DynGFN: Towards Bayesian Inference of Gene Regulatory Networks with GFlowNets

One of the grand challenges of cell biology is inferring the gene regulatory network (GRN) which describes interactions between genes and their products that control gene expression and cellular function. We can treat this as a causal discovery problem but with two non-standard challenges: (1) regulatory networks are inherently cyclic so we should not model a GRN as a directed acyclic graph (DAG), and (2) observations have significant measurement noise, so for typical sample sizes there will always be a large equivalence class of graphs that are likely given the data, and we want methods that capture this uncertainty. Existing methods either focus on challenge (1), identifying cyclic structure from dynamics, or on challenge (2) learning complex Bayesian posteriors over DAGs, but not both. In this paper we leverage the fact that it is possible to estimate the"velocity"of gene expression with RNA velocity techniques to develop an approach that addresses both challenges. Because we have access to velocity information, we can treat the Bayesian structure learning problem as a problem of sparse identification of a dynamical system, capturing cyclic feedback loops through time. Since our objective is to model uncertainty over discrete structures, we leverage Generative Flow Networks (GFlowNets) to estimate the posterior distribution over the combinatorial space of possible sparse dependencies. Our results indicate that our method learns posteriors that better encapsulate the distributions of cyclic structures compared to counterpart state-of-the-art Bayesian structure learning approaches.

Paper

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Peer review

Reviewer 2wco6/10 · confidence 3/52023-07-05

Summary

In this work, the authors propose an extension of GFlowNets to enable posterior sampling of cyclic graph structures and apply their approach to infer gene regulatory networks (GRN). While existing methods are able to infer cyclic graph structures or sample from Bayesian posteriors over DAGS, they cannot do both simultaneously. The authors formulate the problem as a dynamical systems identification problem which can be factored into the priors and model likelihood. The priors are learned using either linear or non-linear Hyper Networks while the model likelihood is estimated using the GFlowNets framework with detailed balance loss. The authors show their model is robust to several parameters including edge sparsity and time intervals between data points. Furthermore, they introduce several optimization ideas enabling a tractable search over admissible graph structures. Finally, they benchmark their approach against several baseline methods and apply their approach to an RNA velocity dataset to show their method is able to recover known and putative gene-gene interactions.

Strengths

The paper addresses an interesting/important question which has been the center of much research for decades: posterior sampling over directed graphs, this time with cycles and not DAGs. The authors appear knowledgable in this field and they take advantage of GFlowNets, building a method to assess posterior structures with cycles as they unfold them across time.

Weaknesses

Our biggest grievance is that the paper is just not reader friendly for anyone who is not an expert in that specific line of work. As noted the work seems technically very solid, one that offers specific new solution for the stated problem. Still, we as researchers not in this specific field, remain unsure after reading it of the overall innovation/significance of creating posterior sampling of structures over GFlowNets. We add more details below. (1) The termination criteria of Algorithm 1 is not clear. Under what conditions does the transition probability become null? Although one of the main innovations of the paper is sampling from the posterior of GFlowNet structures, the authors don’t reference any standard Bayesian analysis. How are posterior samples summarized (for example, how is a point estimate of p obtained from posterior samples in Figure 3)? How and when does Algorithm 1 converge to the posterior? Is the approach sensitive to variable initializations? (2) Although the authors compare their approach to sensible baselines using sensible metrics, it is not clear the approach has a clear advantage over the original GFlowNet in this applied setting. For example, how does the MAP parameter estimate (or equivalent) of this method compare to the original GFlowNet under the AUC metric (these are presumably comparable). Does GFlowNet produce different insight on gene-gene interactions in the RNA velocity dataset? (3) The authors assume causal sufficiency as a criteria for model identifiability. Is this a reasonable assumption given the sparsity of single cell data and limitations in the scalability of the model? In other words, how can we be reasonably certain that all casual gene measurements are observed given the model effectively only functions on a gene space of size 20 or lower? (4) Section 5 on graph augmentation is very abstruse. Efforts could be made to improve readability/accessibility for an audience that doesn’t work directly in this field. The related Figure 2 is also very cryptic. (5) It is not clear or obvious how the factorization into the “per-node” posterior in section 4.1 leads to the reduction in search space from 2^d^2 to d2^d. A brief explanation would be useful. (6) Line 154: “Previous work has shown GFlowNets are useful in settings with multi-modal posteriors.” This needs a citation. (7) There is inconsistency in the notation in Figure 1 which leads to some confusion. Specifically, it appears the Q variable is overloaded. How is the graph prior (represented as an adjacency matrix) sampled? Is there a hyperprior that is used or are graph structures uniform? (8) The patterns that the authors suggest that are evident in the Figure 3 heatmaps are not convincing. Perhaps it would be better to sort both heatmaps the same way instead of independent clustering.

