CaSPIAN: a causal compressive sensing algorithm for discovering directed interactions in gene networks

Granger causality for compressively sensed sparse signals

Compressed sensing is a scheme that allows for sparse signals to be acquired, transmitted, and stored using far fewer measurements than done by conventional means employing the Nyquist sampling theorem. Since many naturally occurring signals are sparse (in some domain), compressed sensing has rapidly seen popularity in a number of applied physics and engineering applications, particularly in designing signal and image acquisition strategies, e.g., magnetic resonance imaging, quantum state tomography, scanning tunneling microscopy, and analog to digital conversion technologies. Contemporaneously, causal inference has become an important tool for the analysis and understanding of processes and their interactions in many disciplines of science, especially those dealing with complex systems. Direct causal analysis for compressively sensed data is required to avoid the task of reconstructing the compressed data. Also, for some sparse signals, such as for sparse temporal data, it may be difficult to discover causal relations directly using available data-driven or model-free causality estimation techniques. In this work, we provide a mathematical proof that structured compressed sensing matrices, specifically circulant and Toeplitz, preserve causal relationships in the compressed signal domain, as measured by Granger causality (GC). We then verify this theorem on a number of bivariate and multivariate coupled sparse signal simulations which are compressed using these matrices. We also demonstrate a real world application of network causal connectivity estimation from sparse neural spike train recordings from rat prefrontal cortex. In addition to demonstrating the effectiveness of structured matrices for GC estimation from sparse signals, we also show a computational time advantage of the proposed strategy for causal inference from compressed signals of both sparse and regular autoregressive processes as compared to standard GC estimation from original signals.

New insights into binocular rivalry from the reconstruction of evolving percepts using model network dynamics

When the two eyes are presented with highly distinct stimuli, the resulting visual percept generally switches every few seconds between the two monocular images in an irregular fashion, giving rise to a phenomenon known as binocular rivalry. While a host of theoretical studies have explored potential mechanisms for binocular rivalry in the context of evoked model dynamics in response to simple stimuli, here we investigate binocular rivalry directly through complex stimulus reconstructions based on the activity of a two-layer neuronal network model with competing downstream pools driven by disparate monocular stimuli composed of image pixels. To estimate the dynamic percept, we derive a linear input-output mapping rooted in the non-linear network dynamics and iteratively apply compressive sensing techniques for signal recovery. Utilizing a dominance metric, we are able to identify when percept alternations occur and use data collected during each dominance period to generate a sequence of percept reconstructions. We show that despite the approximate nature of the input-output mapping and the significant reduction in neurons downstream relative to stimulus pixels, the dominant monocular image is well-encoded in the network dynamics and improvements are garnered when realistic spatial receptive field structure is incorporated into the feedforward connectivity. Our model demonstrates gamma-distributed dominance durations and well obeys Levelt's four laws for how dominance durations change with stimulus strength, agreeing with key recurring experimental observations often used to benchmark rivalry models. In light of evidence that individuals with autism exhibit relatively slow percept switching in binocular rivalry, we corroborate the ubiquitous hypothesis that autism manifests from reduced inhibition in the brain by systematically probing our model alternation rate across choices of inhibition strength. We exhibit sufficient conditions for producing binocular rivalry in the context of natural scene stimuli, opening a clearer window into the dynamic brain computations that vary with the generated percept and a potential path toward further understanding neurological disorders.

Evolving Transcriptomic Profiles From Single-Cell RNA-Seq Data Using Nature-Inspired Multiobjective Optimization

Transcriptomic profiling plays an important role in post-genomic analysis. Especially, the single-cell RNA-seq technology has advanced our understanding of gene expression from cell population level into individual cell level. Many computational methods have been proposed to decipher transcriptomic profiles from those RNA-seq data. However, most of the related algorithms suffer from realistic restrictions such as high dimensionality and premature convergence. In this paper, we propose and formulate an evolutionary multiobjective blind compressed sensing (EMOBCS) to address those problems for evolving transcriptomic profiles from single-cell RNA-seq data. In the proposed framework, to characterize various gene expression profile models, two objective functions including chi-squared kernel score and euclidean distance of different gene expression profiles are formulated. After that, multiobjective blind compressed sensing based on artificial bee colony is designed to optimize the two objective functions on single-cell RNA-seq data by proposing a rank probability model and two new search strategies into the cooperative convolution framework in an unbiased manner. To demonstrate its effectiveness, extensive experiments have been conducted, comparing the proposed algorithm with 14 algorithms including eight state-of-the-art algorithms and six different EMOBCS algorithms under different search strategies on 10 single-cell RNA-seq datasets and one case study. The experimental results reveal that the proposed algorithm is better than or comparable with those compared algorithms. Furthermore, we also conduct the time complexity analysis, convergence analysis, and parameter analysis to demonstrate various properties of EMOBCS.

