Refine
Year of publication
Document Type
- Article (211)
- ZIB-Report (70)
- In Proceedings (23)
- In Collection (9)
- Book chapter (7)
- Book (2)
- Doctoral Thesis (2)
- Other (2)
- Poster (2)
- Habilitation (1)
Is part of the Bibliography
- no (331)
Keywords
- metastability (8)
- cycle decomposition (3)
- rare events (3)
- reaction coordinate (3)
- Bayesian inference (2)
- Clustering (2)
- DS-MLE (2)
- EM algorithm (2)
- Jeffreys prior (2)
- MPLE (2)
Institute
- Numerical Mathematics (151)
- Modeling and Simulation of Complex Processes (57)
- ZIB Allgemein (29)
- Visual and Data-centric Computing (23)
- Computational Systems Biology (18)
- Computational Molecular Design (13)
- Visual Data Analysis (13)
- Mathematics for Life and Materials Science (5)
- Bioinformatics in Medicine (3)
- AI in Society, Science, and Technology (1)
Applications of the cross-entropy method to importance sampling and optimal control of diffusions
(2014)
We study the cross-entropy method for diffusions. One of the results is a versatile cross-entropy algorithm that can be used to design efficient importance sampling strategies for rare events or to solve optimal control problems. The approach is based on the minimization of a suitable cross-entropy functional, with a parametric family of exponentially tilted probability distributions. We illustrate the new algorithm with several numerical examples and discuss algorithmic issues and possible extensions of the method.
Applications of the cross-entropy method to importance sampling and optimal control of diffusions
(2014)
Many interesting rare events in molecular systems like ligand association, protein folding or con- formational changes happen on timescales that often are not accessible by direct numerical simulation. Therefore rare event approximation approaches like interface sampling, Markov state model building or advanced reaction coordinate based free energy estimation have attracted huge attention recently. In this article we analyze the reliability of such approaches: How precise is an estimate of long relaxation timescales of molecular systems resulting from various forms of rare event approximation methods? Our results give a theoretical answer to this question by relating it with the transfer operator approach to molecular dynamics. By doing so they also allow for understanding deep connections between the different approaches.
High-dimensional metastable molecular dynamics (MD) can often be characterised by a few features of the system, that is, collective variables (CVs). Thanks to the rapid advance in the area of machine learning and deep learning, various deep learning-based CV identification techniques have been developed in recent years, allowing accurate modelling and efficient simulation of complex molecular systems. In this paper, we look at two different categories of deep learning-based approaches for finding CVs, either by computing leading eigenfunctions of transfer operator associated to the underlying dynamics, or by learning an autoencoder via minimisation of reconstruction error. We present a concise overview of the mathematics behind these two approaches and conduct a comparative numerical study of these two approaches on illustrative examples.
In this paper, we consider the eigenvalue PDE problem of the infinitesimal generators of metastable diffusion processes. We propose a numerical algorithm based on training artificial neural networks for solving the leading eigenvalues and eigenfunctions of such high-dimensional eigenvalue problem. The algorithm is useful in understanding the dynamical behaviors of metastable processes on large timescales. We demonstrate the capability of our algorithm on a high-dimensional model problem, and on the simple molecular system alanine dipeptide.
We develop a data-driven method to learn chemical reaction networks from trajectory data. Modeling the reaction system as a continuous-time Markov chain and assuming the system is fully observed,our method learns the propensity functions of the system with predetermined basis functions by maximizing the likelihood function of the trajectory data under l^1 sparse regularization. We demonstrate our method with numerical examples using synthetic data and carry out an asymptotic analysis of the proposed learning procedure in the infinite-data limit.
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.
Two different approaches to parameter estimation (PE) in the context of polymerization are introduced, refined, combined, and applied. The first is classical PE where one is interested in finding parameters which minimize the distance between the output of a chemical model and experimental data. The second is Bayesian PE allowing for quantifying parameter uncertainty caused by experimental measurement error and model imperfection. Based on detailed descriptions of motivation, theoretical background, and methodological aspects for both approaches, their relation are outlined. The main aim of this article is to show how the two approaches complement each other and can be used together to generate strong information gain regarding the model and its parameters. Both approaches and their interplay in application to polymerization reaction systems are illustrated. This is the first part in a two-article series on parameter estimation for polymer reaction kinetics with a focus on theory and methodology while in the second part a more complex example will be considered.
We present a numerical method to model dynamical systems from data. We use the recently introduced method Scalable Probabilistic Approximation (SPA) to project points from a Euclidean space to convex polytopes and represent these projected states of a system in new, lower-dimensional coordinates denoting their position in the polytope. We then introduce a specific nonlinear transformation to construct a model of the dynamics in the polytope and to transform back into the original state space. To overcome the potential loss of information from the projection to a lower-dimensional polytope, we use memory in the sense of the delay-embedding theorem of Takens. By construction, our method produces stable models. We illustrate the capacity of the method to reproduce even chaotic dynamics and attractors with multiple connected components on various examples.
