We thank the Reviewer for recognizing the importance and timeliness of the problem we address in our work. We are glad for the positive feedback regarding the robustness and good grounding of Celcomen and are particularly pleased that the Reviewer finds our results on disentanglement and counterfactual prediction to be promising. We also thank the Reviewer for the actionable feedback that helped us to improve our paper. In particular, we have added new experiments to (1) address weakness 6 below; and (2) provide in-vivo validation of our method. We hope that you will find our added experiments and responses satisfactory, and that you will consider revising your score.

Regarding Q1(structure of cells and DAG): We thank the Reviewer for bringing up the DAG assumption in causality. In Appendix E “Review of Probably Approximately Correct learning” we discuss PAC learning theory. Our model possesses identifiability guarantees within its space of available hypotheses, which follow a DAG structure. If the Bayes optimal hypothesis for spatial transcriptomics (noted as $h^{**}$ in the paper) does not follow a DAG (e.g., because there exist cells i,j and genes a,b,c,d such that $s_{ia}$ -> $s_{jb}$ -> $s_{jc}$ -> $s_{id}$ -> $s_{ia}$) then our algorithm will try to find the closest DAG (noted as $h^*$ in the paper) to approximate the Bayes optimal hypothesis. As explained in (Tejada-Lapuerta et al, arXiv:2310.14935) a way to avoid the problem of the presence of cycles in gene regulation is to adopt causal kinetic models. Combining the causal kinetic models’ mentality with our models of Virtual Tissues is an intriguing direction to explore in the future. In our revised paper, we addressed these comments by adding the discussion given here in the Appendix J “Limitations and Future Directions”. Thank you.

Regarding Q2 (readability): Thank you for the comment. We revised our paper accordingly, particularly we added clarifying statements according to reviewers comments in the Motivation section (Section 3.1). All our changes are marked in blue.

Regarding Q3 (additional subsection to related works): Thank you for the comment. To further distinguish our Celcomen, we added a review of methods relevant for disentanglement and counterfactual predictions in dissociated single cell data to the Related Work section (Section 2). Thank you.

Regarding Q4 (intuition to theorem 2): We appreciate your comment. In our paper, we have discussed the intuition behind Theorem 2 in the second paragraph of the Introduction (Section 1) and the paragraph before the statement of Theorem 2. Nonetheless, we agree with the Reviewer that it can be expanded, therefore we expanded Appendix F, which contains the proof of the identifiability theorem, to include more intuition behind it. Specifically, our revisions include the following discussion:

Identifiability for a model means that a unique configuration of our model fits the observed data. This is very important for causal models since otherwise our model could tell us that A either causes B or it doesn’t cause B, which is an unhelpful tautology. Identifiability is a point of principle of whether the model would be able to definitively decide that A causes B if provided enough data. For this reason most proofs of identifiability happen in the infinite data limit. For instance, imagine that we want to estimate the causal effect A has on B when there is an unobserved confounder between A and B. In this case, the causal effect could never be identified by any model, regardless of how much data we collect.

Regarding Q5 (statistical results): We thank the reviewer for the comment. We have now mentioned the explicit value of correlation in the main text for clarity. We also added more context to make the connection to identifiability more explicit. Specifically, what we aim to say regarding identifiability is that since our model is identifiable a single unique hypothesis fits the data. If a model is not identifiable, then more than one hypothesis fits the data equally well, and as a result those models randomly pick one of them based on statistical noise. This is why lack of identifiability leads to results that are not robust, since those models might transition to a completely different hypothesis. Since our results remain robust across different samples of the same biological tissue, this is in line with our model being identifiable.

We thank the Reviewer for the kind remarks about the strong theoretical underpinnings of Celcomen, such as the identifiability Theorem, and the Lemmas deriving a new mean field theory approximation that allows to speed up inference. We are also grateful for the constructive feedback regarding connections to the underlying biology, which we address one by one in our rebuttal and new experiments below. We hope that in light of our responses and revision of the paper, you will consider revising your score.

Regarding W1 (assumptions): We agree with the Reviewer that a bigger discussion of biology would help the reader. To address your comment, we now elaborate on our assumptions. Specifically, we would like to point out that our assumptions are motivated by biological concepts, as follows:

Assumption 1 (nearest neighbor assumption): Cell-cell communication and ligand-receptor interactions are often reflected in local tissue architecture (Ren et al, Cell Research 2020), which mathematically translates into our nearest neighbor assumption. Several methods are based on similar assumptions that a cell’s ligand affect or at least are correlated with their neighbors’ expression of receptor genes. For instance in (Birk et al bioArxiv 2024) they include a term that tries to regress the value of the receptor gene expression in a cell based on the expression of ligand gene in its neighbors.

Assumption 2 (Batch effect assumption, also known as causal sufficiency): By avoiding explicit batch correction, we require that there are no unobserved confounders (often referred to by biologists as batch effects) such as differences stemming from different experimental study sites, different ages or disease states of donors etc.

