TY - GEN A1 - Gelß, Patrick A1 - Matera, Sebastian A1 - Schütte, Christof T1 - Solving the master equation without kinetic Monte Carlo: tensor train approximations for a CO oxidation model N2 - In multiscale models of heterogeneous catalysis, one crucial point is the solution of a Markovian master equation describing the stochastic reaction kinetics. This usually 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 RuO_2(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 stiffness. T3 - ZIB-Report - 15-41 KW - heterogeneous catalysis KW - master equation KW - kinetic Monte Carlo KW - tensor decompositions KW - tensor train format KW - alternating linear scheme Y1 - 2015 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-55743 SN - 1438-0064 ER - TY - JOUR A1 - Kryven, Ivan A1 - Röblitz, Susanna A1 - Schütte, Christof T1 - Solution of the chemical master equation by radial basis functions approximation with interface tracking JF - BMC Systems Biology N2 - Background. The chemical master equation is the fundamental equation of stochastic chemical kinetics. This differential-difference equation describes temporal evolution of the probability density function for states of a chemical system. A state of the system, usually encoded as a vector, represents the number of entities or copy numbers of interacting species, which are changing according to a list of possible reactions. It is often the case, especially when the state vector is high-dimensional, that the number of possible states the system may occupy is too large to be handled computationally. One way to get around this problem is to consider only those states that are associated with probabilities that are greater than a certain threshold level. Results. We introduce an algorithm that significantly reduces computational resources and is especially powerful when dealing with multi-modal distributions. The algorithm is built according to two key principles. Firstly, when performing time integration, the algorithm keeps track of the subset of states with significant probabilities (essential support). Secondly, the probability distribution that solves the equation is parametrised with a small number of coefficients using collocation on Gaussian radial basis functions. The system of basis functions is chosen in such a way that the solution is approximated only on the essential support instead of the whole state space. Discussion. In order to demonstrate the effectiveness of the method, we consider four application examples: a) the self-regulating gene model, b) the 2-dimensional bistable toggle switch, c) a generalisation of the bistable switch to a 3-dimensional tristable problem, and d) a 3-dimensional cell differentiation model that, depending on parameter values, may operate in bistable or tristable modes. In all multidimensional examples the manifold containing the system states with significant probabilities undergoes drastic transformations over time. This fact makes the examples especially challenging for numerical methods. Conclusions. The proposed method is a new numerical approach permitting to approximately solve a wide range of problems that have been hard to tackle until now. A full representation of multi-dimensional distributions is recovered. The method is especially attractive when dealing with models that yield solutions of a complex structure, for instance, featuring multi-stability. Electronic version: http://www.biomedcentral.com/1752-0509/9/67 Y1 - 2015 U6 - https://doi.org/10.1186/s12918-015-0210-y VL - 9 IS - 67 SP - 1 EP - 12 ER - TY - GEN A1 - Osterland, Marc A1 - Benn, Andreas A1 - Prohaska, Steffen A1 - Schütte, Christof T1 - Single Cell Tracking in Phase-Contrast Microscopy T2 - EMBL Symposium 2015 - Seeing is Believing - Imaging the Processes of Life N2 - 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. Y1 - 2015 ER - TY - JOUR A1 - Banisch, Ralf A1 - Djurdjevac Conrad, Natasa A1 - Schütte, Christof T1 - Reactive flows and unproductive cycles for random walks on complex networks JF - The European Physical Journal Special Topics, vol. 224, iss. 12 (2015) pp. 2369-2387 Y1 - 2015 U6 - https://doi.org/10.1140/epjst/e2015-02417-8 ER - TY - GEN A1 - Banisch, Ralf A1 - Djurdjevac Conrad, Natasa A1 - Schütte, Christof T1 - Reactive flows and unproductive cycles for random walks on complex networks N2 - We present a comprehensive theory for analysis and understanding of transition events between an initial set A and a target set B for general ergodic finite-state space Markov chains or jump processes, including random walks on networks as they occur, e.g., in Markov State Modelling in molecular dynamics. The theory allows us to decompose the probability flow generated by transition events between the sets A and B into the productive part that directly flows from A to B through reaction pathways and the unproductive part that runs in loops and is supported on cycles of the underlying network. It applies to random walks on directed networks and nonreversible Markov processes and can be seen as an extension of Transition Path Theory. Information on reaction pathways and unproductive cycles results from the stochastic cycle decomposition of the underlying network which also allows to compute their corresponding weight, thus characterizing completely which structure is used how often in transition events. The new theory is illustrated by an application to a Markov State Model resulting from weakly damped Langevin dynamics where the unproductive cycles are associated with periodic orbits of the underlying Hamiltonian dynamics. T3 - ZIB-Report - 15-19 KW - Complex networks KW - molecular transition networks KW - transition path theory KW - cycle decomposition KW - reactive trajectories KW - Markow State Methods Y1 - 2015 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-54239 SN - 1438-0064 ER - TY - JOUR A1 - Duwal, Sulav A1 - Winkelmann, Stefanie A1 - Schütte, Christof A1 - von Kleist, Max T1 - Optimal