We investigate opinion dynamics based on an agent-based model and are interested in predicting the evolution of the percentages of the entire agent population that share an opinion. Since these opinion percentages can be seen as an aggregated observation of the full system state, the individual opinions of each agent, we view this in the framework of the Mori–Zwanzig projection formalism. More specifically, we show how to estimate a nonlinear autoregressive model (NAR) with memory from data given by a time series of opinion percentages, and discuss its prediction capacities for various specific topologies of the agent interaction network. We demonstrate that the inclusion of memory terms significantly improves the prediction quality on examples with different network topologies.
Mathematical modeling of spatio-temporal population dynamics and application to epidemic spreading
(2021)
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.
Markov Decision Processes (MDP) or Partially Observable MDPs (POMDP) are
used for modelling situations in which the evolution of a process is partly random and
partly controllable. These MDP theories allow for computing the optimal control policy
for processes that can continuously or frequently be observed, even if only partially.
However, they cannot be applied if state observation is very costly and therefore rare
(in time). We present a novel MDP theory for rare, costly observations and derive the
corresponding Bellman equation. In the new theory, state information can be derived
for a particular cost after certain, rather long time intervals. The resulting information
costs enter into the total cost and thus into the optimization criterion. This approach
applies to many real world problems, particularly in the medical context, where the
medical condition is examined rather rarely because examination costs are high. At the
same time, the approach allows for efficient numerical realization. We demonstrate the
usefulness of the novel theory by determining, from the national economic perspective,
optimal therapeutic policies for the treatment of the human immunodefficiency virus
(HIV) in resource-rich and resource-poor settings. Based on the developed theory and
models, we discover that available drugs may not be utilized efficiently in resource-poor
settings due to exorbitant diagnostic costs.
We present the theory of “Markov decision processes (MDP) with rare state observation” and apply it to optimal treatment scheduling and diagnostic testing to mitigate HIV-1 drug resistance development in resource-poor countries. The developed theory assumes that the state of the process is hidden and can only be determined by making an examination. Each examination produces costs which enter into the considered cost functional so that the resulting optimization problem includes finding optimal examination times. This is a realistic ansatz: In many real world applications, like HIV-1 treatment scheduling, the information about the disease evolution involves substantial costs, such that examination and control are intimately connected. However, a perfect compliance with the optimal strategy can rarely be achieved. This may be particularly true for HIV-1 resistance testing in resource-constrained countries. In the present work, we therefore analyze the sensitivity of the costs with respect to deviations from the optimal examination times both analytically and for the considered application. We
discover continuity in the cost-functional with respect to the examination times. For the HIV-application, moreover, sensitivity towards small deviations from the optimal examination rule depends on the disease state. Furthermore, we compare the optimal rare-control strategy to (i) constant control strategies (one action for the remaining time) and to (ii) the permanent control of the original, fully observed MDP. This comparison is
done in terms of expected costs and in terms of life-prolongation. The proposed rare-control strategy offers a clear benefit over a constant control, stressing the usefulness of medical testing and informed decision making. This indicates that lower-priced medical tests could improve HIV treatment in resource-constrained settings and warrants further investigation.
Markov Decision Processes (MDP) or Partially Observable MDPs (POMDP) are used for modelling situations in which the evolution of a process is partly random and partly controllable. These MDP theories allow for computing the optimal control policy for processes that can continuously or frequently be observed, even if only partially. However, they cannot be applied if state observation is very costly and therefore rare (in time). We present a novel MDP theory for rare, costly observations and derive the corresponding Bellman equation. In the new theory, state information can be derived for a particular cost after certain, rather long time intervals. The resulting information costs enter into the total cost and thus into the optimization criterion. This approach applies to many real world problems, particularly in the medical context, where the medical condition is examined rather rarely because examination costs are high. At the same time, the approach allows for efficient numerical realization. We demonstrate the usefulness of the novel theory by determining, from the national economic perspective, optimal therapeutic policies for the treatment of the human immunodeficiency virus (HIV) in resource-rich and resource-poor settings. Based on the developed theory and models, we discover that available drugs may not be utilized efficiently in resource-poor settings due to exorbitant diagnostic costs.
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.
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.
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.
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.
