The segregation of chromosomes during cell division is accomplished by kinetochore machinery that uses depolymerizing microtubules to pull the chromosomes to opposite poles of the dividing cell. While much is known about molecular motors that pull by walking or push by polymerizing, the mechanism of how a pulling force can be achieved by depolymerization is still unresolved. In this talk, I will describe a new model for the depolymerization motor that is used by eukaryotic cells to segregate chromosomes during mitosis. In the process we will explore the use of Huxley-type models (population models) of protein binding and unbinding to study load-velocity curves of several different motor-like proteins.

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.

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.

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.

The jellyfish has been the subject of numerous mathematical and physical studies ranging from the discovery of reentry phenomenon in electrophysiology to the development of axisymmetric methods for solving fluid-structure interaction problems. In this presentation, we develop and test mathematical models describing the pulsing dynamics and the resulting fluid flow generated by the upside down jellyfish, Cassiopea. The kinematics of contraction and distributions of pulse frequencies were obtained from videos and used as inputs into numerical simulations. Particle image velocimetry was used to obtain spatially and temporally resolved flow fields experimentally. The immersed boundary method was then used to solve the fluid-structure interaction problem and explore how changes in morphology and pulsing dynamics alter the resulting fluid flow. Unlike pelagic (swimming) jellyfish, there is no evidence of the formation of a train of vortex rings. Instead, significant mixing occurs around and directly above the oral arms and secondary mouths. We found good agreement between the numerical simulations and experiments, suggesting that the presence of porous oral arms induce net horizontal flow towards the bell and mixing.

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.

A conference in honor of Harold Layton's 60th Birthday (Morning Session)

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

Two-sample tests are an important class of problems in statistics, with abundant applications ranging from astrophysics to zoology. However, much of the previous art assumes the data samples live in finite dimensional Euclidean space. Here, we consider a foray into two-sample testing when the objects live in a non-Euclidean space, with special emphasis on graph valued observations. Via embedding each graph into Euclidean space, and then learning a manifold along which the reside, we demonstrate the existence of a test such that for a given confidence level alpha, we obtain power > alpha. Simulations and real data applications demonstrate the pragmatic utility of our approach even for very large graphs.

Persistent homology measures geometric structures using topological invariants. When the structures are magnetic resonance images of branching arteries, for example, persistent homology records the connectedness of an increasing subset of the vessels. Although the theory of persistent homology is relatively well developed, and many aspects of its behavior are understood in synthetic examples, only recently have applications to genuine experimental data begun. This talk explains what we have learned about the geometry of blood vessels in aging human brains, as well as lessons this exploration has taught us about applications of persistent homology in general. These lessons inform further potential applications of persistent homology in statistical problems from biological and medical imaging. The main results are joint with Paul Bendich, Steve Marron, Aaron Pieloch, and Sean Skwerer (Math junior faculty, Stat faculty, Math undergrad, and Operations Research grad student). The talk will be accessible to advanced mathematics and statistics undergraduates, medical and biological researchers, statistics and mathematics faculty, and everybody in between.

Many mathematical models in biology can be described by a system of hyperbolic conservation laws with nonlinear and nonlocal coefficients. In order to determine these coefficients one needs to solve auxiliary systems of equations, for example elliptic or parabolic PDEs, which are coupled to the hyperbolic equations. In this talk we give several examples: The growth of tumors, the shrinking of dermal wounds, T cell differentiation, the growth of drug resistant bacteria in hospitals, and the transport of molecules along microtubules in axon. In these examples, the auxiliary systems range from elliptic-parabolic free boundary problems to nonlocal ODEs. Motivated by biological questions, we shall describe mathematical results regarding properties of the solutions of the conservation laws. For example, we shall determine the stability of spherical tumors and the growth of �?fingers;�? we shall discuss conditions for shrinking of the wound; suggest how to reduce the growth of drug resistant bacteria, and derive biologically motivated asymptotic behavior of solutions.

One of the challenges in biology is relating biochemical reactions that occur at the protein nanoparticle size of 1-100 nm to large scale effects on the cell or tissue scale of 0.01-10 mm. The cytoskeleton is a remarkable example with actin polymerization/depolymerization leading to locomotion, metastasis or apoptosis. This talk presents a recently developed multiscale model that captures large-scale effects produced by changes in biochemical reactions. The model is a computational algorithm that determines effective continuum properties of a homogenized cytoskeleton model by concurrent microscopic simulation. Concepts from information theory and optimal transport are applied to link disparate scales in a computationally efficient manner. One of the interesting aspects of this approach is the combination of standard computational modeling techniques (finite volume, numerical stochastic ODEs) with statistical concepts and learning theory.

Despite the impressive performance of rituximab (R) containing regimens like R-CHOP in CD20+ Non-Hodgkin�??s Lymphoma (NHL), 30-60% of R-naïve NHL patients are estimated to be resistant, and approximately 60% of those patients will not respond to subsequent single agent R treatment. Given that antibody dependent cell mediated cytotoxicity (ADCC) and phagocytosis (ADCP) are thought to be the major mechanisms of action of Rituximab, increasing the activation levels of natural killer (NK) and macrophage (MP) cells may be one strategy for overcoming R resistance.

During (and after) the Fields Institute Industrial Problem Solving Workshop in August 2019, academic participants and industry mentors developed and calibrated to literature data a quantitative systems pharmacology (QSP) model of ADCC/ADCP to interrogate which mechanisms of R resistance could be overcome by increased NK or MP activation, and how much effector cell activation would be required to overcome a given degree and mechanism of R resistance.

This work was motivated by a real-world pharmaceutical drug development question, and the academic-industry interactions during and after the workshop resulted in sharknado plots as well as a published QSP model (presented at American Association of Cancer Research Annual Meeting, 2021) that was able to address some of the key questions around overcoming R resistance. The published model was then incorporated into an in-house QSP model supporting the development of a Takeda investigational drug which is being developed to restore R sensitivity in an R-resistant patient population.