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Two-way sparsity for time-varying networks, with applications in genomics
Abstract
We propose a novel way of modelling time-varying networks, by inducing two-way sparsity on local models of node connectivity. This two-way sparsity separately promotes sparsity across time and sparsity across variables (within time). Separation of these two types of sparsity is achieved through a novel prior structure, which draws on ideas from the Bayesian lasso and from copula modelling. We provide an efficient implementation of the proposed model via a Gibbs sampler, and we apply the model to data from neural development. In doing so, we demonstrate that the proposed model is able to identify changes in genomic network structure that match current biological knowledge. Such changes in genomic network structure can then be used by neuro-biologists to identify potential targets for further experimental investigation.
... The literature concerning dynamic networks has appeared only recently (Zhou et al., 2010;Kolar et al., 2010;Monti et al., 2014;Xue et al., 2020;Bartlett et al., 2021), while there is rich literature on estimating a static network (see e.g. Meinshausen and Bühlmann, 2006;Yuan and Lin, 2007;Friedman et al., 2008;Peng et al., 2009), among which the Gaussian Graphical Model (GGM) is particularly useful. ...
... The regularization techniques we use in this paper include both l 1 (Tibshirani, 1996;Tibshirani et al., 2005) and l 2 regularization (Zou and Hastie, 2005), leading to two different algorithms. Our approach of modelling time-varying networks differs from those proposed by Xue et al. (2020) and by Bartlett et al. (2021). Though partial correlations are also employed by Xue et al. (2020) to encode network structure at each time point, they are treated as functions of time via regression splines to capture time-varying network structures. ...
... Due to B-spline bases having local support, sparse networks are obtained by imposing a group LASSO penalty on the coefficient vectors in the regression splines. In contrast to Xue et al. (2020) and our method, Bartlett et al. (2021) propose a Bayesian framework to separate two types of sparsity-sparsity across time and sparsity across variables-when modelling time-varying networks. ...
We model time-varying network data as realizations from multivariate Gaussian distributions with precision matrices that change over time. To facilitate parameter estimation, we require not only that each precision matrix at any given time point be sparse, but also that precision matrices at neighboring time points be similar. We accomplish this with two different algorithms, by generalizing the elastic net and the fused LASSO, respectively. Our main focuses are efficient computational algorithms and convenient degree-of-freedom formulae for choosing tuning parameters. We illustrate our methods with two simulation studies. By applying them to an fMRI data set, we also detect some interesting differences in brain connectivity between healthy individuals and ADHD patients.
... Following earlier work [13], denoting log(gene expression+1) at time t as y t for the target gene i and x t for the p − 1 other genes, the model is defined as: ...
... The first term of , i.e., T t=1 �b t,: � 1 , encourages choosing a smaller number of regulator genes, minimising the number of non-zero entries in b t,: , which is referred to as 'sparsity within time' [13]. The second term of , i.e., T t=2 �b t,: − b t−1,: � 1 , encourages smooth time-variation of b t,: , which is referred to as 'sparsity across time' [13]. ...
... The first term of , i.e., T t=1 �b t,: � 1 , encourages choosing a smaller number of regulator genes, minimising the number of non-zero entries in b t,: , which is referred to as 'sparsity within time' [13]. The second term of , i.e., T t=2 �b t,: − b t−1,: � 1 , encourages smooth time-variation of b t,: , which is referred to as 'sparsity across time' [13]. ...
Background
Network models are well-established as very useful computational-statistical tools in cell biology. However, a genomic network model based only on gene expression data can, by definition, only infer gene co-expression networks. Hence, in order to infer gene regulatory patterns, it is necessary to also include data related to binding of regulatory factors to DNA.
Results
We propose a new dynamic genomic network model, for inferring patterns of genomic regulatory influence in dynamic processes such as development. Our model fuses experiment-specific gene expression data with publicly available DNA-binding data. The method we propose is computationally efficient, and can be applied to genome-wide data with tens of thousands of transcripts. Thus, our method is well suited for use as an exploratory tool for genome-wide data. We apply our method to data from human fetal cortical development, and our findings confirm genomic regulatory patterns which are recognised as being fundamental to neuronal development.
Conclusions
Our method provides a mathematical/computational toolbox which, when coupled with targeted experiments, will reveal and confirm important new functional genomic regulatory processes in mammalian development.
... Following earlier work [13], denoting log(gene expression+1) at time t as y t for the target gene i and x t for the p − 1 other genes, the model is defined as: ...
... The first term of Ψ, i.e., T t=1 b t,: 1 , encourages choosing a smaller number of regulator genes, minimising the number of non-zero entries in b t,: , which is referred to as 'sparsity within time' [13]. The second term of Ψ, i.e., T t=2 b t,: − b t−1,: 1 , encourages smooth time-variation of b t,: , which is referred to as 'sparsity across time' [13]. ...
... The first term of Ψ, i.e., T t=1 b t,: 1 , encourages choosing a smaller number of regulator genes, minimising the number of non-zero entries in b t,: , which is referred to as 'sparsity within time' [13]. The second term of Ψ, i.e., T t=2 b t,: − b t−1,: 1 , encourages smooth time-variation of b t,: , which is referred to as 'sparsity across time' [13]. ...
Background
Network models are well-established as very useful computational-statistical tools in cell biology. However, a genomic network model based only on gene expression data can, by definition, only infer gene co-expression networks. Hence, in order to infer gene regulatory patterns, it is necessary to also include data related to binding of regulatory factors to DNA.
Results
We propose a new dynamic genomic network model, for inferring patterns of genomic regulatory influence in dynamic processes such as development. Our model fuses experiment-specific gene expression data with publicly available DNA-binding data. The method we propose is computationally efficient, and can be applied to genome-wide data with tens of thousands of transcripts. Thus, our method is well suited for use as an exploratory tool for genome-wide data. We apply our method to data from human fetal cortical development, and our findings confirm genomic regulatory patterns which are recognised as being fundamental to neuronal development.
Conclusions
Our method provides a mathematical/computational toolbox which, when coupled with targeted experiments, will reveal and confirm important new functional genomic regulatory processes in mammalian development.
