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University of Colorado
1.
Lavington, Jonathan Wilder.
A Probabilistic Modeling Approach to CRISPR-Cas9.
Degree: MS, 2018, University of Colorado
URL: https://scholar.colorado.edu/appm_gradetds/109 
► CRISPR-Cas, a particular type of microbial immune response system, has in recent years been modified to make precise changes to an organisms DNA. In…
(more)
▼ CRISPR-Cas, a particular type of microbial immune response system, has in recent years been modified to make precise changes to an organisms DNA. In the early 2000s scientists discovered through the study of <i>Streptococcus pyogenes</i>, that a unique CRISPR locus (Cas9) exhibited specific RNA-guided cleavage near short trinucleotide motifs (PAMs). Further research on Cas9 eventually led researchers to create methods that actively edit genomes through Cas9-dependent cleavage and to manipulate transcription of genes through engineered nuclease-deficient Cas9 (dCas9). These techniques have enabled new avenues for analyzing existing gene functions or engineering new ones, manipulating gene expression, gene therapy, and much more. While great strides have been made over the last decade, CRISPR is still prone to inaccuracies which often generate sub-optimal editing efficiency or off-target effects. The primary interest of this thesis is the investigation of targeting efficiency concerning changes in the guide RNA (gRNA) composition. While many different factors affect the ability with which a given gRNA can target a DNA sequence, we have focused our research primarily on the formation of the R-loop: the hybrid structure formed when the Cas9/dCas9:gRNA complex binds to a host DNA site. In our investigation, we have attempted to account for several experimental findings reported in the literature as influential for binding efficiency. These include position dependence, base pair composition dependence, and the effects of runs of consecutive mismatches. Using a Gambler’s Ruin Markov model to mimic the process of R-loop formation, we fit our model to experimental data and show that the match/mismatch configuration between the gRNA and the DNA target allows for accurate predictions of R-loop formation in bacteria.
Advisors/Committee Members: Manuel E. Lladser, Stephen Becker, William Kleiber.
Subjects/Keywords: biological modeling; crispr; markov chains; probability theory; statistics; Applied Mathematics; Statistics and Probability
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APA (6th Edition):
Lavington, J. W. (2018). A Probabilistic Modeling Approach to CRISPR-Cas9. (Masters Thesis). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/109
Chicago Manual of Style (16th Edition):
Lavington, Jonathan Wilder. “A Probabilistic Modeling Approach to CRISPR-Cas9.” 2018. Masters Thesis, University of Colorado. Accessed March 09, 2021.
https://scholar.colorado.edu/appm_gradetds/109.
MLA Handbook (7th Edition):
Lavington, Jonathan Wilder. “A Probabilistic Modeling Approach to CRISPR-Cas9.” 2018. Web. 09 Mar 2021.
Vancouver:
Lavington JW. A Probabilistic Modeling Approach to CRISPR-Cas9. [Internet] [Masters thesis]. University of Colorado; 2018. [cited 2021 Mar 09].
Available from: https://scholar.colorado.edu/appm_gradetds/109.
Council of Science Editors:
Lavington JW. A Probabilistic Modeling Approach to CRISPR-Cas9. [Masters Thesis]. University of Colorado; 2018. Available from: https://scholar.colorado.edu/appm_gradetds/109
University of Colorado
2.
Char, Ian Guo-fan.
Algorithmic Construction and Stochastic Analysis of Optimal Automata for Generalized Strings.
Degree: MS, 2018, University of Colorado
URL: https://scholar.colorado.edu/appm_gradetds/115
► In many applications, the need arises to search a text for appearances of a given set of keywords. As an example, in bioinformatics one…
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▼ In many applications, the need arises to search a text for appearances of a given set of keywords. As an example, in bioinformatics one may wish to search a DNA sequence to find so-called <i>biological motifs</i>. A standard approach to this problem is to leverage a <i>deterministic finite automaton</i> – a graph structure which is traversed as letters of the text are read in. However, depending on the number and length of the keywords being sought in the text, the graph may be too large to fit in computer memory, making this approach fruitless. In this thesis, we first present a novel algorithm that, under the assumption that the keywords take the form of a so-called <i>generalized string</i>, constructs the minimal DFA recognizing those keywords. Importantly, the algorithm is iterative and allows one to build the automaton directly, without any use of buffer memory. Not only does this mean that the algorithm is efficient regarding memory consumption, but it also provides useful insight to help facilitate analysis for the size of such DFA. Using this new algorithm and pairing it with the assumption that the generalized strings are drawn at random from some class of probability distributions, we develop bounds on the size of the minimal automaton that are true with high probability. Furthermore, using synthetic data, we provide evidence that the size of the minimal automaton grows linearly in expectation for many cases.
