GO-Based Clustering for Learning Cluster-Level Causal Gene Regulatory Networks

Azlaan Mustafa Samad; Wei Zhang; Adèle H Ribeiro

cs.LGOct 5, 2026

GO-Based Clustering for Learning Cluster-Level Causal Gene Regulatory Networks

Authors: Azlaan Mustafa Samad, Wei Zhang, Adèle H Ribeiro

Organizations: L3S Research Center, Leibniz University Hannover, Lower Saxony Center for Artificial Intelligence and Causal Methods in Medicine (CAIMed), Hannover, Germany · Institute for Causal and Explainable Artificial Intelligence in Life Sciences, Center for Computational Life Sciences, RWTH Aachen University, Germany

Abstract

Discovery of causal relationships in high-dimensional Gene Regulatory Networks (GRN) is computationally challenging and often difficult to interpret due to dense connections. Therefore, grouping genes together into functional modules can improve tractability and biological interpretability. However, existing cluster level causal discovery methods assume access to a predefined admissible partitions, requiring the graph over clusters to be acyclic. Constructing such partitions is therefore challenging. In this work, we introduce GO-based Clustering for Causal Discovery (GO4CD), an algorithm that uses Gene Ontology (GO) to construct biologically meaningful gene partitions at multiple levels of granularity, while favoring those more likely to be admissible for causal discovery. GO4CD groups together genes participating in a shared biological process, and propagates gene annotations through the ontology hierarchy to achieve different granularity of partitions. Furthermore, we integrate GO4CD with Causal Learning over Clusters (CLOC) algorithm and evaluate recovery of true Markov equivalence class both with an oracle of conditional independencies and on simulated gene expression data using multivariate conditional independence tests. We evaluate GO4CD on multiple E.coli regulatory subnetworks and find that it is inadmissible in 18.1% of the cases, compared with 65.3-82.3% for the semantic-similarity baselines. Our results indicate that GO4CD is substantially better suited to learning causal GRNs defined over biologically meaningful gene clusters.

Figures & tables

		inadmissible instances (%)					C-DAG edges					largest (share of $n$ )				singletons (share of $k$ )
size	instance	G	L	R	W	Rand	G	L	R	W	Rand	G	L	R	W	G	L	R	W
50	40	10.0	35.0	50.0	32.5	77.8	20	32	35	31	44	0.64	0.38	0.37	0.36	0.75	0.42	0.39	0.37
100	38	13.2	44.7	52.6	52.6	80.8	40	90	94	91	108	0.65	0.31	0.30	0.30	0.79	0.44	0.42	0.41
200	43	7.0	46.5	79.1	58.1	77.9	57	131	152	140	161	0.73	0.38	0.34	0.33	0.83	0.51	0.50	0.46
400	43	7.0	81.4	95.3	88.4	80.2	110	261	311	261	297	0.73	0.28	0.27	0.28	0.87	0.41	0.38	0.49
600	43	27.9	74.4	97.7	74.4	82.1	163	368	442	373	413	0.72	0.27	0.23	0.27	0.90	0.46	0.38	0.47

Table 1: Percentage of inadmissible instances across different sizes

n

for all methods, beside the number of C-DAG edges that survive the coarsening and the shape of the partitions being compared. G, L, R, W are GO4CD, Lin, Resnik and Wang. largest is the share of the

n

genes in the biggest cluster and singletons the share of the

k

clusters holding one gene, both means over instances.

size	$k$	inst.	G	L	R	W	size	$k$	inst.	G	L	R	W
50	1–10	18	.000 $\pm$ .000	.002 $\pm$ .010	.001 $\pm$ .005	.000 $\pm$ .000	100	1–10	14	.000 $\pm$ .000	.000 $\pm$ .000	.000 $\pm$ .000	.000 $\pm$ .000
	11–20	9	.000 $\pm$ .000	.023 $\pm$ .047	.038 $\pm$ .092	.019 $\pm$ .039		11–20	4	.000 $\pm$ .000	.000 $\pm$ .000	.004 $\pm$ .007	.000 $\pm$ .000
	21–30	6	.000 $\pm$ .000	.154 $\pm$ .129	.217 $\pm$ .097	.147 $\pm$ .147		21–30	3	.000 $\pm$ .000	.077 $\pm$ .104	.178 $\pm$ .176	.056 $\pm$ .045
	31–40	6	.024 $\pm$ .020	.187 $\pm$ .093	.211 $\pm$ .080	.220 $\pm$ .108		31–40	2	.000 $\pm$ .000	.084 $\pm$ .077	.166 $\pm$ .086	.111 $\pm$ .101
	41–50	1	.103 $\pm$ .000	.086 $\pm$ .000	.086 $\pm$ .000	.128 $\pm$ .000		41–50	4	.000 $\pm$ .000	.294 $\pm$ .085	.333 $\pm$ .089	.292 $\pm$ .072
								51–60	5	.000 $\pm$ .000	.315 $\pm$ .075	.343 $\pm$ .081	.331 $\pm$ .075

