Fluctuation analysis in small chemically reacting systems

A theory of noise fluctuations is developed which is applicable to systems of any size in which unimolecular or bimolecular reactions are occurring. The main difference between small and large reacting systems is that in the former the probability of finding a particle in a particular state does not obey a Gaussian distribution, but satisfies a distribution which reflects the mechanism of the chemical reaction. This difference is reflected in the main result of the theory: an autocorrelation function that is expressible as a sum of exponentials, the amplitudes of which are explicit functions of the moments of the distribution. Thus, by using small systems, the autocorrelation function,in principle, allows the elucidation of reaction mechanisms. Numerical simulations indicate that for reacting systems having ten or fewer particles, the deviation of the autocorrelation function from a single exponential should be easily detectable, and that estimates of the first four moments of the distribution should be possible. Accurate inference of the distribution, however, will require further mathematical and experimental advances.     

Size distribution of submicron particles by dynamic light scattering measurements: analyses considering normalization errors

Errors in the experimental baseline used to normalize dynamic light scattering data can seriously affect the size distribution resulting from the data analysis. A revised method, which incorporates the characteristics of this error into the size distribution algorithm CONTIN (Ruf 1989), is tested with experimental data of high statistical accuracy obtained from a sample of phospholipid vesicles. It is shown that the various commonly used ways of accumulating and normalizing dynamic light scattering data are associated with rather different normalization errors. As a consequence a variety of solutions differing in modality, as well as in width, are obtained on carrying out data analysis in the common way. It is demonstrated that a single monomodal solution is retrieved from all these data sets when the new method is applied, which in addition provides the corresponding baseline errors quantitatively. Furthermore, stable solutions are obtainable with data of lower statistical accuracy which results from measurements of shorter duration. The use of an additional parameter in data inversion reduces the occurrence of spurious peaks. This stabilizing effect is accompanied by larger uncertainties in the width of the size distribution. It is demonstrated that these uncertainties are reduced by nearly a factor of two on using the normalization error function instead of the ‘dust term’ option for the analysis of noisy data sets.     

From Fast Fluorescence Imaging to Molecular Diffusion Law on Live Cell Membranes in a Commercial Microscope

It has become increasingly evident that the spatial distribution and the motion of membrane components like lipids and proteins are key factors in the regulation of many cellular functions. However, due to the fast dynamics and the tiny structures involved, a very high spatio-temporal resolution is required to catch the real behavior of molecules. Here we present the experimental protocol for studying the dynamics of fluorescently-labeled plasma-membrane proteins and lipids in live cells with high spatiotemporal resolution. Notably, this approach doesn’t need to track each molecule, but it calculates population behavior using all molecules in a given region of the membrane. The starting point is a fast imaging of a given region on the membrane. Afterwards, a complete spatio-temporal autocorrelation function is calculated correlating acquired images at increasing time delays, for example each 2, 3, n repetitions. It is possible to demonstrate that the width of the peak of the spatial autocorrelation function increases at increasing time delay as a function of particle movement due to diffusion. Therefore, fitting of the series of autocorrelation functions enables to extract the actual protein mean square displacement from imaging (iMSD), here presented in the form of apparent diffusivity vs average displacement. This yields a quantitative view of the average dynamics of single molecules with nanometer accuracy. By using a GFP-tagged variant of the Transferrin Receptor (TfR) and an ATTO488 labeled 1-palmitoyl-2-hydroxy-sn-glycero-3-phosphoethanolamine (PPE) it is possible to observe the spatiotemporal regulation of protein and lipid diffusion on µm-sized membrane regions in the micro-to-milli-second time range.     

Evaluation of Postspike Changes in Neuronal Excitability by Comparing Ordinary and Shuffled Autocorrelation Functions

Responses of single neurons to tonal signals amplitude-modulated by repeating segments of lowfrequency noise were studied in the dorsal (cochlear) medullary nucleus and midbrain auditory center (torus semicircularis) of the grass frog Rana temporaria. An autocorrelation function of the response to a total presentation and a shuffled autocorrelation function were derived. The latter was obtained by correlating the impulse response to each segment of the modulated signal with responses to all other segments with the exception of the initial one. After the necessary normalization, the function differed from the initial autocorrelation only in lacking postspike changes in excitability. A delay dependence of the ratio of the two functions directly demonstrated the time course of the postspike change in excitability of the studied cell. The majority of second-order neurons, which are in the dorsal nucleus of the medulla oblongata, were characterized only by brief intervals of absolute and relative refractoriness. However, cells with excitability that was markedly facilitated immediately after the refractory period were observed even in this nucleus. Neurons with a complex pattern of postspike changes in excitability were detected in the torus semicircularis. In these cells, a comparatively long postspike decrease in excitability was usually interrupted by intervals in which the neuron sensitivity was significantly higher than normal. The results demonstrate that spike generation has a marked effect on subsequent activity in brainstem auditory units. The effects may play an important role in the formation of the temporal pattern of neuronal responses to auditory signals.     

