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We study a class of semi-linear differential Volterra equations with polynomial-type potentials that incorporates the effects of memory while being subjected to random perturbations via an additive Gaussian noise. We show that for a broad class of non-linear potentials, the system always admits invariant probability measures. However, the presence of memory effects precludes access to compactness in a typical fashion. In this paper, this obstacle is overcome by introducing functional spaces adapted to the memory kernels, thereby allowing one to recover compactness. Under the assumption of sufficiently smooth noise, it is then shown that the statistically stationary states possess higher-order regularity properties dictated by the structure of the nonlinearity. This is established through a control argument that asymptotically transfers regularity onto the solution by exploiting the underlying Lyapunov structure of the system in a novel way. 

Abstract The preponderance of intrinsically disordered proteins (IDPs) in the eukaryotic proteome, and their ability to interact with each other, and with folded proteins, RNA, and DNA for functional purposes, have made it important to quantitatively characterize their biophysical properties. Toward this end, we developed the transferable self‐organized polymer (SOP‐IDP) model to calculate the properties of several IDPs. The values of the radius of gyration () obtained from SOP‐IDP simulations are in excellent agreement (correlation coefficient of 0.96) with those estimated from SAXS experiments. For AP180 and Epsin, the predicted values of the hydrodynamic radii () are in nearly quantitative agreement with those from fluorescence correlation spectroscopy (FCS) experiments. Strikingly, the calculated SAXS profiles for 36 IDPs are also nearly superimposable on the experimental profiles. The dependence of and the mean end‐to‐end distance () on chain length, , follows Flory's scaling law, ( and ), suggesting that globally IDPs behave as synthetic polymers in a good solvent. This finding depends on the solvent quality, which can be altered by changing variables such as pH and salt concentration. The values of and are 0.20 and 0.48 nm, respectively. Surprisingly, finite size corrections to scaling, expected on theoretical grounds, are negligible for and . In contrast, only by accounting for the finite sizes of the IDPs, the dependence of experimentally measurable on can be quantitatively explained using . Although Flory scaling law captures the estimates for , , and accurately, the spread of the simulated data around the theoretical curve is suggestive of of sequence‐specific features that emerge through a fine‐grained analysis of the conformational ensembles using hierarchical clustering. Typically, the ensemble of conformations partitions into three distinct clusters, having different equilibrium populations and structural properties. Without any further readjustments to the parameters of the SOP‐IDP model, we also obtained nearly quantitative agreement with paramagnetic relaxation enhancement (PRE) measurements forα‐synuclein. The transferable SOP‐IDP model sets the stage for several applications, including the study of phase separation in IDPs and interactions with nucleic acids. 

Delta blue intensity is a commonly used method to correct for the heartwood-sapwood color change in blue intensity (BI) measurements. It is based on the assumption that the heartwood-sapwood color change is similar in both earlywood and latewood. This assumption has not been supported physiologically. Furthermore, delta BI may confound the climate signals in earlywood and latewood BI as it is technically a linear combination of the other two. Here, instead of using delta BI, we used change point detection to identify the heartwood-sapwood transition, and corrected for the color change by rescaling the mean and variance of BI measurements after the transition to those immediately before. We tested three different change point detection methods and found that they agreed well with one another. Importantly, our approach preserves the climate signals in both earlywood and latewood BI data, while delta BI causes a total loss of climate signals in our test case. Therefore, we suggest that change point detection should be used instead of delta BI to account for the heartwood-sapwood color change. 

Abstract We study the long time statistics of a walker in a hydrodynamic pilot-wave system, which is a stochastic Langevin dynamics with an external potential and memory kernel. While prior experiments and numerical simulations have indicated that the system may reach a statistically steady state, its long-time behavior has not been studied rigorously. For a broad class of external potentials and pilot-wave forces, we construct the solutions as a dynamics evolving on suitable path spaces. Then, under the assumption that the pilot-wave force is dominated by the potential, we demonstrate that the walker possesses a unique statistical steady state. We conclude by presenting an example of such an invariant measure, as obtained from a numerical simulation of a walker in a harmonic potential. 

