Granger causal inference on DAGs identifies genomic loci regulating transcription

1 INTRODUCTION . Understanding the structure of a multivariate dynamical system often boils down to deciphering the causal relationships between its variables . Since inferring true causality is often infeasible , requiring additional experiments or independent mechanistic insight , statistical analysis of observational data to identify predictive relationships between the system ’ s variables can be very valuable . The framework of Granger causality does exactly that : in a dataset where observations are temporally ordered , a time-dependent variable x ( with value xt at time t ) is said to “ Granger cause ” a second variable y if the history of x at time t ( i.e. , x1 , . . . , xt−1 ) is significantly predictive of y at time t even after accounting for y ’ s own history ( Granger , 1969 ; Shojaie & Fox , 2021 ) . Originating in econometrics , Granger causality has been a powerful tool in many domains including biology , finance , and social sciences ( Fujita et al. , 2007 ; Yao et al. , 2015 ; Benhmad , 2012 ; Rasheed & Tahir , 2012 ) . The prerequisite for applying Granger causality , however , is that there be a clear sequential ordering of observations , i.e. , the data must conform to a total ordering along time . Often , only a partial ordering of the observations is possible . For instance , cell differentiation trajectories may have branches . In text-mining , the citation graph of publications captures the flow of ideas and knowledge . In such cases , the dynamics of the system are more suitably represented as a directed acyclic graph ( DAG ) corresponding to the partial ordering , with nodes of the DAG representing dynamical states and its edges indicating the flow of information between states . The key conceptual advance of this work is extending the applicability of Granger causality to systems with partially ordered observations ( Figure 1a ) . On a DAG where each node takes on multivariate values , we define a variable x as Granger causing another variable y if the latter ’ s value yv at the DAG node v can be significantly predicted by the values of x at the ancestor nodes of v , even after accounting for yv ’ s own ancestral history . We introduce GrID-Net ( Granger Inference on DAGs ) , a graph neural network ( GNN ) framework for Granger causal inference in dynamical systems that can be represented as DAGs ( Figure 1b ) . We modify the standard GNN architecture to enable lagged message-passing on the DAG , enabling us to accumulate past information for a variable according to the DAG ’ s structure . This , combined with a mathematical formulation of Granger causality more amenable to neural networks , allows GrID-Net to recover nonlinear long-range Granger causal dependencies between the variables . While our problem formulation is fully general and GrID-Net is broadly applicable , our motivating application is inferring noncoding genomic loci that influence the expression of a specific gene . We apply GrID-Net to single-cell multimodal studies that assay chromatin accessibility ( ATAC-seq ) and gene expression ( RNA-seq ) in the same cell , seeking to identify Granger causal relationships between the accessibility of individual chromatin regions ( ATAC-seq “ peaks ” ) and the expression of specific genes . The system dynamics are represented by a DAG where each node corresponds to a cell . Edges connect cells in similar dynamic states ( i.e . similar ATAC-seq and RNA-seq profiles ) , with edge directions estimated by a pseudotime analysis ( Haghverdi et al. , 2016 ) ( Figure 1c ) . Our work addresses a pressing need to identify temporally causal relationships between noncoding loci and gene expression , as traditional approaches for doing so are expensive and slow . They include population genetics techniques like expression quantitative trait locus ( eQTL ) studies , in which human genetic variability at a particular genomic locus is associated with expression changes in the gene of interest . Alternatively , perturbation-based approaches directly seek to identify gene expression changes in response to CRISPR alterations of specific noncoding genomic loci ( Fulco et al. , 2016 ) . The emergence of single-cell multimodal assays that profile ATAC-seq and RNA-seq simultaneously offers a new way of discovering such regulatory relationships , on which we capitalize . Rather than requiring the large population sizes of eQTL studies or being limited to specific noncoding loci as in the perturbation-based approaches , these multimodal experiments enable unbiased high-throughput , genome-wide estimations of peak–gene associations . However , leveraging this data will require algorithmic innovations . Over 99.5 % of ATAC-seq observations ( “ peaks ” ) in a cell are reported as empty ( Singh et al. , 2021 ; Zhang et al. , 2019 ) , and this sparsity constrains current approaches . A key limitation of existing methods is also that they are based on correlation estimates and can not capture the dynamics governing the temporally causal relationship between chromatin accessibility and gene expression . For a noncoding locus to influence a gene ’ s expression , the locus typically must first become accessible , upon which regulator proteins act on it to drive a change in gene expression ( Figure 1d ) . This causal mechanism , therefore , entails temporal asynchrony between the ATAC-seq and RNA-seq modalities ( Lara-Astiaso et al. , 2014 ; Ostuni et al. , 2013 ) . GrID-Net is specifically designed to recover this dynamics between peaks and genes while being robust to the noise and sparsity in single-cell data . We apply GrID-Net on three single-cell multimodal datasets that characterize a range of dynamic processes , including cancer drug response and cellular differentiation . We show that GrID-Net substantially outperforms current methods in identifying known peak–gene links , opening a path towards unlocking critical insights about gene regulation at genome scale . To our knowledge , this work presents both the first framework for Granger causal inference on dynamical systems represented as DAGs and the first single-cell method to explicitly model the temporal asynchrony between chromatin accessibility and gene expression . 