@article{LueckeWinkelmannHeitzigetal.2024, author = {L{\"u}cke, Marvin and Winkelmann, Stefanie and Heitzig, Jobst and Molkenthin, Nora and Koltai, P{\´e}ter}, title = {Learning interpretable collective variables for spreading processes on networks}, volume = {109}, journal = {Physical Review E}, number = {2}, arxiv = {http://arxiv.org/abs/2307.03491}, doi = {10.1103/PhysRevE.109.L022301}, pages = {L022301}, year = {2024}, abstract = {Collective variables (CVs) are low-dimensional projections of high-dimensional system states. They are used to gain insights into complex emergent dynamical behaviors of processes on networks. The relation between CVs and network measures is not well understood and its derivation typically requires detailed knowledge of both the dynamical system and the network topology. In this Letter, we present a data-driven method for algorithmically learning and understanding CVs for binary-state spreading processes on networks of arbitrary topology. We demonstrate our method using four example networks: the stochastic block model, a ring-shaped graph, a random regular graph, and a scale-free network generated by the Albert-Barab{\´a}si model. Our results deliver evidence for the existence of low-dimensional CVs even in cases that are not yet understood theoretically.}, language = {en} } @article{ErnstBankowskiJusyteetal.2025, author = {Ernst, Ariane and Bankowski, Anastasia and Jusyte, Meida and Okunola, Toluwani and Petrov, Tino and Walter, Alexander and Winkelmann, Stefanie}, title = {Parameter Optimization for a Neurotransmission Recovery Model}, volume = {87}, journal = {Bulletin of Mathematical Biology}, publisher = {Bulletin of Mathematical Biology}, doi = {10.1007/s11538-025-01486-2}, year = {2025}, abstract = {We assess the empirical applicability of a simplified model for neurotransmitter release that incorporates maturation, fusion, and recovery of both release sites and vesicles. Model parameters are optimized by fitting the model to experimental data obtained from neuromuscular junction synapses of 3rd-instar Drosophila melanogaster larvae. In particular, the mean-squared error between the local extrema of the simulated total junction current and its experimental counterpart is minimized. We compare three estimation approaches, differing in the choice of optimized parameters and the fusion rate function. Despite the model's minimalistic structure, it demonstrates a compelling ability to replicate experimental data, yielding plausible parameter estimates for five different animals. An additional identifiability analysis based on the profile likelihood reveals practical non-identifiabilities for several parameters, highlighting the need for additional constraints or data to improve estimation accuracy.}, language = {en} } @article{StraubeOliconMendezWinkelmannetal.2025, author = {Straube, Arthur and Olic{\´o}n M{\´e}ndez, Guillermo and Winkelmann, Stefanie and H{\"o}fling, Felix and Engel, Maximilian}, title = {Unfolding the geometric structure and multiple timescales of the urea-urease pH oscillator}, arxiv = {http://arxiv.org/abs/2508.07275}, year = {2025}, abstract = {We study a two-variable dynamical system modeling pH oscillations in the urea-urease reaction within giant lipid vesicles - a problem that intrinsically contains multiple, well-separated timescales. Building on an existing, deterministic formulation via ordinary differential equations, we resolve different orders of magnitude within a small parameter and analyze the system's limit cycle behavior using geometric singular perturbation theory (GSPT). By introducing two different coordinate scalings - each valid in a distinct region of the phase space - we resolve the local dynamics near critical fold points, using the extension of GSPT through such singular points due to Krupa and Szmolyan. This framework enables a geometric decomposition of the periodic orbits into slow and fast segments and yields closed-form estimates for the period of oscillation. In particular, we link the existence of such oscillations to an underlying biochemical asymmetry, namely, the differential transport across the vesicle membrane.}, language = {en} } @article{CoomberChewleSeckeretal.2025, author = {Coomber, Celvic and Chewle, Surahit and Secker, Christopher and Fackeldey, Konstantin and Weber, Marcus and Winkelmann, Stefanie and Sch{\"u}tte, Christof and Sunkara, Vikram}, title = {Investigating Endogenous Opioids Unravels the Mechanisms Behind Opioid-Induced Constipation, a Mathematical Modeling Approach}, volume = {26}, journal = {International Journal of Molecular Sciences}, number = {13}, doi = {10.3390/ijms26136207}, year = {2025}, abstract = {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.}, language = {en} } @article{WinkelmannZonkerSchuetteetal.2021, author = {Winkelmann, Stefanie and Zonker, Johannes and Sch{\"u}tte, Christof and Djurdjevac Conrad, Natasa}, title = {Mathematical modeling of spatio-temporal population dynamics and application to epidemic spreading}, volume = {336}, journal = {Mathematical Biosciences}, publisher = {Elsevier}, arxiv = {http://arxiv.org/abs/2205.05000}, doi = {10.1016/j.mbs.2021.108619}, year = {2021}, abstract = {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.}, language = {en} } @article{delRazoFroembergStraubeetal.2022, author = {del Razo, Mauricio and Fr{\"o}mberg, Daniela and Straube, Arthur and Sch{\"u}tte, Christof and H{\"o}fling, Felix and Winkelmann, Stefanie}, title = {A probabilistic framework for particle-based reaction-diffusion dynamics using classical Fock space representations}, volume = {112}, journal = {Letters in Mathematical Physics}, number = {49}, arxiv = {http://arxiv.org/abs/arXiv:2109.13616}, doi = {10.1007/s11005-022-01539-w}, year = {2022}, language = {en} } @article{StraubeWinkelmannSchuetteetal.2021, author = {Straube, Arthur and Winkelmann, Stefanie and Sch{\"u}tte, Christof and H{\"o}fling, Felix}, title = {Stochastic pH oscillations in a model of the urea-urease reaction confined to lipid vesicles}, volume = {12}, journal = {J. Phys. Chem. Lett.}, doi = {10.1021/acs.jpclett.1c03016}, pages = {9888 -- 9893}, year = {2021}, abstract = {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.}, language = {en} } @article{ErnstUngerSchuetteetal.2023, author = {Ernst, Ariane and Unger, Nathalie and Sch{\"u}tte, Christof and Walter, Alexander and Winkelmann, Stefanie}, title = {Rate-limiting recovery processes in neurotransmission under sustained stimulation}, volume = {362}, journal = {Mathematical Biosciences}, arxiv = {http://arxiv.org/abs/2302.01635}, doi = {10.1016/j.mbs.2023.109023}, year = {2023}, abstract = {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.}, language = {de} } @article{StraubeWinkelmannHoefling2023, author = {Straube, Arthur and Winkelmann, Stefanie and H{\"o}fling, Felix}, title = {Accurate reduced models for the pH oscillations in the urea-urease reaction confined to giant lipid vesicles}, volume = {127}, journal = {The Journal of Physical Chemistry B}, number = {13}, arxiv = {http://arxiv.org/abs/arXiv:2212.14503}, doi = {10.1021/acs.jpcb.2c09092}, pages = {2955 -- 2967}, year = {2023}, abstract = {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.}, language = {en} } @misc{StraubeWinkelmannHoefling2022, author = {Straube, Arthur and Winkelmann, Stefanie and H{\"o}fling, Felix}, title = {Accurate reduced models for the pH oscillations in the urea-urease reaction confined to giant lipid vesicles}, issn = {1438-0064}, doi = {10.12752/8817}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-88179}, year = {2022}, abstract = {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.}, language = {en} }