Interaction between gene products forms the basis of essential biological processes like signal transduction, cell metabolism or embryonic development. The variety of interactions between genes, proteins and molecules are well captured by network (graph) representations. Experimental advances in the last decade helped uncover the structure of many molecular-to-cellular level networks, such as protein interaction or metabolic networks. For other types of interaction and regulation inference methods based on indirect measurements have been used to considerable success. These advances mark the first steps toward a major goal of contemporary biology: to map out, understand and model in quantifiable terms the topological and dynamic properties of the various networks that control the behavior of the cell.
This talk will sample recent progress in two directions: intracellular network discovery and integration of different types of regulation (e.g. integration of metabolic and transcriptional networks), and connecting intra-cellular network structure, network dynamics and cellular behavior. A significant trust of the current research is to reveal or predict the topological or dynamic changes in the network responsible for abnormal behavior. This line of research will strenghten in time, and can be a fertile ground for mathematical biologists interested in adapting graph theory or nonlinear dynamical systems theory to biological systems.
Most of the phenomena of life that attract our attention result from interactions among many components in a network. Examples include the interactions among neurons in the brain, among birds in a flock or fish in a school, and even the interactions among amino acids in a single protein. In all these cases there are "emergent" or collective behaviors that are properties of the network but not the individual components. In the physics of systems at thermal equilibrium, we have many examples of such emergent phenomena (some mundane, like the rigidity of solids, others more spectacular, such as superconductivity), and we have a language for describing such phenomena, statistical mechanics. There is a long standing intuition that this same language should be useful in thinking about collective phenomena in biological systems, an idea which is best developed in the context of neural networks, but one has to admit that much of what is done theoretically is not terribly well connected to experiment. I will review the argument that the maximum entropy construction gives us a way of going directly from real data to the more abstract statistical mechanics models, emphasizing the opportunities created by new, larger scale experiments. I'll start with flocks of birds, where the simplest version of these ideas seems remarkably successful. I'll then say a few words about proteins, using recent data on complete antibody repertoires in zebrafish as motivation. Finally, I'll discuss neurons, focusing on the response of the vertebrate retina to natural movies. Along the way I hope to make clear the connections between things that seem natural and interesting in the statistical mechanics context and things that seem relevant for the organism. Most startlingly, in all of these systems we find that the particular models which describe the real systems sit close to critical surfaces in the space of all possible models. I'll explain several different ways of seeing that this is true, why it is surprising, and speculate on why it is important. It certainly suggests that there is something deeper going on here, which we don't yet understand.
General anesthesia is a drug-induced, reversible condition comprised of five behavioral states: unconsciousness, amnesia (loss of memory), analgesia (loss of pain sensation), akinesia (immobility), and hemodynamic stability with control of the stress response. The mechanisms by which anesthetic drugs induce the state of general anesthesia are considered one of the biggest mysteries of modern medicine. We have been using three experimental paradigms to study general anesthesia-induced loss of consciousness in humans: combined fMRI/EEG recordings, high-density EEG recordings and intracranial recordings. By using a wide array of signal processing techniques, these studies are allowing us to establish precise neurophysiological, neuroanatomical and behavioral correlates of unconsciousness under general anesthesia. Combined with our mathematical modeling work on how anesthetics act on neural circuits to produce the state of general anesthesia we are able to offer specific hypotheses as to how changes in level of activity in specific circuits lead to the unconscious state. We will discuss the relation between our findings and two other important altered states of arousal: sleep and coma. Our findings suggest that the state of general anesthesia is not as mysterious as currently believed. Statistical and mathematical analyses have played a key role in deciphering this mystery.
Synthetic biology is bringing together engineers, mathematicians and biologists to model, design and construct biological circuits out of proteins, genes and other bits of DNA, and to use these circuits to rewire and reprogram organisms. These re-engineered organisms are going to change our lives in the coming years, leading to cheaper drugs, "green" means to fuel our car and clean our environment, and targeted therapies to attack "superbugs" and diseases such as cancer. In this talk, we highlight recent efforts to model and create synthetic gene networks and programmable cells, and discuss a variety of synthetic biology applications in biocomputing, biotechnology and biomedicine.
