About the scMagnify

About the scMagnify#

API Flowchart

scMagnify leverages a deep-learning framework to reconstruct and decompose multi-scale dynamic gene regulatory networks (GRNs) from single-cell multi-omic (gene expression and chromatin accessibility) data. The inference is achieved through a nonlinear Granger causality model, implemented as an interpretable multi-scale neural network.

The scMagnify workflow begins by integrating three primary inputs:

  1. A gene expression matrix (\(X\)).

  2. A cell transition matrix (\(T\)), typically derived from pseudotime or trajectory inference, which defines a directed acyclic graph (DAG) of cellular relationships.

  3. A basal TF binding network (\(B\)), which is constructed from peak-to-gene correlations and TF motif scanning. This network acts as a structural constraint, incorporating a priori biological knowledge from chromatin accessibility data.

To capture non-linear dynamics and temporal relationships, scMagnify employs a self-explaining neural network that functions as a non-linear extension of the classical Vector Autoregressive (VAR) framework. Instead of using simple time lags, the model uses a graph diffusion operator (\(\tilde{T}\)) derived from the cell transition matrix to aggregate TF expression information from \(k\)-hop ancestral cells in the trajectory graph.

The model’s core architecture (detailed in Supplementary Fig. 1) predicts the target gene (TG) expression \(\hat{X}_{TG}\) by integrating signals across multiple time scales (lags) using parallel branches, each weighted by a learnable attention mechanism \(\alpha_{k}\). The full model is formulated as:

\[ \hat{X}_{TG}=\sum_{k=1}^{K}[\alpha_{k}\sum_{1\le l\le k}(\Psi_{k}^{l}((\tilde{T})^{l}X_{TF})\odot B\cdot(\tilde{T})^{l}X_{TF})]+\epsilon \]

Here, \(\Psi_{k}^{l}\) represents the neural network for a specific branch and lag, \((\tilde{T})^{l}X_{TF}\) is the graph-diffused TF expression from \(l\)-step ancestors, and \(\odot B\) is the element-wise application of the chromatin-derived structural constraint.

The model is trained by minimizing a penalized loss function using mini-batch gradient descent:

\[ \mathcal{L}=\mathcal{L}_{MSE}+\lambda\mathcal{L}_{sparsity}+\gamma\mathcal{L}_{smooth} \]

This loss combines the standard Mean Squared Error (\(\mathcal{L}_{MSE}\)) with two regularization terms: a Group Elastic Net penalty (\(\mathcal{L}_{sparsity}\)) to enforce GRN sparsity and a temporal smoothness penalty (\(\mathcal{L}_{smooth}\)) to ensure regulatory coefficients evolve smoothly along the trajectory.

After training, the model outputs a multi-scale regulatory coefficient tensor (\(\mathcal{J}_{total}\)). This tensor is filtered and aggregated (via median) to produce the final, robust multi-scale GRN, \(\mathcal{G}_{multi-scale} \in \mathbb{R}^{K\times P\times Q}\), which quantifies regulatory strengths across all time lags (\(K\)), TFs (\(P\)), and TGs (\(Q\)).

RegFactor Decomposition#

A key innovation of scMagnify is the systematic decomposition of the inferred multi-scale GRN to identify combinatorial regulatory logic. This is achieved by applying Tucker decomposition to the third-order \(\mathcal{G}_{multi-scale}\) tensor:

\[ \mathcal{G}_{multi-scale}\approx C\times_{1}U^{(lag)}\times_{2}U^{(TF)}\times_{3}U^{(TG)} \]

This decomposition deconstructs the network into a core tensor (\(C\)) and three factor matrices, which define:

  • Co-regulating TF modules (\(U^{(TF)}\))

  • Shared target gene (TG) modules (\(U^{(TG)}\))

  • Temporal activation profiles (\(U^{(lag)}\))

A specific combination of these three components, linked by the core tensor, constitutes a ‘RegFactor’. This allows for a hierarchical dissection of gene regulation, moving from individual regulators to dynamic, combinatorial modules.

Downstream Applications#

The scMagnify framework includes a suite of integrated downstream analyses built upon the inferred GRNs and RegFactors:

  • Regulatory Activity Inference: The activity of individual TFs or entire RegFactors is quantified from the collective expression of their target regulons (defined by \(G_{activation}\) or \(U^{(TG)}\)) using a multivariate linear model (MLM).

  • Intercellular Communication: scMagnify models signaling-to-transcription cascades by correlating the expression of receptors with the expression (or inferred activity) of their downstream TFs along the pseudotime axis. The significance of these interactions is validated using a permutation test to reveal how extracellular cues are translated into intracellular regulatory programs.

See “Decomposing multi-scale dynamic regulation from single-cell multiomics with scMagnify” for a detailed description of the methods and applications on different biological systems.

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