AAnet Resolves a Continuum of Spatially Localized Cell States to Unveil Intratumoral Heterogeneity

AAnet is a neural network framework that resolves archetypal cancer cell states that map to distinct spatial locations within tumors, uncovering intratumoral heterogeneity across primary and metastatic tissues.

Abstract

Identifying functionally important cell states and structure within heterogeneous tumors remains a significant biological and computational challenge. Current clustering- or trajectory-based models are ill-equipped to address the notion that cancer cells reside along a phenotypic continuum. We present Archetypal Analysis network (AAnet), a neural network that learns archetypal states within a phenotypic continuum in single-cell data. Unlike traditional archetypal analysis, AAnet learns archetypes (AT) in a simplex-shaped neural network latent space. Using preclinical and clinical models of breast cancer, AAnet resolves distinct cell states and processes, including cell proliferation, hypoxia, metabolism, and immune interactions. Primary tumor ATs are recapitulated in matched liver, lung, and lymph node metastases. Spatial transcriptomics reveals archetypal organization within the tumor and intra-archetypal mirroring between cancer and adjacent stromal cells. AAnet identifies GLUT3 within the hypoxic AT that proves critical for tumor growth and metastasis. AAnet is a powerful tool, capturing complex, functional cell states from multimodal data.

Significance:

Defining critical cell states among cells that reside along a phenotypic continuum is a current biological and computational challenge. In this study, we present AAnet, a neural network that learns archetypal cell states of cancer cells. AAnet defines discrete spatially localized ATs that resolve intratumoral heterogeneity.

Introduction

Cancer cells can dynamically change their functional state to facilitate survival (1–3). This creates a dynamic phenotypic continuum of cell states within a tumor. Defining the breadth of cell states and their structural organization is currently a significant biological and computational challenge, yet is likely to reveal critical opportunities to perturb cancer progression.

The two predominant approaches for characterizing cell-state heterogeneity from single-cell transcriptomic data are clustering and trajectory inference (4, 5). Clustering partitions the cellular state space into discrete cell types, and trajectory inference identifies continuous paths that define a pattern of cellular dynamics. However, when there are no clear delineations between cellular states nor clear trajectories or lineage structure on the data manifold, neither approach suffices to map the cellular state space. Thus, to define biological similarities and differences within and between these tumors, there is a need for a method that can dissect cellular heterogeneity at single-cell resolution while maintaining continuous variation along the cell-state continuum. For this purpose, we turn to archetypal analysis (AA), a factor analysis technique first introduced by Cutler and Breiman (6). AA first extracts factors that represent the “archetypes” (ATs), or extreme states, of a dataset. Then, all data points can be described as a convex combination of ATs.

For biological systems, the archetypal framework captures trade-offs between multiple tasks, wherein observations near each AT are specialized for single tasks and observations far from each AT correspond to uncommitted or generalist phenotypes (7). AA has thus been useful for characterizing diversity in a range of settings, from morphology to immune ATs (7–9). In the context of cancer, AA has disentangled tumor diversity and selection (10, 11) and cellular plasticity (12–14). However, one known caveat of existing approaches and applications is that they assume the data are shaped as a simplex and do not infer ATs from the underlying nonlinear geometry (12, 15). In other words, AA models the data as a simplex, in which the extreme points are corners and other points are on the faces or are internal to the simplex (Fig. 1). Viewing this technique geometrically makes it clear that most datasets would not be accurately described by such a simplex in the ambient data space. Indeed, for single-cell data, which have been shown to be highly nonlinear and noisy, new AA techniques are needed to reveal expression programs, especially in the context of cancer (16). In the present work, we consider ATs to be extremal cellular states, analyzed in a designed latent space in which intratumoral cellular heterogeneity is described as a convex combination of latent ATs.

Figure 1.

Overview of experimental design, AAnet, and downstream analyses. In an MDA-MB-231 mouse model, scRNA-seq and spatial RNA-seq data were collected for primary tumors, and scRNA-seq was collected for LN, liver, and lung metastases. AAnet infers ATs by transforming the data into a simplicial latent space for AA. AAnet learns cellular ATs within each tumor and enables downstream analysis of gene set enrichment within and across tissue sites, spatial organizations, and ME associations. AAnet also identifies related ATs in human breast cancer tumors and archetypal similarity across cancer subtypes.

Archetypal Analysis network (AAnet) performs AA by learning a simplicial representation of the data using an autoencoder framework, i.e., a neural network that learns meaningful representations of the given data. Unlike other autoencoder architectures for single-cell analysis (17–21), AAnet is uniquely designed to perform AA and characterize heterogeneity within cellular clusters. Instead of fitting a simplex on the data space, the AAnet encoder learns the optimal transformation from the data space to a latent space bound by a simplex, and the decoder learns the transformation back to the data space. This means that AAnet learns ATs and the representation of each data point as a convex combination of ATs through nonlinear dimensionality reduction. This adds flexibility and accuracy to AT-state detection while preserving data geometry.

In this study, we apply AAnet to study triple-negative breast cancer (TNBC). These tumors lack the three well-known receptors for classification of breast cancer status (estrogen receptor (ER), progesterone receptor (PR), and human epidermal growth factor receptor 2 (HER2), making it a particularly heterogeneous subtype of breast cancer with a paucity of effective targeted therapies for treatment. Here, we use AAnet to deconvolute TNBC heterogeneity into biologically interpretable ATs. Using novel single-cell RNA sequencing (scRNA-seq) and spatial transcriptomics datasets modeling tumor formation and metastasis in vivo, AAnet identifies five ATs in primary tumors, demonstrating they are reproducibly defined by distinct cancer hallmarks. These include a proliferative AT associated with cell-cycle progression; an oxidative AT associated with oxidative phosphorylation (OXPHOS), reactive oxygen species (ROS) production, and adipogenesis; a hypoxic AT enriched for enzymes associated with oxygen-independent glycolysis; a cell-damage/death AT that captures variation introduced by technical factors; and an immunostimulatory AT with enriched expression of HLA genes and cytokines. We show that these ATs are biologically significant in that they (i) are recapitulated in distinct metastases, (ii) are spatially organized, and (iii) colocalize with distinct microenvironmental (ME) cell types and niches. Moreover, in a cohort of 34 human TNBC samples, AAnet reveals significant archetypal heterogeneity between patients. Notably, we identify a subset of patients defined by the hypoxic AT and validate GLUT3 as a critical regulator of that archetypal cell state, wherein metastatic growth is significantly inhibited in the absence of GLUT3. These findings highlight the powerful ability of AAnet to define biological function and organization of cancer cells and their microenvironment, potentially aiding in classifying patients according to the biological similarities within and across patient samples. Furthermore, AAnet defines core transcriptional programs driving distinct ATs, thereby suggesting potential therapeutic opportunities to target them.

Results

Defining In Vivo Cellular Heterogeneity in scRNA-seq Data of TNBC

Overview of AAnet

We, and others, have shown that cancer cells reside along a phenotypic continuum (3). The ability to identify critical cell states and the unique molecular programs that drive them should reveal mechanisms that enable cell-state adaption that is critical to cancer progression. For characterizing cell-state heterogeneity from scRNA-seq data, clustering and trajectory inference are considered the standard part of single-cell workflows and best practices (22). However, we show that when the data lies on a continuum without latent cluster structure (e.g., discrete cell types) or latent lineage structure (e.g., development axis), these approaches lack concordance across methods and are limited in their ability to meaningfully characterize the cellular state space (Fig. 2A; Supplementary Fig. S1).

Figure 2.

Overview of AAnet architecture. A, Primary tissue visualization is a continuum of cells. Clustering the data using standard Leiden clustering 100 times (with no parameters changed) results in shifting boundaries between adjacent clusters. B, (i) AAnet learns transformation of data into a latent space shaped as a simplex. (ii) Archetypal loss enforces that data lie in a latent space shaped like a k-dimensional simplex. (iii) Diffusion extrema loss infers the extrema from data geometry. The diffusion extrema also inform the number of ATs for downstream analysis. C, AAnet is initialized with diffusion extrema and learns affinity to each AT and decodes the data-space ATs. D, Manifold distance preservation score (DEMaP; ref. 24) of cluster representation vs. AAnet representation. E, AAnet latent space can be used to characterize continuous gene trends not easily characterizable with clustering.

AA provides a framework to identify “archetypes” or extreme states in a dataset and characterizes the data points as a convex combination of ATs (6, 7, 15, 23). Currently, the biggest challenge of AA is identifying the correct and relevant ATs. Linear AA methods, such as principal convex hull analysis (PCHA), fit a simplex onto the data in ambient space (6) but fail to correctly identify the extrema when the extrema in the ambient space do not conform to the simplicial data geometry. They are also prone to identify outliers as ATs. These methods thus prove inflexible in modeling complex datasets, such as scRNA-seq data.

To this end, we developed AAnet, a neural network for nonlinear AA (See “Methods”). AAnet learns a low-dimensional latent representation of the data as a regular simplex [Fig. 2B (i)]. This is achieved by regularizing the encoding layer of the neural network to encode points as convex coordinates based on the placement of archetypal points [Fig. 2B (ii)]. The autoencoder (i.e., encoder–decoder) architecture and archetypal regularization together ensure that the model learns an accurate transformation to a simplex representation of the data, and it can decode data from the simplicial space to generate new data. To add further robustness to noise and accuracy to the model, we developed an approach (See “Methods”) to identify extreme points based on the underlying data geometry, termed diffusion extrema [Fig. 2B (iii)]. These diffusion extrema serve as initial estimates of archetypal points, which the neural network further optimizes. As geodesic distances between extrema capture how distinct they are from each other, we use these distances to choose the number of ATs k and initialize the ATs to the first k diffusion extrema at the beginning of training (See “Methods”).

Once the model is trained, the simplicial latent representation can be used for exploration of the dataset. The vertices in the latent space, encoded by standard basis (“1-hot”) vectors, can be decoded to the ATs in the gene space, enabling characterization and comparison of their expression profiles. Furthermore, the archetypal space coordinates provide an interpretable measure of each cell’s affinity to each AT (Fig. 2C). We thus term the cellular affinity for a given AT the “archetypal affinity” of the cell. Importantly, these archetypal affinities retain more information about cellular relationships than clustering while maintaining the interpretability of cell types. We show that the simplicial latent representation better preserves geodesic distances (24) than the cluster representation (Fig. 2D), suggesting it could prove useful in tasks that depend on cluster annotations (25, 26). Finally, we can also represent signals, including gene expression, with respect to archetypal affinities, which allows characterization of continuous signals that is not possible with cluster-based enrichment analysis (Fig. 2E).

A comparison of AAnet with Other Approaches on Simulated Data

To compare AAnet with existing approaches for characterizing cell-state heterogeneity and AA, we designed a framework for simulating nonlinearly transformed, or curved, datasets with “ground truth” ATs (Supplementary Fig. S1A). For comparison, we simulated a curved dataset with four ground truth ATs, and points have been defined by their continuous affinity to each AT (Supplementary Fig. S1B). Trajectory inference approaches fail to capture all relationships within and between the ATs because of the inability to define all four axes of variation (See “Methods”, Supplementary Fig. S1C). Clustering approaches arbitrarily cluster the simplex, thus showing disagreement in the number of clusters and assignment of cells to clusters across algorithms (See “Methods”, Supplementary Fig. S1D). To test the ability to characterize signals defined by archetypal affinity, we simulated a nonlinear gene signal based on true affinity to AT 4 (Supplementary Fig. S1E). Clustering was unable to characterize the continuous signal, finding high mean expression in the other ATs, and trajectory inference showed low correlation between the signal and predicted pseudotime (Supplementary Fig. S1E).

