Ecological systems may exhibit complex dynamics, yet the spatial and temporal scales over which these play out make them difficult to explore experimentally. An alternative approach is to develop models based on detailed biological information about the systems and then fit them to observational data using nonlinear time-series techniques. I will give two examples of this approach, both involving systems with alternative states. The first is the dynamics of midges in Lake Myvatn, Iceland, which show fluctuations with amplitudes >105 yet with irregular period. A nonlinear time-series analysis demonstrates that these dynamics could be caused by the system having two states, a stable point and a stable cycle, with the irregular period caused by the population stochastically jumping from the domain of one state to the other. The second example is the dynamics of salvinia, an aquatic weed, and the salvinia weevil that was introduced into the billabongs of Kakadu to control the weed. Here the alternative states are two environmentally (seasonally) forced cycles, one in which salvinia is kept in check by the weevil and one in which it escapes. Understanding complex ecological dynamics may improve our management of vigorously fluctuating natural systems.
Expression of microRNAs, a new class of noncoding RNAs that hybridize to target messenger RNA and regulate their translation into proteins, has been recently demonstrated to be altered in acute myeloid leukemia (AML). Distinctive patterns of increased expression and/or silencing of multiple microRNAs (microRNA signatures) have been associated with specific cytogenetic and molecular subsets of AML. Changes in the expression of several microRNAs altered in AML have been shown to have functional relevance in leukemogenesis, with some microRNAs acting as oncogenes and others as tumor suppressors. Both microRNA signatures and a single microRNA have been shown to supply prognostic information complementing to that gained from cytogenetics, gene mutations, and altered gene expression. Moreover, it has been demonstrated experimentally that microRNAs contribute with signaling pathways to leukemogeneic networks and the antileukemic effects can be achieved by modulating microRNA expression by pharmacologic agents and/or increasing low endogenous levels of microRNAs with tumor suppressor function by synthetic microRNA oligonucleotides, or down-regulating high endogenous levels of leukemogenic microRNAs by antisense oligonucleotides (antagomirs). Therefore, it is reasonable to predict the development of novel microRNA-based diagnostic, prognostic and therapeutic approaches in AML.
In this talk we use a realistic model to demonstrate how mathematics can be applied in real-life applications. An image-based human heart model with fluid-structure interactions (FSI) was introduced to evaluate human heart cardiac function before and after surgery and optimize human pulmonary valve replacement/insertion (PVR) surgical procedure and patch design. Cardiac Magnetic Resonance (CMR) imaging studies were performed to acquire ventricle geometry, flow velocity and flow rate for healthy volunteers and patients needing right ventricle (RV) remodeling and PVR before and after scheduled surgeries. CMR-based RV/LV/Patch FSI models were constructed to perform mechanical analysis and provide accurate assessment for RV mechanical conditions and cardiac function. These models include a) fluid-structure interactions, b) isotropic and anisotropic material properties, c) two-layer construction with myocardial fiber orientation, and d) active contraction. When validated, the computational modeling approach could be used to replace actual surgical experiments on real patients by "virtual" surgery using computational simulations to optimize surgical outcome.
Acknowledgement: This research was in collaboration with Pedro Del Nido, MD, William E. Ladd Professor of Surgery, Chairman of Cardiac Surgery, and Tal Geva, MD, Director of Cardiac MRI Department, Children's Hospital Boston, Harvard Medical School, USA. It was supported in part by NIH R01 HL089269 (del Nido, Tang, Geva), NIH-R01 HL63095 (PJdN), and NIH- 5P50 HL074734 (Clinical Trial, PI-Geva).
The typical mammalian cortical neuron has about 5000 microns of dendrite. These dendrites are studded with synapses at a density of 1 per micron. The dendrites integrate this spatially distributed transient stimulus and coax, on occasion, the neuron to fire. Although the underlying mechanisms are known and well modeled, the sheer complexity has to date retarded attempts to incorporate such models in large network level simulations. In this talk we will couple structure preserving projection techniques for reducing the dynamics of the weakly excitable dendrites with discrete empirical interpolation near the spike initiation zone and arrive at a model with significantly fewer internal variables without sacrificing accuracy of the cell's input/output map. We will then demonstrate their ability to efficiently capture network level behavior.
