@article{KlusGelssPeitzetal., author = {Klus, Stefan and Gelß, Patrick and Peitz, Sebastian and Sch{\"u}tte, Christof}, title = {Tensor-based dynamic mode decomposition}, series = {Nonlinearity}, volume = {31}, journal = {Nonlinearity}, number = {7}, publisher = {IOP Publishing Ltd \& London Mathematical Society}, doi = {10.1088/1361-6544/aabc8f}, language = {en} } @article{GuptaGramatkeEinspanieretal.2017, author = {Gupta, Pooja and Gramatke, Annika and Einspanier, Ralf and Sch{\"u}tte, Christof and von Kleist, Max and Sharbati, Jutta}, title = {In silico cytotoxicity assessment on cultured rat intestinal cells deduced from cellular impedance measurements}, series = {Toxicology in Vitro}, volume = {41}, journal = {Toxicology in Vitro}, issn = {1438-0064}, pages = {179 -- 188}, year = {2017}, abstract = {Early and reliable identification of chemical toxicity is of utmost importance. At the same time, reduction of animal testing is paramount. Therefore, methods that improve the interpretability and usability of in vitro assays are essential. xCELLigence's real-time cell analyzer (RTCA) provides a novel, fast and cost effective in vitro method to probe compound toxicity. We developed a simple mathematical framework for the qualitative and quantitative assessment of toxicity for RTCA measurements. Compound toxicity, in terms of its 50\% inhibitory concentration IC50 on cell growth, and parameters related to cell turnover were estimated on cultured IEC-6 cells exposed to 10 chemicals at varying concentrations. Our method estimated IC50 values of 113.05, 7.16, 28.69 and 725.15 μM for the apparently toxic compounds 2-acetylamino-fluorene, aflatoxin B1, benzo-[a]-pyrene and chloramphenicol in the tested cell line, in agreement with literature knowledge. IC50 values of all apparent in vivo non-toxic compounds were estimated to be non-toxic by our method. Corresponding estimates from RTCA's in-built model gave false positive (toxicity) predictions in 5/10 cases. Taken together, our proposed method reduces false positive predictions and reliably identifies chemical toxicity based on impedance measurements. The source code for the developed method including instructions is available at https://git.zib.de/bzfgupta/toxfit/tree/master.}, language = {en} } @article{RuedrichSarichSchuette, author = {R{\"u}drich, S. and Sarich, Marco and Sch{\"u}tte, Christof}, title = {Utilizing hitting times for finding metastable sets in non-reversible Markov chains}, series = {Journal of Comp. Dynamics}, journal = {Journal of Comp. Dynamics}, language = {en} } @article{ConradGenzelCvetkovicetal., author = {Conrad, Tim and Genzel, Martin and Cvetkovic, Nada and Wulkow, Niklas and Leichtle, Alexander Benedikt and Vybiral, Jan and Kytyniok, Gitta and Sch{\"u}tte, Christof}, title = {Sparse Proteomics Analysis - a compressed sensing-based approach for feature selection and classification of high-dimensional proteomics mass spectrometry data}, series = {BMC Bioinfomatics}, volume = {18}, journal = {BMC Bioinfomatics}, number = {160}, doi = {10.1186/s12859-017-1565-4}, abstract = {Background: High-throughput proteomics techniques, such as mass spectrometry (MS)-based approaches, produce very high-dimensional data-sets. In a clinical setting one is often interested in how mass spectra differ between patients of different classes, for example spectra from healthy patients vs. spectra from patients having a particular disease. Machine learning algorithms are needed to (a) identify these discriminating features and (b) classify unknown spectra based on this feature set. Since the acquired data is usually noisy, the algorithms should be robust against noise and outliers, while the identified feature set should be as small as possible. Results: We present a new algorithm, Sparse Proteomics Analysis (SPA),based on thet heory of compressed sensing that allows us to identify a minimal discriminating set of features from mass spectrometry data-sets. We show (1) how our method performs on artificial and real-world data-sets, (2) that its performance is competitive with standard (and widely used) algorithms for analyzing proteomics data, and (3) that it is robust against random and systematic noise. We further demonstrate the applicability of our algorithm to two previously published clinical data-sets.