Advisors/Committee Members: Manuel E. Lladser, Jem Corcoran, Anne Dougherty.
Subjects/Keywords: aho-corasick automaton; biological motifs; deterministic finite automata; generalized strings; stochastic analysis; Applied Mathematics; Computer Sciences
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APA ·
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APA (6th Edition):
Char, I. G. (2018). Algorithmic Construction and Stochastic Analysis of Optimal Automata for Generalized Strings. (Masters Thesis). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/115
Chicago Manual of Style (16th Edition):
Char, Ian Guo-fan. “Algorithmic Construction and Stochastic Analysis of Optimal Automata for Generalized Strings.” 2018. Masters Thesis, University of Colorado. Accessed March 09, 2021.
https://scholar.colorado.edu/appm_gradetds/115.
MLA Handbook (7th Edition):
Char, Ian Guo-fan. “Algorithmic Construction and Stochastic Analysis of Optimal Automata for Generalized Strings.” 2018. Web. 09 Mar 2021.
Vancouver:
Char IG. Algorithmic Construction and Stochastic Analysis of Optimal Automata for Generalized Strings. [Internet] [Masters thesis]. University of Colorado; 2018. [cited 2021 Mar 09].
Available from: https://scholar.colorado.edu/appm_gradetds/115.
Council of Science Editors:
Char IG. Algorithmic Construction and Stochastic Analysis of Optimal Automata for Generalized Strings. [Masters Thesis]. University of Colorado; 2018. Available from: https://scholar.colorado.edu/appm_gradetds/115
University of Colorado
3.
Hampton, Jerrad Davis.
Dissimilarity and Optimal Sampling in Urn Ensembles.
Degree: PhD, Applied Mathematics, 2012, University of Colorado
URL: https://scholar.colorado.edu/appm_gradetds/27
► We study an ensemble of urns with unknown compositions inferred from initial samples with replacement from each urn. This model fits diverse situations. For…
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▼ We study an ensemble of urns with unknown compositions inferred from initial samples with replacement from each urn. This model fits diverse situations. For instance, in microbial ecology studies each urn represents an environment, each ball within an urn corresponds to an individual bacterium, and a ball's color represents its taxonomic label. In a different context, each urn could represent a random RNA pool and each colored ball a possible solution to a particular binding site problem over that pool. The main parameter of this study is dissimilarity, which we define as the probability that a draw from one urn is not seen in a sample of size k from a possibly different urn. We estimate this parameter with a U-statistic, shown to be the uniformly minimum variance unbiased estimator (UMVUE) of dissimilarity over a range for k determined by initial sample sizes. Furthermore, despite the non-Markovian nature of our estimator when applied sequentially over k, we provide conditions that guarantee uniformly consistent estimates of variances via a jackknife method, and show uniform convergence in probability as well as approximately normal marginal distributions. We apply our U-statistics and a restricted exponential regression to extrapolate dissimilarity over a range beyond that determined by initial sample sizes, which we use to identify an allocation of draws for subsequent sampling that minimizes a measure of pair-wise dissimilarities over the whole ensemble. This is motivated by the challenge faced by microbiome projects worldwide to effectively allocate additional samples for a more robust and reliable estimation of UniFrac distances between pairs of environments. Similar methods are applied to measures of sample quality of the ensemble derived from alpha-diversity and coverage. We test our methods against simulated data, where we compare optimal and inferred draw allocations when considering these three measures, and analyze 16S ribosomal RNA data from the Human Microbiome Project.
