Convergence of numerical time-averaging and stationary measures via Poisson equations

Numerical approximation of the long time behavior of a stochastic di.erential equation (SDE) is considered. Error estimates for time-averaging estimators are obtained and then used to show that the stationary behavior of the numerical method converges to that of the SDE. The error analysis is based on using an associated Poisson equation for the underlying SDE. The main advantages of this approach are its simplicity and universality. It works equally well for a range of explicit and implicit schemes, including those with simple simulation of random variables, and for hypoelliptic SDEs. To simplify the exposition, we consider only the case where the state space of the SDE is a torus, and we study only smooth test functions. However, we anticipate that the approach can be applied more widely. An analogy between our approach and Stein's method is indicated. Some practical implications of the results are discussed. Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

Effects of Natural Organic Matter on the Dissolution Kinetics and Bioavailability of Metal Oxide Nanoparticles

The rapid development of nanotechnology and wider applications of engineered nanomaterials (ENMs) in the last few decades have generated concerns regarding their environmental and health risks. After release into the environment, ENMs undergo aggregation, transformation, and, for metal-based nanomaterials, dissolution processes, which together determine their fate, bioavailability and toxicity to living organisms in the ecosystems. The rates of these processes are dependent on nanomaterial characteristics as well as complex environmental factors, including natural organic matter (NOM). As a ubiquitous component of aquatic systems, NOM plays a key role in the aggregation, dissolution and transformation of metal-based nanomaterials and colloids in aquatic environments.

The goal of this dissertation work is to investigate how NOM fractions with different chemical and molecular properties affect the dissolution kinetics of metal oxide ENMs, such as zinc oxide (ZnO) and copper oxide (CuO) nanoparticles (NPs), and consequently their bioavailability to aquatic vertebrate, with Gulf killifish (Fundulus grandis) embryos as model organisms.

ZnO NPs are known to dissolve at relatively fast rates, and the rate of dissolution is influenced by water chemistry, including the presence of Zn-chelating ligands. A challenge, however, remains in quantifying the dissolution of ZnO NPs, particularly for time scales that are short enough to determine rates. This dissertation assessed the application of anodic stripping voltammetry (ASV) with a hanging mercury drop electrode to directly measure the concentration of dissolved Zn in ZnO NP suspensions, without separation of the ZnO NPs from the aqueous phase. Dissolved zinc concentration measured by ASV ([Zn]ASV) was compared with that measured by inductively coupled plasma mass spectrometry (ICP-MS) after ultracentrifugation ([Zn]ICP-MS), for four types of ZnO NPs with different coatings and primary particle diameters. For small ZnO NPs (4-5 nm), [Zn]ASV was 20% higher than [Zn]ICP-MS, suggesting that these small NPs contributed to the voltammetric measurement. For larger ZnO NPs (approximately 20 nm), [Zn]ASV was (79±19)% of [Zn]ICP-MS, despite the high concentrations of ZnO NPs in suspension, suggesting that ASV can be used to accurately measure the dissolution kinetics of ZnO NPs of this primary particle size.

Using the ASV technique to directly measure dissolved zinc concentration, we examined the effects of 16 different NOM isolates on the dissolution kinetics of ZnO NPs in buffered potassium chloride solution. The observed dissolution rate constants (kobs) and dissolved zinc concentrations at equilibrium increased linearly with NOM concentration (from 0 to 40 mg-C L-1) for Suwannee River humic acid (SRHA), Suwannee River fulvic acid and Pony Lake fulvic acid. When dissolution rates were compared for the 16 NOM isolates, kobs was positively correlated with certain properties of NOM, including specific ultraviolet absorbance (SUVA), aromatic and carbonyl carbon contents, and molecular weight. Dissolution rate constants were negatively correlated to hydrogen/carbon ratio and aliphatic carbon content. The observed correlations indicate that aromatic carbon content is a key factor in determining the rate of NOM-promoted dissolution of ZnO NPs. NOM isolates with higher SUVA were also more effective at enhancing the colloidal stability of the NPs; however, the NOM-promoted dissolution was likely due to enhanced interactions between surface metal ions and NOM rather than smaller aggregate size.

