Prior to cell division, chromosomes are segregated by mitotic spindle. This molecular machine self-assembles remarkably fast and accurately in an elegant process of search and capture, during which dynamic polymers from two centers grow and shrink rapidly and repeatedly until a contact with chromosomes is established. At the same time, the chromosomes interact with each other. I will show computer simulation and experimental microscopy results illustrating how cell deploys a number of redundant mechanisms optimizing the assembly by keeping it accurate yet rapid. Very similar strategies are used to assemble Golgi apparatus. The talk will illustrate how computer modeling assists experiment in unraveling mysteries of cell dynamics.

The chemotaxis network of bacteria such as E. coli is remarkable for its sensitivity to minute relative changes in chemical concentrations in the environment. Indeed, E. coli cells can detect concentration changes corresponding to only ~3 molecules in the volume of a cell. Much of this acute sensitivity can be traced to the collective behavior of teams of chemoreceptors on the cell surface. Instead of receptors switching individually between active and inactive configurations, teams of 6-20 receptors switch on and off, and bind or unbind ligand, collectively. Similar to the binding and unbinding of oxygen molecules by tetramers of hemoglobin, the result is a sigmoidal binding curve. Coupled with a system for adaptation that tunes the operating point to the steep region of this sigmoidal curve, the advantage for chemotaxis is gain – i.e., small relative changes in chemical concentrations are transduced into large relative changes in signaling activity (specifically, the rate of phosphorylation of the response regulator CheY).  However, something is troubling about this simple explanation: in addition to providing gain, the coupling of receptors into teams also increases noise, and the net result is a decrease in the signal-to-noise ratio of the network. Why then are chemoreceptors observed to form cooperative teams? We present a novel hypothesis that the run-and-tumble chemotactic strategy of bacteria leads to a “noise threshold”, below which noise does not significantly decrease chemotactic velocity, but above which noise dramatically decreases this velocity.

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 – 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 – i.e., strokes that are symmetric with respect to time reversal – 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.

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.

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.

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.

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.

The job of embryogenesis is to diversify cells into the hundreds of specialized cellular functions required by an animal or plant and to place those cells in the correct location in the embryo. To accomplish that complex job transcriptional regulation provides control circuits directing each dichotomy, each cell movement, and the patterning of cellular assemblies so that the animal that emerges from embryogenesis can feed and perform the necessary functions for survival of the species. The questions being addressed in this talk are to ask how gene regulatory networks are assembled, how do they change with time, and how do they accomplish the enormous regulatory task needed for building an embryo.

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.

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

The digital nature of biology is crucial to its functioning as an information system, as well in building hierarchical components in a repeatable way. We explain how protein systems can function as discrete components, despite the importance of non-specific forces due to the hydrophobic effect. That is, we address the question of why proteins bind to ligands predictably and not in a continuous distribution of places, the way grease forms into blobs. We will give a detailed description of how data mining in the PDB can reveal how proteins interact. We highlight the role of the hydrophobic effect, but we see that it works inversely to the usual concept of hydrophobic interaction. Our work suggests the need for a more accurate model of the dielectric effect in the vicinity of a protein surface, and we discuss some advances in this direction. Our research also provides an understanding of how molecular recognition and signaling can evolve. We give an example of the use of our ideas in drug design.

A preliminary examination of a very rich data set consisting of a detailed survey of individuals in male-male sex networks in South India. The motivation for the study is to understand the spread of HIV in male-male sex networks in South India. The data contains survey information from participants, as well as their cell phone contacts and incomplete information on the contacts by participants. We provide predictive models of attributes of contacts given participant attributes, as well as predictive models of the attributes, such as sexual position. We study how model parameters vary as a function of connectedness of individuals and how modeling network interactions has an effect on the model.

Eukaryotic cell microenvironment (inner and outer) is composed in large parts from fluids that interact with solid and elastic bodies, whereas it is the cell cytoplasm, cytoskeleton and basal membrane; the interstitial fluid interpenetrating the stroma and tumor cells; or blood flow carrying the immune or circulating tumor cells. I will discuss the use of two fluid-structure interactions methods, the immersed boundary and the regularized Stokeslets, in applications dealing with the tumor development and treatment. First model operates on the cellular scale and will be used to model various cell processes, such as cell growth, division or death, during the cellular self-organization into a normal mammary acinus, a 3D in vitro structure recapitulating the morphology of breast cysts (acini). I will discuss model development, parameterization and tuning with the experimental data, as well as their subsequent use to investigate the link between morphogenesis of epithelial mutants and molecular alterations of tumor cells. Second model acts on the tissue level, and will be used to investigate the relation between tumor tissue structure and efficacy of anticancer drugs in the context of interstitial fluid flow. I will present simulation results showing non-linear relation between tumor tissue structure and effectiveness of drug penetration. I will also discuss how tumor tissue metabolic state(its oxygenation and acidity) becomes modified due to actions of chemotherapeutic drugs leading to the emergence of tumor zones with potentially drug-resistant cells and/or to tumor areas that are not exposed to drugs at all. Both of these phenomena can contribute to the moderateclinical success of many anticancer drugs.

