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.

Stochastic modeling and analysis can help answer pressing medical questions. In this talk, I will attempt to justify this claim by describing recent work on two problems in medicine. The first problem concerns ovarian tissue cryopreservation, which is a proven tool to preserve ovarian follicles prior to gonadotoxic treatments. Can this procedure be applied to healthy women to delay or eliminate menopause? How can it be optimized? The second problem concerns medication nonadherence. What should you do if you miss a dose of medication? How can physicians design dosing regimens that are robust to missed/late doses? I will describe (a) how stochastics theory offers insights into these questions and (b) the mathematical questions that emerge from this investigation. The first problem is based on joint work with Joshua Johnson (University of Colorado School of Medicine), John Emerson (Yale University), and Kutluk Oktay (Yale School of Medicine).

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.

Systems biologists seek to understand the higher-level organizational properties that a proposed system exhibits from the interactions of its many lower level components. It is then typically beneficial to learn, in a statistical sense, the nature of these interactions from experimental data; this is sometimes called reverse engineering. In this talk I will discuss some methods found in the literature for learning the kinetic parameters from stochastic system trajectories measured at discrete time points. I will also discuss estimating the system topology, or network structure, with the stoichiometric algebraic statistical model and detail its relationship with kinetic parameter estimation. Finally, I will discuss some ongoing work aimed at improving both parameter estimation and system topology estimation.

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.

Assouad-Nagata dimension addresses both large-scale and small-scale behaviors of metric spaces and is a refinement of Gromov’s asymptotic dimension. A metric space is a minor-closed metric if it is defined by the distance function on the vertices of an edge-weighted graph that satisfies a fixed graph property preserved under vertex-deletion, edge-deletion, and edge-contraction. In this talk, we determine the Assouad-Nagata dimension of every minor-closed metric. It is a common generalization of known results about the asymptotic dimension of H-minor free unweighted graphs, about the Assouad-Nagata dimension of complete Riemannian surfaces with finite Euler genus, and about their corollaries on weak diameter coloring of minor-closed families of graphs and asymptotic dimension of minor-excluded groups.

My talk will have two parts. PART I, From Birds to Bacteria: Generalised Velocity Jump Processes. There are various cases of animal movement where behaviour broadly switches between two modes of operation, corresponding to a long distance movement state and a resting or local movement state. In this talk, I will give a mathematical description of this process, adapted from Friedrich et. al. (2006). The approach allows the specification any running or waiting time distribution along with any angular and speed distributions. The resulting system of partial integro-differential equations are tumultuous and therefore it is necessary to both simplify and derive summary statistics. We derive an expression for the mean squared displacement, which shows good agreement with experimental data from the bacterium Escherichia coli and the gull Larus fuscus. Finally a large time diffusive approximation is considered via a Cattaneo approximation (Hillen, 2004). This leads to the novel result that the effective diffusion constant is dependent on the mean and variance of the running time distribution but only on the mean of the waiting time distribution. We also consider the Levy regime where the variance of the running distribution tends to infinity. This leads to a fractional diffusion equation for superdiffusive Levy walks and can be solved analytically. Our theory opens up new perspectives both for the systematic derivation of such equations, and for experimental data analysis of intermittent motion. I will also briefly discuss recent developments (by other researchers) within the field of velocity jump processes. PART II: Modelling Osteocyte Network Formation: Healthy and Cancerous Environments. Advanced prostate, breast, and lung cancer can metastasize to bone. In pathological bone, the highly regulated bone remodeling signaling pathway is disrupted. Within bone dendritic osteocytes form a spatial network allowing communication between osteocytes and the osteoblasts located on the bone surface. This communication network facilitates coordinated bone formation. In the presence of a cancerous microenvironment, the morphology of this network changes. Commonly osteocytes appear to be either overdifferentiated (i.e., there are more dendrites) or underdeveloped (i.e., dendrites do not fully form). In addition to structural changes, preliminary studies measuring the number of osteocytes per unit area using pathology slides show that the number density of osteocytes change from healthy to metastatic prostate and breast cancer xenografted mice. We present a stochastic agent-based model for bone formation incorporating osteoblasts and osteocytes that allows us to probe both network structure and number density of osteocytes in bone. Our model both allows for the simulation of our spatial network model and analysis of mean-field equations in the form of integro-partial differential equations. We consider variations of our model to test specific physiological hypotheses related to osteoblast differentiation; for example we can predict how changing measurable biological parameters, such as rates of bone secretion, rates of dendrite growth and rates of osteoblast differentiation can allow for qualitatively different network morphologies, and vice versa. We thenuse our model to hypothesize reasons for the limited efficacy of zoledronate therapy on metastatic breast cancer.

