An overview will be given of recent research of the speaker on problems in the theory of DNA elasticity and the regulation of gene expression. A brief outline of the theory of the elastic rod model for DNA will focus on methods for solving the equations of mechanical equilibrium in cases when self-contact is present and on conditions for determining the stability of equilibrium configurations. Majority of the talk will be concerned with applications of a base pair level theory of DNA elasticity that enables one to incorporate the effects of nucleotide composition and negative charge of DNA in mesoscale modeling of complex protein DNA assemblies. Examples include models of the Lac repressor mediated DNA loop and the Class I CAP dependent transcription activation complex, which are well supported by available data and yield experimentally verifiable conclusions about the influence of DNA deformability on the mechanism of regulation of the Lac operon by LacR and CAP. The talk will conclude with a discussion of the implications of obtained results for the role of DNA deformability in regulation of transcription.
Imaging of the heart has proven to be one of the most challenging applications of MRI technology. MR data acquisition typically takes on the order of seconds to minutes to obtain high resolution images. Irregular cardiac and respiratory motion during this period can give rise to significant artifacts and can render images non-diagnostic. A number of different strategies have evolved to synchronize MR image acquistion with physiological motion. More recently, imaging times have been reduced to under 100msec, fast enough to freeze most physiological motion and permit real-time imaging without cardiac or respiratory synchronization. Strategies for physiological synchronization and real-time MRI will be reviewed in the presentation.
We study the role of cross-diffusion in the existence of spatially non-constant periodic solutions for a Lotka-Volterra competition system for three species. We will show that by choosing cross-diffusion coefficients in a cyclic way Hopf-bifurcation may arise. We characterize the stability of these solutions when the cross diffusion coefficients are small or large compared with the competition coefficients.
Cardiovascular medicine has witnessed tremendous advances in last few years thanks to high-resolution, multidimensional noninvasive technology. Subha V. Raman, MD, MS will provide an overview of recent advances in these fields particularly as they relate to novel approaches in the diagnosis and treatment of cardiovascular disease. Her research efforts involve clinical applications as well technology development drawing on her expertise in cardiovascular medicine and electrical engineering. She will dicuss some of her work in image analysis and discuss some of the computational and technical challenges that require interdisciplinary solutions to facilitate transfer of technological advances to better patient care.
The Weibull distribution is used to model the vertical distribution of insects under the following activities: natural flight without any artificial stimulus, resting behaviour, and response to trapping involving colour attractants and odour baits. Formulae are derived for determining the mean heights at which insect flight, resting and trapping tend to occur. The model is tested on data from three sources: coleoptera in natural flight over Tallulah, Louisiana; Glossina palpalis palpalis at rest during the dry season in Nigeria; and catches of Glossina spp., in Rwanda, by traps placed at varying heights above the ground. Results show that different species of insects tend to fly, rest or be trapped at heights which are characteristic of the species.
Many patterns of cell and tissue organization are specified during development by gradients of morphogens, substances that assign different cell fates at different concentrations. One of the central questions in cell and developmental biology is to identify mechanisms by which the morphogen gradient systems might achieve robustness to ensure reproducible embryonic patterns despite genetic or environmental fluctuations.
Recently, through computations and analysis of various bio-chemical models and examination of old and new experimental data, we found a set of of new mechanisms for enhancing robustness of cell-cell signaling through non-signaling cell surface molecules (e.g., HSPG). In addition, we examined the roles of diffusive ligands (e.g., Sog) on the formation and robustness of BMP (Bone Morphogenetic Protein) gradients in the Drosophila embryo. In this talk, I shall also discuss some mathematical and computational challenges associated with such study, and present a new class of numerical algorithms for reaction-diffusion equations arising from biological models.
