@misc{ZhangWangHartmannetal., author = {Zhang, Wei and Wang, Han and Hartmann, Carsten and Weber, Marcus and Sch{\"u}tte, Christof}, title = {Applications of the cross-entropy method to importance sampling and optimal control of diffusions}, issn = {1438-0064}, doi = {10.1137/14096493X}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-49720}, abstract = {We study the cross-entropy method for diffusions. One of the results is a versatile cross-entropy algorithm that can be used to design efficient importance sampling strategies for rare events or to solve optimal control problems. The approach is based on the minimization of a suitable cross-entropy functional, with a parametric family of exponentially tilted probability distributions. We illustrate the new algorithm with several numerical examples and discuss algorithmic issues and possible extensions of the method.}, language = {en} } @article{ZhangWangHartmannetal., author = {Zhang, Wei and Wang, Han and Hartmann, Carsten and Weber, Marcus and Sch{\"u}tte, Christof}, title = {Applications of the cross-entropy method to importance sampling and optimal control of diffusions}, series = {Siam Journal on Scientific Computing}, volume = {36}, journal = {Siam Journal on Scientific Computing}, number = {6}, doi = {10.1137/14096493X}, pages = {A2654 -- A2672}, language = {en} } @article{WeberFackeldeySchuette, author = {Weber, Marcus and Fackeldey, Konstantin and Sch{\"u}tte, Christof}, title = {Set-Free Markov State Model Building}, series = {Journal of Chemical Physics}, volume = {146}, journal = {Journal of Chemical Physics}, number = {12}, doi = {10.1063/1.4978501}, language = {en} } @misc{WeberFackeldeySchuette, author = {Weber, Marcus and Fackeldey, Konstantin and Sch{\"u}tte, Christof}, title = {Set-free Markov State Building}, issn = {1438-0064}, doi = {10.1063/1.4978501}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-62167}, abstract = {Molecular dynamics (MD) simulations face challenging problems since the timescales of interest often are much longer than what is possible to simulate and even if sufficiently long simulation are possible the complex nature of the resulting simulation data makes interpretation difficult. Markov State Models (MSMs) help to overcome these problems by making experimentally relevant timescales accessible via coarse grained representations that also allows for convenient interpretation. However, standard set-based MSMs exhibit some caveats limiting their approximation quality and statistical significance. One of the main caveats results from the fact that typical MD trajectories repeatedly re-cross the boundary between the sets used to build the MSM which causes statistical bias in estimating the transition probabilities between these sets. In this article, we present a set-free approach to MSM building utilizing smooth overlapping ansatz functions instead of sets and an adaptive refinement approach. This kind of meshless discretization helps to overcome the recrossing problem and yields an adaptive refinement procedure that allows to improve the quality of the model while exploring state space and inserting new ansatz functions into the MSM.}, language = {en} } @article{ThiesSunkaraRayetal., author = {Thies, Arne and Sunkara, Vikram and Ray, Sourav and Wulkow, Hanna and Celik, M. {\"O}zg{\"u}r and Yerg{\"o}z, Fatih and Sch{\"u}tte, Christof and Stein, Christoph and Weber, Marcus and Winkelmann, Stefanie}, title = {Modelling altered signalling of G-protein coupled receptors in inflamed environment to advance drug design}, series = {Scientific Reports}, volume = {13}, journal = {Scientific Reports}, number = {607}, doi = {10.1038/s41598-023-27699-w}, abstract = {We previously reported the successful design, synthesis and testing of the prototype opioid painkiller NFEPP that does not elicit adverse side effects. The design process of NFEPP was based on mathematical modelling of extracellular interactions between G-protein coupled receptors (GPCRs) and ligands, recognizing that GPCRs function differently under pathological versus healthy conditions. We now present an additional and novel stochastic model of GPCR function that includes intracellular dissociation of G-protein subunits and modulation of plasma membrane calcium channels and their dependence on parameters of inflamed and healthy tissue (pH, radicals). The model is validated against in vitro experimental data for the ligands NFEPP and fentanyl at different pH values and radical concentrations. We observe markedly reduced binding affinity and calcium channel inhibition for NFEPP at normal pH compared to lower pH, in contrast to the effect of fentanyl. For increasing radical concentrations, we find enhanced constitutive G-protein activation but reduced ligand binding affinity. Assessing the different effects, the results suggest that, compared to radicals, low pH is a more important determinant of overall GPCR function in an inflamed environment. Future drug design efforts should take this into account.