Cheap Newton steps for optimal control problems: automatic differentiation and Pantoja's algorithm

Original article can be found at: http://www.informaworld.com/smpp/title~content=t713645924~db=all Copyright Taylor and Francis / Informa.In this paper we discuss Pantoja's construction of the Newton direction for discrete time optimal control problems. We show that automatic differentiation (AD) techniques can be used to calculate the Newton direction accurately, without requiring extensive re-writing of user code, and at a surprisingly low computational cost: for an N-step problem with p control variables and q state variables at each step, the worst case cost is 6(p + q + 1) times the computational cost of a single target function evaluation, independent of N, together with at most p3/3 + p2(q + 1) + 2p(q + 1)2 + (q + l)3, i.e. less than (p + q + l)3, floating point multiply-and-add operations per time step. These costs may be considerably reduced if there is significant structural sparsity in the problem dynamics. The systematic use of checkpointing roughly doubles the operation counts, but reduces the total space cost to the order of 4pN floating point stores. A naive approach to finding the Newton step would require the solution of an Np Np system of equations together with a number of function evaluations proportional to Np, so this approach to Pantoja's construction is extremely attractive, especially if q is very small relative to N. Straightforward modifications of the AD algorithms proposed here can be used to implement other discrete time optimal control solution techniques, such as differential dynamic programming (DDP), which use state-control feedback. The same techniques also can be used to determine with certainty, at the cost of a single Newton direction calculation, whether or not the Hessian of the target function is sufficiently positive definite at a point of interest. This allows computationally cheap post-hoc verification that a second-order minimum has been reached to a given accuracy, regardless of what method has been used to obtain it.Peer reviewe

Mathematical modelling of promoter occupancies in MYC-dependent gene regulation

The human MYC proto-oncogene protein (MYC) is a transcription factor that plays a major role in the regulation of cell proliferation. Deregulation of MYC expression is often found in cancer. In the last years, several hypotheses have been proposed to explain cell type specific MYC target gene expression patterns despite genome wide DNA binding of MYC. In a recent publication, a mathematical modelling approach in combination with experimental data demonstrated that differences in MYC-DNA-binding affinity are sufficient to explain distinct promoter occupancies and allow stratification of distinct MYC-regulated biological processes at different MYC concentrations. Here, we extend the analysis of the published mathematical model of DNA-binding behaviour of MYC to demonstrate that the insights gained in the investigation of the human osteosarcoma cell line U2OS can be generalized to other human cell types

Paracrine and autocrine regulation of gene expression by Wnt-inhibitor Dickkopf in wild-type and mutant hepatocytes

BACKGROUND: Cells are able to communicate and coordinate their function within tissues via secreted factors. Aberrant secretion by cancer cells can modulate this intercellular communication, in particular in highly organised tissues such as the liver. Hepatocytes, the major cell type of the liver, secrete Dickkopf (Dkk), which inhibits Wnt/β-catenin signalling in an autocrine and paracrine manner. Consequently, Dkk modulates the expression of Wnt/β-catenin target genes. We present a mathematical model that describes the autocrine and paracrine regulation of hepatic gene expression by Dkk under wild-type conditions as well as in the presence of mutant cells. RESULTS: Our spatial model describes the competition of Dkk and Wnt at receptor level, intra-cellular Wnt/β-catenin signalling, and the regulation of target gene expression for 21 individual hepatocytes. Autocrine and paracrine regulation is mediated through a feedback mechanism via Dkk and Dkk diffusion along the porto-central axis. Along this axis an APC concentration gradient is modelled as experimentally detected in liver. Simulations of mutant cells demonstrate that already a single mutant cell increases overall Dkk concentration. The influence of the mutant cell on gene expression of surrounding wild-type hepatocytes is limited in magnitude and restricted to hepatocytes in close proximity. To explore the underlying molecular mechanisms, we perform a comprehensive analysis of the model parameters such as diffusion coefficient, mutation strength and feedback strength. CONCLUSIONS: Our simulations show that Dkk concentration is elevated in the presence of a mutant cell. However, the impact of these elevated Dkk levels on wild-type hepatocytes is confined in space and magnitude. The combination of inter- and intracellular processes, such as Dkk feedback, diffusion and Wnt/β-catenin signal transduction, allow wild-type hepatocytes to largely maintain their gene expression

Modeling Wnt/β-catenin target gene expression in APC and Wnt gradients under wild type and mutant conditions

The Wnt/β-catenin pathway is involved in the regulation of a multitude of physiological processes by controlling the differential expression of target genes. In certain tissues such as the adult liver, the Wnt/β-catenin pathway can attain different levels of activity due to gradients of Wnt ligands and/or intracellular pathway components like APC. How graded pathway activity is converted into regionally distinct patterns of Wnt/β-catenin target gene expression is largely unknown. Here, we apply a mathematical modeling approach to investigate the impact of different regulatory mechanisms on target gene expression within Wnt or APC concentration gradients. We develop a minimal model of Wnt/beta-catenin signal transduction and combine it with various mechanisms of target gene regulation. In particular, the effects of activation, inhibition, and an incoherent feedforward loop (iFFL) are compared. To specify activation kinetics, we analyze experimental data that quantify the response of β-catenin/TCF reporter constructs to different Wnt concentrations, and demonstrate that the induction of these constructs occurs in a cooperative manner with Hill coefficients between 2 and 5. In summary, our study shows that the combination of specific gene regulatory mechanisms with a time-independent gradient of Wnt or APC is sufficient to generate distinct target gene expression patterns as have been experimentally observed in liver. We find that cooperative gene activation in combination with a TCF feedback can establish sharp borders of target gene expression in Wnt or APC gradients. In contrast, the iFFL renders gene expression independent of gradients of the upstream signaling components. Our subsequent analysis of carcinogenic pathway mutations reveals that their impact on gene expression is determined by the gene regulatory mechanism and the APC concentration of the cell in which the mutation occurs

IL-2 Expression in Activated Human Memory FOXP3+ Cells Critically Depends on the Cellular Levels of FOXP3 as Well as of Four Transcription Factors of T Cell Activation

The human CD4+FOXP3+ T cell population is heterogeneous and consists of various subpopulations which remain poorly defined. Anergy and suppression are two main functional characteristics of FOXP3+Treg cells. We used the anergic behavior of FOXP3+Treg cells for a better discrimination and characterization of such subpopulations. We compared IL-2-expressing with IL-2-non-expressing cells within the memory FOXP3+ T cell population. In contrast to IL-2-non-expressing FOXP3+ cells, IL-2-expressing FOXP3+ cells exhibit intermediate characteristics of Treg and Th cells concerning the Treg cell markers CD25, GITR, and Helios. Besides lower levels of FOXP3, they also have higher levels of the transcription factors NFATc2, c-Fos, NF-κBp65, and c-Jun. An approach combining flow cytometric measurements with statistical interpretation for quantitative transcription factor analysis suggests that the physiological expression levels not only of FOXP3 but also of NFATc2, c-Jun, c-Fos, and NF-κBp65 are limiting for the decision whether IL-2 is expressed or not in activated peripheral human memory FOXP3+ cells. These findings demonstrate that concomitant high levels of NFATc2, c-Jun, c-Fos, and NF-κBp65 lead in addition to potential IL-2 expression in those FOXP3+ cells with low levels of FOXP3. We hypothesize that not only the level of FOXP3 expression but also the amounts of the four transcription factors studied represent determining factors for the anergic phenotype of FOXP3+ Treg cells.Peer Reviewe