The development of mathematical models for chemical systems based on, for example, density functional theory, have provided key insight into the manner in which chemical transformations occur and the nature of intermediates that form in the course of the reaction. In enzymology, these approaches have proven very successful not only in establishing the chemical sequence of events involved in converting substrate to product but, importantly, how the enzyme accomplishes its critical task physiologically of making the reaction go so fast - typically 1012 to 1014 times faster than the uncatalyzed reaction. Here we dissect the reaction mechanism of the enzyme xanthine oxidase using the tools of density functional theory, and examine the basis for rate acceleration in the context of the enzyme's structure.
Bacteria are the most abundant organisms on Earth and they significantly influence carbon cycling and sequestration, decomposition of biomass, and transformation of contaminants in the environment. This motivates our study of the basic principles of bacterial behavior and its control. We have conducted analytical, numerical and experimental studies of suspensions of swimming bacteria. In particular, our studies reveal that active swimming of bacteria drastically alters the material properties of the suspension: the experiments with bacterial suspensions confined in thin films indicate a 7-fold reduction of the effective viscosity and a 10-fold increase of the effective diffusivity of the oxygen and other constituents of the suspending fluid. The principal mechanism behind these unique macroscopic properties is self-organization of the bacteria at the microscopic level - a multiscale phenomenon. Understanding the mechanism of self-organization in general is a fundamental issue in the study of biological and inanimate system. Our work in this area includes
Collaborators: PSU students S. Ryan and B. Haines, and DOE scientists I. Aronson and D. Karpeev (both Argonne Nat. Lab)
Clustering data into groups of similarity is well recognized as an important step in many diverse applications, including biomedical imaging, data mining and bioinformatics. Well known clustering methods, dating to the 70's and 80's, include the K-means algorithm and its generalization, the Fuzzy C-means (FCM) scheme, and hierarchical tree decompositions of various sorts. More recently, spectral techniques have been employed to much success. However, with the inundation of many types of data sets into virtually every arena of science, it makes sense to introduce new clustering techniques which emphasize geometric aspects of the data, the lack of which has been somewhat of a drawback in most previous algorithms.
In this talk, we focus on a slate of "random-walk" distances arising in the context of several weighted graphs formed from the data set, in a comprehensive generalized FCM framework, which allow to assign "fuzzy" variables to data points which respect in many ways their geometry. The method we present groups together data which are in a sense "well-connected", as in spectral clustering, but also assigns to them membership values as in FCM. We demonstrate the effectiveness and robustness of our method on several standard synthetic benchmarks and other standard data sets such as the IRIS and the YALE face data sets. This is joint work with Sijia Liu and Sunder Sethuraman.
We review winner-loser models, the currently popular explanation for the occurrence of linear dominance hierarchies, via a three-part approach. 1) We isolate the two most significant components of the mathematical formulation of three of the most widely-cited models and rigorously evaluate the components' predictions against data collected on hierarchy formation in groups of hens. 2) We evaluate the experimental support in the literature for the basic assumptions contained in winner-loser models. 3) We apply new techniques to the hen data to uncover several behavioral dynamics of hierarchy formation not previously described. The mathematical formulations of these models do not show satisfactory agreement with the hen data, and key model assumptions have either little, or no conclusive, support from experimental findings in the literature. In agreement with the latest experimental results concerning social cognition, the new behavioral dynamics of hierarchy formation discovered in the hen data suggest that members of groups are intensely aware both of their own interactions as well as interactions occurring among other members of their group. We suggest that more adequate models of hierarchy formation should be based upon behavioral dynamics that reflect more sophisticated levels of social cognition.
Numerical models and observational data are critical in modern science and engineering. Since both of these information sources involve uncertainty, the use of statistical, probabilistic methods play a fundamental role. I discuss a general Bayesian framework for combining uncertain information and indicate how various approaches (ensemble forecasting, UQ, etc.) fit in this framework. A paleoclimate analysis illustrates the use of simple physics and statistical modeling to produce inferences. A second example involves glacial dynamics and illustrates how updating models and data can lead to estimates of model error. A third example involves the extraction of information from multi-model ensembles in climate projection studies.
Sequencing data are often obtained by biologists wishing to explore details of a system with which they are extremely familiar; however, analysis techniques exclude these experts and often rely on assumptions that may not be relevant to the experimental design. While biologists can manually explore their data using newer, high-capacity genome browsers, and can often suggest relevant hypotheses for statistical testing, fully informed and thorough data exploration is impossible to do by eye. We have created a biologically-based and statistically grounded tool for determining the correlation of genomewide data with other datasets or known biological features, intended to guide biological exploration of high-dimensional datasets and to act as a hypothesis generator (not intended to provide "answers"). The software enables several biologically motivated approaches to these data; in fact, each analytical approach was inspired by our own work. Our models and statistics are implemented in an R package that efficiently calculates the spatial correlation between two sets of genomic intervals (data and/or annotated features), for use as a metric of functional interaction. The software is accessible from the command line, through a Tk interface, and through a Galaxy plugin, and is intended to guide biologists and statisticians more quickly to the significant features of high-dimensional datasets.