... These successful methods of GRN inference are typically based on the target-gene approach [4][5][6] (Equations 1 and 2), which can find important structure in 'omic networks such as feedback loops [7]. Here, we propose target-gene GRN inference methodology that builds on our previous work to infer networks from epigenomic data such as DNA methylation (DNAme) data [8,9], and using advanced regression methods with scRNA-seq (single-cell RNA-seq) and multimodal data [10][11][12]. The methodology that we propose here infers the GRN using advanced regression methods with scRNA-seq (gene-expression) data following estimation of an epigenomic prior network. ...
... GRNs have been inferred successfully using methods based on random forests (such as GENIE3 / GRNboost2 [5,6] as part of the SCENIC / SCENIC+ pipelines [1][2][3]), as well as alternative approaches based on mutual information [12,[16][17][18][19] and recent extensions [20]. However, a genomic network that has been inferred only from gene expression data (such as scRNAseq data) is more accurately referred to as a gene co-expression network, unless data is included on the interactions between gene-products and DNA [10,21,22]. Hence, in order to infer gene regulatory networks, data must be included in the inference procedure that allows more precise inference about the interaction of the gene product of the regulating gene (i.e., a TF) with the DNA near the regulated gene (cis-regulation) [1,3,11,12]. ...
We show much-improved accuracy of inference of GRN (gene regulatory network) structure resulting from the use of an epigenomic prior network. We also find that DNAme data are very effective for inferring the epigenomic prior network, recapitulating known epigenomic network structure found previously from chromatin accessibility data, and in some cases providing potential TF cis-regulations for eight times as many genes compared to chromatin accessibility data. When our proposed methodology is applied to real datasets from human embryonic development and from women at risk of breast cancer, we find patterns of differential cis-regulation that are in line with expectations under appropriate biological models.
... This assumption is quite restrictive in practice and hardly plausible for many real-world applications, such as gene regulatory networks, social networks, and stocking market, where the underlying data generating mechanisms are often dynamic. On the other hand, dynamic random networks have been extensively studied from the perspective of large random graphs, such as community detection and edge probability estimation for dynamic stochastic block models (DSBMs) [17][18][19][20][21][22][23][24][25][26][27][28][29][30]. Such approaches do not model the sampling distributions of the error (or noise), since the "true" networks are connected with random edges sampled from certain probability models, such as the Erdős-Rényi graphs [31] and random geometric graphs [32]. ...
... To see (29), it suffices to show ...
This paper is concerned with the estimation of time-varying networks for high-dimensional nonstationary time series. Two types of dynamic behaviors are considered: structural breaks (i.e., abrupt change points) and smooth changes. To simultaneously handle these two types of time-varying features, a two-step approach is proposed: multiple change point locations are first identified on the basis of comparing the difference between the localized averages on sample covariance matrices, and then graph supports are recovered on the basis of a kernelized time-varying constrained L 1 -minimization for inverse matrix estimation (CLIME) estimator on each segment. We derive the rates of convergence for estimating the change points and precision matrices under mild moment and dependence conditions. In particular, we show that this two-step approach is consistent in estimating the change points and the piecewise smooth precision matrix function, under a certain high-dimensional scaling limit. The method is applied to the analysis of network structure of the S&P 500 index between 2003 and 2008.
... In the remaining plots, the colours indicate the average log-expression levels of sets of marker genes defined for specific cell-types that are relevant to each data-set. These sets of marker genes have been used previously by us in other studies relating to human neural development [14], human embryonic development [15], and breast cancer initiation [16]. The cell-type validation plots shown in Fig.3 illustrate that the UMAP projection of the data-points in the Laplacian eigenspace (UMAP-LE) make biological sense. ...
We propose a novel way of representing and analysing single-cell genomic count data, by modelling the observed data count matrix as a network adjacency matrix. This perspective enables theory from stochastic networks modelling to be applied in a principled way to this type of data, providing new ways to view and analyse these data, and giving first-principles theoretical justification to established, successful methods. We show the success of this approach in the context of three cell-biological contexts, from the epiblast/epithelial/neural lineage. New technology has made it possible to gather genomic data from single cells at unprecedented scale, and this brings with it new challenges to deal with much higher levels of heterogeneity than expected between individual cells. Novel, tailored, computational-statistical methodology is needed to make the most of these new types of data, involving collaboration between mathematical and biomedical scientists.
... However, all these methods only produce a one-way network inference. [11] proposed a Bayesian model with a prior having decoupled two-way sparsity to infer a dynamic network structure through time, however, the method still depends on a pre-inferred or known ordering of time. Our method extends the Gaussian Copula transformation to enable a two-way network inference, where the structure in both dimensions is to be inferred simultaneously. ...
Classically, statistical datasets have a larger number of data points than features (). The standard model of classical statistics caters for the case where data points are considered conditionally independent given the parameters. However, for or such models are poorly determined. Kalaitzis et al. (2013) introduced the Bigraphical Lasso, an estimator for sparse precision matrices based on the Cartesian product of graphs. Unfortunately, the original Bigraphical Lasso algorithm is not applicable in case of large p and n due to memory requirements. We exploit eigenvalue decomposition of the Cartesian product graph to present a more efficient version of the algorithm which reduces memory requirements from to . Many datasets in different application fields, such as biology, medicine and social science, come with count data, for which Gaussian based models are not applicable. Our multi-way network inference approach can be used for discrete data. Our methodology accounts for the dependencies across both instances and features, reduces the computational complexity for high dimensional data and enables to deal with both discrete and continuous data. Numerical studies on both synthetic and real datasets are presented to showcase the performance of our method.
... This assumption is quite restrictive in practice and hardly plausible for many real-world applications such as gene regulatory networks, social networks, and stocking market, where the underlying data generating mechanisms are often dynamic. On the other hand, dynamic random networks have been extensively studied from the perspective of large random graphs such as community detection and edge probability estimation for dynamic stochastic block models (DSBMs) [38,49,32,15,23,22,30,18,20,47,48,7,4,29]. ...
This paper is concerned with the estimation of time-varying networks for high-dimensional nonstationary time series. Two types of dynamic behaviors are considered: structural breaks (i.e., abrupt change points) and smooth changes. To simultaneously handle these two types of time-varying features, a two-step approach is proposed: multiple change point locations are first identified based on comparing the difference between the localized averages on sample covariance matrices, and then graph supports are recovered based on a kernelized time-varying CLIME estimator on each segment. We derive the rates of convergence for estimating the change points and precision matrices under mild moment and dependence conditions. In particular, we show that this two-step approach is consistent in estimating the change points and the piecewise smooth precision matrix function, under certain high-dimensional scaling limit. The method is applied to the analysis of network structure of the S\&P 500 index between 2003 and 2008.
