TY - JOUR A1 - Heida, Martin A1 - Sikorski, Alexander A1 - Weber, Marcus T1 - Consistency and order 1 convergence of cell-centered finite volume discretizations of degenerate elliptic problems in any space dimension JF - SIAM Journal on Numerical Analysis N2 - We study consistency of cell-centered finite difference methods for elliptic equations with degenerate coefficients in any space dimension $d \geq 2$. This results in order of convergence estimates in the natural weighted energy norm and in the weighted discrete $L^2$-norm on admissible meshes. The cells of meshes under consideration may be very irregular in size. We particularly allow the size of certain cells to remain bounded from below even in the asymptotic limit. For uniform meshes we show that the order of convergence is at least 1 in the energy semi-norm, provided the discrete and continuous solutions exist and the continuous solution has $H^2$ regularity. Y1 - 2022 U6 - https://doi.org/10.20347/WIAS.PREPRINT.2913 ER - TY - JOUR A1 - Fernandes, Rita A1 - Chaowdhary, Suvrat A1 - Saleh, Noureldin A1 - Mikula, Natalia A1 - Kanevche, Katerina A1 - Berlepsch, Hans A1 - Hosogi, Naoki A1 - Heberle, Joachim A1 - Weber, Marcus A1 - Böttcher, Christoph A1 - Koksch, Beate T1 - Cyanine Dye Coupling Mediates Self-assembly of a pH Sensitive Peptide into Novel 3D Architectures JF - Angewandte Chemie N2 - A conjugated Cy5 dye-peptide system reveals the formation of two novel and structurally distinct supramolecular assemblies with photo-physical characteristics of H-type dimers or tetramers, respectively. The molecular ultrastructures are triggered by the complementary interplay of mutual chromophore coupling and pH induced changes in the peptide charge pattern. Y1 - 2022 VL - 61 IS - 48 SP - e202208647 ER - TY - JOUR A1 - Donati, Luca A1 - Weber, Marcus T1 - Assessing transition rates as functions of environmental variables JF - The Journal of Chemical Physics N2 - We present a method to estimate the transition rates of molecular systems under different environmental conditions which cause the formation or the breaking of bonds and require the sampling of the Grand Canonical Ensemble. For this purpose, we model the molecular system in terms of probable "scenarios", governed by different potential energy functions, which are separately sampled by classical MD simulations. Reweighting the canonical distribution of each scenario according to specific environmental variables, we estimate the grand canonical distribution, then we use the Square Root Approximation (SqRA) method to discretize the Fokker-Planck operator into a rate matrix and the robust Perron Cluster Cluster Analysis (PCCA+) method to coarse-grain the kinetic model. This permits to efficiently estimate the transition rates of conformational states as functions of environmental variables, for example, the local pH at a cell membrane. In this work we formalize the theoretical framework of the procedure and we present a numerical experiment comparing the results with those provided by a constant-pH method based on non-equilibrium Molecular Dynamics Monte Carlo simulations. The method is relevant for the development of new drug design strategies which take into account how the cellular environment influences biochemical processes. Y1 - 2022 U6 - https://doi.org/10.1063/5.0109555 VL - 157 IS - 22 SP - 224103-1 EP - 224103-14 PB - AIP Publishing ER - TY - JOUR A1 - Sechi, Renata A1 - Fackeldey, Konstantin A1 - Chewle, Surahit A1 - Weber, Marcus T1 - SepFree NMF: A Toolbox for Analyzing the Kinetics of Sequential Spectroscopic Data JF - Algorithms N2 - This work addresses the problem of determining the number of components from sequential spectroscopic data analyzed by non-negative matrix factorization without separability assumption (SepFree NMF). These data are stored in a matrix M of dimension “measured times” versus “measured wavenumbers” and can be decomposed to obtain the spectral fingerprints of the states and their evolution over time. SepFree NMF assumes a memoryless (Markovian) process to underline the dynamics and decomposes M so that M=WH, with W representing the components’ fingerprints and H their kinetics. However, the rank of this decomposition (i.e., the number of physical states in the process) has to be guessed from pre-existing knowledge on the observed process. We propose a measure for determining the number of components with the computation of the minimal memory effect resulting from the decomposition; by quantifying how much the obtained factorization is deviating from the Markovian property, we are able to score factorizations of a different number of