}, language = {en} } @article{VegaSchuetteConrad, author = {Vega, Iliusi and Sch{\"u}tte, Christof and Conrad, Tim}, title = {Finding metastable states in real-world time series with recurrence networks}, series = {Physica A: Statistical Mechanics and its Applications}, volume = {445}, journal = {Physica A: Statistical Mechanics and its Applications}, doi = {10.1016/j.physa.2015.10.041}, pages = {1 -- 17}, abstract = {In the framework of time series analysis with recurrence networks, we introduce a self-adaptive method that determines the elusive recurrence threshold and identifies metastable states in complex real-world time series. As initial step, we introduce a way to set the embedding parameters used to reconstruct the state space from the time series. We set them as the ones giving the maximum Shannon entropy of the diagonal line length distribution for the first simultaneous minima of recurrence rate and Shannon entropy. To identify metastable states, as well as the transitions between them, we use a soft partitioning algorithm for module finding which is specifically developed for the case in which a system shows metastability. We illustrate our method with a complex time series example. Finally, we show the robustness of our method for identifying metastable states. Our results suggest that our method is robust for identifying metastable states in complex time series, even when introducing considerable levels of noise and missing data points.}, language = {en} } @article{GelssMateraSchuette, author = {Gelß, Patrick and Matera, Sebastian and Sch{\"u}tte, Christof}, title = {Solving the master equation without kinetic Monte Carlo: Tensor train approximations for a CO oxidation model}, series = {Journal of Computational Physics}, volume = {314}, journal = {Journal of Computational Physics}, doi = {10.1016/j.jcp.2016.03.025}, pages = {489 -- 502}, abstract = {In multiscale modeling of heterogeneous catalytic processes, one crucial point is the solution of a Markovian master equation describing the stochastic reaction kinetics. Usually, this is too high-dimensional to be solved with standard numerical techniques and one has to rely on sampling approaches based on the kinetic Monte Carlo method. In this study we break the curse of dimensionality for the direct solution of the Markovian master equation by exploiting the Tensor Train Format for this purpose. The performance of the approach is demonstrated on a first principles based, reduced model for the CO oxidation on the RuO2(110) surface. We investigate the complexity for increasing system size and for various reaction conditions. The advantage over the stochastic simulation approach is illustrated by a problem with increased}, language = {en} } @article{KlusKoltaiSchuette, author = {Klus, Stefan and Koltai, Peter and Sch{\"u}tte, Christof}, title = {On the numerical approximation of the Perron-Frobenius and Koopman operator}, series = {Journal of Computational Dynamics}, volume = {3}, journal = {Journal of Computational Dynamics}, number = {1}, doi = {10.3934/jcd.2016003}, pages = {51 -- 77}, abstract = {Information about the behavior of dynamical systems can often be obtained by analyzing the eigenvalues and corresponding eigenfunctions of linear operators associated with a dynamical system. Examples of such operators are the Perron-Frobenius and the Koopman operator. In this paper, we will review di� fferent methods that have been developed over the last decades to compute � infinite-dimensional approximations of these in� finite-dimensional operators - in particular Ulam's method and Extended Dynamic Mode Decomposition (EDMD) - and highlight the similarities and di� fferences between these approaches. The results will be illustrated using simple stochastic di� fferential equations and molecular dynamics examples.}, language = {en} } @article{HartmannSchuetteZhang, author = {Hartmann, Carsten and Sch{\"u}tte, Christof and Zhang, Wei}, title = {Model reduction algorithms for optimal control and importance sampling of diffusions}, series = {Nonlinearity}, volume = {29}, journal = {Nonlinearity}, number = {8}, doi = {10.1088/0951-7715/29/8/2298}, pages = {2298 -- 2326}, abstract = {We propose numerical algorithms for solving optimal control and importance sampling problems based on simplified models. The algorithms combine model reduction techniques for multiscale diffusions and stochastic optimization tools, with the aim of reducing the original, possibly high-dimensional problem to a lower dimensional representation of the dynamics, in which only a few relevant degrees of freedom are controlled or biased. Specifically, we study situations in which either a reaction coordinate onto which the dynamics can be projected is known, or situations in which the dynamics shows strongly localized behavior in the small noise regime. No explicit assumptions about small parameters or scale separation have to be made. We illustrate the approach with simple, but paradigmatic numerical examples.