hp-adaptive Galerkin Time Stepping Methods for Nonlinear Initial Value Problems

This work is concerned with the derivation of an a posteriori error estimator for Galerkin approximations to nonlinear initial value problems with an emphasis on finite-time existence in the context of blow-up. The structure of the derived estimator leads naturally to the development of both h and hp versions of an adaptive algorithm designed to approximate the blow-up time. The adaptive algorithms are then applied in a series of numerical experiments, and the rate of convergence to the blow-up time is investigated

Maintaining RNA integrity in a homogeneous population of mammary epithelial cells isolated by Laser Capture Microdissection

Background: Laser-capture microdissection (LCM) that enables the isolation of specific cell populations from complex tissues under morphological control is increasingly used for subsequent gene expression studies in cell biology by methods such as real-time quantitative PCR (qPCR), microarrays and most recently by RNA-sequencing. Challenges are i) to select precisely and efficiently cells of interest and ii) to maintain RNA integrity. The mammary gland which is a complex and heterogeneous tissue, consists of multiple cell types, changing in relative proportion during its development and thus hampering gene expression profiling comparison on whole tissue between physiological stages. During lactation, mammary epithelial cells (MEC) are predominant. However several other cell types, including myoepithelial (MMC) and immune cells are present, making it difficult to precisely determine the specificity of gene expression to the cell type of origin. In this work, an optimized reliable procedure for producing RNA from alveolar epithelial cells isolated from frozen histological sections of lactating goat, sheep and cow mammary glands using an infrared-laser based Arcturus Veritas LCM (Applied Biosystems®) system has been developed. The following steps of the microdissection workflow: cryosectioning, staining, dehydration and harvesting of microdissected cells have been carefully considered and designed to ensure cell capture efficiency without compromising RNA integrity.[br/] Results: The best results were obtained when staining 8 μm-thick sections with Cresyl violet® (Ambion, Applied Biosystems®) and capturing microdissected cells during less than 2 hours before RNA extraction. In addition, particular attention was paid to animal preparation before biopsies or slaughtering (milking) and freezing of tissue blocks which were embedded in a cryoprotective compound before being immersed in isopentane. The amount of RNA thus obtained from ca.150 to 250 acini (300,000 to 600,000 μm2) ranges between 5 to 10 ng. RNA integrity number (RIN) was ca. 8.0 and selectivity of this LCM protocol was demonstrated through qPCR analyses for several alveolar cell specific genes, including LALBA (α-lactalbumin) and CSN1S2 (αs2-casein), as well as Krt14 (cytokeratin 14), CD3e and CD68 which are specific markers of MMC, lymphocytes and macrophages, respectively.[br/] Conclusions: RNAs isolated from MEC in this manner were of very good quality for subsequent linear amplification, thus making it possible to establish a referential gene expression profile of the healthy MEC, a useful platform for tumor biomarker discovery

One-Step Preservation of Phosphoproteins and Tissue Morphology at Room Temperature for Diagnostic and Research Specimens