The form of the amino acid (AA) table, and the relationship between genotype and phenotype that it implies, are at the basis of all processes of evolution. We have developed a network view of the AA table in which every codon is a node and every edge is a mutation. We have used the measures this view generates to study the process of affinity maturation in the immune reaction. In this enclosed process of selection, B lymphocytes triggered by a pathogen undergo rapid mutation, proliferation and death over a short period of time. This process leads to the selection of those cells which produce high affinity receptors to the pathogen. Due to the short time scale we expect the process to be dependent on the connectivity of the AA network. We looked at the germline DNA of two light chain types that undergo mutation and selection, and as a control at CD8 - a light chain homologue that does not mutate. Our results suggest three new ideas about selection: First, the chemical properties shared by groups of AA (i.e. "traits") and the potential to change them are a meaningful signal for selection. Second, we found that while all light chains have evolved to generate variable progeny under high rates of mutation. k and l gene families differ in the extent to which they will risk their potential viability. Finally, the existence of a transition bias in mutations means that not all movements on the AA network are equal, dividing it into Transition Neighborhoods, the codons of which tend to mutate into each other. We have found an over expression of codons belonging to a single neighborhood in those regions of the light chain that contact antigen. This is another method to balance viability and variability as it constrains the extent to which mutations will change the structure of the light chain.
Although in the broadly defined genetic algebra, multiplication suggests a forward direction from parents to progeny, when looking from the reverse direction, it also suggests to us a new algebraic structure - coalgebraic structure, which we call genetic coalgebras. It is not the dual coalgebraic structure and can be used in the construction of phylogenetic trees. Mathematically, to construct phylogenetic trees means we need to solve equations x^[n]=a, or x^(n)=a. It is generally impossible to solve these equations in algebras. However, we can solve them in coalgebras in the sense of tracing back for their ancestors. A thorough exploration of coalgebraic structure in genetics is apparently necessary. Here, we develop a theoretical framework of the coalgebraic structure of genetics. From biological viewpoint, we defined various fundamental concepts and examined their elementary properties that contain genetic significance. Mathematically, by genetic coalgebra, we mean any coalgebra that occurs in genetics. They are generally noncoassociative and without counit; and in the case of non-sex-linked inheritance, they are cocommutative. Each coalgebra with genetic realization has a baric property. We have also discussed the methods to construct new genetic coalgebras, including cocommutative duplication, the tensor product, linear combinations and the skew linear map, which allow us to describe complex genetic traits. We also put forward certain theorems that state the relationship between gametic coalgebra and gametic algebra. By Brower's theorem in topology, we prove the existence of equilibrium state for the in-evolution operator. (The paper is available http://math.asu.edu/~mbe/, Vol.1, 2. pp.243-266)
Note: Joint work with Bai-Lian Li, Department of Botany and Plant Sciences, University of California, Riverside.
The typical large scale behavior of an asymmetric particle system is described by a Hamilton-Jacobi equation, in the sense that the random evolution converges to a deterministic solution of such an equation in a space-time scaling limit. This talk describes such limits and the fluctuations from the limit.
It turns out that for asymmetric systems dynamical noise occurs at a scale smaller than the diffusive scale that is common in central limit type results. Specific models from the field of interacting particle systems discussed here are the exclusion process, Hammersley's process, independent random walks, and the random average process.
This talk will concentrate on the fluctuation part of the general picture given by Timo Seppalainen in the preceding talk. A class of systems will be considered, including the exclusion process, independent random walks, the zero range process, and a deposition model. The fluctuation of the current of particles is computable in the diffusive (i.e. square root of time-) scaling within this class.
The results are strongly connected to the behavior of the so-called second class particle, an object coming from probabilistic coupling of two processes. Therefore some properties of this particle are also derived. The arguments support the idea that fluctuations are transported from the initial configuration, while the dynamical noise is not present on this diffusive time-scale.
Phylogenetic trees are representations of the evolutionary history of groups of organisms. The leaves of these graphs represent biological species (or a higher level taxonomic unit) and the internal nodes are interpreted as hypothetical evolutionary ancestors. Although it was considered relevant only to taxonomic and evolutionary studies, phylogenetics is becoming a critical tool for numerous disciplines in biology and medicine, providing a unique organizing framework for biological variation and predictive analysis.
Several phylogenetic methods aim to find the optimal phylogenetic trees from the space of all possible trees, evaluating the hypotheses with an objective function. Thus, this combinatorial optimization problem (phylogenetic tree search) is compute bound and must be approached through heuristics for large and biologically interesting datasets. Large phylogenetic problems are of interest to biologists because they provide a rich context of phenotypes and genotypes. Here, I will approach the problem of analyzing datasets with large number of species (between several hundreds and several thousands) using recently developed tree search algorithms and diverse parallelization strategies using Beowulf clusters for parallel computing.
