Introduction
With the advent of single cell genomics, we now have access to high-resolution, genome-wide transcriptomic measurements at the resolution of single cells. This unprecedented level of detail allows us to explore how genes are expressed across diverse cell types, states, and molecular contexts.
One key objective in single-cell RNA sequencing (scRNA-seq) analysis is to discover a collection of genes that are jointly expressed across diverse cell types and molecular contexts. We often call it as gene modules, sets of genes that exhibit coordianted expression patterns.
To assess the biological relevance of these gene modules, researchers commonly perform ontology$\dagger$ enrichment analysis, to support the functional coherence of collected genes.
$\dagger$ Notable advancement in gene ontology; from GO terms, molecular signatures to large-scale model-based embeddings - I’ll discuss these topics later.
Gene Modules from Pairwise Gene Expression
We naturally expect functional relevance across gene collections inferred from gene expression patterns, and one key principle shared by diverse module detection algorithms is the measurement of pairwise relationships. Common approaches in gene module detection, principal component analysis(PCA) and nonnegative matrix factorization(NMF), both leverage this pricniple, interpreting gene module as a linear combination of individual gene instances.
- In PCA, For normalized gene-feature matrix $X$, we compute $X^TX$; which represents gene-gene graph where edge weight corresponds to the inner product of two feature vectors.
- NMF(nonnegative matrix factorization) has equivalence with K-means clustering on the bipartite graph(which nodes correspond to both cells and genes), applying relaxation by allowing soft assignments to the K clusters[1].
- Other non-linear algorithm, Hotspot[2], explicitly computes local autocorrelation on the defined neighborhood graph for all selected feature pairs.
Despite differences in methodology, these approaches share a conceptual foundation: modeling gene-gene relationships as a graph structure. This observation naturally leads to a question: how useful are the detected gene modules in a biological context? An elegant framework addressing this question is sc-linker[3]. This method identifies gene modules contrastively—focusing on cell-type and disease-specific patterns—and then links these modules to complex trait associations using GWAS summary statistics. By doing so, sc-linker not only discovers expression-driven modules but also prioritizes those that are enriched for trait heritability, effectively nominating them as functional categories relevant to disease biology.
A case of sc-linker
sc-linker define gene module $M$ as a linear combination of gene $x_i$:
\(P=\sum_{i=1}^N w_ix_i\)
where $x_i$ is the expression of gene $i$ and $w_i$ is the corresponding weight. Definition of gene module is roughly categorized into three types:
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$M_{cell}$(cell-type specific): \(M_{cell}=\sum w_ix_i \text{ where } w_i=\sigma(\chi_i) \text{ for } \chi_i=-2log(P_i)~\chi_2^2\). Here, $P_i$ is a p-value after DE test comparing specific cell type $C$ to all others, and $\sigma(\cdot)$ is a min-max scaling function.
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$M_{dis}$(disease dependent): Defined similarly to $M_{cell}$, but the p-values are derived from disease vs. healthy comparisons.
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$M_{proc}$(cellular process): Obatined using contrastive NMF, which learns shared and condition-specific components from healthy and diseased scRNA-seq data.
Contrastive(case-control) NMF for Module identification:
Given two scRNA-seq feature matrices:
- $H_{P\times N_1}$ for healthy samples
- $D_{P\times N_2}$ for diseased samples
Conventional NMF decompose each matrix into:
\[H_{P\times N_1}\approx [L^{CH}L^{UH}]F^H, D_{P\times N_2}\approx [L^{CD}L^{UD}]F^D\],where $L^H=[L^{CH} L^{UH}], L^D=[L^{CD}L^{UD}]$ contain shared($L^{CH}, L^{CD}$) component and unique components (\(L^{UH}, L^{UD}\)).