We describe new distances between pairs of two-dimensional surfaces (embedded in three-dimensional space) that use both local structures and global information in the surfaces.
These are motivated by the need of biological morphologists to compare different phenotypical structures. At present, scientists using physical traits to study evolutionary relationships among living and extinct animals analyze data extracted from carefully defined anatomical correspondence points (landmarks). Identifying and recording these landmarks is time consuming and can be done accurately only by trained morphologists. This necessity renders these studies inaccessible to nonmorphologists and causes phenomics to lag behind genomics in elucidating evolutionary patterns.
Unlike other algorithms presented for morphological correspondences, our approach does not require any preliminary marking of special features or landmarks by the user. It also differs from other seminal work in computational geometry in that our algorithms are polynomial in nature and thus faster, making pairwise comparisons feasible for significantly larger numbers of digitized surfaces.
We illustrate our approach using three datasets representing teeth and different bones of primates and humans, and show that it leads to highly accurate results.
The 20th century revolution in statistics focused on measurement, experimental design, modeling and computational issues in a world of "small" data where the number of observations and/or variables were typically limited and information available in single sources. Scientists face very different challenges in the current age where data is often streamed in real time, and the number of inputs, outputs or confounders are often massive. This presents challenges for reliable inference about "old" questions, while providing opportunities to investigate much more subtle issues about mechanisms of action, while reducing our reliance on unnecessary assumptions. We describe briefly some recent advances in data measurement, cleaning, and analysis that reflect these ideas, focusing finally on two applications (i) determining gene expression signatures of benzene exposure, and (ii) examining the influence of bisphenol A (BPA) in utero on patterns of weight gain in children.
The brain produces electrical activity whose spectral structure is highly correlated with cognitive state. Yet how rhythms participate in cognition, and how changes in rhythms in pathological states affect cognition, is just beginning to be explored. This talk will address several case studies comparing dynamics in normal and altered states, giving insight into the loss of consciousness, pathological rhythms in Parkinson's disease, and mechanisms for selective attention with implications for diseased states. It places these phenomena in the larger context of the multiple interactions of experimental neuroscience and mathematics, interactions that are certain to grow in the future.
The subject of mathematical ecology is one of the oldest in mathematical biology, having its formal roots a century ago in the work of the great mathematician Vito Volterra, with links, some long before, to demography, epidemiology and genetics. Classical challenges remain in understanding the dynamics of populations and connections to the structure of ecological communities. However, the scales of integration and scope for interdisciplinary work have increased dramatically in recent years. Metagenomic studies have provided vast stores of information on the microscopic level, which cry out for methods to allow scaling to the macroscopic level of ecosystems, and for understanding biogeochemical cycles and broad ecosystem patterns as emergent phenomena; indeed, global change has pushed that mandate well beyond the ecosystem to the level of the biosphere. Secondly, the recognition of the importance of collective phenomena, from the formation of biofilms to the dynamics of vertebrate flocks and schools to collective decision-making in human populations poses important and exciting opportunities for mathematicians and physicists to shed light. Finally, from behavioral and evolutionary perspectives, these collectives display conflict of purpose or fitness across levels, leading to game-theoretic problems in understanding how cooperation emerges in Nature, and how it might be realized in dealing with problems of the Global Commons. This lecture will attempt to weave these topics together and both survey recent work, and offer challenges for how mathematics can contribute to open problems.
The collective movement of cells in tissue is vital for normal development but also occurs in abnormal development, such as in cancer. We will review three different models: (i) A vertex-based model to describe cell motion in the early mouse embryo; (ii) A individual-based model for neural crest cell invasion; (iii) A model for acid-mediated tumour invasion.
In each case we shall use the model to answer important issues concerning biology. For example, in (i) we shall propose a role for rosette formation, in (ii) we propose that two cell types are necessary for successful invasion, and in (iii) we shall show how the model suggests possible therapeutic strategies for tumour control.