Beyond clustering and trajectory inference paradigms, existing AA methods also show an inability to characterize nonlinear archetypal datasets. We compared AAnet with four other AA approaches in settings with two, three, and four ground truth ATs. Only AAnet captures the ground truth ATs (Supplementary Fig. S2A). Hypothesizing that this discrepancy was because of the nonlinear transformation, we tested performance of archetypal prediction for datasets of increasing nonlinearity (from linear to highly nonlinear). In all three settings, other AA methods mostly perform well for linear datasets but show worsening performance with increasing nonlinear transformation (Supplementary Fig. S2B). By contrast, AAnet is able to infer the true vertices with a strong performance at all levels of nonlinearity (${\gamma }_{extrema}\mathrm{=}5$, Supplementary Fig. S2B). Without the diffusion extrema loss (${\gamma }_{extrema}\mathrm{=}0$), AAnet lacks robustness at high degrees of nonlinearity (Supplementary Fig. S2B).

We next compared our method to infer the number of ATs with previous approaches. The first diffusion extrema are defined by the global structure of the dataset, in which subsequent extrema are points that are geodesically distant from previous extrema (See “Methods”, Supplementary Fig. S2C). Our approach to choose the number of ATs calculates the mean geodesic distance to all previous extrema and takes the knee point of this curve (27). This captures the correct number of ATs for each simulated dataset and shows improvement over an approach that uses the PCHA and explained variance (15), which fails wherein nonlinearity artificially adds variance, and an approach that uses the elbow point of the reconstruction loss of a held-out set (28), which identifies the correct number of ATs but requires retraining the model in addition to calculating diffusion extrema. Despite the dependence on the choice of the number of ATs, when the incorrect number is chosen, AAnet robustly identifies ATs that are close to the true vertices (See “Methods”, Supplementary Fig. S2D). Finally, beyond the predicted ATs, AAnet captures interpretable archetypal affinities. Visualizing the simulated gene signal against the inferred AT 4 affinity shows AAnet is able to recapitulate the nonlinear relationship with high correlation (Supplementary Fig. S2E).

Validation of AAnet on an Antigen-Specific CD8⁺ T-cell Dataset

To validate our method on real biological data, we leveraged a published single-cell dataset of tumor-specific CD8⁺ T cells (Supplementary Fig. S3A; ref. 29). Using a mouse model of lung adenocarcinoma designed to express neoantigens, Connolly and colleagues (29) identified a reservoir of antigen-specific stem-like T cells in the tumor-draining lymph node (dLN), which then migrated to the site of the tumor to terminally differentiate. Tumor-specific CD8⁺ T cells from early (8 weeks post infection) and late (17 weeks post infection) dLNs (top left), early and late tumors (top right), early dLNs and early tumors (bottom left), and late dLNs and late tumors (bottom right) were co-embedded with PHATE (Supplementary Fig. S3A; ref. 24). “Flags” (white, green, and blue) were hand-annotated on the basis of continuous expression of immune-related marker genes in distinct regions of the embeddings, wherein white corresponds to putative naive T cells (Sell, Lef1, and Ccr7), green corresponds to putative stem-like T cells (Xcl1, Slamf6, and Tcf7), and blue corresponds to putative exhausted T cells (Pdcd1, Havcr2, and Cd101) (Supplementary Fig. S3A). We refer to these flags as ATs.

We ran AAnet separately on each paired embedding to determine if we could recapitulate these ATs (Supplementary Fig. S3B). First, we compared marker gene expression for archetypal characterization. One AT in each paired embedding expressed markers of naive T cells; one AT in three paired embeddings expressed stem-like markers; and one AT in three paired embeddings expressed exhaustion markers (Supplementary Fig. S3C). Three ATs (LN AT2, week 8 AT2, and lung AT3) did not strongly express any markers, and these ATs correspond to regions of the manifold wherein the authors suggest there is weak evidence for prominent naive, stem-like, and exhausted ATs (29).

With the ATs expressing the markers of interest, we computed pairwise cosine similarity across all measured genes to determine if the cell-state classification captures a similarity beyond the marker genes of interest. This reveals three archetypal groups corresponding to naive, exhausted, and stem-like T-cell responses (Supplementary Fig. S3D). This bolsters the finding of Connolly and colleagues (29) by suggesting that these cell states share broader transcriptomic similarity not limited to the nine markers known to be associated with CD8⁺ T-cell states.

Finally, to highlight the use of the latent space learned by AAnet, we visualize marker expression versus the archetypal affinity for each AT in week-17 embedding (Supplementary Fig. S3E). In all cases, the corresponding marker genes are enriched, and the other marker genes are de-enriched. This also reveals the nonlinear dynamics of gene patterns corresponding to the distance in the latent space.

Together, this analysis corroborates the use of AAnet for characterizing datasets with continuous and nonlinear structure with respect to ATs. Furthermore, it validates the ability to compare ATs across datasets to identify unified signatures of response.

AAnet Identifies Five Major Archetypal Expression States in Primary Tumors of a TNBC Model

Having demonstrated the power of AAnet to deconstruct single-cell data into meaningful latent space ATs, we sought to use AAnet to address a critical question in cancer biology: How does the cell-state landscape change across primary and metastatic tumors? Resolving this question may lead to new strategies to prevent nongenetic adaptation that facilitates cancer progression.

To answer this, we generated a new scRNA-seq dataset using an in vivo model of TNBC. Highly metastatic MDA-MB-231 breast cancer cells were injected into the mammary fat pad of NOD/SCID IL2Rg^null (NSG) mice and left to grow (6–8 weeks, female, n = 4; Fig. 1). At 12 weeks, the mice underwent survival surgery to remove primary tumors and allow metastases to develop further. Primary tumors were dissociated into single cells, sorted by flow cytometry to capture the human cancer cells only (CD298⁺ cells; ref. 30), and immediately captured for scRNA-seq. After an additional 4 weeks in vivo, lungs, liver, and lymph nodes (LN) were harvested and CD298⁺ human tumor cells were sorted and captured for scRNA-seq. This hybrid human–mouse model has a number of important strengths that are well suited for this study: (i) the confident delineation of tumor cells from the surrounding microenvironment, (ii) the capture of matched tumors and metastases, and (iii) a homogeneous starting population to control for cell-intrinsic differences such as genetic clonality. With this model, we used AAnet to deconvolute the ATs underlying heterogeneity in primary tumors. A total of 28,478 cells were analyzed from four primary tumors (5,118–8,163 cells) after quality control (QC; See “Methods”).

We first examined the ATs contributing to cellular heterogeneity within primary tumors (Fig. 3A). Each primary tumor was analyzed with AAnet individually (tumor 1–4) and for all primary tumors combined (all). As hypothesized, each dataset showed a continuum of cellular expression states with multiple extrema, rather than discrete clusters or unidirectional trajectories (Fig. 3B; Supplementary Fig. S4A and S4B). Five ATs were identified in each dataset, representing distinct biological roles (Fig. 3B; Supplementary Fig. S4A and S4B). To elucidate the biology underlying these ATs, we identified differentially enriched genes at each AT versus all other ATs and uncommitted cells (Supplementary Table S1; See “Methods”). These genes were used to identify signature gene sets upregulated in each AT and conserved between replicates (FDR <0.05; Fig. 3C; Supplementary Table S1, See “Methods”). These genes and gene sets summarize the biology of each AT as follows:

Figure 3.

A, Experimental approach used to identify tumor ATs in TNBC with AAnet. MDA-MB-231 breast cancer cells were injected into the mammary fat pad of NSG mice and left to grow. At 12 weeks, primary tumors were resected, and human cancer cells were sequenced. AAnet identified ATs for downstream analysis. B, Embedding of TNBC tumor cells (all tumors combined) with five ATs indicated using colored circles. C, Enriched Hallmark gene sets associated with each AT in each tumor (colors indicate AT and numbers indicate tumor). P value is FDR-corrected; gene sets where P > 0.05 are shown in gray. Enrichment represents log₂ fold change. D, Heatmap of cosine similarity between archetypal expression vectors determined in each individual tumor (numbered) and all tumors combined (all). Orthologous ATs between samples are indicated using colors. E, Percentage of cells in each sample committed to each AT (colored) or uncommitted (UC) to an AT (gray).

Proliferative AT (blue): This AT is enriched for hallmarks of cell proliferation (Hallmark gene sets G2/M checkpoint and E2F targets) and growth (MYC targets V1/2 and mTORC1 signaling). Indeed, high expression of cell proliferation and hyperactivated MYC is a hallmark of clinical TNBC (31). The top markers of this cluster include CDC20, CDK1, and CDK4, genes that regulate cell-cycle phase transitions (32). Concomitantly, cell-cycle analysis revealed that >95% of cells were in either the S or G2/M phase (Supplementary Fig. S4C and S4D).

Oxidative/adipogenic AT (yellow): This AT is associated with hallmarks of oxidative metabolism and stress. OXPHOS is the most strongly enriched Hallmark gene set across all replicates, driven by electron transport chain (ETC) components that couple ATP synthesis to oxygen availability in the mitochondria (Supplementary Table S1; ref. 33). Notably, OXPHOS has been identified as the top canonical pathway associated with resistance to neoadjuvant chemotherapy and worse outcome in tumors isolated from patients with TNBC (34). MYC targets are also overrepresented in this AT (34), including many nuclear genes involved in mitogenesis. Genes in the ROS pathway are enriched in this AT, including the 4/6 peroxiredoxin family of antioxidant enzymes (PRDX1/2/4/6; ref. 35). Genes involved in adipogenesis are also significantly overrepresented in all replicates, suggesting fatty acid synthesis may contribute to the metabolic phenotype of this AT. Indeed, fatty acid oxidation is a metabolic vulnerability of MYC-driven TNBC (36).

Hypoxic AT (orange): This AT was significantly associated with the hypoxia hallmark in all replicates. In the clinical setting, a hypoxic phenotype is highly associated with TNBC compared with other breast cancer subtypes and is associated with cellular senescence and poor response to chemotherapy (37, 38). Under hypoxic conditions, most eukaryotic cells switch their metabolic pathways from mitochondrial OXPHOS to glycolysis. Indeed, this AT is also enriched for glycolytic enzymes (the glycolysis hallmark enriched in 3/4 replicates). These include class 1 glucose transporters SLC2A1 and SLC2A3, as well as genes linked to oxygen-independent energy production in cancer: ENO2, HK2, LDHA, and GAPDH (39). Ribosomal subunits also feature prominently among marker genes, suggesting a relationship between ribosome biogenesis and hypoxia. These data suggest that the cells of this AT resided in a low oxygen environment and used glycolytic metabolism.

Cell damage/death AT (amber): The top markers for this AT were genes encoded on the mitochondrial genome associated with cell damage or death (Supplementary Table S1; ref. 22). Analysis of the cells most strongly influenced by this AT showed high expression of mitochondrially-encoded components of the ETC, yet limited expression of somatically-encoded ETC components (Supplementary Fig. S4E). This associates this AT with damaged and dying cells, some of which could potentially arise from technical variables.

Immune-stimulatory AT (purple): AAnet identifies a unique cancer-cell AT in which cancer cells express high levels of immunostimulatory mRNAs, including CXCL1, IFITM2/3, BST2, HLA-A/B/C, B2M, and ICAM1 (Supplementary Table S1). Concordantly, Hallmark analysis showed enriched gene sets related to immune signaling (IFNG, IFNA, TNFA, and TGFB pathways) and apoptosis (TP53). The presence of tumor-infiltrating lymphocytes indicates good prognosis in TNBC (40). Therefore, immune signaling factors secreted by cells in this AT could promote tumor-infiltrating lymphocyte infiltration, as well as infiltration of pro-inflammatory macrophages and neutrophils, to mount an antitumor immune response and improve patient outcome (41). Analysis of cell cycle among the cells most strongly influenced by this AT showed 96% of cells in G1, perhaps suggestive of a G1 arrest (Supplementary Fig. S4C and S4D). These analyses indicate that this AT describes an apoptotic, immunostimulatory expression pattern in tumor cells.

ATs were highly conserved between primary tumors (Fig. 3D; Supplementary Fig. S4F). All ATs had an orthologous AT in each individual primary tumor and the combined dataset, with a high mean cosine similarity of 0.77 to 0.97. By contrast, individual ATs showed low pairwise cosine similarity (between −0.15 and −0.2), consistent with their classification of representing distinct cell states.

