Quantifying the effect of experimental perturbations at single-cell resolution

The authors found cases where subsets within a cluster had divergent perturbation responses that published cluster-level analyses missed.

“We demonstrate cases where subsets of a cluster exhibit divergent responses to a perturbation that were missed in published analysis that was limited to clusters derived using data geometry alone.”

The sample-associated density estimate for two conditions can be computed on 50,000 cells in under 8 minutes in a free Google Colaboratory notebook.

“The sample-associated density estimate for two conditions can be calculated on a dataset of 50,000 cells in less than 8 min in a free Google Colaboratory notebook”

The authors generated and characterized a human pancreatic islet dataset cultured for 24 hours with and without IFN-γ.

“we generated and characterized a dataset of human pancreatic islet cells cultured for 24 h with and without interferon-gamma (IFN-γ)”

The responsive-mid cluster matched a transitional population with a structured distribution of input sample signals.

“The responsive–mid cluster matches our characterization of a transitional population with a structured distribution of input sample signals.”

The VFC-based approach recovers gene signatures more accurately than directly comparing two samples.

“we show in the following section that this approach is capable of recovering gene signatures more accurately than directly comparing two samples.”

MELD is robust to the number of captured cells, with only a 10% performance decrease when 65% of cells in the T cell dataset were removed.

“We also determined that the MELD algorithm is robust to the number of cells captured in the experiment, with only a 10% decrease in performance when 65% of the cells in the T cell dataset were removed (Supplementary Fig. 5).”

Twelve of the 28 published clusters warranted further VFC subclustering, yielding 50 final cluster labels.

“We found that 12 of the 28 published clusters warranted further subclustering with VFC, resulting in a total of 50 final cluster labels (Suppplementary Fig. 10j).”

Knockout of negative TCR regulators including PTPN6, PTPN11, and EGR3 slightly increased the relative likelihood of the stimulation condition.

“We also found a slight increase in relative likelihood of the stimulation condition in cells in which negative regulators of TCR activation are knocked out, including PTPN6, PTPN11 and EGR3.”

VFC analysis found that several clusters contained biologically distinct subpopulations with divergent responses to chd knockout.

“Using VFC analysis, we found that several of these clusters contained biologically distinct subpopulations of cells with divergent responses to chd knockout.”

Two beta cell clusters had intermediate IFN-γ relative likelihood values.

“Notably, we found two clusters of beta cells with intermediate IFN-γ relative likelihood values.”

Established marker genes identified three major islet populations corresponding to alpha, beta, and delta cells.

“We then used established marker genes of islet cells35 to identify three major populations of cells corresponding to alpha, beta and delta cells”

• cites [35] Muraro, M. J. et al. A single-cell transcriptome atlas of the human pancreas. Cell Syst. 3, 385–394 (2016).Not checked yet

  Not checked in the free run.

One beta cluster labeled non-responsive had high-frequency input sample signals indicative of a non-responsive population.

“Examining the distribution of input sample signals values in these intermediate cell types, we find that one cluster, which we label as non-responsive, exhibits high-frequency input sample signals indicative of a population of cells that does not respond to an experimental treatment.”

A rank sum test identified 911 genes differentially expressed in all three endocrine cell types between clusters with highest and lowest IFN-γ relative likelihood.

“Using a rank sum test to identify genes that change the most between the clusters with highest and lowest IFN-γ relative likelihood values within each endocrine population, we identified 911 genes differentially expressed in all three cell types.”

Insulin showed the greatest expression difference, approximately 2.5-fold increased in non-responsive cells.

“The gene with the greatest difference in expression was insulin, the major hormone produced by beta cells, which is approximately 2.5-fold increased in the non-responsive cells (Fig. 6f).”

Applying VFC to each endocrine cell type identified nine clusters in total.

“We next applied VFC to each of the three endocrine cell types and identified a total of nine clusters.”

The T cell activation signature contained 2,335 genes with q < 0.05 by rank sum test with Benjamini-Hochberg correction.

“We identified 2,335 genes with a q value < 0.05 as measured by a rank sum test with a Benjamini–Hochberg false discovery rate correction27.”

