TY - JOUR A1 - Erban, Radek A1 - Winkelmann, Stefanie T1 - Multi-grid reaction-diffusion master equation: applications to morphogen gradient modelling JF - Bulletin of Mathematical Biology N2 - The multi-grid reaction-diffusion master equation (mgRDME) provides a generalization of stochastic compartment-based reaction-diffusion modelling described by the standard reaction-diffusion master equation (RDME). By enabling different resolutions on lattices for biochemical species with different diffusion constants, the mgRDME approach improves both accuracy and efficiency of compartment-based reaction-diffusion simulations. The mgRDME framework is examined through its application to morphogen gradient formation in stochastic reaction-diffusion scenarios, using both an analytically tractable first-order reaction network and a model with a second-order reaction. The results obtained by the mgRDME modelling are compared with the standard RDME model and with the (more detailed) particle-based Brownian dynamics simulations. The dependence of error and numerical cost on the compartment sizes is defined and investigated through a multi-objective optimization problem. Y1 - 2025 U6 - https://doi.org/10.1007/s11538-024-01377-y VL - 87 SP - 6 ER - TY - JOUR A1 - Wehlitz, Nathalie A1 - Sadeghi, Mohsen A1 - Montefusco, Alberto A1 - Schütte, Christof A1 - Pavliotis, Grigorios A. A1 - Winkelmann, Stefanie T1 - Approximating particle-based clustering dynamics by stochastic PDEs JF - SIAM Journal on Applied Dynamical Systems N2 - This work proposes stochastic partial differential equations (SPDEs) as a practical tool to replicate clustering effects of more detailed particle-based dynamics. Inspired by membrane mediated receptor dynamics on cell surfaces, we formulate a stochastic particle-based model for diffusion and pairwise interaction of particles, leading to intriguing clustering phenomena. Employing numerical simulation and cluster detection methods, we explore the approximation of the particle-based clustering dynamics through mean-field approaches. We find that SPDEs successfully reproduce spatiotemporal clustering dynamics, not only in the initial cluster formation period, but also on longer time scales where the successive merging of clusters cannot be tracked by deterministic mean-field models. The computational efficiency of the SPDE approach allows us to generate extensive statistical data for parameter estimation in a simpler model that uses a Markov jump process to capture the temporal evolution of the cluster number. Y1 - 2025 U6 - https://doi.org/10.1137/24M1676661 VL - 24 IS - 2 SP - 1231 EP - 1250 ER - TY - JOUR A1 - Berns, Manon A1 - Yildiz, Mirza A1 - Winkelmann, Stefanie A1 - Walter, Alexander T1 - Independently engaging protein tethers of different length enhance synaptic vesicle trafficking to the plasma membrane JF - The Journal of Physiology N2 - Synaptic vesicle (SV) trafficking toward the plasma membrane (PM) and subsequent SV maturation are essential for neurotransmitter release. These processes, including SV docking and priming, are coordinated by various proteins, such as SNAREs, Munc13, and Synaptotagmin (Syt), which connect—tether—the SV to the PM. Here, we investigated how tethers of varying lengths mediate SV docking using a simplified mathematical model. The heights of the three tether types—estimated from the structures of the SNARE complex, Munc13, and Syt—defined the SV-to-PM distance ranges for tether formation. Geometric considerations linked SV-to-PM distances to the probability and rate of tether formation. We assumed that SV tethering constrains SV motility and that multiple tethers associate by independent interactions. The model predicted that forming multiple tethers favors shorter SV-to-PM distances. Although tethers acted independently in the model, their geometrical properties often caused sequential assembly, from longer ones (Munc13/Syt), that accelerated SV movement towards the PM, to shorter ones (SNAREs) which stabilized PM-proximal SVs. Modifying tether lengths or numbers affected SV trafficking. The independent implementation of tethering proteins enabled their selective removal to mimic gene knockout situations. This showed that simulated SV-to-PM distance distributions qualitatively aligned with published EM studies upon removal of SNARE and Syt tethers, while Munc13 knockout data were best approximated when assuming additional disruption of SNARE tethers. Thus, while salient features of SV docking can be accounted for by independent tethering alone, our results suggest