Having deconvoluted tumor cell heterogeneity into five biologically meaningful ATs, we then determined the association of each cell with each AT using AAnet. AAnet encodes a representation of each cell based on its relative association to each AT, in which the coordinates of each cell are nonnegative and sum to 1. We define cells with archetypal affinity for one AT greater than the sum of all other affinities (i.e., affinity >0.5) as “committed.” Cells that do not surpass this affinity threshold for any AT can be considered “uncommitted”; uncommitted cells seem to be viable, differentiated cells with a “generalist” transcriptional profile (Supplementary Fig. S4G). Using archetypal commitment to analyze the cellular composition of primary tumors, the abundance of cells committed to an AT was consistently >60% in all tumors (62%–67.8%), yet the abundance of committed cells varied between ATs (Fig. 3E). The proliferative AT was the most abundant AT in the combined analysis and in two of four individual tumors. The immunostimulatory AT was the second most abundant AT in the combined analysis and in two of four individual tumors. The oxidative/adipogenic AT represented a minor fraction of tumor cells across all samples, and the hypoxic AT was among the most variable. Together, these analyses show that AAnet deconvolutes cancer-cell heterogeneity into biologically meaningful ATs that are reproducibly detected in discrete tumors. It is thus possible to begin to unravel the archetypal complexity, interactions, and potential vulnerabilities of intratumoral heterogeneity.

AAnet Reveals Conserved and De Novo Heterogeneity across Distinct Metastatic Sites

Next, we used AAnet to identify ATs in metastases. scRNA-seq was performed on the matched LN, liver, and lung metastases that were collected 4 weeks after the resectioning of the primary tumor (Fig. 4A–C). A total of 42,250 metastatic cells were analyzed from the LNs (n = 46,604–19,224 cells per tumor), 42,647 cells from the liver (n = 47,136–18,671 cells per tumor), and 17,687 cells from the lungs (n = 35,599–6,379 cells per tumor) after the removal of one outlier lung sample during QC (See “Methods”, Supplementary Fig. S5A). We combined data from metastases per tissue separately and defined ATs in each site using AAnet.

Figure 4.

A, Approach to define tumor ATs in TNBC metastases from the LNs, liver, and lungs. MDA-MB-231 breast cancer cells were injected into the mammary fat pad of NSG mice and left to grow. At 12 weeks, primary tumors were resected, and human cancer cells were sequenced. At 16 weeks, the LN, liver, and lung metastases were resected and human cancer cells were sequenced. AAnet identified ATs for downstream analysis. B, Embedding of tumor cells from each tissue with similar ATs indicated using colored circles. C, Representative histologic sections of tumors and metastases taken for single-cell sequencing. D, Enriched Hallmark gene sets associated with ATs from each tissue (indicated using colors). P value is FDR-corrected; gene sets in which P > 0.1 are shown in gray. Enrichment represents log₂ fold change. LI, liver; LU, lungs; P, primary. E, Stacked bar plot showing the percentage of tumor cells in each tissue committed to each AT (colored) or uncommitted to an AT (gray). F, Stacked bar plot showing the cell-cycle phase of cells committed to each AT.

Cellular heterogeneity in metastases followed a phenotypic continuum akin to that observed in primary tumors (Fig. 4B). To characterize this continuum, AAnet robustly identified five ATs in the LNs, four in the liver, and five in the lungs, where all ATs detected in the primary had an ortholog in at least two metastatic sites with high cosine similarity, and uncommitted cells showed no differences in viability and differentiation to committed cells (Supplementary Fig. S5B–S5G). Moreover, a highly conserved pattern of hallmark enrichment was observed in orthologous ATs from different sites (Fig. 4D; Supplementary Table S2). These data demonstrate that the factors of variation contributing to cancer-cell heterogeneity were largely recapitulated in primary tumors and metastases.

Although orthologous ATs were identified in many different metastases, the analysis of archetypal affinity indicated that their relative contribution to cellular heterogeneity in each tissue was somewhat distinct.

LN metastases showed the strongest resemblance to primary tumors. All five ATs identified in the LNs had an ortholog in the primary tumor (Fig. 4D–F). The proportions of cells committed to each AT were also highly conserved, with the proliferative and immunostimulatory ATs being the most abundant in both tissues (Fig. 4E). Moreover, the changes in the relative abundance of oxidative, hypoxic, and cell-death ATs were marginal, collectively indicating cell heterogeneity is very similar between primary tumors and LN metastases.

Liver metastases differed from the primary tumors and LNs in the number and abundance of ATs. Only four of the five ATs identified in the primary tumors were detected in the liver, and more cells were committed to an AT than were in other tissues (Fig. 4D–F). The absence of cells committed to the oxidative AT was replaced by a greater proportion of cells committed to the proliferative and immunostimulatory ATs, both nearly double their proportions in the primary tumor (32.94% and 23.54%, respectively). Overall, although the ATs were somewhat conserved with primary tumors and LN metastases, their relative contribution to cellular heterogeneity distinguished liver metastases from the other tissues.

Lung metastases showed the greatest difference from primary tumors and the other metastases. Five ATs were detected in the lungs, four of which were orthologous to ATs in the primary tumor (Fig. 4D–F). Although still the most abundant, the proportion of cells committed to the proliferative AT was significantly lower than in other tissues (18.36%). The oxidative AT, enriched for OXPHOS and ROS hallmarks, was the second most abundant and almost equal to the proliferative AT (17.72%). Conversely, the hypoxic AT was not detected in the lung metastases. This finding is consistent with the notion that lung metastatic cellular heterogeneity is shaped by the oxygen-rich lung microenvironment.

AAnet identified a fifth AT unique to the lungs. Markers of this AT were significantly enriched for cancer hallmarks of an oxidative metabolism (OXPHOS and ROS pathway) driven by genes encoding ETC components (ATP5F1A and ATP5F1B; NDUFA4, NDUFA8, and NDUFA9; and COX7C) and antioxidant enzymes (PRDX2, PRDX4, and PRDX6), respectively (Supplementary Table S2). Additionally, hallmarks of TNF signaling via NF-κB, apoptosis, hypoxia, protein secretion, and coagulation were overrepresented, a profile of pathways indicating a similarity to the immunostimulatory AT. Importantly, enrichment of hypoxia was driven by a separate subset of glycolytic enzymes to those associated with the hypoxic AT in other tissues and did not include markers of HIF1A induction such as CITED2 (Supplementary Table S2).

Together, these analyses show that AAnet-defined ATs in primary tumors are also identified in metastases. This suggests that factors influencing cellular heterogeneity are highly conserved between primary tumors and metastases. Differences between tissues are potentially due to interactions with the metastatic microenvironment.

Spatial Transcriptomics Reveals a Distinct and Structured Spatial Organization of AAnet-Derived ATs within the Tumor

To further validate the significance and biology of the ATs identified by AAnet, we sought to investigate their structural organization within tumors. Thus, we performed spatial transcriptomics (Visium from 10× Genomics) on tissue sections from two primary tumors that were grown in vivo for 8 weeks. These samples were collected from two of the tumors used for scRNA-seq, with parts of these tumors frozen in Optimal Cutting Temperature (OCT) compound for sectioning. Gene expression was measured at 2,275 spatially distributed voxels in the two samples after QC (1,170 and 1,105 voxels, respectively), with each voxel-assaying expression in an area of approximately 3 to 10 cells in size (Fig. 5A).

Figure 5.

A, Histologic sections of TNBC xenografts used for spatial transcriptome sequencing (n = 2). B, Spatial transcriptomics data from primary tumor (consisting of human tumor and mouse microenvironment) are aligned to scRNA-seq primary tumor data using scMMGAN. Aligned spatial RNA-seq voxels are input into trained AAnet primary tumor model to determine archetypal affinity of each spot. C, Spatial voxels with a strong affinity (>0.3) for each AT (colored) as determined by scMMGAN and AAnet. ATs show spatial organization within primary tumor. D, Spatial plots colored by marker gene set score for AT. Gene set scores show spatial organization within primary tumor. E, DREMI score between archetypal affinity and marker gene set scores shows high correspondence, meaning that biology defining each AT is maintained in spatial map.

To map the ATs from the scRNA-seq to the spatial transcriptomic data, we first had to overcome the intrinsic differences in the data generated by these modalities. Raw data from these technologies are not directly comparable as they have very different sample processing protocols (digestion and droplets vs. freezing and sectioning) and biological resolutions (single cells vs. multiple cells). These differences create a batch effect, in which noise introduced in the process of data generation dominates the biological differences between samples. This batch effect was evident when spatial voxels were embedded with the scRNA-seq data from the primary tumor, as there was little alignment between the modalities despite being generated from matched biological samples (Supplementary Fig. S6A).

To address this, we leveraged our single-cell multimodal generative adversarial network (scMMGAN; See “Methods”; ref. 42). This approach uses adversarial learning to align data between modalities, enabling integrated downstream analysis while preserving data geometry (Fig. 5B). We used scMMGAN to generate an scRNA-seq measurement for each spatial voxel (scVoxels) that is aligned to the distribution of cells in our primary scRNA-seq dataset. This allows for direct comparison of spots and single cells without prior deconvolution, that is a spot with a high proportion of cells from one AT would map near the AT in the single-cell space with high affinity. We can then input the aligned spatial data into the AAnet encoder trained on the primary scRNA-seq data. This provides an archetypal representation for each spatial voxel (scVoxel) based on our previously characterized primary ATs (Fig. 5C; Supplementary Fig. S6B). The ability to extend AA to previously unseen data through the neural network framework is an important feature of AAnet versus the existing AA approaches.

With spatial transcriptomic data embedded as spatial voxels (scVoxels) in the AAnet latent space, we calculated their affinity to each AT and mapped back to their corresponding spatial location in the primary tumor (See “Methods”, Supplementary Fig. S6C). Archetypal affinities and marker gene set expression scores shared high mutual information across voxels (See “Methods”, Fig. 5D and E), indicating that the biology defining each AT had been retained in the spatial mapping process.

Crucially, we uncovered unique spatial organization of primary tumor ATs:

Proliferative AT (blue): Voxels with high affinity for the proliferative AT (affinity >0.3) formed the bulk of the tumor yet were markedly absent from the central areas of each tumor section. Areas with high affinity for this AT were enriched for cycling cells (Supplementary Fig. S6D). These findings were consistent with the scRNA-seq commitment analysis (Fig. 3E) indicating that proliferation was a dominant factor of cell heterogeneity in the primary samples.

Oxidative/adipogenic AT (yellow): Areas of the tumor with high affinity for the oxidative AT were located in close proximity to the proliferative AT (Fig. 5D). The affinity scores and expression of marker gene sets were also significantly correlated (Fig. 5E), indicating a relationship between oxidative metabolism and the proliferative cell state.

Hypoxic AT (orange): Strikingly, areas with a strong affinity for the hypoxic AT were localized to the central and peripheral regions of primary tumors, devoid of the proliferative and oxidative ATs (Fig. 5C). These areas showed enriched expression of markers associated with oxygen-independent glycolysis and ribosomal subunits (Fig. 5D and E).

Cell damage/death AT (amber): The cell-death AT was localized to areas with high expression of mitochondrial genes and was strongly correlated with the proliferative and oxidative ATs (Fig. 5C–E). Although preferential enrichment of mitochondrially-encoded genes is often used as a marker of cell death, mitochondrial genes encode critical components of the ETC necessary for oxidative energy production. Consequently, the association between this AT and the proliferative/oxidative ATs may identify oxygen-rich areas within primary tumors rather than cell death.

Immunostimulatory AT (purple): Affinity for the immunostimulatory AT was the highest surrounding the hypoxic AT within the tumor. These areas showed enriched expression (Fig. 5C) of cytokines and antigen-presenting proteins, such as CXCL1 and HLA-A/B/C/B2M (Fig. 5D and E), consistent with the hallmark associations of this AT. Notably, the immunostimulatory AT demarcates the hypoxic and proliferative cancer cell ATs.