• cites [27] Fischer, D. Theislab/diffxpy. Theis Lab https://github.com/theislab/diffxpy (2020).Not checked yet

  Not checked in the free run.

The authors' results agree with bulk RNA-seq validation showing strongest TCR response depletion with LCK, ZAP70, and LAT knockout.

“Encouragingly, our results agree with the bulk RNA sequencing validation experiment in ref.13, showing strongest depletion of TCR response with knockout of kinases LCK and ZAP70 and adaptor protein LAT.”

• cites [13] Datlinger, P. et al. Pooled CRISPR screening with single-cell transcriptome readout. Nat. Methods 14, 297–301 (2017). sourceNot checked yet

  Not checked in the free run.

A noticeable donor-ID batch effect was observed, driven by maximum glucagon, insulin, and somatostatin expression in alpha, beta, and delta cells.

“Examining the expression of marker genes for major cell types of the pancreas, we observed a noticeable batch effect associated with the donor ID, driven by the maximum expression of glucagon, insulin and somatostatin in alpha, beta and delta cells, respectively (Supplementary Fig. 11a).”

The consensus IFN-γ signature included JAK-STAT pathway genes such as STAT1 and IRF1 and antiviral response genes such as MX1, OAS3, ISG20, and RSAD2.

“This consensus signature includes activation of genes in the JAK-STAT pathway, including STAT1 and IRF1 (ref.38), and in the IFN-mediated antiviral response, including MX1, OAS3, ISG20 and RSAD2 (refs.39–41).”

• cites [38] Ramana, C. V., Gil, M. P., Schreiber, R. D. & Stark, G. R. Stat1-dependent and -independent pathways in IFN-γ-dependent signaling. Trends Immunol. 23, 96–101 (2002). sourceNot checked yet

  Not checked in the free run.

• cites [39] Sadler, A. J. & Williams, B. R. G. Interferon-inducible antiviral effectors. Nat. Rev. Immunol. 8, 559–568 (2008). sourceNot checked yet

  Not checked in the free run.

• cites [40] Fitzgerald, K. A. The interferon inducible gene: viperin. J. Interferon Cytokine Res. 31, 131–135 (2011). sourceNot checked yet

  Not checked in the free run.

• cites [41] Zheng, Z., Wang, L. & Pan, J. Interferon-stimulated gene 20-kDa protein (ISG20) in infection and disease: review and outlook. Intractable Rare Dis. Res. 6, 35–40 (2017). sourceNot checked yet

  Not checked in the free run.

Several clusters had a wide range of chordin-relative likelihood values, suggesting cells within them had different perturbation responses.

“However, we observed that several clusters had a wide range of chordin-relative likelihood values, suggesting that there are cells in these clusters with different perturbation responses.”

Manual annotation of the three largest VFC-subdivided clusters revealed previously unreported effects of chd loss of function.

“To determine the biological relevance of the VFC clusters, we manually annotated each of the three largest clusters subdivided by VFC, revealing previously unreported effects of chd loss of function within this dataset.”

For the GO terms highlighted in ref. 13, the MELD gene list had the highest combined score among all examined terms.

“Considering the Gene Ontology (GO) terms highlighted in ref.13, we found that the MELD gene list has the highest combined score in all of the gene terms we examined (Fig. 4d).”

• cites [13] Datlinger, P. et al. Pooled CRISPR screening with single-cell transcriptome readout. Nat. Methods 14, 297–301 (2017). sourceNot checked yet

  Not checked in the free run.

The most highly activated alpha cell cluster was enriched in donor 2.

“Furthermore, the most highly activated alpha cell cluster is enriched in donor 2.”

The MELD/VFC analysis framework outperforms the current best practice of clustering cells by gene expression and then computing differential abundance and differential expression within clusters.

“We have shown that our analysis framework improves over the current best practice of clustering cells based on gene expression and calculating differential abundance and differential expression within clusters.”

The α-decaying kernel is susceptible to batch effects.

“Although the kernel in Equation (4) provides an effective way of capturing neighborhood structure in data, it is susceptible to batch effects.”

