@article{DonatiWeber2022, author = {Donati, Luca and Weber, Marcus}, title = {Assessing transition rates as functions of environmental variables}, volume = {157}, journal = {The Journal of Chemical Physics}, number = {22}, publisher = {AIP Publishing}, doi = {10.1063/5.0109555}, pages = {224103-1 -- 224103-14}, year = {2022}, abstract = {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.}, language = {en} } @article{SechiFackeldeyChewleetal.2022, author = {Sechi, Renata and Fackeldey, Konstantin and Chewle, Surahit and Weber, Marcus}, title = {SepFree NMF: A Toolbox for Analyzing the Kinetics of Sequential Spectroscopic Data}, volume = {15}, journal = {Algorithms}, number = {9}, doi = {10.3390/a15090297}, pages = {297}, year = {2022}, abstract = {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.}, language = {en} } @article{DonatiWeberKeller2022, author = {Donati, Luca and Weber, Marcus and Keller, Bettina G.}, title = {A review of Girsanov Reweighting and of Square Root Approximation for building molecular Markov State Models}, volume = {63}, journal = {Journal of Mathematical Physics}, number = {12}, publisher = {AIP Publishing}, doi = {10.1063/5.0127227}, pages = {123306-1 -- 123306-21}, year = {2022}, abstract = {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.}, language = {en} } @article{RayFackeldeySteinetal.2023, author = {Ray, Sourav and Fackeldey, Konstantin and Stein, Christoph and Weber, Marcus}, title = {Coarse Grained MD Simulations of Opioid interactions with the ยต-opioid receptor and the surrounding lipid membrane}, volume = {3}, journal = {Biophysica}, number = {2}, doi = {10.3390/biophysica3020017}, pages = {263 -- 275}, year = {2023}, abstract = {In our previous studies, a new opioid (NFEPP) was developed to only selectively bind to the ๐œ‡-opoid receptor (MOR) in inflamed tissue and thus avoid the severe side effects of fentanyl. We know that NFEPP has a reduced binding affinity to MOR in healthy tissue. Inspired by the modelling and simulations performed by Sutcliffe et al., we present our own results of coarse-grained molecular dynamics simulations of fentanyl and NFEPP with regards to their interaction with the ๐œ‡-opioid receptor embedded within the lipid cell membrane. For technical reasons, we have slightly modified Sutcliffe's parametrisation of opioids. The pH-dependent opioid simulations are of interest because while fentanyl is protonated at the physiological pH, NFEPP is deprotonated due to its lower pKa value than that of fentanyl. Here, we analyse for the first time whether pH changes have an effect on the dynamical behaviour of NFEPP when it is inside the cell membrane. Besides these changes, our analysis shows a possible alternative interaction of NFEPP at pH 7.4 outside the binding region of the MOR. The interaction potential of NFEPP with MOR is also depicted by analysing the provided statistical molecular dynamics simulations with the aid of an eigenvector analysis of a transition rate matrix. In our modelling, we see differences in the XY-diffusion profiles of NFEPP compared with fentanyl in the cell membrane.}, language = {en} } @article{RaharinirinaFackeldeyWeber2022, author = {Raharinirina, N. Alexia and Fackeldey, Konstantin and Weber, Marcus}, title = {Qualitative Euclidean embedding of Disjoint Sets of Points}, year = {2022}, abstract = {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).}, language = {en} } @article{BoegeFritzeGoergenetal.2023, author = {Boege, Tobias and Fritze, Ren{\ยดe} and G{\"o}rgen, Christiane and Hanselmann, Jeroen and Iglezakis, Dorothea and Kastner, Lars and Koprucki, Thomas and Krause, Tabea and Lehrenfeld, Christoph and Polla, Silvia and Reidelbach, Marco and Riedel, Christian and Saak, Jens and Schembera, Bj{\"o}rn and Tabelow, Karsten and Weber, Marcus}, title = {Research-Data Management Planning in the German Mathematical Community}, volume = {130}, journal = {Eur. Math. Soc. Mag.}, doi = {10.4171/mag/152}, pages = {40 -- 47}, year = {2023}, abstract = {In this paper we discuss the notion of research data for the field of mathematics and report on the status quo of research-data management and planning. A number of decentralized approaches are presented and compared to needs and challenges faced in three use cases from different mathematical subdisciplines. We highlight the importance of tailoring research-data management plans to mathematicians' research processes and discuss their usage all along the data life cycle.}, language = {en} } @misc{ReidelbachWeber2022, author = {Reidelbach, Marco and Weber, Marcus}, title = {MaRDI - The mathematical Research Data Initiative}, journal = {Aktionstag Forschungsdaten}, doi = {10.5281/zenodo.7397588}, year = {2022}, language = {en} } @article{SikorskiNiknejadWeberetal.2024, author = {Sikorski, Alexander and Niknejad, Amir and Weber, Marcus and Donati, Luca}, title = {Tensor-SqRA: Modeling the transition rates of interacting molecular systems in terms of potential energies}, volume = {160}, journal = {Journal of Chemical Physics}, arxiv = {http://arxiv.org/abs/2311.09779}, doi = {10.1063/5.0187792}, pages = {104112}, year = {2024}, abstract = {Estimating the rate of rare conformational changes in molecular systems is one of the goals of molecular dynamics simulations. In the past few decades, a lot of progress has been done in data-based approaches toward this problem. In contrast, model-based methods, such as the Square Root Approximation (SqRA), directly derive these quantities from the potential energy functions. In this article, we demonstrate how the SqRA formalism naturally blends with the tensor structure obtained by coupling multiple systems, resulting in the tensor-based Square Root Approximation (tSqRA). It enables efficient treatment of high-dimensional systems using the SqRA and provides an algebraic expression of the impact of coupling energies between molecular subsystems. Based on the tSqRA, we also develop the projected rate estimation, a hybrid data-model-based algorithm that efficiently estimates the slowest rates for coupled systems. In addition, we investigate the possibility of integrating low-rank approximations within this framework to maximize the potential of the tSqRA.