Together, this analysis shows AAnet ATs have unique and distinct spatial organization within the tumor. Moreover, the organization of the AAnet-derived ATs is consistent with a model whereby the local microenvironment may play a critical role in determining the organization of cell states in primary tumors.

ATs Are Associated with Distinct Cell Types and ME Niches

With clear spatial organization of cancer cell ATs derived by AAnet within the tumor, we next sought to determine if, beyond spatial location, the tumor microenvironment may also be playing a role in archetypal development and commitment. We therefore assessed if ME cells were spatially structured into meaningful ME-ATs (Fig. 6A). The xenograft model enabled the separation of expression at each voxel into tumor and ME data based on alignment to the human or mouse genome, respectively. Then, we used AAnet to deconvolute the murine data into ME-ATs to investigate AT-specific cell types and biological processes.

Figure 6.

A, Approach to analyze tumor microenvironment from spatial transcriptomics data from primary tumor (consisting of human tumor and mouse microenvironment), wherein key ATs are identified using AAnet, characterized via cell-type enrichment, biological processes, archetypal associations, and signaling interactions. B, AAnet identifies six ATs in the tumor microenvironment. ME-AT affinity scores visualized for each spatial voxel. C, PanglaoDB enrichment of cell types associated with each ME-AT. P value is FDR-corrected. Enrichment represents log₂ fold change. D, Enrichment of top two biological processes from the Gene Ontology (GO) associated with each ME-AT. Processes are ranked by FDR (−log₁₀). E, ME-ATs (y-axis) significantly associated with each tumor AT (x-axis). Color reflects tumor AT; enrichment represents log₂ fold change. F, Top ranked ligand–receptor pairs expressed in spatial voxels with a strong affinity for each AT and their colocalized microenvironments. Capitalized gene symbols indicate genes with expression in tumor cells (human). Title case symbols indicate genes with expression in ME cells (mouse). Pairs are ranked by FDR (−log₁₀). G, Ligand–receptor interaction score across spatial voxels for top pairs associated with each AT.

AAnet defined six ME-ATs with unique patterns of spatial organization (Fig. 6B, See “Methods”). As each ME spot consists of multiple diverse, dynamic, and coordinating cell types and structures, to characterize the role of ME-AT in the organization and development of cancer-cell ATs, we assessed the enrichment of Gene Ontology biological processes and PanglaoDB cell type signatures for each ME-AT (Fig. 6C and D). These include a ductal ME-AT with high enrichment of mammary epithelial cell markers in voxels overlaying to the breast ducts and adipocyte ME-AT enriched for adipocyte markers and most highly expressed on the margins of the tumor sections with residual mammary fat pad. This correspondence with underlying histologic features provides orthogonal validation for AAnet. AAnet also identified a stromal ME-AT, enriched for fibroblasts, capillary-lining pericytes, and stromal cells, as well as biological processes related to ECM organization, OXPHOS, and angiogenesis. NSG mice are immunocompromised with only some innate immune activity (no mature T or B cells, deficient in NK cells, but do have macrophages, dendritic cells, and neutrophils). Notably, AAnet revealed distinct immune-associated ME-ATs including a neutrophil ME-AT, enriched for neutrophils and biological processes related to glycolysis, leukocyte chemotaxis, and hypoxia; a myeloid ME-AT enriched for markers of monocytes, macrophages, and dendritic cells, with an enriched response to IFN-γ, immune effector processes, and leukocyte-mediated cytotoxicity; and an antigen response ME-AT enriched for macrophages and genes involved in antigen processing and presentation of exogenous peptide antigen via MHC class II and immunoglobulin-mediated immune response (Fig. 6C and D). Therefore, AAnet deconvolutes the microenvironment into stromal and immune components.

Next, we explored the spatial relationship of the ME-ATs to the cancer-cell ATs and uncovered specific associations (Fig. 6E, See “Methods”). Areas of the tumor with a high affinity for the proliferative AT were strongly associated with the myeloid and stromal ME-ATs, whereas the oxidative/adipogenic AT showed preferential enrichment for the stromal ME-AT. This suggests that these ATs colocalize with highly metabolic and vascularized microenvironments. Both of these ATs were also associated with ductal and fat pad ME-ATs, concordant with the localization of these normal mammary gland structures in histologic sections. Notably, the hypoxic AT colocalized specifically with the neutrophil ME-AT. These interactions occurred at internal parts of the tumor section, indicating that these ATs are associated with decreased oxygen availability and increased neutrophil chemotaxis. The cell-death AT was strongly associated with the antigen ME-AT, likely indicating an active presentation of cancer cells to the immune system, cancer cell death, and phagocytosis of dead and dying cells by macrophages. Interestingly, the immunostimulatory cancer-cell AT, defined by its enrichment for gene sets related to immune signaling, was associated with the antigen and neutrophil ME-ATs, indicating a strong overlap in signaling and cell phenotype between the cancer and ME cells. Thus, we observed multiple examples of spatially localized phenotypic similarity between cancer and ME cells, most notably in metabolic and immune signaling.

To investigate direct cell–cell signaling mechanisms for phenotypic similarity, we analyzed ligand–receptor pairs (LR-pairs) for evidence of paracrine interactions (See “Methods”). We again used our hybrid model system to delineate between tumor (human) and ME (mouse) expression, allowing us to establish the direction of signaling. Specifically, LR-pairs with a ligand expressed in the human data and its cognate receptor expressed in the mouse data indicate signaling from the tumor to the microenvironment, and vice versa (Fig. 6F). We first determined the coexpression of annotated LR-pairs across all voxels and calculated their enrichment in areas with a high affinity for each AT. For all ATs, the proportion of ligands originating from the tumor and the microenvironment was approximately 50%. However, the strongest LR-pairs that localized to high-affinity regions differed between ATs.

Tumor cells expressed stromal growth factors in areas with a high affinity for proliferative and oxidative/adipogenic ATs. These included platelet-derived growth factor (AT ligand: PDGFC; ME receptor: Pdgfrb) and fibroblast growth factor (AT ligand: FGF12; ME receptor: Fgfr1) pathways that promote blood vessel formation (43). The hypoxic AT was characterized by interactions between CD44 expressed in tumor cells and a variety of ligands expressed in the microenvironment, including Mif, which may enhance neutrophil accumulation within the hypoxic AT. Examples of cancer ME cross-talk in cell-death AT are evidenced by APP-Cd74, wherein Cd74 is an MHC class II molecule, and Sema7a-ITGA1, wherein Sema7a is a potent immunomodulator. We also observe strong overlap in LR-pairs regulating immune–cancer cell cross-talk in the immunostimulatory AT including MMP9–tIgam, IL1A–Ilr1rap, and Hmgb1–TLR2. (Fig. 6G).

Given the prominence of immune cells to some ME–cancer-cell archetypal interactions, we explored those findings in a published scRNA-seq dataset of an immunocompetent MMTV-PyMT mouse model of TNBC (See “Methods”; ref. 44). AAnet was applied to single-cell data derived from tumors at week 17. We identified four ATs that show enrichment for cancer hallmarks in concordance with the human ATs identified in the MDA-MB-231 model (Supplementary Fig. S7A). In particular, murine AT1 shows enrichment for Slc2a1 and a hypoxia signature, matching the human hypoxic (orange) AT; murine AT2 shows enrichment for adipogenesis and OXPHOS, matching human oxidative (yellow) AT; murine AT3 is enriched for epithelial–mesenchymal transition and IFN-α response, matching human immunostimulatory (purple) AT; and murine AT4 is enriched for Pdha1 and proliferation/cell-cycle gene signatures, including E2F targets and MYC targets V1, matching human proliferative (blue) AT (Supplementary Fig. S7B). Of the committed cells, the hypoxic and oxidative ATs were the most abundant (Supplementary Fig. S7C).

At week 17, the cancer microenvironment in the MMTV-PyMT model primarily consists of myeloid cells, NK cells, fibroblasts, and T cells, annotated on the basis of marker genes delineated in the original work (Supplementary Fig. S7D). To assess how the stromal and immune compartments interact with the cancer ATs, we performed cell–cell communication analysis using CellPhoneDB (45) and visualized the count of significant interactions between cancer ATs and the microenvironment (Supplementary Fig. S7E). Despite murine AT3 (immunostimulatory) being the least abundant cancer AT, it shows the most abundant number of interactions with the cancer microenvironment, including with myeloid cells (as identified in the MDA-MB-231 model) and fibroblasts. This suggests this AT plays a functional role to maintain spatial organization and regulate the inflammatory microenvironment. Also as predicted, murine AT4 (proliferative) shows the highest number of interactions with fibroblasts, and murine AT1 (hypoxic) shows a high number of interactions with the myeloid compartment. Thus, these findings suggest that the ATs and related immunogenic processes are upheld in an immunocompetent setting.

Together, these results highlight the spatial colocalization of distinct cancer ATs with unique microenvironments, in which paracrine interactions that may enhance phenotypic similarity are an important determinant of intratumoral heterogeneity.

Intratumoral Metabolic Heterogeneity in TNBC and Alignment to Human Breast Cancers

Notably, the OXPHOS-enriched proliferative AT (blue) and the glycolytic-enriched hypoxic AT (orange) showed distinct spatial localizations within the tumor, which is suggestive of different functional roles (Figs. 3D, 4F, and 6D). This led us to ask if targeting a specific metabolic program might impact tumor growth. We first compared the metabolic state of the microenvironments colocalized with each AT and showed significant correlation between the cancer AT and microenvironment (Fig. 7A). Enzymes in the tricarboxylic acid (TCA) cycle, a pathway that requires oxygen to generate energy, were uniformly highly expressed in the tumor and ME of the proliferative, oxidative, and cell-death ATs. By contrast, glycolytic enzymes were clearly enriched in the hypoxic AT and associated ME cells (Fig. 7A). Enzymes most highly enriched in the hypoxic area were associated with oxygen-independent glycolysis in tumors (39) and included PDK1, an inhibitor for the entry of pyruvate into the TCA cycle (Fig. 7B). This indicates that the metabolic heterogeneity in primary tumor ATs is mirrored in their local microenvironments.

Figure 7.

A, Heatmap showing expression of metabolic genes (glycolysis and OXPHOS) across cancer ATs and associated ME cells. Bar graph shows correlation between ATs and ME-ATs and the hypoxic gene signature enriched in orange AT and its associated ME-AT. B, Schematic of glycolysis and TCA cycle pathways with the hypoxia gene component of the glycolysis AT highlighted in orange. C, Proton efflux rate derived from basal glycolysis (GlycoPER) for MDA-MB-231, HCC38-CD44^Hi, and HCC1806 cells treated with either control siRNA (siCTRL) or siRNA targeting SLC2A3 (siSLC2A3-1 and siSLC2A3-2) over 48 hours after knockdown (n = 3 independent experiments, triplicate wells analyzed per condition). Statistical significance defined by two-way ANOVA with Dunnett multiple-comparison post hoc analysis (**** adj. p-value ≤ 0.001, *** adj. p-value ≤ 0.003). D, Quantitation of tumorsphere-forming capacity in MDA-MB-231, HCC38-CD44^Hi, and HCC1806 cells treated with control (siCTRL) or SLC2A3 knockdown (siSLC2A3-1 or siSLC2A3-2; n = 3 independent experiments, 12 wells analyzed per condition). Statistical significance defined by ordinary one-way ANOVA with Dunnett multiple-comparison post hoc analysis. E, Metastatic seeding capacity of control (siCTRL) and SLC2A3 knockdown (siSLC2A3-1 or siSLC2A3-2) MDA-MB-231-luc cells injected retro-orbitally in NSG mice, measured using bioluminescence imaging of the lungs over days 0 to 10 (n = 5/cohort, statistical significance measured using two-way ANOVA, Dunnett multiple-comparison test, ** adj. p-value ≤ 0.003). F, Representative images showing anti-GLUT3 (Alexa Fluor 488), anti-HIF1a (Cy3), anti-CD31 (Alexa Fluor 647), and DAPI staining in MDA-MB-231-luc xenograft, HCI-010, and HCI-001 PDX tissues (n = 4 sections). Scale bar, 50 μm (top); scale bar, 20 μm (inset, bottom). G, Conditional-DREMI scores correlating immunofluorescence intensity values of HIF1a and GLUT3 with conditional Density Rescaled Visualization heatmaps across TNBC models in F. H, Diagrammatic representation of single-cell analysis of 34 human breast cancer samples from four cohorts. Datasets are prefiltered to high-quality cancer epithelial cells, and ATs are identified for each sample using AAnet. I, Heatmap showing expression of metabolic genes from (A) across 26 human cancer ATs, 18 (left) associated with the hypoxic AT and 8 (right) associated with the proliferative AT. ATs are from breast cancer samples across three different breast cancer subtypes.