Questions

see above.

Rating

6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.

Confidence

3: You are fairly confident in your assessment. It is possible that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.

Soundness

3 good

Presentation

2 fair

Contribution

3 good

Limitations

see above.

Area Chair 6Upb2023-08-17

what did you think of the authors' response?

The authors have provided detailed responses to your questions and comments. Please revise the text and score of your review to reflect how their responses have changed your perspective on their submission, and please acknowledge that you have read the authors' carefully written response.

Reviewer uDCE6/10 · confidence 3/52023-07-06

Summary

The authors tackle the well-known problem of Gene Regulatory Network (GRN) inference from time series data. Starting from an estimate of RNA velocity, GRN inference is framed as a causal discovery task, with two specificities: the inferred graphs may be cyclic, and a probabilistic distribution over candidate graphs is sought rather than a point estimate, so as to account for the high amounts of noise inherently present in biological data. The method is articulated around two assumptions: - a dynamic structural model $\frac{dx_i(t)}{dt} = f_i(\text{Pa}(x_i), \epsilon_i)$ for each gene. $\text{Pa}(x_i)$ is directly obtained from the regulatory graph $G$, and the structural causal model (SCM) $f_i$ is parametrized by $\theta$ - a factorisation of the joint distribution of the graph $G$, the structural causal model parameters $\theta$, and the observed data $\mathcal{D}$ as $p(G,\theta,\mathcal{D}) = p(\mathcal{D}|G,\theta)p(\theta|G)p(G)$. The graph sampler $p(G)$ is modeled as a GFlowNets. Next, since the parameters of the SCM depend on the graph structure, they are computed by a HyperNetwork, i.e. $p(\theta|G)$ is substituted to a neural network which takes as input a graph $G$ and outputs the structural equation model parameters $\theta$: $p(\theta|G) = \delta(\theta|G)$. Next, the authors evaluate the proposed method, DynGFN, on several synthetic examples as well as a real-world one involving single-cell RNA velocity data. The authors introduce other baselines by using different approaches than a GFlowNets for the graph sampler $p(G)$. This being said, DynGFN offers substantial improvements when it comes to jointly recovering the ground truth structure while characterizing the uncertainty around it.

Strengths

The big picture of the method is well explained. The work described here is likely to have a significant impact for the GRN inference community, as I believe that not many probabilistic methods are available in this field. I also believe that DynGFN could be used as a starting point for the more general problem of physical/biological dynamical system discovery, where one is not only concerned with the graph adjacency matrix but with providing a mechanistic description of the structural causal model $f_i$ [1,2] [1] Discovering governing equations from data by sparse identification of nonlinear dynamical systems, PNAS, 2016 [2] Identification of dynamic mass-action biochemical reaction networks using sparse Bayesian methods, PLoS Comp. Biol., 2022

