@misc{Winkelmann2016, author = {Winkelmann, Stefanie}, title = {Markov Control with Rare State Observation: Average Optimality}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-60981}, year = {2016}, abstract = {This paper investigates the criterion of long-term average costs for a Markov decision process (MDP) which is not permanently observable. Each observation of the process produces a fixed amount of \textit{information costs} which enter the considered performance criterion and preclude from arbitrarily frequent state testing. Choosing the \textit{rare} observation times is part of the control procedure. In contrast to the theory of partially observable Markov decision processes, we consider an arbitrary continuous-time Markov process on a finite state space without further restrictions on the dynamics or the type of interaction. Based on the original Markov control theory, we redefine the control model and the average cost criterion for the setting of information costs. We analyze the constant of average costs for the case of ergodic dynamics and present an optimality equation which characterizes the optimal choice of control actions and observation times. For this purpose, we construct an equivalent freely observable MDP and translate the well-known results from the original theory to the new setting.}, language = {en} } @misc{WinkelmannSchuette2016, author = {Winkelmann, Stefanie and Sch{\"u}tte, Christof}, title = {The spatiotemporal master equation: approximation of reaction-diffusion dynamics via Markov state modeling}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-60999}, year = {2016}, abstract = {Accurate modeling and numerical simulation of reaction kinetics is a topic of steady interest. We consider the spatiotemporal chemical master equation (ST-CME) as a model for stochastic reaction-diffusion systems that exhibit properties of metastability. The space of motion is decomposed into metastable compartments and diffusive motion is approximated by jumps between these compartments. Treating these jumps as first-order reactions, simulation of the resulting stochastic system is possible by the Gillespie method. We present the theory of Markov state models (MSM) as a theoretical foundation of this intuitive approach. By means of Markov state modeling, both the number and shape of compartments and the transition rates between them can be determined. We consider the ST-CME for two reaction-diffusion systems and compare it to more detailed models. Moreover, a rigorous formal justification of the ST-CME by Galerkin projection methods is presented.}, language = {en} } @article{Winkelmann2017, author = {Winkelmann, Stefanie}, title = {Markov Control with Rare State Observation: Average Optimality}, volume = {23}, journal = {Markov Processes and Related Fields}, pages = {1 -- 34}, year = {2017}, abstract = {This paper investigates the criterion of long-term average costs for a Markov decision process (MDP) which is not permanently observable. Each observation of the process produces a fixed amount of information costs which enter the considered performance criterion and preclude from arbitrarily frequent state testing. Choosing the rare observation times is part of the control procedure. In contrast to the theory of partially observable Markov decision processes, we consider an arbitrary continuous-time Markov process on a finite state space without further restrictions on the dynamics or the type of interaction. Based on the original Markov control theory, we redefine the control model and the average cost criterion for the setting of information costs. We analyze the constant of average costs for the case of ergodic dynamics and present an optimality equation which characterizes the optimal choice of control actions and observation times. For this purpose, we construct an equivalent freely observable MDP and translate the well-known results from the original theory to the new setting.}, language = {en} } @article{WinkelmannSchuette2016, author = {Winkelmann, Stefanie and Sch{\"u}tte, Christof}, title = {The Spatiotemporal Master Equation: Approximation of Reaction-Diffusion Dynamics via Markov State Modeling}, volume = {145}, journal = {Journal of Chemical Physics}, number = {21}, doi = {10.1063/1.4971163}, year = {2016}, abstract = {Accurate modeling and numerical simulation of reaction kinetics is a topic of steady interest.We consider the spatiotemporal chemical master equation (ST-CME) as a model for stochastic reaction-diffusion systems that exhibit properties of metastability. The space of motion is decomposed into metastable compartments and diffusive motion is approximated by jumps between these compartments. Treating these jumps as first-order reactions, simulation of the resulting stochastic system is possible by the Gillespie method. We present the theory of Markov state models (MSM) as a theoretical foundation of this intuitive approach. By means of Markov state modeling, both the number and shape of compartments and the transition rates between them can be determined. We consider the ST-CME for two reaction-diffusion systems and compare it to more detailed models. Moreover, a rigorous formal justification of the ST-CME by Galerkin projection methods is presented.