We model time-varying network data as realizations from multivariate Gaussian distributions with precision matrices that change over time. To facilitate parameter estimation, we require not only that each precision matrix at any given time point be sparse, but also that precision matrices at neighboring time points be similar. We accomplish this with two different algorithms, by generalizing the elastic net and the fused LASSO, respectively. While similar approaches in the literature for modeling time-varying networks have predominantly extended the graphical LASSO of Friedman et al. (Biostatistics 9(3):432-441, 2008), we extend the regression approach of Meinshausen and Bühlmann (Ann Stat 34(3):1436–1462, 2006) and subsequently of Peng et al. (J Am Stat Assoc 104(486):735–746, 2009). This allows us to explicitly focus on and work with the partial correlation coefficients, which are more directly meaningful and interpretable parameters for the biological sciences. We develop efficient algorithms and convenient degree-of-freedom formulae for choosing tuning parameters. The proposed methods are demonstrated through simulation studies. By applying them to an hourly temperature dataset, we detect interesting time-varying connectivity among thirteen Canadian cities. Supplementary materials accompanying this paper appear online.
Recent advances in single-cell omics have transformed characterisation of cell types in challenging-to-study biological contexts. In contexts with limited single-cell samples, such as the early human embryo inference of transcription factor-gene regulatory network (GRN) interactions is especially difficult. Here, we assessed application of different linear or non-linear GRN predictions to single-cell simulated and human embryo transcriptome datasets. We also compared how expression normalisation impacts on GRN predictions, finding that transcripts per million reads outperformed alternative methods. GRN inferences were more reproducible using a non-linear method based on mutual information (MI) applied to single-cell transcriptome datasets refined with chromatin accessibility (CA) (called MICA), compared with alternative network prediction methods tested. MICA captures complex non-monotonic dependencies and feedback loops. Using MICA, we generated the first GRN inferences in early human development. MICA predicted co-localisation of the AP-1 transcription factor subunit proto-oncogene JUND and the TFAP2C transcription factor AP-2γ in early human embryos. Overall, our comparative analysis of GRN prediction methods defines a pipeline that can be applied to single-cell multi-omics datasets in especially challenging contexts to infer interactions between transcription factor expression and target gene regulation.
Recent advances in single cell -omics have been transformative in the characterisation of challenging to study biological contexts, including when the source material is precious, such as the early human embryo. Single cell datasets are technically challenging to infer transcription factor-gene regulatory interactions, due to low read- depth leading to zero inflated data. Here we have systematically assessed the application of four different machine learning linear or non-linear gene regulatory network prediction strategies to single cell simulated and human embryo transcriptome datasets. We have also compared how the method of gene expression normalisation impacts on regulatory network predictions. Integrating chromatin accessibility together with transcript expression datasets improved the reproducibility of the predicted gene regulatory networks. We found that the application of a non-linear network prediction method based on mutual information (MI) to single cell transcriptome datasets refined with chromatin accessibility (CA) (called MICA), exhibited higher reproducibility compared to the alternative network interference methods tested. Moreover, MICA was used to make predictions about GRNs in the preimplantation human embryo, which were supported by previous findings in other developmental and stem cell contexts. Based on the gene regulatory networks predicted by MICA, we discovered co- localisation of the AP-1 transcription factor subunit proto-oncogene JUND and the TFAP2C transcription factor AP-2g in human preimplantation embryos. Overall, our comparative analysis of gene regulatory network prediction methods defines a pipeline that can be applied to single-cell multi-omics datasets to infer interactions between transcription factor expression and target gene regulation, which can then be functionally tested.
In the developing human neocortex, progenitor cells generate diverse cell types prenatally. Progenitor cells and newborn neurons respond to signaling cues, including neurotransmitters. While single-cell RNA sequencing has revealed cellular diversity, physiological heterogeneity has yet to be mapped onto these developing and diverse cell types. By combining measurements of intracellular Ca ²⁺ elevations in response to neurotransmitter receptor agonists and RNA sequencing of the same single cells, we show that Ca ²⁺ responses are cell-type-specific and change dynamically with lineage progression. Physiological response properties predict molecular cell identity and additionally reveal diversity not captured by single-cell transcriptomics. We find that the serotonin receptor HTR2A selectively activates radial glia cells in the developing human, but not mouse, neocortex, and inhibiting HTR2A receptors in human radial glia disrupts the radial glial scaffold. We show highly specific neurotransmitter signaling during neurogenesis in the developing human neocortex and highlight evolutionarily divergent mechanisms of physiological signaling.
About half of mammalian miRNA genes lie within introns of protein-coding genes, yet little is known about functional interactions between miRNAs and their host genes. The intronic miRNA miR-128 regulates neuronal excitability and dendritic morphology of principal neurons during mouse cerebral cortex development. Its conserved host genes, R3hdm1 and Arpp21, are predicted RNA-binding proteins. Here we use iCLIP to characterize ARPP21 recognition of uridine-rich sequences with high specificity for 3'UTRs. ARPP21 antagonizes miR-128 activity by co-regulating a subset of miR-128 target mRNAs enriched for neurodevelopmental functions. Protein-protein interaction data and functional assays suggest that ARPP21 acts as a positive post-transcriptional regulator by interacting with the translation initiation complex eIF4F. This molecular antagonism is reflected in inverse activities during dendritogenesis: miR-128 overexpression or knockdown of ARPP21 reduces dendritic complexity; ectopic ARPP21 leads to an increase. Thus, we describe a unique example of convergent function by two products of a single gene.
Although single-cell RNA-seq is revolutionizing biology, data interpretation remains a challenge. We present SCENIC for the simultaneous reconstruction of gene regulatory networks and identification of cell states. We apply SCENIC to a compendium of single-cell data from tumors and brain, and demonstrate that the genomic regulatory code can be exploited to guide the identification of transcription factors and cell states. SCENIC provides critical biological insights into the mechanisms driving cellular heterogeneity.