Identifying evolutionarily conserved blocks in orthologous genomic sequences is an effective way to detect regulatory elements. In this study, with the aim of elucidating the architecture of the regulatory network, we systematically estimated the degree of conservation of the upstream sequences of 3,750 humanmouse ortholoogue pairs along 8-kb stretches. We found that the genes with high upstream conservation are predominantly transcription factor (TF) genes. In particular, developmental process-related TF genes showed significantly higher conservation of the upstream sequences than other TF genes. Such extreme upstream conservation of the developmental process-related TF genes suggests that the regulatory networks involved with developmental processes have been evolutionarily well conserved in both human and mouse lineages.
Work done in collaboration with Hisakazu Iwama and Takashi Gojobori.
In the last twenty years, coalescent theory has been developed into a powerful analytical tool for population genetics. This theory is especially significant with the rapid accumulation of DNA sequence data. First formulation in the seminal work of Kingman in 1982, coalescent theory offers various sample-based and highly efficient statistical methods for analyzing molecular data such as DNA sequence samples. Mathematically, coalescent theory studies stochastic processes leading to the most recent common ancestor (MRCA) from a sample under various coalescent models. If one thinks of the more commonly studied branching processes as stochastic models of generating random trees from their roots, coalescent processes can be viewed as the inverse processes which recover random trees from their leaves. In more elaborated versions crucial for population genetics, coalescent processes are usually superimposed with mutation processes. These mutation processes can be viewed as independent Poisson processes running over the random trees generated by the coalescent processes with the edge lengths of the random trees serving as the time scale for the mutation processes. In our this research, we introduce a colored coalescent process which recovers random colored genealogical trees. Here a color genealogical tree has its vertices colored black or white. Moving backward along the colored genealogical tree, the color of vertices may change only when two vertice coalesce. The rule that governs the change of color involves a parameter x. When this parameter takes value of one half, the colored coalescent process can be derived from a variant of the Wright-Fisher model for a haploid population in population genetics. Explicit computations of the expectation and the cumulative distribution function of the coalescent time are carried out. For example, our calculation shows that when x=1/2, for a sample of n colored individuals, the expected time for the colored coalescent process to reach a black MRAC or a white MRAC, respectively, is 3 - 2/n. On the other hand, the expected time for the colored coalescent process to reach a MRAC, either black or white, is 2 - 2/n, which is the same as that for the standard Kingman coalescent process. This colored coalescent process with a color mutation process superimposed is also studied in explicit details. (The paper is available at arXiv:math.PR/0410514 v1)
Note: Joint work with Xiao-Song Lin, Department of Mathematics, University of California, Riverside.
Age-related slowing hypotheses were evaluated with 305 participants, ranging in age from 4 to 95 years. Various perception and motor tasks, including spontaneous and synchronize-continue tapping were employed to assess different models derived, respectively, from interval time and entrainment theory. Spontaneous motor tapping and judgments of preferred sequence tempi showed different age-related regions of preferred tapping, with younger participants favoring faster tempi than adults. Accuracy and variability of continuation tapping also varied systematically with age in a manner consistent with age-related slowing, especially in children. These findings were in accord with the entrainment hypothesis that people rely on preferred internal periods which change over the lifespan. This approach correctly predicts age-related changes in error and variability and leads to a modification of Weber's Law, the Restricted Weber Function.
A long-standing issue in statistical approaches to X-ray crystallography phase estimation is to solve a set of entropy maximization problems efficiently during the estimation. Each of these entropy maximization problems is a semi-infinite convex program and can be solved in a finite dual space by using a standard Newton method. However, the Newton method is too expensive since it requires O (n3) floating-point operations per iteration, where n corresponds to the number of the phases to be estimated. Other less expensive methods have been used but they cannot guarantee fast convergence. In this talk, I will describe a fast Newton method my colleagues and I have recently developed for solving the entropy maximization problems. The method uses the Sherman-Morrison-Woodbury Formula and the Fast Fourier Transform to compute the Newton step and requires only O (n log n) floating-point operations per iteration. On the other hand, it can converge in the same rate as the standard Newton. I will show how the method works and present some numerical results.
