DIMACS REU 2020

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Project Description

The human genome contains billions of base pairs that stretch out to be meters long. As such, DNA is highly folded in order to fit within cell nuclei. In order to do this, DNA wraps into nucleosomes. The stretches of DNA between nucleosomes is called free DNA and it has been modeled using Bezier curves and B-splines for their smooth nature and ability to be determined by relatively few control points.

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Research Log

Week 1:

This week, I met Dr. Olson and was introduced members of her lab. They described the projects that they've been working on and recommended papers related to the lab's research. The goal of this week was to familiarize myself with the scientific background of my project and the math that I will be using this summer. Through reading several papers, I have gained a great breadth of knowledge about DNA folding.

Week 2:

This week, I used python to create a program that plots closed B-Spline curves given any number of controlling points in two or three dimensional space. Later in the week, I modified the program to rescale the models to biologically relevant sizes given a desired contour length or the number of base pairs on the modeled curve. In addition, I computed unit tangent, normal, and binormal vectors along the curve and used them to attach Frenet-Serret frames at the locations of each base pair along the modeled B-Spline curves. The next step will be to plot the DNA base pairs in these Frenet-Serret frames with the information provided by the nucleic acid data base.

Week 3:

This week, I attached DNA base pairs to each Frenet-Serret frame and introduced twist to the model. These steps have enabled me to turn B-splines into three dimensional models of DNA. In addition to this, I worked on debugging code to make the models more accurate. To this end, I've worked on ensuring that the placement of the base pairs is such that they are separated by 3.4 Angstroms. Furthermore, I have begun to replicate this process to create a similar model based upon Bezier Curves. Next week, I hope to further refine the B-Spline script and use it to model various types of DNA, included nucleosomal DNA. In addition, I hope to further my efforts in creating the Bezier curve based model.

Week 4:

This week, I worked on creating a model of DNA pathways based on Bezier curves. This model functions similarly to the aforementioned B-Spline model with a few differences. Most notably, this model creates open loops as opposed to closed curves with fixed tangent values at their end points. Next week, I would like to attach these curves to models of nucleosomal DNA.

Week 5:

I spent my time this week debating the merits of different methods of attaching Bezier curves to models of nucleosomal DNA. To this end, I worked on various methods of rescaling bezier curves to fit the desired number of DNA base pairs while maintaining a fixed distance between endpoints, the analytical integrity of the curve, and staying as close to the original shape as possible.

Week 6:

At the beginning of this week, I finalized the creation of the rescaling method which I spent week 5 contemplating. With this method, I attached various curves of DNA to various structures including the nucleosomal structure 1kx5. By doing this, I was able to explore the effects of slight changes in nucleosomal entrance and exit angle on the larger DNA structure.

Week 7:

Over the course of this week, I worked on editing and cleaning my python script for making curves of DNA. I made edits which helped to ensure the orthogonality of the reference frame axes, and with the help of Stefi, a postdoc in Dr. Olson's lab, I have started to work on optimizing the twist parameters of the reference frames on the structures which my code generates. In addition to this, I have begun to work on modeling DNA using polynomial curves. Going forward, we would like to use the models that I have created to explore the effects of changing linking number in topoisomers on the structures' resting energy states.

Week 8:

With the use of Stefi's optimization program, I was able to create DNA structures of topoisomers. These structures were constructed with changes in linking number that were solely the result of manipulations of twist. In B-Spline structures, changes in twist were uniformly applied to all reference frames. In models which connected Bezier curves to pre-existing structures, changes in twist only effected the Bezier reference frames. Through the use of this process, many topoisomer structures were generated for later analysis.

Week 9:

This week, I worked on finalizing my DIMACS report, and handed over my code to others in Dr. Olson's lab.

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Additional Information

My Mentor: Dr. Wilma Olson