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Natural killer (NK) cells belong to the first line of host defense against infection and cancer. Cytokines, including interleukin-15 (IL-15), critically regulate NK cell activity, resulting in recognition and direct killing of transformed and infected target cells. NK cells have to adapt and respond in inflamed and often hypoxic areas. Cellular stabilization and accumulation of the transcription factor hypoxia-inducible factor-1α (HIF-1α) is a key mechanism of the cellular hypoxia response. At the same time, HIF-1α plays a critical role in both innate and adaptive immunity. While the HIF-1α hydroxylation and degradation pathway has been recently described with the help of mathematical methods, less is known concerning the mechanistic mathematical description of processes regulating the levels of HIF-1α mRNA and protein. In this work we combine mathematical modeling with experimental laboratory analysis and examine the dynamic relationship between HIF-1α mRNA, HIF-1α protein, and IL-15-mediated upstream signaling events in NK cells from human blood. We propose a system of non-linear ordinary differential equations with positive and negative feedback loops for describing the complex interplay of HIF-1α regulators. The experimental design is optimized with the help of mathematical methods, and numerical optimization techniques yield reliable parameter estimates. The mathematical model allows for the investigation and prediction of HIF-1α stabilization under different inflammatory conditions and provides a better understanding of mechanisms mediating cellular enrichment of HIF-1α. Thanks to the combination of experimental data and predictions we identified the mammalian target of rapamycin (mTOR), the nuclear factor-κB (NF-κB), and the signal transducer and activator of transcription 3 (STAT3) as central regulators of HIF-1α accumulation. We hypothesize that the regulatory pathway proposed here for NK cells can be extended to other types of immune cells. Understanding the molecular mechanisms involved in the dynamic regulation of the HIF-1α pathway in immune cells is of central importance to the immune cell function and could be a promising strategy in the design of treatments for human inflammatory diseases and cancer.

We extend recent work on nonlinear optimal control problems with integer restrictions on some of the control functions (mixed-integer optimal control problems, MIOCP). We improve a theorem (Sager et al. in Math Program 118(1): 109–149, 2009) that states that the solution of a relaxed and convexified problem can be approximated with arbitrary precision by a solution fulfilling the integer requirements. Unlike in previous publications the new proof avoids the usage of the Krein-Milman theorem, which is undesirable as it only states the existence of a solution that may switch infinitely often. We present a constructive way to obtain an integer solution with a guaranteed bound on the performance loss in polynomial time. We prove that this bound depends linearly on the control discretization grid. A numerical benchmark example illustrates the procedure. As a byproduct, we obtain an estimate of the Hausdorff distance between reachable sets. We improve the approximation order to linear grid size h instead of the previously known result with order (Häckl in Reachable sets, control sets and their computation, augsburger mathematisch-naturwissenschaftliche schriften. Dr. Bernd Wißner, Augsburg, 1996). We are able to include a Special Ordered Set condition which will allow for a transfer of the results to a more general, multi-dimensional and nonlinear case compared to the Theorems in Pietrus and Veliov in (Syst Control Lett 58:395–399, 2009).

In clinical diagnostics, both array comparative genomic hybridization (array CGH) and single nucleotide polymorphism (SNP) genotyping have proven to be powerful genomic technologies utilized for the evaluation of developmental delay, multiple congenital anomalies, and neuropsychiatric disorders. Differences in the ability to resolve genomic changes between these arrays may constitute an implementation challenge for clinicians: which platform (SNP vs array CGH) might best detect the underlying genetic cause for the disease in the patient? While only SNP arrays enable the detection of copy number neutral regions of absence of heterozygosity (AOH), they have limited ability to detect single-exon copy number variants (CNVs) due to the distribution of SNPs across the genome. To provide comprehensive clinical testing for both CNVs and copy-neutral AOH, we enhanced our custom-designed high-resolution oligonucleotide array that has exon-targeted coverage of 1860 genes with 60,000 SNP probes, referred to as Chromosomal Microarray Analysis - Comprehensive (CMA-COMP). Of the 3240 cases evaluated by this array, clinically significant CNVs were detected in 445 cases including 21 cases with exonic events. In addition, 162 cases (5.0%) showed at least one AOH region >10 Mb. We demonstrate that even though this array has a lower density of SNP probes than other commercially available SNP arrays, it reliably detected AOH events >10 Mb as well as exonic CNVs beyond the detection limitations of SNP genotyping. Thus, combining SNP probes and exon-targeted array CGH into one platform provides clinically useful genetic screening in an efficient manner.

