Background: Endogenous opioids, such as endomorphin-2, are key regulators of the body’s pain pathways and mediate analgesia by engaging the μ-opioid receptor. This class of opioids are distinguished by their transient activation of the μ-opioid receptor, which is attributed to rapid enzymatic degradation. Methods: To understand how degradation of endomorphin-2 by the enzyme DPP IV affects its interaction with the μ-opioid receptor, we analyzed the ligand–receptor conformational dynamics and interaction patterns of molecular dynamics simulations data of morphine, fentanyl and endomorphin-2 and one degradation product Phe-Phe-NH2, using molecular fingerprints and the mathematical framework ISOKANN. Results: Our analyses revealed that both the clinically relevant opioids, morphine and fentanyl, as well as the endogenous opioid endomorphin-2, adopt a set of recurring binding conformations within the μ-opioid receptor binding pocket, maintaining overlapping interaction motifs throughout the simulations. In contrast, Phe-Phe-NH2 failed to maintain a persistent binding mode over the simulated timescale. This instability arises from the dipeptidyl peptidase IV mediated cleavage of endomorphin-2, which generates Phe-Phe-NH2 and removes critical proline and tyrosine residues, thereby leading to the loss of stabilizing hydrophobic contacts with receptor residues Tyr1503,33, Val2385,43 and Val3026,55. Conclusion: By mapping structural interaction motifs essential for stable μ-opioid receptor binding, this study provides mechanistic insights into how endogenous degradation reshapes ligand–receptor interactions.

Time-resolved spectroscopy is a widely used tool for the investigation of physical and chemical processes. Analysis of the results is often challenging due to the inherent complexity of the data, encoding the chemical nature and time evolution of multiple species involved in the reaction. Many existing analytical methods are unsatisfactory as they introduce bias by relying on unjustified mathematical or mechanistic assumptions about the studied process. Here, we introduce a generalized analytical strategy based on non-negative matrix factorization. The methodology builds on a bottom-up model-free approach, in which physically grounded mathematical constraints can be introduced by active choice, allowing for an unbiased analysis of complex time series of spectroscopic data. The strength of this strategy is demonstrated by successful deconvolution of synthetic data mimicking different types of chemical reactions and typical challenges encountered in time-resolved Raman spectroscopy.

A series of lanthanoid-doped bismuth oxido nanoclusters (BiO-NCs) of the type [Bi38O45(NO3)24(dmso)y]:Ln (C-1:Ln; Ln3+ = La - Lu, ≠ Pm; y = 26-28) including isostructural host and doped structures is reported. Successful doping with ≈1 ω% of the BiO-NCs was demonstrated by ICP-OES, ESI-MS, and exemplarily by SC XRD analysis for C-1:Gd and C-1:Dy. The dopants are statistically albeit nonuniformly distributed on the lattice position of Bi3+ and change the optical and magnetic properties of the BiO-NC host material. Altered optical properties (UV-vis and PL) were demonstrated with special focus on C-1:Er, C-1:Yb, and C-2d:Dy showing dual Bi3+ and the Ln3+ specific PL emissions, resulting in characteristic emission combinations over a wide wavelength range from visible to the NIR region and lifetimes up to 0.5 ms. Starting from C-1:Gd and C-1:Dy the methacrylate-substituted BiO-NCs [Bi38O45(OMc)24(EtOH)14]:Gd (C-2E:Gd) and [Bi38O45(OMc)24(EtOH)14]:Dy (C-2E:Dy) were synthesized to finally give solvate-free [Bi38O45(OMc)24]:Ln (C-2d:Gd, C-2d:Dy) after drying. The introduction of paramagnetic behavior in the BiO-NC diamagnetic host structure was confirmed by SQUID measurements. We further investigated the suitability of Gd3+ in C-2d:Gd as a polarization agent for dynamic nuclear polarization (DNP)-enhanced MAS NMR experiments, which resulted in a proton enhancement factor of approximately 10.