that functional tether interactions not yet featured in our model are crucial for biological function. Y1 - 2025 U6 - https://doi.org/10.1113/JP286651 VL - 603 IS - 20 SP - 6107 EP - 6134 ER - TY - JOUR A1 - Montefusco, Alberto A1 - Helfmann, Luzie A1 - Okunola, Toluwani A1 - Winkelmann, Stefanie A1 - Schütte, Christof T1 - Partial mean-field model for neurotransmission dynamics JF - Mathematical Biosciences N2 - This article addresses reaction networks in which spatial and stochastic effects are of crucial importance. For such systems, particle-based models allow us to describe all microscopic details with high accuracy. However, they suffer from computational inefficiency if particle numbers and density get too large. Alternative coarse-grained-resolution models reduce computational effort tremendously, e.g., by replacing the particle distribution by a continuous concentration field governed by reaction-diffusion PDEs. We demonstrate how models on the different resolution levels can be combined into hybrid models that seamlessly combine the best of both worlds, describing molecular species with large copy numbers by macroscopic equations with spatial resolution while keeping the stochastic-spatial particle-based resolution level for the species with low copy numbers. To this end, we introduce a simple particle-based model for the binding dynamics of ions and vesicles at the heart of the neurotransmission process. Within this framework, we derive a novel hybrid model and present results from numerical experiments which demonstrate that the hybrid model allows for an accurate approximation of the full particle-based model in realistic scenarios. Y1 - 2024 U6 - https://doi.org/10.1016/j.mbs.2024.109143 VL - 369 ER - TY - JOUR A1 - Coomber, Celvic A1 - Chewle, Surahit A1 - Secker, Christopher A1 - Fackeldey, Konstantin A1 - Weber, Marcus A1 - Winkelmann, Stefanie A1 - Schütte, Christof A1 - Sunkara, Vikram T1 - Investigating Endogenous Opioids Unravels the Mechanisms Behind Opioid-Induced Constipation, a Mathematical Modeling Approach JF - International Journal of Molecular Sciences N2 - 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. Y1 - 2025 U6 - https://doi.org/10.3390/ijms26136207 VL - 26 IS - 13 ER - TY - JOUR A1 - Winkelmann, Stefanie A1 - Zonker, Johannes A1 - Schütte, Christof A1 - Djurdjevac Conrad, Natasa T1 - Mathematical modeling of spatio-temporal population dynamics and application to epidemic spreading JF - Mathematical Biosciences N2 - Agent based models (ABMs) are a useful tool for modeling spatio-temporal population dynamics, where many details can be included in the model description. Their computational cost though is very high and for stochastic ABMs a lot of individual simulations are required to sample quantities of interest. Especially, large numbers of agents render the sampling infeasible. Model reduction to a metapopulation model leads to a significant gain in computational efficiency, while preserving important dynamical properties. Based on a precise mathematical description of spatio-temporal ABMs, we present two different metapopulation approaches (stochastic and piecewise deterministic) and discuss the approximation steps between the different models within this framework. Especially, we show how the stochastic metapopulation model results from a Galerkin projection of the underlying ABM onto a finite-dimensional ansatz space. Finally, we utilize our modeling framework to provide a conceptual model for the spreading of COVID-19 that can be scaled to real-world scenarios. Y1 - 2021 U6 - https://doi.org/10.1016/j.mbs.2021.108619 VL - 336 PB - Elsevier ER - TY - JOUR A1 - Tierno, Pietro A1 - Johansen, Tom H. A1 - Straube, Arthur T1 - Thermally active nanoparticle clusters enslaved by engineered domain wall traps JF - Nature Commun. N2 - The stable assembly of fluctuating nanoparticle clusters on a surface represents a technological challenge of widespread interest for both fundamental and applied research. Here we demonstrate a technique to stably confine in two dimensions clusters of interacting nanoparticles via size-tunable, virtual magnetic traps. We use cylindrical Bloch walls arranged to form a triangular lattice of ferromagnetic domains within an epitaxially grown ferrite garnet film. At each domain, the magnetic stray field generates an effective harmonic potential with a field tunable stifness. The experiments are combined with theory to show that the magnetic confinement is effectively harmonic and pairwise interactions are of dipolar nature, leading to