}, language = {en} } @article{KlusSchuette, author = {Klus, Stefan and Sch{\"u}tte, Christof}, title = {Towards tensor-based methods for the numerical approximation of the Perron-Frobenius and Koopman operator}, series = {Journal of Computational Dynamics}, volume = {3}, journal = {Journal of Computational Dynamics}, number = {2}, doi = {10.3934/jcd.2016007}, pages = {139 -- 161}, abstract = {The global behavior of dynamical systems can be studied by analyzing the eigenvalues and corresponding eigenfunctions of linear operators associated with the system. Two important operators which are frequently used to gain insight into the system's behavior are the Perron-Frobenius operator and the Koopman operator. Due to the curse of dimensionality, computing the eigenfunctions of high-dimensional systems is in general infeasible. We will propose a tensor-based reformulation of two numerical methods for computing finite-dimensional approximations of the aforementioned infinite-dimensional operators, namely Ulam's method and Extended Dynamic Mode Decomposition (EDMD). The aim of the tensor formulation is to approximate the eigenfunctions by low-rank tensors, potentially resulting in a significant reduction of the time and memory required to solve the resulting eigenvalue problems, provided that such a low-rank tensor decomposition exists. Typically, not all variables of a high-dimensional dynamical system contribute equally to the system's behavior, often the dynamics can be decomposed into slow and fast processes, which is also reflected in the eigenfunctions. Thus, the weak coupling between different variables might be approximated by low-rank tensor cores. We will illustrate the efficiency of the tensor-based formulation of Ulam's method and EDMD using simple stochastic differential equations.}, 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{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} } @misc{KlebanovSikorskiSchuetteetal., author = {Klebanov, Ilja and Sikorski, Alexander and Sch{\"u}tte, Christof and R{\"o}blitz, Susanna}, title = {Empirical Bayes Methods for Prior Estimation in Systems Medicine}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-61307}, abstract = {One of the main goals of mathematical modelling in systems medicine related to medical applications is to obtain patient-specific parameterizations and model predictions. In clinical practice, however, the number of available measurements for single patients is usually limited due to time and cost restrictions. This hampers the process of making patient-specific predictions about the outcome of a treatment. On the other hand, data are often available for many patients, in particular if extensive clinical studies have been performed. Therefore, before applying Bayes' rule separately to the data of each patient (which is typically performed using a non-informative prior), it is meaningful to use empirical Bayes methods in order to construct an informative prior from all available data. We compare the performance of four priors - a non-informative prior and priors chosen by nonparametric maximum likelihood estimation (NPMLE), by maximum penalized lilelihood estimation (MPLE) and by doubly-smoothed maximum likelihood estimation (DS-MLE) - by applying them to a low-dimensional parameter estimation problem in a toy model as well as to a high-dimensional ODE model of the human menstrual cycle, which represents a typical example from systems biology modelling.