BACKGROUND: There is an urgent need to measure phosphorylated cell signaling proteins in cancer tissue for the individualization of molecular targeted kinase inhibitor therapy. However, phosphoproteins fluctuate rapidly following tissue procurement. Snap-freezing preserves phosphoproteins, but is unavailable in most clinics and compromises diagnostic morphology. Formalin fixation preserves tissue histomorphology, but penetrates tissue slowly, and is unsuitable for stabilizing phosphoproteins. We originated and evaluated a novel one-step biomarker and histology preservative (BHP) chemistry that stabilizes signaling protein phosphorylation and retains formalin-like tissue histomorphology with equivalent immunohistochemistry in a single paraffin block. RESULTS: Total protein yield extracted from BHP-fixed, routine paraffin-embedded mouse liver was 100% compared to snap-frozen tissue. The abundance of 14 phosphorylated proteins was found to be stable over extended fixation times in BHP fixed paraffin embedded human colon mucosa. Compared to matched snap-frozen tissue, 8 phosphoproteins were equally preserved in mouse liver, while AMPKβ1 Ser108 was slightly elevated after BHP fixation. More than 25 tissues from mouse, cat and human specimens were evaluated for preservation of histomorphology. Selected tissues were evaluated in a multi-site, independent pathology review. Tissue fixed with BHP showed equivalent preservation of cytoplasmic and membrane cytomorphology, with significantly better nuclear chromatin preservation by BHP compared to formalin. Immunohistochemical staining of 13 non-phosphorylated proteins, including estrogen receptor alpha, progesterone receptor, Ki-67 and Her2, was equal to or stronger in BHP compared to formalin. BHP demonstrated significantly improved immunohistochemical detection of phosphorylated proteins ERK Thr202/Tyr204, GSK3-α/β Ser21/Ser9, p38-MAPK Thr180/Tyr182, eIF4G Ser1108 and Acetyl-CoA Carboxylase Ser79. CONCLUSION: In a single paraffin block BHP preserved the phosphorylation state of several signaling proteins at a level comparable to snap-freezing, while maintaining the full diagnostic immunohistochemical and histomorphologic detail of formalin fixation. This new tissue fixative has the potential to greatly facilitate personalized medicine, biobanking, and phospho-proteomic research

Stochastic Differential Systems with Memory: Theory, Examples and Applications

The purpose of this article is to introduce the reader to certain aspects of stochastic differential systems, whose evolution depends on the past history of the state. Chapter I begins with simple motivating examples. These include the noisy feedback loop, the logistic time-lag model with Gaussian noise , and the classical ``heat-bath model of R. Kubo , modeling the motion of a ``large molecule in a viscous fluid. These examples are embedded in a general class of stochastic functional differential equations (sfde\u27s). We then establish pathwise existence and uniqueness of solutions to these classes of sfde\u27s under local Lipschitz and linear growth hypotheses on the coefficients. It is interesting to note that the above class of sfde\u27s is not covered by classical results of Protter, Metivier and Pellaumail and Doleans-Dade. In Chapter II, we prove that the Markov (Feller) property holds for the trajectory random field of a sfde. The trajectory Markov semigroup is not strongly continuous for positive delays, and its domain of strong continuity does not contain tame (or cylinder) functions with evaluations away from zero. To overcome this difficulty, we introduce a class of quasitame functions. These belong to the domain of the weak infinitesimal generator, are weakly dense in the underlying space of continuous functions and generate the Borel $\sigma$-algebra of the state space. This chapter also contains a derivation of a formula for the weak infinitesimal generator of the semigroup for sufficiently regular functions, and for a large class of quasitame functions. In Chapter III, we study pathwise regularity of the trajectory random field in the time variable and in the initial path. Of note here is the non-existence of the stochastic flow for the singular sdde $dx(t)= x(t-r) dW(t)$ and a breakdown of linearity and local boundedness. This phenomenon is peculiar to stochastic delay equations. It leads naturally to a classification of sfde\u27s into regular and singular types. Necessary and sufficient conditions for regularity are not known. The rest of Chapter III is devoted to results on sufficient conditions for regularity of linear systems driven by white noise or semimartingales, and Sussman-Doss type nonlinear sfde\u27s. Building on the existence of a compacting stochastic flow, we develop a multiplicative ergodic theory for regular linear sfde\u27s driven by white noise, or general helix semimartingales (Chapter IV). In particular, we prove a Stable Manifold Theorem for such systems. In Chapter V, we seek asymptotic stability for various examples of one-dimensional linear sfde\u27s. Our approach is to obtain upper and lower estimates for the top Lyapunov exponent. Several topics are discussed in Chapter VI. These include the existence of smooth densities for solutions of sfde\u27s using the Malliavin calculus, an approximation technique for multidimensional diffusions using sdde\u27s with small delays, and affine sfde\u27s