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.

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.

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

It is well-known that genomes encode ancestry through replication errors - on average the greater the numbers of differences between two genomes, the greater the time since they shared a common ancestor ("molecular clock hypothesis"). This approach is commonly used to infer ancestries of species and populations, and these same tools can be applied to somatic cell evolution, in particular to better infer ancestries of normal and neoplastic tissues. For example, by sampling from opposite sides of the same human colorectal tumor, one can coalesce back to the earliest tumor cells. Such studies reveal that many human colorectal tumors are simple single "Big Bang" expansions, with evidence of neutral evolution during growth. It may be possible to understand in detail what is impossible to directly observe - the first few divisions of human tumors.

Birds, insects, and fish all exploit the fact that flexible wings or fins generally perform better than their rigid counterparts. Given the task of designing an optimal wing, though, it is not clear how to best distribute the flexibility: Should the wing be uniformly flexible along its length, or could some advantage be gained by making certain sections more rigid than others? I will discuss this question by using a 2D small-amplitude model for the fluid-structure interaction combined with an efficient Chebyshev PDE solver. Numerical optimization shows that concentrating flexibility near the leading edge of the wing maximizes thrust production, an arrangement that resembles the torsional-joint flexibility mechanism found in insect wings. I will discuss the possibility of extending into three dimensions to address the question of optimal wing architecture more generally.

In this talk, we will introduce a deterministic malaria model for determining the drug administration protocol that leads to the smallest first malaria episodes during the wet season. To explore the effects of administering the malaria drug on different days during wet season while minimizing the potential harmful effects of drug overdose, we will define 40 drug administration protocols. Our results fit well with the clinical studies of Coulibaly et al. at a site in Mali. In addition, we will provide protocols that lead to small number of first malaria episodes during the wet season than the protocol of Coulibaly et al. In the second part of the talk, we will use our malaria model to "capture" the 2013 Centers of Disease Control and Prevention (CDC) reported data on the 2011 number of imported malaria cases in the US. Furthermore; we will use our "fitted" malaria models for the top 20 countries of malaria acquisition by US residents to study the impact of protecting US residents from malaria infection when they travel to malaria endemic areas, the impact of protecting residents of malaria endemic regions from mosquito bites and the impact of killing mosquitoes in those endemic areas on the 2013 CDC malaria surveillance data.

I will describe a range of models, from the cellular to cortical scales, that illuminate how we perceive stimuli and make decisions. Large networks composed of individual spiking neurons can capture biophysical details of neuromodulation and synaptic transmission, but their complexity renders them opaque to analysis. Employing methods of mean field and dynamical systems theory, I will argue that these high-dimensional stochastic differential equations can be reduced to simple drift-diffusion processes used by cognitive psychologists to fit behavioral data. This allows us to relate them to optimal methods from statistical decision theory, and prompts new questions on why we fail to make good choices.

Targeted cancer drugs attack pathway specific phenotypes and can lead to very positive outcomes when a particular phenotype dominates the population of a specific tumor. However, these drugs often fail because not all cells express the targeted phenotype to the same degree. This leads to a heterogeneous response to treatment, and ultimate recurrence of the cancer as sensitive cells die off and resistant cells take over. We explore how treatment strategies informed by a tumorÂ?s phenotypic mix, can help slow the emergence of resistance and stave off tumor recurrence. We use an off-lattice agent-based model that incorporates inheritance of two phenotypes Â? proliferation rate and migration speed Â? and is modulated by a space limiting selection force. We find how and when distinct distributions of phenotypes require different treatment strategies.