}, language = {en} } @article{SikorskiWeberSchuette, author = {Sikorski, Alexander and Weber, Marcus and Sch{\"u}tte, Christof}, title = {The Augmented Jump Chain}, series = {Advanced Theory and Simulations}, volume = {4}, journal = {Advanced Theory and Simulations}, number = {4}, publisher = {Wiley-VCH}, doi = {10.1002/adts.202000274}, abstract = {Modern methods of simulating molecular systems are based on the mathematical theory of Markov operators with a focus on autonomous equilibrated systems. However, non-autonomous physical systems or non-autonomous simulation processes are becoming more and more important. A representation of non-autonomous Markov jump processes is presented as autonomous Markov chains on space-time. Augmenting the spatial information of the embedded Markov chain by the temporal information of the associated jump times, the so-called augmented jump chain is derived. The augmented jump chain inherits the sparseness of the infinitesimal generator of the original process and therefore provides a useful tool for studying time-dependent dynamics even in high dimensions. Furthermore, possible generalizations and applications to the computation of committor functions and coherent sets in the non-autonomous setting are discussed. After deriving the theoretical foundations, the concepts with a proof-of-concept Galerkin discretization of the transfer operator of the augmented jump chain applied to simple examples are illustrated.}, language = {en} } @article{SeckerFackeldeyWeberetal., author = {Secker, Christopher and Fackeldey, Konstantin and Weber, Marcus and Ray, Sourav and Gorgulla, Christoph and Sch{\"u}tte, Christof}, title = {Novel multi-objective affinity approach allows to identify pH-specific μ-opioid receptor agonists}, series = {Journal of Cheminformatics}, volume = {15}, journal = {Journal of Cheminformatics}, doi = {10.1186/s13321-023-00746-4}, abstract = {Opioids are essential pharmaceuticals due to their analgesic properties, however, lethal side effects, addiction, and opioid tolerance are extremely challenging. The development of novel molecules targeting the μ-opioid receptor (MOR) in inflamed, but not in healthy tissue, could significantly reduce these unwanted effects. Finding such novel molecules can be achieved by maximizing the binding affinity to the MOR at acidic pH while minimizing it at neutral pH, thus combining two conflicting objectives. Here, this multi-objective optimal affinity approach is presented, together with a virtual drug discovery pipeline for its practical implementation. When applied to finding pH-specific drug candidates, it combines protonation state-dependent structure and ligand preparation with high-throughput virtual screening. We employ this pipeline to characterize a set of MOR agonists identifying a morphine-like opioid derivative with higher predicted binding affinities to the MOR at low pH compared to neutral pH. Our results also confirm existing experimental evidence that NFEPP, a previously described fentanyl derivative with reduced side effects, and recently reported β-fluorofentanyls and -morphines show an increased specificity for the MOR at acidic pH when compared to fentanyl and morphine. We further applied our approach to screen a >50K ligand library identifying novel molecules with pH-specific predicted binding affinities to the MOR. The presented differential docking pipeline can be applied to perform multi-objective affinity optimization to identify safer and more specific drug candidates at large scale.