Accurate simulation of THz generation with finite-element time domain methods.

We investigate the accurate full broadband simulation of complex nonlinear optical processes. A mathematical model and numerical simulation techniques in the time domain are developed to simulate complex nonlinear optical processes without the usual used slowly varying envelope approximation. We illustrate the accuracy by numerical simulations. Furthermore, they are used to elucidate THz generation in periodically poled Lithium Niobate (PPLN) including optical harmonic generation.

Error analysis for discretizations of parabolic problems using continuous finite elements in time and mixed finite elements in space.

Variational time discretization schemes are getting of increasing importance for the accurate numerical approximation of transient phenomena. The applicability and value of mixed finite element methods in space for simulating transport processes have been demonstrated in a wide class of works. We consider a family of continuous Galerkin-Petrov time discretization schemes that is combined with a mixed finite element approximation of the spatial variables. The existence and uniqueness of the semidiscrete approximation and of the fully discrete solution are established. For this, the Banach-Necas-Babuska theorem is applied in a non-standard way. Error estimates with explicit rates of convergence are proved for the scalar and vector-valued variable. An optimal order estimate in space and time is proved by duality techniques for the scalar variable. The convergence rates are analyzed and illustrated by numerical experiments, also on stochastically perturbed meshes.

Glypican1/2/4/6 and sulfated glycosaminoglycans regulate the patterning of the primary body axis in the cnidarian Nematostella vectensis.

Glypicans are members of the heparan sulfate (HS) subfamily of proteoglycans that can function in cell adhesion, cell crosstalk and as modulators of the major developmental signalling pathways in bilaterians. The evolutionary origin of these multiple functions is not well understood. In this study we investigate the role of glypicans in the embryonic and larval development of the sea anemone Nematostella vectensis, a member of the non-bilaterian clade Cnidaria. Nematostella has two glypican (gpc) genes that are expressed in mutually exclusive ectodermal domains, NvGpc1/2/4/6 in a broad aboral domain, and NvGpc3/5 in narrow oral territory. The endosulfatase NvSulf (an extracellular modifier of HS chains) is expressed in a broad oral domain, partially overlapping with both glypicans. Morpholino-mediated knockdown of NvGpc1/2/4/6 leads to an expansion of the expression domains of aboral marker genes and a reduction of oral markers at gastrula stage, strikingly similar to knockdown of the Wnt receptor NvFrizzled5/8. We further show that treatment with sodium chlorate, an inhibitor of glycosaminoglycan (GAG) sulfation, phenocopies knockdown of NvGpc1/2/4/6 at gastrula stage. At planula stage, knockdown of NvGpc1/2/4/6 and sodium chlorate treatment result in alterations in aboral marker gene expression that suggest additional roles in the fine-tuning of patterning within the aboral domain. These results reveal a role for NvGpc1/2/4/6 and sulfated GAGs in the patterning of the primary body axis in Nematostella and suggest an ancient function in regulating Frizzled-mediated Wnt signalling.

Development of the aboral domain in Nematostella requires ß-catenin and the opposing activities of Six3/6 and Frizzled5/8.

The development of the oral pole in cnidarians and the posterior pole in bilaterians is regulated by canonical Wnt signaling, whereas a set of transcription factors, including Six3/6 and FoxQ2, controls aboral development in cnidarians and anterior identity in bilaterians. However, it is poorly understood how these two patterning systems are initially set up in order to generate correct patterning along the primary body axis. Investigating the early steps of aboral pole formation in the sea anemone Nematostella vectensis, we found that, at blastula stage, oral genes are expressed before aboral genes and that Nvß-catenin regulates both oral and aboral development. In the oral hemisphere, Nvß-catenin specifies all subdomains except the oral-most, NvSnailA-expressing domain, which is expanded upon Nvß-catenin knockdown. In addition, Nvß-catenin establishes the aboral patterning system by promoting the expression of NvSix3/6 at the aboral pole and suppressing the Wnt receptor NvFrizzled5/8 at the oral pole. NvFrizzled5/8 expression thereby gets restricted to the aboral domain. At gastrula stage, NvSix3/6 and NvFrizzled5/8 are both expressed in the aboral domain, but they have opposing activities, with NvSix3/6 maintaining and NvFrizzled5/8 restricting the size of the aboral domain. At planula stage, NvFrizzled5/8 is required for patterning within the aboral domain and for regulating the size of the apical organ by modulation of a previously characterized FGF feedback loop. Our findings suggest conserved roles for Six3/6 and Frizzled5/8 in aboral/anterior development and reveal key functions for Nvß-catenin in the patterning of the entire oral-aboral axis of Nematostella.