Gels formed from mixtures of polymers and solvent are ubiquitous in physiological systems. The structure and properties of a gel can change dramatically in response to chemical modulation. Two examples of the influence of chemistry on gel properties will be discussed. The structure of fibrin gels that form during blood clotting is strongly influenced by the concentration of the enzyme thrombin that produces the fibrin monomers from which the gel is built. Presumably at higher thrombin concentrations, fibrin monomers are produced more rapidly from the precursor fibrinogen molecules. I will present an analysis of a possible mechanism of fibrin branching that can explain the sensitivity of fibrin structure to the rate of supply of monomers. Mucin gel is released from vesicles in goblet cells. During this exocytotic process, the polyelectrolyte mucin gel swells to many times its original volume at a very rapid rate. I will present a model in which this swelling is triggered by an exchange of divalent calcium ions in the vesicle and monovalent sodium ions in the extracellular space, and in which the ion concentrations and the rheological properties of the mucin gel determine its equilibrium size and the dynamics of its swelling.

Cracking the neural code is one of the central challenges of neuroscience. Typically, this has been understood as finding the relationship between single neurons and the stimuli they represent. More generally, neural activity must also reflect relationships between stimuli, such as proximity between locations in an environment. Convex codes, comprised of activity patterns for neurons with classical receptive fields, may be the brain's solution to this problem. These codes have been observed in many areas, including sensory cortices and the hippocampus. What makes a code convex? Using algebra, we can uncover intrinsic signatures of convexity and dimension in neural codes. I will report on some recent results by multiple authors, including participants in my 2014 AMS Math Research Community.

In the first part of the talk we present a review of our work on PDE models of swimming bacteria. First we introduce a stochastic PDE model for a dilute suspension of self-propelled bacteria and obtain an explicit asymptotic formula for the effective viscosity (E.V.) that explains the mechanisms of the drastic reduction of E.V.. Next, we introduce a model for semi-dilute suspensions with pairwise interactions and excluded volume constraints. We compute E.V. analytically (based on a kinetic theory approach) and numerically. Comparison with the dilute case leads to a phenomenon of stochasticity arising from a deterministic system. We develop a ODE/PDE model that captures the phase transition, an appearance of correlations and large scale structures due to interbacterial interactions. Collaborators: S. Ryan, B. Haines, (PSU students); I. Aronson, A. Sokolov, D. Karpeev (Argonne); In the second part of the talk we discuss a system of two parabolic PDEs arising in modeling of motility of eukaryotic cells on substrates. The two key properties of this system are (i) presence of gradients in the coupling terms (gradient coupling) and (ii) mass (volume) preservation constraints. We derive the equation of the motion of the cell boundary, which is the mean curvature motion perturbed by a novel nonlinear term and prove that the sharp interface property of initial conditions is preserved in time. This novel term leads to surprising features of the motion of the interface such as discontinuities of the interface velocity and hysteresis. This is joint work with V. Rybalko and M. Potomkin.

Biological projectiles range from a 10-micrometer spore to a 1-meter leaping mammal. Pre-launch accelerations scale inversely with length, with that of the smallest projectile approaching a million times gravity. These projectiles follow Borelli's rule, that all jumpers should jump to the same height. Nonetheless, his rationale is wrong on at least two accounts. For one thing, it presumes a muscular engine operating with no energy storage, often far from the case. For another, it ignores drag, critical for small projectiles, which operate in an overwhelmingly drag-dominated rather than gravity- dominated domain and whose optimal trajectories look decidedly unfamiliar. But the rule can be given quite a differentÂ?and more generalÂ?basis. And a simple dimensionless index helps us anticipate best launch angles and path lengths, these latter illustrated with a simple computer simulation.

We discuss the question: What properties of solutions to coupled cell network systems are invariant under changes of coordinates that preserve network structure? This question was motivated by trying to understand the biological phenomenon of homeostasis in a mathematically satisfactory way. In its simplest mathematical form homeostasis can be described as follows. Given a stable equilibrium $x(lambda)$ of a system that depends on an input parameter $lambda$: When is some coordinate (say $x_j(lambda)$) approximately constant? First, we translate approximately constant to derivative approximately 0. This allows us to search for regions of homeostasis in a model using bifurcation theory like formulas. Second, we claim that there is a sense in which homeostasis can be thought of as a network invariant. This is joint work with Ian Stewart.