Large-scale nonlinear Granger causality for inferring directed dependence from short multivariate time-series data

A key challenge to gaining insight into complex systems is inferring nonlinear causal directional relations from observational time-series data. Specifically, estimating causal relationships between interacting components in large systems with only short recordings over few temporal observations remains an important, yet unresolved problem. Here, we introduce large-scale nonlinear Granger causality (lsNGC) which facilitates conditional Granger causality between two multivariate time series conditioned on a large number of confounding time series with a small number of observations. By modeling interactions with nonlinear state-space transformations from limited observational data, lsNGC identifies casual relations with no explicit a priori assumptions on functional interdependence between component time series in a computationally efficient manner. Additionally, our method provides a mathematical formulation revealing statistical significance of inferred causal relations. We extensively study the ability of lsNGC in inferring directed relations from two-node to thirty-four node chaotic time-series systems. Our results suggest that lsNGC captures meaningful interactions from limited observational data, where it performs favorably when compared to traditionally used methods. Finally, we demonstrate the applicability of lsNGC to estimating causality in large, real-world systems by inferring directional nonlinear, causal relationships among a large number of relatively short time series acquired from functional Magnetic Resonance Imaging (fMRI) data of the human brain.

Inference of phenotype-relevant transcriptional regulatory networks elucidates cancer type-specific regulatory mechanisms in a pan-cancer study

Reconstruction of transcriptional regulatory networks (TRNs) is a powerful approach to unravel the gene expression programs involved in healthy and disease states of a cell. However, these networks are usually reconstructed independent of the phenotypic (or clinical) properties of the samples. Therefore, they may confound regulatory mechanisms that are specifically related to a phenotypic property with more general mechanisms underlying the full complement of the analyzed samples. In this study, we develop a method called InPheRNo to identify "phenotype-relevant" TRNs. This method is based on a probabilistic graphical model that models the simultaneous effects of multiple transcription factors (TFs) on their target genes and the statistical relationship between the target genes' expression and the phenotype. Extensive comparison of InPheRNo with related approaches using primary tumor samples of 18 cancer types from The Cancer Genome Atlas reveals that InPheRNo can accurately reconstruct cancer type-relevant TRNs and identify cancer driver TFs. In addition, survival analysis reveals that the activity level of TFs with many target genes could distinguish patients with poor prognosis from those with better prognosis.

Parallel residual projection: a new paradigm for solving linear inverse problems

A grand challenge to solve a large-scale linear inverse problem (LIP) is to retain computational efficiency and accuracy regardless of the growth of the problem size. Despite the plenitude of methods available for solving LIPs, various challenges have emerged in recent times due to the sheer volume of data, inadequate computational resources to handle an oversized problem, security and privacy concerns, and the interest in the associated incremental or decremental problems. Removing these barriers requires a holistic upgrade of the existing methods to be computationally efficient, tractable, and equipped with scalable features. We, therefore, develop the parallel residual projection (PRP), a parallel computational framework involving the decomposition of a large-scale LIP into sub-problems of low complexity and the fusion of the sub-problem solutions to form the solution to the original LIP. We analyze the convergence properties of the PRP and accentuate its benefits through its application to complex problems of network inference and gravimetric survey. We show that any existing algorithm for solving an LIP can be integrated into the PRP framework and used to solve the sub-problems while handling the prevailing challenges.

Time series experimental design under one-shot sampling: The importance of condition diversity

Many biological data sets are prepared using one-shot sampling, in which each individual organism is sampled at most once. Time series therefore do not follow trajectories of individuals over time. However, samples collected at different times from individuals grown under the same conditions share the same perturbations of the biological processes, and hence behave as surrogates for multiple samples from a single individual at different times. This implies the importance of growing individuals under multiple conditions if one-shot sampling is used. This paper models the condition effect explicitly by using condition-dependent nominal mRNA production amounts for each gene, it quantifies the performance of network structure estimators both analytically and numerically, and it illustrates the difficulty in network reconstruction under one-shot sampling when the condition effect is absent. A case study of an Arabidopsis circadian clock network model is also included.

Review of Causal Discovery Methods Based on Graphical Models

A fundamental task in various disciplines of science, including biology, is to find underlying causal relations and make use of them. Causal relations can be seen if interventions are properly applied; however, in many cases they are difficult or even impossible to conduct. It is then necessary to discover causal relations by analyzing statistical properties of purely observational data, which is known as causal discovery or causal structure search. This paper aims to give a introduction to and a brief review of the computational methods for causal discovery that were developed in the past three decades, including constraint-based and score-based methods and those based on functional causal models, supplemented by some illustrations and applications.

