. Additionally, we propose a new modeling framework capable of simultaneous clustering and ordination of ecological abundance data that allows for estimation of the number of clusters present in the latent ordination space. This modeling framework is also extended to accommodate hierarchical sampling designs. The proposed modeling framework is applied to two data sets and code to fit our model is provided. The software and statistical methodology proposed in this dissertation illustrate the flexibility of hierarchical latent variable modeling to accommodate a variety of data types.
A biofilm model that avoids a tragedy of the commons
(Montana State University - Bozeman, College of Letters & Science, 2021) Dayutis, Seth Aaron; Chairperson, Graduate Committee: Jack D. Dockery
The study of competition between multiple species is of great significance in biology. Competitive behavior is often observed to occur in biofilms and understanding cooperation between multiple species in a single biofilm is the center of much research. The species that grow in biofilms are frequently studied in chemostats, which have a rich history in mathematical modeling. In this thesis, a review of a mathematical chemostat model is presented in which a tragedy of the commons occurs. The chemostat model is then developed into a biofilm model to see if a tragedy occurs in a biofilm under similar conditions. The biofilm and chemostat model consist of two species, a cooperator and a cheater. The cooperator produces an enzyme that combines with a substrate to produce a nutrient. The nutrient is then consumed by the cooperator and cheater. The cooperator is at a disadvantage since it must allocate some of its nutrient uptake towards enzyme production. A one dimensional biofilm model is developed with reaction advection equations governing the behavior of the species and reaction-diffusion equations governing the behavior of the substrate, nutrient ,and enzyme. A set of numerical methods is then outlined on how to solve the system of equations. It is found that a tragedy of the commons is avoided in the biofilm and both species can persist when numerical simulations are run for a finite amount of time. It is then argued that the cooperative behavior exhibited by the two species is a stable equilibrium by approximating the steady state solutions. Further evidence is provided for the existence of a stable equilibrium by perturbing the system and finding that the perturbed system tends back to the equilibrium. Finally, the eigenvalues of the discretized linear system are computed and the results suggest that either the equilibrium is stable or moves away from the equilibrium slowly.
A mathematical model of a biphasic DNA amplification reaction
(Montana State University - Bozeman, College of Letters & Science, 2019) Ciesielski, Danielle Kristine; Chairperson, Graduate Committee: Tomas Gedeon
Isothermal DNA amplification reactions have many applications ranging from analyte detection to DNA circuits. EXPonential Amplification Reaction (EXPAR) is a popular isothermal DNA amplification method that exponentially amplifies short DNA oligonucleotides. A recent modification of this technique using an energetically stable looped template with palindromic binding regions demonstrated unexpected biphasic amplification and much higher DNA yield than EXPAR. This Ultrasensitive DNA Amplification Reaction (UDAR) shows high-gain, switch-like DNA output from low concentrations of DNA input. Here we present the first mathematical model of UDAR based on four reaction mechanisms. We show that the model can reproduce the experimentally observed biphasic behavior. Furthermore, we show that three of these mechanisms are necessary to reproduce biphasic experimental results. The reaction mechanisms are (i) positively cooperative multistep binding caused by two palindromic trigger binding sites on the template; (ii) gradual template deactivation; (iii) recycling of deactivated templates into active templates; and (iv) polymerase sequestration. Understanding of these mechanisms also illuminates behavior of EXPAR and other nucleic acid amplification reactions. For a deeper understanding of the roles these mechanisms play in DNA amplification reactions, we apply dynamical systems analysis to the model. We first consider the long term behavior of partial models that lack key reaction mechanisms described above to see how their omission impacts the system's overall behavior. Then we use perturbation theory to examine the time scales on which these mechanisms operate and how their interaction leads to biphasic growth. We find that mechanisms (i) and (ii) together create a stable equilibrium reminiscent of EXPAR reactions, but the addition of mechanism (iii) changes the stability of this equilibrium and generates UDAR's characteristic high amplification. Finally, mechanism (iv) introduces a second stable equilibrium that indicates that polymerase sequestration is the mechanism that ends the second fast amplification phase. In addition, throughout this work we identify which rate constants shape different parts of the biphasic growth. These results can guide future work in rational design of molecular detection assays.