Abnormal development of ventral midbrain (VM) dopaminergic (DA) pathways, essential for motor and cognitive function, may underpin a number of neurological disorders and thereby highlight the importance of understanding the birth and connectivity of the associated neurons. While a number of regulators of VM DA neurogenesis are known, processes involved in later developmental events, including terminal differentiation and axon morphogenesis, are less well understood. Recent transcriptional analysis studies of the developing VM identified genes expressed during these stages, including the cell adhesion molecule with homology to L1 (Chl1). Here, we map the temporal and spatial expression of CHL1 and assess functional roles of substrate-bound and soluble-forms of the protein during VM DA development. Results showed early CHL1 in the VM, corresponding with roles in DA progenitor migration and differentiation. Subsequently, we demonstrated roles for CHL1 in both axonal extension and repulsion, selectively of DA neurons, suggestive of a role in guidance towards forebrain targets and away from hindbrain nuclei. In part, CHL1 mediates these roles through homophilic CHL1-CHL1 interactions. Collectively, these findings enhance our knowledge of VM DA pathways development, and may provide new insights into understanding DA developmental conditions such as autism spectrum disorders.
Motivation: The analysis of RNA-Seq data from individual differentiating cells enables us to reconstruct the differentiation process and the degree of differentiation (in pseudo-time) of each cell. Such analyses can reveal detailed expression dynamics and functional relationships for differentiation. To further elucidate differentiation processes, more insight into gene regulatory networks is required. The pseudo-time can be regarded as time information and, therefore, single-cell RNA-Seq data are time-course data with high time resolution. Although time-course data are useful for inferring networks, conventional inference algorithms for such data suffer from high time complexity when the number of samples and genes is large. Therefore, a novel algorithm is necessary to infer networks from single-cell RNA-Seq during differentiation.
Results: In this study, we developed the novel and efficient algorithm SCODE to infer regulatory networks, based on ordinary differential equations. We applied SCODE to three single-cell RNA-Seq datasets and confirmed that SCODE can reconstruct observed expression dynamics. We evaluated SCODE by comparing its inferred networks with use of a DNaseI-footprint based network. The performance of SCODE was best for two of the datasets and nearly best for the remaining dataset. We also compared the runtimes and showed that the runtimes for SCODE are significantly shorter than for alternatives. Thus, our algorithm provides a promising approach for further single-cell differentiation analyses.
Availability: The R source code of SCODE is available at https://github.com/hmatsu1226/SCODE.
Contact:hirotaka.matsumoto@riken.jp
Supplementary information
:
Supplementary data
are available at Bioinformatics online.
Our focus is on realistically modeling and forecasting dynamic
networks of face-to-face contacts among individuals. Important as-
pects of such data that lead to problems with current methods in-
clude the tendency of the contacts to move between periods of slow
and rapid changes, and the dynamic heterogeneity in the actors' con-
nectivity behaviors. Motivated by this application, we develop a novel
method for Locally Adaptive DYnamic (LADY) network inference.
The proposed model relies on a dynamic latent space representation
in which each actor's position evolves in time via stochastic di�eren-
tial equations. Using a state space representation for these stochastic
processes and P�olya-gamma data augmentation, we develop an e�-
cient MCMC algorithm for posterior inference along with tractable
procedures for online updating and forecasting of future networks.We
evaluate performance in simulation studies, and consider an applica-
tion to face-to-face contacts among individuals in a primary school.
In the present paper we consider a dynamic stochastic network model. The objective is estimation of the tensor of connection probabilities when it is generated by a Dynamic Stochastic Block Model (DSBM) or a dynamic graphon. In particular, in the context of DSBM, we derive penalized least squares estimator of and show that satisfies an oracle inequality and also attains minimax lower bounds for the risk. We extend those results to estimation of when it is generated by a dynamic graphon function. The estimators constructed in the paper are adaptive to the unknown number of blocks in the context of DSBM or of the smoothness of the graphon function. The technique relies on the vectorization of the model and leads to to much simpler mathematical arguments than the ones used previously in the stationary set up. In addition, all our results are non-asymptotic and allow a variety of extensions.
We study a broad class of stochastic process models for dynamic networks that satisfy the minimal regularity conditions of (i) exchangeability and (ii) cadlag sample paths. Our main theorems characterize these processes through their induced behavior in the space of graph limits. Under the assumption of time-homogeneous Markovian dependence, we classify the dis-continuities of these processes into three types, prove bounded variation of the sample paths in graph limit space and express the process as a mixture of time-inhomogeneous, exchangeable Markov processes with cadlag sample paths.
Single-cell RNA-seq data allows insight into normal cellular function and various disease states through molecular characterization of gene expression on the single cell level. Dimensionality reduction of such high-dimensional data sets is essential for visualization and analysis, but single-cell RNA-seq data are challenging for classical dimensionality-reduction methods because of the prevalence of dropout events, which lead to zero-inflated data. Here, we develop a dimensionality-reduction method, (Z)ero (I)nflated (F)actor (A)nalysis (ZIFA), which explicitly models the dropout characteristics, and show that it improves modeling accuracy on simulated and biological data sets.
Electronic supplementary material
The online version of this article (doi:10.1186/s13059-015-0805-z) contains supplementary material, which is available to authorized users.
Network studies of large-scale brain connectivity have demonstrated that highly connected areas, or “hubs,” are a key feature of human functional and structural brain organization. We use resting-state functional MRI data and connectivity clustering to identify multi-network hubs and show that while hubs can belong to multiple networks their degree of integration into these different networks varies dynamically over time. The extent of the network variation was related to the connectedness of the hub. In addition, we found that these network dynamics were inversely related to positive self-generated thoughts reported by individuals and were further decreased with older age. Moreover, the left caudate varied its degree of participation between a default mode subnetwork and a limbic network. This variation was predictive of individual differences in the reports of past-related thoughts. These results support an association between ongoing thought processes and network dynamics and offer a new approach to investigate the brain dynamics underlying mental experience.
