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Barrier crossing in the presence of multi-exponential memory functions with unequal friction amplitudes and memory times
Get PDFWe study the non-Markovian Langevin dynamics of a massive particle in a onedimensional
double-well potential in the presence of multi-exponential memory by simulations.
We consider memory functions as the sum of two or three exponentials with different friction
amplitudes γi and different memory times τi and confirm the validity of a previously suggested
heuristic formula for the mean first-passage time τMFP. Based on the heuristic formula, we derive a
general scaling diagram that features a Markovian regime for short memory times, an asymptotic
long-memory-time regime where barrier crossing is slowed down and τMFP grows quadratically
with the memory time, and a non-Markovian intermediate regime where barrier crossing is slightly
accelerated or slightly slowed down, depending primarily on the particle mass. The relative weight
of different exponential memory contributions is described by the scaling variable γi/τ 2
i , i.e.,
memory contributions with long memory times or small amplitudes are negligible compared to
other memory contributions
Predictive modeling of long non-coding RNA chromatin (dis-)association
No full textLong non-coding RNAs (lncRNAs) are involved in gene expression regulation in cis and trans. Although enriched in the chromatin cell fraction, to what degree this defines their broad range of functions remains unclear. In addition, the factors that contribute to lncRNA chromatin tethering, as well as the molecular basis of efficient lncRNA chromatin dissociation and its functional impact on enhancer activity and target gene expression, remain to be resolved. Here, we combine pulse-chase metabolic labeling of nascent RNA with chromatin fractionation and transient transcriptome sequencing to follow nascent RNA transcripts from their co-transcriptional state to their release into the nucleoplasm. By incorporating functional and physical characteristics in machine learning models, we find that parameters like co-transcriptional splicing contributes to efficient lncRNA chromatin dissociation. Intriguingly, lncRNAs transcribed from enhancer-like regions display reduced chromatin retention, suggesting that, in addition to splicing, lncRNA chromatin dissociation may contribute to enhancer activity and target gene expression
Dictionary Learning for transcriptomics data reveals type-specific gene modules in a multi-class setting
No full textExtracting information from large biological datasets is a challenging task, due to the large data size, high-dimensionality, noise, and errors in the data. Gene expression data contains information about which gene products have been formed by a cell, thus representing which genes have been read to activate a particular biological process. Understanding which of these gene products can be related to which processes can for example give insights about how diseases evolve and might give hints about how to fight them.
The Next Generation RNA-sequencing method emerged over a decade ago and is nowadays state-of-the-art in the field of gene expression analyses. However, analyzing these large, complex datasets is still a challenging task. Many of the existing methods do not take into account the underlying structure of the data.
In this paper, we present a new approach for RNA-sequencing data analysis based on dictionary learning. Dictionary learning is a sparsity enforcing method that has widely been used in many fields, such as image processing, pattern classification, signal denoising and more. We show how for RNA-sequencing data, the atoms in the dictionary matrix can be interpreted as modules of genes that either capture patterns specific to different types, or else represent modules that are reused across different scenarios. We evaluate our approach on four large datasets with samples from multiple types. A Gene Ontology term analysis, which is a standard tool indicated to help understanding the functions of genes, shows that the found gene-sets are in agreement with the biological context of the sample types. Further, we find that the sparse representations of samples using the dictionary can be used to identify type-specific differences
Stochastic Dynamics in Computational Biology
No full textThe aim of this book is to provide a well-structured and coherent overview of existing mathematical modeling approaches for biochemical reaction systems, investigating relations between both the conventional models and several types of deterministic-stochastic hybrid model recombinations. Another main objective is to illustrate and compare diverse numerical simulation schemes and their computational effort. Unlike related works, this book presents a broad scope in its applications, from offering a detailed introduction to hybrid approaches for the case of multiple population scales to discussing the setting of time-scale separation resulting from widely varying firing rates of reaction channels. Additionally, it also addresses modeling approaches for non well-mixed reaction-diffusion dynamics, including deterministic and stochastic PDEs and spatiotemporal master equations. Finally, by translating and incorporating complex theory to a level accessible to non-mathematicians, this book effectively bridges the gap between mathematical research in computational biology and its practical use in biological, biochemical, and biomedical systems
Singular Value Decomposition of Operators on Reproducing Kernel Hilbert Spaces
No full textAbstract
Reproducing kernel Hilbert spaces (RKHSs) play an important role in many statistics and machine learning applications ranging from support vector machines to Gaussian processes and kernel embeddings of distributions. Operators acting on such spaces are, for instance, required to embed conditional probability distributions in order to implement the kernel Bayes rule and build sequential data models. It was recently shown that transfer operators such as the Perron–Frobenius or Koopman operator can also be approximated in a similar fashion using covariance and cross-covariance operators and that eigenfunctions of these operators can be obtained by solving associated matrix eigenvalue problems. The goal of this paper is to provide a solid functional analytic foundation for the eigenvalue decomposition of RKHS operators and to extend the approach to the singular value decomposition. The results are illustrated with simple guiding examples