Hypoxic niches provide a permissive environment for the maintenance of both pluripotent stem-cell and cancer stem-cell populations (46–48). AAnet identified that SLC2A1 (GLUT1) and SLC2A3 (GLUT3) are enriched in cancer cells that reside in the hypoxic niche. Given that GLUT1 is ubiquitously expressed in normal cells throughout the body, we asked if ablating GLUT3 could eradicate the aggressive phenotype of the cancer cells in the hypoxic niche. This approach is supported by the fact that GLUT3 is largely confined to the brain and sperm in normal tissues (49); is increased in TNBC and associated with metastasis and poor prognosis (50); and has approximately threefold higher affinity for glucose than GLUT1 (51), suggesting GLUT3 would be highly beneficial to cancer-cell populations with poor access to blood flow and nutrients. We treated three TNBC cancer stem cell–enriched cell lines (MDA-MB-231, HCC1806, and HCC38-CD44^Hi) with a control siRNA (siCTRL) or two independent siRNAs targeting SLC2A3 (siSLC2A3-1 and siSLC2A3-2). Efficient knockdown of SLC2A3 RNA and GLUT3 protein was confirmed (Supplementary Fig. S8A–S8E). Under SLC2A3 knockdown conditions, glucose uptake and basal glycolysis were significantly reduced (Fig. 7C; Supplementary Fig. S8F–S8J), leading to a significant reduction in proliferation in all cell lines tested (Supplementary Fig. S8K–S8M). These data show a strong in vitro dependence of cancer stem-cell populations on basal glycolysis.

Next, we confirmed that SLC2A3 knockdown significantly inhibited tumorsphere formation, an in vitro surrogate assay for in vivo tumor-initiating ability, under normoxic (Fig. 7D) and hypoxic (Supplementary Fig. S8N–S8P) conditions. Moreover, SLC2A3 knockdown led to a significant reduction in in vivo tumorigenicity by inhibiting metastatic growth in the lungs after retro-orbital intravenous implantation (Fig. 7E, Supplementary Fig. S8Q) and subcutaneous implantation (Supplementary Fig. S8R).

To determine if GLUT3 is expressed in hypoxic niches in vivo, we used immunofluorescent staining to determine if HIF1A expression (to identify hypoxic cells) colocalizes with GLUT3 expression in MBA-MB-231 xenografts and two TNBC patient-derived xenograft (PDX) models (HCI-010 and HCI-001; Fig. 7F). We computed the median distance of a GLUT3+ cell to either a HIF1A+ cell or CD31⁺ endothelial cell. These data show significantly smaller distances between GLUT3+ and HIF1A+ cells than between GLUT3+ and CD31⁺ cells, suggesting that GLUT3+ cells are enriched in hypoxic areas distant to blood vessels (Supplementary Fig. S8S–S8U). To investigate the relationship between HIF1A and GLUT3 mean expression per cell, we generated conditional Density Resampled Estimate of Mutual Information (DREMI) scores, alongside conditional Density Rescaled Visualization heatmaps, for each model (Fig. 7G). In all cases, we observed a positive dependency between HIF1A and GLUT3 expression. DREMI scores were higher for GLUT3 conditioned on HIF1A rather than HIF1A conditioned on GLUT3, indicating a stronger dependency of GLUT3 on HIF1A compared to the inverse. The asymmetric relationship suggests that HIF1A expression precedes and potentially regulates GLUT3.

Together, these results show that GLUT3 is important for TNBC growth and metastasis, and its expression is increased in response to HIF1A expression in vivo. These data suggest that GLUT3 expression is upregulated in hypoxic environments enabling metabolic plasticity that invokes glycolysis and survival of cancer-cell populations in the hypoxic niche.

To determine if the metabolic ATs are reproduced in human disease, we analyzed single-cell transcriptomes from human breast cancer cells across four distinct studies, corresponding to 34 samples from three major breast cancer subtypes: ER⁺, HER2⁺, and TNBC (Fig. 7H; Supplementary Fig. S9A; refs. 52–54). After filtering, AAnet identified 155 ATs across the 34 samples (15 ER⁺, 13 TNBC, and six HER2⁺) and showed interesting similarities and differences across samples, cohorts, and breast cancer subtypes (See “Methods”, Supplementary Fig. S9B). Although two samples from patients were treated with neoadjuvant therapy (54) and two samples contained BRCA1 mutations (53), ATs from these samples did not substantially differ from other ATs based on overall cosine similarity of expression. On average, AAnet identified 4.56 ATs per patient, with no significant differences in the number of ATs between subtypes [Kolmogorov–Smirnov (KS) test P > 0.05 for all tests; See “Methods”, Supplementary Fig. S9A). Uncommitted cells constituted 42.34% of cells and 12.83% of cells were committed to one AT on average across all samples, with no significant differences between subtypes (KS test P > 0.05 for all tests; See “Methods”, Supplementary Fig. S9A). To further investigate the association of human ATs with the proliferative (blue) and hypoxic (orange) primary tumor ATs (Figure 3B), we identified 18 ATs across all human tumors that had transcriptomic profiles modestly similar to the hypoxic AT and strongly dissimilar to the proliferative AT because of anticipated differences between the human tumors and cell line data. We also identified eight ATs modestly similar to proliferative AT and strongly dissimilar to hypoxic AT (See “Methods”, Fig. 7I). These metabolic ATs are common in breast cancer, represented 20 human tumors. Similar to the xenograft model, six tumors contain both a hypoxic AT and proliferative AT within the same sample. Together, these results show correspondence between the metabolic profiles of ATs identified by our model and human breast cancer tumors, and further, the identified ATs are relevant across breast cancer subtypes.

Importantly, the human patient sample ATs associated with hypoxia showed high enrichment of metabolic markers of glycolysis, including GLUT3, and low TCA cycle expression (Fig. 7I; Supplementary Fig. S9C), similar to that observed in our xenograft hypoxic AT data (Fig. 7A). Conversely, the human patient proliferative ATs showed low hypoxic gene expression and high TCA cycle genes, which are also concordant with our xenograft proliferative AT (Fig. 7A). Finally, several patient sample ATs also show enrichment of immune-related markers IFITM2/3, BST2, HLA-A/B/C, and B2M, wherein these ATs are also modestly similar to the immunostimulatory AT (purple) from the xenograft model (Supplementary Fig. S9D).

Notably, there was no significant enrichment for a particular cancer subtype or association with the hypoxic or proliferative primary tumor ATs (KS test P > 0.05 for all tests) nor was there a significant difference between cancer subtypes in the proportion of cells committed to the hypoxic or proliferative ATs (Wilcoxon rank-sum test P > 0.05 for all tests). Indeed, we did not find significant enrichment for a particular cancer subtype with any AT, suggesting that AAnet can be used to identify phenotypic similarities across cancer subtypes and thereby offers a functional method, beyond hormone and molecular subtyping, to classify breast cancers according to intratumoral archetypal heterogeneity.

Discussion

Nongenetic programs (e.g., epigenetic, transcriptional, and translational) are a major driver of tumor heterogeneity (55). Their dynamic and reversible nature favors rapid evolution of cancer cell states (e.g., seconds, minutes, or hours) to enable survival in unfavorable microenvironments encountered throughout the metastatic cascade and in response to therapy. Thus, as single-cell technologies continue to resolve the breadth and structure of nongenetic heterogeneity in cancer and stromal cells within and across patient tumors, developing strategies to identify and validate the specific cell states and molecular mechanisms that fuel cancer progression remains a significant technological and biological challenge.

To address this knowledge gap, we developed AAnet, an AA method to identify archetypal cell states within and between samples and their associated biological processes. AA is a framework to describe a dataset as a convex combination of extreme, or archetypal, observations. In contrast to other unsupervised approaches to characterize such data, AA is aptly suited for both identifying key cell states reflecting distinct biological processes and analyzing the cellular state space as a continuum of cells committed to these processes. However, identifying the archetypal states remains a fundamental challenge of AA. In particular, nonlinearities can worsen the performance of existing AA tools, as the extreme states of data geometry do not conform to the extreme states of data space.

AAnet solves this problem by learning a transformation of the data into a simplex rather than fitting a simplex on the data directly. The latent space of the autoencoder thus preserves relationships between cells and characterizes cells by their commitment to each AT. We further regularize the latent space to initialize the ATs to diffusion extrema (inferred from the cell–cell affinity graph) for improved accuracy and robustness.

Applied here to single-cell data from preclinical and clinical breast cancer samples, we have shown that AAnet enabled the discovery of biologically and functionally distinct ATs within a phenotypic continuum of cell states within the tumor and captured their associated molecular drivers. Interestingly, we show that the number and distribution of cells committed to ATs in primary tumors is remarkably similar to those found in LN metastases, yet liver metastases differ by the loss of one AT and lung metastases deviate from primary LN and liver metastases via the emergence of one new AT. These analyses demonstrate that AAnet can reveal the emergence of new cell states, the number of cells committed to a specific AT, and the underlying biology that facilitate site-specific metastatic adaptation. Critically, we validate the significance of the ATs identified by mapping the scRNA-seq data to matched spatial transcriptomic data via scMMGAN (42). These data confirm that AAnet-defined ATs resolve into distinct spatially-localized regions within the tumor and are enriched with distinct stromal cell types. Thus, AAnet robustly identifies functional and spatially distinct cellular ATs within a tumor.

This expands upon prior work analyzing cancer ATs from bulk RNA sequencing (RNA-seq) data (10). Beyond reducing each patient to their dominant AT or identifying common pan-cancer cell states, our work highlights the complexity of intratumoral heterogeneity that AAnet can define and the existence of cellular ATs, as well as unique archetypal compositions specific to given tumors. This represents a key advancement in the characterization of cancer hallmarks.

Of note, we uncovered metabolic heterogeneity not seen before in TNBC, in which cells with discrete spatial archetypal localization use distinct metabolic programs. We have previously analyzed bulk RNA-seq data (METABRIC and The Cancer Genome Atlas) and shown that TNBC exhibits a unique highly metabolic gene expression phenotype, upregulating a range of pathways, including glycolysis, compared with other breast cancer subtypes such as luminal A (56). Our AA provides further granularity to these data, showing the individual contributions of each AT to this unique TNBC metabolic signature. For example, the hypoxic AT clearly contributes to the high expression of SLC2A1, SLC2A3 (GLUT3), HK1, HK2, ALDOA, ALDOC, TPI, GAPDH, PGK1, ENO1, PDK1, and LDHA in the bulk RNA-seq data, with contributions also from the immunostimulatory AT but not from the most abundant proliferative AT. By comparison, these findings reveal that different regions of the tumor uniquely invoke glycolysis or OXPHOS, which could not be determined when analyzing existing bulk RNA-seq data. Furthermore, we show that we could use our identified ATs to predict novel therapeutic targets within distinct cellular subsets, such as the glucose transporter GLUT3 in stem cells within the hypoxic AT. The demonstration that GLUT3 drives glucose uptake, basal glycolysis, and increase in response to HIF1A expression in vivo explains why targeting GLUT3 significantly reduces TNBC growth and metastasis. These data build significantly upon prior works that show a role for GLUT3 in TNBC in regulating the inflammatory microenvironment (50), epithelial–mesenchymal transition and migration (57, 58), and glucose uptake in the brain (59). Strikingly, we also discovered that the distinct metabolic phenotypes of two ATs are reflected in the ME cells associated with these ATs. We also confirm that the human ATs and stromal enrichments are present in an immunocompetent murine model. Of note, Slc2a3 is poorly expressed in the murine model, highlighting the complementarity of using both human and murine models. Together, these data underscore that intratumoral metabolic heterogeneity could prove critical to cancer progression.