Sequencing studies, such as targeted, whole exome and whole genome sequencing studies, are increasingly being conducted to identify rare variants that are associated with complex traits. Design and analysis of such population based sequencing association studies face many challenges. The talk has three parts. I will first provide an overview of several methods for studying rare variant effects, including burden tests, SKAT and optimal unified tests. Analysis pipelines for whole exome sequencing association studies, such as filtering criteria and small sample adjustments of statistical methods, will be discussed. In the second part of the talk, I will discuss designs of sequencing association studies, such as sample size and power calculations, and pros and cons of extreme phenotype sampling and analysis strategies for extreme phenotype sequencing studies. In the last part of the talk, I will discuss the performance of imputation using GWAS data for studying rare variants effects. Simulation studies and real data will be used to illustrate the results.
A number of phenomena in visual perception involve wave-like propagation dynamics. Examples include perceptual filling-in, migraine aura, and the expansion of illusory contours. Another important example is the wave-like propagation of perceptual dominance during binocular rivalry. Binocular rivalry is the phenomenon where perception switches back and forth between different images presented to the two eyes. The resulting fluctuations in perceptual dominance and suppression provide a basis for non-invasive studies of the human visual system and the identification of possible neural mechanisms underlying conscious visual awareness. In this talk we present a neural field model of binocular rivalry waves in visual cortex. For each eye we consider a one-dimensional network of neurons that respond maximally to a particular feature of the corresponding image such as the orientation of a grating stimulus. Recurrent connections within each one-dimensional network are assumed to be excitatory, whereas connections between the two networks are taken to be inhibitory (cross-inhibition). Slow adaptationis incorporated into the model by taking the network connections to exhibit synaptic depression. We derive an analytical expression for the speed of a binocular rivalry wave as a function of various neurophysiological parameters, and show how properties of the wave are consistent with the wave-like propagation of perceptual dominance observed in recent psychophysical experiments. In addition to providing an analytical framework for studying binocular rivalry waves, we show how neural field methods provide insights into the mechanisms underlying the generation of the waves. In particular, we highlight the important role of slow adaptation in providing a "symmetry breaking mechanism" that allows waves to propagate. We end by discussing recent work on the effects of noise.
Odors are important cues for identification of many types of objects that animals need for survival. Natural odors are typically mixtures of up to a few dozen chemical components. Important information about odor 'objects' is often encoded in the ratio of components in the mixture. However, this odor mixture problem is complicated by two factors. First, many times the information channel is composed of a submixture of the overall mixture composition. Second, the ratios of components in the submixture can vary from one object to the next, which means that animals must learn to 'generalize' across a range of variation among objects that mean the same thing. Floral odors, for example, are important for honey bees to locate nectar and pollen sources that their colony needs for survival. Honey bees need to learn about the range of variation in odor composition so that they can optimally include flowers that have resources and exclude flowers with similar odors but which do not have nectar or pollen. I argue that nonassociative and associative plasticity in neural networks involved in early sensory coding is critical for extracting the relevant features of an odor mixture and setting up categories of odor objects. This plasticity can enhance decisions about pattern matching and multimodal associations in later processing in the brain.
The genetic differences that separate humans from other great apes seem modest, yet we are a prominent phenotypic outlier in several regards. Although little is known about the genetic and molecular bases underlying uniquely human traits, changes in gene regulation are likely an important component. My group is currently investigating changes in the transcriptomes of multiple tissues that accompanied human origins. We use genome-wide functional assays to identify the molecular mechanisms that mediate evolutionary changes in transcription, including the role of chromatin configuration and the function of noncoding RNAs. These assays highlight genomic regions of particular interest, where we have carried out focused functional analyses that provide insights into the evolution of human diet and brain size.
Systems biology aims to explain how a biological system functions by investigating the interactions of its individual components from a systems perspective. Modeling is a vital tool as it helps to elucidate the underlying mechanisms of the system. Many discrete model types can be translated into the framework of polynomial dynamical systems (PDS), that is, time- and state-discrete dynamical systems over a finite field where the transition function for each variable is given as a polynomial. This allows for using a range of theoretical and computational tools from computer algebra, which results in a powerful computational engine for model construction, parameter estimation, and analysis methods.