Weaknesses

- The paper builds on GFlowNets for the graph sampler $P(G)$ and Hypernetworks to generate the structural causal model parameters $\theta$ given a sampled graph $G$. I found it slightly difficult to understand everything $in$ $detail$ given that these tools are each introduced only in a small paragraph. - Even though the method focuses on Bayesian Inference, it would have been interesting to assess DynGFN's performances compared to SOTA methods for GRN inference from time series data such as DynGENIE3 [1], BINGO [2] or others. Lastly, these methods use time-resolved gene expression data, from which one can get an estimate of the RNA velocity using for instance finite differences, which may be a quite rough estimate for sparsely sampled measurements. Do I understand right that the RNA velocity data employed in this submission somehow constitutes a ``more principled'' way to estimate the RNA velocity? [1] dynGENIE3: dynamical GENIE3 for the inference of gene networks from time series expression data, Scientific Reports, 2018 [2] Gene regulatory network inference from sparsely sampled noisy data, Nature Communications, 2020

Questions

- Any insights on why $h$-DynGFN yields significantly better results than $\ell$-DynGFN on the linear system presented in table 1? - In the single cell experiment, where does the ground truth network (used to compute AUC / Bayes-SHD) come from? Could you explain a little more what does "correlation over cell cycle time" means in Section 6.3? I have read the rebuttal done by the authors.

Rating

6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.

Confidence

3: You are fairly confident in your assessment. It is possible that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.

Soundness

2 fair

Presentation

2 fair

Contribution

3 good

Limitations

The authors mention as main limitation the difficulties to scale to large gene regulatory networks due to the combinatorial explosion. Causal sufficiency is also assumed here, which means that all relevant variables are observed. This assumption hardly ever holds in real-world biological problems. This issue is also mentioned by the authors in the Appendix.

Area Chair 6Upb2023-08-17

what did you think of the authors' response?

The authors have provided detailed responses to your questions and comments. Please revise the text and score of your review to reflect how their responses have changed your perspective on their submission, and please acknowledge that you have read the authors' carefully written response.

Reviewer b9zX6/10 · confidence 4/52023-07-07

Summary

Authors introduced a principled methodology called DynGFN, which effectively identified cyclic structures and concurrently modeled the Bayesian posteriors over directed acyclic graphs (DAGs). Leveraging RNA velocity, the authors formulated a dynamic system that unveiled the underlying gene regulatory networks. DynGFN was meticulously designed with three modules, and its superior performance was demonstrated through synthetic experiments and real-world analysis.

Strengths

The real biological systems can be rarely formulated as DAGs and there are always feedback loops to make the system work. DynGFN was appropriately motivated with a real biological thinking.

Weaknesses

The overall manuscript was hard to follow as there was too much content included. There were still a lot of technical details that needed to be clarified. It would improve substantially if the manuscript can be carefully restructured.

Questions

Major concerns 1. The synthetic experiments and real analysis did not explicitly present any results related to feedback loop identification, which serves as one of the motivations behind DynGFN. Could you elaborate on this point? 2. How many epochs did it take to stabilize the training of l-DynGFN? 3. Given the inference of RNA velocity involves multiple free parameters, how do these parameters impact the inference results obtained from DynGFN? 4. In the synthetic experiments simulating dynamic systems, how were the ground truth regulatory networks constructed based on the simulated dynamic systems? 5. Appendix B.3 mentions the construction of the validation and test sets to fine-tune the hyperparameters. Could you explain how these sets were constructed for the experiment? 6. Regarding Table 4, could you clarify what the ground truth graph was used for calculating Bayes-SHD and AUC? Was it formulated based on external knowledge? 7. In Figure 3, it is important to note that gene regulation is not solely determined by the direct interaction between corresponding proteins. In fact, it has been reported that "CDK1 targets MCM2-7 complex..." [1]. [1] Enserink, Jorrit M., and Richard D. Kolodner. "An overview of Cdk1-controlled targets and processes." Cell division 5.1 (2010): 1-41. Minor concerns 1. Figure 1 lacks definitions for numerous parameters, which may cause confusion in understanding the illustration.

Rating

6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.

Confidence

4: You are confident in your assessment, but not absolutely certain. It is unlikely, but not impossible, that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work.

