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.

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.

Experiments on macaque monkeys show that spike-triggered stimulation performed by a Bidirectional Brain-Computer-Interfaces (BBCI) can artificially strengthen synaptic connections between distant neural sites in Motor Cortex (MC) and even between MC and spinal cord, with changes that last several days. Here, a neural implant records from some neurons in MC and electrically stimulates others after set delays. The working hypothesis is that this stimulation procedure, which interacts with the very fast spiking activity of cortical circuits (on the order of milliseconds), induces changes mediated by synaptic plasticity mechanisms on much longer timescales (hours and days). The field of online, closed-loop BBCI's is rapidly evolving, with applications ranging from a science-oriented tool to clinical treatments of motor injuries. However, with the enhanced capability of novel devices that can record and stimulate an ever-growing number of neural sites comes growing complexity. It is therefore crucial to develop a theoretical understanding of the effects of closed-loop artificial stimulation in the highly recurrent neural circuits found in cortex, and how such protocols affect functional cotex-to-muscle mappings across a range of timescales. In parallel with ongoing experiments, we are developing a mathematical model of recurrent MC networks with probabilistic spiking mechanisms and spike-time-dependent plastic synapses (STDP) capable of capturing both neural and synaptic activity statistics relevant to BBCI protocols. This model successfully reproduces key experimental results and we use analytical derivations to predict optimal operational regimes for BBCIs. We make experimental predictions concerning the efficacy of spike-triggered stimulation in different regimes of cortical activity such as awake behaving states or sleep. Importantly, this work provides a first step toward a theoretical framework aimed at the design and development of next-generations applications of BBCI's.

The notochord, the defining feature of chordates, is a pressurized tube which actuates elongation of the chordate embryo. The zebrafish notochord consists of large vacuolated cells surrounded by a thin sheath. We characterized the patterns of the cells’ packing, and their relationship to the known regular patterns from the study of foams, and irregular patterns in a gel bead system. Disruption of the wild type packing pattern leads to developmental defects. We characterize the bifurcations between the relevant regular patterns in terms of nondimensional geometrical and mechanical ratios, and suggest an important developmental role for the eccentric "staircase" pattern.

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.

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.

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.

A major focus of synthetic biology is to engineer gene circuits to perform user-defined functions. These gene circuits can serve as well-defined models to probe basic biological questions of broad significance. In this talk, I will discuss our efforts along this line of research, whereby we have engineered gene circuits to program bacterial dynamics in time and space, guided by quantitative modeling and experiments. Insights learnt from these circuits have implications for developing new strategies to combat bacterial pathogens or to fabricate new materials.

The use of delay differential equations (DDEs) to study biological phenomena has a long history, when the rate of change of model variables depends their previous history. Today, DDEs occupy a central place in models of infectious disease dynamics, epidemiology, ecology and tumor growth. In this talk, I will present a delayed partial differential equation (PDE) model of tumor growth and treatment. The model accounts for cell cycle arrest and cell death induced by chemotherapy, and explicitly includes intracellular signaling pathways relevant to drug action. The model is simplified to give a 1D hyperbolic PDE, which is further reduced to a nonlinear, non-autonomous DDE by projecting along characteristics. Necessary and sufficient conditions for the global stability of the cancer-free equilibrium are derived and conditions under which the system evolves to periodic solutions are determined. This has clinical implications since it leads to a lower bound for the amount of therapy required to affect a cure. Finally, I will present a clinical application of the model, by applying it to the treatment of ovarian cancers. Two types of drugs are considered – platinum-based chemotherapeutic agents that are the current standard of care for most solid tumors, and small molecule cell death inducers that are currently under development. The model is calibrated versus in vitro experimental results, and is then used to predict optimal doses and administration time scheduling for the treatment of a tumor growing in vivo.

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

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.

It has long been accepted that type 1 diabetes results from a lack of insulin, as the insulin-secreting pancreatic beta cells are destroyed by an autoimmune process. In contrast, the cause of type 2 diabetes (T2D) is less clear. Most people with pre-diabetes or in the early stages of T2D have abnormally high plasma insulin concentrations, and insulin rises before glucose does. We show that these difficulties are resolved by a mathematical model in which the onset of T2D is represented by the crossing of a threshold. The threshold is atypical in some respects and requires consideration of the slow manifolds to avoid incorrect conclusions.

Certain neurons can, in isolation, generate a bursting rhythm, in which phases of active spiking alternative repetitively with phases of quiescence. This behavior is itself mathematically interesting, and neurons with this capability have been found in the mammalian respiratory brain stem, suggesting that they might drive the respiratory rhythm. In this talk, I will survey some mathematical and computational work that runs counter to this suggestion. The methods involved include slow-fast decomposition and associated bifurcation analysis in single-neuron and few-neuron ODE models as well as a genetic algorithm applied to larger network models. The larger network results may have general implications for networks of nodes with heterogeneous dynamics, coupled in small-world, scale-free, and other architectures

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.

