We report the results of our integrated experimental and theoretical investigations of biological systems dominated by interfacial effects. Particular attention is given to elucidating natural strategies for water-repellency, walking on water, underwater breathing, and drinking.

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 different—and more general—basis. And a simple dimensionless index helps us anticipate best launch angles and path lengths, these latter illustrated with a simple computer simulation.

John Hopfield pointed out, in his seminal paper on kinetic proofreading, that if a biochemical system operates at thermodynamic equilibrium there is a barrier to how well it can achieve high-fidelity in transcription and translation. Hopfield showed that the only way to bypass this barrier is to dissipate energy and maintain the system away from equilibrium. Eukaryotic gene regulation uses dissipative mechanisms, such as nucleosome remodelling, DNA methylation and post-translational modification of histones, which are known to play a critical regulatory role but have been largely ignored in quantitative treatments. I will describe joint work with my colleague Angela DePace in which we use the recently-developed, graph-theoretic “linear framework” to show that the sharpness with which a gene is turned “on” or “off” in response to an upstream transcription factor is limited if the regulatory system operates at equilibrium, even with arbitrary degrees of higher-order cooperativity. In contrast, if the regulatory system is maintained away from equilibrium, substantially higher degrees of sharpness can be achieved. We suggest that achieving sharpness in gene regulation exhibits a Hopfield Barrier, and uncover, along the way, a new interpretation for the ubiquitously used, but poorly justified, Hill function.

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.

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.

Population geneticists study the dynamics of alternative genetic types in a replicating population. Most theoretical work rests on a simple Markov chain, called the Wright-Fisher model, to describe how an allele's frequency changes from one generation to the next. We have introduced a broad class of Markov models that share the same mean and variance as the Wright-Fisher model, but may otherwise differ. Even though these models all have the same variance effective population size, they encode a rich diversity of alternative forms of genetic drift, with significant consequences for allele dynamics. We have characterized the behavior of standard population-genetic quantities across this family of generalized models. The generalized population models can produce startling phenomena that differ qualitatively from classical behavior -- such as assured fixation of a new mutant despite the presence of genetic drift. We have derived the forward-time continuum limits of the generalized processes, analogous to Kimura's diffusion limit of the Wright-Fisher process. Finally, we have shown that some of these exotic models are more likely than the Wright-Fisher model itself, given empirical data on genetic variation in Drosophila populations. Joint work with Ricky Der and Charlie Epstein.

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.

Considering cancer as an evolutionary disease, we aim at understanding the means by which cancer cell populations develop resistance mechanisms to drug therapies, in order to circumvent them by using optimised therapeutic combinations. Rather than focusing on molecular mechanisms such as overexpression of intracellular drug processing enzymes or ABC transporters that are responsible for resistance at the individual cell level, we propose to introduce abstract phenotypes of resistance structuring cancer cell populations. The models we propose rely on continuous adaptive dynamics of cell populations, and are amenable to predict asymptotic evolution of these populations with respect to the phenotypic traits of interest. Drug-induced drug resistance, the question we are tackling from a theoretical and experimental point of view, may be due to biological mechanisms of different natures, mere local regulation, epigenetic modifications (reversible, nevertheless inheritable) or genetic mutations (irreversible), according to the extent to which the genome of the cells in the population is affected. In this respect, the models we develop are more likely to be biologically corresponding to epigenetic modifications, although eventual induction of emergent resistant cell clones due to mutations under drug pressure is not to be completely excluded. From the biologist's point of view, we study phenotypically heterogeneous, but genetically homogeneous, cancer cell populations under stress by drugs. According to the cell populations at stake and to the exerted drug pressure, is drug resistance in cancer a permanently acquired phenotypic trait or is it reversible? Can it be avoided or overcome by rationally (model-guided) designed combinations of drugs? These are some of the questions we will try to answer in a collaboration between a team of mathematicians and another one of biologists, both dealing with cancer and Darwinian - possibly also Lamarckian - evolution of cell populations.

              "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.

Mechanistic mathematical models are important tools for understanding the processes that shape ecological systems. Models have been used to describe life cycles of individuals, population dynamics, behavior, and more. However, in order for these models to reach their full potential as both tools for understanding and for prediction we must be able to link modeled quantities to data and infer model parameters. However, general methods of parameter inference for many of these models, including Individual Based Models (IBMs) or their component models, are not available. In addition, some models include components that are unmeasurable or poorly known, which can impact parameter inference and thus prediction. Here I discuss two examples of ecological models of these types. First is a bioenergetic model of individual growth and reproduction in a dynamic environment. This example highlights how input mis-specification can affect inference, and the consequences for prediction for both individuals and populations. The second example uses an example of an IBM developed to describe the spread of Chytridiomycosis in a population of frogs. This case study shows how one can perform inference for IBMs that exhibit certain characteristics with a traditional likelihood-based approach.

