CycleGRN: Inferring Gene Regulatory Networks from Cyclic Flow Dynamics in Single-Cell RNA-seq

CycleGRN is a novel framework that infers gene regulatory networks from single-cell RNA-seq data by modeling cyclic biological processes as stochastic differential equations and utilizing flow-aligned directed graphs to accurately recover oscillatory and directional gene interactions without requiring temporal binning or splicing dynamics.

The Gist

Imagine you are trying to understand the rules of a complex dance, but you only have a single, frozen photograph of the entire dance floor. In this photo, you see hundreds of dancers in different poses, but you don't know who is leading, who is following, or the order in which they move.

This is exactly the problem scientists face when studying cells. Cells are constantly changing, dividing, and making decisions. To understand how they work, scientists need to map out the "Gene Regulatory Network" (GRN)—a giant wiring diagram showing which genes turn other genes on or off.

The problem? Most single-cell experiments take a "snapshot" of cells at one random moment. It's like taking that frozen photo of the dance floor. Without knowing the sequence of moves, it's incredibly hard to figure out the choreography (the cause-and-effect relationships).

Enter CycleGRN.

Here is how this new method works, explained through simple analogies:

1. The "Invisible Clock" (The Cell Cycle)

Many cells, like those in our skin or blood, go through a repeating loop called the cell cycle (birth, growth, division, death). Think of this as a clock that never stops ticking.

• The Old Way: Scientists usually tried to ignore this clock, thinking it was just "noise" or a distraction. They would try to erase the cell cycle data to study other things.
• The CycleGRN Way: The authors say, "Wait! That clock is actually the key!" They treat the cell cycle like a metronome. Even though they don't have a video of the dance, they know the dancers are moving in a circle. By focusing on the genes that control this "clock," they can figure out the direction of the flow.

2. Reconstructing the Flow (The River Analogy)

Imagine the cells are leaves floating down a river that forms a perfect circle.

• You can't see the water moving, but you can see where the leaves are clustered.
• CycleGRN uses a mathematical trick (like a smart computer simulation) to ask: "If I draw a current here, would the leaves naturally settle into the pattern I see in the photo?"
• It learns the invisible current (the flow field) that pushes the cells around the circle. Once it knows the direction of the river, it knows which leaf came before which.

3. Predicting the Future (The Lie Derivative)

Now that the computer knows the direction of the river, it can guess what happens next.

• It looks at a specific gene (a specific instruction in the cell's manual) and asks: "If this cell moves one tiny step forward along the river, does this gene's activity go up or down?"
• This is called a Lie derivative. Think of it as a speedometer for every single gene in every single cell. It tells you not just what the gene is doing now, but how fast it is changing as the cell moves through its life cycle.

4. The "Time-Lagged" Connection (The Domino Effect)

Finally, the method connects the dots.

• If Gene A starts speeding up, and 10 seconds later Gene B starts speeding up, Gene A probably caused Gene B to speed up.
• Because CycleGRN knows the "river flow," it can measure these time delays without ever having a stopwatch. It builds a map showing who is the leader (the cause) and who is the follower (the effect).

Why is this a Big Deal?

• No Time Travel Needed: You don't need to wait days to watch cells grow. You can take a single snapshot and reconstruct the whole movie.
• Better Accuracy: Traditional methods often get confused by the circular nature of the cell cycle. CycleGRN embraces the circle, making it much better at finding the true "wiring" of the cell.
• Real-World Proof: The authors tested this on simulated data (where they knew the answer) and real mouse eye cells. In both cases, CycleGRN found the correct connections better than existing tools, even identifying which genes were driving the cell to divide and which were telling it to stop.

The Bottom Line

CycleGRN is like a detective who can look at a crime scene with only one photo and perfectly reconstruct the entire sequence of events, simply by understanding the rhythm of the "dance" the suspects were doing. It turns a static picture of cells into a dynamic movie, revealing the hidden rules that control life, disease, and development.

Technical Summary

1. Problem Statement

Oscillatory biological processes, such as the cell cycle and Notch signaling, are critical for cell fate determination and disease progression. However, standard Gene Regulatory Network (GRN) inference methods often fail to capture these dynamics effectively for several reasons:

• Static Assumptions: Most methods treat single-cell RNA-seq (scRNA-seq) data as static snapshots, ignoring the temporal ordering inherent in cyclic processes.
• Confounder Removal: Common practice involves regressing out cell cycle genes to remove "noise," inadvertently discarding valuable temporal information needed to infer causal regulatory relationships.
• Limitations of Existing Velocity Methods: Current approaches like RNA velocity rely on spliced/unspliced ratios (often unavailable) or require discrete temporal binning, which fails to capture the continuous, periodic nature of oscillatory systems.
• Lack of Ground Truth: In real datasets, explicit time labels are absent, making it difficult to establish the directionality of regulatory interactions.

Goal: To develop a framework that infers directed, signed gene regulatory networks by leveraging the invariant cyclic dynamics of oscillatory processes directly from unlabelled scRNA-seq data.

2. Methodology: The CycleGRN Framework

CycleGRN models the expression of oscillatory genes as an invariant measure of a stochastic differential equation (SDE). The pipeline consists of three main stages:

A. Learning Invariant Cyclic Flow (PDE-Constrained Optimization)

Instead of inferring a pseudotime trajectory, the method learns a continuous vector field (flow) $\nu(x)$ in the subspace of known cycle genes ($G_{inv}$).