Abstract The identification of suitable lead molecules in the vast chemical space is a critical and challenging task in drug discovery campaigns. Recently, it has been demonstrated that large-scale virtual screening provides a powerful approach to accelerate the identification of novel drug candidates by screening ever increasing virtual ligand libraries, which have reached magnitudes of > 10 20 compounds. However, this desirable increase in potentially bioactive molecules poses a new challenge as enumerating and virtually screening such huge compound libraries is computationally prohibitive. Consequently, advanced approaches to navigate ultra-large chemical spaces and to identify suitable candidate molecules therein are urgently needed. Here, we present an evolutionary algorithm framework using molecular generative AI, reaction-based substructure searching, and iterative model fine-tuning for a targeted and efficient exploration of chemical fragment spaces. Combining this approach with large-scale virtual screening we are able to identify target-specific candidate molecules within the commercially available Enamine REAL Space (∼10 15 ). We demonstrate the applicability of the approach by successfully identifying and biochemically validating pH-specific ligands of the µ -opioid receptor. Our results demonstrate that integrating generative AI with evolutionary algorithms provides a promising route to explore ultra-large chemical spaces for the discovery of novel, synthetically accessible lead molecules.

Sepsis remains a diagnostic challenge due to its heterogeneous molecular signatures and complex immune responses. In this study, we develop a logical data analysis framework based on Boolean polynomial rings. This method constructs an ideal ℐ of selection criteria that isolate empty subsets of previously analyzed patient data. This approach enables the derivation of interpretable classification rules based on biomarker profiles. We demonstrate that logical data analysis identifies distinct logical patterns for positive and negative sepsis classification. For instance, elevated levels of GLP-1 and MyD88 are associated with septic states in our dataset, whereas high TRAIL and low MyD88 concentrations may suggest a non-septic condition. Importantly, a new way to integrate expert knowledge to filter out potential overfitting or dataset-specific artifacts is shown. Our findings highlight the utility of logics in generating transparent, biologically plausible rules for a data-based and expert-based understanding of sepsis. Moreover, we show how data analysis can benefit from algebraic structures.

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.

Structural and optical characterization of Eu3+ doped atomically precise bismuth oxido nanoclusters (BiO-NCs) of about 2 nm in size is reported. The BiO-NC [Bi38O45(NO3)24(dmso)28]:Eu (C-1D:Eu) is transformed into soluble methacrylate (-OMc) coordinated [Bi38O45(OMc)24(dmso)4]:Eu·4dmso·4H2O (C-2D:Eu) and [Bi38O45(OMc)24(EtOH)10]:Eu·4EtOH (C-2E:Eu) by ligand exchange and without changing the amount of Eu3+ doping (ω ≈ 1%). Based on SC XRD analysis of C-1D:Eu and C-2E:Eu partial substitution of the bismuth atoms with preference in the inner {Bi6O9} core of the {Bi38O45} cluster is suggested. Temperature dependent photoluminescence studies show bismuth- and europium-based dual emission at low temperatures, whereas only the latter is preserved at room temperature. As a result of the low-symmetry environment of Eu3+ a remarkable intensity of the 5D0→7F0 (electric dipole) transition is observed, accompanied by lifetimes of about 2 ms and quantum yields ranging from 53% (C-2D:Eu) to 65 % (C-1D:Eu). The solubility of C-2E:Eu enables its use for the synthesis of organic-inorganic hybrid materials. Thus, transparent hybrid materials were prepared by radical copolymerization of methyl methacrylate, 2-hydroxyethyl methacrylate and C-2E:Eu, showing distinct photoluminescence characteristics even with a low Eu3+ content of 2.7·10-5 mol%.

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 the interpretability of deep-learned reaction coordinates by connecting them to atomistic mechanisms, without requiring any a priori knowledge of collective variables, pathways, or end points. Here, AMORE-MD employs the ISOKANN algorithm to learn a neural membership function χ representing the dominant slow process, from which transition pathways are reconstructed as minimum-energy paths aligned with the gradient of χ, 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 Müller-Brown potential, alanine dipeptide, and the elastin-derived hexapeptide VGVAPG.