I will present the problem of adequate data subsampling for asymptotically consistent parametric estimation of unobservable stochastic differential equations (SDEs) when data are generated by multiscale dynamic systems approximated by these SDEs. The challenge is that the approximation accuracy is scale dependent, and degrades at very small scales. Data from multiscale dynamics systems, namely the Additive Triad model, will be used to illustrate this subsampling problem. I will also indicate the general framework for estimation under this indirect observability and present practical numerical techniques to identify the correct subsampling regime to construct bias-corrected estimators.
In this talk, I will discuss how to discretize space in the stochastic model for chemical reaction-diffusion networks based on the chemical master equation. A system with reaction and diffusion is modeled using a continuous time Markov jump process. Diffusion is described as a jump to the neighboring computational cell with proper spatial discretization. Considering the steady-state mean and variance of the number of molecules of each species in each computational cell, an upper bound for the computational cell size for spatial discretization will be suggested.
Then, I will show conditions for the exponential convergence of concentration to its uniform solution in the corresponding PDE model for chemical reaction-diffusion networks. Conditions obtained from the PDE model give an estimate for the maximal compartment size for space discretization in the stochastic model.
This is a joint work with Hans G. Othmer and Likun Zheng at the University of Minnesota.
Proton transport plays an important role in biological energy transduction and sensory systems. A multi-scale model is introduced to study the proton transport in membrane channels. Quantum dynamics in utilized to model the motion of target particle (protons), while multi-scale treatments are given to surrounding environments in classical mechanics, upon the priorities of interests and importance. All the system components are assembled on an equal footing in a total energy framework, from which the generalized Poisson-Boltzmann, Kohn-Sham and Laplace-Beltrami equations are derived. Simulations are implemented based on the coupled governing equations and compared with experimental results.
The cytoskeleton of dividing cells is highly dynamic with microtubules stochastically transitioning between states of growth and shortening. In this dynamic environment "primitive" polymeric machines can generate force. In eukaryotic cells, chromosomes move to the cell equator by attaching to multiple dynamic microtubules. Attachment is mediated by complex multi-protein scaffolds called kinetochores. In this talk, we present a mathematical model for force generation at the microtubule/kinetochore interface in eukaryotic cells. Movement is modeled using a jump-diffusion process that incorporates both biased diffusion due to microtubule lattice binding by kinetochore elements as well as thermal ratchet forces due to microtubule polymerization against the kinetochore plate. A key result is that kinetochore motors obey nonlinear force-velocity relations. Finally, time permitting, we extend our modeling to explore how polymeric assemblies might facilitate the motility of the circular chromosome of Caulobacter Crescentus.
A new approach to anti-cancer therapy modeling is presented, that reconciles existing observations for the combined action of carboplatin (a Pt-based chemotherapeutic agent) and ABT-737 (a small molecule inhibitor of Bcl-2/xL) against ovarian cancers. To accurately simulate the action of these compounds, an age-structure together with a delay is imposed on proliferating cancer cells, and detailed biochemistry of Bcl-xL-mediated apoptotic pathways is incorporated. 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. The age-structured model gives rise to a 1D hyperbolic Partial Differential Equation which can be reduced to a nonlinear, non-autonomous Delay Differential Equation by projecting along the characteristics. I prove the existence of periodic solutions and derive conditions for their stability. This has clinical implications since it leads to a lower bound for the amount of therapy required to effect a cure.
Waterborne diseases cause over 3.5 million deaths annually, with cholera alone responsible for 3-5 million cases/year and over 100,000 deaths/year. Many waterborne diseases exhibit multiple characteristic timescales or pathways of infection, which can be modeled as direct and indirect transmission. A major public health issue for waterborne diseases involves understanding the modes of transmission in order to improve control and prevention strategies. One question of interest is: given data for an outbreak, can we determine the role and relative importance of direct vs. environmental/waterborne routes of transmission? We examine these issues by exploring the identifiability and parameter estimation of a differential equation model of waterborne disease transmission dynamics. We use a novel differential algebra approach together with several numerical approaches to examine the theoretical and practical identifiability of a waterborne disease model and establish if it is possible to determine the transmission rates from outbreak case data (i.e. whether the transmission rates are identifiable).