Although AAnet revealed some common signatures between the MDA-MB-231 model and clinical breast cancer samples, there are indeed important differences between cell lines and human samples (including tumor stage/grade at the time of analysis, prior treatment history, exposure to cytotoxic agents, and distinct mutational backgrounds), which may hinder deeper functional comparison between AAnet-derived ATs and human tumor–derived ATs. Nonetheless, such analysis demonstrates the broad applicability of AAnet and the ability to identify meaningful ATs in clinical cohorts.

Tumor heterogeneity remains a significant clinical challenge for diagnosis and therapeutic management. With AAnet, it is now possible to segment tumors into functionally distinct regions comprising cancer cell states associated with unique cellular microenvironments. Thus, it is possible to resolve tumor heterogeneity to a level not yet achieved with previous computational tools. In the future, classifying patients according to biological ATs with tools like AAnet is likely to improve tumor subclassifications. Applied to samples before and after specific treatment, we can begin to learn how ATs change over time, in response to specific therapies, and in different metastatic sites. Moreover, this approach will reveal the molecular programs driving each cellular AT, as well as when and how they emerge. Ultimately, these tools may deliver the knowledge to enable the development of improved and effective therapeutic strategies.

AAnet has the potential for widespread use because of its ability to broadly characterize cellular heterogeneity in diverse contexts, including cancer, immunology, and development. Additionally, the flexibility of the approach demonstrates that it can be used both independently and as a part of the large single-cell analysis pipeline to interpret the archetypal distribution underlying any single-cell dataset. Given its general-purpose design to characterize cellular heterogeneity, AAnet is a valuable tool for the single-cell community.

Methods

Background on Cell-State Heterogeneity Analysis

Clustering-Based Approaches Unreliably Identify Clusters When Data Are a Continuum of Cells

Clustering is the most commonly used technique for characterizing cell-state heterogeneity in single-cell data and considered a standard part of single-cell workflows and best practices (22). However, clustering can be a nontrivial task, both with computational challenges and challenges with interpretation and annotation (60). This is in part due to the fact that clustering assumes that data are composed of biologically distinct groups, such as discrete cell types.

After embedding the primary scRNA-seq data into three dimensions, it is evident that for the primary tumor, the cells form a connected manifold along the cellular state space rather than separate into clusters (Fig. 2A). After running Leiden clustering 100 times with default parameters, the cluster assignments changed at the boundaries, indicating that these cells are not strongly committed to one cluster.

We hypothesized that the cells at cluster boundaries are intermediate cells between the more distal extreme states. To test the ability of clustering-based analysis to characterize such datasets, we simulated a curved three-simplex (tetrahedron), wherein the data points are defined as a continuum between the vertices (Supplementary Fig. S1A). Clustering the simulated data with 10 different clustering algorithms reveals the lack of concordance across clustering methods due to the lack of latent cluster structure in the dataset (Supplementary Fig. S1D). We also defined a signal on the tetrahedron as a function of the affinity to AT4. Clustering was limited in its ability to characterize continuous signals, finding significantly higher expression in ATs 1–3, despite no defined association of the signal with these ATs (Supplementary Fig. S1E).

Trajectory-Based Approaches Enforce Lineage Structure That Do Not Accurately Capture Simulated Signal

Alternatively, trajectory inference methods are commonly used to identify continuous paths in datasets in order to define pseudotemporal ordering of cells, often for learning developmental decisions (61–64). We show that, without clear lineage structure in the dataset, trajectory-based methods are not able to learn an intelligible ordering of cells (Supplementary Fig. S1C), finding curves/ordering with high correlation with AT1 and AT2, but negative correlation with AT3 and none with AT4. Because of this inability to characterize cell affinity with respect to AT4, trajectory-based approaches cannot meaningfully characterize the simulated signal (Supplementary Fig. S1E).

Background on AA

AA Overview

AA is an unsupervised learning method that aims to find extremal points, called archetypes (ATs), such that every point in a dataset can be approximated as a mixture of these ATs (6). Given a dataset $X\mathrm{=}\left\{x_1\mathrm{,}\mathrm{\dots }\mathrm{,}{x}_N\right\}\mathrm{\subset }{\mathbb{R}}^n$, the ATs $\left\{z_1\mathrm{,}\mathrm{\dots }\mathrm{,}{z}_k\right\}\mathrm{\subset }{\mathbb{R}}^n$ are chosen so that for each data point $x_i$, there exists ${\alpha }_{i\mathrm{,}1}\mathrm{,}\mathrm{\dots }\mathrm{,}{\alpha }_{i\mathrm{,}k}\mathrm{\in }\left[0\mathrm{,}1\right]$ such that

$${\displaystyle \mathrm{\sum }}_{j\mathrm{=}1}^k{\alpha }_{i\mathrm{,}j}{z}_k\mathrm{\approx }{x}_i$$ (A)

$${\displaystyle \mathrm{\sum }}_{j\mathrm{=}1}^k{\alpha }_{i\mathrm{,}j}\mathrm{=}1$$ (B)

This type of linear combination wherein the coefficients are nonnegative and sum to 1 is called a convex combination. The set of all such convex combinations of the ATs $\left\{z_1\mathrm{,}\mathrm{\dots }\mathrm{,}{z}_k\right\}$ is a $\left(k\mathrm{-}1\right)$simplex. Note that the ATs are not constrained to be points from the dataset.

Principal Convex Hull Analysis

One of the first AA algorithms was principal convex hull analysis (PCHA), proposed in the work of Cutler and Breiman (6). PCHA constrains the ATs to be convex combinations of the input data points. It finds these ATs through the following optimization problem:

$$\begin{array}{c}\underset{\mathbf{W}\mathrm{,}\mathbf{H}}{\min }{\mathrm{\parallel }{\mathbf{X}}^T\mathrm{-}{\mathbf{X}}^T\mathbf{\text{WH}}\mathrm{\parallel }}_F^2\mathrm{\hspace{0.17em}}{\mathrm{s}\mathrm{.}\mathrm{t}\mathrm{.}\mathrm{\hspace{0.17em}}\mathbf{1}}^T\mathbf{W}\mathrm{=}\mathbf{1}\mathrm{,}\mathrm{\hspace{0.17em}}{\mathbf{1}}^T\mathbf{H}\mathrm{=}\mathbf{1}\mathrm{,}\mathrm{\hspace{0.17em}}\mathbf{C}\mathrm{\ge }0\mathrm{,}\mathrm{\hspace{0.17em}}\mathbf{W}\mathrm{\ge }0\mathrm{\hspace{0.17em}}\end{array}$$ (C)

where $\mathbf{W}\mathrm{\in }{\mathbb{R}}^{N\mathrm{\times }p}$ maps the data to the ATs and $\mathbf{H}\mathrm{\in }{\mathbb{R}}^{p\mathrm{\times }N}$ contains the coordinates of the ATs in the feature space. The constraints on $\mathbf{W}$ guarantee that the ATs are convex combinations of the data points, whereas the constraints on $\mathbf{H}$ guarantee that the data points are convex combinations of the ATs. Cutler and Breiman (6) define an optimization algorithm for the expression above using alternating convex least squares. This algorithm seeks solutions that satisfy the constraints ${\mathbf{1}}^T\mathbf{W}\mathrm{=}\mathbf{1}$ and ${\mathbf{1}}^T\mathbf{H}\mathrm{=}\mathbf{1}$ by adding auxiliary terms to the objective function. Mørup and Hansen (65) build on this work by modifying the optimization algorithm to use projected gradient in the alternating optimization steps to find solutions that satisfy the constraints.

Alternative AA Variants

Mørup and colleagues (65) also propose kernel PCHA (kPCHA), which is analogous to kernel principal component analysis (PCA). The PCHA is rotation equivariant, meaning that applying a rotation to the input dataset effectively results in the same rotation being applied to the output features and ATs. This implies that the output of the PCHA only depends on the kernel matrix ${\mathbf{X}}^T\mathbf{X}$ rather than on the actual input dataset $\mathbf{X}$. kPCHA takes advantage of this fact by computing this kernel matrix in a possibly infinite-dimensional reproducing kernel Hilbert space and then running the PCHA on this kernel matrix instead of ${\mathbf{X}}^T\mathbf{X}$. The mapping from the input feature space into the reproducing kernel Hilbert space is typically nonlinear, allowing kPCHA to potentially take advantage of the manifold geometry of the underlying dataset.

Several other works have also extended the algorithm proposed by Cutler and Breiman (6). In the work of Javadi and Montanari (66), the requirement that the ATs are convex combinations of input data is relaxed. The ATs are found by optimizing an objective function with two terms. The first term is similar to the objective function in Eq. (C) and captures how well the convex hull of the ATs aligns with the dataset. The second term reflects how close the ATs are to the convex hull of the dataset. Meanwhile, Chen and colleagues (67) propose an algorithm to optimize Eq. (C) using an active-set approach.

Background on Machine Learning

Autoencoders

An autoencoder is a type of neural network that is used to learn compressed representations of data. Autoencoders are composed of two separate networks: an encoder and a decoder. The encoder network maps the input data into a low-dimensional feature space or latent space, whereas the decoder tries to reconstruct the original data from this low-dimensional representation. Through minimizing the error between the original data and reconstruction, termed reconstruction loss, autoencoders have been shown to successfully learn the structure of data and have had particular use in capturing a meaningful representation of single-cell data (17–21).

Manifold Learning

Manifold learning is a subfield of machine learning built around the manifold hypothesis, which asserts that high-dimensional datasets are sampled from low-dimensional manifolds that lie in the high-dimensional space. Here, a manifold refers to a space that is locally isomorphic to a Euclidean space. Many methods in unsupervised learning attempt to implicitly or explicitly capture the structure of the underlying data manifold. For instance, the latent space of an autoencoder can be viewed as a parameterization of the data manifold.

AAnet Overview

AAnet Architecture

AAnet is designed to have a flexible number and size of layers depending on the complexity for the task. For our purposes, we found that a two-layer (256-node and 128-node) encoder and a two-layer (128-node and 256-node) decoder worked well. The batch size was 256; the optimizer was ADAM; the learning rate was set to 1e−3; and the weight initialization was Xavier. All hidden layers contain Tanh nonlinear activations besides layers directly before and after the archetypal layer, which are linear so that each point is a linear combination of ATs. The default weight on extrema loss${\gamma }_{extrema}$ is set to 1. To encourage the ATs to be tight, i.e., close to the data, we can add Gaussian noise $\mathrm{\sim }N\left(0\mathrm{,}0.05\right)$ in the latent layer during training. For all datasets, we reduced dimensionality using PCA before running AAnet and inverse-transformed the learned ATs to the ambient space.

