Evaluation of reference-based two-color methods for measurement of gene expression ratios using spotted cDNA microarrays

Abstract Background Spotted cDNA microarrays generally employ co-hybridization of fluorescently-labeled RNA targets to produce gene expression ratios for subsequent analysis. Direct comparison of two RNA samples in the same microarray provides the highest level of accuracy; however, due to the number of combinatorial pair-wise comparisons, the direct method is impractical for studies including large number of individual samples (e.g., tumor classification studies). For such studies, indirect comparisons using a common reference standard have been the preferred method. Here we evaluated the precision and accuracy of reconstructed ratios from three indirect methods relative to ratios obtained from direct hybridizations, herein considered as the gold-standard. Results We performed hybridizations using a fixed amount of Cy3-labeled reference oligonucleotide (RefOligo) against distinct Cy5-labeled targets from prostate, breast and kidney tumor samples. Reconstructed ratios between all tissue pairs were derived from ratios between each tissue sample and RefOligo. Reconstructed ratios were compared to (i) ratios obtained in parallel from direct pair-wise hybridizations of tissue samples, and to (ii) reconstructed ratios derived from hybridization of each tissue against a reference RNA pool (RefPool). To evaluate the effect of the external references, reconstructed ratios were also calculated directly from intensity values of single-channel (One-Color) measurements derived from tissue sample data collected in the RefOligo experiments. We show that the average coefficient of variation of ratios between intra- and inter-slide replicates derived from RefOligo, RefPool and One-Color were similar and 2 to 4-fold higher than ratios obtained in direct hybridizations. Correlation coefficients calculated for all three tissue comparisons were also similar. In addition, the performance of all indirect methods in terms of their robustness to identify genes deemed as differentially expressed based on direct hybridizations, as well as false-positive and false-negative rates, were found to be comparable. Conclusion RefOligo produces ratios as precise and accurate as ratios reconstructed from a RNA pool, thus representing a reliable alternative in reference-based hybridization experiments. In addition, One-Color measurements alone can reconstruct expression ratios without loss in precision or accuracy. We conclude that both methods are adequate options in large-scale projects where the amount of a common reference RNA pool is usually restrictive.

Finite element methods for the Stokes system based on a Zienkiewicz type N-simplex

Hermite interpolation is increasingly showing to be a powerful numerical solution tool, as applied to different kinds of second order boundary value problems. In this work we present two Hermite finite element methods to solve viscous incompressible flows problems, in both two- and three-dimension space. In the two-dimensional case we use the Zienkiewicz triangle to represent the velocity field, and in the three-dimensional case an extension of this element to tetrahedra, still called a Zienkiewicz element. Taking as a model the Stokes system, the pressure is approximated with continuous functions, either piecewise linear or piecewise quadratic, according to the version of the Zienkiewicz element in use, that is, with either incomplete or complete cubics. The methods employ both the standard Galerkin or the Petrov–Galerkin formulation first proposed in Hughes et al. (1986) [18], based on the addition of a balance of force term. A priori error analyses point to optimal convergence rates for the PG approach, and for the Galerkin formulation too, at least in some particular cases. From the point of view of both accuracy and the global number of degrees of freedom, the new methods are shown to have a favorable cost-benefit ratio, as compared to velocity Lagrange finite elements of the same order, especially if the Galerkin approach is employed.