Our results show that both direct and environmental transmission routes are identi?able, though they become practically unidenti?able with fast water dynamics. Adding measurements of pathogen shedding or water concentration can improve identi?ability and allow more accurate estimation of waterborne transmission parameters, as well as the basic reproduction number. Parameter estimation for a recent outbreak in Angola suggests that both transmission routes are needed to explain the observed cholera dynamics. I will also discuss some ongoing applications to the current cholera outbreak in Haiti.
Alternans, a long-short alternation of cardiac action potential durations, emerges as a period-doubling bifurcation under rapid pacing. Detecting alternans or bifurcation of the cardiac restitution has been a major task in prevent heart disease. We developed a new stochastic protocol and a regression method to approximate the full dynamics in a time interval. We also discuss the propagation of alternans in 1D cardiac fiber.
In a given population there are usually more than two configurations of sex chromosomes vs. phenotypical gender differentiation. A common configuration is XX chromosomes for females and XY chromosomes for males; variations of this theme (e.g. XY female) occur naturally at low frequencies. There is a vast family of such variations as a result of environmental intervention. In this talk I will present a general formulation for multi-sexual populations using hypermatrices. I will present a method to compare the asymptotic behavior (i.e. who goes to extinction first, if at all) of these competitive dynamical systems of different dimension under certain conditions of biological relevance.
In this two part talk I will summarize work from my Ph.D. thesis, then introduce some ongoing projects as an MBI Postdoctoral Fellow and part of OSU's Aquatic Ecology Laboratory (AEL). The first part of this talk will focus on an infectious disease in house finches (Carpodacus mexicanus) and other wild birds caused by the pathogen Mycoplasma gallisepticum. After introducing the biological system, I will present results from a mathematical model of the immune-pathogen interaction which address the immune system's role in mediating disease symptoms and controlling infection. For the second part of the talk, we will shift gears and consider population dynamics in the context of simple aquatic food webs. I will start off with a brief but general introduction of the biology. I will then present results from model that combines consumer-resource (predator-prey) and host-parasite interactions. These results describe the consequences of some unexpected connections between consumer-resource and host-parasite interactions, as motivated by recent empirical findings from the study of Daphnia (a kind of freshwater zooplankton) their parasites and Daphnia's algal food source. The last part of the talk will introduce two ongoing projects with Stuart Ludsin and others at the AEL. The first of these focuses on the role of hypoxia in shaping disease risk among fish. The second investigates the importance of an aquatic larval insect (phantom midges; family Chaoboridae) in freshwater lakes and reservoirs in Ohio by modelling how they affect the dynamics of those ecosystems.
In this study, we use multi-stage cell lineages model, which include stem cell and multiple progenitor cell stages, to study how feedback regulation from different growth controls homeostasis of tissue growth and generation of a robust spatial stratification. ODE and PDE models have been presented for the multi-stage cell lineages. Our analysis shows how negative feedbacks enhance the stability of steady states and inter-regulation among different growth factors are responsible for developing spatial stratification. We also showed that the feedback on cell cycle from the growth factor is important for forming temporary "stem cell niche" during the development of the tissue.
Infant apnea, defined as a pause in breathing for more than 20 seconds, can lead to oxygen desaturation and the need for resuscitation and assisted ventilation. A recent study has demonstrated that continuously applied stochastic (randomly fluctuating) somatosensory stimulation stabilizes breathing patterns in preterm infants and can reduce apnea by approximately 65%. I hypothesize that stochastic inputs to the respiratory central pattern generator (CPG) increase the dynamic range of the breathing rhythm in neonates. In this talk, I will discuss a proposal to test this hypothesis by using a combination of in vitro electrophysiology and computational modeling to understand the role of noise in the immature respiratory CPG.
Complement Receptor 3 (CR3) and Toll-like Receptor 2 (TLR2) are pattern recognition receptors ex- pressed on the surface of human macrophages. Although these receptors are essential components of the innate immune system, pathogen coordinated crosstalk between them can suppress the production of protective cytokines and promote infection. I will discuss a mathematical model of TLR2/CR3 crosstalk in the context of Francisella tularensis infection.
