Nina Golyandina

Gene Expression Noise in Spatial Patterning: hunchback Promoter Structure Affects Noise Amplitude and Distribution in Drosophila Segmentation

Positional information in developing embryos is specified by spatial gradients of transcriptional regulators. One of the classic systems for studying this is the activation of the hunchback (hb) gene in early fruit fly (Drosophila) segmentation by the maternally-derived gradient of the Bicoid (Bcd) protein. Gene regulation is subject to intrinsic noise which can produce variable expression. This variability must be constrained in the highly reproducible and coordinated events of development. We identify means by which noise is controlled during gene expression by characterizing the dependence of hb mRNA and protein output noise on hb promoter structure and transcriptional dynamics. We use a stochastic model of the hb promoter in which the number and strength of Bcd and Hb (self-regulatory) binding sites can be varied. Model parameters are fit to data from WT embryos, the self-regulation mutant hb 14F, and lacZ reporter constructs using different portions of the hb promoter. We have corroborated model noise predictions experimentally. The results indicate that WT (self-regulatory) Hb output noise is predominantly dependent on the transcription and translation dynamics of its own expression, rather than on Bcd fluctuations. The constructs and mutant, which lack self-regulation, indicate that the multiple Bcd binding sites in the hb promoter (and their strengths) also play a role in buffering noise. The model is robust to the variation in Bcd binding site number across a number of fly species. This study identifies particular ways in which promoter structure and regulatory dynamics reduce hb output noise. Insofar as many of these are common features of genes (e.g. multiple regulatory sites, cooperativity, self-feedback), the current results contribute to the general understanding of the reproducibility and determinacy of spatial patterning in early development.

Multivariate and 2D Extensions of Singular Spectrum Analysis with the Rssa Package

Implementation of multivariate and 2D extensions of singular spectrum analysis (SSA) by means of the R package Rssa is considered. The extensions include MSSA for simultaneous analysis and forecasting of several time series and 2D-SSA for analysis of digital images. A new extension of 2D-SSA analysis called shaped 2D-SSA is introduced for analysis of images of arbitrary shape, not necessary rectangular. It is shown that implementation of shaped 2D-SSA can serve as a basis for implementation of MSSA and other generalizations. Efficient implementation of operations with Hankel and Hankel-block-Hankel matrices through the fast Fourier transform is suggested. Examples with code fragments in R, which explain the methodology and demonstrate the proper use of Rssa, are presented.

Variations of singular spectrum analysis for separability improvement: non-orthogonal decompositions of time series

Singular spectrum analysis (SSA) as a nonparametric tool for decomposition of an observed time series into sum of interpretable components such as trend, oscillations and noise is considered. The separability of these series components by SSA means the possibility of such decomposition. Two variations of SSA, which weaken the separability conditions, are proposed. Both proposed approaches consider inner products corresponding to oblique coordinate systems instead of the conventional Euclidean inner product. One of the approaches performs iterations to obtain separating inner products. The other method changes contributions of the components by involving the series derivative to avoid component mixing. Performance of the suggested methods is demonstrated on simulated and real-life data.

New approaches to nonparametric density estimation and selection of smoothing parameters

The application of Singular Spectrum Analysis (SSA) to the empirical distribution function sampled at a grid of points spanning the range of the sample leads to a novel and promising method for the computer-intensive nonparametric estimation of both the distribution function and the density function. SSA yields a data-adaptive filter, whose length is a parameter that controls the smoothness of the filtered series. A data-adaptive algorithm for the automatic selection of a general smoothing parameter is introduced, which controls the number of modes of the estimated density. Extensive computer simulations demonstrate that the new automatic bandwidth selector improves on other popular methods for various densities of interest. A general uniform error bound is proved for the proposed SSA estimate of the distribution function, which ensures its uniform consistency. The simulation results indicate that the SSA density estimate with the automatic choice of the filter length outperforms the kernel density estimate in terms of the mean integrated squared error and the Kolmogorov-Smirnov distance for various density shapes. Two applications to problems arising in photovoltaic quality control and economic market research are studied to illustrate the benefits of SSA estimation.

