Designing and sample size calculation in presence of heterogeneity in biological studies involving high-throughput data.

The designing and determination of sample size are important for conducting high-throughput biological experiments such as proteomics experiments and RNA-Seq expression studies, thus leading to better understanding of complex mechanisms underlying various biological processes. The variations in the biological data or technical approaches to data collection lead to heterogeneity for the samples under study. We critically worked on the issues of technical and biological heterogeneity. The quantitative measurements based on liquid chromatography (LC) coupled with mass spectrometry (MS) often suffer from the problem of missing values (MVs) and data heterogeneity. We considered a proteomics data set generated from human kidney biopsy material to investigate the technical effects of sample preparation and the quantitative MS. We studied the effect of tissue storage methods (TSMs) and tissue extraction methods (TEMs) on data analysis. There are two TSMs: frozen (FR) and FFPE (formalin-fixed paraffin embedded); and three TEMs: MAX, TX followed by MAX and SDS followed by MAX. We assessed the impact of different strategies to analyze the data while considering heterogeneity and MVs. We found that the FFPE is better than that of FR for tissue storage. We also found that the one-step TEM (MAX) is better than those of two-steps TEMs. Furthermore, we found the imputation method is a better approach than excluding the proteins with MVs or using unbalanced design. We introduce a web application, PWST (Proteomics Workflow Standardization Tool) to standardize the proteomics workflow. The tool will be helpful in deciding the most suitable choice for each step and studying the variability associated with technical steps as well as the effects of continuous variables. We have used the special cases of general linear model - ANCOVA and ANOVA with fixed effects to study the effects due to various sources of variability. We introduce an interactive tool, “SATP: Statistical Analysis Tool for Proteomics”, for analyzing proteomics expression data that is scalable to large clinical proteomic studies. The user can perform differential expression analysis of proteomics data either at the protein or peptide level using multiple approaches. We have developed statistical approaches for calculating sample size for proteomics experiments under allocation and cost constraints. We have developed R programs and a shiny app “SSCP: Sample Size Calculator for Proteomics Experiment” for computing sample sizes. We have proposed statistical approaches for calculating sample size for RNA-Seq experiments considering allocation and cost. We have developed R programs and shiny apps to calculate sample size for conducting RNA-Seq experiments

Application of Graph Theory in Computer Science

The field of mathematics have important roll in various fields. One of the important area in mathematics is Graph Theory. Which used in structural modeling in many area’s. The structural arrangements of various objects or technologies lead to new inventions and modification in the existing environment for enhancement in those field. The field of graph theory started from problem of Konigsberg bridge in 1735. This paper given an overview of the application of graph theory in heterogeneous field to some extent but mainly focuses on computer science application but uses graph theoretical concepts

Comprative Study Kirchoff’s and Tutte Matrix Theorem

In the mathematical field of graph theory, there are so many topic. In general the Cayley’s formula which provides the number of spanning tree in complete graph. There are two theorem Kirchoff’s Matrix theorem and Tutte Martrix theorem are also used to count spanning tree of a graph. In this paper we focus to count number of spanning in a graph with help of both theorem. At last we draw conclusion that Tutte Matrix theorem approach to compute the number of spanning arborescence of graph also

Study of Different algorithm in Euler Graph

In this paper we discuss about the algorithm to constructing an Euler Path in Euler Graph .There are many algorithm available to constructing an Path in Euler Graph, but in this paper we study mainly two algorithm i.e. Fleury’s & Hierholzer’s. Both are very important algorithm for determination of Euler path in Euler graph. Also both algorithms are different and more effective than simple algorithm

Lattice Boltzmann method to study the contraction of a viscous ligament

We employ a recently formulated axisymmetric version of the multiphase Shan-Chen (SC) lattice Boltzmann method (LBM) [Srivastava et al, in preparation (2013)] to simulate the contraction of a liquid ligament. We compare the axisymmetric LBM simulation against the slender jet (SJ) approximation model [T. Driessen and R. Jeurissen, IJCFD {\bf 25}, 333 (2011)]. We compare the retraction dynamics of the tail-end of the liquid ligament from the LBM simulation, the SJ model, Flow3D simulations and a simple model based on the force balance (FB). We find good agreement between the theoretical prediction (FB), the SJ model, and the LBM simulations

Axisymmetric multiphase lattice Boltzmann method

A lattice Boltzmann method for axisymmetric multiphase flows is presented and validated. The method is capable of accurately modeling flows with variable density. We develop the classic Shan-Chen multiphase model [ Phys. Rev. E 47 1815 (1993)] for axisymmetric flows. The model can be used to efficiently simulate single and multiphase flows. The convergence to the axisymmetric Navier-Stokes equations is demonstrated analytically by means of a Chapmann-Enskog expansion and numerically through several test cases. In particular, the model is benchmarked for its accuracy in reproducing the dynamics of the oscillations of an axially symmetric droplet and on the capillary breakup of a viscous liquid thread. Very good quantitative agreement between the numerical solutions and the analytical results is observed