Defining the transcriptional dynamics of a temporal process such as cell differentiation is challenging owing to the high variability in gene expression between individual cells. Time-series gene expression analyses of bulk cells have difficulty distinguishing early and late phases of a transcriptional cascade or identifying rare subpopulations of cells, and single-cell proteomic methods rely on a priori knowledge of key distinguishing markers. Here we describe Monocle, an unsupervised algorithm that increases the temporal resolution of transcriptome dynamics using single-cell RNA-Seq data collected at multiple time points. Applied to the differentiation of primary human myoblasts, Monocle revealed switch-like changes in expression of key regulatory factors, sequential waves of gene regulation, and expression of regulators that were not known to act in differentiation. We validated some of these predicted regulators in a loss-of function screen. Monocle can in principle be used to recover single-cell gene expression kinetics from a wide array of cellular processes, including differentiation, proliferation and oncogenic transformation.
Significant efforts have gone into the development of statistical models for
analyzing data in the form of networks, such as social networks. Most existing
work has focused on modeling static networks, which represent either a single
time snapshot or an aggregate view over time. There has been recent interest in
statistical modeling of dynamic networks, which are observed at multiple points
in time and offer a richer representation of many complex phenomena. In this
paper, we propose a state-space model for dynamic networks that extends the
well-known stochastic blockmodel for static networks to the dynamic setting. We
then propose a procedure to fit the model using a modification of the extended
Kalman filter augmented with a local search. We apply the procedure to analyze
a dynamic social network of email communication.
Penalized regression methods for simultaneous variable selection and coefficient estimation, especially those based on the lasso of Tibshirani (1996), have received a great deal of attention in recent years, mostly through frequen-tist models. Properties such as consistency have been studied, and are achieved by different lasso variations. Here we look at a fully Bayesian formulation of the problem, which is flexible enough to encompass most versions of the lasso that have been previously considered. The advantages of the hierarchical Bayesian for-mulations are many. In addition to the usual ease-of-interpretation of hierarchical models, the Bayesian formulation produces valid standard errors (which can be problematic for the frequentist lasso), and is based on a geometrically ergodic Markov chain. We compare the performance of the Bayesian lassos to their fre-quentist counterparts using simulations, data sets that previous lasso papers have used, and a difficult modeling problem for predicting the collapse of governments around the world. In terms of prediction mean squared error, the Bayesian lasso performance is similar to and, in some cases, better than, the frequentist lasso.
We propose a non-parametric link prediction algorithm for a sequence of graph
snapshots over time. The model predicts links based on the features of its
endpoints, as well as those of the local neighborhood around the endpoints.
This allows for different types of neighborhoods in a graph, each with its own
dynamics (e.g, growing or shrinking communities). We prove the consistency of
our estimator, and give a fast implementation based on locality-sensitive
hashing. Experiments with simulated as well as five real-world dynamic graphs
show that we outperform the state of the art, especially when sharp
fluctuations or non-linearities are present.
Dorsal spinal cord neurons receive and integrate somatosensory information provided by neurons located in dorsal root ganglia. Here we demonstrate that dorsal spinal neurons require the Krüppel-C(2)H(2) zinc-finger transcription factor Bcl11a for terminal differentiation and morphogenesis. The disrupted differentiation of dorsal spinal neurons observed in Bcl11a mutant mice interferes with their correct innervation by cutaneous sensory neurons. To understand the mechanism underlying the innervation deficit, we characterized changes in gene expression in the dorsal horn of Bcl11a mutants and identified dysregulated expression of the gene encoding secreted frizzled-related protein 3 (sFRP3, or Frzb). Frzb mutant mice show a deficit in the innervation of the spinal cord, suggesting that the dysregulated expression of Frzb can account in part for the phenotype of Bcl11a mutants. Thus, our genetic analysis of Bcl11a reveals essential functions of this transcription factor in neuronal morphogenesis and sensory wiring of the dorsal spinal cord and identifies Frzb, a component of the Wnt pathway, as a downstream acting molecule involved in this process.
In this report, we introduce the concept of co-community structure in
time-varying networks. We propose a novel optimization algorithm to rapidly
detect co-community structure in these networks. Both theoretical and numerical
results show that the proposed method not only can resolve detailed
co-communities, but also can effectively identify the dynamical phenomena in
these networks.
The ability to analyze multiple single-cell parameters is critical for understanding cellular heterogeneity. Despite recent advances in measurement technology, methods for analyzing high-dimensional single-cell data are often subjective, labor intensive and require prior knowledge of the biological system. To objectively uncover cellular heterogeneity from single-cell measurements, we present a versatile computational approach, spanning-tree progression analysis of density-normalized events (SPADE). We applied SPADE to flow cytometry data of mouse bone marrow and to mass cytometry data of human bone marrow. In both cases, SPADE organized cells in a hierarchy of related phenotypes that partially recapitulated well-described patterns of hematopoiesis. We demonstrate that SPADE is robust to measurement noise and to the choice of cellular markers. SPADE facilitates the analysis of cellular heterogeneity, the identification of cell types and comparison of functional markers in response to perturbations.
Deciphering the complex mechanisms by which regulatory networks control gene expression remains a major challenge. While some studies infer regulation from dependencies between the expression levels of putative regulators and their targets, others focus on measured physical interactions.
Here, we present Physical Module Networks, a unified framework that combines a Bayesian model describing modules of co-expressed genes and their shared regulation programs, and a physical interaction graph, describing the protein-protein interactions and protein-DNA binding events that coherently underlie this regulation. Using synthetic data, we demonstrate that a Physical Module Network model has similar recall and improved precision compared to a simple Module Network, as it omits many false positive regulators. Finally, we show the power of Physical Module Networks to reconstruct meaningful regulatory pathways in the genetically perturbed yeast and during the yeast cell cycle, as well as during the response of primary epithelial human cells to infection with H1N1 influenza.
The PMN software is available, free for academic use at http://www.compbio.cs.huji.ac.il/PMN/.
aregev@broad.mit.edu; nirf@cs.huji.ac.il.
Biological networks are highly dynamic in response to environmental and physiological cues. This variability is in contrast to conventional analyses of biological networks, which have overwhelmingly employed static graph models which stay constant over time to describe biological systems and their underlying molecular interactions.
To overcome these limitations, we propose here a new statistical modelling framework, the ARTIVA formalism (Auto Regressive TIme VArying models), and an associated inferential procedure that allows us to learn temporally varying gene-regulation networks from biological time-course expression data. ARTIVA simultaneously infers the topology of a regulatory network and how it changes over time. It allows us to recover the chronology of regulatory associations for individual genes involved in a specific biological process (development, stress response, etc.).