Based on the above results, we designed experiments to quantitatively link the dissolution kinetics and bioavailability of CuO NPs to Gulf killifish embryos under the influence of NOM. The CuO NPs dissolved to varying degrees and at different rates in diluted 5‰ artificial seawater buffered to different pH (6.3-7.5), with or without selected NOM isolates at various concentrations (0.1-10 mg-C L-1). NOM isolates with higher SUVA and aromatic carbon content (such as SRHA) were more effective at promoting the dissolution of CuO NPs, as with ZnO NPs, especially at higher NOM concentrations. On the other hand, the presence of NOM decreased the bioavailability of dissolved Cu ions, with the uptake rate constant negatively correlated to dissolved organic carbon concentration ([DOC]) multiplied by SUVA, a combined parameter indicative of aromatic carbon concentration in the media. When the embryos were exposed to CuO NP suspension, changes in their Cu content were due to the uptake of both dissolved Cu ions and nanoparticulate CuO. The uptake rate constant of nanoparticulate CuO was also negatively correlated to [DOC]×SUVA, in a fashion roughly proportional to changes in dissolved Cu uptake rate constant. Thus, the ratio of uptake rate constants from dissolved Cu and nanoparticulate CuO (ranging from 12 to 22, on average 17±4) were insensitive to NOM type or concentration. Instead, the relative contributions of these two Cu forms were largely determined by the percentage of CuO NP that was dissolved.

Overall, this dissertation elucidated the important role that dissolved NOM plays in affecting the environmental fate and bioavailability of soluble metal-based nanomaterials. This dissertation work identified aromatic carbon content and its indicator SUVA as key NOM properties that influence the dissolution, aggregation and biouptake kinetics of metal oxide NPs and highlighted dissolution rate as a useful functional assay for assessing the relative contributions of dissolved and nanoparticulate forms to metal bioavailability. Findings of this dissertation work will be helpful for predicting the environmental risks of engineered nanomaterials.

Information-theoretic analysis of a stimulated-Brillouin-scattering-based slow-light system.

We use an information-theoretic method developed by Neifeld and Lee [J. Opt. Soc. Am. A 25, C31 (2008)] to analyze the performance of a slow-light system. Slow-light is realized in this system via stimulated Brillouin scattering in a 2 km-long, room-temperature, highly nonlinear fiber pumped by a laser whose spectrum is tailored and broadened to 5 GHz. We compute the information throughput (IT), which quantifies the fraction of information transferred from the source to the receiver and the information delay (ID), which quantifies the delay of a data stream at which the information transfer is largest, for a range of experimental parameters. We also measure the eye-opening (EO) and signal-to-noise ratio (SNR) of the transmitted data stream and find that they scale in a similar fashion to the information-theoretic method. Our experimental findings are compared to a model of the slow-light system that accounts for all pertinent noise sources in the system as well as data-pulse distortion due to the filtering effect of the SBS process. The agreement between our observations and the predictions of our model is very good. Furthermore, we compare measurements of the IT for an optimal flattop gain profile and for a Gaussian-shaped gain profile. For a given pump-beam power, we find that the optimal profile gives a 36% larger ID and somewhat higher IT compared to the Gaussian profile. Specifically, the optimal (Gaussian) profile produces a fractional slow-light ID of 0.94 (0.69) and an IT of 0.86 (0.86) at a pump-beam power of 450 mW and a data rate of 2.5 Gbps. Thus, the optimal profile better utilizes the available pump-beam power, which is often a valuable resource in a system design.

Refractive changes after descemet stripping endothelial keratoplasty: a simplified mathematical model.

PURPOSE: To develop a mathematical model that can predict refractive changes after Descemet stripping endothelial keratoplasty (DSEK). METHODS: A mathematical formula based on the Gullstrand eye model was generated to estimate the change in refractive power of the eye after DSEK. This model was applied to four DSEK cases retrospectively, to compare measured and predicted refractive changes after DSEK. RESULTS: The refractive change after DSEK is determined by calculating the difference in the power of the eye before and after DSEK surgery. The power of the eye post-DSEK surgery can be calculated with modified Gullstrand eye model equations that incorporate the change in the posterior radius of curvature and change in the distance between the principal planes of the cornea and lens after DSEK. Analysis of this model suggests that the ratio of central to peripheral graft thickness (CP ratio) and central thickness can have significant effect on refractive change where smaller CP ratios and larger graft thicknesses result in larger hyperopic shifts. This model was applied to four patients, and the average predicted hyperopic shift in the overall power of the eye was calculated to be 0.83 D. This change reflected in a mean of 93% (range, 75%-110%) of patients' measured refractive shifts. CONCLUSIONS: This simplified DSEK mathematical model can be used as a first step for estimating the hyperopic shift after DSEK. Further studies are necessary to refine the validity of this model.

Differential Apaf-1 levels allow cytochrome c to induce apoptosis in brain tumors but not in normal neural tissues.

