In butterflies and moths, wing scales provide the cellular basis for coloration. Many times these scales are filled with pigments providing colors such as black, brown, red, and yellow. In some cases, however, color is generated structurally, which is often the case for green and blue coloration. Optical physicists have done a remarkable job defining how wing nanostructures create structural color, but we know little about the developmental basis for creating the patterned nanostructures. I will describe our work to understand the developmental and cellular basis for scale formation and the generation of structural color, focusing on a group of butterflies known as the Achillides swallowtails. The scales of these butterflies use a combination of multilayer reflection and scale geometry to create a range of colors. Developmentally, the scale geometry appears to be controlled by cytoskeletal reorganization, and evolutionary changes in geometry appear to contribute to variation between species, between populations, and between seasonal variants.

A major feature of biological sciences in the 21st Century is their transition from phenomenological and descriptive disciplines to quantitative and predictive ones. However, the emergence of complexity in self-organizing biological systems poses fabulous challenges to their quantitative description because of the excessively high dimensionality. A crucial question is how to reduce the number of degrees of freedom, while preserving the fundamental physics in complex biological systems. We discuss a multiscale multiphysics and multidomain paradigm for biomolecular systems. We describe macromolecular system, such as protein, DNA, ion channel, membrane, molecular motors etc., by a number of approaches, including macroscopic electrostatics and elasticity and/or microscopic molecular mechanics and quantum mechanics; while treating the aqueous environment as a dielectric continuum or electrolytic fluids. We use differential geometry theory of surfaces to couple various microscopic and macroscopic domains on an equal footing. Based on the variational principle, we derive the coupled Poisson-Boltzmann, Nernst-Planck, Kohn-Sham, Laplace-Beltrami, Newton, elasticity and/or Navier-Stokes equations for the structure, function, dynamics and transport of protein, protein-ligand binding and ion-channel systems.

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.

In order to better understand wound contraction fibroblast populated collagen lattices have been studied for many years.  In this talk I will discuss mathematical models for lattice contraction.  The models are formulated with components at the cellular and sub cellular level with the goal of understanding the macroscopic behavior of the lattice.

The macroscopic diffusion constant for small ions in biological environments is in part dependent on the volume excluded by diffusional barriers and by long-range interactions between those barriers and the ion. Increasing excluded volume reduces diffusive transport of the solute, while long-range interactions can either increase or decrease diffusivity, depending on the nature of the potential. My colleagues and I have developed a computational methodology for studying these effects via a homogenized finite element method for the Smoluchowski equation. I will introduce the relevant background, both biological and mathematical, and present our recent results. This work is joint with Peter Kekenes-Huskey (U. Kentucky) and J. Andrew McCammon (UC San Diego).

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 “snapshot generator” 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.

Glutathione (GSH), a tripeptide formed from glutamate, cysteine, and
glycine, is arguably the most important antioxidant in the body. NAPQI, a
byproduct of acetaminophen (APAP) metabolism which is toxic to liver
cells, is neutralized by GSH. Although produced in great quantity by the
liver, in cases of APAP overdose demand for GSH can outstrip supply,
causing liver failure. Currently, patients presenting to the ER with APAP
overdose are given an infusion of cysteine since it is believed to be the
rate-limiting amino acid in GSH synthesis, however, there is evidence that
under some circumstances glutamate can become rate-limiting. Complicating
the issue is that in most hepatocytes, glutamate is not absorbable from
blood plasma but is formed from glutamine, which is produced in large
amounts by the skeletal muscle. In order to develop better rescue
protocols for APAP overdose, we have developed a mathematical model of
glutamate and glutamine metabolism in the rat. We have also investigated
how model parameters should change in the case of increased cortisol
production, such as occurs during sepsis, trauma, burns, and other
pathological states; the cortisol-stressed state has been studied in rats
by giving them dexamethasone. We compare model predictions with
experimental data for the normal, healthy rat and dexamethasone-stressed
rat. Biological parameters are taken from the literature wherever possible.

Stimulus tuning in a reduced model for neural competition leads to mixed-mode oscillations (MMOs), a temporal pattern featuring alternating small- and large-amplitude oscillations. Key ingredients for the generation of MMOs are mutual inhibition, slow negative-feedback in the form of adaptation, and nonlinearity of the gain function. Exploiting a normal form calculation, we show that MMOs occur due to the interaction of canard and singular Hopf mechanisms as the stimulus strength approaches a critical regime. We show then that local excitation and lateral inhibition-type coupling combined with the underlying MMOs produces complex spatio-temporal patterns in larger networks.

