This presentation will explore in some detail group-based models and their invariants. (The content is drawn from the joint work with Sonja Petrovic, UIC , email@example.com)
We develop mathematical models that describe the interaction between the immune cells and the Human Immunodeficiency Virus (HIV). First, we consider a model that includes drug treatment whose efficacy determines the prognosis of the disease in terms of the parameters that describe the threshold and actual number of virions produced. We shall determine the model efficacy and show that when the efficacy is below the model efficacy, the CD4 T cell count decreases to a low level that cannot sustain an effective immune response. On the other hand, at drug efficacy levels greater than or equal to the model efficacy, the CD4 T cell count increases to levels sufficient to support an effective immune response but this state is unstable and a small residual infection remains in the body which is suppressed by continuously taking the medication. Secondly, we investigate the interaction between two infectious HIV strains and show how this affects disease progression.
Spatial segregation among life cycle stages has been observed in many stage-structured species, both in homogeneous and heterogeneous environments. We investigate density dependent dispersal of life cycle stages as a mechanism responsible for this separation by using stage-structured, integrodifference equation (IDE) models that incorporate density dependent dispersal kernels. After investigating mechanisms that can lead to spatial patterns in two dimensional Juvenile-Adult IDE models, we construct spatial models to describe the population dynamics of the flour beetle species T. castaneum, T. confusum and T. brevicornis and use them to assess density dependent dispersal mechanisms that are able to explain spatial formations observed in these species.
The goal of this talk is to describe the analysis of a specific aspect of transient dynamics not covered by previous theory. The question addressed is whether one component of a perturbed solution to a system of differential equations can overtake the corresponding component of a reference solution as both converge to a stable node at the origin, given that the perturbed solution was initially farther away and that both solutions are nonnegative for all time. We call this phenomenon tolerance, for its relation to a biological effect.
Using geometric arguments it is shown that tolerance will exist in generic linear systems with a complete set of eigenvectors and in excitable nonlinear systems. A notion of inhibition is also defined that may constrain the regions in phase space where the possibility of tolerance arises in general systems. However, these general existence theorems do not yield an assessment of tolerance for specific initial conditions. To address that issue, some analytical tools were developed to determine if particular perturbed and reference solution initial conditions will exhibit tolerance.
Thyroid hormone regulation is a classic example of biological feedback control, and thyroid disorders such as hypothyroidism affect more than 300 million people worldwide. We developed a physiologically based, ordinary differential equation model of the human hypothalamic-pituitary-thyroid axis, in order to address several clinical applications. The model is broken into two major components-- the thyroid and brain submodels, each quantified from human clinical data. We combined these two submodels to form a complete closed loop model, which we validated using additional independent clinical data. Using the closed-loop model, we address several applications in replacement thyroid hormone (L-T4) bioequivalence (equivalence between different brands/preparations of L-T4), circadian rhythms, and thyroid cancer.
While alternans in a single cardiac cell appears through a simple period-doubling bifurcation, in extended tissue the exact nature of the bifurcation is unclear. In particular, the phase of alternans can exhibit wave-like spatial dependence, either stationary or travelling, which is known as discordant alternans. We study these phenomena in simple cardiac models through a modulation equation proposed by Echebarria-Karma. We perform a bifurcation analysis for their modulation equation. We also find that for some extreme range of parameters, there are chaotic solutions. Chaotic waves in recent years have been regarded to be closely related to dreadful cardiac arrhythmia. Proceeding work illustrates some chaotic phenomena in two- or three-dimensional space, for instance spiral and scroll waves. We show the existence of chaotic waves in one dimension, which may provide a different mechanism accounting for the instabilities in cardiac dynamics.
The transcription factor NF-kB is critical to the control of responses to cellular stress, inter- and intracellular signaling, cell growth, survival and apoptosis. At rest, NF-kB is sequestered by its inhibitor IkB in the cytoplasm. Upon stimulation, such as tumor necrosis factor $alpha$ (TNF$alpha$), NF-kB gets released from IkB and translocates to the nucleus and regulates genes transcription, including regulating transcription of gene IkB. Then the newly synthesized IkB, on the other hand, removes NF-kB from the nucleus. Hence, NF-kB and IkB form a negative feedback loop. Negative feedback loop is often associated to oscillations. Indeed, oscillations of the concentration of nuclear NF-kB has been observed both at population and single cell levels by Hoffmann et al. and Nelson et al. respectively. Ashall et al. recently reported that different frequencies of the oscillations leads to different gene expression. It has been reported in many works that NF-kB signaling pathway may interact with many other signaling pathways, including P53 signaling pathway. So it is important to understand that how the frequencies of NF-kB oscillations may be influenced by its interacting signals. However, the existence and mechanism of those potential interactions are not clear. In this talk, I study this issue by considering the pathway subjected to two types of putative signals: sinusoid and pulsate signals. A rich variety of nonlinear dynamics can be observed. In addition, we consider possible cell-cell communication by secretion of TNF$alpha$.