We demonstrate that the ARTIVA approach generates detailed insights into the function and dynamics of complex biological systems and exploits efficiently time-course data in systems biology. In particular, two biological scenarios are analyzed: the developmental stages of Drosophila melanogaster and the response of Saccharomyces cerevisiae to benomyl poisoning.
ARTIVA does recover essential temporal dependencies in biological systems from transcriptional data, and provide a natural starting point to learn and investigate their dynamics in greater detail.
Graphical models are frequently used to explore networks, such as genetic networks, among a set of variables. This is usually carried out via exploring the sparsity of the precision matrix of the variables under consideration. Penalized likelihood methods are often used in such explorations. Yet, positive-definiteness constraints of precision matrices make the optimization problem challenging. We introduce nonconcave penalties and the adaptive LASSO penalty to attenuate the bias problem in the network estimation. Through the local linear approximation to the nonconcave penalty functions, the problem of precision matrix estimation is recast as a sequence of penalized likelihood problems with a weighted penalty and solved using the efficient algorithm of Friedman et al. [Biostatistics 9 (2008) 432--441]. Our estimation schemes are applied to two real datasets. Simulation experiments and asymptotic theory are used to justify our proposed methods. Comment: Published in at http://dx.doi.org/10.1214/08-AOAS215 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org)
The balance between self-renewal and differentiation of neural progenitor cells is an absolute requirement for the correct formation of the nervous system. Much is known about both the pathways involved in progenitor cell self-renewal, such as Notch signaling, and the expression of genes that initiate progenitor differentiation. However, whether these fundamental processes are mechanistically linked, and specifically how repression of progenitor self-renewal pathways occurs, is poorly understood. Nuclear factor I A (Nfia), a gene known to regulate spinal cord and neocortical development, has recently been implicated as acting downstream of Notch to initiate the expression of astrocyte-specific genes within the cortex. Here we demonstrate that, in addition to activating the expression of astrocyte-specific genes, Nfia also downregulates the activity of the Notch signaling pathway via repression of the key Notch effector Hes1. These data provide a significant conceptual advance in our understanding of neural progenitor differentiation, revealing that a single transcription factor can control both the activation of differentiation genes and the repression of the self-renewal genes, thereby acting as a pivotal regulator of the balance between progenitor and differentiated cell states.
Neurons have the remarkable ability to polarize even in symmetrical in vitro environments. Although recent studies have shown that asymmetric intracellular signals can induce neuronal polarization, it remains unclear how these polarized signals are organized without asymmetric cues. We describe a novel protein, named shootin1, that became up-regulated during polarization of hippocampal neurons and began fluctuating accumulation among multiple neurites. Eventually, shootin1 accumulated asymmetrically in a single neurite, which led to axon induction for polarization. Disturbing the asymmetric organization of shootin1 by excess shootin1 disrupted polarization, whereas repressing shootin1 expression inhibited polarization. Overexpression and RNA interference data suggest that shootin1 is required for spatially localized phosphoinositide-3-kinase activity. Shootin1 was transported anterogradely to the growth cones and diffused back to the soma; inhibiting this transport prevented its asymmetric accumulation in neurons. We propose that shootin1 is involved in the generation of internal asymmetric signals required for neuronal polarization.
We consider the problem of estimating sparse graphs by a lasso penalty applied to the inverse covariance matrix. Using a coordinate
descent procedure for the lasso, we develop a simple algorithm—the graphical lasso—that is remarkably fast: It solves a 1000-node problem (∼500000 parameters) in at most a minute and is 30–4000 times faster
than competing methods. It also provides a conceptual link between the exact problem and the approximation suggested by Meinshausen
and Bühlmann (2006). We illustrate the method on some cell-signaling data from proteomics.
Emerging evidence suggests that myocyte enhancer factor 2 (MEF2) transcription factors act as effectors of neurogenesis in the brain, with MEF2C the predominant isoform in developing cerebrocortex. Here, we show that conditional knockout of Mef2c in nestin-expressing neural stem/progenitor cells (NSCs) impaired neuronal differentiation in vivo, resulting in aberrant compaction and smaller somal size. NSC proliferation and survival were not affected. Conditional null mice surviving to adulthood manifested more immature electrophysiological network properties and severe behavioral deficits reminiscent of Rett syndrome, an autism-related disorder. Our data support a crucial role for MEF2C in programming early neuronal differentiation and proper distribution within the layers of the neocortex.
• neurogenesis
• synaptogenesis
• autism
• Rett syndrome
In this paper we propose a Bayesian nonparametric approach to modelling sparse time-varying networks. A positive parameter is associated to each node of a network, which models the sociability of that node. Sociabilities are assumed to evolve over time, and are modelled via a dynamic point process model. The model is able to capture long term evolution of the sociabilities. Moreover, it yields sparse graphs, where the number of edges grows subquadratically with the number of nodes. The evolution of the sociabilities is described by a tractable time-varying generalised gamma process. We provide some theoretical insights into the model and apply it to three datasets: a simulated network, a network of hyperlinks between communities on Reddit, and a network of co-occurences of words in Reuters news articles after the September attacks.KeywordsBayesian nonparametricsPoisson random measuresNetworksRandom graphsSparsityPoint processes
Building a brain
The human brain is built in an inside-out manner as a series of layers. Although progenitor cells spin off new neurons in a seemingly organized fashion, the devil is in the details. Nowakowski et al. analyzed the transcriptomes of single cells from the developing brain to elucidate the hidden complexity of brain construction. For each cell, its position within the brain matters, as well as what type of neuron is being made at what point during overall development. These individual expression patterns result in organized diversity in the brain's cortex.
Science , this issue p. 1318
The fused lasso penalizes a loss function by the norm for both the regression coefficients and their successive differences to encourage sparsity of both. In this paper, we propose a Bayesian generalized fused lasso modeling based on a normal-exponential-gamma (NEG) prior distribution. The NEG prior is assumed into the difference of successive regression coefficients. The proposed method enables us to construct a more versatile sparse model than the ordinary fused lasso by using a flexible regularization term. We also propose a sparse fused algorithm to produce exact sparse solutions. Simulation studies and real data analyses show that the proposed method has superior performance to the ordinary fused lasso.