Table 2: SHD by

k

bin, sizes 50 and 100. Mean

\pm

std over each bin. The remaining sizes see Table 6 in Appendix D.2 .

			precision $\uparrow$				sepset / $B$ $\downarrow$
size	inst		G	L	R	W	G	L	R	W
50	10	macro	0.711	0.642	0.701	0.662	0.294	0.378	0.492	0.361
		micro	0.689	0.665	0.723	0.685	0.201	0.300	0.388	0.254
100	3	macro	0.796	0.446	0.420	0.293	0.360	0.670	0.757	0.810
		micro	0.750	0.444	0.392	0.289	0.311	0.531	0.661	0.676
200	11	macro	0.784	0.465	0.515	0.433	0.500	0.772	0.784	0.765

Table 3: Precision over the learned cluster skeleton and the share of the worst-case CI budget

B

spent, on the 24 instances. macro averages the per-instance value, micro is the ratio of sums.

Appendix figures & tables13 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 1: Where the inadmissible cells sit in (n, k). Inadmissible cells for different sizes for all the methods. Each of the 308 matched-

k

cells is a point. Shading is

0.148\leq k/n\leq 1

, below 0.148 (lower bound) GO4CD is never cyclic.

Figure 2: Largest-cluster share against

k

inside GO4CD’s biggest cluster, after that cluster is cut out as a network of its own and re-partitioned at every deeper GO depth. Depths are marked inside the plot.

size 50										size 100
		largest (share of $n$ )				singletons (share of $k$ )						largest (share of $n$ )				singletons (share of $k$ )
$k$	instances	G	L	R	W	G	L	R	W	$k$	instances	G	L	R	W	G	L	R	W
1–10	18	0.88	0.58	0.56	0.51	0.62	0.21	0.17	0.11	1–20	18	0.94	0.49	0.49	0.49	0.67	0.16	0.14	0.11
11–20	9	0.59	0.32	0.28	0.32	0.77	0.45	0.40	0.45	21–40	5	0.52	0.26	0.23	0.24	0.81	0.56	0.49	0.56
21–30	6	0.44	0.21	0.23	0.23	0.89	0.62	0.59	0.61	41–60	9	0.42	0.14	0.13	0.13	0.91	0.69	0.70	0.68
31–40	6	0.28	0.12	0.12	0.12	0.92	0.74	0.76	0.73	61–80	6	0.25	0.07	0.07	0.06	0.94	0.79	0.79	0.78

Appendix

Table 4: How the four methods cut the same genes, within each network size and

k

bin. largest is the share of the

n

genes sitting in the biggest cluster, singletons the share of the

k

clusters holding one gene; both are means over the instances in the bin.

k

is identical across the four methods within a instance, so a row holds granularity fixed. Bins with no instances are omitted, as are instances with

k=n

. G, L, R, W are GO4CD, Lin, Resnik and Wang.

size	rep	$d$	$k$	$d^{\prime}$	$k^{\prime}$	size	rep	$d$	$k$	$d^{\prime}$	$k^{\prime}$	size	rep	$d$	$k$	$d^{\prime}$	$k^{\prime}$	size	rep	$d$	$k$	$d^{\prime}$	$k^{\prime}$
50	torD	10	35	9	26	400	trxB	10	301	9	219	600	ttdB	10	452	9	338	1000	gatA	7	292	6	166
50	trxB	10	34	9	24	400	uvrB	10	296	9	212	800	torD	6	127	5	54	1000	gatA	8	460	6	166
50	ttdB	10	32	9	19	600	gatA	10	446	9	334	800	torD	7	232	5	54	1000	torD	6	164	5	72
50	uvrB	10	34	9	22	600	torD	6	89	5	40	800	torD	8	371	5	54	1000	torD	7	283	5	72
100	gatA	10	77	9	61	600	torD	7	168	5	40	800	trxB	6	131	5	56	1000	torD	8	451	5	72
100	torD	10	72	9	57	600	torD	8	270	5	40	800	trxB	7	236	5	56	1000	trxB	7	290	6	165

