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.

Distinct functions of POT1 at telomeres.

The mammalian protein POT1 binds to telomeric single-stranded DNA (ssDNA), protecting chromosome ends from being detected as sites of DNA damage. POT1 is composed of an N-terminal ssDNA-binding domain and a C-terminal protein interaction domain. With regard to the latter, POT1 heterodimerizes with the protein TPP1 to foster binding to telomeric ssDNA in vitro and binds the telomeric double-stranded-DNA-binding protein TRF2. We sought to determine which of these functions-ssDNA, TPP1, or TRF2 binding-was required to protect chromosome ends from being detected as DNA damage. Using separation-of-function POT1 mutants deficient in one of these three activities, we found that binding to TRF2 is dispensable for protecting telomeres but fosters robust loading of POT1 onto telomeric chromatin. Furthermore, we found that the telomeric ssDNA-binding activity and binding to TPP1 are required in cis for POT1 to protect telomeres. Mechanistically, binding of POT1 to telomeric ssDNA and association with TPP1 inhibit the localization of RPA, which can function as a DNA damage sensor, to telomeres.

B-lymphocyte effector functions in health and disease.

B-lymphocytes have traditionally been thought to contribute to immunity and autoimmune disease through terminal differentiation into plasma cells that secrete antibody. However, studies in mice and recent clinical studies have demonstrated that genetically altered B-cell function and B-cell-targeted therapies can significantly affect autoimmune diseases that were predominantly thought to be T-cell-mediated. B-cell depletion in mouse models of disease has also led to the identification of alternative B-cell effector functions that regulate normal immune responses and autoimmune disease. This review highlights multiple B-cell effector mechanisms, including the promotion of cellular immunity, the negative regulation of immune responses, and the production of pathogenic antibodies.

Typical delay determines waiting time on periodic-food schedules: Static and dynamic tests.

Pigeons and other animals soon learn to wait (pause) after food delivery on periodic-food schedules before resuming the food-rewarded response. Under most conditions the steady-state duration of the average waiting time, t, is a linear function of the typical interfood interval. We describe three experiments designed to explore the limits of this process. In all experiments, t was associated with one key color and the subsequent food delay, T, with another. In the first experiment, we compared the relation between t (waiting time) and T (food delay) under two conditions: when T was held constant, and when T was an inverse function of t. The pigeons could maximize the rate of food delivery under the first condition by setting t to a consistently short value; optimal behavior under the second condition required a linear relation with unit slope between t and T. Despite this difference in optimal policy, the pigeons in both cases showed the same linear relation, with slope less than one, between t and T. This result was confirmed in a second parametric experiment that added a third condition, in which T + t was held constant. Linear waiting appears to be an obligatory rule for pigeons. In a third experiment we arranged for a multiplicative relation between t and T (positive feedback), and produced either very short or very long waiting times as predicted by a quasi-dynamic model in which waiting time is strongly determined by the just-preceding food delay.

Scale-wise evolution of rainfall probability density functions fingerprints the rainfall generation mechanism

© 2010 by the American Geophysical Union.The cross-scale probabilistic structure of rainfall intensity records collected over time scales ranging from hours to decades at sites dominated by both convective and frontal systems is investigated. Across these sites, intermittency build-up from slow to fast time-scales is analyzed in terms of heavy tailed and asymmetric signatures in the scale-wise evolution of rainfall probability density functions (pdfs). The analysis demonstrates that rainfall records dominated by convective storms develop heavier-Tailed power law pdfs toward finer scales when compared with their frontal systems counterpart. Also, a concomitant marked asymmetry build-up emerges at such finer time scales. A scale-dependent probabilistic description of such fat tails and asymmetry appearance is proposed based on a modified q-Gaussian model, able to describe the cross-scale rainfall pdfs in terms of the nonextensivity parameter q, a lacunarity (intermittency) correction and a tail asymmetry coefficient, linked to the rainfall generation mechanism.

Resonant transmission near nonrobust periodic slab modes.