Additionally, we note that, without interventional data, it is mathematically impossible (without extra assumptions) to infer anything more than the Markov equivalence of the underlying DAG, which is the undirected graph plus information about colliders. Indeed our method obeys this mathematical bound, and Theorem 2 guarantees the identifiability of the undirected DAG.

The assumptions were added to our revised paper in Appendix J “Limitations and future directions”. Thank you.

Regarding W2 (cells in tissues): We thank the reviewer for their comments and the exciting prospect of including cell type information. We think that the raised idea can be incorporated both in the generative and the inference modules of our Celcomen. For instance, in the inference module, we can infer not just gene-gene matrices but gene-gene-CellType-CellType matrices so that we can allow different gene-gene interactions for each pair of cell types. In the generative module, we could take cell types into account when generating the counterfactuals. This amounts to imposing a prior probability for each cell/node in the graph.

On the other hand, incorporating cell type information might also create problems when cell type continua are arbitrarily clustered into separate cell types, which is why it merits a separate future study as an important and difficult task. Moreover, the 4D tensor required for gene-gene-CellType-CellType forces might not require modifications to Theorems 1 and 2, which form the main novelty and motivation of the paper. All those directions are indeed exciting, and we will pursue them in future work. We added part of this discussion in the Appendix J “Limitations and future directions”. Thank you.

We thank the Reviewer for the positive review and the kind words. We agree with the Reviewer, that Celcomen, to the best of our knowledge, is novel in terms of an identifiable method for spatial transcriptomics analysis, and Celcomen can be valuable to the community. We also agree that an important feature of Celcomen is that its architecture is mathematically derived from biologically motivated assumptions according to Theorem 1.

We are also thankful for the constructive feedback that helped us improve the paper by adding one new in-vivo experiment and refining the clarity of the text. Below we provide our responses to each of your comments. We hope that you find them satisfactory, and that you will consider revising your score.

Regarding W1 (notation): We thank the reviewer for pointing out these typos. We addressed them in the text to increase clarity. For example, $s_{ia}$ is now corrected to $s_i^a$.

Regarding W2 (Baselines and Ground truth): We agree with the Reviewer that there is a lack of what we consider models of Virtual Tissues to compare against. There are models of Virtual Cells like Biolord (Piran et al, Nat Biotech 2024) and GEARS (Roohani et at, Nat Biotech 2024) that predict the response of a cell given a perturbation of its environment or its genes. What we aim to do here is essentially the converse: determine not only how the cell is affected by its environment, but also how it affects its environment. Moreover, most methods of virtual cells are supervised, whereas our virtual model of tissues is unsupervised allowing for unbiased discovery not limited by current knowledge. We added this important distinction in the introduction to the paper.

Furthermore, we agree with the reviewer that although the results of the perturbation seem sensible, 1) more quantitative assessment against ground truth real data and 2) comparison to baselines would consolidate our arguments. To directly provide a benchmarking of our model to ground truth experimental data, we validate Celcomen on in vivo data, see new Figure 4 in the main text. Moreover to directly address the reviewer’s question about more comparisons to baselines we compare our method to a random baseline, see new Figure 4 in the main text. To our knowledge, the only dataset that has measured gene knockouts via spatial transcriptomics is Perturb-map (Dhainaut et al, Cell 2022) which leverages the non-HD version of the 10x Genomics Visium technology; notably, this is not single-cell resolution because each spot contains around seven cells on average and thus represents cell communities. Thus, our model is disentangling intra- and inter-spot gene regulation, namely intracellular+juxtacrine+paracrine (short range) regulation from paracrine (long range) regulation, rather than intra- and inter-cellular gene regulation. When we challenged our model with this ground truth we observed very promising results with in silico and in vivo perturbed transcriptomes appearing well correlated and well beyond what a random baseline could achieve. The discussions above were added to the revised paper section 4.3. Thank you.

Regarding W3 (real-life experiment): Thank you for the inspiring suggestion. The often large difference between synthetic and real data is a relevant point raised by the Reviewer. Even more to the reviewer’s point, there are also large differences between in-vitro experiments (performed in controlled laboratory settings, such as organoids, petri dishes or test tubes) and in-vivo experiments which are conducted in the context of a full living organism. To address your comment, we validate our predictions on real data using the Perturb-Map public dataset (Dhainaut et al, Cell 2022) of whole transcriptome profiling for in-vivo spatial gene knock-outs. Moreover, we benchmark our performance against a random baseline. As described above, across all five lesions, we observe a 0.3-0.5 Spearman correlation between our in-silico perturbation changes and the measured in-vivo perturbation changes, which is far above what can be explained by random baseline (p=0.0079). In summary, in our simulation, our real data experiment and our in-vivo experiment we confirm celcomen’s ability to predict spatial counterfactuals. The results on the new in-vivo experiment are presented in figure 4 and discussed in section 4.3.