Real World networks often exhibit a significant number of vertices which are sparsely and irregularly connected to other vertices in the network. For clustering theses networks with a model based algorithm, we propose the Stochastic Block Model with Irrelevant Vertices (SBMIV) for weighted net- works. We propose an original Variational Bayesian Expectation Maximiza- tion inference algorithm for the SBMIV which is an advanced version of our Blockloading algorithm for the Stochastic Block Model. We introduce a model selection criterion for the number of clusters of the SBMIV which is based on the lower variational bound of the model likelihood. We propose a fully Bayesian inference process, based on plausible informative priors, which is independent of other algorithms for preprocessing start values for the cluster assignment of vertices. Our inference methods allow for a multi level identification of irrelevant vertices which are hard to cluster reliably ac- cording to the SBM. We demonstrate that our methods improve on the normal Stochastic Block model by applying it to to Earthquake Networks which are an example of networks with a large number of sparsely and irregularly con- nected vertices.
We propose the Blockloading algorithm for the clustering of large and complex graphs with tens of thousands of vertices according to a Stochastic Block Model (SBM). Blockloading is based on generalized Variational Bayesian EM (VBEM) schemes and works for weighted and unweighted graphs. Existing Variational (Bayesian) EM methods have to consider each possible number of clusters sepa- rately to determine the optimal number of clusters and are prone to converge to local optima making multiple restarts necessary. These factors impose a severe restriction on the size and complexity of graphs these methods can handle. In con- trast, the Blockloading algorithm restricts restarts to subnetworks in a way that provides error correction of an existing cluster assignment. The number of clusters need not be specified in advance because Blockloading will return it as a result. We show that Blockloading outperforms all other variational methods regarding reliability of the results and computational efficiency.
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.
In this paper, we investigate the conformational dynamics of alanine dipeptide under an external electric field by nonequilibrium molecular dynamics simulation. We consider the case of a constant and of an oscillatory field. In this context, we propose a procedure to implement the temperature control, which removes the irrelevant thermal effects of the field. For the constant field different time-scales are identified in the conformational, dipole moment, and orientational dynamics. Moreover, we prove that the solvent structure only marginally changes when the external field is switched on. In the case of oscillatory field, the conformational changes are shown to be as strong as in the previous case, and nontrivial nonequilibrium circular paths in the conformation space are revealed by calculating the integrated net probability fluxes.
Recent years have seen an increased interest in non-equilibrium molecular dynamics (NEMD) simulations, especially for molecular systems with periodic forcing by external fields, e.g., in the context of studying effects of electromagnetic radiation on the human body tissue. Lately, an NEMD methods with local thermostating has been proposed that allows for studying non-equilibrium processes in a statistically reliable and thermodynamically consistent way. In this article, we demonstrate how to construct Markov State Models (MSMs) for such NEMD simulations. MSM building has been well-established for systems in equilibrium where MSMs with just a few (macro-)states allow for accurate reproduction of the essential kinetics of the molecular system under consideration. Non-equilibrium MSMs have been lacking so far. The article presents how to construct such MSMs and illustrates their validity and usefulness for the case of conformation dynamics of alanine dipeptide in an external electric field.
Grand-canonical-like molecular-dynamics simulations by using an adaptive-resolution technique
(2013)
Linear response theory and optimal control for a molecular system under nonequilibrium conditions
(2013)
In this paper, we propose a straightforward generalization of linear
response theory to systems in nonequilibrium that are subject to
nonequilibrium driving. We briefly revisit the standard linear response
result for equilibrium systems, where we consider Langevin dynamics
as a special case, and then give an alternative derivation using a
change-of-measure argument that does not rely on any stationarity or
reversibility assumption. This procedure moreover easily enables us
to calculate the second order correction to the linear response formula
(which may or may not be useful in practice). Furthermore, we outline
how the novel nonequilibirum linear response formula can be used to
compute optimal controls of molecular systems for cases in which one
wants to steer the system to maximize a certain target expectation
value. We illustrate our approach with simple numerical examples.
Linear response theory and optimal control for a molecular system under nonequilibrium conditions
(2013)
The transition mechanism of jump processes between two different subsets in state space reveals important dynamical information of the processes and therefore has attracted considerable attention in the past years. In this paper, we study the first passage path ensemble of both discrete-time and continuous-time jump processes on a finite state space. The main approach is to divide each first passage path into nonreactive and reactive segments and to study them separately. The analysis can be applied to jump processes which are non-ergodic, as well as continuous-time jump processes where the waiting time distributions are non-exponential. In the particular case that the jump processes are both Markovian and ergodic, our analysis elucidates the relations between the study of the first passage paths and the study of the transition paths in transition path theory. We provide algorithms to numerically compute statistics of the first passage path ensemble. The computational complexity of these algorithms scales with the complexity of solving a linear system, for which efficient methods are available. Several examples demonstrate the wide applicability of the derived results across research areas.
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.