Having puzzled the scientists for decades, the problem of protein folding still remains a grand challenge of modern science. While it is a fundamental problem in biology, its solution requires knowledge beyond the traditional field of biology and has appealed research across many other disciplines including mathematics, computer science, physics, and chemistry. In this talk, I will discuss some computational approaches to protein folding including the minimum energy principle and the initial and boundary value problems for fold simulation. I will review some most recent results in the field and discuss related mathematical and computational issues.
George Calin: Micro RNAs (miRNA genes) are a large family of highly conserved non-coding genes thought to be involved in temporal and tissue specific gene regulation. MiRs are transcribed as short hairpin precursors (~70nt) and are processed into active 21-22 nucleotides RNAs by Dicer, a ribonuclease that recognizes target mRNAs via base-pairing interactions. We reported that miR15a and miR16-1 are located at chromosome 13q14, a region deleted in more than half of B cell chronic lymphocytic leukemias (B-CLL), a disorder characterized by increased survival. Detailed deletion and expression analysis shows that miR15 and miR16 are located within a 30 kb region of loss in CLL and that both genes are deleted or down-regulated in the majority (approximately 68%) of CLL cases.
To further investigate the possible involvement of miRNAs in human cancers on a genome-wide basis we have mapped 186 miRNAs and compared their location to the location of previous reported non-random genetic alterations. We show that miRNA genes are frequently located at fragile sites, as well as in minimal regions of loss of heterozygosity (minimal LOH), minimal regions of amplification (minimal amplicons), or common breakpoint regions. Overall, 98 of 186 (52.5%) of miR genes are in cancer-associated genomic regions (CAGR) or in fragile sites. Much more, by Northern blotting we have shown that several miRNAs located in deleted regions have low levels of expression in cancer samples. These data provide a catalogue of miRNA genes that may have roles in cancer and argue that the miRNome (defined as the full complement of miRNAs in a genome) may be extensively involved in cancers.
Little is known about miRNA expression levels or function in normal and neoplastic cells. We identified, using genome-wide expression profiling of miRNAs with an oligonucleotide microarray, two distinct clusters of human B-CLL samples associated with the presence or the absence of Zap-70 expression, a predictor of early disease progression. Two miRNA signatures were associated with presence or absence of mutations in the expressed immunoglobulin variable-region genes or with deletions at 13q14 respectively. These data suggest that miRNA expression patterns have relevance to the biological and clinical behaviour of this leukemia.
Chang-gong Liu: MicroRNAs (miRNAs) are a class of small non-coding RNA genes recently found to be abnormally expressed in several types of cancer. We described a novel methodology for miRNA gene expression profiling based on the development of a microchip containing oligonucleotides corresponding to 245 miRNAs from human and mouse genomes. Using these microarrays, we obtained highly reproducible results that revealed tissue-specific miRNA expression signatures, data confirmed by assessment of expression by Northern blots, real-time PCR and literature search. The microchip oligolibrary can be expanded to include an increasing number of miRNAs discovered in various species, and is useful for the analysis of normal and disease states.
Behind the phenomena of genetics and stochastic processes, we find there is an intrinsic algebraic structure. We call this algebraic structure --- evolution algebra. Evolution algebras are non-associative (non-power-associative) Banach algebras and have many connections with other mathematical fields including graph theory, group theory, Markov chains, dynamic systems, knot theory, 3-manifold and the study of the Riemann-zeta function. In the present talk, we will give the foundation of the theory of evolution algebras and establish a hierarchical structure theorem for evolution algebras. One of the unusual features of an evolution algebra is that it possesses an evolution operator. This evolution operator reveals the dynamic information of an evolution algebra. However, what makes the theory of evolution algebras different from the classical theory of algebras is that in an evolution algebra, we can have two different kinds of generators: algebraically persistent generators and algebraically transient generators. The basic notions of algebraic persistency and algebraic transiency, and their relative versions, lead to a hierarchical structure on an evolution algebra. Dynamically, this hierarchical structure displays the direction of the flow induced by the evolution operator. Algebraically, this hierarchical structure is given in the form of a sequence of semi-direct-sum decompositions of a general evolution algebra. Thus, this hierarchical structure demonstrates that an evolution algebra is a mixed algebraic and dynamic object. The algebraic nature of this hierarchical structure allows us to have a rough skeleton-shape classification of evolution algebras. On the other hand, the dynamic nature of this hierarchical structure is what makes the notion of an evolution algebra applicable to the study of stochastic processes and many other objects in different fields. For example, when we apply our structure theorem to evolution algebras induced by Markov chains, we see that any general Markov chain has a dynamic hierarchy and the probabilistic flow is moving with invariance on this hierarchy, and that all Markov chains can be classified by the skeleton-shape classification of their evolution algebras. There is a bunch of open problems in this direction.