Appendix

Table 5: All 52 instances where GO4CD’s C-DAG is cyclic, in four column blocks.

d

and

k

are the cyclic depth and its cluster count;

d^{\prime}

and

k^{\prime}

are the deepest shallower depth whose C-DAG is acyclic and the cluster count there.

size	$k$	G	L	R	W	size	$k$	G	L	R	W
200	1–10	.000 $\pm$ .000	.000 $\pm$ .000	.000 $\pm$ .000	.000 $\pm$ .000	600	1–10	.000 $\pm$ .000	.000 $\pm$ .000	.000 $\pm$ .000	.000 $\pm$ .000
	11–20	.000 $\pm$ .000	.000 $\pm$ .000	.005 $\pm$ .010	.002 $\pm$ .003		11–20	.000 $\pm$ .000	.000 $\pm$ .000	.011 $\pm$ .012	.000 $\pm$ .000
	21–30	.000 $\pm$ .000	.000 $\pm$ .000	.006 $\pm$ .006	.001 $\pm$ .001		31–40	.000 $\pm$ .000	.001 $\pm$ .001	.011 $\pm$ .008	.000 $\pm$ .001
	31–40	.000 $\pm$ .000	.000 $\pm$ .000	.001 $\pm$ .001	.003 $\pm$ .001		41–50	.000 $\pm$ .000	.000 $\pm$ .000	.013 $\pm$ .005	.000 $\pm$ .000
	51–60	.000 $\pm$ .000	.004 $\pm$ .002	.066 $\pm$ .012	.004 $\pm$ .003	800	1–10	.000 $\pm$ .000	.000 $\pm$ .000	.006 $\pm$ .013	.000 $\pm$ .000
400	1–10	.000 $\pm$ .000	.000 $\pm$ .000	.000 $\pm$ .000	.000 $\pm$ .000		11–20	.000 $\pm$ .000	.000 $\pm$ .000	.011 $\pm$ .004	.000 $\pm$ .000

Appendix

Table 6: SHD by

k

bin, sizes 200, 400 and 600, 800, 1000. Mean

\pm

std over each bin.

		share of charge (%)
method	SHD/instance	FP	reversed	2-cycle
GO4CD	1.13	92.0	3.5	4.5
Lin	27.31	88.6	10.0	1.4
Resnik	35.05	87.4	10.5	2.1
Wang	28.91	87.8	10.7	1.5

Appendix

Table 7: Share of each method’s SHD coming from false positives, reversed directions and 2-cycles, summed over the 176 instances. Edge is never missed so FN is always zero.

method	edges	correct	undirected	reversed
GO4CD	3058	1341 (43.9%)	1710 (55.9%)	7 (0.2%)
Lin	4780	2687 (56.2%)	1614 (33.8%)	479 (10.0%)
Resnik	5726	3071 (53.6%)	2009 (35.1%)	646 (11.3%)
Wang	4791	2734 (57.1%)	1511 (31.5%)	546 (11.4%)

Appendix

Table 8: Breakdown of the directed edges in the true cluster graph, for all instances. Correct, undirected and reversed refer to CLOC’s orientation of each edge.

GO4CD vs Lin — 147 instances
method	edges	correct	undirected	reversed
GO4CD	1808	453 (25.1%)	1355 (74.9%)	0 (0.0%)
Lin	1965	528 (26.9%)	1430 (72.8%)	7 (0.4%)
GO4CD vs Resnik — 117 instances
GO4CD	832	169 (20.3%)	663 (79.7%)	0 (0.0%)
Resnik	998	167 (16.7%)	826 (82.8%)	5 (0.5%)

Appendix

Table 9: Breakdown of CLOC’s output on instances with no FP.

GO4CD vs Lin — 95 instances
method	edges	oriented	undirected
GO4CD	808	207 (25.6%)	601 (74.4%)
Lin	887	215 (24.2%)	672 (75.8%)
GO4CD vs Resnik — 44 instances
GO4CD	151	10 (6.6%)	141 (93.4%)
Resnik	159	0 (0.0%)	159 (100.0%)

Appendix

Table 10: Breakdown of the true cluster graph’s directed edges on the admissible, FP

=0

instances. Every oriented edge here is correct, since CLOC is sound and complete in this regime.