SAIMeR: Self-adapted method for the identification of metastable states in real-world time series
(2014)
In the framework of time series analysis with recurrence networks, we introduce SAIMeR, a heuristic self-adapted method that determines the elusive recurrence threshold and identifies metastable states in complex time series. To identify metastable states as well as the transitions between them, we use graph theory concepts and a fuzzy partitioning clustering algorithm. We illustrate SAIMeR by applying it to three real-world time series and show that it is able to identify metastable states in real-world data with noise and missing data points. Finally, we suggest a way to choose the embedding parameters used to construct the state space in which this method is performed, based on the analysis of how the values of these parameters affect two recurrence quantitative measurements: recurrence rate and entropy.
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.
Due to the increase in accessibility and robustness of sequencing technology, single cell RNA-seq (scRNA-seq) data has become abundant. The technology has made significant contributions to discovering novel phenotypes and heterogeneities of cells. Recently, there has been a push for using single-- or multiple scRNA-seq snapshots to infer the underlying gene regulatory networks (GRNs) steering the cells' biological functions. To date, this aspiration remains unrealised.
In this paper, we took a bottom-up approach and curated a stochastic two gene interaction model capturing the dynamics of a complete system of genes, mRNAs, and proteins. In the model, the regulation was placed upstream from the mRNA on the gene level. We then inferred the underlying regulatory interactions from only the observation of the mRNA population through~time.
We could detect signatures of the regulation by combining information of the mean, covariance, and the skewness of the mRNA counts through time. We also saw that reordering the observations using pseudo-time did not conserve the covariance and skewness of the true time course. The underlying GRN could be captured consistently when we fitted the moments up to degree three; however, this required a computationally expensive non-linear least squares minimisation solver.
There are still major numerical challenges to overcome for inference of GRNs from scRNA-seq data. These challenges entail finding informative summary statistics of the data which capture the critical regulatory information. Furthermore, the statistics have to evolve linearly or piece-wise linearly through time to achieve computational feasibility and scalability.
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.
The Augmented Jump Chain
(2021)
Modern methods of simulating molecular systems are based on the mathematical theory of Markov operators with a focus on autonomous equilibrated systems. However, non-autonomous physical systems or non-autonomous simulation processes are becoming more and more important. A representation of non-autonomous Markov jump processes is presented as autonomous Markov chains on space-time. Augmenting the spatial information of the embedded Markov chain by the temporal information of the associated jump times, the so-called augmented jump chain is derived. The augmented jump chain inherits the sparseness of the infinitesimal generator of the original process and therefore provides a useful tool for studying time-dependent dynamics even in high dimensions. Furthermore, possible generalizations and applications to the computation of committor functions and coherent sets in the non-autonomous setting are discussed. After deriving the theoretical foundations, the concepts with a proof-of-concept Galerkin discretization of the transfer operator of the augmented jump chain applied to simple examples are illustrated.
Collaborative comparisons and combinations of epidemic models are used as policy-relevant evidence during epidemic outbreaks. In the process of collecting multiple model projections, such collaborations may gain or lose relevant information. Typically, modellers contribute a probabilistic summary at each time-step. We compared this to directly collecting simulated trajectories. We aimed to explore information on key epidemic quantities; ensemble uncertainty; and performance against data, investigating potential to continuously gain information from a single cross-sectional collection of model results. Methods We compared July 2022 projections from the European COVID-19 Scenario Modelling Hub. Five modelling teams projected incidence in Belgium, the Netherlands, and Spain. We compared projections by incidence, peaks, and cumulative totals. We created a probabilistic ensemble drawn from all trajectories, and compared to ensembles from a median across each model’s quantiles, or a linear opinion pool. We measured the predictive accuracy of individual trajectories against observations, using this in a weighted ensemble. We repeated this sequentially against increasing weeks of observed data. We evaluated these ensembles to reflect performance with varying observed data. Results. By collecting modelled trajectories, we showed policy-relevant epidemic characteristics. Trajectories contained a right-skewed distribution well represented by an ensemble of trajectories or a linear opinion pool, but not models’ quantile intervals. Ensembles weighted by performance typically retained the range of plausible incidence over time, and in some cases narrowed this by excluding some epidemic shapes. Conclusions. We observed several information gains from collecting modelled trajectories rather than quantile distributions, including potential for continuously updated information from a single model collection. The value of information gains and losses may vary with each collaborative effort’s aims, depending on the needs of projection users. Understanding the differing information potential of methods to collect model projections can support the accuracy, sustainability, and communication of collaborative infectious disease modelling efforts. Data availability All code and data available on Github: https://github.com/covid19-forecast-hub-europe/aggregation-info-loss
Various feature selection algorithms have been proposed to identify cancer prognostic biomarkers. In recent years, however, their reproducibility is criticized. The performance of feature selection algorithms is shown to be affected by the datasets, underlying networks and evaluation metrics. One of the causes is the curse of dimensionality, which makes it hard to select the features that generalize well on independent data. Even the integration of biological networks does not mitigate this issue because the networks are large and many of their components are not relevant for the phenotype of interest. With the availability of multi-omics data, integrative approaches are being developed to build more robust predictive models. In this scenario, the higher data dimensions create greater challenges. We proposed a phenotype relevant network-based feature selection (PRNFS) framework and demonstrated its advantages in lung cancer prognosis prediction. We constructed cancer prognosis relevant networks based on epithelial mesenchymal transition (EMT) and integrated them with different types of omics data for feature selection. With less than 2.5% of the total dimensionality, we obtained EMT prognostic signatures that achieved remarkable prediction performance (average AUC values above 0.8), very significant sample stratifications, and meaningful biological interpretations. In addition to finding EMT signatures from different omics data levels, we combined these single-omics signatures into multi-omics signatures, which improved sample stratifications significantly. Both single- and multi-omics EMT signatures were tested on independent multi-omics lung cancer datasets and significant sample stratifications were obtained.
