Math 412 - Topology with Applications

Highlights of Data Expedition: • Students explored daily observations of local climate data spanning the past 35 years. • Topological Data Analysis, or TDA for short, provides cutting-edge tools for studying the geometry of data in arbitrarily high dimensions. • Using TDA tools, students discovered intrinsic dynamical features of the data and learned how to quantify periodic phenomenon in a time-series. • Since nature invariably produces noisy data which rarely has exact periodicity, students also considered the theoretical basis of almost-periodicity and even invented and tested new mathematical definitions of almost-periodic functions. Summary The dataset we used for this data expedition comes from the Global Historical Climatology Network. “GHCN (Global Historical Climatology Network)-Daily is an integrated database of daily climate summaries from land surface stations across the globe.” Source: https://www.ncdc.noaa.gov/oa/climate/ghcn-daily/ We focused on the daily maximum and minimum temperatures from January 1, 1980 to April 1, 2015 collected from RDU International Airport. Through a guided series of exercises designed to be performed in Matlab, students explore these time-series, initially by direct visualization and basic statistical techniques. Then students are guided through a special sliding-window construction which transforms a time-series into a high-dimensional geometric curve. These high-dimensional curves can be visualized by projecting down to lower dimensions as in the figure below (Figure 1), however, our focus here was to use persistent homology to directly study the high-dimensional embedding. The shape of these curves has meaningful information but how one describes the “shape” of data depends on which scale the data is being considered. However, choosing the appropriate scale is rarely an obvious choice. Persistent homology overcomes this obstacle by allowing us to quantitatively study geometric features of the data across multiple-scales. Through this data expedition, students are introduced to numerically computing persistent homology using the rips collapse algorithm and interpreting the results. In the specific context of sliding-window constructions, 1-dimensional persistent homology can reveal the nature of periodic structure in the original data. I created a special technique to study how these high-dimensional sliding-window curves form loops in order to quantify the periodicity. Students are guided through this construction and learn how to visualize and interpret this information. Climate data is extremely complex (as anyone who has suffered from a bad weather prediction can attest) and numerous variables play a role in determining our daily weather and temperatures. This complexity coupled with imperfections of measuring devices results in very noisy data. This causes the annual seasonal periodicity to be far from exact. To this end, I have students explore existing theoretical notions of almost-periodicity and test it on the data. They find that some existing definitions are also inadequate in this context. Hence I challenged them to invent new mathematics by proposing and testing their own definition. These students rose to the challenge and suggested a number of creative definitions. While autocorrelation and spectral methods based on Fourier analysis are often used to explore periodicity, the construction here provides an alternative paradigm to quantify periodic structure in almost-periodic signals using tools from topological data analysis.

Enzymatic Polymerization of High Molecular Weight DNA

The use of DNA as a polymeric building material transcends its function in biology and is exciting in bionanotechnology for applications ranging from biosensing, to diagnostics, and to targeted drug delivery. These applications are enabled by DNA’s unique structural and chemical properties, embodied as a directional polyanion that exhibits molecular recognition capabilities. Hence, the efficient and precise synthesis of high molecular weight DNA materials has become key to advance DNA bionanotechnology. Current synthesis methods largely rely on either solid phase chemical synthesis or template-dependent polymerase amplification. The inherent step-by-step fashion of solid phase synthesis limits the length of the resulting DNA to typically less than 150 nucleotides. In contrast, polymerase based enzymatic synthesis methods (e.g., polymerase chain reaction) are not limited by product length, but require a DNA template to guide the synthesis. Furthermore, advanced DNA bionanotechnology requires tailorable structural and self-assembly properties. Current synthesis methods, however, often involve multiple conjugating reactions and extensive purification steps.

The research described in this dissertation aims to develop a facile method to synthesize high molecular weight, single stranded DNA (or polynucleotide) with versatile functionalities. We exploit the ability of a template-independent DNA polymerase−terminal deoxynucleotidyl transferase (TdT) to catalyze the polymerization of 2’-deoxyribonucleoside 5’-triphosphates (dNTP, monomer) from the 3’-hydroxyl group of an oligodeoxyribonucleotide (initiator). We termed this enzymatic synthesis method: TdT catalyzed enzymatic polymerization, or TcEP.

Specifically, this dissertation is structured to address three specific research aims. With the objective to generate high molecular weight polynucleotides, Specific Aim 1 studies the reaction kinetics of TcEP by investigating the polymerization of 2’-deoxythymidine 5’-triphosphates (monomer) from the 3’-hydroxyl group of oligodeoxyribothymidine (initiator) using in situ 1H NMR and fluorescent gel electrophoresis. We found that TcEP kinetics follows the “living” chain-growth polycondensation mechanism, and like in “living” polymerizations, the molecular weight of the final product is determined by the starting molar ratio of monomer to initiator. The distribution of the molecular weight is crucially influenced by the molar ratio of initiator to TdT. We developed a reaction kinetics model that allows us to quantitatively describe the reaction and predict the molecular weight of the reaction products.

Specific Aim 2 further explores TcEP’s ability to transcend homo-polynucleotide synthesis by varying the choices of initiators and monomers. We investigated the effects of initiator length and sequence on TcEP, and found that the minimum length of an effective initiator should be 10 nucleotides and that the formation of secondary structures close to the 3’-hydroxyl group can impede the polymerization reaction. We also demonstrated TcEP’s capacity to incorporate a wide range of unnatural dNTPs into the growing chain, such as, hydrophobic fluorescent dNTP and fluoro modified dNTP. By harnessing the encoded nucleotide sequence of an initiator and the chemical diversity of monomers, TcEP enables us to introduce molecular recognition capabilities and chemical functionalities on the 5’-terminus and 3’-terminus, respectively.

Building on TcEP’s synthesis capacities, in Specific Aim 3 we invented a two-step strategy to synthesize diblock amphiphilic polynucleotides, in which the first, hydrophilic block serves as a macro-initiator for the growth of the second block, comprised of natural and/or unnatural nucleotides. By tuning the hydrophilic length, we synthesized the amphiphilic diblock polynucleotides that can self-assemble into micellar structures ranging from star-like to crew-cut morphologies. The observed self-assembly behaviors agree with predictions from dissipative particle dynamics simulations as well as scaling law for polyelectrolyte block copolymers.

In summary, we developed an enzymatic synthesis method (i.e., TcEP) that enables the facile synthesis of high molecular weight polynucleotides with low polydispersity. Although we can control the nucleotide sequence only to a limited extent, TcEP offers a method to integrate an oligodeoxyribonucleotide with specific sequence at the 5’-terminus and to incorporate functional groups along the growing chains simultaneously. Additionally, we used TcEP to synthesize amphiphilic polynucleotides that display self-assemble ability. We anticipate that our facile synthesis method will not only advance molecular biology, but also invigorate materials science and bionanotechnology.