}, language = {en} } @book{WinkelmannSchuette2020, author = {Winkelmann, Stefanie and Sch{\"u}tte, Christof}, title = {Stochastic Dynamics in Computational Biology}, volume = {8}, journal = {Frontiers in Applied Dynamical Systems: Reviews and Tutorials}, publisher = {Springer International Publishing}, isbn = {978-3-030-62386-9}, doi = {10.1007/978-3-030-62387-6}, year = {2020}, language = {en} } @article{NiemannWinkelmannWolfetal.2021, author = {Niemann, Jan-Hendrik and Winkelmann, Stefanie and Wolf, Sarah and Sch{\"u}tte, Christof}, title = {Agent-based modeling: Population limits and large timescales}, volume = {31}, journal = {Chaos: An Interdisciplinary Journal of Nonlinear Science}, number = {3}, issn = {1438-0064}, doi = {10.1063/5.0031373}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-77309}, year = {2021}, abstract = {Modeling, simulation and analysis of interacting agent systems is a broad field of research, with existing approaches reaching from informal descriptions of interaction dynamics to more formal, mathematical models. In this paper, we study agent-based models (ABMs) given as continuous-time stochastic processes and their pathwise approximation by ordinary and stochastic differential equations (ODEs and SDEs, respectively) for medium to large populations. By means of an appropriately adapted transfer operator approach we study the behavior of the ABM process on long time scales. We show that, under certain conditions, the transfer operator approach allows to bridge the gap between the pathwise results for large populations on finite timescales, i.e., the SDE limit model, and approaches built to study dynamical behavior on long time scales like large deviation theory. The latter provides a rigorous analysis of rare events including the associated asymptotic rates on timescales that scale exponentially with the population size. We demonstrate that it is possible to reveal metastable structures and timescales of rare events of the ABM process by finite-length trajectories of the SDE process for large enough populations. This approach has the potential to drastically reduce computational effort for the analysis of ABMs.}, language = {en} } @misc{WinkelmannSchuette2017, author = {Winkelmann, Stefanie and Sch{\"u}tte, Christof}, title = {Hybrid Models for Chemical Reaction Networks: Multiscale Theory and Application to Gene Regulatory Systems}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-64264}, year = {2017}, abstract = {Well-mixed stochastic chemical kinetics are properly modelled by the chemical master equation (CME) and associated Markov jump processes in molecule number space. If the reactants are present in large amounts, however, corresponding simulations of the stochastic dynamics become computationally expensive and model reductions are demanded. The classical model reduction approach uniformly rescales the overall dynamics to obtain deterministic systems characterized by ordinary differential equations, the well-known mass action reaction rate equations. For systems with multiple scales there exist hybrid approaches that keep parts of the system discrete while another part is approximated either using Langevin dynamics or deterministically. This paper aims at giving a coherent overview of the different hybrid approaches, focusing on their basic concepts and the relation between them. We derive a novel general description of such hybrid models that allows to express various forms by one type of equation. We also check in how far the approaches apply to model extensions of the CME for dynamics which do not comply with the central well-mixed condition and require some spatial resolution. A simple but meaningful gene expression system with negative self-regulation is analysed to illustrate the different approximation qualities of some of the hybrid approaches discussed.