• AssumptionRests on an assumption: Batch effects can make within-batch affinities much higher than between-batch affinities, creating separation by batch rather than biological state.

In the zebrafish dataset, published clusters contained biologically relevant subpopulations with divergent responses to the experimental perturbation.

“In our analysis of the zebrafish dataset, we showed that the published clusters contained biologically relevant subpopulations of cells with divergent responses to the experimental perturbation.”

Gene set enrichment analysis of the signature genes found strong enrichment for IFN signaling pathway terms.

“To confirm the validity of our gene signatures, we use EnrichR28 to perform gene set enrichment analysis on the signature genes and found strong enrichment for terms associated with IFN signaling pathways (Supplementary Fig. 11d).”

• cites [28] Kuleshov, M. V. et al. Enrichr: a comprehensive gene set enrichment analysis web server 2016 update. Nucleic Acids Res. 44, W90–W97 (2016). sourceNot checked yet

  Not checked in the free run.

In quantitative comparisons, MELD recovers the true conditional likelihood more accurately than simpler label-smoothing methods.

“As we show in our quantitative comparisons, this formulation performs better at deriving the true conditional likelihood in quantitative comparisons than simpler label-smoothing algorithms.”

Gene signatures extracted using VFC-derived clusters outperform signatures from direct two-sample comparison or traditional clustering.

“We show that gene signatures extracted using these clusters outperform those derived from direct comparison of two samples or traditional clustering approaches.”

Laplacian regularization acts as a low-pass filter over Laplacian eigenvalues, with β setting the cutoff.

“Supplementary Fig. 13 shows that this function is a low-pass filter on the Laplacian eigenvalues with cutoff parameterized by β.”

Depending on α and ρ, the manifold heat filter can act as low-pass, band-pass, or high-pass.

“The parameters α and ρ change the filter from low pass to band pass or high pass.”

A previously unreported subpopulation of beta cells in pancreatic islets stimulated with IFN-γ appeared unresponsive to stimulation.

“We also described a previously unpublished dataset of pancreatic islet cells stimulated with IFN-γ and characterized a previously unreported subpopulation of beta cells that appeared unresponsive to stimulation.”

The best β values change as a function of the number of cells generated by Splatter when the underlying geometry is held constant.

“Supplementary Fig. 14b shows how the optimal β values vary as a function of the number of cells generated using Splatter while keeping the underlying geometry the same.”

A coarse grid search over MELD parameters found that α = 0 and ρ = 1 gave the best performance irrespective of the β parameter.

“A coarse-grained grid search revealed that setting α = 0 and ρ = 1 performed best regardless of the β parameter.”

The frequency composition of sample labels and relative likelihood scores can serve as the basis for vertex frequency clustering.

“Second, we show that the frequency composition of the sample label and the relative likelihood scores can be used as the basis for a clustering algorithm that we call vertex frequency clustering (VFC).”

The heat kernel uses diffusion to adapt locally in regions of varying data density, as seen in single-cell data.

“This kernel uses diffusion to provide local adaptivity in regions of varying data density21, such as is observed in single-cell data.”

• cites [21] Botev, Z. I., Grotowski, J. F. & Kroese, D. P. Kernel density estimation via diffusion. Ann. Stat. 38, 2916–2957 (2010). sourceNot checked yet

  Not checked in the free run.

Nearly all prior work comparing single-cell datasets first partitions the data discretely before downstream analysis.

“Almost all previous work quantifying differences between single-cell datasets relies on discrete partitioning of the data before downstream analysis9–16.”

• cites [9] Patel, A. P. et al. Single-cell RNA-seq highlights intratumoral heterogeneity in primary glioblastoma. Science 344, 1396–1401 (2014). sourceNot checked yet

  Not checked in the free run.

• cites [10] Finak, G. et al. MAST: a flexible statistical framework for assessing transcriptional changes and characterizing heterogeneity in single-cell RNA sequencing data. Genome Biol. 16, 278 (2015). sourceNot checked yet

  Not checked in the free run.