Most phytoplankton movement is passive and occurs through either sinking/ floating (depending on their density relative to water) or through turbulent diffusion. As they move vertically in the water column, phytoplankton experience gradients in critical environmental factors, such as light intensity and nutrient concentrations. The rate at which phytoplankton move across these gradients can be critical to their persistence and vertical distribution. Grazing can also play a critical role in dictating where in the water column phytoplankton are found. However, theoretical models of critical sinking and diffusion rates either do not explicitly consider grazing loss or treat it as vertically homogenous, thus making it independent of movement. In nature, however, grazing intensity is often vertically heterogeneous. Despite its common occurrence, how such grazing heterogeneity influences critical rates of phytoplankton movement is not well understood. Here we put forth some basic predictions regarding phytoplankton persistence and spatial heterogeneity of grazing, using a reaction-diffusion-advection model. We introduce some new ideas to investigate the combined effects of advection, diffusion, and heterogeneous grazing pressure on the persistence of phytoplankton and to determine the unique number of critical sinking/buoyant rates that are specified by the inclusion of depth dependent mortality that is a result of heterogeneous predation.
PhyloPTE/P (Phylogeny with Path to Event, in People) is a method to bridge the gap between large, gene sequencing based (and, in the near future, other *omic based) studies and phylogenetically-driven approaches developed in other fields, for example epistatic-effect detection using comparison of phylogenetic tree reconstructions for different genes. PhyloPTE/P should be of interest to a wide audience of investigators, including those in biomedical informatics or medical genomics, as well as those in systems biology or evolutionary biology, and serve as a software platform to foster collaboration between the two areas.
Cholera, a waterborne diarrheal disease, is a major public health threat in many parts of the world. It is spread via direct contact with infected individuals as well as indirectly through a contaminated water source. Cholera dynamics can be described by the SIWR model, a modified SIR model incorporating an equation to track the concentration of the pathogen in the water (W) and the additional water transmission pathway. Factors affecting both transmission rates are likely to vary among different populations. Here we consider a multi-patch SIWR model, specifically a system of non-mixing patches sharing a common water source, and explore the effect of heterogeneity in transmission on the spread of the disease, as well as the implications for control.
We formulate and analyze a Markov process modeling the motion of DNA nanomechanical walking devices. We consider a molecular biped restricted to a well-defined one-dimensional track and study its asymptotic behavior. Our analysis allows for the biped legs to be of different molecular composition, and thus to contribute differently to the dynamics. Our main result is a functional central limit theorem for the biped with an explicit formula for the effective diffusivity coefficient in terms of the parameters of the model. A law of large numbers, a recurrence/transience characterization and large deviation estimates are also obtained. Our approach is applicable to a variety of other biological motors such as myosin and motor proteins on polymer filaments. This is joint work with Iddo Ben-Ari and Alexander Roitershtein.
In my talk I will discuss how predator-prey population dynamics can be altered by predator adaptive foraging behavior and/or avoidance strategies of prey. I will consider the Lotka-Volterra predator-prey population model which assumes that interaction strength between predator and prey is fixed. Increasing empirical evidence, however, indicates prey and/or predators change their behavior in response to the presence of the other species. For example, prey decrease their activity, become vigilant, or move to a refuge to avoid predators. Similarly, predator foraging behavior (e.g., prey switching) depends on prey densities. These observations clearly show that interaction strength in the Lotka-Volterra model are not fixed, but is itself a function of population densities. As behavioral effects often operate on a short time scale when compared to a population time scale, it is also not clear if behavioral effects attenuate at the population time scale or not. In my talk, I will show how the games predator and prey play can change predictions of the Lotka-Volterra model.
Molecular motors are either proteins or macromolecular complexes which move along filamentous tracks utilizing some form of input energy. In contrast to their macroscopic counterparts, these natural nano-machines are (i) made of soft matter, (ii) driven by isothermal engines, (iii) far from thermodynamic equilibrium, and (iv) their dynamics is dominated by viscous forces and thermal noise. Mathematical models based on master equation (or, Langevin equation) are the most appropriate for a quantitative theory of their stochastic kinetics. In this talk I'll begin with a brief discussion on the basic theoretical and experimental techniques that are used for studying molecular motors. One characteristic feature of their "directed", albeit noisy, movements is an alternating sequence of pause and translocation. The main aim of this talk is to show how important "hidden" information on the kinetics of such motors can be extracted from the statistics of the durations of pause+translocation. I'll present our recent results on dwell-time distributions of two motors, namely, a member of the kinesin superfamily and the ribosome. I'll also mention the nature of collective spatio-temporal organization of the motors on the track and the effects of their crowding on the dwell time distribution.
Classical population genetics begins with a Markov chain model for the genetic types of the individuals in a finite population and then replaces the discrete model by a diffusion approximation under the assumption that the population is large. "Lookdown" constructions of these models, introduced in work with Peter Donnelly, allow one to retain discrete individuals in the diffusion limit and, in particular, obtain population genealogies coupled to the diffusion approximations. These constructions will be described along with related constructions for spatially distributed populations.
