Title: Preconditioning for Accurate Solutions of Linear Systems and Eigenvalue Problems

Abstract: This paper develops the preconditioning technique as a method to address the accuracy issue caused by ill-conditioning. Given a preconditioner M for an ill-conditioned linear system Ax=b, we show that, if the inverse of the preconditioner can be applied to vectors accurately, then the linear system can be solved accurately. A stability concept called inverse-equivalent accuracy is introduced to describe higher accuracy that is achieved and an error analysis will be presented. As an application, we use the preconditioning approach to accurately compute a few smallest eigenvalues of certain ill-conditioned matrices. Numerical examples are presented to illustrate the error analysis and the performance of the methods.

Title: Algebraic Statistics Applications in Epidemiology



Abstract: Interactions between single nucleotide polymorphisms (SNPs) and complex diseases have been an important topic throughout epidemiological studies. Previous studies have mostly focused on gene variables at a single locus. In this talk, I will discuss a focused candidate gene study to test the interaction of multiple SNPs with the risk of different types of cancer.

We will exemplify the fact that traditional asympotic results in statistical analysis do not apply in our setting. This is due mainly to the fact that we have a relatively small fixed data set.  In our work we develop a new statistical approach using techniques from the field of algebraic statistics. Algebraic statistics focuses on mathematical aspects of statistical models, where algebraic, geometric and combinatorial insights can be useful to study behavior of statistical procedures.



Using the R package algstat, developed by Kahle, Garcia Puente, and Yoshida, we implemented an algebraic statistics method that can test for independence between several variables and the desease. We applied our methods to the study of gene-gene interaction on cancer data obtained from the European case-control study Gen-Air extending previous work by Ricceri, Fassino, Matullo, Roggero, Torrente, Vineis, and Terracini.

Title: Structural and Functional Characterization of Expected and Aberrant Metal Ion Coordination in Proteins

Abstract: Metalloproteins bind and utilize metal ions for a variety of biological purposes.   Due to the ubiquity of metalloprotein involvement throughout these processes across all domains of life, how proteins coordinate metal ions for different biochemical functions is of great relevance to understanding the implementation of these biological processes. Towards these ends, we have improved our methodology for structurally and functionally characterizing metal binding sites in metalloproteins.  Our new ligand detection method is statistically much more robust, producing estimated false positive and false negative rates of ~0.11% and ~1.2%, respectively.  Additional improvements expand both the range of metal ions and their coordination number that can be effectively analyzed.  Also, the inclusion of many additional quality control filters has significantly improved structure-function Spearman correlations as demonstrated by rho values greater than 0.90 for several metal coordination analyses and even one rho value above 0.95.     Also, improvements in bond-length distributions have revealed bond-length modes specific to chemical functional groups involved in multidentation.  Using these improved methods, we analyzed all single metal ion binding sites with Zn, Mg, Ca, Fe, and Na ions in wwPDB, producing statistically rigorous results supporting the existence of both a significant number of unexpected compressed angles and subsequent aberrant metal ion coordination geometries (CGs) within structurally known metalloproteins.  By recognizing these aberrant CGs in our clustering analyses, high correlations are achieved between structural and functional descriptions of metal ion coordination.  Moreover, distinct biochemical functions are associated with aberrant CGs versus non-aberrant CGs.

TITLE: Synchrony in a Boolean network model of the L-arabinose operon



ABSTRACT: In genetics, an operon is a segment of DNA that contains several co-transcribed genes, which together form a functional regulatory unit. Operons have primarily been studied in prokaryotes, with both the lactose and tryptophan operons in E. Coli having been classically modeled with differential equations and more recently, with Boolean networks. The L-arabinose operon in E. coli encodes proteins that function in the catabolism of arabinose. This operon has several complex features, such as a protein that acts both as an activator, a DNA looping repressing mechanism, and the lack of inducer exclusion by glucose. In this talk, I will propose a Boolean network model of the ara operon, and then show how computational algebra in Sage establishes that for 11 of the 12 choices of initial conditions, the state space contains a single fixed point that correctly predicts the biology. The final initial condition describes the case where there are medium levels of arabinose and no glucose, and it successfully predicts bistability of the system. Finally, I will compare the state space under synchronous and asynchronous update, and show how the former has several artificial cycles that go away under a general asynchronous update.

Title: Insight into Molecular through Subcellular Calcium Signaling via Multi-Scale Simulation

Abstract: Calcium is critical to a wide range of physiological processes, including neurological function, immune responses, and muscle contraction. Calcium-dependent signaling pathways enlist a variety of proteins and channels that must rapidly and selectively bind calcium against thousand-fold higher cationic concentrations. Frequently these pathways further require the co-localization of these proteins within specialized subcellular structures to function properly. Our lab has developed multi-scale simulation tools to elucidate how protein structure and co-localization facilitate intracellular calcium signaling. Developments include combining molecular simulations with a statistical mechanical model of ion binding, a homogenization theory to upscale molecular interactions into micron-scale diffusion models, and reaction-diffusion simulations that leverage sub-micron microscopy data. In this seminar, I will describe these tools and their applications toward molecular mechanisms of calcium-selective recognition and cross-talk between co-localized calcium binding proteins inside the cell.