Visual processing begins in the outer plexiform layer of the retina, where
bipolar, horizontal, and photoreceptor cells interact. In vertebrates, the
onset of dim backgrounds can enhance small spot flicker responses of
retinal horizontal cells. This flicker response is called background-
induced flicker enhancement. The underlying mechanism for the feedback
is unclear but competing hypotheses have been proposed. One is the GABA
hypothesis, which states that the inhibitory neurotransmitter GABA,
released from horizontal cells, mediates the feedback by blocking calcium
channels. Another is the ephaptic hypothesis, which contends that calcium
entry is regulated by changes in the electrical potential within the
intersynaptic space between cones and horizontal cells. In this study, a
continuum spine model of cone-horizontal cell synaptic circuitry is
formulated. The model captures two spatial scales - the scale of an
individual synapse and the scale of the receptive field involving hundreds
to thousands of synapses. We show that the ephaptic mechanism produces
reasonable qualitative agreement with the temporal dynamics exhibited by
flicker enhancement experiments. We find that although GABA produces
enhancement, this mechanism alone is insufficient to reproduce the
experimental results. We view this multiscale continuum approach as a
first step in formulating a multi-layer mathematical model of retinal
circuitry, which would include the other ‘brain nuclei’ within the retina:
the inner plexiform layer where bipolar, amacrine, interplexiform, and
ganglion cells interact.

Mathematical modeling has become a widely-used tool for integrating
biological data, designing experiments, and ultimately understanding
biological systems. In recent years two important challenges for the
successful use of mathematical models have become apparent. One is that
models contain parameters that determine the behavior of the model, and
the values of these parameters are often hard to determine from the
available biological data. The other challenge is that many biological
systems exhibit a great deal of heterogeneity in behavior, so even if the
model parameters could be perfectly calibrated by pooling cell behaviors
to produce an “average cell model”, this model may not provide a good
description of any single cell in the population. In this seminar I will
describe a technique that we are using to integrate mathematical modeling
into experimental studies in a way that addresses both of these challenges.
We study endocrine pituitary cells that release a variety of hormones into
the blood, and our aim is to develop an approach for modeling the
behaviors of these cells with enough accuracy so that we can use the
models to make and test predictions in real time.

              "Perhaps a proper understanding of the complex
              regulatory networks making up cellular systems
              like the cell cycle will require a shift from common
              sense thinking...to a more abstract world, more
              readily analyzable in terms of mathematics."
              (Paul Nurse, Cell, 7 January 2000)

The cell cycle is a striking example of the necessity of systems-level
thinking in 21st century molecular cell biology. The resolute reductionism
of the last century, albeit necessary for identifying the molecular
components of cellular control systems and their interactions, has failed
to provide a comprehensive, integrative understanding of the molecular
basis of cell physiology. Putting the pieces back together requires new
ways of thinking about and doing molecular biology--an approach now
known as molecular systems biology. In this lecture I will show how
systems-level thinking reveals deep and unexpected principles of cell
cycle regulation.

In recent years it has become increasingly clear that stochasticity plays
an important role in many biological processes. Examples include bistable
genetic switches, noise enhanced robustness of oscillations, and fluctuation
enhanced sensitivity or "stochastic focusing." Numerous cellular systems
rely on spatial stochastic noise for robust performance. We examine the
need for stochastic models, report on the state of the art of algorithms and
software for modeling and simulation of stochastic biochemical systems, and
identify some computational challenges.

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.

Tissues grow, change shape, and differentiate, function normally or abnormally, get diseased or injured, repair themselves, and sometimes atrophy. This complex suite of behaviors is governed by a complex suite of controls. Nonetheless, we can identify some general principles at work in the dynamics of tissues. Our goal is to understand how a tissue’s mechanics and biology regulate each other. Our models use a biologically-based framework to track the mechanics, biology, and mechanobiology of the component cells, fluids, signaling molecules, and extracellular matrix materials. The presentation will describe our modeling approach, reveal some of the general principles we have identified, and discuss some of the questions our findings have raised about specific morphogenetic systems such as the lung.

This talk presents a work in progress with Sylvain Billard, Regis Ferriere and Chi Viet Tran. How the neutral diversity is affected by selection and adaptation is investigated in an eco-evolutionary framework. In our model, we study a finite population in continuous time, where each individual is characterized by a trait under selection and a completely linked neutral marker. The dynamics is ruled by births and deaths, mutations at birth and competition between individuals. The ecological phenomena depend only on the trait values but we expect that these effects influence the generation and maintenance of neutral variation. Considering a large population limit with rare mutations, but where the marker mutates faster than the trait, we prove the convergence of our stochastic individual-based process to a new measure-valued diffusive process with jumps that we call Substitution Fleming-Viot Process. This process restricted to the trait space is the Trait Substitution Sequence introduced by Metz et al. (1996). During the invasion of a favorable mutation, the marker associated with this favorable mutant is hitchhiked, creating a genetical bottleneck. The hitchhiking effect and how the neutral diversity is restored afterwards are studied. We show that the marker distribution is approximated by a Fleming-Viot distribution between two trait substitutions and that time-scale separation phenomena occur. The SFVP has important and relevant implications that are discussed and illustrated by simulations. We especially show that after a selective sweep, the neutral diversity restoration depend on mutations, ecological parameters and trait values.