Emergence of cooperative behavior in microbial consortia
(Montana State University - Bozeman, College of Letters & Science, 2018) Schepens, Diana Ruth; Chairperson, Graduate Committee: Tomas Gedeon
Cooperative microbial communities and their impact are ubiquitous in nature. The complexities of the cross-feeding interactions within such communities invite the application of mathematical models as a tool which can be used to investigate key influences in the emergence of cooperative behavior and increased productivity of the community. In this work, we develop and investigate a differential equation model of competition within a chemostat between four microbial strains utilizing a substrate to produce two necessary metabolites. The population of our chemostat includes a wild type strain that generalizes in producing both metabolites, two cross-feeding cooperator strains that each specialize in producing one of the two metabolites, and a cheater strain that produces neither metabolite. Using numerical methods we consider three key characteristics of the microorganisms and investigate the impact on the emergence of mutual cross-feeding in the community. First, we investigate the impact that substrate input concentration and the rate and type (active vs. passive) of metabolite transport between cells has on the emergence of cooperation and multi-stabilities resulting from the competition. Second, we investigate the role that resource allocation within metabolic pathways plays in the results of the competition between cells with different metabolite production strategies. Introducing metabolite production cost into the model leads to new outcomes of the competition, including stable coexistence between different strains. Lastly, we examine the effect that an initial population of a non-cooperative cheater strain has on the outcome of competition. Our results show that the emergence of a cross-feeding consortia relies on the availability and efficient use of resources, ease of transport of metabolites between cells, and limited existence of cheaters.
The mathematics in mathematical modeling
(Montana State University - Bozeman, College of Letters & Science, 2017) Fulton, Elizabeth Anne White; Chairperson, Graduate Committee: Elizabeth Burroughs
The purpose of this research was to investigate how teachers interact with mathematics while teaching mathematical modeling to elementary students. To conduct this study, I used a case study approach with four elementary teachers. Each teacher participated in professional development on mathematical modeling prior to the study and incorporated mathematical modeling into their classroom. Modeling task lessons were observed and teachers participated in interviews before and after each lesson. I qualitatively explored what mathematical decisions teachers made while teaching mathematical modeling and how students' mathematical contributions influenced the modeling cycle. This analysis took place through three analytical lenses: the mathematics used, the teacher's interactions with their student's mathematical ideas, and as compared to components of the mathematical modeling cycle. Findings indicate that students engaged in meaningful mathematics to explore real-world problems. Across all cases, teachers prepared students to use mathematics by creating tasks with mathematical opportunities and by orienting the students towards using mathematics to investigate the problems. Each teacher allowed their students to introduce most of the mathematical ideas used to investigate the modeling questions. Each task became a mathematical modeling task by the way it was implemented, through teachers' and students' contributions to the activity.
Statistics in the presence of cost : cost-considerate variable selection and MCMC convergence diagnostics
(Montana State University - Bozeman, College of Letters & Science, 2016) Lerch, Michael David; Chairperson, Graduate Committee: Steve Cherry
The overarching objective of this research is to address and recognize the cost-benefit trade-off inherent in much of statistics. We identify two places where such a balance is present for researchers: variable selection and Markov chain Monte Carlo (MCMC) sampling. An easily identifiable source of cost in science occurs when taking measurements. Researchers measure variables to estimate another quantity based on a model. When model building, researchers may have access to a large number of variables to include in the model and may consider using a subset of the variables so that future uses of the model need only measure this subset rather than all variables. The researchers are incentivized to proceed in this manner if some variables are prohibitively expensive to measure for future uses of the model. In this research, we present a new algorithm for cost-considerate variable selection in linear modeling when confronted with this problem. Since overfitting may be a danger when many variables at the disposal of the researcher, we build on the LARS and Lasso algorithms to perform cost-based variable selection in concert with model regularization. In MCMC sampling for Bayesian statistics, the cost-benefit trade-off is unavoidable. Researchers sampling from a posterior distribution must run a sampler for some number of iterations before finally stopping the sampler to make inference on the finite number of samples drawn. In this situation, the cost to be reduced is time to run the sampler while realizing the longer the sampler is run, the better the convergence. Time may not be as tangible a cost as a dollar figure, but increased wait time to perform analyses incurs the cost of running a computer and any negative effects associated with a delay as the researcher waits until the sampler has finished running. In this research, we introduce new convergence assessment tools in a diagnostic and plot. Unlike commonly used convergence diagnostics, these new tools focus explicitly on posterior quantiles and probabilities which are common inferential objectives in Bayesian statistics. Additionally, we introduce equivalence testing to the convergence assessment domain by using it as the framework of the diagnostic.

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