}, language = {en} } @misc{KlebanovSikorskiSchuetteetal., author = {Klebanov, Ilja and Sikorski, Alexander and Sch{\"u}tte, Christof and R{\"o}blitz, Susanna}, title = {Empirical Bayes Methods, Reference Priors, Cross Entropy and the EM Algorithm}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-61230}, abstract = {When estimating a probability density within the empirical Bayes framework, the non-parametric maximum likelihood estimate (NPMLE) usually tends to overfit the data. This issue is usually taken care of by regularization - a penalization term is subtracted from the marginal log-likelihood before the maximization step, so that the estimate favors smooth solutions, resulting in the so-called maximum penalized likelihood estimation (MPLE). The majority of penalizations currently in use are rather arbitrary brute-force solutions, which lack invariance under transformation of the parameters(reparametrization) and measurements. This contradicts the principle that, if the underlying model has several equivalent formulations, the methods of inductive inference should lead to consistent results. Motivated by this principle and using an information-theoretic point of view, we suggest an entropy-based penalization term that guarantees this kind of invariance. The resulting density estimate can be seen as a generalization of reference priors. Using the reference prior as a hyperprior, on the other hand, is argued to be a poor choice for regularization. We also present an insightful connection between the NPMLE, the cross entropy and the principle of minimum discrimination information suggesting another method of inference that contains the doubly-smoothed maximum likelihood estimation as a special case.}, language = {en} } @misc{WinkelmannSchuette, 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}, 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{DuwalWinkelmannSchuetteetal., author = {Duwal, Sulav and Winkelmann, Stefanie and Sch{\"u}tte, Christof and von Kleist, Max}, title = {Optimal Treatment Strategies in the Context of 'Treatment for Prevention' against HIV/1 in Resource-Poor Settings}, series = {PloS Computational Biology}, volume = {11}, journal = {PloS Computational Biology}, number = {4}, doi = {10.1371/journal.pcbi.1004200}, abstract = {An estimated 2.7 million new HIV-1 infections occurred in 2010. `Treatment-for-prevention' may strongly prevent HIV-1 transmission. The basic idea is that immediate treatment initiation rapidly decreases virus burden, which reduces the number of transmittable viruses and thereby the probability of infection. However, HIV inevitably develops drug resistance, which leads to virus rebound and nullifies the effect of `treatment-for-prevention' for the time it remains unrecognized. While timely conducted treatment changes may avert periods of viral rebound, necessary treatment options and diagnostics may be lacking in resource-constrained settings. Within this work, we provide a mathematical platform for comparing different treatment paradigms that can be applied to many medical phenomena. We use this platform to optimize two distinct approaches for the treatment of HIV-1: (i) a diagnostic-guided treatment strategy, based on infrequent and patient-specific diagnostic schedules and (ii) a pro-active strategy that allows treatment adaptation prior to diagnostic ascertainment. Both strategies are compared to current clinical protocols (standard of care and the HPTN052 protocol) in terms of patient health, economic means and reduction in HIV-1 onward transmission exemplarily for South Africa. All therapeutic strategies are assessed using a coarse-grained stochastic model of within-host HIV dynamics and pseudo-codes for solving the respective optimal control problems are provided. Our mathematical model suggests that both optimal strategies (i)-(ii) perform better than the current clinical protocols and no treatment in terms of economic means, life prolongation and reduction of HIV-transmission. The optimal diagnostic-guided strategy suggests rare diagnostics and performs similar to the optimal pro-active strategy. Our results suggest that 'treatment-for-prevention' may be further improved using either of the two analyzed treatment paradigms.}, language = {en} } @article{WinkelmannSchuette, author = {Winkelmann, Stefanie and Sch{\"u}tte, Christof}, title = {The Spatiotemporal Master Equation: Approximation of Reaction-Diffusion Dynamics via Markov State Modeling}, series = {Journal of Chemical Physics}, volume = {145}, journal = {Journal of Chemical Physics}, number = {21}, doi = {10.1063/1.4971163}, 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} } @misc{ZhangHartmannSchuette, author = {Zhang, Wei and Hartmann, Carsten and Sch{\"u}tte, Christof}, title = {Effective Dynamics Along Given Reaction Coordinates, and Reaction Rate Theory}, issn = {1438-0064}, doi = {10.1039/C6FD00147E}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-59706}, abstract = {In molecular dynamics and related fields one considers dynamical descriptions