central, strictly repulsive forces. For clusters of magnetic nanoparticles, the stationary collective states arise from the competition between repulsion, confinement and the tendency to fill the central potential well. Using a numerical simulation model as a quantitative map between the experiment and theory we explore the field-induced crystallization process for larger clusters and unveil the existence of three different dynamical regimes. The present method provides a model platform for investigations of the collective phenomena emerging when strongly confined nanoparticle clusters are forced to move in an idealized, harmonic-like potential. Y1 - 2021 U6 - https://doi.org/10.1038/s41467-021-25931-7 VL - 12 SP - 5813 ER - TY - JOUR A1 - del Razo, Mauricio A1 - Frömberg, Daniela A1 - Straube, Arthur A1 - Schütte, Christof A1 - Höfling, Felix A1 - Winkelmann, Stefanie T1 - A probabilistic framework for particle-based reaction–diffusion dynamics using classical Fock space representations JF - Letters in Mathematical Physics Y1 - 2022 U6 - https://doi.org/10.1007/s11005-022-01539-w VL - 112 IS - 49 ER - TY - JOUR A1 - Straube, Arthur A1 - Winkelmann, Stefanie A1 - Schütte, Christof A1 - Höfling, Felix T1 - Stochastic pH oscillations in a model of the urea–urease reaction confined to lipid vesicles JF - J. Phys. Chem. Lett. N2 - The urea-urease clock reaction is a pH switch from acid to basic that can turn into a pH oscillator if it occurs inside a suitable open reactor. We numerically study the confinement of the reaction to lipid vesicles, which permit the exchange with an external reservoir by differential transport, enabling the recovery of the pH level and yielding a constant supply of urea molecules. For microscopically small vesicles, the discreteness of the number of molecules requires a stochastic treatment of the reaction dynamics. Our analysis shows that intrinsic noise induces a significant statistical variation of the oscillation period, which increases as the vesicles become smaller. The mean period, however, is found to be remarkably robust for vesicle sizes down to approximately 200 nm, but the periodicity of the rhythm is gradually destroyed for smaller vesicles. The observed oscillations are explained as a canard-like limit cycle that differs from the wide class of conventional feedback oscillators. Y1 - 2021 U6 - https://doi.org/10.1021/acs.jpclett.1c03016 VL - 12 SP - 9888 EP - 9893 ER - TY - JOUR A1 - Montefusco, Alberto A1 - Sharma, Upanshu A1 - Tse, Oliver T1 - Fourier-Cattaneo equation: stochastic origin, variational formulation, and asymptotic limits N2 - We introduce a variational structure for the Fourier-Cattaneo (FC) system which is a second-order hyperbolic system. This variational structure is inspired by the large-deviation rate functional for the Kac process which is closely linked to the FC system. Using this variational formulation we introduce appropriate solution concepts for the FC equation and prove an a priori estimate which connects this variational structure to an appropriate Lyapunov function and Fisher information, the so-called FIR inequality. Finally, we use this formulation and estimate to study the diffusive and hyperbolic limits for the FC system. Y1 - 2022 ER - TY - JOUR A1 - Ernst, Ariane A1 - Unger, Nathalie A1 - Schütte, Christof A1 - Walter, Alexander A1 - Winkelmann, Stefanie T1 - Rate-limiting recovery processes in neurotransmission under sustained stimulation JF - Mathematical Biosciences N2 - At chemical synapses, an arriving electric signal induces the fusion of vesicles with the presynaptic membrane, thereby releasing neurotransmitters into the synaptic cleft. After a fusion event, both the release site and the vesicle undergo a recovery process before becoming available for reuse again. Of central interest is the question which of the two restoration steps acts as the limiting factor during neurotrans-mission under high-frequency sustained stimulation. In order to investigate this question, we introduce a novel non-linear reaction network which involves explicit recovery steps for both the vesicles and the release sites, and includes the induced time-dependent output current. The associated reaction dynamics are formulated by means of ordinary differential equations (ODEs), as well as via the associated stochastic jump process. While the stochastic jump model describes a single release site, the average over many release sites is close to the ODE solution and shares its periodic structure. The reason for this can be traced back to the insight