The three-dimensional higher-order structure of biomolecules determines their functionality. While assessing primary structure is fairly easily accessible, reconstruction of higher order structure is costly. It often requires elaborate correction of atomic clashes, frequently not fully successful. Using RNA data, we describe a purely statistical method, learning error correction, drawing power from a two-scale approach. Our microscopic scale describes single suites by dihedral angles of individual atom bonds; here, addressing the challenge of torus principal component analysis (PCA) leads to a fundamentally new approach to PCA building on principal nested spheres by Jung et al. (2012). Based on an observed relationship with a mesoscopic scale, landmarks describing several suites, we use Fréchet means for angular shape and size-and-shape, correcting within-suite-backbone-to-backbone clashes. We validate this method by comparison to reconstructions obtained from simulations approximating biophysical chemistry and illustrate its power by the RNA example of SARS-CoV-2.

This is joint work with Benjamin Eltzner, Kanti V. Mardia and Henrik Wiechers.

Literature:

Eltzner, B., Huckemann, S. F., Mardia, K. V. (2018): Torus principal component analysis with applications to RNA structure. Ann. Appl. Statist. 12(2), 1332?1359.

Jung, S., Dryden, I. L., Marron, J. S. (2012): Analysis of principal nested spheres. Biometrika, 99 (3), 551-568

Mardia, K. V., Wiechers, H., Eltzner, B., Huckemann, S. F. (2022). Principal component analysis and clustering on manifolds. Journal of Multivariate Analysis, 188, 104862, https://www.sciencedirect.com/science/article/pii/S0047259X21001408

Wiechers, H., Eltzner, B., Mardia, K. V., Huckemann, S. F. (2021). Learning torus PCA based classification for multiscale RNA backbone structure correction with application to SARS-CoV-2. To appear in the Journal of the Royal Statistical Society, Series C, bioRxiv https://doi.org/10.1101/2021.08.06.455406

Circadian rhythms are biological oscillations that align our physiology and behavior with the 24-hour environmental cycles conferred by the Earth’s rotation. In this talk, I will discuss two projects that focus on circadian clock cells in the brain and the entrainment of circadian rhythms to the light-dark cycle. Most of what we know about the electrical activity of circadian clock neurons comes from studies of nocturnal (night-active) rodents, hindering the translation of this knowledge to diurnal (day-active) humans. In the first part of the talk, we use data assimilation and patch-clamp recordings from the diurnal rodent Rhabdomys pumilio to build the first mathematical models of the electrophysiology of circadian neurons in a day-active species. We find that the electrical activity of circadian neurons is similar overall between nocturnal and diurnal rodents but that there are some interesting differences in their responses to inhibition. In the second part of the talk, we use tools from dynamical systems theory to study the reentrainment of a model of the human circadian pacemaker following perturbations that simulate jet lag. We show that the reentrainment dynamics are organized by invariant manifolds of fixed points of a 24-hour stroboscopic map and use these manifolds to explain a rapid reentrainment phenomenon that occurs under certain jet lag scenarios.

We present two models of perpetrators' decision-making in extracting resources from a protected area. It is assumed that the authorities conduct surveillance to counter the extraction activities, and that perpetrators choose their post-extraction paths to balance the time/hardship of travel against the expected losses from a possible detection. In our first model, the authorities are assumed to use ground patrols and the protected resources are confiscated as soon as the extractor is observed with them. The perpetrators' path-planning is modeled using the optimal control of randomly-terminated process. In our second model, the authorities use aerial patrols, with the apprehension of perpetrators and confiscation of resources delayed until their exit from the protected area. In this case the path-planning is based on multi-objective dynamic programming. Our efficient numerical methods are illustrated on several examples with complicated geometry and terrain of protected areas, non-uniform distribution of protected resources, and spatially non-uniform detection rates due to aerial or ground patrols.

The notochord, the defining feature of chordates, is a pressurized tube which actuates elongation of the chordate embryo. The zebrafish notochord consists of large vacuolated cells surrounded by a thin sheath. We characterized the patterns of the cells’ packing, and their relationship to the known regular patterns from the study of foams, and irregular patterns in a gel bead system. Disruption of the wild type packing pattern leads to developmental defects. We characterize the bifurcations between the relevant regular patterns in terms of nondimensional geometrical and mechanical ratios, and suggest an important developmental role for the eccentric "staircase" pattern.

Many patterns in Nature and industry arise from the system minimizing an appropriate energy. Torn plastic sheets and growing leaves provide striking examples of pattern forming systems which can transition from single wavelength geometries (leaves) to complex fractal-like shapes (lettuce). These fractal-like patterns seem to have many length scales, i.e. the same amount of extra detail can be seen when looking closer (“statistical self-similarity”). It is a mystery how such complex patterns could arise from energy minimization alone. In this talk I will address this puzzle by showing that such patterns naturally arise from the sheet adopting a hyperbolic non-Euclidean geometry. However, there are many different hyperbolic geometries that the growing leaf could select. I will show using techniques from analysis, differential geometry and numerical optimization that the fractal like patterns are indeed the natural minimizers for the system. I will also discuss the implications of our work to developing shape changing soft matter which can be implemented in soft machines.