of complex systems in full (atomic) detail. In order to reduce the overwhelming complexity of realistic systems (high dimension, large timescale spread, limited computational resources) the projection of the full dynamics onto some reaction coordinates is examined in order to extract statistical information like free energies or reaction rates. In this context, the effective dynamics that is induced by the full dynamics on the reaction coordinate space has attracted considerable attention in the literature. In this article, we contribute to this discussion: We first show that if we start with an ergodic diffusion processes whose invariant measure is unique then these properties are inherited by the effective dynamics. Then, we give equations for the effective dynamics, discuss whether the dominant timescales and reaction rates inferred from the effective dynamics are accurate approximations of such quantities for the full dynamics, and compare our findings to results from approaches like Zwanzig-Mori, averaging, or homogenization. Finally, by discussing the algorithmic realization of the effective dynamics, we demonstrate that recent algorithmic techniques like the "equation-free" approach and the "heterogeneous multiscale method" can be seen as special cases of our approach.}, language = {en} } @misc{OsterlandBennProhaskaetal., author = {Osterland, Marc and Benn, Andreas and Prohaska, Steffen and Sch{\"u}tte, Christof}, title = {Single Cell Tracking in Phase-Contrast Microscopy}, series = {EMBL Symposium 2015 - Seeing is Believing - Imaging the Processes of Life}, journal = {EMBL Symposium 2015 - Seeing is Believing - Imaging the Processes of Life}, abstract = {In this work, we developed an automatic algorithm to analyze cell migration in chemotaxis assays, based on phase-contrast time-lapse microscopy. While manual approaches are still widely used in recent publications, our algorithm is able to track hundreds of single cells per frame. The extracted paths are analysed with traditional geometrical approaches as well as diffusion-driven Markov state models (MSM). Based on these models, a detailed view on spatial and temporal effects is possible. Using our new approach on experimental data, we are able to distinguish between directed migration (e.g. towards a VEGF gradient) and random migration without favored direction. A calculation of the committor probabilities reveals that cells of the whole image area are more likely to migrate directly towards the VEGF than away from it during the first four hours. However, in absence of a chemoattractant, cells migrate more likely to their nearest image border. These conclusions are supported by the spatial mean directions. In a next step, the cell-cell interaction during migration and the migration of cell clusters will be analyzed. Furthermore, we want to observe phenotypical changes during migration based on fluorescence microscopy and machine learning. The algorithm is part of a collaborative platform which brings the experimental expertise of scientists from life sciences and the analytical knowledge of computer scientists together. This platform is built using web-based technologies with a responsive real-time user interface. All data, including raw and metadata as well as the accompanying results, will be stored in a secure and scalable compute cluster. The compute cluster provides sufficient space and computational power for modern image-based experiments and their analyses. Specific versions of data and results can be tagged to keep immutable records for archival.}, language = {en} } @inproceedings{GuptaKrauseRikeitetal., author = {Gupta, Pooja and Krause, Carola and Rikeit, Paul and R{\"o}blitz, Susanna and Knaus, Petra and Sch{\"u}tte, Christof}, title = {Modeling of the BMP mediated co-regulation of the Smad and Non-Smad pathways in the context of cell density}, series = {10th International BMP conference, 2014, Berlin, Germany}, booktitle = {10th International BMP conference, 2014, Berlin, Germany}, language = {en} } @misc{GuptaRoeblitzKrauseetal., author = {Gupta, Pooja and R{\"o}blitz, Susanna and Krause, Carola and Knaus, Petra and Sch{\"u}tte, Christof}, title = {Mathematical modeling of the Smad and Non-Smad BMP signaling pathways in context of cell density}, series = {Computational Models in biology and medicine, 2013, Dresden, Germany}, journal = {Computational Models in biology and medicine, 2013, Dresden, Germany}, language = {en} }