Theses and Dissertations at Montana State University (MSU)
Permanent URI for this collectionhttps://scholarworks.montana.edu/handle/1/733
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Item 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 GedeonIsothermal 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.Item The emergence of collective behavior on social and biological networks(Montana State University - Bozeman, College of Letters & Science, 2018) Wilander, Adam Troy Charles; Chairperson, Graduate Committee: Scott McCallla; Dissertation contains an article of which Adam Troy Charles Wilander is not the main author.In this thesis, we broadly examine collective behaviors in various social and biological contexts. Aggregation, for instance, is a natural phenomenon that occurs in a variety of contexts; it is observed in schools of fish, flocks of birds, and colonies of bacteria, among others. This behavior can be found in some agent-based models, where it is typically assumed every pair of individuals interact according to a simple set of rules. In the first half of this thesis, we study a particular, well-understood aggregation model upon relaxation of the assumption that every individual interacts with every other. We review prior results on this topic -- when the underlying structure of interactions is an Erdos-Renyi graph. Seeking to incorporate community structure into the network, we establish the analogous problem under a class of networks called stochastic block graphs; a particular aspect of the system's metastable dynamics is explored upon varying the graph's connection densities. Finally, we evaluate the potential to leverage this system's dynamics in order to recover community structure (given a known graph as input). In the second half of this thesis, we similarly explore the aggregate behaviors of synchronization and desynchronization, appearing in diverse settings such as the study of metabolic oscillations and cell behaviors over time, respectively. Previous studies have leveraged a model in which repressilator entities are connected by a diffusive quorum sensing mechanism; these have shown (numerically) that the complex composition of observable behaviors depends upon the insertion point of the upregulating protein in the feedback loop. We rigorously prove a version of this; for negative feedback, negative signaling (Nf-Ns) systems we find only a unique stable equilibrium or a stable oscillation is possible. Additionally, we observe (numerically) the complex multistable dynamics that arise when a positive signal is included in the feedback loop and characterize this shift as a saddle node bifurcation of a cubic curve.Item The topological complexity of Cr-diffeomorphisms with homoclinic tangency(Montana State University - Bozeman, College of Letters & Science, 2001) Martensen, Brian FarleyItem Cataloging the global behavior of dynamical systems : adaptively searching parameter space using the Conley-Morse database(Montana State University - Bozeman, College of Letters & Science, 2013) Spendlove, Kelly Tulare; Chairperson, Graduate Committee: Tomas GedeonThe aim of this thesis is to build upon a combinatorial-topological framework to global dynamics of multiparameter dynamical systems. A combinatorial multivalued map of the dynamics for each subset of the parameter range is computed using rigorous numerical methods and is represented via a directed graph. The dynamics is then decomposed into the recurrent and gradient-like parts by graph theoretic algorithms using an adaptive computation. The novelty of this thesis is to introduce a similar adaptive scheme in parameter space. Furthermore, it is proven that this scheme produces an output which is naturally coarser than the output of an original computation. Incorporating previous results, we make an estimate for the savings achieved by this adaptive scheme in the setting of a saddle-node bifurcation. Furthermore, we make an empirical comparison of how well our scheme approximates an original computation.Item Computing multifractal spectra via simplicial measures(Montana State University - Bozeman, College of Letters & Science, 2011) Berwald, Jesse James; Chairperson, Graduate Committee: Tomas GedeonComplex dynamical systems occur on many scales in the natural world, and serve as rich subjects of study. Examples include ecosystems, physiological systems, and financial markets. Simplified versions of these system can be described by dynamical systems. As such, understanding the qualitative behavior of dynamical systems provides an important window into real-world phenomena. In this manuscript we focus on the qualitative behavior described by the measure concentrated on the attractor of a dynamical system. A common way to study such complicated measures is through their multifractal spectra. We will describe a new method, developed to approximate the Sinai-Bowen-Ruelle measure on an attractor, that is based on the Vietoris-Rips complex. We use it to approximate various measures concentrated on a number of example sets, and demonstrate its efficacy by computing the corresponding multifractal spectra.