2 METHODS . 2.1 BACKGROUND ON GRANGER CAUSALITY : STANDARD FORMULATIONS . In time series analyses , the classical Granger causality framework uses the vector autoregressive ( VAR ) model , which can be expressed as follows ( Lütkepohl , 2005 ) . yt = L∑ ` =1 a ( ` ) 1 yt− ` + L∑ ` =1 a ( ` ) 2 xt− ` + t ( 1 ) Here , yt , xt ∈ R are values at time t for two stationary time series y and x , respectively ; a ( ` ) 1 and a ( ` ) 2 are coefficients that specify how lag ` affects the future of time series y ; and t is a zero-mean noise term . In this model , yt is assumed to be a linear combination of the L most recent values of y and x each , with x said to Granger cause time series y if and only if a ( ` ) 2 6= 0 for all ` . A more generalized formulation — the one we follow — for performing Granger causal inference is to consider two related models for forecasting time series y . A full model considers the past values of both y and x to forecast y , while a reduced model excludes the effect of x , only containing terms related to the past values of y. yt = f ( full ) ( yt−1 , ... , yt−L ; xt−1 , ... , xt−L ) + t ( 2 ) yt = f ( reduced ) ( yt−1 , ... , yt−L ) + t ( 3 ) Here , f ( full ) ( · ) and f ( reduced ) ( · ) are generalized functions that specify how the value of yt depends on past values of y and x . The predictions of the two models are then compared , upon which a Granger causal relationship is declared if the full model ’ s predictions are significantly more accurate . 2.2 GRANGER CAUSALITY ON A DAG : GRAPH NEURAL NETWORK FORMULATION . Let the data be represented by a DAG G = ( V , E ) with n = |V| nodes ( i.e. , observations ) and directed edges E indicating the information flow or partial order between these observations . For instance , when applied to the case of standard Granger causal inference on time series data , G would be the linear graph corresponding to the time series . Let y , x ∈ Rn correspond to the values of variables y and x on the nodes in V , with x putatively Granger-causing y . To infer Granger causality , we compare the full and reduced models as above : y = g ( h̃ ( full ) y + ch̃ ( full ) x ) + ( 4 ) y = g ( h̃ ( reduced ) y ) + ( 5 ) Here , h̃ ( full ) y , h̃ ( reduced ) y , h̃ ( full ) x ∈ Rn represent the historical information of y or x ( as denoted by the subscript ) that is aggregated by graph neural network ( GNN ) layers . For example , at each node v ∈ V , h̃ [ v ] is the information accumulated from v ’ s ancestors in G. While h̃ ( full ) y and h̃ ( reduced ) y are outputs determined from similar architectures , their learned weights and bias terms end up being different since h̃ ( full ) y is influenced by its interaction with h̃x . The coefficient c mediates this interaction and describes the effect of x ’ s history on y . In addition , ∈ Rn is a zero-mean noise term . We also include g ( · ) as an optional link function that maps its input to the support of the distribution of variable y . In our single-cell application , we set g as the exponential function since the target variable ( y ) represents normalized transcript counts , which are non-negative . For brevity , we only describe how h̃ ( full ) x is computed , with h̃ ( full ) y and h̃ ( reduced ) y being independently computed analogously . Also , sincex occurs only in the full model , for notational convenience we drop its superscript ( full ) below , writing just h̃x . We express h̃x as the mean of the outputs of L consecutive GNN layers , denoting the layerwise outputs as h ( ` ) x : h̃x = 1 L L∑ ` =1 h ( ` ) x ( 6 ) h ( ` ) x = { σ ( w ( ` ) x AT+h ( ` −1 ) x + b ( ` ) x ) if ` > 1 σ ( w ( ` ) x ATx+ b ( ` ) x ) if ` = 1 ( 7 ) Here , w ( ` ) x , b ( ` ) x ∈ R are the per-layer weight and bias terms , respectively . σ ( · ) represents the nonlinear activation in each of the GNN layers , chosen here to be the hyperbolic tangent function . A and A+ ∈ Rn×n are matrices defined by the DAG structure , designed to introduce a one-step lag in the first GNN layer that is propagated by subsequent layers . A is the column-normalized adjacency matrix of G , with Aij = 1dj if edge ( i , j ) ∈ E and 0 otherwise , where dj is the in-degree of node j . We note that a DAG does not have self-loops , so the diagonal terms of A are zero . In A+ , we include these diagonal terms , setting them to the same value as others in the column ( i.e . dj is incremented by 1 inA+ to preserve normalization ) . Together , A andA+ allow us to extend the key intuition of Granger causality to DAGs : predicting a value at node v from the values of its ancestors that are within L steps . The first layer introduces the lag , using information at the parents of v but not the information at v itself . Each subsequent layer introduces the preceding set of parents , with the diagonal term in A+ ensuring that information already stored at v is integrated with the new information propagated via v ’ s parents . The sequence of GNN layers therefore reflects the successive aggregation of past information by traversing L steps backward along the DAG for each node in the graph . Thus , we are able to conceptually match time series-based formulations ( e.g. , Eqn . 1 ) but with the crucial advantage of leveraging the richness of a DAG ’ s structure in directing the information flow and aggregating sparse data .

Summary: The paper proposes a modification of Granger causality so that it is applicable in partially ordered observations. Authors achieve that by adapting the architecture of GNNs to enable lagged-passed information in the DAG. Finally they apply it on a real data problem with the goal to infer non-coding genomic loci that causally influence the expression of a specific gene. In comparison with other three methods, the proposed method GrID net outperforms them on an open dataset.