that recovery dynamics of vesicles and release sites are statistically almost independent. A sensitivity analysis on the recovery rates based on the ODE formulation reveals that neither the vesicle nor the release site recovery step can be identified as the essential rate-limiting step but that the rate- limiting feature changes over the course of stimulation. Under sustained stimulation the dynamics given by the ODEs exhibit transient dynamics leading from an initial depression of the postsynaptic response to an asymptotic periodic orbit, while the individual trajectories of the stochastic jump model lack the oscillatory behavior an asymptotic periodicity of the ODE-solution. Y1 - 2023 U6 - https://doi.org/10.1016/j.mbs.2023.109023 VL - 362 ER - TY - JOUR A1 - Secker, Christopher A1 - Fackeldey, Konstantin A1 - Weber, Marcus A1 - Ray, Sourav A1 - Gorgulla, Christoph A1 - Schütte, Christof T1 - Novel multi-objective affinity approach allows to identify pH-specific μ-opioid receptor agonists JF - Journal of Cheminformatics N2 - 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. Y1 - 2023 U6 - https://doi.org/10.1186/s13321-023-00746-4 VL - 15 ER - TY - JOUR A1 - Straube, Arthur A1 - Winkelmann, Stefanie A1 - Höfling, Felix T1 - Accurate reduced models for the pH oscillations in the urea-urease reaction confined to giant lipid vesicles JF - The Journal of Physical Chemistry B N2 - This theoretical study concerns a pH oscillator based on the urea-urease reaction confined to giant lipid vesicles. Under suitable conditions, differential transport of urea and hydrogen ion across the unilamellar vesicle membrane periodically resets the pH clock that switches the system from acid to basic, resulting in self-sustained oscillations. We analyse the structure of the phase flow and of the limit cycle, which controls the dynamics for giant vesicles and dominates the pronouncedly stochastic oscillations in small vesicles of submicrometer size. To this end, we derive reduced models, which are amenable to analytic treatments that are complemented by numerical solutions, and obtain the period and amplitude of the oscillations as well as the parameter domain, where oscillatory behavior persists. We show that the accuracy of these predictions is highly sensitive to the employed reduction scheme. In particular, we suggest an accurate two-variable model and show its equivalence to a three-variable model that admits an interpretation in terms of a chemical reaction network. The faithful modeling of a single pH oscillator appears crucial for rationalizing experiments and understanding communication of vesicles and synchronization of rhythms. Y1 - 2023 U6 - https://doi.org/10.1021/acs.jpcb.2c09092 VL - 127 IS - 13 SP - 2955 EP - 2967 ER - TY - JOUR A1 - Öttinger, Hans Christian A1 - Montefusco, Alberto A1 - Peletier, Mark A. T1 - A Framework of Nonequilibrium Statistical Mechanics. I. Role and Types of Fluctuations JF - Journal of Non-Equilibrium Thermodynamics N2 - Understanding the fluctuations by which phenomenological evolution equations with thermodynamic structure can be enhanced is the key to a general framework of nonequilibrium statistical mechanics. These fluctuations provide an idealized representation of microscopic details. We consider fluctuation-enhanced equations associated with Markov processes and elaborate the general recipes for evaluating dynamic material properties, which characterize force-flux constitutive laws, by statistical mechanics. Markov processes with continuous trajectories are conveniently characterized by stochastic differential equations and lead to Green–Kubo-type formulas for dynamic material properties. Markov processes with discontinuous jumps include transitions over energy barriers with the rates calculated by Kramers. We describe a unified approach to Markovian fluctuations and demonstrate how the appropriate type of fluctuations (continuous versus discontinuous) is reflected in the mathematical structure of the phenomenological equations. Y1 - 2021 U6 - https://doi.org/10.1515/jnet-2020-0068 VL - 46 IS - 1 SP - 1 EP - 13 PB - De Gruyter ER - TY - JOUR A1 - Montefusco, Alberto A1 - Peletier, Mark A. A1 - Öttinger, Hans Christian T1 - A Framework of Nonequilibrium Statistical Mechanics. II. Coarse-Graining JF - Journal of Non-Equilibrium Thermodynamics N2 - For a given thermodynamic system, and a given choice of coarse-grained state variables, the knowledge of a force-flux constitutive law is the basis for any nonequilibrium modeling. In the first paper of this series we established