Transcriptome-Enabled Network Inference Revealed the GmCOL1 Feed-Forward Loop and Its Roles in Photoperiodic Flowering of Soybean

Photoperiodic flowering, a plant response to seasonal photoperiod changes in the control of reproductive transition, is an important agronomic trait that has been a central target of crop domestication and modern breeding programs. However, our understanding about the molecular mechanisms of photoperiodic flowering regulation in crop species is lagging behind. To better understand the regulatory gene networks controlling photoperiodic flowering of soybeans, we elucidated global gene expression patterns under different photoperiod regimes using the near isogenic lines (NILs) of maturity loci (E loci). Transcriptome signatures identified the unique roles of the E loci in photoperiodic flowering and a set of genes controlled by these loci. To elucidate the regulatory gene networks underlying photoperiodic flowering regulation, we developed the network inference algorithmic package CausNet that integrates sparse linear regression and Granger causality heuristics, with Gaussian approximation of bootstrapping to provide reliability scores for predicted regulatory interactions. Using the transcriptome data, CausNet inferred regulatory interactions among soybean flowering genes. Published reports in the literature provided empirical verification for several of CausNet's inferred regulatory interactions. We further confirmed the inferred regulatory roles of the flowering suppressors GmCOL1a and GmCOL1b using GmCOL1 RNAi transgenic soybean plants. Combinations of the alleles of GmCOL1 and the major maturity locus E1 demonstrated positive interaction between these genes, leading to enhanced suppression of flowering transition. Our work provides novel insights and testable hypotheses in the complex molecular mechanisms of photoperiodic flowering control in soybean and lays a framework for de novo prediction of biological networks controlling important agronomic traits in crops.

Inference of Gene Regulatory Networks Using Bayesian Nonparametric Regression and Topology Information

Gene regulatory networks (GRNs) play an important role in cellular systems and are important for understanding biological processes. Many algorithms have been developed to infer the GRNs. However, most algorithms only pay attention to the gene expression data but do not consider the topology information in their inference process, while incorporating this information can partially compensate for the lack of reliable expression data. Here we develop a Bayesian group lasso with spike and slab priors to perform gene selection and estimation for nonparametric models. B-spline basis functions are used to capture the nonlinear relationships flexibly and penalties are used to avoid overfitting. Further, we incorporate the topology information into the Bayesian method as a prior. We present the application of our method on DREAM3 and DREAM4 datasets and two real biological datasets. The results show that our method performs better than existing methods and the topology information prior can improve the result.

Compressive sensing reconstruction of feed-forward connectivity in pulse-coupled nonlinear networks

Utilizing the sparsity ubiquitous in real-world network connectivity, we develop a theoretical framework for efficiently reconstructing sparse feed-forward connections in a pulse-coupled nonlinear network through its output activities. Using only a small ensemble of random inputs, we solve this inverse problem through the compressive sensing theory based on a hidden linear structure intrinsic to the nonlinear network dynamics. The accuracy of the reconstruction is further verified by the fact that complex inputs can be well recovered using the reconstructed connectivity. We expect this Rapid Communication provides a new perspective for understanding the structure-function relationship as well as compressive sensing principle in nonlinear network dynamics.

Exact reconstruction of gene regulatory networks using compressive sensing

Background

We consider the problem of reconstructing a gene regulatory network structure from limited time series gene expression data, without any a priori knowledge of connectivity. We assume that the network is sparse, meaning the connectivity among genes is much less than full connectivity. We develop a method for network reconstruction based on compressive sensing, which takes advantage of the network’s sparseness.

Results

For the case in which all genes are accessible for measurement, and there is no measurement noise, we show that our method can be used to exactly reconstruct the network. For the more general problem, in which hidden genes exist and all measurements are contaminated by noise, we show that our method leads to reliable reconstruction. In both cases, coherence of the model is used to assess the ability to reconstruct the network and to design new experiments. We demonstrate that it is possible to use the coherence distribution to guide biological experiment design effectively. By collecting a more informative dataset, the proposed method helps reduce the cost of experiments. For each problem, a set of numerical examples is presented.

Conclusions

The method provides a guarantee on how well the inferred graph structure represents the underlying system, reveals deficiencies in the data and model, and suggests experimental directions to remedy the deficiencies.

Electronic supplementary material

The online version of this article (doi:10.1186/s12859-014-0400-4) contains supplementary material, which is available to authorized users.