Cell fate specification relies on the action of critical transcription factors that become available at distinct stages of embryonic development. One such factor is NeuroD1, which is essential for eliciting the neuronal development program and possesses the ability to reprogram other cell types into neurons. Given this capacity, it is important to understand its targets and the mechanism underlying neuronal specification. Here, we show that NeuroD1 directly binds regulatory elements of neuronal genes that are developmentally silenced by epigenetic mechanisms. This targeting is sufficient to initiate events that confer transcriptional competence, including reprogramming of transcription factor landscape, conversion of heterochromatin to euchromatin, and increased chromatin accessibility, indicating potential pioneer factor ability of NeuroD1. The transcriptional induction of neuronal fate genes is maintained via epigenetic memory despite a transient NeuroD1 induction during neurogenesis. NeuroD1 also induces genes involved in the epithelial-to-mesenchymal transition, thereby promoting neuronal migration. Our study not only reveals the NeuroD1-dependent gene regulatory program driving neurogenesis but also increases our understanding of how cell fate specification during development involves a concerted action of transcription factors and epigenetic mechanisms.
Statistical node clustering in discrete time dynamic networks is an emerging
field that raises many challenges. Here, we explore statistical properties and
deterministic inference in a model that combines a stochastic block model (SBM)
for its static part with independent Markov chains for the evolution of the
nodes groups through time. We model binary data as well as weighted dynamic
random graphs (with discrete or continuous edges values). Our approach
particularly focuses on the control for label switching issues across the
different time steps. We study identifiability of the model parameters, propose
an inference procedure based on a variational expectation maximization
algorithm as well as a model selection criterion to select for the number of
groups. We carefully discuss our initialization strategy which plays an
important role in the method and compare our procedure with existing ones on
synthetic datasets. We also illustrate our approach on a real data set of
encounters among high school students and provide an implementation of the
method into a R package called dynsbm.
Networks provide a powerful mathematical framework for analyzing the
structure and dynamics of complex systems (1-3). The study of group behavior
has deep roots in the social science literature (4,5) and community detection
is a central part of modern network science. Network communities have been
found to be highly overlapping and organized in a hierarchical structure (6-9).
Recent technological advances have provided a toolset for measuring the
detailed social dynamics at scale (10,11). In spite of great progress, a
quantitative description of the complex temporal behavior of social groups-with
dynamics spanning from minute-by-minute changes to patterns expressed on the
timescale of years-is still absent. Here we uncover a class of fundamental
structures embedded within highly dynamic social networks. On the shortest
time-scale, we find that social gatherings are fluid, with members coming and
going, but organized via a stable core of individuals. We show that cores
represent social contexts (9), with recurring meetings across weeks and months,
each with varying degrees of regularity. In this sense, cores provide a
vocabulary which quantifies the patterns of social life. The simplification is
so powerful that participation in social contexts is predictable with high
precision. Our results offer new perspectives for modeling and understanding of
processes in social systems, with applications including epidemiology, social
contagion, urban planning, digital economy, and quantitative sociology
This paper presents necessary and sufficient conditions under which a random variable X may be generated as the ratio Z/V where Z and V are independent and Z has a standard normal distribution. This representation is useful in Monte Carlo calculations. It is established that when 1/2V2 is exponential, X is double exponential; and that when 1/2V has the asymptotic distribution of the Kolmogorov distance statistic, X is logistic.
Single-cell data provide a means to dissect the composition of complex tissues and specialized cellular environments. However, the analysis of such measurements is complicated by high levels of technical noise and intrinsic biological variability. We describe a probabilistic model of expression-magnitude distortions typical of single-cell RNA-sequencing measurements, which enables detection of differential expression signatures and identification of subpopulations of cells in a way that is more tolerant of noise.
Developmental fate decisions are dictated by master transcription factors (TFs) that interact with cis-regulatory elements to direct transcriptional programs. Certain malignant tumors may also depend on cellular hierarchies reminiscent of normal development but superimposed on underlying genetic aberrations. In glioblastoma (GBM), a subset of stem-like tumor-propagating cells (TPCs) appears to drive tumor progression and underlie therapeutic resistance yet remain poorly understood. Here, we identify a core set of neurodevelopmental TFs (POU3F2, SOX2, SALL2, and OLIG2) essential for GBM propagation. These TFs coordinately bind and activate TPC-specific regulatory elements and are sufficient to fully reprogram differentiated GBM cells to “induced” TPCs, recapitulating the epigenetic landscape and phenotype of native TPCs. We reconstruct a network model that highlights critical interactions and identifies candidate therapeutic targets for eliminating TPCs. Our study establishes the epigenetic basis of a developmental hierarchy in GBM, provides detailed insight into underlying gene regulatory programs, and suggests attendant therapeutic strategies.
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Noncoding RNAs (ncRNAs) accomplish a remarkable variety of biological functions. They regulate gene expression at the levels of transcription, RNA processing, and translation. They protect genomes from foreign nucleic acids. They can guide DNA synthesis or genome rearrangement. For ribozymes and riboswitches, the RNA structure itself provides the biological function, but most ncRNAs operate as RNA-protein complexes, including ribosomes, snRNPs, snoRNPs, telomerase, microRNAs, and long ncRNAs. Many, though not all, ncRNAs exploit the power of base pairing to selectively bind and act on other nucleic acids. Here, we describe the pathway of ncRNA research, where every established "rule" seems destined to be overturned.
We study full Bayesian procedures for high-dimensional linear regression
under sparsity constraints. The prior is a mixture of point masses at zero and
continuous distributions. Under compatibility conditions on the design matrix
the posterior distribution is shown to contract at the optimal rate for
recovery of the unknown sparse vector, and to give optimal prediction of the
response vector. It is also shown to select the correct sparse model, or at
least the coefficients that are significantly different from zero. The
asymptotic shape of the posterior distribution is characterized, and employed
to the construction and study of credible sets for uncertainty quantification.
This paper proposes a new approach to sparsity, called the horseshoe estimator, which arises from a prior based on multivariate-normal
scale mixtures. We describe the estimator’s advantages over existing approaches, including its robustness, adaptivity to different
sparsity patterns and analytical tractability. We prove two theorems: one that characterizes the horseshoe estimator’s tail
robustness and the other that demonstrates a super-efficient rate of convergence to the correct estimate of the sampling density
in sparse situations. Finally, using both real and simulated data, we show that the horseshoe estimator corresponds quite
closely to the answers obtained by Bayesian model averaging under a point-mass mixture prior.