We present a precise theoretical explanation and prediction of certain resonant peaks and dips in the electromagnetic transmission coefficient of periodically structured slabs in the presence of nonrobust guided slab modes. We also derive the leading asymptotic behavior of the related phenomenon of resonant enhancement near the guided mode. The theory applies to structures in which losses are negligible and to very general geometries of the unit cell. It is based on boundary-integral representations of the electromagnetic fields. These depend on the frequency and on the Bloch wave vector and provide a complex-analytic connection in these parameters between generalized scattering states and guided slab modes. The perturbation of three coincident zeros-those of the dispersion relation for slab modes, the reflection constant, and the transmission constant-is central to calculating transmission anomalies both for lossless dielectric materials and for perfect metals.

Desensitization, internalization, and signaling functions of beta-arrestins demonstrated by RNA interference.

Beta-arrestins bind to activated G protein-coupled receptor kinase-phosphorylated receptors, which leads to their desensitization with respect to G proteins, internalization via clathrin-coated pits, and signaling via a growing list of "scaffolded" pathways. To facilitate the discovery of novel adaptor and signaling roles of beta-arrestins, we have developed and validated a generally applicable interfering RNA approach for selectively suppressing beta-arrestins 1 or 2 expression by up to 95%. Beta-arrestin depletion in HEK293 cells leads to enhanced cAMP generation in response to beta(2)-adrenergic receptor stimulation, markedly reduced beta(2)-adrenergic receptor and angiotensin II receptor internalization and impaired activation of the MAP kinases ERK 1 and 2 by angiotensin II. This approach should allow discovery of novel signaling and regulatory roles for the beta-arrestins in many seven-membrane-spanning receptor systems.

A tale of three functions: The self-reported uses of autobiographical memory

Theories hold that autobiographical memory serves several broad functions (directive, self, and social). In the current study, items were derived from the theoretical literature to create the Thinking About Life Experiences (TALE) questionnaire to empirically assess these three functions. Participants (N = 167) completed the TALE. To examine convergent validity, they also rated their overall tendency to think about and to talk about the past and completed the Reminiscence Functions Scale (Webster, 1997). The results lend support to the existence of these theoretical functions, but also offer room for refinements in future thinking about both the breadth and specificity of the functions that autobiographical memory serves.

The inoculum effect and band-pass bacterial response to periodic antibiotic treatment.

The inoculum effect (IE) refers to the decreasing efficacy of an antibiotic with increasing bacterial density. It represents a unique strategy of antibiotic tolerance and it can complicate design of effective antibiotic treatment of bacterial infections. To gain insight into this phenomenon, we have analyzed responses of a lab strain of Escherichia coli to antibiotics that target the ribosome. We show that the IE can be explained by bistable inhibition of bacterial growth. A critical requirement for this bistability is sufficiently fast degradation of ribosomes, which can result from antibiotic-induced heat-shock response. Furthermore, antibiotics that elicit the IE can lead to 'band-pass' response of bacterial growth to periodic antibiotic treatment: the treatment efficacy drastically diminishes at intermediate frequencies of treatment. Our proposed mechanism for the IE may be generally applicable to other bacterial species treated with antibiotics targeting the ribosomes.

Mix it and fix it: functions of composite olfactory signals in ring-tailed lemurs.

Animals communicating via scent often deposit composite signals that incorporate odorants from multiple sources; however, the function of mixing chemical signals remains understudied. We tested both a 'multiple-messages' and a 'fixative' hypothesis of composite olfactory signalling, which, respectively, posit that mixing scents functions to increase information content or prolong signal longevity. Our subjects-adult, male ring-tailed lemurs (Lemur catta)-have a complex scent-marking repertoire, involving volatile antebrachial (A) secretions, deposited pure or after being mixed with a squalene-rich paste exuded from brachial (B) glands. Using behavioural bioassays, we examined recipient responses to odorants collected from conspecific strangers. We concurrently presented pure A, pure B and mixed A + B secretions, in fresh or decayed conditions. Lemurs preferentially responded to mixed over pure secretions, their interest increasing and shifting over time, from sniffing and countermarking fresh mixtures, to licking and countermarking decayed mixtures. Substituting synthetic squalene (S)-a well-known fixative-for B secretions did not replicate prior results: B secretions, which contain additional chemicals that probably encode salient information, were preferred over pure S. Whereas support for the 'multiple-messages' hypothesis underscores the unique contribution from each of an animal's various secretions, support for the 'fixative' hypothesis highlights the synergistic benefits of composite signals.