Understanding and predicting cancer progression requires detailed interacting models of tumor and stromal cells, all calibrated to experimental data. Work to date has been limited by a lack of standardization of data representations of multicellular systems, though this is now being addressed through MultiCellDS (MultiCellular Data Standard) and digital cell lines, which are standardized representations of microenvironment-dependent cell phenotypes. Computational cancer modelers require biologically and mathematically consistent initialization routines to seed simulations with cells defined in digital cell lines. In this talk, we will briefly introduce a 3-D agent-based model designed for use in integrative computational biology. We introduce a Â?snapshot generatorÂ? that can take a digital cancer cell line and produce for the agent-based model an initial cell arrangement and a phenotypic state based upon analyses of the digital cell line data elements. We demonstrate 2-D monolayer and 3-D hanging drop simulations up to 500k MCF7 cells, a common breast cancer cell line. We additionally demonstrate the production of digital snapshots, standardized simulation output that will facilitate computational model comparison with a common core of analytical tools. With an early version of these tools, we assess the match between simulations and in vitro experiments. In the future, this work will be used to create and simulate combinations of tumor and stromal cells from appropriate digital cell lines in realistic tissue environments in order to understand, predict, and eventually control cancer progression in individual patients.

In an experimental study of single enzyme reactions, it has been proposed that the rate constants of the enzymatic reactions fluctuate randomly, according to a given distribution. To quantify the uncertainty arising from random rate constants, it is necessary to investigate how one can simulate such a biochemical system. To do this, we will take the Gillespie's stochastic simulation algorithm for simulation the evolution of the state of a chemical system, and study a modification of the algorithm that incorporates the random rate constants, using in part the Metropolis-Hastings algorithm to enact the distribution on the random rate constants. This modified algorithm, when applied to the single enzyme reaction system, produces simulation outputs that are corroborated by the experimental results. This project is in its early stages, and it is hoped that it can subsequently be used as a tool for the estimation or calibration of parameters in the system using experimental data.

Advances in high-throughput genomics have facilitated the development of pooled population sequencing techniques which involve the en masse sequencing of tens to hundreds of individual genomes in a single sequencing reaction. Pooled population sequencing methods have numerous applications in quantitative, population and evolutionary genetics. I will discuss some of the statistical and computational challenges associated with the analysis of pooled sequence data in the context of quantitative trait locus (QTL) mapping and detecting selection during experimental evolution.

RNA-Seq is a new technology for measuring the content of a transcriptome using high-throughput sequencing technology. I will provide a self-contained introduction to the technology, and proceed to discuss some interesting mathematical questions we have had to address in order to realize the potential of "comparative transcriptomics" for comparing and contrasting transcriptomes. We will start with the "freshman's dream", and proceed to examine issues related to maximum matching, the (phylogenetic) space of trees and Simpson's paradox. This is joint work with my current and former students Natth Bejraburnin, Nicolas Bray, Adam Roberts, Cole Trapnell and Meromit Singer.

Circadian clocks persist with a constant period (~24-hour) even after a significant change of the expression level of clock genes. To study the biochemical mechanisms of timekeeping, we develop the most accurate mathematical model of mammalian intracellular timekeeping, as well as a simplified model amenable to mathematical analysis. This modeling work raises interesting questions about existence and uniqueness of models given knowledge of their solutions. Although much is known about cellular circadian timekeeping, little is known about how these rhythms are sustained with a constant period. Here, we show how a universal motif of circadian timekeeping, where repressors bind activators rather than directly binding to DNA, can generate oscillations when activators and repressors are in stoichiometric balance. Furthermore, we find that, even in the presence of large changes in gene expression levels, an additional slow negative feedback loop keeps this stoichiometry in balance and maintains oscillations with a fixed period. These results explain why the network structure found naturally in circadian clocks can generate ~24-hour oscillations in many conditions.

