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In contrast to regular ordinary differential equations, the problem of accurately setting initial conditions just emerges in the context of differential-algebraic equations where the dynamic degree of freedom of the system is smaller than the absolute dimension of the described process, and the actual lower-dimensional configuration space of the system is deeply implicit. For linear higher-index differential-algebraic equations, we develop an appropriate numerical method based on properties of canonical subspaces and on the so-called geometric reduction. Taking into account the fact that higher-index differential-algebraic equations lead to ill-posed problems in naturally given norms, we modify this approach to serve as transfer conditions from one time-window to the next in a time stepping procedure and combine it with window-wise overdetermined least-squares collocation to construct the first fully numerical solvers for higher-index initial-value problems.

To develop new therapeutic options for the treatment of disc degeneration we tested the possibility of overexpression of active growth and differentiation factor (GDF) 5 and of transforming growth factor (TGF) beta(1) by adenoviral gene transfer and characterized its effect on cell proliferation and matrix synthesis of cultured rabbit and human intervertebral disc cells. Recombinant adenovirus encoding for GDF-5 or TGF-beta(1) was developed and transgene expression characterized by RT-PCR, western blot and ELISA. Growth and matrix synthesis of transduced cells was measured by [(3)H]thymidine or [(35)S]sulfate incorporation. Disc cells expressed the receptors BMPR1A, BMPR1B, and BMPR2, which are relevant for GDF-5 action. Adenovirus efficiently transferred the GDF-5 gene or the TGF-beta(1) gene to rabbit and human intervertebral disc cells. About 50 ng GDF-5 protein/10(6 )cells per 24 h or 7 ng TGF-beta(1) protein/10(6 )cells per 24 h was produced. According to western blotting, two GDF-5 forms, with molecular weights consistent with the activated GDF-5 dimer and the proform, were secreted over the 3 weeks following gene transfer. Overexpressed GDF-5 and TGF-beta(1) were bioactive and promoted growth of rabbit disc cells in monolayer culture. Our results suggest that ex vivo gene delivery of GDF-5 and TGF-beta(1) is an attractive approach for the release of mature and pre-GDF-5 in surrounding tissue. This leads us to hope that it will prove possible to improve the treatment of degenerative disc disease by means of ex vivo gene transfer of single or multiple growth factors.

We introduce a direct numerical treatment of nonlinear higher-index differential-algebraic equations by means of overdetermined polynomial least-squares collocation. The procedure is not much more computationally expensive than standard collocation methods for regular ordinary differential equations. The numerical experiments show impressive results. In contrast, the theoretical basic concept turns out to be considerably challenging. So far, quite recently convergence proofs for linear problems have been published. In the present paper we come up to a first convergence result for nonlinear problems. (Michael Hanke), maerz@math.hu-berlin.de (Roswitha März) 1 The number of unknowns equals the number of equations. 2 More precisely: Essentially ill-posed in Tichonov's sense, that is, the related operators feature nonclosed ranges.

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