• cites [11] Tirosh, I. et al. Dissecting the multicellular ecosystem of metastatic melanoma by single-cell RNA-seq. Science 352, 189–196 (2016). sourceNot checked yet

  Not checked in the free run.

• cites [12] Jaitin, D. A. et al. Dissecting immune circuits by linking CRISPR-pooled screens with single-cell RNA-seq. Cell 167, 1883–1896 (2016). sourceNot checked yet

  Not checked in the free run.

• cites [13] Datlinger, P. et al. Pooled CRISPR screening with single-cell transcriptome readout. Nat. Methods 14, 297–301 (2017). sourceNot checked yet

  Not checked in the free run.

• cites [14] Gao, X., Hu, D., Gogol, M. & Li, H. ClusterMap: comparing analyses across multiple single cell RNA-seq profiles. Bioinformatics 35, 3038–3045 (2018). sourceNot checked yet

  Not checked in the free run.

• cites [15] Wagner, D. E. et al. Single-cell mapping of gene expression landscapes and lineage in the zebrafish embryo. Science 360, 981–987 (2018). sourceNot checked yet

  Not checked in the free run.

• cites [16] Farrell, J. A. et al. Single-cell reconstruction of developmental trajectories during zebrafish embryogenesis. Science 360, eaar3131 (2018). sourceNot checked yet

  Not checked in the free run.

The authors' method measures perturbation effects per cell using a continuous quantity across the transcriptomic space.

“Here we quantify the effects of perturbations at the single-cell level using a continuous measure of the effect of a perturbation across the transcriptomic space.”

The Gaussian kernel density estimator can be generalized to a manifold by using the eigenbasis of the Laplace operator as the kernel basis.

“However, this basis is not well defined for a Riemannian manifold, so we, instead, use the eigenbasis of the Laplace operator as our kernel basis.”

Analyzing experimental effects only at the cluster level loses power available from single-cell data.

“However, reducing experimental analysis to the level of clusters sacrifices the power of single-cell data.”

The sample-associated relative likelihood can identify cell states that are most and least affected by an experimental treatment.

“First, the relative likelihoods of each condition can be used to identify the cell states most and least affected by an experimental treatment.”

Sample-associated density estimates allow identification of cells most and least affected by a perturbation.

“With the sample-associated density estimates, it is now possible to identify the cells that are most and least affected by an experimental perturbation.”

Sample-associated relative likelihoods can be applied to scRNA-seq perturbation studies with different experimental designs.

“The sample-associated relative likelihoods can be used to analyze scRNA-seq perturbation studies of varying experimental designs.”

VFC identifies populations with similar perturbation responses at the granularity of the response.

“VFC identifies populations of cells that are similarly affected (enriched, depleted or unchanged) between conditions at the level of granularity of the perturbation response.”

MELD and VFC are suitable for characterizing a biological process such as TCR activation in a complex Cas9 knockout screen.

“Together, these results show that the MELD and VFC algorithms are suitable for characterizing a biological process, such as TCR activation in the context of a complex Cas9 knockout screen.”

Incomplete penetrance is expected in the zebrafish dataset because Cas9 mutagenesis is mosaic.

“We expect incomplete penetrance of the perturbation in this dataset because of the mosaic nature of Cas9 mutagenesis29.”

• cites [29] Yen, S.-T. et al. Somatic mosaicism and allele complexity induced by CRISPR/Cas9 RNA injections in mouse zygotes. Dev. Biol. 393, 3–9 (2014). sourceNot checked yet

  Not checked in the free run.

The VFC feature basis combines frequency information, relative likelihood magnitude, and the cell similarity graph to identify phenotypically similar populations with uniform perturbation response.

“The VFC feature basis combines this frequency information with the magnitude of the sample-associated relative likelihood and the cell similarity graph to identify phenotypically similar populations of cells with uniform response to a perturbation.”

Clustering cells by transcriptome alone and then testing differential representation is problematic because cluster sizes may not match responsive population sizes.