how, by a generalization of the classical fluctuation-dissipation theorem (FDT), the structure of a constitutive law is directly related to the distribution of the fluctuations of the state variables. When these fluctuations can be expressed in terms of diffusion processes, one may use Green–Kubo-type coarse-graining schemes to find the constitutive laws. In this paper we propose a coarse-graining method that is valid when the fluctuations are described by means of general Markov processes, which include diffusions as a special case. We prove the success of the method by numerically computing the constitutive law for a simple chemical reaction A⇄B. Furthermore, we show that, for such a system, one cannot find a consistent constitutive law by any Green–Kubo-like scheme. Y1 - 2021 U6 - https://doi.org/10.1515/jnet-2020-0069 VL - 46 IS - 1 SP - 15 EP - 33 PB - De Gruyter ER - TY - JOUR A1 - Mielke, Alexander A1 - Montefusco, Alberto A1 - Peletier, Mark A. T1 - Exploring families of energy-dissipation landscapes via tilting: three types of EDP convergence JF - Continuum Mechanics and Thermodynamics N2 - This paper revolves around a subtle distinction between two concepts: passing to the limit in a family of gradient systems, on one hand, and deriving effective kinetic relations on the other. The two concepts are strongly related, and in many examples they even appear to be the same. Our main contributions are to show that they are different, to show that well-known techniques developed for the former may give incorrect results for the latter, and to introduce new tools to remedy this. The approach is based on the Energy-Dissipation Principle that provides a variational formulation to gradient-flow equations that allows one to apply techniques from Γ-convergence of functional on states and functionals on trajectories. Y1 - 2021 U6 - https://doi.org/10.1007/s00161-020-00932-x VL - 33 SP - 611 EP - 637 PB - Springer ER - TY - GEN A1 - Straube, Arthur A1 - Winkelmann, Stefanie A1 - Höfling, Felix T1 - Accurate reduced models for the pH oscillations in the urea-urease reaction confined to giant lipid vesicles N2 - Our theoretical study concerns an urea-urease-based pH oscillator confined to giant lipid vesicles. Under suitable conditions, differential transport of urea and hydrogen ion across the unilamellar vesicle membrane periodically resets the pH clock that switches the system from acid to basic, resulting in self-sustained oscillations. We analyse the structure of the limit cycle, which controls the dynamics for giant vesicles and dominates the strongly stochastic oscillations in small vesicles of submicrometer size. To this end, we derive reduced models, amenable to analytic treatments, and show that the accuracy of predictions, including the period of oscillations, is highly sensitive to the choice of the reduction scheme. In particular, we suggest an accurate two-variable model and show its equivalence to a three-variable model that admits an interpretation in terms of a chemical reaction network. The accurate description of a single pH oscillator appears crucial for rationalizing experiments and understanding communication of vesicles and synchronization of rhythms. T3 - ZIB-Report - 22-21 Y1 - 2022 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-88179 SN - 1438-0064 ER - TY - JOUR A1 - del Razo, Mauricio A1 - Winkelmann, Stefanie A1 - Klein, Rupert A1 - Höfling, Felix T1 - Chemical diffusion master equation: formulations of reaction-diffusion processes on the molecular level JF - Journal of Mathematical Physics N2 - The chemical diffusion master equation (CDME) describes the probabilistic dynamics of reaction--diffusion systems at the molecular level [del Razo et al., Lett. Math. Phys. 112:49, 2022]; it can be considered the master equation for reaction--diffusion processes. The CDME consists of an infinite ordered family of Fokker--Planck equations, where each level of the ordered family corresponds to a certain number of particles and each particle represents a molecule. The equations at each level describe the spatial diffusion of the corresponding set of particles, and they are coupled to each other via reaction operators --linear operators representing chemical reactions. These operators change the number of particles in the system, and thus transport probability between different levels in the family. In this work, we present three approaches to formulate the CDME and show the relations between them. We further deduce the non-trivial combinatorial factors contained in the reaction operators, and