Reconstruction Loss

The main loss function for the autoencoder seeks to minimize the difference between the original input fed into the encoder $z\mathrm{=}E\left(x\right)$ and the reconstructed input produced by the decoder $\stackrel{\sim }{x}\mathrm{=}D\left(z\right)$, termed the reconstruction loss. Standardly, autoencoders use the mean squared error (MSE), quantifying the difference of these two terms:

$$\text{Reconstruction}\mathrm{\hspace{0.17em}}\text{MSE}\mathrm{=}{\mathbb{E}}_{x\mathrm{\in }X}\left[\mathrm{\parallel }x\mathrm{-}\stackrel{\sim }{x}{\mathrm{\parallel }}^2\right]\mathrm{=}{\mathbb{E}}_{x\mathrm{\in }X}\mathrm{[}\mathrm{\parallel }x\mathrm{-}D\mathrm{(}E\left(x\right){\mathrm{\parallel }}^2]$$

Archetypal Loss

In addition to the reconstruction loss, we want to enforce the latent space of the autoencoder learns the structure of the data with respect to the ATs. To this end, we convert the coordinates from Cartesian to barycentric after the encoder learns the transformation. The barycentric coordinate system, related to Cartesian coordinates, is a system in which each point is specified by reference to a simplex. When coordinates are normalized to sum to 1, the vertices of the simplex are denoted by k one-hot vectors of length k for a $k\mathrm{-}1$-simplex. For example, a triangle is a two-simplex with three vertices, wherein the three vertices are (1,0,0), (0,1,0), and (0,0,1). All coefficients of point P are positive if and only if P is inside the simplex.

As this coordinate system describes points with respect to a simplex, it is well suited to be the latent space for learning archetypes.

To enable interpretation of points as convex combinations of ATs, we enforce each point to stay within the simplex by adding an archetypal loss term, the mean squared error of the negative coefficients:

$$\text{Archetypal}\mathrm{\hspace{0.17em}}\text{MSE}\mathrm{=}{\mathbb{E}}_{x\mathrm{\in }neg\mathrm{.}\mathrm{\hspace{0.17em}}coefficients}\left[\mathrm{\parallel }x{\mathrm{\parallel }}^2\right]$$

Extrema Loss

We developed a novel method to identify $k$ plausible ATs prior to model training. This method, explained in detail below, builds a graph from the data and then uses the eigenvectors of the Laplacian matrix to find extreme points in the datasets; these points will be referred to as diffusion extrema. We then include an extrema loss term that penalizes large distances in the latent space between the diffusion extrema and the vertices of the simplex. If the diffusion extrema and standard basis vectors in the latent space ${\mathbb{R}}^k$ are labeled as $\mathrm{\{}{\mathcal{l}}_i{\mathrm{\}}}_{i\mathrm{=}1}^k$ and $\mathrm{\{}{e}_i{\mathrm{\}}}_{i\mathrm{=}1}^k$, respectively, then this loss term can be calculated as

$$\text{Diffusion}\mathrm{\hspace{0.17em}}\text{extrema}\mathrm{\hspace{0.17em}}\text{MSE}\mathrm{=}\frac{1}{k}{\displaystyle \mathrm{\sum }}_{i\mathrm{=}1}^k{\mathrm{\parallel }E\left({\mathcal{l}}_i\right)\mathrm{-}{e}_i\mathrm{\parallel }}_2^2$$

Let $\left\{x_1\mathrm{,}\mathrm{\dots }\mathrm{,}{x}_n\right\}\mathrm{\in }{\mathbb{R}}^m$ be the points in the dataset. Then, the procedure for finding these diffusion extrema is as follows:

• 1. Construct a graph $G$ from the dataset. This can be done by computing a symmetrized $k$-nearest neighbors graph from the dataset and then weighting the edges with a Gaussian kernel, as was done by Moon and colleagues (24).

• 2. Let ${\mathbf{\text{ψ}}}_1\mathrm{,}\mathrm{\dots }\mathrm{,}{\mathbf{\text{ψ}}}_n$ denote the eigenvectors of the combinatorial Laplacian $\mathbf{L}$ of $G$ with corresponding eigenvalues ${\lambda }_1\mathrm{\le }\mathrm{\cdots }\mathrm{\le }{\lambda }_n$. Compute

$$i_1\mathrm{=}{\arg \mathrm{ }\max }_{i\mathrm{\in }\left\{1\mathrm{,}\mathrm{\dots }\mathrm{,}n\right\}}\left|{\mathbf{\text{ψ}}}_2\left(i\right)\right|$$

where ${\mathbf{\text{ψ}}}_2\left(i\right)$ is the $i$th entry of ${\mathbf{\text{ψ}}}_2$.

• 3. Let ${\mathbf{L}}^{\left(i_1\right)}$ denote $\mathbf{L}$ with the entries in the $i$th row and $i$th column be replaced by zeros. Likewise let ${\mathbf{\text{ψ}}}_1^{\left(i_1\right)}\mathrm{,}\mathrm{\dots }\mathrm{,}{\mathbf{\text{ψ}}}_n^{\left(i_1\right)}$ denote the eigenvectors of ${\mathbf{L}}^{\left(i_1\right)}$, again ordered in an ascending fashion by corresponding eigenvalues. Compute

$$i_2\mathrm{=}{\arg \mathrm{ }\max }_{i\mathrm{\in }\left\{1\mathrm{,}\mathrm{\dots }\mathrm{,}n\right\}}\left|{\mathbf{\text{ψ}}}_2^{\left(i_1\right)}\left(i\right)\right|$$

• 4. Let ${\mathbf{L}}^{\left(i_1\mathrm{,}\mathrm{\dots }\mathrm{,}{i}_m\right)}$ denote $\mathbf{L}$ with the entries in the $i_1$th, $\mathrm{\dots }$, $i_{m\mathrm{-}1}$st, and $i_m$th rows and columns replaced by zeros. Likewise let ${\mathbf{\text{ψ}}}_1^{\left(i_1\mathrm{,}\mathrm{\dots }\mathrm{,}{i}_m\right)}\mathrm{,}\mathrm{\dots }\mathrm{,}{\mathbf{\text{ψ}}}_n^{\left(i_1\mathrm{,}\mathrm{\dots }\mathrm{,}{i}_m\right)}$ denote the eigenvectors of ${\mathbf{L}}^{\left(i_1\mathrm{,}\mathrm{\dots }\mathrm{,}{i}_m\right)}$. Iteratively, for each $j\mathrm{=}3\mathrm{,}\mathrm{\dots }\mathrm{,}k$, compute

$$i_j\mathrm{=}{\arg \mathrm{ }\max }_{i\mathrm{\in }\left\{1\mathrm{,}\mathrm{\dots }\mathrm{,}n\right\}}\left|{\mathbf{\text{ψ}}}_j^{\left(i_1\mathrm{,}\mathrm{\dots }\mathrm{,}{i}_{j\mathrm{-}1}\right)}\left(i\right)\right|$$

• 5. The diffusion extrema are $x_{i_1}\mathrm{,}\mathrm{\dots }\mathrm{,}{x}_{i_k}$.

The intuition behind this algorithm comes from an application of the Courant–Fischer theorem for symmetric matrices. Given an $n\mathrm{\times }n$ symmetric matrix $\mathbf{A}$ with eigenvectors ${\mathbf{a}}_1\mathrm{,}\mathrm{\dots }\mathrm{,}{\mathbf{a}}_k$, Courant–Fischer tells us that

$${\mathbf{a}}_1\mathrm{=}{\arg \mathrm{ }\min }_{\mathrm{\parallel }\mathbf{x}\mathrm{\parallel }\mathrm{=}1}{\mathbf{x}}^T\mathbf{\text{Ax}}\mathit{ }\mathit{ }\text{and}\mathit{ }\mathit{ }{\mathbf{a}}_2\mathrm{=}{\arg \mathrm{ }\min }_{\begin{array}{c}\mathrm{\parallel }\mathbf{x}\mathrm{\parallel }\mathrm{=}1\\ \langle \mathbf{x}\mathrm{,}{\mathbf{a}}_1\rangle \mathrm{=}0\end{array}}{\mathbf{x}}^T\mathbf{\text{Ax}}$$

If $\mathbf{A}$ is the Laplacian matrix $\mathbf{L}$ for some weighted graph $G\mathrm{=}\left(V\mathrm{,}E\mathrm{,}w\right)$, then

$${\mathbf{x}}^T\mathbf{\text{Lx}}\mathrm{=}{\displaystyle \mathrm{\sum }}_{\left(i\mathrm{,}j\right)\mathrm{\in }E}{w}_{i\mathrm{,}j}{\left(\mathbf{x}\left(i\right)\mathrm{-}\mathbf{x}\left(j\right)\right)}^2$$

Intuitively, the quadratic form ${\mathbf{x}}^T\mathbf{\text{Lx}}$ captures how smoothly $\mathbf{x}$ varies over the edges of $G$. Hence, ${\mathbf{\text{ψ}}}_1$ is the (normalized) constant vector, and ${\mathbf{\text{ψ}}}_1$ can be viewed as a smooth signal on $G$ that is orthogonal to the constant vector. Now, if we consider the matrix ${\mathbf{L}}^{\left(i\right)}$, we see that minimizing the quadratic form ${\mathbf{x}}^T{\mathbf{L}}^{\left(i\right)}\mathbf{x}$ can be recast as minimizing ${\mathbf{x}}^T\mathbf{\text{Lx}}$ with the additional constraint that $\mathbf{x}\left(i\right)\mathrm{=}0$, that is

$${\mathbf{\text{ψ}}}_2^{\left(i\right)}\mathrm{=}\underset{\begin{array}{c}\mathrm{\parallel }\mathbf{x}\mathrm{\parallel }\mathrm{=}1\\ \langle \mathbf{x}\mathrm{,}\mathbf{1}\rangle \mathrm{=}0\end{array}}{\min }{\mathbf{x}}^T{\mathbf{L}}^{\left(i\right)}\mathbf{x}\mathrm{=}\underset{\begin{array}{c}\mathrm{\parallel }\mathbf{x}\mathrm{\parallel }\mathrm{=}1\\ \mathbf{x}\left(i\right)\mathrm{=}0\end{array}}{\min }{\mathbf{x}}^T{\mathbf{L}}^{\left(i\right)}\mathbf{x}$$

This is because ${\mathbf{\text{ψ}}}_1^{\left(i\right)}\mathrm{=}{e}_i$, the $i$th standard basis vector. Extending this reasoning to ${\mathbf{L}}^{\left(i_1\mathrm{,}\mathrm{\dots }\mathrm{,}{i}_m\right)}$, where $i_1\mathrm{,}\mathrm{\dots }\mathrm{,}{i}_m\mathrm{\in }\mathrm{\{}1\mathrm{,}\mathrm{\dots }\mathrm{,}|V\left|\mathrm{\right\}}$ are unique but arbitrary, we see that ${\mathbf{\text{ψ}}}_j^{\left(i_1\mathrm{,}\mathrm{\dots }\mathrm{,}{i}_m\right)}\mathrm{=}{e}_{i_j}$ for $1\mathrm{\le }j\mathrm{\le }m$. Hence,

$${\mathbf{\text{ψ}}}_{m\mathrm{+}1}^{\left(i_1\mathrm{,}\mathrm{\dots }\mathrm{,}{i}_m\right)}\mathrm{=}\underset{\begin{array}{c}\mathrm{\parallel }\mathbf{x}\mathrm{\parallel }\mathrm{=}1\\ \mathbf{x}\left(i_1\right)\mathrm{=}\mathrm{\cdots }\mathrm{=}\mathbf{x}\left(i_m\right)\mathrm{=}0\end{array}}{\min }{\mathbf{x}}^T{\mathbf{L}}^{\left(i_1\mathrm{,}\mathrm{\dots }\mathrm{,}{i}_m\right)}\mathbf{x}$$

This tells us that the $m$th eigenvector of ${\mathbf{L}}^{\left(i_1\mathrm{,}\mathrm{\dots }\mathrm{,}{l}_m\right)}$ is a signal on $G$ that is as smooth as possible while also having the value $0$ on vertices $i_1\mathrm{,}\mathrm{\dots }\mathrm{,}{i}_m$. Then, we should expect ${\mathbf{L}}^{\left(i_1\mathrm{,}\mathrm{\dots }\mathrm{,}{l}_m\right)}$ to attain its largest absolute value at a vertex that is far from the vertices $i_1\mathrm{,}\mathrm{\dots }\mathrm{,}{i}_m$. This is the guiding principle behind the above diffusion extrema method. Step (2) takes advantage of the fact that ${\mathbf{\text{ψ}}}_2$ is smooth on $G$ and therefore is likely to attain its maximum absolute value at extremal points in the graph. The vertex at which ${\mathbf{\text{ψ}}}_2$ has its largest absolute value is chosen to be the first diffusion extremum. Steps (3) and (4) then use the properties of ${\mathbf{\text{ψ}}}_{m\mathrm{+}1}^{\left(i_1\mathrm{,}\mathrm{\dots }\mathrm{,}{i}_m\right)}$ discussed above to iteratively find vertices in the graph that are far away from the extrema that have already been chosen.