Migration and differentiation potential of stem cells in the cnidarian Hydractinia analysed in eGFP-transgenic animals and chimeras.

To analyse cell migration and the differentiation potential of migratory stem cells in Hydractinia, we generated animals with an eGFP reporter gene stably expressed and transmitted via the germline. The transgene was placed under the control of two different actin promoters and the promoter of elongation factor-1α. One actin promoter (Act-II) and the EF-1α promoter enabled expression of the transgene in all cells, the other actin promoter (Act-I) in epithelial and gametogenic cells, but not in the pluripotent migratory stem cells. We produced chimeric animals consisting of histocompatible wild type and transgenic parts. When the transgene was under the control of the epithelial cell specific actin-I promoter, non-fluorescent transgenic stem cells immigrated into wild type tissue, stopped migration and differentiated into epithelial cells which then commenced eGFP-expression. Migratory stem cells are therefore pluripotent and can give rise not only to germ cells, nematocytes and nerve cells, but also to epithelial cells. While in somatic cells expression of the act-I promoter was restricted to epithelial cells it became also active in gametogenesis. The act-I gene is expressed in spermatogonia, oogonia and oocytes. In males the expression pattern showed that migratory stem cells are the precursors of both the spermatogonia and their somatic envelopes. Comparative expression studies using the promoters of the actin-II gene and the elongation factor-1α gene revealed the potential of transgenic techniques to trace the development of the nervous system.

A model describing the effect of enzymatic degradation on drug release from collagen minirods.

A drug delivery system, named minirod, containing insoluble non-cross-linked collagen was prepared to investigate the release of model drug compounds. To characterise the complete drug release process properly, a mathematical model was developed. Previously, a mathematical model describing water penetration, matrix swelling and drug release by diffusion from dense collagen matrices has been introduced and tested. However, enzymatic matrix degradation influences the drug release as well. Based on experimental data, a model was developed which describes drug release by collagenolytic matrix degradation based on enzyme diffusion, adsorption and cleavage. Data for swelling, collagen degradation and FITC dextran release from insoluble equine collagen type I minirods were collected. Sorption studies demonstrated a tight sorption of collagenase on collagen surfaces that follows a Freundlich sorption isotherm and results in a degradation constant of 3.8x10(-5) mol/l for the minirods. The diffusion coefficients of FITC dextran 20 and 70 (3x10(-3) and 2.4x10(-3) cm2/h) in water were analyzed by fluorescence correlation spectroscopy (FCS). Using these data, the mathematical model was verified by two-dimensional simulations. The numerical results agreed well with the measurements.

Modeling of drug release from collagen matrices.

Drug release from collagen matrices is in most cases governed by diffusion from swollen matrices but also enzymatic matrix degradation or hydrophobic drug/collagen interactions may contribute. To reduce water uptake and to prolong the release, insoluble collagen matrices have been chemically or dehydrothermally crosslinked. Assuming Fickian diffusion a one-dimensional model was developed and tested that allows description of water penetration, swelling and drug release and that may be expanded considering a subsequent erosion process or interactions. Swelling is described by a volume balance. For dry collagen matrices crosslinked by thermal treatment the existence of a moving front separating the polymer from a gel phase was considered, and a convective term induced by the volume expansion was incorporated. The resulting moving boundary problem was solved using a method based on biquadratic finite elements in both space and time that is stable, shows high accuracy, and is suitable for solving problems with a singularity at the initial time point. The model was verified for insoluble collagen matrices at different crosslinking degrees for both chemical and thermal treatment. For constant diffusion coefficients a close form of the solution was derived yielding equivalent results to the numerical approach.