}, language = {en} } @article{WinkelmannSchuette2017, author = {Winkelmann, Stefanie and Sch{\"u}tte, Christof}, title = {Hybrid models for chemical reaction networks: Multiscale theory and application to gene regulatory systems}, volume = {147}, journal = {The Journal of Chemical Physics}, number = {11}, doi = {10.1063/1.4986560}, pages = {114115-1 -- 114115-18}, year = {2017}, abstract = {Well-mixed stochastic chemical kinetics are properly modeled by the chemical master equation (CME) and associated Markov jump processes in molecule number space. If the reactants are present in large amounts, however, corresponding simulations of the stochastic dynamics become computationally expensive and model reductions are demanded. The classical model reduction approach uniformly rescales the overall dynamics to obtain deterministic systems characterized by ordinary differential equations, the well-known mass action reaction rate equations. For systems with multiple scales, there exist hybrid approaches that keep parts of the system discrete while another part is approximated either using Langevin dynamics or deterministically. This paper aims at giving a coherent overview of the different hybrid approaches, focusing on their basic concepts and the relation between them. We derive a novel general description of such hybrid models that allows expressing various forms by one type of equation. We also check in how far the approaches apply to model extensions of the CME for dynamics which do not comply with the central well-mixed condition and require some spatial resolution. A simple but meaningful gene expression system with negative self-regulation is analysed to illustrate the different approximation qualities of some of the hybrid approaches discussed. Especially, we reveal the cause of error in the case of small volume approximations.}, language = {en} } @article{ThiesSunkaraRayetal.2023, 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}, volume = {13}, journal = {Scientific Reports}, number = {607}, doi = {10.1038/s41598-023-27699-w}, year = {2023}, 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{MontefuscoSchuetteWinkelmann2023, author = {Montefusco, Alberto and Sch{\"u}tte, Christof and Winkelmann, Stefanie}, title = {A route to the hydrodynamic limit of a reaction-diffusion master equation using gradient structures}, volume = {83}, journal = {SIAM Journal on Applied Mathematics}, number = {2}, arxiv = {http://arxiv.org/abs/2201.02613}, doi = {10.1137/22M1488831}, pages = {837 -- 861}, year = {2023}, abstract = {The reaction-diffusion master equation (RDME) is a lattice-based stochastic model for spatially resolved cellular processes. It is often interpreted as an approximation to spatially continuous reaction-diffusion models, which, in the limit of an infinitely large population, may be described by means of reaction-diffusion partial differential equations. Analyzing and understanding the relation between different mathematical models for reaction-diffusion dynamics is a research topic of steady interest. In this work, we explore a route to the hydrodynamic limit of the RDME which uses gradient structures. Specifically, we elaborate on a method introduced in [J. Maas and A. Mielke, J. Stat. Phys., 181 (2020), pp. 2257-2303] in the context of well-mixed reaction networks by showing that, once it is complemented with an appropriate limit procedure, it can be applied to spatially extended systems with diffusion. Under the assumption of detailed balance, we write down a gradient structure for the RDME and use the method in order to produce a gradient structure for its hydrodynamic limit, namely, for the corresponding RDPDE.}, language = {en} } @article{ErnstFalkenhagenWinkelmann2023, author = {Ernst, Ariane and Falkenhagen, Undine and Winkelmann, Stefanie}, title = {Model reduction for calcium-induced vesicle fusion dynamics}, volume = {23}, journal = {Proceedings in Applied Mathematics \& Mechanics}, number = {4}, doi = {10.1002/pamm.202300184}, year = {2023}, abstract = {In this work, we adapt an established model for the Ca2+-induced fusion dynamics of synaptic vesicles and employ a lumping method to reduce its complexity. In the reduced system, sequential Ca2+-binding steps are merged to a single releasable state, while keeping the important dependence of the reaction rates on the local Ca2+ concentration. We examine the feasibility of this model reduction for a representative stimulus train over the physiologically relevant site-channel distances. Our findings show that the approximation error is generally small and exhibits an interesting nonlinear and non-monotonic behavior where it vanishes for very low distances and is insignificant at intermediary distances. Furthermore, we give expressions for the reduced model's reaction rates and suggest that our approach may be used to directly compute effective fusion rates for assessing the validity of a fusion model, thereby circumventing expensive simulations.}, language = {en} }