This talk provides some fundamentals of statistical techniques for data on non-Euclidean spaces. Such data occur in the analysis of shape of geometrical objects, e.g. in applications studying biological growth. Naturally, shape is modeled on a manifold quotient (e.g. unit size objects) under a Lie group action (e.g. translations and rotations) which can be given a manifold structure, possibly with singularities. We show how this scenario allows for one and two sample tests as well as principal component analysis.

Cartilage physiology is regulated by a single population of specialized cells called chondrocytes. The chondroyctes are sparsely distributed within the extracellular matrix (ECM) and maintain a state of homeostasis in healthy tissue. ECM degeneration due to osteoarthritis can lead to compete degradation of cartilage surfaces, necessitating total joint replacement. Chondrocytes can be utilized to regenerate cartilage via tissue engineering approaches in which these cells are seeded in biocompatible and degradable biopolymer or hydrogel scaffolds. In such systems, biosynthetic activity of the cells in response to their non-native environment results in regeneration and accumulation of ECM constituents concurrent with degradation of the surrounding scaffold material. In this talk, mixture models are presented for interactions between biosynthesis of ECM constituents and ECM linking in cell-seeded scaffolds. Both ODE-based (temporal) models for evolution of average apparent densities and PDE-based (spatio-temporal) models will be presented for variables including unlinked ECM, linked ECM and scaffold. Model extensions accounting for cell proliferation will also be discussed. Of particular interest are model predictions for the evolution of solid phase apparent density, which is correlated with the compressive elastic modulus of the tissue construct. These models provide a quantitative framework for assessing and optimizing the design of engineered cell-scaffold systems and guiding strategies for articular cartilage tissue engineering.

ABSTRACT: The past decade has seen a number of engineering innovations that make construction of devices of micro- and even nanometric dimensions feasible. Hence, there is a growing interest in exploring new and efficient ways to generate propulsion at these small scales. Here we explore optimization of one particular type of low Reynolds number propulsion mechanism Â? flagella. Beyond the general challenges associated with optimization, there are a number of issues that are unique to swimming at low Reynolds numbers. At small scales, the fluid equations of motion are linear and time-reversible, hence reciprocal motion Â? i.e., strokes that are symmetric with respect to time reversal Â? cannot generate any net translation (a limitation commonly referred to as the Scallop Theorem). One possible way to break this symmetry is through carefully chosen morphologies and kinematics. One symmetry-breaking solution commonly employed by eukaryotic microorganisms is to select nonreciprocal stroke patterns by actively generating torques at fixed intervals along the organism. Hence, we will address the question: For a given morphology, what are the optimal kinematics? In this talk we present optimal stroke patterns using biologically inspired geometries such as single-tailed spermatozoa and the double-tail morphology of Chlamydomonas, a genus of green alga widely considered to be a model system in molecular biology.

G protein-coupled receptors (GPCRs) are membrane receptors that play a pivotal role in the regulation of reproduction and behavior in humans. Upon binding to specific ligands, they trigger a local cAMP production. Activated receptor are then internalized to different endosomal compartments where they can continue signaling before being recycled or destroyed. Recent studies showed that the different pools of cAMP have different effect on the cell.
In the first part of the talk, I will present a piecewise deterministic Markov process (PDMP) of intracellular signaling. The stochastic part of the model accounts for formation, coagulation, fragmentation and recycling of intracellular vesicles which contain the receptor, whereas the deterministic part of the model represents evolution of chemical reactions due to signaling activity of the receptor. We are interested in the existence of and convergence to a stationary measure. I will present different cases for which we were able to obtain results in this direction.
In the second part of the talk, I will present the numerical workflow (SBML, PEtab and PyPESTO) we use to fit ODEs model of GPCR signaling to longitudinal measure of chemical concentrations (BRET data).