Discovering meaningful patterns in the wealth of data produced by gene expression experiments and genome sequencing projects requires rigorous statistical methods. Multiple testing problems arise whenever one wishes to perform statistical tests for each of many genes or genomic regions. Identifying differently expressed genes from microarray experiments is a typical example. We have derived a general characterization of the null distribution for multiple testing that asymptotically controls type I error rates without conditions such as subset pivotality. This characterization is novel, because it utilizes the distribution of the test statistics rather than a data null distribution. A simple bootstrap estimator of this distribution is presented. With a statistically significant subset in hand, clustering methods assist in the identification of patterns in the data. We have developed a hybrid clustering algorithm called HOPACH, which combines the strengths of both partitioning and agglomerative hierarchical clustering methods. Using this algorithm as an example, I demonstrate how the bootstrap can be employed as a statistical method in cluster analysis to establish the reproducibility of the clusters and the overall variability of the followed procedure. Applications to microarray data and comparative genomics illustrate the methodologies.
The dopaminergic neuron ordinarily will not fire faster than about 10 Hz when depolarized in slices. In vivo, much higher rates are briefly attained, for example after an unexpected reward. Using a biophysically based model, we suggest a mechanism for the burst generation and show that the mechanism is influenced by dendritic geometry. Our model represents the neuron as a number of electrically coupled oscillators (compartments) with different natural frequencies, corresponding to the soma and parts of the dendrite. We show that the coupled system, but none of the individual compartments, has oscillatory transient dynamics. Moreover, the transient frequency in such a system can be higher than the frequency of the fastest of the oscillators in isolation. We study dynamical mechanisms that lead to a substantial frequency difference between the transient and steady-state oscillations. Finally, we study the role of dendritic geometry for the burst generation, and show that branching and long thin sections help to create the transients. An implication of this is that slicing itself may explain the absence of the burst firing in vitro: slicing cuts off distal dendrites and so reduces the number of fast compartments.
This seminar is a presentation of a project in progress studying the dynamic properties of the QT interval of the EKG. Lengthening of the QT interval is thought to be a precursor of some types of ventricular arrhythmias. However, from beat to beat in live subjects, the QT interval is highly variable, and results are often averaged in order to produce a single measure of QT length. In this project, we are attempting to construct a model that can account for the mechanism and give a framework for understanding beat to beat QT dynamics. This mathematical model will (at minimum) include the electrical activity of the heart, the baroreceptor reflex, and a component that relates heart activity to blood pressure. Ideally, it is hoped that a better understanding of QT dynamics through this model will lead to a better quantification of QT lengthening and it's relation to ventricular arrhythmias.
Recent work by Drover et al. on a network of Hodgkin-Huxley neurons coupled by excitatory synapses showed significant slowing of the firing rate of the synchronized network. The slowing of the firing rate is accompanied by subthreshold oscillations near the action potential threshold. I will explain these observations using methods from dynamical systems theory. Especially, i show that so called `Canards' (french for `duck') are responsible for the delay and line out the geometric explanation for this phenomenon. General conditions for neuronal models to possess canards are given.
Binary-splitting or bifurcating models are concerned with modeling tree-indexed time series data, e.g., each individual in one generation gives rise to two offspring in the next generation. Cell lineage data (e.g. Powell(1955)) are typically of this kind.
In this talk, bifurcating autoregressive (BAR) models are introduced. Exact and asymptotic distributions of the maximum likelihood estimator of the autoregressive parameter in a BAR(1) model with exponential innovation are derived. The limit distributions for the stationary, critical and explosive cases are unified via a single random pivot. The pivot is shown to be asymptotically exponential for all values of the autoregressive parameter. Some simulation results will be discussed.
