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The characterization and numerical solution of two non-smooth optimal control problems governed by a Fokker–Planck (FP) equation are investigated in the framework of the Pontryagin maximum principle (PMP). The two FP control problems are related to the problem of determining open- and closed-loop controls for a stochastic process whose probability density function is modelled by the FP equation. In both cases, existence and PMP characterisation of optimal controls are proved, and PMP-based numerical optimization schemes are implemented that solve the PMP optimality conditions to determine the controls sought. Results of experiments are presented that successfully validate the proposed computational framework and allow to compare the two control strategies.
The signal modelling framework JimenaE simulates dynamically Boolean networks. In contrast to SQUAD, there is systematic and not just heuristic calculation of all system states. These specific features are not present in CellNetAnalyzer and BoolNet. JimenaE is an expert extension of Jimena, with new optimized code, network conversion into different formats, rapid convergence both for system state calculation as well as for all three network centralities. It allows higher accuracy in determining network states and allows to dissect networks and identification of network control type and amount for each protein with high accuracy. Biological examples demonstrate this: (i) High plasticity of mesenchymal stromal cells for differentiation into chondrocytes, osteoblasts and adipocytes and differentiation-specific network control focusses on wnt-, TGF-beta and PPAR-gamma signaling. JimenaE allows to study individual proteins, removal or adding interactions (or autocrine loops) and accurately quantifies effects as well as number of system states. (ii) Dynamical modelling of cell–cell interactions of plant Arapidopsis thaliana against Pseudomonas syringae DC3000: We analyze for the first time the pathogen perspective and its interaction with the host. We next provide a detailed analysis on how plant hormonal regulation stimulates specific proteins and who and which protein has which type and amount of network control including a detailed heatmap of the A.thaliana response distinguishing between two states of the immune response. (iii) In an immune response network of dendritic cells confronted with Aspergillus fumigatus, JimenaE calculates now accurately the specific values for centralities and protein-specific network control including chemokine and pattern recognition receptors.
Machine learning techniques are excellent to analyze expression data from single cells. These techniques impact all fields ranging from cell annotation and clustering to signature identification. The presented framework evaluates gene selection sets how far they optimally separate defined phenotypes or cell groups. This innovation overcomes the present limitation to objectively and correctly identify a small gene set of high information content regarding separating phenotypes for which corresponding code scripts are provided. The small but meaningful subset of the original genes (or feature space) facilitates human interpretability of the differences of the phenotypes including those found by machine learning results and may even turn correlations between genes and phenotypes into a causal explanation. For the feature selection task, the principal feature analysis is utilized which reduces redundant information while selecting genes that carry the information for separating the phenotypes. In this context, the presented framework shows explainability of unsupervised learning as it reveals cell-type specific signatures. Apart from a Seurat preprocessing tool and the PFA script, the pipeline uses mutual information to balance accuracy and size of the gene set if desired. A validation part to evaluate the gene selection for their information content regarding the separation of the phenotypes is provided as well, binary and multiclass classification of 3 or 4 groups are studied. Results from different single-cell data are presented. In each, only about ten out of more than 30000 genes are identified as carrying the relevant information. The code is provided in a GitHub repository at https://github.com/AC-PHD/Seurat_PFA_pipeline.
A sequential quadratic Hamiltonian (SQH) scheme for solving different classes of non-smooth and non-convex PDE optimal control problems is investigated considering seven different benchmark problems with increasing difficulty. These problems include linear and nonlinear PDEs with linear and bilinear control mechanisms, non-convex and discontinuous costs of the controls, L\(^1\) tracking terms, and the case of state constraints.
The SQH method is based on the characterisation of optimality of PDE optimal control problems by the Pontryagin's maximum principle (PMP). For each problem, a theoretical discussion of the PMP optimality condition is given and results of numerical experiments are presented that demonstrate the large range of applicability of the SQH scheme.
A mathematical optimal-control tumor therapy framework consisting of radio- and anti-angiogenesis control strategies that are included in a tumor growth model is investigated. The governing system, resulting from the combination of two well established models, represents the differential constraint of a non-smooth optimal control problem that aims at reducing the volume of the tumor while keeping the radio- and anti-angiogenesis chemical dosage to a minimum. Existence of optimal solutions is proved and necessary conditions are formulated in terms of the Pontryagin maximum principle. Based on this principle, a so-called sequential quadratic Hamiltonian (SQH) method is discussed and benchmarked with an “interior point optimizer―a mathematical programming language” (IPOPT-AMPL) algorithm. Results of numerical experiments are presented that successfully validate the SQH solution scheme. Further, it is shown how to choose the optimisation weights in order to obtain treatment functions that successfully reduce the tumor volume to zero.
Mathematical optimization framework allows the identification of certain nodes within a signaling network. In this work, we analyzed the complex extracellular-signal-regulated kinase 1 and 2 (ERK1/2) cascade in cardiomyocytes using the framework to find efficient adjustment screws for this cascade that is important for cardiomyocyte survival and maladaptive heart muscle growth. We modeled optimal pharmacological intervention points that are beneficial for the heart, but avoid the occurrence of a maladaptive ERK1/2 modification, the autophosphorylation of ERK at threonine 188 (ERK\(^{Thr188}\) phosphorylation), which causes cardiac hypertrophy. For this purpose, a network of a cardiomyocyte that was fitted to experimental data was equipped with external stimuli that model the pharmacological intervention points. Specifically, two situations were considered. In the first one, the cardiomyocyte was driven to a desired expression level with different treatment strategies. These strategies were quantified with respect to beneficial effects and maleficent side effects and then which one is the best treatment strategy was evaluated. In the second situation, it was shown how to model constitutively activated pathways and how to identify drug targets to obtain a desired activity level that is associated with a healthy state and in contrast to the maleficent expression pattern caused by the constitutively activated pathway. An implementation of the algorithms used for the calculations is also presented in this paper, which simplifies the application of the presented framework for drug targeting, optimal drug combinations and the systematic and automatic search for pharmacological intervention points. The codes were designed such that they can be combined with any mathematical model given by ordinary differential equations.
Once biological systems are modeled by regulatory networks, the next step is to include external stimuli, which model the experimental possibilities to affect the activity level of certain network’s nodes, in a mathematical framework. Then, this framework can be interpreted as a mathematical optimal control framework such that optimization algorithms can be used to determine external stimuli which cause a desired switch from an initial state of the network to another final state. These external stimuli are the intervention points for the corresponding biological experiment to obtain the desired outcome of the considered experiment. In this work, the model of regulatory networks is extended to controlled regulatory networks. For this purpose, external stimuli are considered which can affect the activity of the network’s nodes by activation or inhibition. A method is presented how to calculate a selection of external stimuli which causes a switch between two different steady states of a regulatory network. A software solution based on Jimena and Mathworks Matlab is provided. Furthermore, numerical examples are presented to demonstrate application and scope of the software on networks of 4 nodes, 11 nodes and 36 nodes. Moreover, we analyze the aggregation of platelets and the behavior of a basic T-helper cell protein-protein interaction network and its maturation towards Th0, Th1, Th2, Th17 and Treg cells in accordance with experimental data.