The 2D Boussinesq system is potentially relevant to the study of atmospheric and oceanographic turbulence, as well as other astrophysical situations where rotation and stratification play a dominant role. In fluid mechanics, the 2D Boussinesq system is commonly used in the field of buoyancy-driven flow. It describes the motion of incompressible inhomogeneous viscous fluid subject to convective heat transfer under the influence of gravitational force. It is well-known that the 2D Boussinesq equations are closely related to 3D Euler or Navier-Stokes equations for incompressible flow, and it shares a similar vortex stretching effect as that in the 3D incompressible flow. In fact, in vortex formulation, the 2D inviscid Boussinesq equations are formally identical to the 3D incompressible Euler equations for axisymmetric swirling flow. Therefore, the qualitative behaviors of the solutions to the two systems are expected to be identical. Better understanding of the 2D Boussinesq system will undoubtedly shed light on the understanding of 3D flows. In this talk, I will discuss some recent results concerning global existence, uniqueness and asymptotic behavior of classical solutions to initial boundary value problems for 2D Boussinesq equations with partial viscosity terms on bounded domains for large initial data.
Antibiotic resistant organisms (ARO) pose an increasing serious threat in hospitals. Factors which contribute to the spread of ARO in hospitals are poor immune system of most patients, close living quarters, and contact with health care workers (HCWs). One of the most life threatening ARO is methicillin-resistant staphylococcus aureus (MRSA).
In this talk, we will introduce a new mathematical model which focuses on the evolution of two MRSA bacterial strains: drug- resistant and non-drug resistant within population of patients and HCWs in a single hospital. We will introduce two important quantities: the threshold at which time the drug treatment is administered to patients, and time duration of drug-treatment. We will investigate the role of the amount, threshold and time duration of drug treatment on reducing the non-resistant bacteria in patients.
Simulations of the model show that as the amount of drug given to the patient is increased the drug-resistant bacteria significantly decreases, and as the treatment period is increased from one week to two weeks, the drug-resistant bacteria also decreases. Furthermore, we will demonstrate that the choice of the threshold has little influence on the outcome level of drug-resistant bacteria.
Bayesian hierarchical modeling pervades modern statistical research. In this talk, I will discuss two examples of these models in molecular evolution. The first describes the problem of non-vertical evolution and how to adequately model this process in a formal statistical framework. In particular, I will describe methods to hierarchically model branch length and topological incongruence between vertical trees inferred from multiple loci. After this, I will describe a hierarchical model to infer the population dynamic characteristics of large North American mammal populations.
Sequence variations altering protein function are a fundamental driving force in evolution. While the rapid proliferation of whole-genome sequence data should provide unprecedented insight into the evolution of protein function, especially in bacterial organisms for which thousands of complete genome sequences will soon be available, there are practical obstacles to achieving this potential. New methods to assess the functional similarity between proteins are needed to overcome these obstacles. We have evaluated an orthology-based method to group bacterial proteins based on likely similarity in biochemical function. The foundation of this method involves using the occurrence of multiple homologous proteins in a single microbial organism as evidence of functional diversification among those homologs. The resulting groups of functionally similar proteins are called Classes of Reciprocal Sequence Homologs (CRSHs). Different CRSHs vary tremendously in their degree of sequence conservation in widely diverged organisms (ranging from 25-70% identity). However, once this variation is taken into account, a simple model using only the mean evolutionary distance between pairs of microbial organisms accounts for the vast majority of the sequence differences within each CRSH. The likely functional similarity of the proteins in each CRSH is also supported by preservation of gene neighborhood in remotely related microbial organisms, which in turn is strongly correlated with transcriptional co-regulation in the model bacterium E. coli. Furthermore, a CRSH-based metric achieves 30% accuracy in predicting manually validated physical inter-protein interactions in E. coli. A webserver at www.orthology.org provides access to the CRSHs along with related quality-control, gene-neighborhood, and annotation information.