Novel multi-objective affinity approach allows to identify pH-specific μ-opioid receptor agonists
(2023)
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.
Models for occupation dynamics in discrete quantum systems lead to large or even infinite systems of ordinary differential equations. Some new mathematical techniques, developed for the simulation of chemical processes, make a numerical solution of countable systems of ordinary differential equations possible. Both, a basic physical concept for the construction of such systems and the structure of the numerical tools for solving them are presented. These conceptual aspects are illustrated by a simulation of an occupation process from spectroscopy. In this example the structures of rotation spectra observed in infrared spectroscopy are explained and some possibilities for an extension of the model are shown.
Markov State Modelling as a concept for a coarse grained description of the essential kinetics of a molecular system in equilibrium has gained a lot of atten- tion recently. The last 10 years have seen an ever increasing publication activity on how to construct Markov State Models (MSMs) for very different molecular systems ranging from peptides to proteins, from RNA to DNA, and via molecu- lar sensors to molecular aggregation. Simultaneously the accompanying theory behind MSM building and approximation quality has been developed well be- yond the concepts and ideas used in practical applications. This article reviews the main theoretical results, provides links to crucial new developments, outlines the full power of MSM building today, and discusses the essential limitations still to overcome.
Markov State Modelling as a concept for a coarse grained description of the essential kinetics of a molecular system in equilibrium has gained a lot of atten- tion recently. The last 10 years have seen an ever increasing publication activity on how to construct Markov State Models (MSMs) for very different molecular systems ranging from peptides to proteins, from RNA to DNA, and via molecu- lar sensors to molecular aggregation. Simultaneously the accompanying theory behind MSM building and approximation quality has been developed well be- yond the concepts and ideas used in practical applications. This article reviews the main theoretical results, provides links to crucial new developments, outlines the full power of MSM building today, and discusses the essential limitations still to overcome.
Conformation Dynamics
(2009)
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.
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 ex-
periments, protein function and protein folding , DNA and RNA, and
ligand-receptor interaction in drug design and more complicated multi-
valent 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 proper-
ties 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 en-
ergy landscapes are located differently and an alchemical transformation
of butane to pentane where the dimension of the state space is changed.
In molecular dynamics applications there is a growing interest in mixed quantum-classical models. The article is concerned with the so-called QCMD model. This model describes most atoms of the molecular system by the means of classical mechanics but an important, small portion of the system by the means of a wavefunction. We review the conditions under which the QCMD model is known to approximate the full quantum dynamical evolution of the system. In most quantum-classical simulations the {\em Born-Oppenheimer model} (BO) is used. In this model, the wavefunction is adiabatically coupled to the classical motion which leads to serious approximation deficiencies with respect to non-adiabatic effects in the fully quantum dynamical description of the system. In contrast to the BO model, the QCMD model does include non-adiabatic processes, e.g., transitions between the energy levels of the quantum system. It is demonstrated that, in mildly non-adiabatic scenarios, so-called {\em surface hopping} extensions of QCMD simulations yield good approximations of the non-adiabatic effects in full quantum dynamics. The algorithmic strategy of such extensions of QCMD is explained and the crucial steps of its realization are discussed with special emphasis on the numerical problems caused by highly oscillatory phase effects.
One of the main challenges in molecular dynamics is overcoming the “timescale barrier”, a phrase used to describe that in many realistic molecular systems, biologically important rare transitions occur on timescales that are not accessible to direct numerical simulation, not even on the largest or specifically dedicated supercomputers. This article discusses how to circumvent the timescale barrier by a collection of transfer operator-based techniques that have emerged from dynamical systems theory, numerical mathematics, and machine learning over the last two decades. We will focus on how transfer operators can be used to approximate the dynamical behavior on long timescales, review the introduction of this approach into molecular dynamics, and outline the respective theory as well as the algorithmic development from the early numerics-based methods, via variational reformulations, to modern data-based techniques utilizing and
improving concepts from machine learning. Furthermore, its relation to rare event simulation techniques will be explained, revealing a broad equivalence of variational principles for long-time quantities in MD. The article will mainly take a mathematical perspective and will leave the application to real-world molecular systems to the more than 1000 research articles already written on this subject.
