COnstraint-Based Reconstruction and Analysis (COBRA) provides a molecular mechanistic framework for integrative analysis of experimental data and quantitative prediction of physicochemically and biochemically feasible phenotypic states. The COBRA Toolbox is a comprehensive software suite of interoperable COBRA methods. It has found widespread applications in biology, biomedicine, and biotechnology because its functions can be flexibly combined to implement tailored COBRA protocols for any biochemical network. Version 3.0 includes new methods for quality controlled reconstruction, modelling, topological analysis, strain and experimental design, network visualisation as well as network integration of chemoinformatic, metabolomic, transcriptomic, proteomic, and thermochemical data. New multi-lingual code integration also enables an expansion in COBRA application scope via high-precision, high-performance, and nonlinear numerical optimisation solvers for multi-scale, multi-cellular and reaction kinetic modelling, respectively. This protocol can be adapted for the generation and analysis of a constraint-based model in a wide variety of molecular systems biology scenarios. This protocol is an update to the COBRA Toolbox 1.0 and 2.0. The COBRA Toolbox 3.0 provides an unparalleled depth of constraint-based reconstruction and analysis methods. ]); 61 | The MUST sets are the sets of reactions that must increase or decrease their flux in order to achieve the desired phenotype in the mutant strain. As shown in Figure 6, the first order MUST sets are MustU and MustL while second order MUST sets are denoted as MustUU, MustLL, and MustUL. After parameters and constraints are defined, the functions findMustL and findMustU are run to determine the mustU and mustL sets, respectively. Define an ID of the run with:Each time the MUST sets are determined, folders are generated to read inputs and store outputs, i.e., reports. These folders are located in the directory defined by the uniquely defined runID.62 | In order to find the first order MUST sets, constraints should be defined: >> constrOpt = struct('rxnList', {{'EX_gluc', 'R75', 'EX_suc'}}, 'values', [-100; 0; 155.5]); 63 | The first order MUST set MustL is determined by running: >> [mustLSet, pos_mustL] = findMustL(model, minFluxesW, maxFluxesW, ... 'constrOpt', constrOpt, 'runID', runID);If runID is set to 'TestoptForceL', a folder TestoptForceL is created, in which two additional folders InputsMustL and OutputsMustL are created. The InputsMustL folder contains all the inputs required to run the function findMustL, while the OutputsMustL folder contains the mustL set found and a report that summarises all the inputs and outputs. In order to maintain a chronological order of computational experiments, the report is timestamped.64 | Display the reactions that belong to the mustL set using: >> disp(mustLSet) 65 | The first order MUST set MustU is determined by running: >> [mustUSet, pos_mustU] = findMustU(model, minFluxesW, maxFluxesW, ... 'constrOpt', constrOpt, 'runID', runID);...

We describe and analyse Levenberg-Marquardt methods for solving systems of nonlinear equations. More specifically, we propose an adaptive formula for the Levenberg-Marquardt parameter and analyse the local convergence of the method under Hölder metric subregularity of the function defining the equation and Hölder continuity of its gradient mapping. Further, we analyse the local convergence of the method under the additional assumption that the Lojasiewicz gradient inequality holds. We finally report encouraging numerical results confirming the theoretical findings for the problem of computing moiety conserved steady states in biochemical reaction networks. This problem can be cast as finding a solution of a system of nonlinear equations, where the associated mapping satisfies the Lojasiewicz gradient inequality assumption.Keywords Nonlinear equation · Levenberg-Marquardt method · Local convergence rate · Hölder metric subregularity · Lojasiewicz inequality Mathematics Subject Classification (2010) 65K05 · 65K10 · 90C26 · 92C42 F.J.

Abstract. The paper concerns the study of variational systems described by parameterized generalized equations/variational conditions important for many aspects of nonlinear analysis, optimization, and their applications. Focusing on the fundamental properties of metric regularity and Lipschitzian stability, we establish various qualitative and quantitative relationships between these properties for multivalued parts/fields of parametric generalized equations and the corresponding solution maps for them in the framework of arbitrary Banach spaces of decision and parameter variables.

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Abstract. For a version of Newton's method applied to a generalized equation with a parameter, we extend the paradigm of the Lyusternik-Graves theorem to the framework of a mapping acting from the pair "parameter-starting point" to the set of corresponding convergent Newton sequences. Under ample parameterization, metric regularity of the mapping associated with convergent Newton sequences becomes equivalent to the metric regularity of the mapping associated with the generalized equation. We also discuss an inexact Newton method and present an application to discretized optimal control.