}, language = {en} } @inproceedings{SikorskiRabbenChewleetal.2025, author = {Sikorski, Alexander and Rabben, Robert Julian and Chewle, Surahit and Weber, Marcus}, title = {Capturing the Macroscopic Behaviour of Molecular Dynamics with Membership Functions}, booktitle = {Mathematical Optimization for Machine Learning: Proceedings of the MATH+ Thematic Einstein Semester 2023}, editor = {Fackeldey, K.}, publisher = {De Gruyter}, arxiv = {http://arxiv.org/abs/2404.10523}, doi = {10.1515/9783111376776-004}, pages = {41 -- 58}, year = {2025}, abstract = {Markov processes serve as foundational models in many scientific disciplines, such as molecular dynamics, and their simulation forms a common basis for analysis. While simulations produce useful trajectories, obtaining macroscopic information directly from microstate data presents significant challenges. This paper addresses this gap by introducing the concept of membership functions being the macrostates themselves. We derive equations for the holding times of these macrostates and demonstrate their consistency with the classical definition. Furthermore, we discuss the application of the ISOKANN method for learning these quantities from simulation data. In addition, we present a novel method for extracting transition paths based on the ISOKANN results and demonstrate its efficacy by applying it to simulations of the ๐œ‡-opioid receptor. With this approach we provide a new perspective on analyzing the macroscopic behaviour of Markov systems.}, language = {en} } @article{ErlekamZumbansenWeber2022, author = {Erlekam, Franziska and Zumbansen, Maximilian and Weber, Marcus}, title = {Parameter estimation on multivalent ITC data sets}, volume = {12}, journal = {Scientific Reports}, doi = {10.1038/s41598-022-17188-x}, pages = {13402}, year = {2022}, abstract = {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.}, language = {en} } @article{BennerBurgerGoeddekeetal.2022, author = {Benner, Peter and Burger, Michael and G{\"o}ddeke, Dominik and G{\"o}rgen, Christiane and Himpe, Christian and Heiland, Jan and Koprucki, Thomas and Ohlberger, Mario and Rave, Stephan and Reidelbach, Marco and Saak, Jens and Sch{\"o}bel, Anita and Tabelow, Karsten and Weber, Marcus}, title = {Die mathematische Forschungsdateninitiative in der NFDI: MaRDI (Mathematical Research Data Initiative)}, journal = {GAMM Rundbrief}, number = {1}, pages = {40 -- 43}, year = {2022}, language = {de} } @misc{WitzigBeckenbachEifleretal.2016, author = {Witzig, Jakob and Beckenbach, Isabel and Eifler, Leon and Fackeldey, Konstantin and Gleixner, Ambros and Grever, Andreas and Weber, Marcus}, title = {Mixed-Integer Programming for Cycle Detection in Non-reversible Markov Processes}, issn = {1438-0064}, doi = {10.1137/16M1091162}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-60353}, year = {2016}, abstract = {In this paper, we present a new, optimization-based method to exhibit cyclic behavior in non-reversible stochastic processes. While our method is general, it is strongly motivated by discrete simulations of ordinary differential equations representing non-reversible biological processes, in particular molecular simulations. Here, the discrete time steps of the simulation are often very small compared to the time scale of interest, i.e., of the whole process. In this setting, the detection of a global cyclic behavior of the process becomes difficult because transitions between individual states may appear almost reversible on the small time scale of the simulation. We address this difficulty using a mixed-integer programming model that allows us to compute a cycle of clusters with maximum net flow, i.e., large forward and small backward probability. For a synthetic genetic regulatory network consisting of a ring-oscillator with three genes, we show that this approach can detect the most productive overall cycle, outperforming classical spectral analysis methods. Our method applies to general non-equilibrium steady state systems such as catalytic reactions, for which the objective value computes the effectiveness of the catalyst.