Freie Universität Berlin
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Slicing and Dicing: Optimal Coarse-Grained Representation to Preserve Molecular Kinetics
No full textThe aim of molecular coarse-graining approaches is to recover relevant physical properties of the molecular system via a lower-resolution model that can be more efficiently simulated. Ideally, the lower resolution still accounts for the degrees of freedom necessary to recover the correct physical behavior. The selection of these degrees of freedom has often relied on the scientist’s chemical and physical intuition. In this article, we make the argument that in soft matter contexts desirable coarse-grained models accurately reproduce the long-time dynamics of a system by correctly capturing the rare-event transitions. We propose a bottom-up coarse-graining scheme that correctly preserves the relevant slow degrees of freedom, and we test this idea for three systems of increasing complexity. We show that in contrast to this method existing coarse-graining schemes such as those from information theory or structure-based approaches are not able to recapitulate the slow time scales of the system
A route to the hydrodynamic limit of a reaction-diffusion master equation using gradient structures
No full textThe reaction-diffusion master equation (RDME) is a lattice-based stochastic model for spatially resolved cellular processes. It is often interpreted as an approximation to spatially continuous reaction-diffusion models, which, in the limit of an infinitely large population, may be described by means of reaction-diffusion partial differential equations (RDPDEs). Analyzing and understanding the relation between different mathematical models for reaction-diffusion dynamics is a research topic of steady interest. In this work, we explore a route to the hydrodynamic limit of the RDME which uses gradient structures. Specifically, we elaborate on a method introduced in [J. Maas, A. Mielke: Modeling of chemical reactions systems with detailed balance using gradient structures. J. Stat. Phys. (181), 2257-2303 (2020)] in the context of well-mixed reaction networks by showing that, once it is complemented with an appropriate limit procedure, it can be applied to spatially extended systems with diffusion. Under the assumption of detailed balance, we write down a gradient structure for the RDME and use the method to produce a gradient structure for its hydrodynamic limit, namely, for the corresponding RDPDE
Understanding microbiome dynamics via interpretable graph representation learning
No full textLarge-scale perturbations in the microbiome constitution are strongly correlated, whether as a driver or a consequence, with the health and functioning of human physiology. However, understanding the difference in the microbiome profiles of healthy and ill individuals can be complicated due to the large number of complex interactions among microbes. We propose to model these interactions as a time-evolving graph whose nodes are microbes and edges are interactions among them. Motivated by the need to analyse such complex interactions, we develop a method that learns a low-dimensional representation of the time-evolving graph and maintains the dynamics occurring in the high-dimensional space. Through our experiments, we show that we can extract graph features such as clusters of nodes or edges that have the highest impact on the model to learn the low-dimensional representation. This information can be crucial to identify microbes and interactions among them that are strongly correlated with clinical diseases. We conduct our experiments on both synthetic and real-world microbiome datasets
Continuous-time extensions of discrete-time cocycles
No full textWe consider linear cocycles taking values in driven by homeomorphic transformations of a smooth manifold, in discrete and continuous time. We show that any discrete-time cocycle can be extended to a continuous-time cocycle, while preserving its characteristic properties. We provide a necessary and sufficient condition under which this extension is natural in the sense that the base is extended to an associated suspension flow and that the dimension of the cocycle does not change. Further, we refine our general result for the case of (quasi-)periodic driving. As an example, we present a discrete-time cocycle due to Michael Herman. The Furstenberg--Kesten limits of this cocycle do not exist everywhere and its Oseledets splitting is discontinuous. Our results on the continuous-time extension of discrete-time cocycles allow us to construct a continuous-time cocycle with analogous properties
Contamination detection and microbiome exploration with GRIMER
No full textBackground:
Contamination detection is a important step that should be carefully considered in early stages when designing and performing microbiome studies to avoid biased outcomes. Detecting and removing true contaminants is challenging, especially in low-biomass samples or in studies lacking proper controls. Interactive visualizations and analysis platforms are crucial to better guide this step, to help to identify and detect noisy patterns that could potentially be contamination. Additionally, external evidence, like aggregation of several contamination detection methods and the use of common contaminants reported in the literature, could help to discover and mitigate contamination.
Results:
We propose GRIMER, a tool that performs automated analyses and generates a portable and interactive dashboard integrating annotation, taxonomy, and metadata. It unifies several sources of evidence to help detect contamination. GRIMER is independent of quantification methods and directly analyzes contingency tables to create an interactive and offline report. Reports can be created in seconds and are accessible for nonspecialists, providing an intuitive set of charts to explore data distribution among observations and samples and its connections with external sources. Further, we compiled and used an extensive list of possible external contaminant taxa and common contaminants with 210 genera and 627 species reported in 22 published articles.