“Existing methods cluster cells by transcriptome alone and then attempt to quantify the degree to which these clusters are differentially represented in the two conditions. However, this is problematic because the granularity, or sizes, of these clusters might not correspond to the sizes of the cell populations that respond similarly to experimental treatment.”

The non-responsive beta cluster resembles a recently described extreme beta cell population with elevated insulin mRNA that is more abundant in diabetic mice.

“This cluster of cells bears resemblance to a recently described ‘extreme’ population of beta cells that exhibit elevated insulin messenger RNA (mRNA) levels and are found to be more abundant in diabetic mice36,37.”

• cites [36] Xin, Y. et al. Pseudotime ordering of single human β-cells reveals states of insulin production and unfolded protein response. Diabetes 67, 1783–1794 (2018). sourceNot checked yet

  Not checked in the free run.

• cites [37] Farack, L. et al. Transcriptional heterogeneity of beta cells in the intact pancreas. Dev. Cell 48, 115–125 (2019). sourceNot checked yet

  Not checked in the free run.

MELD and VFC outperform existing methods for analyzing multiple scRNA-seq datasets from different experimental conditions.

“These results confirm that MELD and VFC outperform existing methods for analyzing multiple scRNA-seq datasets from different experimental conditions.”

The published clusters' resolution is too coarse because chordin-relative likelihood values are very widely distributed in several clusters.

“We infer that the resolution of the published clusters is too coarse because the distribution of chordin-relative likelihood values is very large for several of the clusters.”

VFC provides a feature basis for clustering based on the spectrogram of sample indicator signals.

“The VFC algorithm provides a feature basis for clustering based on the spectrogram23 of the sample indicator signals, which can be thought of as a histogram of frequency components of graph signals.”

• cites [23] Shuman, D. I., Ricaud, B. & Vandergheynst, P. Vertex-frequency analysis on graphs. Applied Comput. Harmon. Anal. 40, 260–291 (2016). sourceNot checked yet

  Not checked in the free run.

The magnitude of the relative likelihood alone does not fully describe differences in perturbation response.

“However, the magnitude of the relative likelihood does not give a complete picture of differences in response to a perturbation.”

Distinguishing transitional from even-mixture cases requires considering both the relative likelihood magnitude and the frequency of input sample indicator signals over the manifold.

“To differentiate between such scenarios, we must consider not only the magnitude of the sample-associated relative likelihood but also the frequency of the input sample indicator signals over the manifold.”

The gene signature from MELD and VFC outperformed standard differential expression analyses for a real-world experimental perturbation.

“Furthermore, the gene signature identified using the MELD and VFC outperformed standard differential expression analyses to identify the signature of a real-world experimental perturbation.”

Clustering cells before quantifying compositional differences can miss divergent responses of subpopulations within a cluster.

“Clustering before quantifying compositional differences can fail to identify the divergent responses of subpopulations of cells within a cluster.”

Sample-associated relative likelihood can quantify differences among more than two conditions.

“Although most of the analysis here focuses on two-condition experiments, we show that it is possible to use the sample-associated relative likelihood to quantify the differences between more than two conditions.”

The MNN kernel adjustment alleviates batch effects in graph construction.

“To alleviate such effects, we adjust the kernel construction using an approach inspired by recent work in ref.49 on the MNN kernel.”

• cites ref.49Not checked yet

  Not checked in the free run.

• AssumptionRests on an assumption: Batch effects can make within-batch affinities much higher than between-batch affinities, creating separation by batch rather than biological state.

For one parameter choice, MELD's density estimate matches Gaussian KDE on the graph.

“showing that our density estimate with a specific parameter set is equivalent to the Gaussian density estimate on the graph.”

• AssumptionThe authors state this as an assumption: Single-cell data lie on a low-dimensional manifold embedded in ambient space.

Normalizing for density permits density estimation on the underlying manifold even with non-uniform sampling.

“This correction allows us to estimate density over the underlying geometry of the manifold even in the case where data are not uniformly sampled.”

• AssumptionThe authors state this as an assumption: Single-cell data lie on a low-dimensional manifold embedded in ambient space.
• AssumptionRests on an assumption: Real single-cell data are rarely uniformly sampled from a manifold.