Brain tumors are typically resistant to conventional chemotherapeutics, most of which initiate apoptosis upstream of mitochondrial cytochrome c release. In this study, we demonstrate that directly activating apoptosis downstream of the mitochondria, with cytosolic cytochrome c, kills brain tumor cells but not normal brain tissue. Specifically, cytosolic cytochrome c is sufficient to induce apoptosis in glioblastoma and medulloblastoma cell lines. In contrast, primary neurons from the cerebellum and cortex are remarkably resistant to cytosolic cytochrome c. Importantly, tumor tissue from mouse models of both high-grade astrocytoma and medulloblastoma display hypersensitivity to cytochrome c when compared with surrounding brain tissue. This differential sensitivity to cytochrome c is attributed to high Apaf-1 levels in the tumor tissue compared with low Apaf-1 levels in the adjacent brain tissue. These differences in Apaf-1 abundance correlate with differences in the levels of E2F1, a previously identified activator of Apaf-1 transcription. ChIP assays reveal that E2F1 binds the Apaf-1 promoter specifically in tumor tissue, suggesting that E2F1 contributes to the expression of Apaf-1 in brain tumors. Together, these results demonstrate an unexpected sensitivity of brain tumors to postmitochondrial induction of apoptosis. Moreover, they raise the possibility that this phenomenon could be exploited therapeutically to selectively kill brain cancer cells while sparing the surrounding brain parenchyma.

Convergent differential regulation of parvalbumin in the brains of vocal learners.

Spoken language and learned song are complex communication behaviors found in only a few species, including humans and three groups of distantly related birds--songbirds, parrots, and hummingbirds. Despite their large phylogenetic distances, these vocal learners show convergent behaviors and associated brain pathways for vocal communication. However, it is not clear whether this behavioral and anatomical convergence is associated with molecular convergence. Here we used oligo microarrays to screen for genes differentially regulated in brain nuclei necessary for producing learned vocalizations relative to adjacent brain areas that control other behaviors in avian vocal learners versus vocal non-learners. A top candidate gene in our screen was a calcium-binding protein, parvalbumin (PV). In situ hybridization verification revealed that PV was expressed significantly higher throughout the song motor pathway, including brainstem vocal motor neurons relative to the surrounding brain regions of all distantly related avian vocal learners. This differential expression was specific to PV and vocal learners, as it was not found in avian vocal non-learners nor for control genes in learners and non-learners. Similar to the vocal learning birds, higher PV up-regulation was found in the brainstem tongue motor neurons used for speech production in humans relative to a non-human primate, macaques. These results suggest repeated convergent evolution of differential PV up-regulation in the brains of vocal learners separated by more than 65-300 million years from a common ancestor and that the specialized behaviors of learned song and speech may require extra calcium buffering and signaling.

Can Genetics Predict Response to Complex Behavioral Interventions? Evidence from a Genetic Analysis of the Fast Track Randomized Control Trial.

Early interventions are a preferred method for addressing behavioral problems in high-risk children, but often have only modest effects. Identifying sources of variation in intervention effects can suggest means to improve efficiency. One potential source of such variation is the genome. We conducted a genetic analysis of the Fast Track randomized control trial, a 10-year-long intervention to prevent high-risk kindergarteners from developing adult externalizing problems including substance abuse and antisocial behavior. We tested whether variants of the glucocorticoid receptor gene NR3C1 were associated with differences in response to the Fast Track intervention. We found that in European-American children, a variant of NR3C1 identified by the single-nucleotide polymorphism rs10482672 was associated with increased risk for externalizing psychopathology in control group children and decreased risk for externalizing psychopathology in intervention group children. Variation in NR3C1 measured in this study was not associated with differential intervention response in African-American children. We discuss implications for efforts to prevent externalizing problems in high-risk children and for public policy in the genomic era.

Differential predictability of four dimensions of affect intensity.

Individual differences in affect intensity are typically assessed with the Affect Intensity Measure (AIM). Previous factor analyses suggest that the AIM is comprised of four weakly correlated factors: Positive Affectivity, Negative Reactivity, Negative Intensity and Positive Intensity or Serenity. However, little data exist to show whether its four factors relate to other measures differently enough to preclude use of the total scale score. The present study replicated the four-factor solution and found that subscales derived from the four factors correlated differently with criterion variables that assess personality domains, affective dispositions, and cognitive patterns that are associated with emotional reactions. The results show that use of the total AIM score can obscure relationships between specific features of affect intensity and other variables and suggest that researchers should examine the individual AIM subscales.