}, language = {en} } @article{SchuetteNielsenWeber, author = {Sch{\"u}tte, Christof and Nielsen, Adam and Weber, Marcus}, title = {Markov State Models and Molecular Alchemy}, series = {Molecular Physics}, volume = {113}, journal = {Molecular Physics}, number = {1}, doi = {10.1080/00268976.2014.944597}, pages = {69 -- 78}, abstract = {In recent years Markov State Models (MSMs) have attracted a consid- erable amount of attention with regard to modelling conformation changes and associated function of biomolecular systems. They have been used successfully, e.g., for peptides including time-resolved spectroscopic experiments, protein function and protein folding , DNA and RNA, and ligand-receptor interaction in drug design and more complicated multivalent scenarios. In this article a novel reweighting scheme is introduced that allows to construct an MSM for certain molecular system out of an MSM for a similar system. This permits studying how molecular properties on long timescales differ between similar molecular systems without performing full molecular dynamics simulations for each system under con- sideration. The performance of the reweighting scheme is illustrated for simple test cases including one where the main wells of the respective energy landscapes are located differently and an alchemical transformation of butane to pentane where the dimension of the state space is changed.}, language = {en} } @misc{SchuetteNielsenWeber, author = {Sch{\"u}tte, Christof and Nielsen, Adam and Weber, Marcus}, title = {Markov State Models and Molecular Alchemy}, issn = {1438-0064}, doi = {10.1080/00268976.2014.944597}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-46718}, abstract = {In recent years Markov State Models (MSMs) have attracted a consid- erable amount of attention with regard to modelling conformation changes and associated function of biomolecular systems. They have been used successfully, e.g., for peptides including time-resolved spectroscopic ex- periments, protein function and protein folding , DNA and RNA, and ligand-receptor interaction in drug design and more complicated multi- valent scenarios. In this article a novel reweighting scheme is introduced that allows to construct an MSM for certain molecular system out of an MSM for a similar system. This permits studying how molecular proper- ties on long timescales differ between similar molecular systems without performing full molecular dynamics simulations for each system under con- sideration. The performance of the reweighting scheme is illustrated for simple test cases including one where the main wells of the respective en- ergy landscapes are located differently and an alchemical transformation of butane to pentane where the dimension of the state space is changed.}, language = {en} } @misc{SchuetteDeuflhardNoeetal., author = {Sch{\"u}tte, Christof and Deuflhard, Peter and No{\´e}, Frank and Weber, Marcus}, title = {Design of functional molecules}, series = {MATHEON-Mathematics for Key Technologies}, volume = {1}, journal = {MATHEON-Mathematics for Key Technologies}, editor = {Deuflhard, Peter and Gr{\"o}tschel, Martin and H{\"o}mberg, Dietmar and Horst, Ulrich and Kramer, J{\"u}rg and Mehrmann, Volker and Polthier, Konrad and Schmidt, Frank and Sch{\"u}tte, Christof and Skutella, Martin and Sprekels, J{\"u}rgen}, publisher = {European Mathematical Society}, pages = {49 -- 65}, language = {en} } @misc{RayThiesSunkaraetal., author = {Ray, Sourav and Thies, Arne and Sunkara, Vikram and Wulkow, Hanna and Celik, {\"O}zg{\"u}r and Yerg{\"o}z, Fatih and Sch{\"u}tte, Christof and Stein, Christoph and Weber, Marcus and Winkelmann, Stefanie}, title = {Modelling altered signalling of G-protein coupled receptors in inflamed environment to advance drug design}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-82797}, abstract = {Initiated by mathematical modelling of extracellular interactions between G-protein coupled receptors (GPCRs) and ligands in normal versus diseased (inflamed) environments, we previously reported the successful design, synthesis and testing of the prototype opioid painkiller NFEPP that does not elicit adverse side effects. Uniquely, this design recognised that GPCRs function differently under pathological versus healthy conditions. We now present a novel stochastic model of GPCR function that includes intracellular dissociation of G-protein subunits and modulation of plasma membrane calcium channels associated with parameters of inflamed tissue (pH, radicals). By means of molecular dynamics simulations, we also assessed qualitative changes of the