Eusociality is an advanced form of social organization, where some individuals reduce their reproductive potential to raise the offspring of others. Eusociality is rare but hugely successful: only about 2% of insects are eusocial but they represent 50% of the insect biomass. The biomass of ants alone exceeds that of all terrestrial non-human vertebrates combined. I will present a simple model for the origin of eusociality. In the solitary life style all offspring leave to reproduce. In the primitively eusocial life style some offspring stay and help raise further offspring. A standard natural selection equation determines which of those two reproductive strategies wins for a given ecology. The model makes simple and testable predictions without any need to evoke inclusive fitness theory. More generally, I will discuss the limitations of inclusive fitness theory. I will argue: once fitness is calculated in a standard model of natural selection every aspect of relatedness is included.
The modern era of human genomics began ten years ago with the launch of the HapMap project following the publication of the first draft of the human genome. Although the sequencing of the genome was a major scientific achievement, it has become clear that naive analysis of sequence will not be sufficient to address the fundamental challenge in genomics: determination of the function of genes and the prediction of their regulatory dynamics.
We will discuss modern "Star-Seq" technologies that leverage cheap sequencing technology to enable high-throughput molecular biology and that are revealing, for the first time, the complexities of the genome and its dynamics at full resolution. The development, analysis and interpretation of the assays is based on a number of computational, statistical and mathematical primitives that we will survey.
The sequencing of the first vertebrate genomes coincided with the founding of the Mathematical Biosciences Institute, and we will highlight the huge impact that the marriage of mathematics and genomics has had on biology, with a view towards the exciting possibilities in the decade ahead.
My research work is in two directions: (1) studying conducting and selectivity functions of ion channels and (2) exploring tumor growth under cancer-immune system interactions and therapeutic treatments. These complicated biological systems usually consists of composite materials and involve intensive interactions among components. The former is at molecular level: multiscale treatments (atomic and continuum) and multiphysics (classical and quantum) are applied to different components (water molecules, channel proteins, membranes, and mobile ions, etc) according to biological importances of objects and computational efficiency. The solute-solvent surface serves as a free boundary to couple the discrete and continuum scales. The cancer research is macroscopic: interactions pathways of a large amounts of cells and cytokines are outlined from experimental observations and modeled by a systems of governing equations; the tumor is described as a moving domain with free boundary and has obviously different phase-field from normal tissues.
Both of the two topics require advanced numerical techniques for solving partial differential equations and simulations are validated by experimental data. Analysis, such as existence and uniqueness of the solutions are performed to these free boundary problems.
Cholera is a waterborne intestinal infection which causes profuse, watery diarrhea, vomiting, and dehydration. It can be transmitted via contaminated water as well as person to person, with 3-5 million cases/year and over 100,000 deaths/year. An important public health question involves understanding the modes of cholera transmission in order to improve control and prevention strategies. One issue of interest is: given data for an outbreak, can we determine the role and relative importance of waterborne vs. person-to-person routes of transmission? To examine this issue, we explored the identifiability and parameter estimation of a differential equation model of cholera transmission dynamics. We used a computational algebra approach to establish whether it is possible to determine the transmission rates from outbreak case data (i.e. whether the transmission rates are identifiable), and then applied the model to a recent cholera outbreak in Angola which resulted in over 80,000 cases and over 3000 deaths. Our results show that both water and person-to-person transmission routes are identifiable, although they become practically unidentifiable with fast water dynamics. Using these results, parameter estimation for the Angola outbreak suggests that both water and person-to-person transmission are needed to explain the observed cholera dynamics. We will also discuss some ongoing work using this model, including modeling the spatial spread of outbreaks, public health interventions and control strategies, and applications to the ongoing cholera outbreak in Haiti.
The availability of great amounts of genetic sequence data promises deep insight into the relationship between individual molecular changes and the phenotypic diversity of living organisms, especially among humans and among human pathogens. In complement to this, sequence data can be used to explore the genetic changes that arise in response to environmental pressures, host-parasite relationships, or medication. However, the covariance between natural observations, arising from such phenomena as population structure, family structure or a history of speciation, can be a significant confounder in interpreting this rich data source. Here we present a novel method, Phylogeny with Path to Event for Viruses (PhyloPTE/V) to detect associations between individual molecular changes and phenotypic outcomes or environmental/medical triggers under a wide range of covariance patterns, and scalable to the large data sets produced by modern methods.