A Statistical Physics View of Pitch Fluctuations in the Classical Music from Bach to Chopin: Evidence for Scaling

Because classical music has greatly affected our life and culture in its long history, it has attracted extensive attention from researchers to understand laws behind it. Based on statistical physics, here we use a different method to investigate classical music, namely, by analyzing cumulative distribution functions (CDFs) and autocorrelation functions of pitch fluctuations in compositions. We analyze 1,876 compositions of five representative classical music composers across 164 years from Bach, to Mozart, to Beethoven, to Mendelsohn, and to Chopin. We report that the biggest pitch fluctuations of a composer gradually increase as time evolves from Bach time to Mendelsohn/Chopin time. In particular, for the compositions of a composer, the positive and negative tails of a CDF of pitch fluctuations are distributed not only in power laws (with the scale-free property), but also in symmetry (namely, the probability of a treble following a bass and that of a bass following a treble are basically the same for each composer). The power-law exponent decreases as time elapses. Further, we also calculate the autocorrelation function of the pitch fluctuation. The autocorrelation function shows a power-law distribution for each composer. Especially, the power-law exponents vary with the composers, indicating their different levels of long-range correlation of notes. This work not only suggests a way to understand and develop music from a viewpoint of statistical physics, but also enriches the realm of traditional statistical physics by analyzing music.     

Autocorrelation function of finite-length data in fluorescence correlation spectroscopy

The experimental autocorrelation function of fluorescence correlation spectroscopy calculated from finite-length data is a biased estimator of the theoretical correlation function. This study presents a new theoretical framework that explicitly accounts for the data length to allow for unbiased analysis of experimental autocorrelation functions. To validate our theory, we applied it to experiments and simulations of diffusion and characterized the accuracy and precision of the resulting parameter estimates. Because measurements in living cells are often affected by instabilities of the fluorescence signal, autocorrelation functions are typically calculated on segmented data to improve their robustness. Our reformulated theory extends the range of usable segment times down to timescales approaching the diffusion time. This flexibility confers unique advantages for live-cell data that contain intensity variations and instabilities. We describe several applications of short segmentation to analyze data contaminated with unwanted fluctuations, drifts, or spikes in the intensity that are not suited for conventional fluorescence correlation analysis. These results demonstrate the potential of our theoretical framework to significantly expand the experimental systems accessible to fluorescence correlation spectroscopy.     

Computing the self-diffusion coefficient for TIP4P water

A molecular dynamics study was made for the TIP4P model of liquid water. Thermal dependences of water density and radial distribution functions were calculated for model verification. Different methods were used to calculate the self-diffusion coefficient, and assessed for sensitivity to molecular system size and trajectory length. The Green-Kubo formula deriving the diffusion coefficient from the velocity autocorrelation function is preferable in short MD simulations with a high sampling rate, whereas the Einstein equation for diffusion is the method of choice in long simulations. The latter approach yields more stable and reliable results, especially at very short times and for a small number of molecules, if the diffusion coefficient is estimated not from the limit ratio of mean squared displacement to time, but from the slope of the time plot of mean squared displacement.     

Equilibrating errors: reliable estimation of information transmission rates in biological systems with spectral analysis-based methods

Shannon’s seminal approach to estimating information capacity is widely used to quantify information processing by biological systems. However, the Shannon information theory, which is based on power spectrum estimation, necessarily contains two sources of error: time delay bias error and random error. These errors are particularly important for systems with relatively large time delay values and for responses of limited duration, as is often the case in experimental work. The window function type and size chosen, as well as the values of inherent delays cause changes in both the delay bias and random errors, with possibly strong effect on the estimates of system properties. Here, we investigated the properties of these errors using white-noise simulations and analysis of experimental photoreceptor responses to naturalistic and white-noise light contrasts. Photoreceptors were used from several insect species, each characterized by different visual performance, behavior, and ecology. We show that the effect of random error on the spectral estimates of photoreceptor performance (gain, coherence, signal-to-noise ratio, Shannon information rate) is opposite to that of the time delay bias error: the former overestimates information rate, while the latter underestimates it. We propose a new algorithm for reducing the impact of time delay bias error and random error, based on discovering, and then using that size of window, at which the absolute values of these errors are equal and opposite, thus cancelling each other, allowing minimally biased measurement of neural coding.     