reaction rates due to additional disulfide bridges inside the GPCR binding pocket and used these rates for stochastic simulations of the corresponding reaction jump process. The modelling results were validated with in vitro experiments measuring calcium currents and G-protein activation. We found markedly reduced G-protein dissociation and calcium channel inhibition induced by NFEPP at normal pH, and enhanced constitutive G-protein activation but lower probability of ligand binding with increasing radical concentrations. These results suggest that, compared to radicals, low pH is a more important determinant of overall GPCR function in an inflamed environment. Future drug design efforts should take this into account.}, language = {en} } @article{RaySunkaraSchuetteetal.2020, author = {Ray, Sourav and Sunkara, Vikram and Sch{\"u}tte, Christof and Weber, Marcus}, title = {How to calculate pH-dependent binding rates for receptor-ligand systems based on thermodynamic simulations with different binding motifs}, series = {Molecular Simulation}, volume = {46}, journal = {Molecular Simulation}, number = {18}, publisher = {Taylor and Francis}, doi = {10.1080/08927022.2020.1839660}, pages = {1443 -- 1452}, year = {2020}, abstract = {Molecular simulations of ligand-receptor interactions are a computational challenge, especially when their association- ('on'-rate) and dissociation- ('off'-rate) mechanisms are working on vastly differing timescales. One way of tackling this multiscale problem is to compute the free-energy landscapes, where molecular dynamics (MD) trajectories are used to only produce certain statistical ensembles. The approach allows for deriving the transition rates between energy states as a function of the height of the activation-energy barriers. In this article, we derive the association rates of the opioids fentanyl and N-(3-fluoro-1-phenethylpiperidin-4-yl)-N-phenyl propionamide (NFEPP) in a μ-opioid receptor by combining the free-energy landscape approach with the square-root-approximation method (SQRA), which is a particularly robust version of Markov modelling. The novelty of this work is that we derive the association rates as a function of the pH level using only an ensemble of MD simulations. We also verify our MD-derived insights by reproducing the in vitro study performed by the Stein Lab.}, language = {en} } @misc{RaySunkaraSchuetteetal., author = {Ray, Sourav and Sunkara, Vikram and Sch{\"u}tte, Christof and Weber, Marcus}, title = {How to calculate pH-dependent binding rates for receptor-ligand systems based on thermodynamic simulations with different binding motifs}, issn = {1438-0064}, doi = {10.1080/08927022.2020.1839660}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-78437}, abstract = {Molecular simulations of ligand-receptor interactions are a computational challenge, especially when their association- (``on''-rate) and dissociation- (``off''-rate) mechanisms are working on vastly differing timescales. In addition, the timescale of the simulations themselves is, in practice, orders of magnitudes smaller than that of the mechanisms; which further adds to the complexity of observing these mechanisms, and of drawing meaningful and significant biological insights from the simulation. One way of tackling this multiscale problem is to compute the free-energy landscapes, where molecular dynamics (MD) trajectories are used to only produce certain statistical ensembles. The approach allows for deriving the transition rates between energy states as a function of the height of the activation-energy barriers. In this article, we derive the association rates of the opioids fentanyl and N-(3-fluoro-1-phenethylpiperidin-4-yl)- N-phenyl propionamide (NFEPP) in a \$\mu\$-opioid receptor by combining the free-energy landscape approach with the square-root-approximation method (SQRA), which is a particularly robust version of Markov modelling. The novelty of this work is that we derive the association rates as a function of the pH level using only an ensemble of MD simulations. We also verify our MD-derived insights by reproducing the in vitro study performed by the Stein Lab, who investigated the influence of pH on the inhibitory constant of fentanyl and NFEPP (Spahn et al. 2017). MD simulations are far more accessible and cost-effective than in vitro and in vivo studies. Especially in the context of the current opioid crisis, MD simulations can aid in unravelling molecular functionality and assist in clinical decision-making; the approaches presented in this paper are a pertinent step forward in this direction.