we elucidate the relation to the original formulation of the CDME, which is based on creation and annihilation operators acting on many-particle probability density functions. Finally we discuss applications to multiscale simulations of biochemical systems among other future prospects. Y1 - 2023 U6 - https://doi.org/10.1063/5.0129620 VL - 64 IS - 1 ER - TY - JOUR A1 - Boltz, Horst-Holger A1 - Sirbu, Alexei A1 - Stelzer, Nina A1 - de Lanerolle, Primal A1 - Winkelmann, Stefanie A1 - Annibale, Paolo T1 - The Impact of Membrane Protein Diffusion on GPCR Signaling JF - Cells N2 - Spatiotemporal signal shaping in G protein-coupled receptor (GPCR) signaling is now a well-established and accepted notion to explain how signaling specificity can be achieved by a superfamily sharing only a handful of downstream second messengers. Dozens of Gs-coupled GPCR signals ultimately converge on the production of cAMP, a ubiquitous second messenger. This idea is almost always framed in terms of local concentrations, the differences in which are maintained by means of spatial separation. However, given the dynamic nature of the reaction-diffusion processes at hand, the dynamics, in particular the local diffusional properties of the receptors and their cognate G proteins, are also important. By combining some first principle considerations, simulated data, and experimental data of the receptors diffusing on the membranes of living cells, we offer a short perspective on the modulatory role of local membrane diffusion in regulating GPCR-mediated cell signaling. Our analysis points to a diffusion-limited regime where the effective production rate of activated G protein scales linearly with the receptor–G protein complex’s relative diffusion rate and to an interesting role played by the membrane geometry in modulating the efficiency of coupling Y1 - 2022 U6 - https://doi.org/10.3390/cells11101660 VL - 11 IS - 10 SP - 1660 ER - TY - JOUR A1 - Engel, Maximilian A1 - Olicón-Méndez, Guillermo A1 - Wehlitz, Nathalie A1 - Winkelmann, Stefanie T1 - Synchronization and random attractors in reaction jump processes JF - Journal of Dynamics and Differential Equations N2 - This work explores a synchronization-like phenomenon induced by common noise for continuous-time Markov jump processes given by chemical reaction networks. Based on Gillespie’s stochastic simulation algorithm, a corresponding random dynamical system is formulated in a two-step procedure, at first for the states of the embedded discrete-time Markov chain and then for the augmented Markov chain including random jump times. We uncover a time-shifted synchronization in the sense that—after some initial waiting time—one trajectory exactly replicates another one with a certain time delay. Whether or not such a synchronization behavior occurs depends on the combination of the initial states. We prove this partial time-shifted synchronization for the special setting of a birth-death process by analyzing the corresponding two-point motion of the embedded Markov chain and determine the structure of the associated random attractor. In this context, we also provide general results on existence and form of random attractors for discrete-time, discrete-space random dynamical systems. Y1 - 2024 U6 - https://doi.org/10.1007/s10884-023-10345-4 ER - TY - JOUR A1 - Wehlitz, Nathalie A1 - Pavliotis, Grigorios A1 - Schütte, Christof A1 - Winkelmann, Stefanie T1 - Data-driven Reduction of Transfer Operators for Particle Clustering Dynamics N2 - We develop an operator-based framework to coarse-grain interacting particle systems that exhibit clustering dynamics. Starting from the particle-based transfer operator, we first construct a sequence of reduced representations: the operator is projected onto concentrations and then further reduced by representing the concentration dynamics on a geometric low-dimensional manifold and an adapted finite-state discretization. The resulting coarse-grained transfer operator is finally estimated from dynamical simulation data by inferring the transition probabilities between the Markov states. Applied to systems with multichromatic and Morse interaction potentials, the reduced model reproduces key features of the clustering process, including transitions between cluster configurations and the emergence of metastable states. Spectral analysis and transition-path analysis of the estimated operator reveal implied time scales and dominant transition pathways, providing an interpretable and efficient description of particle-clustering dynamics. Y1 - 2026 ER -