					$\|E\|$				$g_{*}$				precision				recall
size	rep	depth	$k$	cyc	G	L	R	W	G	L	R	W	G	L	R	W	G	L	R	W
50	uvrB	8	16	R	15	22	25	22	29	16	14	16	0.88	0.85	0.89	0.88	1.00	1.00	1.00	1.00
50	torD	6	14	R	13	13	24	13	25	16	16	16	0.72	0.62	0.75	0.62	1.00	1.00	1.00	1.00
50	torD	7	19	L R W	18	19	33	19	23	15	15	15	0.67	0.66	0.89	0.68	1.00	1.00	1.00	1.00
50	trxB	5	8	R	7	7	9	7	35	22	23	21	0.54	0.47	0.50	0.41	1.00	1.00	1.00	1.00
50	trxB	6	10	L R	9	13	12	11	22	21	22	20	0.50	0.62	0.50	0.58	1.00	1.00	1.00	1.00

Appendix

Table 11: Per-instance results on the 24 instances at

k\leq 20

where at least one method’s projected true C-DAG carries a cycle, named in the cyc column.

|E|

is that graph’s edge count,

g_{*}

the largest cluster’s gene count, and precision and recall are over the learned skeleton. G / L / R / W = GO4CD / Lin / Resnik / Wang. Each size block ends with macro and micro rows: the mean of the per-instance values and the ratio of sums, the latter undefined for

g_{*}

depth	$k$	G	L	R	W	G	L	R	W	G	L	R	W
		$g_{*}$				singletons				precision
6	10	38	25	27	24	0.70	0.30	0.50	0.20	1.00	0.60	0.67	0.58
8	13	36	25	26	24	0.85	0.46	0.54	0.38	0.86	0.60	0.68	0.67
9	19	30	24	22	24	0.89	0.74	0.63	0.68	0.83	0.75	0.73	0.78

Appendix

Table 12: Size 50, replicate 3. One replicate’s depth ladder, so

k

and

g_{*}

move only with depth.

g_{*}

is the largest cluster’s gene count and singletons the share of the

k

clusters holding one gene.

depth	$k$	G	L	R	W	G	L	R	W	G	L	R	W
		$g_{*}$				singletons				precision
2	3	198	197	152	140	0.67	0.33	0.33	0.33	1.00	1.00	1.00	0.67
3	4	197	139	152	139	0.75	0.25	0.25	0.50	1.00	0.75	1.00	0.60
4	8	193	58	114	63	0.88	0.12	0.12	0.38	0.64	0.45	0.47	0.41
5	18	183	57	109	59	0.94	0.33	0.50	0.33	0.63	0.33	0.38	0.23

Appendix

Table 13: Size 200, replicate 3. The same ladder at the size where the methods separate.

g_{*}

is the largest cluster’s gene count and singletons the share of the

k

clusters holding one gene.

depth	$k$	G	L	R	W	G	L	R	W	G	L	R	W
		$g_{*}$				singletons				precision
2	3	198	143	146	142	0.67	0.33	0.33	0.33	1.00	0.67	0.67	0.67
4	6	195	56	108	74	0.83	0.17	0.17	0.33	0.83	0.33	0.46	0.45
5	13	188	56	102	57	0.92	0.31	0.38	0.31	0.75	0.26	0.35	0.26

Appendix

Table 14: Size 200, replicate 4.

g_{*}

is the largest cluster’s gene count and singletons the share of the

k

clusters holding one gene.

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Department of Informatics, King’s College London, Bush House, 30 Aldwych, WC2B 4BG, London, United Kingdom

Causal Discovery Methods Regulatory Networks Causal Downstream Reasoning

Open paper AskPDF arXiv

Biological systems exhibit a hierarchical structure, characterised by directed flow from upstream regulators to downstream effects. Although this ordering provides a natural scaffold for causal inference, most causal discovery and GRN methods either ignore the tiered organisation or condition on all upstream variables, which becomes infeasible for high-dimensional omics data. We present ASCEND (Ancestral Scalable Causal discovEry via iNherited Descent), a constraint-based framework that leverages known two-tiered structure to enable genome-scale causal discovery. ASCEND introduces a divide-and-conquer strategy that maintains dynamically updated ancestral conditioning sets for each downstream variable, dramatically reducing the number of conditional independence tests required, and achieves polynomial-time complexity where traditional approaches face exponential blow-up. Through extensive simulations and real biological data, we demonstrate that ASCEND accurately recovers ancestral relationships, scales properly and much faster, and outperforms existing gene regulatory network inference methods in both causal precision and computational efficiency. The algorithm's ability to resolve directionality makes it particularly suited for integrating multi-omic data where upstream regulators (e.g., SNPs, methylation sites) and downstream responses (e.g., gene expression) are measured jointly.

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