• Objective: Find a flow $\nu_\theta$ such that its steady-state invariant distribution ($\rho_{\nu_\theta}$) matches the observed empirical density of cells ($\rho^*$) in the cycle gene subspace.
• Optimization: The problem is formulated as minimizing a distance metric $D(\rho_{\nu_\theta}, \rho^*)$ (e.g., Wasserstein distance or KL divergence).
• Physics Constraint: The steady-state density is derived from the Fokker-Planck equation: $\nabla \cdot (\rho_\nu(x)\nu(x)) = D\nabla^2 \rho_\nu(x)$.
• Implementation: A neural network parameterizes the flow $\nu$. The steady-state solution is computed using a finite-volume numerical solver, and gradients are backpropagated via the adjoint-state method to update the network parameters.
• Identifiability: The flow is unique up to global orientation and time reparameterization. Orientation is fixed using minimal biological priors (e.g., known S/G2M phase markers).

B. Cycle-Aware Lie Derivatives (Velocity Estimation)

Once the flow is learned in the cycle subspace, it is extended to the full transcriptome ($G$) to estimate gene velocities.

• Transition Matrix: A directed $K$-nearest neighbor graph is constructed where edges are weighted by the alignment of cell displacement vectors with the local flow direction $\nu$. This creates a transition matrix $L$.
• Discrete Lie Derivative: The velocity of any gene $g$ in cell $c$ is estimated as the expected forward change along the flow:
  $$V_g = X(L^\top - I)$$
  where $X$ is the gene expression matrix. This effectively computes a discrete Lie derivative, estimating the rate of change for all genes based on the cyclic flow of the cycle genes.

C. Time-Lagged Correlation for GRN Inference

To infer directional regulatory interactions, the method defines a time-lagged correlation operator on the flow-aligned graph.

• Propagation Matrix: A lagged neighborhood operator $P_\alpha = (I - \alpha L)^{-1}L$ is constructed, representing multi-step propagation along the flow with exponentially decaying weights.
• Correlation Metric: For any gene pair $(g_1, g_2)$, the correlation is defined as:
  $$C_\alpha(g_1, g_2) = \langle V_{g_1}, P_\alpha V_{g_2} \rangle$$
• Output: This yields a signed, asymmetric matrix. Asymmetry allows for the inference of directionality (causality), while the sign indicates activation or repression.

3. Key Contributions

1. Invariant Flow Learning: Introduces a PDE-constrained optimization approach to learn latent cyclic dynamics directly from scRNA-seq data without time labels or splicing information.
2. Flow-Aligned Lie Derivatives: Develops a method to extend the learned cyclic dynamics from a subset of cycle genes to the entire transcriptome, providing cell-specific velocity estimates.
3. Directional Correlation Operator: Defines a novel time-lagged correlation metric that respects the intrinsic geometry of the data manifold, enabling the inference of signed, directed regulatory edges.
4. Robustness: The method operates on raw expression data, avoiding the need for discrete temporal binning or the assumption of a unidirectional trajectory.

4. Results

The authors evaluated CycleGRN on four synthetic datasets (HARISSA models with known ground truth) and a real mouse retinal progenitor cell (RPC) dataset.

• Synthetic Data:

  • CycleGRN consistently ranked as a top-performing method across all metrics (AUPR for directed and signed networks).
  • It outperformed or matched mechanistic models (HARISSA, CARDAMOM) that had access to the true simulation parameters, despite CycleGRN using only expression data and a list of cycle genes.
  • It successfully recovered temporal ordering even when "experimental time" was misaligned with the internal cell cycle clock.

• Real Data (Mouse Retinal Progenitors):

  • Flow Recovery: Successfully recovered a coherent cyclic flow in early, late, and neurogenic RPCs, whereas Optimal Transport (OT) methods failed to find a consistent cyclic structure even with Tricycle coordinates.
  • Gene Velocity: Correctly identified phase-specific expression patterns (e.g., Top2a peaking in mid-cycle).
  • Network Inference: Outperformed GENIE3, GRNBoost2, and SINCERITIES in AUPR ratios, particularly in cell types with strong oscillatory signals.
  • Case Study (Top2a): CycleGRN correctly predicted the directional edge from Top2a to Cenpa (a downstream G2/M gene), which static methods missed.
  • Perturbation Analysis: In Nfia/b/x triple-knockout cells, the method detected specific network rewiring: a reinforced self-sustaining proliferative module and a collapse of differentiation targets, accurately reflecting the biological phenotype.

5. Significance and Future Directions

• Paradigm Shift: CycleGRN challenges the convention of treating cell cycle effects as noise, instead utilizing them as a "latent clock" to resolve causal regulatory relationships.
• Applicability: While focused on the cell cycle, the framework is applicable to any recurrent biological process (e.g., circadian rhythms, Notch oscillations).
• Limitations & Future Work:
  • Current implementation relies on 2D embeddings, which may struggle with high heterogeneity; future work aims to extend to higher dimensions.
  • The flow is not strictly unique; future iterations may incorporate time-delay coordinates or surrogate constraints to improve identifiability.
  • The method is biased toward oscillatory genes; extending it to multi-scale inference (combining fast oscillations with slow differentiation) is a key goal.

In summary, CycleGRN provides a mathematically rigorous, data-driven approach to uncovering the temporal logic of gene regulation in cyclic systems, offering superior performance over existing static and time-binned methods.