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.

Stochastic evolutionary models of biological sequences are widely used for phylogenetic inference and ancestral reconstruction. However, at long divergence times sequences enter the "twilight zone" of homology detection and reconstruction becomes very difficult. We describe a stochastic evolutionary model for protein 3D structure using elements of shape theory. This model significantly resolves this uncertainty and stabilizes evolutionary inferences. We also provide theoretical bounds on inferring evolutionary divergence times via connections to the probabilistic "cutoff phenomenon", in which a Markov chain remains far equilibrium for an extended period followed by a rapid transition into equilibrium. We show that this cutoff explains several previously reported problems with common default priors for Bayesian phylogenetic analysis, and suggest a new class of priors to address these problems.

In the biopharma industry of drug development, figuring out the best doses to use is considered a high priority. It can mean the difference between having an effective drug and having one that gives no benefit. It can reduce toxicities that otherwise could prevent the drug from being used. And proposing a dose that later turns out to be excessive not only looks bad, it can also mean substantial revenue loss.

The type of control theory used in academia for many decades is now being looked at by industry as a potential way to address the problem of dose selection. The problem becomes even harder when a drug will be used in combination with one or more other drugs. I will explain some of the mathematics and show examples of how control theory can be used to optimize dose regimens.

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 benthic upside down jellyfish, Cassiopea spp., and the pelagic moon jellyfish, Aurelia spp. 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. For Cassiopea, 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. For Aurelia, maximum swim speeds are generated when the elastic bell is resonating at its natural frequency. Alternating vortex rings can also enhance swimming speed and efficiency.

Frequencies of k-mers in sequences are sometimes used as a basis for inferring phylogenetic trees without first obtaining a multiple sequence alignment. We show that a standard approach of using the squared-Euclidean distance between k-mer vectors to approximate a tree metric can be statistically inconsistent. To remedy this, we derive model-based distance corrections for orthologous sequences without gaps, which lead to consistent tree inference. The identifiability of model parameters from k-mer frequencies is also studied. Finally, we report simulations showing the corrected distance out-performs many other k-mer methods, even when sequences are generated with an insertion and deletion process. These results have implications for multiple sequence alignment as well, since k-mer methods are usually the first step in constructing a guide tree for such algorithms. This is joint work with Elizabeth Allman and John Rhodes.

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.

Cancer emerges due to an evolutionary process in somatic tissue. The fundamental laws of evolution can best be formulated as exact mathematical equations. Therefore, the process of cancer initiation and progression is amenable to mathematical investigation. Of special importance are changes that occur early during malignant transformation because they may result in oncogene addiction and represent promising targets for therapeutic intervention. Here we describe a mathematical approach, called Retracing the Evolutionary Steps in Cancer (RESIC), to deduce the temporal sequence of genetic events during tumorigenesis from crosssectional genomic data of tumors at their fully transformed stage. When applied to a dataset of 70 advanced colorectal cancers, our algorithm accurately predicts the sequence of APC, KRAS, and TP53 mutations previously defined by analyzing tumors at different stages of colon cancer formation. We further validate the method with glioblastoma and leukemia sample data and then apply it to complex integrated genomics databases, finding that high-level EGFR amplification appears to be a late event in primary glioblastomas. RESIC represents the first evolutionary mathematical approach to identify the temporal sequence of mutations driving tumorigenesis and may be useful to guide the validation of candidate genes emerging from cancer genome surveys.

Elastic capsules occur in nature in the form of cells and vesicles and are manufactured for biomedical applications. They are widely modeled but there are few analytical results. In this talk, complex variable techniques are used to derive semi-analytic solutions for the steady-state response and time-dependent evolution of elastic capsules in 2D Stokes flow. The analysis is complemented by spectrally accurate numerical simulations of the time-dependent evolution. One motivation for this work is to provide analytical solutions to help validate the accuracy of numerical methods for elastic membranes in flow. A second motivation is to clarify the steady response of capsules in some canonical flows. Finally, we investigate the formation of finite-time cusp singularities, of which there are only a few examples in interfacial Stokes flow, and where none involve elastic interfaces. This is joint work with Michael Booty and Michael Higley.

              "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.

Cochlear implants are neural prostheses that restore a sense of hearing to individuals with severe to profound deafness. Two fundamental theoretical questions that we face are: How does the auditory nerve respond to electrical stimulation? And how is sound information represented in the spike trains of auditory nerve fibers? We will discuss model-based efforts to investigate these questions. I will focus on the development of reduced models that incorporate essential biological features of this complicated system, and remain useful tools for analyzing neural coding.

Using a point process model of the auditory nerve, I simulate amplitude modulation detection, a common test of temporal resolution. I find that the temporal information in the simulated spike trains does not limit modulation sensitivity in cochlear implant users, and discuss how the point process framework can be extended to include additional biophysical mechanisms. Next, I illustrate how spatial spread of excitation and neural degeneration can lead to of within- and across-patient variability in listening outcomes. This points toward an important goal of computational modeling: to develop patient-specific models that can be used to optimize stimulation strategies for individual cochlear implant users.

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.

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.

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.