One of the main challenges in molecular dynamics is overcoming the ‘timescale barrier’: in many realistic molecular systems, biologically important rare transitions occur on timescales that are not accessible to direct numerical simulation, even on the largest or specifically dedicated supercomputers. This article discusses how to circumvent the timescale barrier by a collection of transfer operator-based techniques that have emerged from dynamical systems theory, numerical mathematics and machine learning over the last two decades. We will focus on how transfer operators can be used to approximate the dynamical behaviour on long timescales, review the introduction of this approach into molecular dynamics, and outline the respective theory, as well as the algorithmic development, from the early numerics-based methods, via variational reformulations, to modern data-based techniques utilizing and improving concepts from machine learning. Furthermore, its relation to rare event simulation techniques will be explained, revealing a broad equivalence of variational principles for long-time quantities in molecular dynamics. The article will mainly take a mathematical perspective and will leave the application to real-world molecular systems to the more than 1000 research articles already written on this subject.
The article surveys the development of novel mathematical concepts and algorithmic approaches based thereon in view of their possible applicability to biomolecular design. Both a first deterministic approach, based on the Frobenius-Perron operator corresponding to the flow of the Hamiltonian dynamics, and later stochastic approaches, based on a spatial Markov operator or on Langevin dynamics, can be subsumed under the unified mathematical roof of the transfer operator approach to effective dynamics of molecular systems. The key idea of constructing specific transfer operators especially taylored for the purpose of conformational dynamics appears as the red line throughout the paper. Different steps of the algorithm are exemplified by a trinucleotide molecular system as a small representative of possible RNA drug molecules.
The function of many important biomolecules is related to their dynamic properties and their ability to switch between different {\em conformations}, which are understood as {\em almost invariant} or {\em metastable} subsets of the positional state space of the system. Recently, the present authors and their coworkers presented a novel algorithmic scheme for the direct numerical determination of such metastable subsets and the transition probability between them. Although being different in most aspects, this method exploits the same basic idea as {\sc Dellnitz} and {\sc Junge} in their approach to almost invariance in discrete dynamical systems: the almost invariant sets are computed via certain eigenvectors of the Markov operators associated with the dynamical behavior. In the present article we analyze the application of this approach to (high--friction) Langevin models describing the dynamical behavior of molecular systems coupled to a heat bath. We will see that this can be related to theoretical results for (symmetric) semigroups of Markov operators going back to {\sc Davies}. We concentrate on a comparison of our approach in respect to random perturbations of dynamical systems.
An analysis of relaxation oscillations in local Er-doped optically pumped lasers is reported. It is based on a time dependent rate equation model for a quasi-two-level-system with wavelength dependent emission- and absorption cross-sections. For the first time a numerically reliable simulation of the characteristic laser behaviour was possible: the onset and decay of the oscillations, the time-dependent repetition period and the steady state signal output power. The characteristic waveguide parameters, as the erbium-concentration profile, the polarization dependent pump- and signal mode intensity profiles, the scattering losses, the cavity length and the front and rear reflectivities were all taken into account. The basic formulas are general and can also be used for Er-doped fiber lasers. Mathematically the problem can be characterized as a large boundary value problem, which can approximately be replaced by a stiff initial value problem of ordinary differential equations. The used algorithmic replacement procedure is motivated and discussed in detail. Here, pump- and signal evolution versus time are presented for an planar Er-diffused $\rm Ti$:$\rm LiNbO_{3}$ waveguide laser. The numerically obtained results show a nearly quantitative agreement with experimental investigations. Simultanously they supply knowledge about non-measureable (space-dependent population dynamic of the Er-atoms) and till today not measured data (dynamical response of the laser by a sharp peak in the external pump).
Recently, a novel concept for the computation of essential features of the dynamics of Hamiltonian systems (such as molecular dynamics) has been proposed. The realization of this concept had been based on subdivision techniques applied to the Frobenius--Perron operator for the dynamical system. The present paper suggests an alternative but related concept that merges the conceptual advantages of the dynamical systems approach with the appropriate statistical physics framework. This approach allows to define the phrase ``conformation'' in terms of the dynamical behavior of the molecular system and to characterize the dynamical stability of conformations. In a first step, the frequency of conformational changes is characterized in statistical terms leading to the definition of some Markov operator $T$ that describes the corresponding transition probabilities within the canonical ensemble. In a second step, a discretization of $T$ via specific hybrid Monte Carlo techniques is shown to lead to a stochastic matrix $P$. With these theoretical preparations, an identification algorithm for conformations is applicable. It is demonstrated that the discretization of $T$ can be restricted to few essential degrees of freedom so that the combinatorial explosion of discretization boxes is prevented and biomolecular systems can be attacked. Numerical results for the n-pentane molecule and the triribonucleotide adenylyl\emph{(3'-5')}cytidylyl\emph{(3'-5')}cytidin are given and interpreted.