The functionalization of atomically precise bismuth oxido nanoclusters (BiO-NCs) by partial substitution of bismuth with cerium, chiral modification by ligand substitution, and their 2D-supramolecular self-assembly on surfaces such as Au(111) and HOPG(0001) was studied. Starting from [Bi38O45(NO3)20(dmso)28](NO3)4·4dmso (C-1) and its cerium doped counterpart [Bi38O45(NO3)24(dmso)28]:Ce·1.5dmso (C-1:Ce), soluble BiO-NCs with a size of 2 nm are obtained by reaction with methacrylate and chiral carboxylates to give [Bi38O45(L)24] and [Bi38O45(L)24]:Ce with L = McO- (C-2 and C-2:Ce), Boc-L-Phe-O- (C-3 and C-3:Ce), and Boc-L-Ala-O- (C-4 and C-4:Ce). The cerium doping content determined using ICP-OES and the oxidation state of cerium determined via XP- and EPR spectroscopy were studied, showing Ce(III) for the carboxylate functionalized BiO-NCs, whereas mixed valency Ce(III)/Ce(IV) was detected for the nitrate C-1:Ce. However, doping of BiO-NCs with cerium results in a change of absorption from the UV to the visible light region for all compounds, whereby the red-shift is concluded to originate from the MLCT from Ce(III) to oxygen. The self-assembly of chiral BiO-NCs [Bi38O45(Boc-L-Phe-O)24] (C-3) and [Bi38O45(Boc-L-Phe-O)24]:Ce (C-3:Ce) on Au(111) and HOPG(0001) was analyzed. STM images of BiO-NCs revealed a stepwise one-dimensional arrangement of BiO-NCs and STS analysis proved significant changes in the electronic energy gap as a result of ≈1 ω% Ce(III) doping (C-3Eg = 3.5 eV vs.C-3:CeEg = 2.6 eV).

BACKGROUND:The dynamics of many gene regulatory networks (GRNs) is characterized by the occurrence of metastable phenotypes and stochastic phenotype switches. The chemical master equation (CME) is the most accurate description to model such stochastic dynamics, whereby the long-time dynamics of the system is encoded in the spectral properties of the CME operator. Markov State Models (MSMs) provide a general framework for analyzing and visualizing stochastic multistability and state transitions based on these spectral properties. Until now, however, this approach is either limited to low-dimensional systems or requires the use of high-performance computing facilities, thus limiting its usability. RESULTS:We present a domain decomposition approach (DDA) that approximates the CME by a stochastic rate matrix on a discretized state space and projects the multistable dynamics to a lower dimensional MSM. To approximate the CME, we decompose the state space via a Voronoi tessellation and estimate transition probabilities by using adaptive sampling strategies. We apply the robust Perron cluster analysis (PCCA+) to construct the final MSM. Measures for uncertainty quantification are incorporated. As a proof of concept, we run the algorithm on a single PC and apply it to two GRN models, one for the genetic toggle switch and one describing macrophage polarization. By comparing the results with reference solutions, we demonstrate that our approach correctly identifies the number and location of metastable phenotypes with adequate accuracy and uncertainty bounds. We show that accuracy mainly depends on the total number of Voronoi cells, whereas uncertainty is determined by the number of sampling points. CONCLUSIONS:A DDA enables the efficient computation of MSMs with quantified uncertainty. Since the algorithm is trivially parallelizable, it can be applied to larger systems, which will inevitably lead to new insights into cell-regulatory dynamics.

Molecular Dynamics simulations are indispensable tools for comprehending the dynamic behavior of biomolecules, yet extracting meaningful molecular pathways from these simulations remains challenging due to the vast amount of high dimensional data. In this work, we present Molecular Kinetics via Topology (MoKiTo), a novel approach that combines the ISOKANN algorithm to determine the membership function of a molecular system with a topological analysis tool inspired by the Mapper algorithm. Our strategy efficiently identifies and characterizes distinct molecular pathways, enabling the detection and visualization of critical conformational transitions and rare events. This method offers deeper insights into molecular mechanisms, facilitating the design of targeted interventions in drug discovery and protein engineering.

Stochastic dynamical systems like gene regulatory networks (GRNs) often exhibit behavior characterized by metastable sets (representing cellular phenotypes), in which trajectories remain for long times, whereas switches between these sets in the phase space are rare events. One way to capture these rare events is to infer the system's long-term behavior from the spectral characteristics (eigenvalues and eigenvectors) of its Koopman operator. For GRNs, the Koopman operator is based on the chemical master equation (CME), which provides a precise mathematical modeling framework for stochastic GRNs. Since the CME is typically analytically intractable, methods based on discretizing the CME operator have been developed. However, determining the number and location of metastable sets in the phase space as well as the transition rates between them remains computationally challenging, especially for large GRNs with many genes and interactions. A promising alternative method, called ISOKANN (invariant subspaces of Koopman operators with artificial neural networks) has been developed in the context of molecular dynamics. ISOKANN uses a combination of the power iteration and neural networks to learn the basis functions of an invariant subspace of the Koopman operator. In this paper, we extend the application of ISOKANN to the CME operator and apply it to two small GRNs: a genetic toggle switch model and a model for macrophage polarization. Our work opens a new field of application for the ISOKANN algorithm and demonstrates the potential of this algorithm for studying large GRNs.