components. In this way, we estimate the number of different entities which contribute to the observed system, and we can extract kinetic information without knowing the characteristic spectra of the single components. This manuscript provides the mathematical background as well as an analysis of computer generated and experimental sequentially measured Raman spectra. Y1 - 2022 U6 - https://doi.org/10.3390/a15090297 VL - 15 IS - 9 SP - 297 ER - TY - JOUR A1 - Donati, Luca A1 - Weber, Marcus A1 - Keller, Bettina G. T1 - A review of Girsanov Reweighting and of Square Root Approximation for building molecular Markov State Models JF - Journal of Mathematical Physics N2 - Dynamical reweighting methods permit to estimate kinetic observables of a stochastic process governed by a target potential U(x) from trajectories that have been generated at a different potential V(x). In this article, we present Girsanov reweighting and Square Root Approximation (SqRA): the first method reweights path probabilities exploiting the Girsanov theorem and can be applied to Markov State Models (MSMs) to reweight transition probabilities; the second method was originally developed to discretize the Fokker-Planck operator into a transition rate matrix, but here we implement it into a reweighting scheme for transition rates. We begin by reviewing the theoretical background of the methods, then present two applications relevant to Molecular Dynamics (MD), highlighting their strengths and weaknesses. Y1 - 2022 U6 - https://doi.org/10.1063/5.0127227 VL - 63 IS - 12 SP - 123306-1 EP - 123306-21 PB - AIP Publishing ER - TY - JOUR A1 - Raharinirina, N. Alexia A1 - Fackeldey, Konstantin A1 - Weber, Marcus T1 - Qualitative Euclidean embedding of Disjoint Sets of Points N2 - We consider two disjoint sets of points with a distance metric, or a proximity function, associated with each set. If each set can be separately embedded into separate Euclidean spaces, then we provide sufficient conditions for the two sets to be jointly embedded in one Euclidean space. In this joint Euclidean embedding, the distances between the points are generated by a specific relation-preserving function. Consequently, the mutual distances between two points of the same set are specific qualitative transformations of their mutual distances in their original space; the pairwise distances between the points of different sets can be constructed from an arbitrary proximity function (might require scaling). Y1 - 2022 UR - https://arxiv.org/abs/2212.00058 ER - TY - GEN A1 - Reidelbach, Marco A1 - Weber, Marcus T1 - MaRDI - The mathematical Research Data Initiative T2 - Aktionstag Forschungsdaten Y1 - 2022 U6 - https://doi.org/10.5281/zenodo.7397588 ER - TY - JOUR A1 - Erlekam, Franziska A1 - Zumbansen, Maximilian A1 - Weber, Marcus T1 - Parameter estimation on multivalent ITC data sets JF - Scientific Reports N2 - The Wiseman fitting can be used to extract binding parameters from ITC data sets, such as heat of binding, number of binding sites, and the overall dissociation rate. The classical Wiseman fitting assumes a direct binding process and neglects the possibility of intermediate binding steps. In principle, it only provides thermodynamic information and not the kinetics of the process. In this article we show that a concentration dependent dissociation constant could possibly stem from intermediate binding steps. The mathematical form of this dependency can be exploited with the aid of the Robust Perron Cluster Cluster Analysis method. Our proposed extension of the Wiseman fitting rationalizes the concentration dependency, and can probably also be used to determine the kinetic parameters of intermediate binding steps of a multivalent binding process. The novelty of this paper is to assume that the binding rate varies per titration step due to the change of the ligand concentration and to use this information in the Wiseman fitting. We do not claim to produce the most accurate values of the binding parameters, we rather present a novel method of how to approach multivalent bindings from a different angle. Y1 - 2022 U6 - https://doi.org/10.1038/s41598-022-17188-x VL - 12 SP - 13402 ER - TY - JOUR A1 - Benner, Peter A1 - Burger, Michael A1 - Göddeke, Dominik A1 - Görgen, Christiane A1 - Himpe, Christian A1 - Heiland, Jan A1 - Koprucki, Thomas A1 - Ohlberger, Mario A1 - Rave, Stephan A1 - Reidelbach, Marco A1 - Saak, Jens A1 - Schöbel, Anita A1 - Tabelow, Karsten A1 - Weber, Marcus T1 - Die mathematische Forschungsdateninitiative in der NFDI: MaRDI (Mathematical Research Data Initiative) JF - GAMM Rundbrief Y1 - 2022 IS - 1 SP - 40 EP - 43 ER -