The sample-associated relative likelihood algorithm's key innovation is quantifying perturbation effects at single-cell resolution using manifold learning theory.

“The key innovation in the sample-associated relative likelihood algorithm is quantifying the effect of a perturbation at the resolution of single cells using theory from manifold learning.”

MELD analysis can reveal biological insights in a clinically relevant experiment.

“Together, these results demonstrate the utility of MELD analysis to reveal biological insights in a clinically relevant biological experiment.”

With infinite samples, MELD's density estimate should approach the density on the continuous data manifold.

“This density estimate, as the number of samples goes to infinity, should converge to the density estimate along a continuous manifold formed by the data.”

• AssumptionThe authors state this as an assumption: Single-cell data lie on a low-dimensional manifold embedded in ambient space.

MELD analysis identified a beta cell population with high insulin expression that appears unaffected by IFN-γ stimulation.

“Here, we applied MELD analysis to identify the signature of IFN-γ stimulation across alpha, beta and delta cells, and we identified a population of beta cells with high insulin expression that appears unaffected by IFN-γ stimulation.”

A squared Laplacian allows the manifold heat filter to simplify to Laplacian regularization.

“In our results, we demonstrate a manifold heat filter that may be reduced to Laplacian regularization using a squared Laplacian.”

For data drawn from a smooth density on the manifold, recovering manifold geometry requires dividing out that density.

“When the data are, instead, sampled from a smooth density X~q(x) over the manifold, then the density must be normalized out to recover the geometry of the manifold.”

• cites ref.50Not checked yet

  Not checked in the free run.

• AssumptionThe authors state this as an assumption: Single-cell data lie on a low-dimensional manifold embedded in ambient space.
• AssumptionRests on an assumption: Real single-cell data are rarely uniformly sampled from a manifold.

Unlike previous Laplacian filters, MELD parameters allow analysis of signals that combine several underlying changes at various frequencies.

“In contrast to previous works using Laplacian filters, our parameters allow analysis of signals that are combinations of several underlying changes occurring at various frequencies.”

Setting β = tλmax, α = 0, and φ = 1 makes the graph heat-equation solution match MELD.

“This is equivalent to MELD when β = tλmax, α = 0 and ϕ = 1.”

• AssumptionThe authors state this as an assumption: Single-cell data lie on a low-dimensional manifold embedded in ambient space.

When ρ is an integer, columns of L^ρ are localized to ρ-hop neighborhoods.

“Each column of Lk is k − hoplocalized, meaning that Lijk is non-zero if and only if the there exists a path length k between vertex i and vertex j”

• cites ref.56, Section 5.2Not checked yet

  Not checked in the free run.

Combining samples from several distributions improves estimation of the shared manifold by using all data.

“This allows us to use samples from multiple distributions, in our case distributions over cellular states, which allows a better estimate of underlying manifold using all available data.”

• AssumptionThe authors state this as an assumption: Single-cell data lie on a low-dimensional manifold embedded in ambient space.

Under uniform manifold sampling and a standard Gaussian graph kernel, the heat-equation construction converges to Gaussian KDE on the manifold.

“When data X are sampled uniformly from the manifold M and the standard Gaussian kernel is used to construct the graph, then Theorem 2.1 of ref.70, which proves the convergence of the eigenvalues of the discrete graph Laplacian to the continuous Laplacian and implies (20), converges to the Gaussian KDE on the manifold.”

• cites ref.70Not checked yet

  Not checked in the free run.

• AssumptionThe authors state this as an assumption: Single-cell data lie on a low-dimensional manifold embedded in ambient space.

For low-pass filtering, Laplacian regularization is a special case of the manifold heat filter.

“Laplacian regularization is a sub-problem of the manifold heat filter that we will discuss for low-pass filtering.”

Frequency trends of sample indicator signals allow inference of movements in cellular state space during experimental perturbation.

“The frequency trends of the sample indicator signals are important because they allow us to infer movements in the cellular state space that occur during experimental perturbation.”

The graph-filter frequency response for Laplacian regularization is hreg(λ) = (1+βλ)−1.