Discrete biological models are often intuitive to construct and able to reveal the qualitative dynamics of a complex system. Sensitivity analysis, which provides insight toward the effect of perturbations and uncertainty in a system, is typically regarded as essential in the modeling process. While methods for performing sensitivity analysis of continuous models have been studied extensively, far fewer analogous methods exist for discrete models. In this presentation, a novel method is proposed for performing sensitivity analysis on discrete models based on analogous continuous model techniques. The method of quantifying sensitivity is based on artificially introducing unknown parameters to the model and comparing the resulting dynamics to the original model. A mathematical framework, namely polynomial dynamical systems, is used to algebraically compute the dynamics of models with unknown parameters, a computation that might otherwise be computationally infeasible without the developed theory. The algorithm was applied to published gene regulatory networks to provide a benchmark for the sensitivity of discrete biological models. An implementation of the algorithm is publicly available as a Macaulay2 package.
A biochemically motivated model of the treatment of ovarian cancers that accounts for cell cycle arrest and cell death induced by chemotherapy is developed. The actions of carboplatin (a platinum-based drug), and ABT-737 (a small molecule inhibitor of the cell-death regulating intracellular proteins Bcl-2/xL), are simulated in order to elucidate the molecular basis of synergy between the two drugs. This information is then used to predict optimal dosing and scheduling of the drugs, and to investigate treatment options targeting the emergence of resistance to carboplatin.
Somitogenesis is the whole process for the development of somites which are segmental structure lies along the anterior-posterior (AP) axis of vertebrate embryos. The pattern of somites is governed by clock gene expressions which undergo synchronous oscillation over adjacent cells with period about 30 minutes in the tail bud of presomitic mesoderm (PSM), oscillation slowing down and traveling wave pattern in PSM, and the oscillation-arrested with high and low expression levels in the anterior of PSM. In order to investigate this scenario in zebrafish, we consider mathematical models which depict the kinetics of the zebrafish segmentation clock genes subject to direct autorepression by their own products under time delay, and cell-to-cell interaction through Delta-Notch signaling.
First, we consider two-cell model. We employ a sequential-contracting technique to derive the criteria for the global convergence to the equilibrium, which corresponds to the oscillation-arrested for the cells at the anterior of PSM. Applying the delay Hopf bifurcation theory, the center manifold theorem, and the normal form method, we obtain the conditions for the existence of stable synchronous oscillations for the cells at the tail bud of the PSM. In addition, our analysis provides the basic parameter regimes and delay magnitudes for stable synchronous and oscillation-arrested. Next, we generate the two-cell model to N-cell model and extend these analytical methods to the N-cell model to investigate the oscillation-arrested and synchronous oscillations. Furthermore, with suitable gradients of degradation rates and delays along the AP axis, the N-cell model can generate synchronous oscillation with period about 30 minutes, normal traveling wave pattern, oscillation slowing-down, and oscillation-arrested, in each corresponding region in the embryo.
Cell polarity is induced through the localization of specific molecules to proper location of the cell membrane. Here we propose a generic model consisting of the particle density of membrane bound molecules undergoing polarization to study the mechanisms for different polarized site selection patterns, including random polarized site selection in the absence of a pre-localized signal, axial polarized pattern, bipolar polarization pattern and adjacent positioning of polarization.
Work done in collaboration with Ching-Shan Chou and Hay-Oak Park.
In several species, wake bout durations have a power-law distribution. While it has been observed that sleep and wake states are governed by competitive interactions between two mutually inhibitory neuronal networks, the biological mechanisms underlying subexponential wake bout durations remain unknown. For a single network, a power-law degree distribution can promote some forms of power law dynamics. Here we ask what role neuronal network architecture may play in the activity dynamics generated by competitive interaction between the sleep and wake networks. We explore this question via examination of three levels of stochastic models: 1) a network model, 2) drift-diffusion dynamics of collective variables, and 3) a toy model of regulated Brownian motion.
A class of Shigesada-Kawasaki-Teramoto type reaction-cross diffusion models with vanishing random diffusion coefficients are considered. For homogeneous Dirichlet boundary conditions, it is shown that non-trivial non-negative smooth solutions do not exist globally in time. Numerical simulations suggest the possibility of finite-time extinction.