Regression Contour Cluster Analysis could be applied in a wide variety of scientific fields where the goal is to find the regions in the input space with relatively high (low) values of the target variable. Often there are problems where a decision maker can in a sense choose the values or ranges of the input variables so as to optimize the values of the target variable. For example, in Clinical Trials, the goal may be to find variables that describe patients with an extremely bad or good prognosis, which is useful for detecting risk potentials or judging new therapies. In medical research, physicians may be interested in identifying groups of patients with different prognosis in order to take different treatment. These tasks can be done using Regression Contour Cluster Analysis.
In this talk, the excess mass approach will be first introduced and extended from density to regression for regression contour cluster analysis. This is accomplished without prior estimation of the regression function itself. The basic idea is that measuring and comparing mass concentrations might be a fruitful concept when dealing with higher dimensional situations. Then a new data mining method called Patient Rule Induction Method (PRIM) will be presented and applied to regression contour cluster estimation. The modified PRIM called Ratio-Controlled PRIM will be developed. Simulation results and applications to Classification for census income data will also be presented.
Stochasticity is often viewed as a nuisance. However, stochastic processes can impart structural features in the dynamics of biological systems that would not exist in a purely deterministic system. When applying inverse modeling techniques to these systems it is essential to be able to reliably sample from a population of stochastic realizations. In this talk, I'll focus on techniques for adaptive numerical solution of stochastic differential equations using a new JAVA-based numerical library.
The BIAcore is an ingenious device that allows the measurement of rate constants for binding processes without disturbing the system. The BIAcore is used mainly to analyze biochemical systems, and its geometry and fluid dynamics closely mimic real biological applications. In order to estimate the rate constants, an accurate mathematical model is needed to interpret the raw data correctly. This talk will discuss the latest enhancements to these models, namely flow inside the reacting receptor layer. By using asymptotics and perturbation methods, simple expressions may be obtained which are valid for a wide range of experimental parameters. These solutions, which provide corrections to the rate constants measured in the BIAcore, are interpreted physically.
This lecture is a presentation of some of the basic biology and modeling involved in cardiac physiology. Topics will include the structure and function of the heart, ECG interpretation, cardiac arrhythmias, and the heart's neural regulation system.
Flows driven by pumping without valves are observed, motivated by biomedical applications: cardiopulmonary resuscitation (CPR) for the thoracic pump model and the human fetus before the development of the heart valves. Although the mechanism of valveless pumping (VP) has been discussed over the centuries, lots of phenomena of VP are still remained mysterious. In this talk, we present the two different mathematical models of VP in order to explain the mechanism of VP: One is a lumped parameter circuit model with time-dependent resistances and compliances governing by an ODE system. The other is a two-dimensional computational model with a full Navier-Stokes system solved by the Immersed Boundary Method. The flow mechanism around a loop of tubing that is composed of the two different compliant sections is investigated when an asymmetric force is applied in the valveless circulatory system. In both models, we present that the direction and magnitude of a net flow around the loop of tubing are dependent on the parameters, such as frequency, amplitude, and compression duration.
Gillies et al.  have experimentally shown the existence of oscillation in the theta frequency range (8-10 Hz) in slices of the CA1 hippocampal area of the brain when phasic excitation is blocked. In the presence of phasic excitation, the predominant frequency is in the gamma range (~ 40 Hz). Two different types of inhibitory neurons are involved in the mechanism of generating these rhythms: fast spiking interneurons and OLM cells; the latter have a hyperpolarization activated inward current (Ih) in addition to the standard Hodgkin-Huxley currents. They also have longer lasting inhibitory postsynaptic potentials (IPSP) in comparison with the former.
We present a biophysically plausible mathematical model that successfully reproduces experimental findings. This model focuses on the activity of O-LM (O), cells producing slow IPSPs, and other inhibitory neurons (I), each modeled as a single compartment. In addition to standard Hodgkin-Huxley currents, we model the Ih current in the O cells; blockade of Ih has been shown to destroy the rhythmicity both experimentally and in simulations.
We propose a mechanism by which coherent theta oscillations are created, due to the interaction of the I and O cells via the fast and slow inhibition. Using numerical and analytical techniques we demonstrate the effect that the I cells exert on the O cells due to the presence of Ih.
 Gillies, M. J., Traub, R. D., Le Bleau, F. E. N, Davis, C. H., Gloveli, T., Buhl, E. H., et al. (2002). A model of atropine-resistant theta oscillations in rat hippocampal area CA1. J Physiol (Lond), 543, 779-793.