Cancer cell dynamics in presence of telomerase inhibitors: analysis of in vitro data

The inhibition of telomerase activity in actively dividing cells leads to suppression of cell growth after a time delay (inhibitory delay) required to reach a threshold telomeric DNA size. We developed a mathematical model of the dynamics of telomere size distribution and cell growth in the presence of telomere inhibitors that allowed quantification of the inhibitory delay. The model based on the solution of a system of differential equations described quantitatively recent experimental data on dynamics of cultured cells in presence of telomerase inhibitors. The analysis of the data by this model suggested the existence of at least two distinct subpopulations of cells with different proliferative activity. Size distribution of telomeres, fraction of proliferating cells, and tumor doubling times are of critical importance for the dynamics of cancer cells growth in presence of telomerase inhibitors. Rapidly growing cells with large telomeric DNA heterogeneity and small proliferating fractions as well as those with very short homogeneous telomeres would be the most sensitive to telomerase inhibitors.     

芯片数据标准化方法比较研究

在基因芯片实验中,基因表达水平之间的相关性在推断基因间相互关系时起到非常重要的作用.未经标准化处理的芯片数据基因之间往往都呈现出很强的相关性,这些高相关性一部分是由基因表达水平变化引起的,而另外一部分是由系统偏差引起的.对芯片数据进行标准化处理的目的之一是消除系统偏差引起的高相关性,同时保留由真正生物学原因引起的基因表达水平高相关性.虽然目前对标准化方法已经有了不少比较研究,但还较少有人研究标准化方法对基因之间相关系数的影响,以及哪种方法最有利于恢复基因之间的相关性结构.通过对基因表达水平数据的模拟,具体比较了几种常用标准化方法的效果,从而给出最有利于恢复基因之间相关性结构的那种标准化方法.     

OLIN: optimized normalization, visualization and quality testing of two-channel microarray data

SUMMARY: Microarray data are generated in complex experiments and frequently compromised by a variety of systematic errors. Subsequent data normalization aims to correct these errors. Although several normalization methods have recently been proposed, they frequently fail to account for the variability of systematic errors within and between microarray experiments. However, optimal adjustment of normalization procedures to the underlying data structure is crucial for the efficiency of normalization. To overcome this restriction of current methods, we have developed two normalization schemes based on iterative local regression combined with model selection. The schemes have been demonstrated to improve considerably the quality of normalization. They are implemented in a freely available R package. Additionally, functions for visualization and detection of systematic errors in microarray data have been incorporated in the software package. A graphical user interface is also available. AVAILABILITY: The R package can be downloaded from http://itb.biologie.hu-berlin.de/~futschik/software/R/OLIN. It underlies the GPL version 2. CONTACT: m.futschik@biologie.hu-berlin.de SUPPLEMENTARY INFORMATION: Further information about the methods used in the OLIN software package can be found at http://itb.biologie.hu-berlin.de/~futschik/software/R/OLIN.     

Biophysical models of ciliary activity: Gaussian frequency distributions

A model of a freely rotating exended scatterer is proposed to describe light scattering from beating cilia. Gaussian rotation frequency distributions, characterized by a mean angular frequency and a standard deviation, are introduced in order to simulate intensity autocorrelation functions and to fit the model to experimental data. Thus the ciliary beats are characterized by a mean beat frequency and a standard deviation of the beat frequency distribution. The standard deviation influences the damping of the intensity autocorrelation function of light scattered from cilia. The calculated intensity autocorrelation function shows a more prominent oscillating behaviour the smaller the standard deviation of the beat frequency. The validity of the model is supported by experimental data in two ways: 1) The model fits very well to experimental data in computer evaluations, 2) Neither the model nor information obtained from measurements are dependent on the measuring angle.The contents were presented in part at the 9th International Biophysics Congress in Jerusalem, Israel, August 23–28, 1987 Offprint requests to: P. Thyberg     

The description of spatial pattern using two-dimensional spectral analysis

Two-dimensional spectral analysis is a general interrogative technique for describing spatial patterns. Not only is it able to detect all possible scales of pattern which can be present in the data but it is also sensitive to directional components. Four functions are described: the autocorrelation function; the periodogram; and, the R- and Θ-spectra which respectively summarize the periodogram in terms of scale and directional components of pattern. The use of these functions is illustrated by their application to a simple wave pattern, a wave pattern with added noise, and patterns simulating competition and invasion processes.     