Phylogenetic methods are useful for finding the origin and tracing the character evolution of infectious disease. Here I will present case studies from H5N1 bird flu and H1N1 swine flu, where non-parametric maximum parsimony methods have been used to determine the geographic spread of bird flu and the geographic and animal host origins of the 2009 H1N1 swine flu.
Chronic wounds represent a major public health problem affecting 6.5 million people in the United States. Ischemia, primarily caused by peripheral artery diseases, represents a major complicating factor in cutaneous wound healing. In this talk, we present a mathematical model of ischemic dermal wounds. The model consists of a coupled system of partial differential equations in the partially healed region, with the wound boundary as a free boundary. The extracellular matrix (ECM) is assumed to be viscoelastic, and the free boundary moves with the velocity of the ECM at the boundary. The model equations involve the concentration of oxygen, PDGF and VEGF, the densities of macrophages, fibroblasts, capillary tips and sprouts, and the density and velocity of the ECM. Simulations of the model demonstrate how ischemic conditions may limit macrophage recruitment to the wound-site and impair wound closure. The results are in general agreement with experimental findings.
Deciphering a transcriptional regulatory code: modeling short- range repression in the Drosophila embryo
Ovarian cancers remain difficult to treat due to the emergence of drug resistance, which may be conferred in part by the expression of anti-apoptotic members of the Bcl-family of proteins. ABT-737 is a recently developed small molecule inhibitor of these proteins, currently in stages I/II of clinical trial. In recent experiments, ABT-737 co-administered with Carboplatin, a Pt-based chemotherapeutic drug used to treat ovarian carcinomas, was found to act in a synergistic manner on cancer cells in vitro. Here we develop a mathematical model to investigate the molecular basis of this synergism. The model is built up of two modules, simulating treatment by each compound as a single agent, and is calibrated versus in vitro cell growth inhibition data. These two components are then integrated to represent combination therapy. Numerical simulations indicate that Carboplatin sensitizes the cells to ABT-737 therapy, due to a diminished ability of cells to withstand DNA damage under lowered Bcl-xL levels. The model predicts the existence of a threshold, so that if intracellular Bcl-xL falls below this, cells with relatively low DNA damage are unable to evade apoptosis. Further, simulations indicate that co-treatment and post-treatment with ABT-737 is an optimal strategy to exploit the synergism of the two drugs. Pre-treatment however displays poor results in comparison, due to their proposed mechanism of action. Such modeling, if developed in conjunction with experimentation, can thus have far reaching effects in the field of anti-cancer drug development.
In this talk, we discuss the statistical properties of such approximation and their influence on the accuracy of various spectral clustering algorithms. We found that the perturbation of spectrum due to subsampling could lead to large discrepancy among clustering results based on different subsamples. In order to provide accurate and stable clustering results for large datasets, we propose a method to combine multiple sub-samples using data spectroscopic clustering and the Nystrom extension. In addition, we propose a sparse approximation of the eigenvectors to further speed up the computation. Simulation and experiments on real data sets showed that this multi-sample approach is fast and accuracy.
Present docking methodologies simulate only one single ligand at a time during docking process. In reality, the molecular recognition process always involves multiple molecular species. Typical protein-ligand interactions are, for example, substrate and cofactor in catalytic cycle; metal ion coordination together with ligand(s); and ligand binding with water molecules. In order to simulate the real molecular binding processes, we propose a novel multiple ligand simultaneous docking (MLSD) strategy which can deal with all the above processes, vastly improving docking sampling and binding free energy scoring. The work also compares two search strategies: Lamarckian Genetic Algorithm and Particle Swarm Optimization, which have respective advantages depending on the specific systems. The methodology proves robust through systematic testing against several diverse model systems: E. coli PNP complex with two substrates, SHP2NSH2 complex with two peptides and Bcl-xL complex with ABT-737 fragments. In all cases, the final correct docking poses and relative binding free energies were obtained. In PNP case, the simulations also capture the binding intermediates and reveal the binding dynamics during the recognition processes, which is consistent with the proposed enzymatic mechanism. In the other two cases, conventional single ligand docking fails due to energetic and dynamic coupling among ligands, whereas MLSD results in the correct binding modes. These three cases also represent potential applications in the areas of exploring enzymatic mechanism, interpreting noisy X-ray crystallographic maps, and aiding fragment-based drug design, respectively.