Soundness

4 excellent

Presentation

2 fair

Contribution

3 good

Limitations

The authors mentioned two limitations: 1) scalability issues concerning larger systems and 2) hyperparameter tuning. To address these challenges, the authors suggested employing more informative priors or grouping genes together as nodes.

Area Chair 6Upb2023-08-17

what did you think of the authors' response?

The authors have provided detailed responses to your questions and comments. Please revise the text and score of your review to reflect how their responses have changed your perspective on their submission, and please acknowledge that you have read the authors' carefully written response.

Reviewer PVQy7/10 · confidence 4/52023-07-24

Summary

The authors propose to learn dynamical systems with cyclic dependencies. They factorise the generative model using a variant of the GFlowNet model, a HyperNetwork and MLPs.

Strengths

Positives include: 1. extending the formulation from DAGs to cyclic graphs 2. introducing a per-node posterior formulation to improve the computational complexity 3. using GFlowNets to address the multimodality issues

Weaknesses

Negatives include: 1. due to the high density the paper is hard to understand: a lot is explained and detailed in the Appendix 2. it is hard to gauge the quality of the results: even for the artificial case it is hard to assess intuitively if the approach solves the problem in a satisfactory way. Perhaps the authors should use simple systems with few nodes, with and without cycles, and show how well the true dependencies are recovered. 3. it is unclear how important it is to being able to model cyclic structures: where are the cyclic dependencies in the examples shown in Fig. 2 and Fig. 3? The authors should show a clear example where failing to model the cyclic dependencies results in a poor fit. 4. it is unclear how to translate the declared advantage of being able to model multi-modal posteriors in practical terms: how would we make use of this representation capacity when explaining the cause dependencies in a gene regulatory network?

Questions

To illustrate the advantages of the proposed approach: 1. use simple systems with few nodes, with and without cycles, and show how well the true dependencies are recovered. 2. show a clear example where failing to model the cyclic dependencies results in a poor fit. 3. show how to use the multimodal representation capacity when explaining the cause dependencies in a gene regulatory network.

Rating

7: Accept: Technically solid paper, with high impact on at least one sub-area, or moderate-to-high impact on more than one areas, with good-to-excellent evaluation, resources, reproducibility, and no unaddressed ethical considerations.

Confidence

4: You are confident in your assessment, but not absolutely certain. It is unlikely, but not impossible, that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work.

Soundness

2 fair

Presentation

2 fair

Contribution

2 fair

Limitations

Yes.

Area Chair 6Upb2023-08-17

what did you think of the authors' response?

The authors have provided some responses to your questions and comments. Please revise the text and score of your review to reflect how their responses have changed your perspective on their submission, and please acknowledge that you have read the authors' carefully written response.

Reviewer PVQy2023-08-18

The authors have provided a satisfactory answer to the main concerns we raised. The high density and not being a self contained paper is still an issue. I will therefore correspondently raise the score.

Area Chair 6Upb2023-08-17

acknowledge and discuss the authors' detailed response

The authors have provided a detailed response, which includes answers to reviewers' questions and more exposition of the technical material. **Drop a comment in this thread with how their response changed your perspective on the submission.** Remember to also update your scores/reviews to reflect how your views have changed in light of the authors' response, and respond to comments/questions the authors have left underneath individual reviews.

Reviewer 2wco2023-08-17

Feedback for the rebuttal

We have read through the authors response and appreciate the effort made to fix/clarify points we raised. As noted, the "causal sufficiency assumption" is a very strong assumption and as the authors admit with current scale (~20 genes) far from realistic. This limits the actual applicability of the work and makes it more of a computational/methodological contribution. Nonetheless, we appreciate the work/contribution and retain our (positive) score.

Reviewer b9zX2023-08-21

I have read through the responses from authors and I appreciate the efforts to further improve the manuscript. NOTEARS is inherently not a suitable method to identify cyclic relations in the data. An alternative approach will be preferred. Therefore I keep my original score unchanged.

Program Chairsdecision2023-09-21

Decision

Accept (poster)

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