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Chebfun is a collection of algorithms and an open-source software system in object-oriented Matlab that extends familiar powerful methods of numerical computation involving numbers to continuous or piecewise-continuous functions. The success of this strategy is based on the mathematical fact that smooth functions can be represented very efficiently by polynomial

Chebfun is a collection of algorithms and an open-source software system in object-oriented Matlab that extends familiar powerful methods of numerical computation involving numbers to continuous or piecewise-continuous functions. The success of this strategy is based on the mathematical fact that smooth functions can be represented very efficiently by polynomial interpolation at Chebyshev points or by trigonometric interpolation at equispaced points for periodic functions. More recently, the system has been extended to handle bivariate functions and vector fields. These two new classes of objects are called Chebfun2 and Chebfun2v, respectively. We will show that Chebfun2 and Chebfun2v, and can be used to accurately and efficiently perform various computations on parametric surfaces in two or three dimensions, including path trajectories and mean and Gaussian curvatures. More advanced surface computations such as mean curvature flows are also explored. This is also the first work to use the newly implemented trigonometric representation, namely Trigfun, for computations on surfaces.
ContributorsPage-Bottorff, Courtney Michelle (Author) / Platte, Rodrigo (Thesis director) / Kostelich, Eric (Committee member) / School of Mathematical and Statistical Sciences (Contributor) / Barrett, The Honors College (Contributor)
Created2016-05
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Description
Using object-oriented programming in MATLAB, a collection of functions, named Fourfun, has been created to allow quick and accurate approximations of periodic functions with Fourier expansions. To increase efficiency and reduce the number of computations of the Fourier transform, Fourfun automatically determines the number of nodes necessary for representations that

Using object-oriented programming in MATLAB, a collection of functions, named Fourfun, has been created to allow quick and accurate approximations of periodic functions with Fourier expansions. To increase efficiency and reduce the number of computations of the Fourier transform, Fourfun automatically determines the number of nodes necessary for representations that are accurate to close to machine precision. Common MATLAB functions have been overloaded to keep the syntax of the Fourfun class as consistent as possible with the general MATLAB syntax. We show that the system can be used to efficiently solve several differential equations. Comparisons with Chebfun, a similar system based on Chebyshev polynomial approximations, are provided.
ContributorsMcleod, Kristyn Noelle (Author) / Platte, Rodrigo (Thesis director) / Gelb, Anne (Committee member) / Barrett, The Honors College (Contributor) / School of Mathematical and Statistical Sciences (Contributor) / School of International Letters and Cultures (Contributor)
Created2014-05
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Description
Predicting the binding sites of proteins has historically relied on the determination of protein structural data. However, the ability to utilize binding data obtained from a simple assay and computationally make the same predictions using only sequence information would be more efficient, both in time and resources. The purpose of

Predicting the binding sites of proteins has historically relied on the determination of protein structural data. However, the ability to utilize binding data obtained from a simple assay and computationally make the same predictions using only sequence information would be more efficient, both in time and resources. The purpose of this study was to evaluate the effectiveness of an algorithm developed to predict regions of high-binding on proteins as it applies to determining the regions of interaction between binding partners. This approach was applied to tumor necrosis factor alpha (TNFα), its receptor TNFR2, programmed cell death protein-1 (PD-1), and one of its ligand PD-L1. The algorithms applied accurately predicted the binding region between TNFα and TNFR2 in which the interacting residues are sequential on TNFα, however failed to predict discontinuous regions of binding as accurately. The interface of PD-1 and PD-L1 contained continuous residues interacting with each other, however this region was predicted to bind weaker than the regions on the external portions of the molecules. Limitations of this approach include use of a linear search window (resulting in inability to predict discontinuous binding residues), and the use of proteins with unnaturally exposed regions, in the case of PD-1 and PD-L1 (resulting in observed interactions which would not occur normally). However, this method was overall very effective in utilizing the available information to make accurate predictions. The use of the microarray to obtain binding information and a computer algorithm to analyze is a versatile tool capable of being adapted to refine accuracy.
ContributorsBrooks, Meilia Catherine (Author) / Woodbury, Neal (Thesis director) / Diehnelt, Chris (Committee member) / Ghirlanda, Giovanna (Committee member) / Department of Psychology (Contributor) / School of Molecular Sciences (Contributor) / Barrett, The Honors College (Contributor)
Created2018-05
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Description

The field of biomedical research relies on the knowledge of binding interactions between various proteins of interest to create novel molecular targets for therapeutic purposes. While many of these interactions remain a mystery, knowledge of these properties and interactions could have significant medical applications in terms of understanding cell signaling

The field of biomedical research relies on the knowledge of binding interactions between various proteins of interest to create novel molecular targets for therapeutic purposes. While many of these interactions remain a mystery, knowledge of these properties and interactions could have significant medical applications in terms of understanding cell signaling and immunological defenses. Furthermore, there is evidence that machine learning and peptide microarrays can be used to make reliable predictions of where proteins could interact with each other without the definitive knowledge of the interactions. In this case, a neural network was used to predict the unknown binding interactions of TNFR2 onto LT-ɑ and TRAF2, and PD-L1 onto CD80, based off of the binding data from a sampling of protein-peptide interactions on a microarray. The accuracy and reliability of these predictions would rely on future research to confirm the interactions of these proteins, but the knowledge from these methods and predictions could have a future impact with regards to rational and structure-based drug design.

ContributorsPoweleit, Andrew Michael (Author) / Woodbury, Neal (Thesis director) / Diehnelt, Chris (Committee member) / Chiu, Po-Lin (Committee member) / School of Molecular Sciences (Contributor, Contributor) / Barrett, The Honors College (Contributor)
Created2021-05