Mechanisms of Airborne Infection via Evaporating and Sedimenting Droplets Produced by Speaking
No full textFor estimating the infection risk from virus-containing airborne droplets, it is crucial to consider the interplay of all relevant physical-chemical effects that affect droplet evaporation and sedimentation times. For droplet radii in the range 70 nm < R < 60 μm, evaporation can be described in the stagnant-flow approximation and is diffusion-limited. Analytical equations are presented for the droplet evaporation rate, the time-dependent droplet size, and the sedimentation time, including evaporation cooling and solute osmotic-pressure effects. Evaporation makes the time for initially large droplets to sediment much longer and thus significantly increases the viral air load. Using recent estimates for SARS-CoV-2 concentrations in sputum and droplet production rates while speaking, a single infected person that constantly speaks without a mouth cover produces a total steady-state air load of more than 104 virions at a given time. In a midsize closed room, this leads to a viral inhalation frequency of at least 2.5 per minute. Low relative humidity, as encountered in airliners and inside buildings in the winter, accelerates evaporation and thus keeps initially larger droplets suspended in air. Typical air-exchange rates decrease the viral air load from droplets with an initial radius larger than 20 μm only moderately
On the convergence of the Laplace approximation and noise-level-robustness of Laplace-based Monte Carlo methods for Bayesian inverse problems
No full textThe Bayesian approach to inverse problems provides a rigorous framework for the incorporation and quantification of uncertainties in measurements, parameters and models. We are interested in designing numerical methods which are robust w.r.t. the size of the observational noise, i.e., methods which behave well in case of concentrated posterior measures. The concentration of the posterior is a highly desirable situation in practice, since it relates to informative or large data. However, it can pose a computational challenge for numerical methods based on the prior measure. We propose to employ the Laplace approximation of the posterior as the base measure for numerical integration in this context. The Laplace approximation is a Gaussian measure centered at the maximum a-posteriori estimate and with covariance matrix depending on the logposterior density. We discuss convergence results of the Laplace approximation in terms of the Hellinger distance and analyze the efficiency of Monte Carlo methods based on it. In particular, we show that Laplace-based importance sampling and Laplace-based quasi-Monte-Carlo methods are robust w.r.t. the concentration of the posterior for large classes of posterior distributions and integrands whereas prior-based importance sampling and plain quasi-Monte Carlo are not. Numerical experiments are presented to illustrate the theoretical findings
A weak characterization of slow variables in stochastic dynamical systems
No full textWe present a novel characterization of slow variables for continuous Markov processes that provably preserve the slow timescales. These slow variables are known as reaction coordinates in molecular dynamical applications, where they play a key role in system analysis and coarse graining. The defining characteristics of these slow variables is that they parametrize a so-called transition manifold, a low-dimensional manifold in a certain density function space that emerges with progressive equilibration of the system's fast variables. The existence of said manifold was previously predicted for certain classes of metastable and slow-fast systems. However, in the original work, the existence of the manifold hinges on the pointwise convergence of the system's transition density functions towards it. We show in this work that a convergence in average with respect to the system's stationary measure is sufficient to yield reaction coordinates with the same key qualities. This allows one to accurately predict the timescale preservation in systems where the old theory is not applicable or would give overly pessimistic results. Moreover, the new characterization is still constructive, in that it allows for the algorithmic identification of a good slow variable. The improved characterization, the error prediction and the variable construction are demonstrated by a small metastable system
Evaluation of Shock Dividers using Numerical and Experimental Methods
No full textMitigation of pressure pulsations in the exhaust of a pulse detonation combustor is crucial for operation with a downstream turbine. For this purpose, a device termed the shock divider is designed and investigated. The intention of the divider is to split the leading shock wave into two weaker waves that propagate along separated ducts with different cross-sections, allowing the shock waves to travel with different velocities along different paths. The separated shock waves redistribute the energy of the incident shock wave. The shock dynamics inside the divider are investigated using numerical simulations. A second order dimensional split finite volume MUSCL-scheme is used to solve the compressible Euler equations. Furthermore, low-cost simulations are performed using Geometrical Shock Dynamics to predict the shock wave propagation inside the divider. The numerical simulations are compared to high-speed schlieren images and time-resolved total pressure recording. For the latter, a high frequency pressure probe is placed at the divider outlet, which is shown to resolve the transient total pressure during the shock passage. Moreover, the separation of the shock waves is investigated and found to grow as the divider duct width ratio increases. The numerical and experimental results allow for a better understanding of the dynamic evolution of the flow inside the divider and inform its capability to reduce the pressure pulsations at the exhaust of the pulse detonation combustor