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.

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.

Many patterns in Nature and industry arise from the system minimizing an appropriate energy. Torn plastic sheets and growing leaves provide striking examples of pattern forming systems which can transition from single wavelength geometries (leaves) to complex fractal-like shapes (lettuce). These fractal-like patterns seem to have many length scales, i.e. the same amount of extra detail can be seen when looking closer (“statistical self-similarity”). It is a mystery how such complex patterns could arise from energy minimization alone. In this talk I will address this puzzle by showing that such patterns naturally arise from the sheet adopting a hyperbolic non-Euclidean geometry. However, there are many different hyperbolic geometries that the growing leaf could select. I will show using techniques from analysis, differential geometry and numerical optimization that the fractal like patterns are indeed the natural minimizers for the system. I will also discuss the implications of our work to developing shape changing soft matter which can be implemented in soft machines.

Recent studies of cholesterol-rich membrane microdomains, called caveolae, reveal that caveolae are reservoirs of recruitable sodium ion channels. Caveolar channels constitute a substantial and previously unrecognized source of sodium current in cardiac cells. In this talk, I will present a family of DE and PDE models to investigate caveolar sodium currents and their contributions to cardiac action potential morphology. We show that the b-agonist-induced opening of caveolae may have substantial impacts on peak overshoot, maximum upstroke velocity, and conduction velocity. Additionally, we show that prolonged action potentials and the formation of potentially arrhythmogenic afterdepolarizations, can arise if caveolae open intermittently throughout the action potential.

We discuss two types of partial differential equation models of fishery harvesting problems. We consider steady state spatial models and diffusive spatial-temporal models. We characterize the distribution of harvest effort which maximizes the harvest yield, and in the steady state case, also minimizes the cost of the effort. We show numerical results to illustrate various cases. The results inform ongoing debate about the use of reserves (regions where fishing is not allowed), and are increasingly relevant as technology enables enforcement of spatially structured harvest constraints.

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.

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.

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.

Many pulse-coupled dynamical systems possess synchronous attracting states. Even stochastically driven model networks of Integrate and Fire neurons demonstrate synchrony over a large range of parameters. We study the interplay between fluctuations which de-synchronize and synaptic coupling along the network connections that synchronize the network by calculating the probability to see repeated cascading total firing events, during which all the neurons in the network fire at once. Using this characterization of synchrony we investigate the significance of the local network topology and of more physiological additions to the model on the model neuronal networks ability to synchronize. The mean time between total firing events characterizes the perfectly synchronous state, and we compute this from a first-passage time problem in terms of a Fokker-Planck equation for a single neuron.

The ability of animals to self-organize into remarkable patterns of movement is seen throughout nature from herds of large mammals, to flocks of birds, schools of fish, and swarms of insects. Remarkably, patterns of collective movement can be seen even in the simplest forms of life such as bacteria. M. xanthus are common soil bacteria that are among the most “social" bacteria in nature. In this talk clustering mechanism of swarming M. xanthus will be described using combination of experimental movies and stochastic model simulations. Continuous limits of discrete stochastic dynamical systems simulating cell aggregation will be described in the form of reaction-diffusion and nonlinear diffusion equations. Surface motility such as swarming is thought to precede biofilm formation during infection. Population of bacteria P. aeruginosa, major infection in hospitals, will be shown to efficiently propagate as high density waves that move symmetrically as rings within swarms towards the extending tendrils. Multi-scale model simulations suggest a mechanism of wave propagation as well as branched tendril formation at the edge of the population that depend upon competition between the changing viscosity of the bacterial liquid suspension and the liquid film boundary expansion caused by Marangoni forces. This collective mechanism of cell- cell coordination was recently shown to moderate swarming direction of individual bacteria to avoid a toxic environment. In the last part of the talk a three-dimensional multiscale modeling approach will be described for studying fluid–viscoelastic cell interaction during blood clot formation.

I consider two kinds of biological novelty using shape of the Drosophila wing as a focal phenotype. Quantitative novelty is the evolution of more or less of the same elements already present in the ancestor, or evolution within one topological space. Qualitative novelty is gain of elements not present in the ancestor, or perhaps the loss of such elements, making a transition from one topological space to another. A quantitative genetic approach allows us to studying the major determinants of quantitative novelty, standing genetic variation and mutational variation. I will present measurements of both mutation and standing variation for a multivariate phenotype, the fly wing. Mathematical and statistical challenges in quantitative novelty involve the identification of subspaces within a topological space with genetic variation, and their relationship to the subspaces found in other samples. Qualitative novelty is easy to identify among species, but its evolution is difficult to study, as by definition it is absent in the ancestor. The process of development, where qualitatively different structures are progressively introduced during most life cycles, provides a framework to understand the evolution of novelty. Developmental studies show that continuous changes in gene expression precede transitions to a new shape topology. Gene expression state within a shape is more complex object that shape itself, adding dimensions that may allow us to identify regions of one shape topology that are neighbors to other topologies.