Recently, a novel concept for the computation of essential features of Hamiltonian systems (such as those arising in molecular dynamics) has been proposed. The realization of that concept was based on subdivision techniques applied to the Frobenius--Perron operator for the dynamical system. The present paper suggests an alternative but related concept based on statistical mechanics, which allows to attack realistic molecular systems. In a first step, the frequency of conformational changes is characterized in statistical terms leading to the definition of some Markov operator $T$ that describes the corresponding transition probabilities within the canonical ensemble. In a second step, a discretization of $T$ via hybrid Monte Carlo techniques (based on short term subtrajectories only) is shown to lead to a stochastic matrix $P$. With these theoretical preparations, an identification algorithm for conformations is applicable (to be presented elsewhere). Numerical results for the n-pentane molecule are given and interpreted.
A theoretical investigation of the dynamic properties of integrated optical Er--doped waveguide lasers is presented. It includes the construction of a physical model and of numerical techniques which allow reliable simulations of the dynamical behaviour of the laser signal depending on essential parameters of the laser device and on its external, time--dependent pump radiation. Therefore, a physical theory is developed which describes the propagation of light and its interaction with the active substrate in the laser cavity. This is realized in two steps. First, a {\em fundamental model} based on Maxwell's equations and on rate equations for the transitions in the active medium is constructed. Since this turns out to prohibit reliable simulations, it is, in a second step, reformulated via averaging in time and space which suppresses the fluctuations on the fastest time scales but represents them correctly. For this {\em reduced model} reliable and efficient simulation techniques using adaptive control schemes are designed and implemented. We apply the linear--implicit Euler discretization with extrapolation in time and a multilevel quadrature scheme in space. Finally, the model is justified in comparison with experimental observations in four cases of technological relevance.
In molecular dynamics applications there is a growing interest in including quantum effects for simulations of larger molecules. This paper is concerned with {\em mixed quantum-classical} models which are currently discussed: the so-called QCMD model with variants and the time-dependent Born-Oppenheimer approximation. All these models are known to approximate the full quantum dynamical evolution---under different assumptions, however. We review the meaning of these assumptions and the scope of the approximation. In particular, we characterize those typical problematic situations where a mixed model might largely deviate from the full quantum evolution. One such situation of specific interest, a non-adiabatic excitation at certain energy level crossings, can promisingly be dealt with by a modification of the QCMD model that we suggest.
{\footnotesize In classical Molecular Dynamics a molecular system is modelled by classical Hamiltonian equations of motion. The potential part of the corresponding energy function of the system includes contributions of several types of atomic interaction. Among these, some interactions represent the bond structure of the molecule. Particularly these interactions lead to extremely stiff potentials which force the solution of the equations of motion to oscillate on a very small time scale. There is a strong need for eliminating the smallest time scales because they are a severe restriction for numerical long-term simulations of macromolecules. This leads to the idea of just freezing the high frequency degrees of freedom (bond stretching and bond angles) via increasing the stiffness of the strong part of the potential to infinity. However, the naive way of doing this via holonomic constraints mistakenly ignores the energy contribution of the fast oscillations. The paper presents a mathematically rigorous discussion of the limit situation of infinite stiffness. It is demonstrated that the average of the limit solution indeed obeys a constrained Hamiltonian system but with a {\em corrected soft potential}. An explicit formula for the additive potential correction is given via a careful inspection of the limit energy of the fast oscillations. Unfortunately, the theory is valid only as long as the system does not run into certain resonances of the fast motions. Behind those resonances, there is no unique limit solution but a kind of choatic scenario for which the notion ``Takens chaos'' was coined. For demonstrating the relevance of this observation for MD, the theory is applied to a realistic, but still simple system: a single butan molecule. The appearance of ``Takens chaos'' in smoothed MD is illustrated and the consequences are discussed.}
In Quantum Chemistry the field of Laser--Assisted Molecular Control'' has received a considerable amount of attention recently. One key problem in this new field is the simulation of the dynamical reaction of a molecule subjected to external radiation. This problem is described by the Schrödinger equation, which, after eigenfunction expansion, can be written in the form of a large system of ordinary differential equations, the solutions of which show a highly oscillatory behaviour. The oscillations with high frequencies and small amplitudes confine the stepsizes of any numerical integrator -- an effect, which, in turn, blows up the simulation time. Larger stepsizes can be expected by averaging these fast oscillations, thus smoothing the trajectories. Standard smoothing techniques (averaging, filtering) would kill the whole process and thus, lead to wrong numerical results. To avoid this unwanted effect and nevertheless speed up computations, this paper presents a quasiresonant smoothing algorithm (QRS). In QRS, a natural splitting parameter $\delta$ controls the smoothing properties. An adaptive QRS--version (AQRS) is presented which includes an error estimation scheme for choosing this parameter $\delta$ in order to meet a given accuracy requirement. In AQRS $\delta$ is permanently adapted to the solution properties for computing the chemically necessary information'' only. The performance of AQRS is demonstrated in several test problems from the field Laser--Assisted Selective Excitation of Molecules'' in which the external radiation is a picosecond laser pulse. In comparison with standard methods speedup factors of the order of $10^2$ are observed.