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.

Certain strains of the influenza virus form coherent single, globally circulating viral populations. Multiple models of viral evolution have considered the virus on this level, identifying immune and structural factors underlying the observed changes in viral genotype. In this talk, I outline an alternative approach for studying viral evolution, considering events at the level of within-host viral growth and transmission. In particular, I describe statistical approaches to infer the extent to which natural selection acts upon the viral population within the course of a single infection, on the basis of genome sequencing data from Sanger sequencing, and next-generation technologies. Examining data from evolutionary experiments conducted with a reassortant H5N1 influenza virus, I discuss the potential for using data from evolutionary experiments to predict the potential evolution of this virus in a human population.

Elevations in systolic blood pressure are believed to be closely linked to the pathogenesis and progression of renal diseases. It has been hypothesized that the afferent arteriole (AA) protects the glomerulus from the damaging effects of hypertension by sensing increases in systolic blood pressure and responding with a compensatory vasoconstriction. To investigate this hypothesis, we developed a mathematical model of the myogenic response of an AA smooth muscle cell, based on an arteriole model by Gonzalez-Fernandez and Ermentrout (Math Biosci 1994). renal hemodynamic regulation. The model incorporates ionic transport, cell membrane potential, contraction of the AA smooth muscle cell, and the mechanics of a thick-walled cylinder. The model represents a myogenic response based on a pressure-induced shift in the voltage dependence of calcium channel openings: with increasing transmural pressure, model vessel diameter decreases; and with decreasing pressure, vessel diameter increases. Further, the model myogenic mechanism includes a rate-sensitive component that yields constriction and dilation kinetics similar to behaviors observed in vitro. A parameter set is identified based on physical dimensions of an AA in a rat kidney. Model results suggest that the interaction of Ca2+ and K+ fluxes mediated by voltage-gated and voltage-calcium-gated channels, respectively, gives rise to periodicity in the transport of the two ions. This results in a time-periodic cytoplasmic calcium concentration, myosin light chains phosphorylation, and crossbridges formation with the attending muscle stress. Further, the model predicts myogenic responses that agree with experimental observations, most notably those which demonstrate that the renal AA constricts in response to increases in both steady and systolic blood pressures. The myogenic model captures these essential functions of the renal AA, and it may prove useful as a fundamental component in a multi-scale model of the renal microvasculature suitable for investigations of the pathogenesis of hypertensive renal diseases.

Parkinson’s disease has been traditionally thought of as a dopaminergic (DA) disease in which cells of the substantia nigra pars compacta (SNc) die. However, accumulating evidence implies an important role for the serotonergic (5HT) system in Parkinson’s disease. We use a mathematical model to investigate the consequences of levodopa therapy on the serotonergic system and on the pulsatile release of DA from dopaminergic and serotonergic terminals in the striatum. We compute the time course of DA release in the striatum from both 5HT and DA neurons and show how the time course changes as more and more SNc cells die. This enables us to explain the shortening of the therapeutic time window for the efficacy of levodopa as Parkinson’s disease progresses. Finally, we study the effects 5HT1a and 5HT1b autoreceptor agonists and explain why they have a synergistic effect and why they lengthen the therapeutic time window for LD therapy. Our results are consistent with and help explain results in the experimental literature and provide new predictions that can be tested experimentally.

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).

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.

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.

One of the first immunologic responses against HIV infection is the presence of neutralizing antibodies that seem able to inactivate several HIV strains. Moreover, in vitro studies have shown the existence of monoclonal antibodies that exhibit broad crossclade neutralizing potential. Yet their number is low and slow to develop in vivo. In this paper, we investigate the potential benefits of inducing poly-specific neutralizing antibodies in vivo throughout immunization. We develop a mathematical model that considers the activation of families of B lymphocytes producing poly-specific and strain-specific antibodies and use it to demonstrate that, even if such families are successful in producing neutralizing antibodies, competition between them may limit the poly-specific response allowing the virus to escape. We modify this model to account for viral evolution under the pressure of antibody responses in natural HIV infection as well as the need to neutralize more than one viral spike. The model can reproduce viral escape under certain conditions of B lymphocyte competition. Using these models we provide explanations for the observed antibody failure in controlling natural infection and predict quantitative measures that need to be satisfied for long-term control of HIV infection.

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.