Conclusion:
GRIMER enables visual data exploration and analysis, supporting contamination detection in microbiome studies. The tool and data presented are open source and available at https://gitlab.com/dacs-hpi/grimer
Maximum a posteriori estimators in ℓp are well-defined for diagonal Gaussian priors
No full textWe prove that maximum a posteriori estimators are well-defined for diagonal Gaussian priors μ on ℓp under common assumptions on the potential Φ. Further, we show connections to the Onsager--Machlup functional and provide a corrected and strongly simplified proof in the Hilbert space case p=2, previously established by Dashti et al (2013) and Kretschmann (2019).
These corrections do not generalize to the setting 1≤p<∞, which requires a novel convexification result for the difference between the Cameron--Martin norm and the p-norm
Model reduction for Ca²+ -induced vesicle fusion dynamics
No full textIn this work, we adapt an established model for the Ca²+ -induced fusion dynamics of synaptic vesicles and employ a lumping method to reduce its complexity. In the reduced system, sequential Ca²+ -binding steps are merged to a single releasable state, while keeping the important dependence of the reaction rates on the local Ca²+ concentration. We examine the feasibility of this model reduction for a representative stimulus train over the physiologically relevant site-channel distances. Our findings show that the approximation error is generally small and exhibits an interesting nonlinear and non-monotonic behavior where it vanishes for very low distances and is insignificant at intermediary distances. Furthermore, we give expressions for the reduced model's reaction rates and suggest that our approach may be used to directly compute effective fusion rates for assessing the validity of a fusion model, thereby circumventing expensive simulations
Optimal Reaction Coordinates: Variational Characterization and Sparse Computation
No full textReaction coordinates (RCs) are indicators of hidden, low-dimensional mechanisms that govern the long-term behavior of high-dimensional stochastic processes. We present a novel and general variational characterization of optimal RCs and provide conditions for their existence. Optimal RCs are minimizers of a certain loss function, and reduced models based on them guarantee a good approximation of the statistical long-term properties of the original high-dimensional process. We show that for slow-fast systems, metastable systems, and other systems with known good RCs, the novel theory reproduces previous insight. Remarkably, for reversible systems, the numerical effort required to evaluate the loss function scales only with the variability of the underlying, low-dimensional mechanism, and not with that of the full system. The theory provided lays the foundation for an efficient and data-sparse computation of RCs via modern machine learning techniques
Optimal Rates for Regularized Conditional Mean Embedding Learning
No full textWe address the consistency of a kernel ridge regression estimate of the conditional mean embedding (CME), which is an embedding of the conditional distribution of Y given X into a target reproducing kernel Hilbert space HY. The CME allows us to take conditional expectations of target RKHS functions, and has been employed in nonparametric causal and Bayesian inference. We address the misspecified setting, where the target CME is in the space of Hilbert-Schmidt operators acting from an input interpolation space between HX and L2, to HY. This space of operators is shown to be isomorphic to a newly defined vector-valued interpolation space. Using this isomorphism, we derive a novel and adaptive statistical learning rate for the empirical CME estimator under the misspecified setting. Our analysis reveals that our rates match the optimal O(logn/n) rates without assuming HY to be finite dimensional. We further establish a lower bound on the learning rate, which shows that the obtained upper bound is optimal
Revisiting the storage capacity limit of graphite battery anodes: spontaneous lithium overintercalation at ambient pressure
No full textThe market quest for fast-charging, safe, long-lasting and performant batteries drives the exploration of new energy storage materials, but also promotes fundamental investigations of materials already widely used. Presently, revamped interest in anode materials is observed -- primarily graphite electrodes for lithium-ion batteries. Here, we focus on the upper limit of lithium intercalation in the morphologically quasi-ideal highly oriented pyrolytic graphite (HOPG), with a LiC6 stoichiometry corresponding to 100\% state of charge (SOC). We prepared a sample by immersion in liquid lithium at ambient pressure and investigated it by static 7Li nuclear magnetic resonance (NMR). We resolved unexpected signatures of superdense intercalation compounds, LiC6−x. These have been ruled out for decades, since the highest geometrically accessible composition, LiC2, can only be prepared under high pressure. We thus challenge the widespread notion that any additional intercalation beyond LiC6 is not possible under ambient conditions. We monitored the sample upon calendaric aging and employed ab initio calculations to rationalise the NMR results. The computed relative stabilities of different superdense configurations reveal that non-negligible overintercalation does proceed spontaneously beyond the currently accepted capacity limit