Abstract Most amino acids are encoded by multiple codons, making the genetic code degenerate. Synonymous mutations affect protein translation and folding, but their impact on RNA itself is often neglected. We developed a genetic algorithm that introduces synonymous mutations to control the diversity of structures sampled by an mRNA. The behavior of the designed mRNAs reveals a physical code layered in the genetic code. We find that mRNA conformational heterogeneity directs physical properties and functional outputs of RNA-protein complexes and biomolecular condensates. The role of structure and disorder of proteins in biomolecular condensates is well appreciated, but we find that RNA conformational heterogeneity is equally important. This feature of RNA enables both evolution and engineers to build cellular structures with specific material and responsive properties. 

Abstract This manuscript describes the development of a resource module that is part of a learning platform named ‘NIGMS Sandbox for Cloud-based Learning’ (https://github.com/NIGMS/NIGMS-Sandbox). The module delivers learning materials on Cloud-based Consensus Pathway Analysis in an interactive format that uses appropriate cloud resources for data access and analyses. Pathway analysis is important because it allows us to gain insights into biological mechanisms underlying conditions. But the availability of many pathway analysis methods, the requirement of coding skills, and the focus of current tools on only a few species all make it very difficult for biomedical researchers to self-learn and perform pathway analysis efficiently. Furthermore, there is a lack of tools that allow researchers to compare analysis results obtained from different experiments and different analysis methods to find consensus results. To address these challenges, we have designed a cloud-based, self-learning module that provides consensus results among established, state-of-the-art pathway analysis techniques to provide students and researchers with necessary training and example materials. The training module consists of five Jupyter Notebooks that provide complete tutorials for the following tasks: (i) process expression data, (ii) perform differential analysis, visualize and compare the results obtained from four differential analysis methods (limma, t-test, edgeR, DESeq2), (iii) process three pathway databases (GO, KEGG and Reactome), (iv) perform pathway analysis using eight methods (ORA, CAMERA, KS test, Wilcoxon test, FGSEA, GSA, SAFE and PADOG) and (v) combine results of multiple analyses. We also provide examples, source code, explanations and instructional videos for trainees to complete each Jupyter Notebook. The module supports the analysis for many model (e.g. human, mouse, fruit fly, zebra fish) and non-model species. The module is publicly available at https://github.com/NIGMS/Consensus-Pathway-Analysis-in-the-Cloud. This manuscript describes the development of a resource module that is part of a learning platform named ``NIGMS Sandbox for Cloud-based Learning'' https://github.com/NIGMS/NIGMS-Sandbox. The overall genesis of the Sandbox is described in the editorial NIGMS Sandbox [1] at the beginning of this Supplement. This module delivers learning materials on the analysis of bulk and single-cell ATAC-seq data in an interactive format that uses appropriate cloud resources for data access and analyses. 

Repeat RNA sequences self-associate to form condensates. Simulations of a coarse-grained single-interaction site model for (CAG)n (n = 30 and 31) show that the salt-dependent free energy gap, ΔGS, between the ground (perfect hairpin) and the excited state (slipped hairpin (SH) with one CAG overhang) of the monomer for (n even) is the primary factor that determines the rates and yield of self-assembly. For odd n, the free energy (GS) of the ground state, which is an SH, is used to predict the self-association kinetics. As the monovalent salt concentration, CS, increases, ΔGS and GS increase, which decreases the rates of dimer formation. In contrast, ΔGS for shuffled sequences, with the same length and sequence composition as (CAG)31, is larger, which suppresses their propensities to aggregate. Although demonstrated explicitly for (CAG) polymers, the finding of inverse correlation between the free energy gap and RNA aggregation is general.