Differential equation models that describe the dynamic changes of biochemical signaling states are important tools to understand cellular behavior. An essential task in building such representations is to infer the affinities, rate constants, and other parameters of a model from actual measurement data. However, intuitive measurement protocols often fail to generate data that restrict the range of possible parameter values. Here we utilized a numerical method to iteratively design optimal live-cell fluorescence microscopy experiments in order to reveal pharmacological and kinetic parameters of a phosphatidylinositol 3,4,5-trisphosphate (PIP(3)) second messenger signaling process that is deregulated in many tumors. The experimental approach included the activation of endogenous phosphoinositide 3-kinase (PI3K) by chemically induced recruitment of a regulatory peptide, reversible inhibition of PI3K using a kinase inhibitor, and monitoring of the PI3K-mediated production of PIP(3) lipids using the pleckstrin homology (PH) domain of Akt. We found that an intuitively planned and established experimental protocol did not yield data from which relevant parameters could be inferred. Starting from a set of poorly defined model parameters derived from the intuitively planned experiment, we calculated concentration-time profiles for both the inducing and the inhibitory compound that would minimize the predicted uncertainty of parameter estimates. Two cycles of optimization and experimentation were sufficient to narrowly confine the model parameters, with the mean variance of estimates dropping more than sixty-fold. Thus, optimal experimental design proved to be a powerful strategy to minimize the number of experiments needed to infer biological parameters from a cell signaling assay.

An efficient Newton-type scheme for the approximate on-line solution of optimization problems as they occur in optimal feedback control is presented. The scheme allows a fast reaction to disturbances by delivering approximations of the exact optimal feedback control which are iteratively refined during the runtime of the controlled process. The contractivity of this real-time iteration scheme is proven, and a bound on the loss of optimality---compared with the theoretical optimal solution---is given. The robustness and excellent real-time performance of the method is demonstrated in a numerical experiment, the control of an unstable system, namely, an airborne kite that shall fly loops.

Point mutations and genomic deletions of the CDKL5 (STK9) gene on chromosome Xp22 have been reported in patients with severe neurodevelopmental abnormalities, including Rett-like disorders. To date, only larger-sized (8-21 Mb) duplications harboring CDKL5 have been described. We report seven females and four males from seven unrelated families with CDKL5 duplications 540-935 kb in size. Three families of different ethnicities had identical 667kb duplications containing only the shorter CDKL5 isoform. Four affected boys, 8-14 years of age, and three affected girls, 6-8 years of age, manifested autistic behavior, developmental delay, language impairment, and hyperactivity. Of note, two boys and one girl had macrocephaly. Two carrier mothers of the affected boys reported a history of problems with learning and mathematics while at school. None of the patients had epilepsy. Similarly to CDKL5 mutations and deletions, the X-inactivation pattern in all six studied females was random. We hypothesize that the increased dosage of CDKL5 might have affected interactions of this kinase with its substrates, leading to perturbation of synaptic plasticity and learning, and resulting in autistic behavior, developmental and speech delay, hyperactivity, and macrocephaly.

Many practical optimal control problems include discrete decisions. These may be either time-independent parameters or time-dependent control functions as gears or valves that can only take discrete values at any given time. While great progress has been achieved in the solution of optimization problems involving integer variables, in particular mixed-integer linear programs, as well as in continuous optimal control problems, the combination of the two is yet an open field of research. We consider the question of lower bounds that can be obtained by a relaxation of the integer requirements. For general nonlinear mixed-integer programs such lower bounds typically suffer from a huge integer gap. We convexify (with respect to binary controls) and relax the original problem and prove that the optimal solution of this continuous control problem yields the best lower bound for the nonlinear integer problem. Building on this theoretical result we present a novel algorithm to solve mixed-integer optimal control problems, with a focus on discrete-valued control functions. Our algorithm is based on the direct multiple shooting method, an adaptive refinement of the underlying control discretization grid and tailored heuristic integer methods. Its applicability is shown by a challenging application, the energy optimal control of a subway train with discrete gears and velocity limits.

In nature, there are many processes that are assumed to run optimally, for example, human motion or, in particular, human gait. In this paper, we consider models of such processes, which can be described by optimal control problems. These models are characterized by quantities that are not known and cannot be measured (examples are system parameters or the constitution of the optimality criterion). Our goal is to determine these unknown quantities from measurement data, which is equivalent to solving a parameter estimation problem based on an optimal control model. We investigate two approaches for finding the solution of this hierarchical optimization problem; both follow the idea of replacing the underlying optimal control problem by its optimality conditions. The first approach is based on Pontryagin's maximum principle, whereas the second approach uses direct multiple shooting and the corresponding Karush--Kuhn--Tucker conditions. Both approaches pose numerical challenges that are discussed in this paper. Finally, we investigate the performance of our methods by means of a benchmark problem. [PUBLICATION ABSTRACT]

An adolescent with Hurler-Scheie syndrome is reported. This now 15 year-old-young woman was initially diagnosed at age 4. She was assessed for neurocognitive functioning at ages 5, 13, and 15 years. Results show a significant decline in intellectual functioning from the superior range to the average range from age 5 to age 13, and then no change from age 13 to age 15. The relationship between Hurler-Scheie syndrome, premorbid intellectual functioning, and cognitive-behavioral interventions are discussed in light of the longitudinal neurocognitive effects of this disease.