Normalization strategies for microRNA profiling experiments: a ‘normal’ way to a hidden layer of complexity?

MicroRNA (miRNA) profiling is a first important step in elucidating miRNA functions. Real time quantitative PCR (RT-qPCR) and microarray hybridization approaches as well as ultra high throughput sequencing of miRNAs (small RNA-seq) are popular and widely used profiling methods. All of these profiling approaches face significant introduction of bias. Normalization, often an underestimated aspect of data processing, can minimize systematic technical or experimental variation and thus has significant impact on the detection of differentially expressed miRNAs. At present, there is no consensus normalization method for any of the three miRNA profiling approach. Several normalization techniques are currently in use, of which some are similar to mRNA profiling normalization methods, while others are specifically modified or developed for miRNA data. The characteristic nature of miRNA molecules, their composition and the resulting data distribution of profiling experiments challenges the selection of adequate normalization techniques. Based on miRNA profiling studies and comparative studies on normalization methods and their performances, this review provides a critical overview of commonly used and newly developed normalization methods for miRNA RT-qPCR, miRNA hybridization microarray, and small RNA-seq datasets. Emphasis is laid on the complexity, the importance and the potential for further optimization of normalization techniques for miRNA profiling datasets.     

Development of a local size hierarchy causes regular spacing of trees in an even-aged Abies forest: analyses using spatial autocorrelation and the mark correlation function

Background and Aims

During the development of an even-aged plant population, the spatial distribution of individuals often changes from a clumped pattern to a random or regular one. The development of local size hierarchies in an Abies forest was analysed for a period of 47 years following a large disturbance in 1959.

Methods

In 1980 all trees in an 8 × 8 m plot were mapped and their height growth after the disturbance was estimated. Their mortality and growth were then recorded at 1- to 4-year intervals between 1980 and 2006. Spatial distribution patterns of trees were analysed by the pair correlation function. Spatial correlations between tree heights were analysed with a spatial autocorrelation function and the mark correlation function. The mark correlation function was able to detect a local size hierarchy that could not be detected by the spatial autocorrelation function alone.

Key Results

The small-scale spatial distribution pattern of trees changed from clumped to slightly regular during the 47 years. Mortality occurred in a density-dependent manner, which resulted in regular spacing between trees after 1980. The spatial autocorrelation and mark correlation functions revealed the existence of tree patches consisting of large trees at the initial stage. Development of a local size hierarchy was detected within the first decade after the disturbance, although the spatial autocorrelation was not negative. Local size hierarchies that developed persisted until 2006, and the spatial autocorrelation became negative at later stages (after about 40 years).

Conclusions

This is the first study to detect local size hierarchies as a prelude to regular spacing using the mark correlation function. The results confirm that use of the mark correlation function together with the spatial autocorrelation function is an effective tool to analyse the development of a local size hierarchy of trees in a forest.Key words: Abies, local size hierarchy, mark correlation function, pair correlation function, regenerating forest, regular spacing, spatial autocorrelation     

Minimizing errors in identifying Lévy flight behaviour of organisms

1. Lévy flights are specialized random walks with fundamental properties such as superdiffusivity and scale invariance that have recently been applied in optimal foraging theory. Lévy flights have movement lengths chosen from a probability distribution with a power-law tail, which theoretically increases the chances of a forager encountering new prey patches and may represent an optimal solution for foraging across complex, natural habitats. 2. An increasing number of studies are detecting Lévy behaviour in diverse organisms such as microbes, insects, birds, and mammals including humans. A principal method for detecting Lévy flight is whether the exponent (micro) of the power-law distribution of movement lengths falls within the range 1 < micro < or = 3. The exponent can be determined from the histogram of frequency vs. movement (step) lengths, but different plotting methods have been used to derive the Lévy exponent across different studies. 3. Here we investigate using simulations how different plotting methods influence the micro-value and show that the power-law plotting method based on 2(k) (logarithmic) binning with normalization prior to log transformation of both axes yields low error (1.4%) in identifying Lévy flights. Furthermore, increasing sample size reduced variation about the recovered values of micro, for example by 83% as sample number increased from n = 50 up to 5000. 4. Simple log transformation of the axes of the histogram of frequency vs. step length underestimated micro by c.40%, whereas two other methods, 2(k) (logarithmic) binning without normalization and calculation of a cumulative distribution function for the data, both estimate the regression slope as 1-micro. Correction of the slope therefore yields an accurate Lévy exponent with estimation errors of 1.4 and 4.5%, respectively. 5. Empirical reanalysis of data in published studies indicates that simple log transformation results in significant errors in estimating micro, which in turn affects reliability of the biological interpretation. The potential for detecting Lévy flight motion when it is not present is minimized by the approach described. We also show that using a large number of steps in movement analysis such as this will also increase the accuracy with which optimal Lévy flight behaviour can be detected.     