}, language = {en} } @article{MetznerWeberSchuette2010, author = {Metzner, Ph. and Weber, Marcus and Sch{\"u}tte, Christof}, title = {Observation uncertainty in reversible Markov chains}, series = {Phys. Rev. E}, volume = {82}, journal = {Phys. Rev. E}, number = {3}, publisher = {American Physical Society}, doi = {10.1103/PhysRevE.82.031114}, pages = {031114}, year = {2010}, language = {en} } @article{HartmannSchuetteWeberetal., author = {Hartmann, Carsten and Sch{\"u}tte, Christof and Weber, Marcus and Zhang, Wei}, title = {Importance sampling in path space for diffusion processes with slow-fast variables}, series = {Probability Theory and Related Fields}, journal = {Probability Theory and Related Fields}, doi = {10.1007/s00440-017-0755-3}, pages = {1 -- 52}, abstract = {Importance sampling is a widely used technique to reduce the variance of a Monte Carlo estimator by an appropriate change of measure. In this work, we study importance sampling in the framework of diffusion process and consider the change of measure which is realized by adding a control force to the original dynamics. For certain exponential type expectation, the corresponding control force of the optimal change of measure leads to a zero-variance estimator and is related to the solution of a Hamilton-Jacobi-Bellmann equation. We focus on certain diffusions with both slow and fast variables, and the main result is that we obtain an upper bound of the relative error for the importance sampling estimators with control obtained from the limiting dynamics. We demonstrate our approximation strategy with an illustrative numerical example.}, language = {en} } @article{DonatiSchuetteWeber, 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}, series = {Molecular Physics}, journal = {Molecular Physics}, number = {Ciccotti Special Issue (by invitation only)}, publisher = {Taylor \& Francis}, doi = {10.1080/00268976.2024.2356748}, pages = {e2356748}, 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{DjurdjevacConradWeberSchuette, author = {Djurdjevac Conrad, Natasa and Weber, Marcus and Sch{\"u}tte, Christof}, title = {Finding dominant structures of nonreversible Markov processes}, series = {Multiscale Modeling and Simulation}, volume = {14}, journal = {Multiscale Modeling and Simulation}, number = {4}, doi = {10.1137/15M1032272}, pages = {1319 -- 1340}, language = {en} } @misc{DjurdjevacConradWeberSchuette, author = {Djurdjevac Conrad, Natasa and Weber, Marcus and Sch{\"u}tte, Christof}, title = {Finding dominant structures of nonreversible Markov processes}, issn = {1438-0064}, doi = {10.1137/15M1032272}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-55739}, abstract = {Finding metastable sets as dominant structures of Markov processes has been shown to be especially useful in modeling interesting slow dynamics of various real world complex processes. Furthermore, coarse graining of such processes based on their dominant structures leads to better understanding and dimension reduction of observed systems. However, in many cases, e.g. for nonreversible Markov processes, dominant structures are often not formed by metastable sets but by important cycles or mixture of both. This paper aims at understanding and identifying these different types of dominant structures for reversible as well as nonreversible ergodic Markov processes. Our algorithmic approach generalizes spectral based methods for reversible process by using Schur decomposition techniques which can tackle also nonreversible cases. We illustrate the mathematical construction of our new approach by numerical experiments.}, language = {en} } @article{DjurdjevacConradFuerstenauGrabundzijaetal., author = {Djurdjevac Conrad, Natasa and Fuerstenau, Daniel and Grabundzija, Ana and Helfmann, Luzie and Park, Martin and Schier, Wolfram and Sch{\"u}tt, Brigitta and Sch{\"u}tte, Christof and Weber, Marcus and Wulkow, Niklas and Zonker, Johannes}, title = {Mathematical modeling of the spreading of innovations in the ancient world}, series = {eTopoi. Journal for Ancient Studies}, volume = {7}, journal = {eTopoi. Journal for Ancient Studies}, issn = {ISSN 2192-2608}, doi = {10.17171/4-7-1}, language = {en} }