Endogenous opioids, such as Endomorphin-2, are not typically associated with severe constipation, unlike pharmaceutical opioids, which induce opioid-induced constipation (OIC) by activating μ-opioid receptors in the gastrointestinal tract. In this study, we present a mathematical model, which integrates the serotonergic and opioid pathways, simulating the interaction between serotonin and opioid signaling within the enteric nervous system (ENS). The model explores the mechanisms underlying OIC, with a focus on the change in adenylyl cyclase (AC) activity, cAMP accumulation, and the distinct functionalities of Endomorphin-2 compared to commonly used pharmaceutical opioids. We study the effects of Morphine, Fentanyl, and Methadone and contrast them with Endomorphin-2. Our findings reveal that opioids do not perturb the signaling of serotonin, but only the activity of AC, suggesting that serotonin levels have no influence on improving opioid-induced constipation. Furthermore, this study reveals that the primary difference between endogenous and pharmaceutical opioids is their degradation rates. This finding shows that modulating opioid degradation rates significantly improves cAMP recovery. In conclusion, our insights steer towards exploring opioid degrading enzymes, localized to the gut, as a strategy for mitigating OIC.

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 mu-opioid receptor. With this approach we provide a new perspective on analyzing the macroscopic behaviour of Markov systems.

The dominant eigenfunctions of the Koopman operator characterize the metastabilities and slow-timescale dynamics of stochastic diffusion processes. In the context of molecular dynamics and Markov state modeling, they allow for a description of the location and frequencies of rare transitions, which are hard to obtain by direct simulation alone. In this article, we reformulate the eigenproblem in terms of the ISOKANN framework, an iterative algorithm that learns the eigenfunctions by alternating between short burst simulations and a mixture of machine learning and classical numerics, which naturally leads to a proof of convergence. We furthermore show how the intermediate iterates can be used to reduce the sampling variance by importance sampling and optimal control (enhanced sampling), as well as to select locations for further training (adaptive sampling). We demonstrate the usage of our proposed method in experiments, increasing the approximation accuracy by several orders of magnitude.

The synthesis and characterization of twin monomers [Ti(OCH2-2-MeO-C6H4)(4)(HOCH2-2-MeO-C6H4)](2) (3) and [Al(OCH2-2-MeO-C6H4)(3)](4) (5) by reacting HOCH2-2-MeO-C6H4 (1) with Ti((OPr)-Pr-i)(4) (2), or 1 with AlMe3 (4) is discussed. Single crystal X-ray structure analysis of 3 shows a dimeric structure with two alkoxide ligands bridging the titanium ions, while the others are terminal bonded. The respective phenolic resin / metal oxide hybrid materials HM_Ti and HM_Al were obtained in moderate (HM_Ti) to excellent (HM_Al) yields using typical base-catalyzed twin polymerization conditions (230 degrees C, 2 h). Nuclear magnetic resonance and infrared spectroscopy as well as scanning electron microscopy and scanning transmission electron microscopy combined with energ-dispersive X-ray spectroscopy proved the formation of inorganic-organic hybrid materials consisting of resin and MxOy materials (HM_Ti, TiO2; HM_Al, Al2O3) containing interpenetrating phase nano-domains with sizes of < 5 nm, as is charcteristic for twin polymerization processes. Oxidation of HM_Ti and HM_Al yielded the respective oxide materials Ox_Ti (TiO2) and Ox_Al (Al2O3), which possess low surface areas of A(BET) = 53 m(2)/g and 76 m(2)/g, respectively.