In the contrastive setting, we enforce similarity between shared components by introducing a penalty term $\Vert L^{CH}-L^{CD}\Vert $. This term is added to conventional NMF obejctive thus it leads to the minimzation of objective $Q$:
\[Q=\frac{1}{2} \Vert{H-L^HF^H}\Vert _F^2+\frac{1}{2}\Vert{D-L^DF^D}\Vert _F^2+\frac{\mu}{2}(\Vert{L^H} \Vert _F^2+\Vert{L^D}\Vert _F^2)+\frac{\gamma}{2}(\Vert{L^{CH}-L^{CD}}\Vert _F^2)\]Compuptation of gradient $\nabla Q(L^H), \nabla Q(L^D), \nabla Q(F^H), \nabla Q(F^D)$ yields a multiplicative update rule, derived from splitting the gradient into positive and negative components[4]:
\[\nabla Q(L)=Q_+ - Q_-, L \leftarrow L \circ \frac{Q_-}{Q_+}\]This encourages shared components to capture common structure while allowing disease-specific modules to emerge.
Linking Gene Modules to GWAS via s-LDSC
Once modules are defined, sc-linker treats each module as a functional category and uses stratified LDSC (s-LDSC)[5] to quantify its contribution to trait heritability.
It is a long journey to introduce s-LDSC from scratch, but some key concepts include:
- regressing GWAS summary statistics(chi-square statistic) with LD score(sum of LD $r^2$ values between that SNP and all others in the region) yields a estimate of hearitability and confounding factors such as population structure.
- s-LDSC extends LDSC accounting for the functional category of SNPs
- in
sc-linkerstudy, genes comprising the module are mapped to SNPs via enhancer-gene mapping(Roadmap-ABC)[6-10] strategy thus translates gene functional categories to a collection of mapped SNPs.
These consecutive steps in identifying gene programs and linking with GWAS summary statistics, provide a foundational framework in identifying and applying a collection of genes to interpret complex, context-dependent hierarchy across variant-enhancer-gene-trait.
Reference
[1] Ding, C., Li, T., & Peng, W. (2008). On the equivalence between non-negative matrix factorization and probabilistic latent semantic indexing. Computational Statistics & Data Analysis, 52(8), 3913-3927.
[2] DeTomaso, D., & Yosef, N. (2021). Hotspot identifies informative gene modules across modalities of single-cell genomics. Cell systems, 12(5), 446-456.
[3] Jagadeesh, K. A., Dey, K. K., Montoro, D. T., Mohan, R., Gazal, S., Engreitz, J. M., … & Regev, A. (2022). Identifying disease-critical cell types and cellular processes by integrating single-cell RNA-sequencing and human genetics. Nature genetics, 54(10), 1479-1492.
[4] Lee, Daniel, and H. Sebastian Seung. “Algorithms for non-negative matrix factorization.” Advances in neural information processing systems 13 (2000).
[5] Finucane, Hilary K., et al. “Partitioning heritability by functional annotation using genome-wide association summary statistics.” Nature genetics 47.11 (2015): 1228-1235.
[6] Ernst, J., Kheradpour, P., Mikkelsen, T. S., Shoresh, N., Ward, L. D., Epstein, C. B., … & Bernstein, B. E. (2011). Mapping and analysis of chromatin state dynamics in nine human cell types. Nature, 473(7345), 43-49.
[7] Kundaje, A., Meuleman, W., Ernst, J., Bilenky, M., Yen, A., Kheradpour, P., … & Roadmap Epigenomics Consortium. (2015). Integrative analysis of 111 reference human epigenomes. Nature, 518(7539), 317.
[8] Liu, Y., Sarkar, A., Kheradpour, P., Ernst, J., & Kellis, M. (2017). Evidence of reduced recombination rate in human regulatory domains. Genome biology, 18(1), 193.
[9] Fulco, C. P., Nasser, J., Jones, T. R., Munson, G., Bergman, D. T., Subramanian, V., … & Engreitz, J. M. (2019). Activity-by-contact model of enhancer–promoter regulation from thousands of CRISPR perturbations. Nature genetics, 51(12), 1664-1669.
[10] Nasser, Joseph, et al. “Genome-wide enhancer maps link risk variants to disease genes.” Nature 593.7858 (2021): 238-243.
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