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.

I will discuss some old and new results related to the analysis of stochastic SIR-type epidemics on a configuration model (CM) random graph having a fixed degree distribution p_k. In particular, I will describe the relevant large graph limit result which yields the law of large numbers (LLN) for the edge-based process. I will also discuss the applications of the LLN approximation in building a "network-free" SIR Markov hybrid model which can be used for epidemic parameters inference. The hybrid model idea appears particularly relevant in the context of the recent Ebola and the Zika epidemics.

I will present a proof-of-principle model for auto-regulation of liquid volume in the lung airway surface layer (ASL). The model couples biochemical kinetics, transient ASL volume, and homeostatic mechanical stresses. The work is based on the hypothesis that ASL volume is sensed through soluble mediators and phasic stresses generated by beating cilia and global percussive and air drag forces. Simulations demonstrate that homeostatic volume regulation is a natural consequence of the model and the underlying hypotheses. The model successfully reproduces many experimental results and passes qualitative sensitivity tests. These results compel an advanced airway hydration model in which detailed kinetics of multiple molecular pathways, feedback to ASL viscoelasticity properties, and stress signaling from the ASL to the cilia and epithelial cells are taken into account.

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.

I present an overview of models of the Darwinian evolution resulting from the interplay of phenotypic variation and natural selection through ecological interactions. Starting from a stochastic birth and death process, several types of deterministic or probabilistic models can be derived. When advantageous mutations are rare and small, a time scale separation takes place in the models leading to the concentration of the population along a few selected traits. Those then evolve according to a PDE of Hamilton-Jacobi type or to the canonical equation of adaptive dynamics.

Carcinogenesis is the transformation of normal cells into cancer cells. This process has been shown to be of a multistage nature, with stem cells that go through a series of (stochastic) genetic and epigenetic changes that eventually lead to a malignancy. Since the origins of the multistage theory in the 1950s, mathematical modeling has played a prominent role in the investigation of the mechanisms of carcinogenesis. In particular, two stochastic (mechanistic) models, the Armitage-Doll and the two-stage clonal expansion (TSCE) model, have been widely used in the past for cancer risk assessment and for the analysis of cancer population and experimental data. In this talk, I will introduce some of the biological and mathematical concepts behind the theory of multistage carcinogenesis, and discuss in detail the use of these models in cancer epidemiology and cancer prevention and control. Recent applications of multistage and state-transition Markov models to assess the potential impact of lung cancer screening in the US will be reviewed.

Indirect measurements are ubiquitous in the life sciences because it is often impossible or impractical to directly measure the process of interest. I will show how dynamic mathematical models of biological systems can aid interpretation of biological markers, focusing on the dynamics of acetaminophen (APAP, Tylenol) overdose and stable isotope signatures. APAP is one of the most common drugs on the planet. While safe in therapeutic doses, APAP is the leading cause of acute liver failure in the United States. I will present a mathematical model of APAP overdose that can be used to estimate time since overdose, overdose amount, and need for liver transplant from measurable markers of liver injury at the time of hospital admission. Similarly, stable isotopes are used by ecologists and forensic scientists as markers of diet and movement patterns. I will show how, with suitable mathematical models, stable isotope ratios of the hair of a murder victim were used to reconstruct the victim's movement history in the time preceding death.