One of the major challenges facing researchers studying complex biological systems is integration of data from omics platforms. Omic-scale data include DNA variations, transcriptom profiles, and RAomics. Selection of an appropriate approach for a data integration task is problem dependent, primarily dictated by the information contained in the data. In situations where modeling of multiple raw data sets jointly might be extremely challenging due to their vast differences, rankings from each data set would provide a commonality based on which results could be integrated. Because the underlying spaces of genes (elements) from which each ranked list come from are likely to be different, taking the underlying spaces into consideration is paramount, as failure to do so would lead to inefficient use of data and might render biases and/or sub-optimal results. However, this important aspect is usually overlooked in the literature on rank-based integration methods for omic-scale data. Nevertheless, although no assumptions about the underlying spaces are explicitly stated, carefully dissections of the algorithms reveal implicit assumptions about the spaces regardless of whether such assumptions are valid for a particular integration problem. In this talk, I will discuss a number of space oriented methods, including Markov chain based heuristic algorithms and optimization based cross entropy Monte Carlo methods for integrating ranking data. Examples will be shown to dissect the methods and to demonstrate the effects of assumptions about the underlying spaces.
Budding yeast "Saccharomyces cerevisiae" is a model system for studying cell polarization, a fundamental symmetry breaking process underlying cell physiology.In this talk, I will introduce mathematical models for the establishment and maintenance of yeast cell polarization induced by mating pheromone. Simulation results including cell morphological changes will be presented and compared to experimental data. Roles of cell membrane dynamics such as endocytosis, exocytosis and the roles the microdomains on cell membrane (lipid raft) will also be discussed.
TRAP (trp RNA-binding Attenuation Protein) forms oligomeric protein rings that can bind to cellular tryptophan (Trp). Once bound to Trp, TRAP becomes activated for binding to a conserved RNA sequence in the 5'-leader region of the trp operon, whose genes encodes a number of enzymes involved in the biosynthesis of the amino acid. Binding of TRAP to RNA prevents transcription of the trp operon, lowering Trp production. In the absence of Trp, TRAP cannot bind RNA, and the 5' leader adopts a conformation that allows for transcription of the gene and increased production of Trp.
Anti-TRAP is an oligomeric protein that can bind to Trp-activated TRAP and prevent it from binding RNA. Anti-TRAP production in cells is also responsive to cellular levels of Trp. Anti-TRAP is found to equilibrate between different oligomeric states: trimers (AT3) and dodecamers (AT12). It is the trimeric form of the protein, AT3, which can bind and inhibits TRAP. The equilibrium between AT3 and AT12 is determined by concentration and pH. The pH dependence of the equilibrium between "active" AT3 and "inactive" AT12 hints to a new mechanism for responding to environmental changes.
Biophysical studies (spectroscopic, thermodynamic, kinetic) of the process and effect of ligand binding to allosteric gene-regulatory macromolecules provide unique as well as universal insights into the role of structure and dynamics in the regulation of gene expression by small molecule ligands.
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Here we show how covariance can be used to exploit pathway structure without biasing selection in favor of known pathways. Starting with a simple model for differences of expression in a paired-subject experiment, we show that, for large, highly coordinated gene networks, the eigenvectors of the covariance matrix may contain substantial information about which genes are relevant to the differential processes. A similar type covariance structure is identified for gene expression at different epochs in the reproductive cycle of rainbow trout, and a robust method for feature selection, called SCOOP, is developed to select genes that naturally describe reproductive processes and the suggest implication of genes with previously unknown function.
This is joint work with Yushi Liu and Bill Hayton.
Cells are living organisms, and it is natural to think that behavioral or shape changes of a cell bear information about the underlying mechanisms that generate these changes. Reading the cell motion, namely, understanding these underlying biophysical and mechanochemical mechanisms of behavioral and shape changes is of paramount importance. This is analogous to the examination of patients, analysis of their symptoms, behavioral changes by physicians for possible causes behind their pathologies. The mathematical models we developed play the role of physicians or physical exams in diagnosis of the pathologies for cells instead of human subjects. Thus, the technique transforms the simplest physical information, i.e. cell position, into practical and clinically usable form.