}, language = {en} } @article{WitzigBeckenbachEifleretal.2018, author = {Witzig, Jakob and Beckenbach, Isabel and Eifler, Leon and Fackeldey, Konstantin and Gleixner, Ambros and Grever, Andreas and Weber, Marcus}, title = {Mixed-Integer Programming for Cycle Detection in Non-reversible Markov Processes}, volume = {16}, journal = {Multiscale Modeling and Simulation}, number = {1}, issn = {1438-0064}, doi = {10.1137/16M1091162}, pages = {248 -- 265}, year = {2018}, abstract = {In this paper, we present a new, optimization-based method to exhibit cyclic behavior in non-reversible stochastic processes. While our method is general, it is strongly motivated by discrete simulations of ordinary differential equations representing non-reversible biological processes, in particular molecular simulations. Here, the discrete time steps of the simulation are often very small compared to the time scale of interest, i.e., of the whole process. In this setting, the detection of a global cyclic behavior of the process becomes difficult because transitions between individual states may appear almost reversible on the small time scale of the simulation. We address this difficulty using a mixed-integer programming model that allows us to compute a cycle of clusters with maximum net flow, i.e., large forward and small backward probability. For a synthetic genetic regulatory network consisting of a ring-oscillator with three genes, we show that this approach can detect the most productive overall cycle, outperforming classical spectral analysis methods. Our method applies to general non-equilibrium steady state systems such as catalytic reactions, for which the objective value computes the effectiveness of the catalyst.}, language = {en} } @article{DonatiSchuetteWeber2025, author = {Donati, Luca and Sch{\"u}tte, Christof and Weber, Marcus}, title = {The Kramers turnover in terms of a macro-state projection on phase space}, volume = {123}, journal = {Molecular Physics}, number = {7-8: Giovanni Ciccotti: A Renaissance Physicist}, publisher = {Taylor \& Francis}, doi = {10.1080/00268976.2024.2356748}, pages = {e2356748}, year = {2025}, abstract = {We have investigated how Langevin dynamics is affected by the friction coefficient using the novel algorithm ISOKANN, which combines the transfer operator approach with modern machine learning techniques. ISOKANN describes the dynamics in terms of an invariant subspace projection of the Koopman operator defined in the entire state space, avoiding approximations due to dimensionality reduction and discretization. Our results are consistent with the Kramers turnover and show that in the low and moderate friction regimes, metastable macro-states and transition rates are defined in phase space, not only in position space.}, language = {en} } @article{KresseSikorskiChewleetal.2026, author = {Kresse, Jakob and Sikorski, Alexander and Chewle, Surahit and Sunkara, Vikram and Weber, Marcus}, title = {Revealing the Atomistic Mechanism of Rare Events in Molecular Dynamics}, volume = {22}, journal = {Journal of Chemical Theory and Computation}, number = {5}, arxiv = {http://arxiv.org/abs/2511.15514}, doi = {10.1021/acs.jctc.5c01906}, pages = {2380 -- 2389}, year = {2026}, abstract = {Interpretable reaction coordinates are essential for understanding rare conformational transitions in molecular dynamics. The Atomistic Mechanism Of Rare Events in Molecular Dynamics (AMORE-MD) framework enhances interpretability of deep-learned reaction coordinates by connecting them to atomistic mechanisms, without requiring any a priori knowledge of collective variables, pathways, or endpoints. Here, AMORE-MD employs the ISOKANN algorithm to learn a neural membership function ฯ‡ representing the dominant slow process, from which transition pathways are reconstructed as minimum-energy paths aligned with the gradient of ฯ‡, and atomic contributions are quantified through gradient-based sensitivity analysis. Iterative enhanced sampling further enriches transition regions and improves coverage of rare events enabling recovery of known mechanisms and chemically interpretable structural rearrangements at atomic resolution for the M{\"u}ller-Brown potential, alanine dipeptide, and the elastin-derived hexapeptide VGVAPG.}, language = {en} } @article{SchuetteSikorskiKresseetal.2026, author = {Sch{\"u}tte, Christof and Sikorski, Alexander and Kresse, Jakob and Weber, Marcus}, title = {On-the-Fly Lifting of Coarse Reaction-Coordinate Paths to Full-Dimensional Transition Path Ensembles}, arxiv = {http://arxiv.org/abs/2603.23266}, doi = {10.48550/arXiv.2603.23266}, year = {2026}, abstract = {Effective dynamics on a low-dimensional collective-variable (CV) or latent space can be simulated far more cheaply than the underlying high-dimensional stochastic system, but exploiting such coarse predictions requires lifting: turning a coarse CV trajectory into dynamically consistent full-dimensional states and path ensembles, without relying on global sampling of invariant or conditional fiber measures. We present a local, on-the-fly lifting strategy based on guided full-system trajectories. First an effective model in CV space is used to obtain a coarse reference trajectory. Then, an ensemble of full-dimensional trajectories is generated from a guided version of the original dynamics, where the guidance steers the trajectory to track the CV reference path. Because guidance biases the path distribution, we correct it via pathwise Girsanov reweighting, yielding a correct-by-construction importance-sampling approximation of the conditional law of the uncontrolled dynamics. We further connect the approach to stochastic optimal control, clarifying how coarse models can inform variance-reducing guidance for rare-event quantities. Numerical experiments demonstrate that inexpensive coarse transition paths can be converted into realistic full-system transition pathways (including barrier crossings and detours) and can accelerate estimation of transition pathways and statistics while providing minimal bias through weighted ensembles.}, language = {en} }