We introduce and analyze a size-structured oocyte population model, with non local nonlinearities on recruitment, growth and mortality rates to take into account interactions between cells. We pay special attention to the form of the recruitment term, and its influence on the asymptotic behavior of the cell population.
This model is well-suited for representing oocyte population dynamics within the fish ovary. The nonlocal nonlinearities enable us to capture the diverse feedback mechanisms acting on the growth of oocytes of varying sizes and on the recruitment of new oocytes.
We firstly investigate the existence and uniqueness of global bounded solutions by transforming the partial differential equation into an equivalent system of integral equations, which can be solved using the Contraction Mapping Principle.
In a second step, we investigate the asymptotic behavior of the model. Under an additional assumption regarding the form of the growth rate, we can, with the use of a classical time-scaling transformation, reduce the study to that of a equation with linear growth speed and nonlinear inflow boundary condition. Using arguments from the theory of abstract semilinear Cauchy problems, we investigate the local stability of stationary solutions of this equation by reducing it to a characteristic equation involving the eigenvalues of the linearized problem around equilibrium states.
When the mortality rate is zero, the study of existence and stability of stationary solutions is simplified. Explicit calculations can be carried out in certain interesting cases.

Dinoflagellates swim due to the action of two eucaryotic flagella - a trailing, longitundinal flagellum that propagates planar waves, and a transverse flagellum that propagates helical waves. The transverse flagellum wraps around the cell in a plane perpendicular to the trailing flagellum, and is thought to provide both forward thrust along with rotational torque. Motivated by the intriguing function of this transverse flagellum, we study the fundamental fluid dynamics of a helically-undulating ring in a viscous fluid.  We contrast this biofluiddynamic study, where the kinematics of the waveform are taken as given, with a model of mammalian sperm hyperactivated motility.  Here, our goal is to examine how the complex interplay of fluid dynamics, biochemistry, and elastic properties of the flagellum give rise to the swimming patterns observed.  

The evolution of resistance remains an elusive problem in the treatment of both cancer and infectious disease, and represents one of the most important medical problems of our time. While the illnesses are different on several non-trivial levels including timescale and complexity, the underlying biological phenomenon is the same: Darwinian evolution. To comprehensively approach these problems, I have focussed my attention on building a broad suite of investigations centered around the causes and consequences of the evolutionary process in these contexts. I will discuss my and my collaborator's efforts to; model the evolutionary process on the genomic scale in both an analytic (Markov process) and stochastic (individual based model and inference) format; to quantify in vitro competition and interaction between cancer cell lines through an evolutionary game theoretic lens using time-lapse microscopy and computer vision; and to understand the evolutionary contingencies inherent in collateral sensitivity in E. coli and ALK mutated non-small cell lung cancer.

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.

Cellular motility arises from the interaction of numerous components: actin filaments, ligands, adhesion complexes to name a few. Each individual component exhibits stochastic behavior, yet overall cooperative behavior is observed, and leads to motility. Linking overall cellular motion to the stochastic behavior of its components is a remarkable mathematical challenge. This talk introduces a method to cyclically extract surrogate motility models from detailed stochastic simulation of the components. The starting point is a sampling of the detailed cytoskeleton dynamics over a short time interval. This detailed information is repeatedly projected onto a lower dimensional statistical manifold to obtain a coarse-grained model of the cytoskeleton, from which a surrogate cell model is obtained. The projection operation corresponds to transport along the geodesics of embedded statistical manifolds. The surrogate model is advanced over a larger time interval and then used to recreate the detailed microscopic cytoskeleton state needed to start the next cycle. The procedure is applied to study the behavior of Listeria monocytogenes bacterium and inert models of the motility behavior of this bacterium

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.

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.

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