Advisors/Committee Members: Manuel E. Lladser, Rob Knight, Jem Corcoran.
Subjects/Keywords: Alpha-Diversity; Coverage; Dissimilarity; Urn Models; Applied Mathematics; Statistics and Probability
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APA (6th Edition):
Hampton, J. D. (2012). Dissimilarity and Optimal Sampling in Urn Ensembles. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/27
Chicago Manual of Style (16th Edition):
Hampton, Jerrad Davis. “Dissimilarity and Optimal Sampling in Urn Ensembles.” 2012. Doctoral Dissertation, University of Colorado. Accessed March 09, 2021.
https://scholar.colorado.edu/appm_gradetds/27.
MLA Handbook (7th Edition):
Hampton, Jerrad Davis. “Dissimilarity and Optimal Sampling in Urn Ensembles.” 2012. Web. 09 Mar 2021.
Vancouver:
Hampton JD. Dissimilarity and Optimal Sampling in Urn Ensembles. [Internet] [Doctoral dissertation]. University of Colorado; 2012. [cited 2021 Mar 09].
Available from: https://scholar.colorado.edu/appm_gradetds/27.
Council of Science Editors:
Hampton JD. Dissimilarity and Optimal Sampling in Urn Ensembles. [Doctoral Dissertation]. University of Colorado; 2012. Available from: https://scholar.colorado.edu/appm_gradetds/27
University of Colorado
4.
Pieper, Jaden K.
Unentangling Quantum Algorithms for Mathematicians and Engineers.
Degree: MS, Applied Mathematics, 2017, University of Colorado
URL: https://scholar.colorado.edu/appm_gradetds/85
► As industry continues to inspire considerable growth in the research and development of quantum computers, it is increasingly worthwhile to familiarize oneself with the…
(more)
▼ As industry continues to inspire considerable growth in the research and development of quantum computers, it is increasingly worthwhile to familiarize oneself with the computational theory of this new and exciting field. In this manuscript, we introduce readers to quantum computing by first developing a quantum intuition through the enlightening results of the so-called Stern-Gerlach experiment. After getting a feeling for quantum physics concepts such as superposition and measurement, and the need of linear algebra and probability to describe quantum phenomena, we move to a discussion of quantum computing. We develop a "quantum toolbox,'' the mathematical tools used to explore quantum computing, before walking through two quantum algorithms, Deutsch's and Grover's algorithms. We finally explore a real quantum computer that is available to the public and show readers how to implement these quantum algorithms on it. We also consider the effect of error due to quantum decoherence, and the limitations such errors pose on quantum algorithms.
Advisors/Committee Members: Manuel E. Lladser, Harvey Segur, Stephen Becker, James Meiss.
Subjects/Keywords: quantum computing; public interface; Grover's algorithm; Deutsch's algorithm; decoherence; Applied Mathematics; Computational Engineering
Record Details
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Record Details
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APA ·
Chicago ·
MLA ·
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CSE |
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to Zotero / EndNote / Reference
Manager
APA (6th Edition):
Pieper, J. K. (2017). Unentangling Quantum Algorithms for Mathematicians and Engineers. (Masters Thesis). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/85
Chicago Manual of Style (16th Edition):
Pieper, Jaden K. “Unentangling Quantum Algorithms for Mathematicians and Engineers.” 2017. Masters Thesis, University of Colorado. Accessed March 09, 2021.
https://scholar.colorado.edu/appm_gradetds/85.
MLA Handbook (7th Edition):
Pieper, Jaden K. “Unentangling Quantum Algorithms for Mathematicians and Engineers.” 2017. Web. 09 Mar 2021.
Vancouver:
Pieper JK. Unentangling Quantum Algorithms for Mathematicians and Engineers. [Internet] [Masters thesis]. University of Colorado; 2017. [cited 2021 Mar 09].
Available from: https://scholar.colorado.edu/appm_gradetds/85.
Council of Science Editors:
Pieper JK. Unentangling Quantum Algorithms for Mathematicians and Engineers. [Masters Thesis]. University of Colorado; 2017. Available from: https://scholar.colorado.edu/appm_gradetds/85
. 
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