Treatment Strategies in the Context of 'Treatment for Prevention' against HIV/1 in Resource-Poor Settings JF - PloS Computational Biology N2 - 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. Y1 - 2015 U6 - https://doi.org/10.1371/journal.pcbi.1004200 VL - 11 IS - 4 ER - TY - JOUR A1 - Djurdjevac Conrad, Natasa A1 - Banisch, Ralf A1 - Schütte, Christof T1 - Modularity of Directed Networks: Cycle Decomposition Approach JF - Journal of Computational Dynamics 2 (2015) pp. 1-24 N2 - The problem of decomposing networks into modules (or clusters) has gained much attention in recent years, as it can account for a coarsegrained description of complex systems, often revealing functional subunits of these systems. A variety of module detection algorithms have been proposed, mostly oriented towards finding hard partitionings of undirected networks. Despite the increasing number of fuzzy clustering methods for directed networks, many of these approaches tend to neglect important directional information. In this paper, we present a novel random walk based approach for finding fuzzy partitions of directed, weighted networks, where edge directions play a crucial role in defining how well nodes in a module are interconnected. We will show that cycle decomposition of a random walk process connects the notion of network modules and information transport in a network, leading to a new, symmetric measure of node communication. Finally, we will use this measure to introduce a communication graph, for which we will show that although being undirected it inherits all necessary information about modular structures from the original network. Y1 - 2015 U6 - https://doi.org/10.3934/jcd.2015.2.1 ER - TY - GEN A1 - Gul, Raheem A1 - Schütte, Christof A1 - Bernhard, Stefan T1 - Mathematical modeling and sensitivity analysis of arterial anastomosis in arm arteries N2 - Cardiovascular diseases are one of the major problems in medicine today and the number of patients increases worldwide. To find the most efficient treatment, prior knowledge about function and dysfunction of the cardiovas- cular system is required and methods need to be developed that identify the disease in an early stage. Mathematical modeling is a powerful tool for prediction and investigation of cardiovascular diseases. It has been shown that the Windkessel model, being based on an analogy between electrical circuits and fluid flow, is a simple but effective method to model the human cardiovascular system. In this paper, we have applied parametric local sensitivity analysis (LSA) to a linear elastic model of the arm arteries, to find and rank sensitive param- eters that may be helpful in clinical diagnosis. A computational model for end-to-side anastomosis (superior ulnar collateral anastomosis with posterior ulnar recurrent, SUC-PUR) is carried out to study the effects of some clinically relevant haemodynamic parameters like blood flow resistance and terminal re- sistance on pressure and flow at different locations of the arm artery. In this context, we also discuss the spatio-temporal dependency of local sensitivities. The sensitivities with respect to cardiovascular parameters reveal the flow resistance and diameter of the vessels as most sensitive parameters. These parameters play a key role in diagnosis of severe stenosis and aneurysms. In contrast, wall thickness and elastic modulus are found to be less sensitive. T3 - ZIB-Report - 15-22 KW - computational cardiovascular model KW - cardiovascular parameters KW - sensitivity analysis KW - anastomosis KW - Windkessel model Y1 - 2015 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-54339 SN - 1438-0064 ER - TY - JOUR A1 - Schütte, Christof A1 - Nielsen, Adam A1 - Weber, Marcus T1 - Markov State Models and Molecular Alchemy JF - Molecular Physics N2 - In recent years Markov State Models (MSMs) have attracted a consid- erable amount of attention with regard to modelling conformation changes and associated function of biomolecular systems. They have been used successfully, e.g., for peptides including time-resolved spectroscopic experiments, protein function and protein folding , DNA and RNA, and ligand-receptor interaction in drug design and more complicated multivalent scenarios. In this article a novel reweighting scheme is introduced that allows to construct an MSM for certain molecular system out of an MSM for a similar system. This permits studying how molecular properties on long timescales differ between similar molecular systems without performing full molecular dynamics simulations for each system under con- sideration. The performance of the reweighting scheme is illustrated for simple test cases including one where the main wells of the respective energy landscapes are located differently and an alchemical transformation of butane to pentane where the dimension of the state space is changed. KW - MSM KW - Reweighting KW - Girsanov Y1 - 2015 U6 - https://doi.org/10.1080/00268976.2014.944597 VL - 113 IS - 1 SP - 69 EP - 78 ER - TY - GEN A1 - Enciso, Marta A1 - Schütte, Christof A1 - Delle Site, Luigi T1 - Influence of pH and sequence in peptide aggregation via molecular simulation N2 - We employ a recently developed coarse-grained model for peptides and proteins where the effect of pH is automatically included. We explore the effect of pH in the aggregation process of the amyloidogenic peptide KTVIIE and two related sequences, using three different pH environments. Simulations using large systems (24 peptides chains per box) allow us to correctly account for the formation of realistic peptide aggregates. We evaluate the thermodynamic and kinetic implications of changes in sequence and pH upon peptide aggregation, and we discuss how a minimalistic coarse- grained model can account for these details. T3 - ZIB-Report - 15-42 KW - coarse graining KW - peptide aggregation dynamics KW - Monte Carlo Y1 - 2015 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-55805 SN - 1438-0064 ER -