A general combined model to describe tree‐diameter distributions within subtropical and temperate forest communities

The size distribution of trees in natural forests is a fundamental attribute of forest structure. Previous attempts to model tree size distributions using simple functions (such as power or Weibull functions) have had limited success, typically overestimating the number of large stems observed. We describe a model which assumes that the dominant mortality process is asymmetric competition when trees are smaller, and size‐independent processes (e.g. disturbance) when trees are larger. This combination of processes leads to a size distribution which takes the form of a power distribution in the small tree phase and a Weibull distribution in the large tree phase. Analyses of data from four large‐scale (≥ 24 ha each) subtropical and temperate forest plots totalling 99 ha and approximately 0.4 million trees provide support for this model in two respects: (a) the combined function provided unbiased predictions and (b) power‐law functions fitted to small trees had exponents that deviated from the universal exponent of –2 predicted by metabolic scaling theory, gradually decreasing from subtropical evergreen to temperate deciduous forests along the latitudinal gradient.     

Testing Pairwise Association between Spatially Autocorrelated Variables: A New Approach Using Surrogate Lattice Data

Background

Independence between observations is a standard prerequisite of traditional statistical tests of association. This condition is, however, violated when autocorrelation is present within the data. In the case of variables that are regularly sampled in space (i.e. lattice data or images), such as those provided by remote-sensing or geographical databases, this problem is particularly acute. Because analytic derivation of the null probability distribution of the test statistic (e.g. Pearson''s r) is not always possible when autocorrelation is present, we propose instead the use of a Monte Carlo simulation with surrogate data.

Methodology/Principal Findings

The null hypothesis that two observed mapped variables are the result of independent pattern generating processes is tested here by generating sets of random image data while preserving the autocorrelation function of the original images. Surrogates are generated by matching the dual-tree complex wavelet spectra (and hence the autocorrelation functions) of white noise images with the spectra of the original images. The generated images can then be used to build the probability distribution function of any statistic of association under the null hypothesis. We demonstrate the validity of a statistical test of association based on these surrogates with both actual and synthetic data and compare it with a corrected parametric test and three existing methods that generate surrogates (randomization, random rotations and shifts, and iterative amplitude adjusted Fourier transform). Type I error control was excellent, even with strong and long-range autocorrelation, which is not the case for alternative methods.

Conclusions/Significance

The wavelet-based surrogates are particularly appropriate in cases where autocorrelation appears at all scales or is direction-dependent (anisotropy). We explore the potential of the method for association tests involving a lattice of binary data and discuss its potential for validation of species distribution models. An implementation of the method in Java for the generation of wavelet-based surrogates is available online as supporting material.     

Photon correlation analysis of cytoplasmic streaming

Laser light scattering has been used to investigate particle movements in a plant cell. Intensity autocorrelation functions are obtained by digital photon correlation of laser light scattered from cells of Nitella opaca both during cytoplasmic streaming and during the transitory cessation of streaming induced by electrical stimulation. The average velocity computed from the periodic oscillation in the intensity autocorrelation function during streaming corresponds to the velocity estimated using light microscopy. An estimate of the distribution of streaming velocities has been obtained from the decay in the amplitude of the envelope of the autocorrelation function derived from a streaming cell.     

Inference in MCMC step selection models

Habitat selection models are used in ecology to link the spatial distribution of animals to environmental covariates and identify preferred habitats. The most widely used models of this type, resource selection functions, aim to capture the steady-state distribution of space use of the animal, but they assume independence between the observed locations of an animal. This is unrealistic when location data display temporal autocorrelation. The alternative approach of step selection functions embed habitat selection in a model of animal movement, to account for the autocorrelation. However, inferences from step selection functions depend on the underlying movement model, and they do not readily predict steady-state space use. We suggest an analogy between parameter updates and target distributions in Markov chain Monte Carlo (MCMC) algorithms, and step selection and steady-state distributions in movement ecology, leading to a step selection model with an explicit steady-state distribution. In this framework, we explain how maximum likelihood estimation can be used for simultaneous inference about movement and habitat selection. We describe the local Gibbs sampler, a novel rejection-free MCMC scheme, use it as the basis of a flexible class of animal movement models, and derive its likelihood function for several important special cases. In a simulation study, we verify that maximum likelihood estimation can recover all model parameters. We illustrate the application of the method with data from a zebra.