Genetic intratumoral heterogeneity is a natural consequence of imperfect DNA replication. Any two randomly selected cells, whether normal or cancerous, are therefore genetically different. I will discuss the extent of genetic heterogeneity within untreated cancers with particular regard to its clinical relevance. While genomic heterogeneity within primary tumors is associated with relapse, heterogeneity among treatment‑naïve metastases has not been comprehensively assessed. We analyzed sequencing data for 76 untreated metastases from 20 patients and inferred cancer phylogenies for breast, colorectal, endometrial, gastric, lung, melanoma, pancreatic, and prostate cancers. We found that within individual patients a large majority of driver gene mutations are common to all metastases. Further analysis revealed that the driver gene mutations that were not shared by all metastases are unlikely to have functional consequences. A mathematical model of tumor evolution and metastasis formation provides an explanation for the observed driver gene homogeneity. Last, we found that individual metastatic lesions responded concordantly to targeted therapies in 91% of 44 patients. These data indicate that the cells within the primary tumors that gave rise to metastases are genetically homogeneous with respect to functional driver gene mutations and suggest that future efforts to develop combination therapies have the capacity to be curative.

The cellular cytoskeleton ensures the dynamic transport, localization, and anchoring of various proteins and vesicles. In the development of egg cells into embryos, messenger RNA (mRNA) molecules bind and unbind to and from cellular roads called microtubules, switching between bidirectional transport, diffusion, and stationary states. Since models of intracellular transport can be analytically intractable, asymptotic methods are useful in understanding effective cargo transport properties as well as their dependence on model parameters. We consider these models in the framework of partial differential equations as well as stochastic processes and derive the effective velocity and diffusivity of cargo at large time for a general class of problems. Including the geometry of the microtubule filaments allows for better prediction of particle localization and for investigation of previously unexplored mechanisms. Our numerical studies incorporating model microtubule structures suggest that anchoring of mRNA-molecular motor complexes may be necessary in localization, to promote healthy development of oocytes into embryos.

Traditional infectious disease epidemiology is built on the foundation of high quality and high accuracy data on disease and behavior. While these data are usually characterized by smallsize, they benefit from designed sampling schemes that make it possible to make population-level inferences. On the other hand, digital infectious disease epidemiology uses existing digital traces, re-purposing them to identify patterns in health-related processes. In this talk, I will discuss our work using data from small epidemiological studies as well as administrative “big data” to understand influenza transmission dynamics and inform disease surveillance and control.

Israel experienced an outbreak of wild poliovirus type 1 (WPV1) in 2013-14, detected through environmental surveillance of the sewage system. No cases of acute flaccid paralysis were reported, and the epidemic subsided after a bivalent oral polio vaccination (bOPV) campaign. As we approach global eradication, polio will increasingly be detected only through environmental surveillance. However, we have lacked the theory to translate environmental surveillance into public health metrics; it is a priori unclear how much environmental surveillance can even say about population-level disease dynamics. We developed a framework to convert quantitative polymerase chain reaction (qPCR) cycle threshold data into scaled WPV1 and OPV1 concentrations for inference within a deterministic, compartmental infectious disease transmission model. We used differential algebra and profile likelihood techniques to perform identifiability analysis, that is, to assess how much information exists in the data for the model, and to quantify inference uncertainty. From the environmental surveillance data, we estimated the epidemic curve and transmission dynamics, determining that the outbreak likely happened much faster than previously thought. Our mathematical modeling approach brings public health relevance to environmental data that, if systematically collected, can guide eradication efforts.

Complex interactions occur between tumor and host immune system during cancer development and treatment, and a weak systemic immune response can be prognostic of poor patient outcomes. We strive to not only better understand the dynamic behavior of circulating immune cell populations before and during cancer therapy, but also to monitor these dynamic changes to facilitate real-time prediction of patient outcomes and potentially therapy adaptation. I will provide examples of both theoretical (mathematical) and data-driven (epidemiological) approaches to incorporating established systemic immune markers into clinical decision-making. First, coupling models of local tumor-immune dynamics and systemic T cell trafficking allows us to simulate the evolution of tumor and immune cell populations in anatomically distant sites following local therapy, in turn identifying the optimal treatment target for maximum reduction of global tumor burden. Second, improved understanding of how circulating immune markers vary both within and between individual patients can allow more accurate risk stratification at diagnosis, and personalized prediction of patient response to therapy. The importance of multi-disciplinary collaborations in making predictive and prognostic models clinically relevant will be discussed.

Cancers are caused by mutations that may be inherited, induced by environmental factors, or result from DNA replication errors (R). We studied the relationship between the number of normal stem cell divisions and the risk of 17 cancer types in 69 countries throughout the world. The data revealed a strong correlation (median = 0.80) between cancer incidence and normal stem cell divisions in all countries, regardless of their environment. The major role of R mutations in cancer etiology was supported by an independent approach, based solely on cancer genome sequencing and epidemiological data, which suggested that R mutations are responsible for two-thirds of the mutations in human cancers. All of these results are consistent with epidemiological estimates of the fraction of cancers that can be prevented by changes in the environment. Moreover, they accentuate the importance of early detection and intervention to reduce deaths from the many cancers arising from unavoidable R mutations.

Understanding the folding of RNA sequences into three-dimensional structures is one of the fundamental challenges in molecular biology. For example, the branching of an RNA secondary structure is an important molecular characteristic yet difficult to predict correctly, especially for sequences on the scale of viral genomes. However, results from enumerative, probabilistic, analytic, and geometric combinatorics yield insights into RNA structure formation, and suggest new directions in viral capsid assembly.

It has long been accepted that type 1 diabetes results from a lack of insulin, as the insulin-secreting pancreatic beta cells are destroyed by an autoimmune process. In contrast, the cause of type 2 diabetes (T2D) is less clear. Most people with pre-diabetes or in the early stages of T2D have abnormally high plasma insulin concentrations, and insulin rises before glucose does. We show that these difficulties are resolved by a mathematical model in which the onset of T2D is represented by the crossing of a threshold. The threshold is atypical in some respects and requires consideration of the slow manifolds to avoid incorrect conclusions.