“Thus, Laplacian regularization is a graph filter with frequency response hreg(λ) = (1+βλ)−1.”

MELD can be interpreted as applying Gaussian KDE on the discrete data manifold.

“The MELD algorithm can be thought of as a Gaussian KDE over the discrete manifold formed by the data.”

• AssumptionThe authors state this as an assumption: Single-cell data lie on a low-dimensional manifold embedded in ambient space.

At intermediate transcriptomic states between the extreme states most enriched in either condition, both low- and middle-frequency components of the sample indicator signal are observed.

“We note that, at intermediate cell transcriptomic states between the extreme states that are most enriched in either condition, we observe both low- and middle-frequency sample indicator signal components; see the blue cell in the cartoon in Fig. 2a.”

Using the actual input signal as a feature lets the clustering use both frequency and sign information from the raw experimental signal.

“Second, using the actual input signal as a feature allows the clustering to consider both frequency and sign information in the raw experimental signal.”

For any function f whose frequency content is concentrated around λ0, the modulated function Mkf is concentrated in frequency around λk.

“Furthermore, for any function f whose frequency content is localized around λ0, (Mkf) is localized in frequency around λk.”

• cites ref.23Not checked yet

  Not checked in the free run.

By using windowed signals that are large and small, VFC can find clusters of both sizes that are similarly affected by an experimental perturbation.

“By considering windowed signals that are large (that is, contain many neighboring points) and small (that is, very proximal on the graph), we can identify clusters both large and small that are similarly affected by an experimental perturbation.”

Trends in the sample indicator signal linked to low-frequency eigenvectors of the graph Laplacian are interpreted as biological transitions between cell states.

“Because the low-frequency eigenvectors of the graph Laplacian identify smoothly varying axes of variance through a graph, we associate trends in the sample indicator signal associated with these low-frequency eigenvectors as biological transitions between cell states.”

• AssumptionRests on an assumption: The graph Laplacian's low-frequency eigenvectors capture smoothly varying directions of variance across a graph.

When a group of cells is completely unaffected by an experimental perturbation, only high-frequency variations in the sample indicator signal are expected, with no low-frequency transition component.

“Here, we expect to find only high-frequency variations in the sample indicator signal and no underlying transition or low-frequency component.”

Translating the normalized heat kernel, multiplying it element-wise with the target signal, and taking the Fourier transform yields a WGFT of f that is localized according to diffusion distance from each vertex to every other vertex.

“By translating this kernel, element-wise multiplying it with our target signal f and taking the Fourier transform of the result, we obtain a WGFT of f that is localized based on the diffusion distance23,57 from each vertex to every other vertex in the graph.”

• cites [23] Shuman, D. I., Ricaud, B. & Vandergheynst, P. Vertex-frequency analysis on graphs. Applied Comput. Harmon. Anal. 40, 260–291 (2016). sourceNot checked yet

  Not checked in the free run.

• cites [57] Perraudin, N., Ricaud, B., Shuman, D. & Vandergheynst, P. Global and local uncertainty principles for signals on graphs. APSIPA Trans. Signal Inform. Process. 7, E3 (2018); https://doi.org/10.1017/ATSIP.2018.2 sourceNot checked yet

  Not checked in the free run.

For non-uniform sampling, density normalization yields an estimate of the underlying geometry that is not biased by density.

“When the density is non-uniformly sampled from the manifold, this allows an estimation of the underlying geometry unbiased by density.”

• AssumptionThe authors state this as an assumption: Single-cell data lie on a low-dimensional manifold embedded in ambient space.
• AssumptionRests on an assumption: Real single-cell data are rarely uniformly sampled from a manifold.

Including the relative likelihood of each sample in the clustering basis lets VFC account for the degree of enrichment of each cell group between conditions.

“Our second contribution is the inclusion of the relative likelihood of each sample in our basis for clustering. This allows VFC to take into account the degree of enrichment of each group of cells between condition.”

The multi-scale approach eliminates the need to choose a window size.

“Foremost, it removes the complexity of picking a window size.”