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In prokaryotes, RNA thermometers regulate a number of heat shock and virulence genes. These temperature sensitive RNA elements are usually located in the 5'-untranslated regions of the regulated genes. They repress translation initiation by base pairing to the Shine–Dalgarno sequence at low temperatures. We investigated the thermodynamic stability of the temperature labile hairpin 2 of the Salmonella fourU RNA thermometer over a broad temperature range and determined free energy, enthalpy and entropy values for the base-pair opening of individual nucleobases by measuring the temperature dependence of the imino proton exchange rates via NMR spectroscopy. Exchange rates were analyzed for the wild-type (wt) RNA and the A8C mutant. The wt RNA was found to be stabilized by the extraordinarily stable G14–C25 base pair. The mismatch base pair in the wt RNA thermometer (A8–G31) is responsible for the smaller cooperativity of the unfolding transition in the wt RNA. Enthalpy and entropy values for the base-pair opening events exhibit linear correlation for both RNAs. The slopes of these correlations coincide with the melting points of the RNAs determined by CD spectroscopy. RNA unfolding occurs at a temperature where all nucleobases have equal thermodynamic stabilities. Our results are in agreement with a consecutive zipper-type unfolding mechanism in which the stacking interaction is responsible for the observed cooperativity. Furthermore, remote effects of the A8C mutation affecting the stability of nucleobase G14 could be identified. According to our analysis we deduce that this effect is most probably transduced via the hydration shell of the RNA.
We analyse statistical and information-theoretical properties of EEG microstate sequences, as seen through the lens of five different clustering algorithms. Microstate sequences are computed for n = 20 resting state EEG recordings during wakeful rest. The input for all clustering algorithms is the set of EEG topographic maps obtained at local maxima of the spatial variance. This data set is processed by two classical microstate clustering algorithms (1) atomize and agglomerate hierarchical clustering (AAHC) and (2) a modified K-means algorithm, as well as by (3) K-medoids, (4) principal component analysis (PCA) and (5) fast independent component analysis (Fast-ICA). Using this technique, EEG topographies can be substituted with microstate labels by competitive fitting based on spatial correlation, resulting in a symbolic, non-metric time series, the microstate sequence. Microstate topographies and symbolic time series are further analyzed statistically, including static and dynamic properties. Static properties, which do not contain information about temporal dependencies of the microstate sequence include the maximum similarity of microstate maps within and between the tested clustering algorithms, the global explained variance and the Shannon entropy of the microstate sequences. Dynamic properties are sensitive to temporal correlations between the symbols and include the mixing time of the microstate transition matrix, the entropy rate of the microstate sequences and the location of the first local maximum of the autoinformation function. We also test the Markov property of microstate sequences, the time stationarity of the transition matrix and detect periodicities by means of time-lagged mutual information. Finally, possible long-range correlations of microstate sequences are assessed via Hurst exponent estimation. We find that while static properties partially reflect properties of the clustering algorithms, information-theoretical quantities are largely invariant with respect to the clustering method used. As each clustering algorithm has its own profile of computational speed, ease of implementation, determinism vs. stochasticity and theoretical underpinnings, our results convey a positive message concerning the free choice of method and the comparability of results obtained from different algorithms. The invariance of these quantities implies that the tested properties are algorithm-independent, inherent features of resting state EEG derived microstate sequences.
We present an open-source Python package to compute information-theoretical quantities for electroencephalographic data. Electroencephalography (EEG) measures the electrical potential generated by the cerebral cortex and the set of spatial patterns projected by the brain's electrical potential on the scalp surface can be clustered into a set of representative maps called EEG microstates. Microstate time series are obtained by competitively fitting the microstate maps back into the EEG data set, i.e., by substituting the EEG data at a given time with the label of the microstate that has the highest similarity with the actual EEG topography. As microstate sequences consist of non-metric random variables, e.g., the letters A–D, we recently introduced information-theoretical measures to quantify these time series. In wakeful resting state EEG recordings, we found new characteristics of microstate sequences such as periodicities related to EEG frequency bands. The algorithms used are here provided as an open-source package and their use is explained in a tutorial style. The package is self-contained and the programming style is procedural, focusing on code intelligibility and easy portability. Using a sample EEG file, we demonstrate how to perform EEG microstate segmentation using the modified K-means approach, and how to compute and visualize the recently introduced information-theoretical tests and quantities. The time-lagged mutual information function is derived as a discrete symbolic alternative to the autocorrelation function for metric time series and confidence intervals are computed from Markov chain surrogate data. The software package provides an open-source extension to the existing implementations of the microstate transform and is specifically designed to analyze resting state EEG recordings.
An information-theoretic approach to numerically determine the Markov order of discrete stochastic processes defined over a finite state space is introduced. To measure statistical dependencies between different time points of symbolic time series, two information-theoretic measures are proposed. The first measure is time-lagged mutual information between the random variables Xn and Xn+k, representing the values of the process at time points n and n + k, respectively. The measure will be termed autoinformation, in analogy to the autocorrelation function for metric time series, but using Shannon entropy rather than linear correlation. This measure is complemented by the conditional mutual information between Xn and Xn+k, removing the influence of the intermediate values Xn+k−1, …, Xn+1. The second measure is termed partial autoinformation, in analogy to the partial autocorrelation function (PACF) in metric time series analysis. Mathematical relations with known quantities such as the entropy rate and active information storage are established. Both measures are applied to a number of examples, ranging from theoretical Markov and non-Markov processes with known stochastic properties, to models from statistical physics, and finally, to a discrete transform of an EEG data set. The combination of autoinformation and partial autoinformation yields important insights into the temporal structure of the data in all test cases. For first- and higher-order Markov processes, partial autoinformation correctly identifies the order parameter, but also suggests extended, non-Markovian effects in the examples that lack the Markov property. For three hidden Markov models (HMMs), the underlying Markov order is found. The combination of both quantities may be used as an early step in the analysis of experimental, non-metric time series and can be employed to discover higher-order Markov dependencies, non-Markovianity and periodicities in symbolic time series.