For a given biochemical network of interest it is often desirable to estimate its reaction constants. I shall discuss several different approaches to rate constants estimation from partial trajectory data. The presentation will discuss the LSE as well Bayesian and MLE approaches as well as possible conditions on the data process which guarantee identifiability and estimators consistency. We shall also consider ways of approximating the likelihood of a partially observed biochemical network with certain other likelihoods (e.g., Gaussian) for which inference problem is simplified.
The focus of this talk is to provide a basic but informative answer to the ecological question: How do habitat disturbances and fragmentation affect species persistence and diversity? In order to answer this question, I will develop and analyze a deterministic metapopulation model that takes into account a time-dependent patchy environment. The variability of the patchy-environment could be thought to be due to environmental changes. I will demonstrate, accordingly to the model, the effects of spatial variations on persistence and coexistence of two competing species. Also, I will compare the analytical results of the deterministic model with simulations of a stochastic version of the model.
The Euler-Poisson system is a fundamental two-fluid model in physics. It describes the motion of ions and electrons coupled through their self-consistent electric field. For the three-dimensional case Guo '98 first constructed a smooth global solution around the constant-density equilibrium by using a Klein-Gordon dispersive effect. It has been long conjectured that same results should also hold in the two-dimensional case. The main issues in 2D are slow dispersion, quasilinearity and certain nonlocal obstructions in the nonlinearity. I will discuss some recent advances which lead to the complete resolution of this conjecture.
Rhythmic behaviors in neural systems often combine features of limit cycle dynamics (stability and periodicity) with features of near heteroclinic or near homoclinic cycle dynamics (extended dwell times in localized regions of phase space). Proximity of a limit cycle to one or more saddle equilibria can have a profound effect on the timing of trajectory components and response to both fast and slow perturbations, providing a possible mechanism for adaptive control of rhythmic motions. Reyn showed that for a planar dynamical system with a stable heteroclinic cycle (or separatrix polygon), small perturbations satisfying a net inflow condition will generically give rise to a stable limit cycle (Reyn, 1980; Guckenheimer and Holmes, 1983). Here we consider the asymptotic behavior of the infinitesimal phase response curve (iPRC) for examples of two systems satisfying Reyn's inflow criterion, (i) a smooth system with a chain of four hyperbolic saddle points and (ii) a piecewise linear system corresponding to local linearization of the smooth system about its saddle points. For system (ii), we obtain exact expressions for the limit cycle and the iPRC as a function of a parameter $\mu>0$ representing the distance from a heteroclinic bifurcation point. In the $\mu\to 0$ limit, we find that perturbations parallel to the unstable eigenvector direction in a piecewise linear region lead to divergent phase response, as previously observed (Brown, Moehlis and Holmes, 2004). In contrast to previous work, we find that perturbations parallel to the stable eigenvector direction can lead to either divergent or convergent phase response, depending on the phase at which the perturbation occurs. In the smooth system (i), we show numerical evidence of qualitatively similar phase specific sensitivity to perturbation. Having the exact expression for the iPRC for the piecewise linear system allows us to investigate its stability under diffusive couplin g. In addition, we qualitatively compare iPRCs obtained for systems (i) and (ii) to iPRCs for the Morris-Lecar equations near a bifurcation from limit cycles to a saddle-homoclinic orbit. Joint work with K. Shaw, Y. Park, and H. Chiel.
In this talk, we shall discuss the stochastic modeling of chemotaxis and derive an anisotropic diffusion chemotaxis models from the Langevin stochastic equations which takes into account the movement persistence. The various techniques, such as mean-filed theory, minimization principle, moment closure and scaling argument, will be used to carry out the results. In addition the stochastic simulations exhibiting the chemotactic behavior will be presented.
Currently, efforts are underway to develop vaccines for several viral infections, including Human Immunodeficiency Virus type 1 (HIV-1) and Herpes Simplex Virus type 2 (HSV-2). In this talk, I will present the results of mathematical models that address vaccination strategies for these viral infections. I will demonstrate the use of these results to predict the impact of prevention efforts as well as to assess the mechanisms of virus-host interactions. I will also show how such studies can guide the development of future vaccines and other therapeutic interventions.