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.

From habitat degradation and climate change to spatial spread of invasive species, dispersal plays a central role in determining how organisms cope with a changing environment. How should organisms disperse “optimally” in heterogeneous environments? I will discuss some recent development on the evolution of dispersal, focusing on evolutionarily stable strategies (ESS) for dispersal.

I will describe three different types of problems inspired by the dynamics of the cytoskeleton: (i) the structural and kinetic aspects of dynamic instability in macromolecular assemblies, (ii) the collective flagella-like dynamics of ordered assemblies of active particles and cells, and (iii) the onset of oscillations and the persistence of strain in disordered aggregates of motors and filaments. In all cases, I will show how simple aspects of geometry, chemical kinetics and statistical and continuum mechanics allow us to explain biological observations in a minimal setting.

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.

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.

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.

Heterogeneity in cancer is an observed fact, both genotypically and phenotypically. Cell-cell variation is seen in almost all aspects of cancer from early development all the way through to invasion and subsequent metastasis. Our current understanding of this heterogeneity has mainly focused at the genetic scale with little information on how this variation translates to actual changes in cell phenotypic behavior. Given that many genotypes can lead to the same cellular phenotype, it is important that we quantify the range and scope of this heterogeneity at the phenotypic scale as ultimately this variability will dictate the aggressiveness of the tumor and its treatability. Central to our understanding of this heterogeneity is how the tumor cells interact with each other and with their microenvironment. Since it is these very interactions that drive selection and that ultimately define the ecology of the tissue in which the tumor is developing. Considering an organ as an ecological system, means that we should view normal tissue homeostasis as an equilibrium that cancer cells must disrupt if they are to be successful. Disruption of this equilibrium is often one of the first events in cancer development, as the normal control mechanisms of the tissue are damaged or ignored. We will discuss the interplay between homeostasis, heterogeneity, evolution and ecology in cancer progression and treatment failure with an emphasis on the metabolism of breast cancer.

Ductal carcinoma in situ (DCIS) is an early stage non-invasive breast cancer that originates in the epithelial lining of the milk ducts, but it can evolve into comedo DCIS and ultimately, into the most common type of breast cancer, invasive ductal carcinoma. Understanding the progression and how to effectively intervene in it presents a major scientific challenge. The extracellular matrix surrounding a duct contains several types of cells and several types of growth factors that are known to individually affect tumor growth, but at present the complex biochemical and mechanical interactions of these stromal cells and growth factors with tumor cells is poorly understood. We will discuss a mathematical model that incorporates the cross-talk between stromal and tumor cells, and which can predict how perturbations of the local biochemical and mechanical state influence tumor evolution. We focus on the EGF and TGF-$\beta$ signaling pathways and show how up- or down-regulation of components in these pathways affects cell growth and proliferation, and describe a hybrid model for the interaction of cells with the tumor microenvironment. The analysis sheds light on the interactions between growth factors, mechanical properties of the ECM, and feedback signaling loops between stromal and tumor cells, and suggests how epigenetic changes in transformed cells affect tumor progression.

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.

It will be explained how the following problems in the diagnosis and treatment of breast cancer have led to mathematical problems: 1. How can one improve the diagnosis of breast cancer? 2. How can one determine the growth rate of a cancer once it has been detected? 3. In which order should drugs be given in order to improve relapse and survival times? The first problem led to the design , construction, and testing of an electrical impedance spectroscopy system combined with an x- ray mammography system. The second problem led to a quantitative model to predict the growth rate of some cancers as a function of the number of Her2 and EGF receptors on the cells involved. The third problem led to quantitative models capable of predicting the outcome of specific chemotherapy regimens used by Bonadonna involving the use of CMF and A (Doxorubicin) for the adjuvant treatment of breast cancer.

Decades of research in cancer have yielded scant results other than highlighting the need for new approaches that could go beyond the tried and tested molecular-based ones. Recent clinical studies show that tumour heterogeneity and selection, the ingredients of Darwinian evolution, can explain cancer progression towards malignancy as well as recurrence after treatment. In this talk I will describe mathematical and computational models that explore cancer evolutionary dynamics and that can explain how the interactions between the tumour with its environment (the tumour ecosystem) can yield a better understanding of cancer biology and lead to better and more efficacious treatments for cancer patients.