The joint analysis of two datasets [Formula: see text] and [Formula: see text] that describe the same phenomena (e.g. the cellular state), but measure disjoint sets of variables (e.g. mRNA vs. protein levels) is currently challenging. Traditional methods typically analyze single interaction patterns such as variance or covariance. However, problem-tailored external knowledge may contain multiple different information about the interaction between the measured variables. We introduce MIASA, a holistic framework for the joint analysis of multiple different variables. It consists of assembling multiple different information such as similarity vs. association, expressed in terms of interaction-scores or distances, for subsequent clustering/classification. In addition, our framework includes a novel qualitative Euclidean embedding method (qEE-Transition) which enables using Euclidean-distance/vector-based clustering/classification methods on datasets that have a non-Euclidean-based interaction structure. As an alternative to conventional optimization-based multidimensional scaling methods which are prone to uncertainties, our qEE-Transition generates a new vector representation for each element of the dataset union [Formula: see text] in a common Euclidean space while strictly preserving the original ordering of the assembled interaction-distances. To demonstrate our work, we applied the framework to three types of simulated datasets: samples from families of distributions, samples from correlated random variables, and time-courses of statistical moments for three different types of stochastic two-gene interaction models. We then compared different clustering methods with vs. without the qEE-Transition. For all examples, we found that the qEE-Transition followed by Ward clustering had superior performance compared to non-agglomerative clustering methods but had a varied performance against ultrametric-based agglomerative methods. We also tested the qEE-Transition followed by supervised and unsupervised machine learning methods and found promising results, however, more work is needed for optimal parametrization of these methods. As a future perspective, our framework points to the importance of more developments and validation of distance-distribution models aiming to capture multiple-complex interactions between different variables.

Modeling-Simulation-Optimization workflows play a fundamental role in applied mathematics. The Mathematical Research Data Initiative, MaRDI, responded to this by developing a FAIR and machine-interpretable template for a comprehensive documentation of such workflows. MaRDMO, a Plugin for the Research Data Management Organiser, enables scientists from diverse fields to document and publish their workflows on the MaRDI Portal seamlessly using the MaRDI template. Central to these workflows are mathematical models. MaRDI addresses them with the MathModDB ontology, offering a structured formal model description. Here, we showcase the interaction between MaRDMO and the MathModDB Knowledge Graph through an algebraic modeling workflow from the Digital Humanities. This demonstration underscores the versatility of both services beyond their original numerical domain.

We analyze how Langevin dynamics is affected by the friction coefficient using an invariant subspace projection of the associated Koopman operator. This provides the friction-dependent metastable macro-states of the dynamical system as well as the transition rates in the entire phase space. We used the algorithm ISOKANN for a wide range of friction coefficient values and reproduced results consistent with the Kramers turnover.

The adsorption of chiral molecules onto metallic surfaces triggers electron spin polarization at the interface, paving the way for applications in chiral opto-spintronics. However, the spin effects sensitively depend on the binding and ordering of the chiral species on surfaces. This study explores the adsorption of chiral thioether-functionalized atomically precise bismuth oxido nanoclusters (BiO-NCs) on gold (Au) surfaces, extending the conventional approach of using thiol-containing molecules and complexes to nanoclusters. Starting from the precursor [Bi38O45(NO3)(20)(dmso)(28)](NO3)(4)4dmso (A), chiral BiO-NCs were synthesized by substituting the nitrates with N-(tert-butoxycarbonyl)-l-methionine (Boc-l-Met-O-) ligands to obtain [Bi38O45(Boc-l-Met-O)(24)] (2). The full exchange of nitrate by the Boc-l-methionine ligand was demonstrated by powder X-ray diffractograms, dynamic light scattering, electrospray ionization mass spectrometry, nuclear magnetic resonance, infrared, circular dichroism, and X-ray photoelectron spectroscopy. Compared to previously reported [Bi38O45(Boc-l-Phe-O)(24)(dmso)(9)] (1), BiO-NC 2 shows differences in the growth mode on a Au surface as revealed by scanning electron microscopy, wherefore a stronger binding of BiO-NC 2 is assumed. Anchoring of BiO-NC 2 to the Au surface through thioether groups induced a discernible change in the optical response of the Au surface analyzed by spectroscopic ellipsometry (SE). From the numerical modeling of the SE parameters, a layer thickness of similar to 2 nm, corresponding to a monolayer of BiO-NC 2, was estimated for the samples prepared by dip coating. Thus, strong adsorption of BiO-NC 2 to the Au surface is concluded, which is an essential prerequisite for chiral-induced interface spin polarization.