[1st paragraph] At first sight, Bayesian inference with Markov Chain Monte Carlo (MCMC) appears to be straightforward. The user defines a full probability model, perhaps using one of the programs discussed in this issue; an underlying sampling engine takes the model definition and returns a sequence of dependent samples from the posterior distribution of the model parameters, given the supplied data. The user can derive any summary of the posterior distribution from this sample. For example, to calculate a 95% credible interval for a parameter α, it suffices to take 1000 MCMC iterations of α and sort them so that α1<α2<...<α1000. The credible interval estimate is then (α25, α975). However, there is a price to be paid for this simplicity. Unlike most numerical methods used in statistical inference, MCMC does not give a clear indication of whether it has converged. The underlying Markov chain theory only guarantees that the distribution of the output will converge to the posterior in the limit as the number of iterations increases to infinity. The user is generally ignorant about how quickly convergence occurs, and therefore has to fall back on post hoc testing of the sampled output. By convention, the sample is divided into two parts: a “burn in” period during which all samples are discarded, and the remainder of the run in which the chain is considered to have converged sufficiently close to the limiting distribution to be used. Two questions then arise: 1. How long should the burn in period be? 2. How many samples are required to accurately estimate posterior quantities of interest? The coda package for R contains a set of functions designed to help the user answer these questions. Some of these convergence diagnostics are simple graphical ways of summarizing the data. Others are formal statistical tests.
We present a path algorithm for the generalized lasso problem. This problem
penalizes the norm of a matrix D times the coefficient vector, and has
a wide range of applications, dictated by the choice of D. Our algorithm is
based on solving the dual of the generalized lasso, which greatly facilitates
computation of the path. For D=I (the usual lasso), we draw a connection
between our approach and the well-known LARS algorithm. For an arbitrary D, we
derive an unbiased estimate of the degrees of freedom of the generalized lasso
fit. This estimate turns out to be quite intuitive in many applications.
The era of genome sequencing has produced long lists of the molecular parts from which cellular machines are constructed. A fundamental goal in systems biology is to understand how cellular behavior emerges from the interaction in time and space of genetically encoded molecular parts, as well as nongenetically encoded small molecules. Networks provide a natural framework for the organization and quantitative representation of all the available data about molecular interactions. The structural and dynamic properties of molecular networks have been the subject of intense research. Despite major advances, bridging network structure to dynamics-and therefore to behavior-remains challenging. A key concept of modern engineering that recurs in the functional analysis of biological networks is modularity. Most approaches to molecular network analysis rely to some extent on the assumption that molecular networks are modular-that is, they are separable and can be studied to some degree in isolation. We describe recent advances in the analysis of modularity in biological networks, focusing on the increasing realization that a dynamic perspective is essential to grouping molecules into modules and determining their collective function.
Stochastic networks are a plausible representation of the relational information among entities in dynamic systems such as living cells or social communities. While there is a rich literature in estimating a static or temporally invariant network from observation data, little has been done toward estimating time-varying networks from time series of entity attributes. In this paper we present two new machine learning methods for estimating time-varying networks, which both build on a temporally smoothed -regularized logistic regression formalism that can be cast as a standard convex-optimization problem and solved efficiently using generic solvers scalable to large networks. We report promising results on recovering simulated time-varying networks. For real data sets, we reverse engineer the latent sequence of temporally rewiring political networks between Senators from the US Senate voting records and the latent evolving regulatory networks underlying 588 genes across the life cycle of Drosophila melanogaster from the microarray time course. Comment: Published in at http://dx.doi.org/10.1214/09-AOAS308 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org)
Satb2 is a DNA-binding protein that regulates chromatin organization and gene expression. In the developing brain, Satb2 is expressed in cortical neurons that extend axons across the corpus callosum. To assess the role of Satb2 in neurons, we analyzed mice in which the Satb2 locus was disrupted by insertion of a LacZ gene. In mutant mice, beta-galactosidase-labeled axons are absent from the corpus callosum and instead descend along the corticospinal tract. Satb2 mutant neurons acquire expression of Ctip2, a transcription factor that is necessary and sufficient for the extension of subcortical projections by cortical neurons. Conversely, ectopic expression of Satb2 in neural stem cells markedly decreases Ctip2 expression. Finally, we find that Satb2 binds directly to regulatory regions of Ctip2 and induces changes in chromatin structure. These data suggest that Satb2 functions as a repressor of Ctip2 and regulatory determinant of corticocortical connections in the developing cerebral cortex.
The lasso penalizes a least squares regression by the sum of the absolute values ("L"1-norm) of the coefficients. The form of this penalty encourages sparse solutions (with many coefficients equal to 0). We propose the 'fused lasso', a generalization that is designed for problems with features that can be ordered in some meaningful way. The fused lasso penalizes the "L"1-norm of both the coefficients and their successive differences. Thus it encourages sparsity of the coefficients and also sparsity of their differences-i.e. local constancy of the coefficient profile. The fused lasso is especially useful when the number of features "p" is much greater than "N", the sample size. The technique is also extended to the 'hinge' loss function that underlies the support vector classifier. We illustrate the methods on examples from protein mass spectroscopy and gene expression data. Copyright 2005 Royal Statistical Society.
The bigraphical lasso
- A Kalaitzis
- J Lafferty
- N Lawrence
- S Zhou
KALAITZIS, A., LAFFERTY, J., LAWRENCE, N. and ZHOU, S. (2013). The bigraphical lasso. In Proceedings of
the 30th International Conference on Machine Learning (ICML-13) 1229-1237.
- Z Linqing
- J Guohua
- L Haoming
- T Xuelei
- Q Jianbing
- T Meiling
LINQING, Z., GUOHUA, J., HAOMING, L., XUELEI, T., JIANBING, Q. and MEILING, T. (2015). RUNX1T1
regulates the neuronal differentiation of radial glial cells from the rat hippocampus. Stem Cells Translational
Medicine 4 110-116.
A dynamic Erdős-Rényi graph model
- S Rosengren
- P Trapman
ROSENGREN, S. and TRAPMAN, P. (2019). A dynamic Erdős-Rényi graph model. Markov Process. Related
Fields 25 275-300. MR3967544