In molecular dynamics applications there is a growing interest in mixed quantum-classical models. The {\em quantum-classical Liouville equation} (QCL) describes most atoms of the molecular system under consideration by means of classical phase space density but an important, small portion of the system by means of quantum mechanics. The QCL is derived from the full quantum dynamical (QD) description by applying the Wigner transform to the classical part'' of the system only. We discuss the conditions under which the QCL model approximates the full QD evolution of the system. First, analysis of the asymptotic properties of the Wigner transform shows that solving the QCL yields a first order approximation of full quantum dynamics. Second, we discuss the adiabatic limit of the QCL. This discussion shows that the QCL solutions may be interpretated as classical phase space densities, at least near the adiabatic limit. Third, it is demonstrated that the QCL yields good approximations of {\em non-adiabatic quantum effects,} especially near so-called {\em avoided crossings} where most quantum-classical models fail.
A quasiresonant smoothing algorithm for the fast analysis of selective vibrational excitation.
(1993)
One key problem in modern chemistry is the simulation of the dynamical reaction of a molecule subjected to external radiation. This is described by the Schrödinger equation, which, after eigenfunction expansion, can be written in form of a system of ordinary differential equations, whose solutions show a highly oscillatory behaviour. The oscillations with high frequencies and small amplitudes confine the stepsizes of any numerical integrator -- an effect, which, in turn, blows up the simulation time. Larger stepsizes can be expected by averaging these fast oscillations, thus smoothing the trajectories. This idea leads to the construction of a quasiresonant smoothing algorithm (QRS). In QRS, a natural and computationally available splitting parameter $\delta$ controls the smoothing properties. The performance of QRS is demonstrated in two applications treating the selective excitation of vibrational states by picosecond laser pulses. In comparison with standard methods a speedup factor of 60--100 is observed. A closer look to purely physically motivated quasiresonant approximations such as WFQRA shows some additional advantages of the above smoothing idea. Among these the possibility of an adaptive formulation of QRS via the parameter $\delta$ is of particular importance.
This paper makes use of statistical mechanics in order to construct effective potentials for Molecular Dynamics for systems with nonstationary thermal embedding. The usual approach requires the computation of a statistical ensemble of trajectories. In the context of the new model the evaluation of only one single trajectory is sufficient for the determination of all interesting quantities, which leads to an enormous reduction of computational effort. This single trajectory is the solution to a corrected Hamiltonian system with a new potential $\tilde{V}$. It turns out that $\tilde{V}$ can be defined as spatial average of the original potential $V$. Therefore, the Hamiltonian dynamics defined by $\tilde{V}$ is smoother than that effected by $V$, i.e. a numerical integration of its evolution in time allows larger stepsizes. Thus, the presented approach introduces a Molecular Dynamics with smoothed trajectories originating from spatial averaging. This is deeply connected to time--averaging in Molecular Dynamics. These two types of {\em smoothed Molecular Dynamics} share advantages (gain in efficiency, reduction of error amplification, increased stability) and problems (necessity of closing relations and adaptive control schemes) which will be explained in detail.
The function of many important biomolecules comes from their dynamic properties and their ability to switch between different {\em conformations}. In a conformation, the large scale geometric structure of the molecule is understood to be conserved, whereas on smaller scales the system may well rotate, oscillate or fluctuate. In a recent article [J. Comp. Phys., 151,1 (1999)], the present author and coworkers demonstrated that (a) conformations can be understood as almost invariant sets of some Markov chain being defined via the Hamiltonian system governing the molecular dynamics and that (b) these sets can efficiently be computed via eigenvectors of the corresponding Markov operator. The persent manuscript reviews the mathematical modelling steps behind the novel concept, includes a rigorous analytical justification of this approach and especially of the numerical details of the algorithm, and illustrates its performance when applied to realistic molecular systems.