My primary goal in this talk will be to provide a summary of some current research interests with the hope of stimulating potential collaborations with other faculty and postdocs during my time at MBI. The general theme will concern the effective statistical description of a complex microbiological system consisting of a number of individual dynamical components with some structural interactions as well as with stochastic noise sources. I will briefly touch on the examples of molecular motors and swimming microorganisms, then describe in some more detail a recent study of synchrony in stochastically driven neuronal networks.
A brief introduction is presented to stochastic differential equations (SDEs) in mathematical biology. In particular, a procedure is described for deriving accurate SDE models for randomly varying biological dynamical systems. Next, several research projects involving SDE models in biology are briefly described. Specifically summarized are: an investigation of a schistosomiasis infection with biological control, a derivation of stochastic partial differential equations for size-and age-structured populations, and the development of SDE models for biological diversity. Finally, current/future work is pointed out.
A mathematical model which incorporates the spatial dispersal and interaction dynamics of mistletoes and birds is derived and studied to gain insights of the spatial heterogeneity in abundance of mistletoes. Fickian diffusion and chemotaxis are used to model the random movement of birds and the aggregation of birds due to the attraction of mistletoes respectively. The spread of mistletoes by birds is expressed by a convolution integral with a dispersal kernel. Two different types of kernel functions are used to study the model, one is Dirac delta function which reflects one extreme case that the spread behavior is local, and the other one is a general non-negative symmetric function which describes the nonlocal spread of mistletoes. When the kernel function is taken as the Dirac delta function, the threshold condition for the existence of mistletoes is given and explored in term of parameters. For the general non-negative symmetric kernel case, we prove the existence and stability of non-constant equilibrium solutions. Numerical simulations are conducted by taking specific forms of kernel functions. Our study shows that the spatial heterogeneous patterns of the mistletoes are related to the specific dispersal pattern of the birds which carry mistletoe seeds.
Remarkable progress in advanced microscopy has yielded unprecedented access to a path-wise observation of the diffusive behavior of bacteria, viruses, organelles and various invasive particulates in biological fluids. Upon inspection of the data one immediately notes, "That's not Brownian motion!" Perhaps not surprisingly, media such as human mucus are highly heterogeneous and exhibit significant viscoelastic properties. In this talk, I will provide a survey of recent experimental observations along with mathematical models that are currently in use. Wherever possible I will point out open problems in this burgeoning area of research.
Classical mathematical formulation of the dynamics of chemical reaction systems involves setting up and analyzing a system of ODEs, or PDEs if spatial effects are considered. However, a system may be sensitive to the stochasticity inherent in the mechanism of chemical reactions, for example due to having small numbers of molecules, or reaction rates which vary over several orders of magnitude. We consider such a reaction system in a cellular environment, and also impose a 'global' cell division mechanism, which adds noise to the concentrations of chemical species along a given lineage, and find parameter regimes for which this produces a qualitative change in the dynamics. We model these reaction and division processes as Jump Markov Processes, and discuss some toy models in which the stochasticity can allow the system to exhibit behavior that is not possible with a deterministic formulation. One such behavior is bistability, for which we find two processes that have similar macroscopic signatures but whose underlying causes are fundamentally different; one such case leads to the Large-Deviation theory of Freidlin and Wentzell. Such bistability is characteristic of many gene expression systems that effectively incorporate an ON/OFF switch, but the framework is very general and is applicable in other areas, such as population genetics, where bistability may represent alternating dominance of allelic types in a population. This is joint work with Lea Popovic of Concordia University (Montreal).
This talk will attempt to provide a synthesis of the topics discussed at the workshop and to distill some central themes that point to opportunities and challenges in the field.
Complex gene regulatory networks underlie many cellular and developmental processes. While a variety of experimental approaches can be used to discover how genes interact, few biological systems have been systematically evaluated to the extent required for an experimental definition of the underlying network. Therefore, the development of computational methods that can use limited experimental data would provide a useful tool to extract relevant information from existing data, identify unexpected regulatory relationships, and prioritize future experiments.
We have developed a hybrid modeling method that combines two existing methods: a recently development tool from algebraic geometry and a traditional statistical tool. We reverse engineered a mathematical model from time-course gene expression data collected from wildtype C. elegans embryos and compared it to an existing knowledge-driven biological model based on the same data set. We show that the mathematical model predicts more interactions observed in subsequent perturbation experiments than does the knowledge-driven model. It provides new insights into the function of a key transcriptional regulator and identifies distinctive activities of two genes previously thought to be redundant. This work provides a strong example that data-driven mathematical models can complement knowledge-driven models to identify non-intuitive network relationships and to guide future experiments.