Measuring Gene Expression Noise in Early Drosophila Embryos: Nucleus-to-nucleus Variability

In recent years the analysis of noise in gene expression has widely attracted the attention of experimentalists and theoreticians. Experimentally, the approaches based on in vivo fluorescent reporters in single cells appear to be straightforward and effective tools for bacteria and yeast. However, transferring these approaches to multicellular organisms presents many methodological problems. Here we describe our approach to measure between-nucleus variability (noise) in the primary morphogenetic gradient of Bicoid (Bcd) in the precellular blastoderm stage of fruit fly (Drosophila) embryos. The approach is based on the comparison of results for fixed immunostained embryos with observations of live embryos carrying fluorescent Bcd (Bcd-GFP). We measure the noise using two-dimensional Singular Spectrum Analysis (2D SSA). We have found that the nucleus-to-nucleus noise in Bcd intensity, both for live (Bcd-GFP) and for fixed immunstained embryos, tends to be signal-independent. In addition, the character of the noise is sensitive to the nuclear masking technique used to extract quantitative intensities. Further, the method of decomposing the raw quantitative expression data into a signal (expression surface) and residual noise affects the character of the residual noise. We find that careful masking of confocal images and use of appropriate computational tools to decompose raw expression data into trend and noise makes it possible to extract and study the biological noise of gene expression.

Shaped Singular Spectrum Analysis for Quantifying Gene Expression, with Application to the Early Drosophila Embryo

In recent years, with the development of automated microscopy technologies, the volume and complexity of image data on gene expression have increased tremendously. The only way to analyze quantitatively and comprehensively such biological data is by developing and applying new sophisticated mathematical approaches. Here, we present extensions of 2D singular spectrum analysis (2D-SSA) for application to 2D and 3D datasets of embryo images. These extensions, circular and shaped 2D-SSA, are applied to gene expression in the nuclear layer just under the surface of the Drosophila (fruit fly) embryo. We consider the commonly used cylindrical projection of the ellipsoidal Drosophila embryo. We demonstrate how circular and shaped versions of 2D-SSA help to decompose expression data into identifiable components (such as trend and noise), as well as separating signals from different genes. Detection and improvement of under- and overcorrection in multichannel imaging is addressed, as well as the extraction and analysis of 3D features in 3D gene expression patterns.

SSA for Multivariate Time Series

In Chap. 4 the problem of simultaneous decomposition, reconstruction, and forecasting for a collection of time series is considered from the viewpoint of SSA; note that individual time series can have different length. The main method of this chapter is usually called either Multichannel SSA or Multivariate SSA, shortly MSSA. The principal idea of MSSA is the same as for the case of one-dimensional time with the difference lying in the way of constructing of the trajectory matrix. The aim of MSSA is to take into consideration the combined structure of a multivariate series to obtain more accurate results.

SSA Analysis of One-Dimensional Time Series

In Chap. 2, the use of SSA for analyzing one-dimensional data is thoroughly examined. In this chapter, the use of models is minimal so that the main techniques can be considered as non-parametric and descriptory. Relations with algorithms of space rotation and many other methods aiming at achieving better separability of signal from noise are outlined. The common problems of smoothing, filtration, and splitting of a time series into identifiable components such as trend, seasonality, and noise are thoroughly discussed and illustrated on case studies with real data. An important issue of automatization of the SSA methods is also considered in detail.

Parameter Estimation, Forecasting, Gap Filling

Chapter 3 is devoted to applications of SSA for one-dimensional series for forecasting, gap filling, low-rank approximation, parameter estimation, and change-point detection. The SSA analysis of time series of Chap. 2 is model-free. Methods of Chap. 3, on the contrary, are model-based. The model is constructed on the base of the approximating subspace built in the process of performing the SSA analysis of Chap. 2. The main parametric model is a linear recurrence relation which the signal should approximately satisfy. Application of methods is illustrated on real-life data.

Analysis and Design in the Problem of Vector Deconvolution

We formulate the problem of deconvolution of a given vector as an optimal design problem and suggest numerical algorithms for solving this problem. We then discuss an important application of the proposed methods for problems of time series analysis and signal processing and also to the low-rank approximation of structured matrices.