While it has not been conclusively proven that the eigenvectors of the Laplacian $\mathbf{L}$ and Laplacian variant ${\mathbf{L}}^{\left(i_1\mathrm{,}\mathrm{\dots }\mathrm{,}{i}_m\right)}$ attain their maximum/minimal values at extremal points in the graph, experiments show this diffusion extrema method to be robust and effective in identifying reasonable ATs in a dataset.

Choosing Number of ATs

To choose the number of ATs, we first define a range of possible AT counts $m\mathrm{=}2\mathrm{,}3\mathrm{,}\mathrm{.}\mathrm{.}\mathrm{.}\mathrm{,}{n}_{at}$. Then, we calculate $n_{at}$ diffusion extrema. For the mth diffusion extremum, we calculate the geodesic distance of that extremum to all previous extremum (first to m − 1th) and take the mean of all the distances. Intuitively, this score tells how transcriptionally distinct AT m is from all existing ATs, and if it is very similar, it is likely not a new AT. Thus, we take the knee point of this score as the number of ATs $k$ for downstream analysis.

$$s_m\mathrm{=}\frac{1}{m}{\displaystyle \mathrm{\sum }}_{i\mathrm{=}1}^{m\mathrm{-}1}geodesic\mathrm{\hspace{0.17em}}distance\left({\mathcal{l}}_m\mathrm{,}{\mathcal{l}}_i\right)\mathrm{\hspace{0.17em}}\text{for}\mathrm{\hspace{0.17em}}m\mathrm{\hspace{0.17em}}\mathrm{=}\mathrm{\hspace{0.17em}}2\mathrm{,}\mathrm{\hspace{0.17em}}3\mathrm{,}\mathrm{\hspace{0.17em}}\mathrm{.}\mathrm{.}\mathrm{.}\mathrm{,}\mathrm{\hspace{0.17em}}{n}_{at}$$

By contrast, an alternative approach uses PCHA and the knee point of the explained variance to choose the number of ATs (15). This fails to predict the correct number of ATs for two simplices, as the nonlinearity artificially adds variance in the ambient space that is not reflected on the underlying manifold, a known limitation described in the original work (Supplementary Fig. S2C). Another approach from van Dijk and colleagues (28) suggests training AAnet for all tested numbers of ATs (e.g., 2–10) and then choosing the number of ATs that is the elbow point of the reconstruction loss for a held-out set. Although this identifies the correct number of ATs (Supplementary Fig. S2C), it requires first calculating the diffusion extrema and then training the model, which can be time-intensive for many runs and number of ATs.

When the incorrect number of ATs is chosen (e.g., $k\mathrm{+}1$ or $k\mathrm{-}1$ ATs for a dataset with $k$ ground truth vertices), AAnet still identifies ATs that are close to the true vertices of the simplex (except in the $k\mathrm{+}1$ case where there exists one AT that is far from all vertices (Supplementary Fig. S2D). This demonstrates that although AAnet is dependent on this parameter, it is robust to small changes.

Comparative Analysis

Simulated Datasets

Many datasets naturally exist with archetypal geometry because of trade-offs between different selective pressures. However, due to the high degree of nonlinearity in single-cell data, we do not expect there to be a linear relationship between the feature space and the archetypal space. Thus, to compare AAnet with other unsupervised tasks and AA approaches, we developed an approach to simulate points sampled from a simplex in which we can tune the amount of nonlinearity to test the association between nonlinearity and method performance.

For this approach, we sampled Euclidean points uniformly from an N-dimensional simplex with a specified radius. Then, we computed the inverse of a stereographic projection, mapping points to (N − 1)-dimensional space on the unit hypersphere. This allows us to obtain datasets with ground truth simplicial structure (for any N) and tunable nonlinearity (wherein the higher radius corresponds to more curvature; Supplementary Fig. S1A).

For comparisons in Supplementary Fig. S1, we sample 10,000 points from a three-simplex (a tetrahedron) projected onto a hypersphere. This results in a curved tetrahedron, wherein the ground truth ATs do not correspond to the extrema in the data space (as they would for a classic tetrahedron). We embed the curved tetrahedron with data visualization approach PHATE (24) and visualize the ground truth ATs and archetypal affinity. Additionally, we simulated a signal, defined by the sine of the ground truth affinity between AT4 and all other points in the tetrahedron. This signal is defined by its nonlinear relation to a ground truth AT and is designed to model biological processes that are enriched with respect to a cellular AT (Supplementary Fig. S1E). For comparison with AA methods, we simulated one (line), two (triangle), and three (tetrahedron) simplices with 10,000 points for each given nonlinearity transformation.

Comparison with Trajectory-Inference Methods

For trajectory inference comparisons, we ran Slingshot because, unlike other pseudotime inference approaches, it automatically infers the start and end cells. Slingshot requires cluster labels and a dimensionality-reduced representation. We used the PHATE embedding and the KMeans cluster labels as the input, as KMeans is a popular method for clustering that produced stable results. Besides these two inputs, all other parameters were default. Slingshot-inferred trajectories do not reveal affinity with AT4 and do not capture the association with the simulated signal (Supplementary Fig. S1C and S1E).

Comparison with Clustering Methods

To compare AA with clustering on the curved tetrahedron, we clustered the data using default parameters for 10 different clustering algorithms in sklearn.cluster. Five methods required the number of clusters to be specified, and for these, we specified four clusters, the ground truth number of vertices for a tetrahedron. We ran clustering using 10 different clustering algorithms: five that require the number of clusters to be specified [KMeans, agglomerative clustering (Ward), agglomerative clustering (single linkage), agglomerative clustering (complete linkage), and spectral clustering] and five that infer the number of clusters from the data (affinity propagation, mean shift, DBSCAN, OPTICS, and BIRCH). The number of clusters across algorithms ranged from 1 to 449. Additionally, with the exception of KMeans and spectral clustering, there was low concordance in cluster assignments (mean adjusted mutual info score = 0.216), suggesting that the partitions made by the algorithm are not robust (Supplementary Fig. S1D). Additionally, clustering approaches are not able to capture association with the simulated signal (Supplementary Fig. S1E).

Comparison with AA Methods

All AA methods require input of the number of ATs, so we specified four ATs for each method, otherwise running with default parameters. To generate MSE calculations between the ground truth vertices and inferred ATs, we ran each method five times and visualized the first run for each method.

By the definition of barycentric coordinates, a point has a value closer to 1 for dimension $k$ if it has a high affinity to the $k$th vertex and a value closer to 0 if it has a low affinity to the $k$th vertex. Therefore, we color the dataset with the latent coordinates for each dimension to determine if the latent space has semantic structure and if AAnet is effectively learning affinity relative to each extremum.

AAnet for Published Antigen-Specific CD8⁺ T Cells

We analyzed the processed data from Connolly and colleagues (29) without further filtering. This resulted in 2,470 cells from the LNs, 1,448 cells from the lungs, 2,379 cells at week 8, and 1,454 cells at week 17. For each paired dataset, we ran TruncatedSVD to reduce the dimensionality to 100 and then ran AAnet with default parameters. The expression levels in Supplementary Fig. S3 were z-score normalized and ATs were clustered on the basis of cosine similarity.

Computational Methods for TNBC

Single-Cell RNA Preprocessing

The Cell Ranger analysis pipeline (v3.0.2) was used to align the sequencing reads (FASTQ) to a pre-built reference genome (10× Genomics) containing both the human and mouse genomes (GRCh38 + the mm10 genome) and gene expression quantified in each cell. Cells from four primary tumors were sequenced (T1–T4) with a total of 33,938 cells sequenced (T1 = 8,707, T2 = 5,951, T3 = 8,934, and T4 = 10,346). The gene expression data were loaded into Python (v3.8), and QC statistics were computed using the scanpy.pp.calculate_qc_metrics function (v1.9.1; ref. 68). To ensure that only human tumor cells were taken for downstream AA, cells with <99% of data aligning to the human genome were removed. Mouse genes were also removed prior to calculating library size. Cells with a total library size between 2,000 and 50,000 unique molecular identifiers (UMIs) and expressing at least 1,000 genes were retained for downstream analysis. A total of 28,478 primary tumor cells passed QC filtering (T1 = 7,606, T2 = 5,118, T3 = 8,163, and T4 = 7,591); 42,250 LN cells (T1 = 6,604, T2 = 8,316, T3 = 19,224, and T4 = 8,106); 17,687 lung cells (T1 = 5,712, T2 = 5,599, and T4 = 6,376); and 42,647 liver cells (T1 = 7,136, T2 = 9,030, T3 = 18,671, and T4 = 7,810). Remaining tumor cells were normalized to 10,000 reads per cell and square root–transformed using the scprep package (v1.2.3, github.com/krishnaswamylab/scprep). To correct for dropout data, it was smoothed with the manifold smoothing method MAGIC (v3.0.0; ref. 69). Cell numbers remaining after QC were visualized using the R package ggplot2 (v3.4.2; ref. 70). Cell-cycle phase assignment was performed using the scanpy.tl.score_genes_cell_cycle function with previously defined S-phase and G2/M-phase gene lists (71).

We constructed a cell–cell graph based on k-nearest neighbors (kNN) and alpha decay kernel using graphtools (v1.5.3, github.com/krishnaswamylab/graphtools). For the combined analysis of all tumors, we used a mutual nearest neighbors (MNN)MNN kernel to build a cellular graph with batch correction between the replicates. We then used MAGIC (v3.0.0; ref. 69) to transform each graph into the gene space and ran TruncatedSVD to reduce the dimensionality to 100 and visualize samples in reduced dimensions.

AT Similarity, Affinity, and Commitment

AAnet was run on both individual samples and a combined dataset using default parameters. Data ATs were defined for each dataset using AAnet, and archetypal expression vectors were generated by transforming AT coordinates back into the gene space. Cosine similarity was calculated between expression vectors and hierarchical clustering was used to identify orthologous expression states between datasets. We compared this approach to the approach of computing pairwise cosine similarity between the nearest cells with each AT, in which the nearest cells are defined from 25 to 150 cells. For both primary replicates and cross-tissue comparisons, the ATs cluster even when using the nearest cells, and this is not sensitive to the number of cells chosen.

The affinity of each cell to each AT was calculated on the basis of the distance of each cell to each AT in the AAnet latent embedding. Given a cell’s total affinity to all ATs is equal to 1, cells with a higher affinity to one AT versus all other ATs combined (i.e. affinity > 0.5 to one AT) are defined as committed to that AT. The archetypal composition of each sample was determined by counting the number of committed cells per AT and assigning cells with affinity for all ATs < 0.5 as uncommitted. We evaluated if uncommitted cells were viable and of high quality by comparing proxy QC metrics total ribosomal counts, total mitochondrial counts, and the number of genes per cell between uncommitted and committed cells. For both primary replicates and cross-tissue comparison, we saw no difference between committed and uncommitted cells in these metrics. We also evaluated if uncommitted cells were less differentiated by comparing scores of stem-cell markers CD44, EPCAM, and ZEB1. Again, we saw no difference with committed cells, suggesting that these are not less differentiated, but instead the display of a generalist transcriptional profile.

Biological Characterization of Archetypal States

Marker genes upregulated in each AT were calculated by comparing gene expression between cells that were the most committed to each AT. The top 125 cells with the strongest affinity for each AT in the combined AAnet latent space were selected from each replicate (500 cells per AT), as well as the 125 cells per replicate furthest from any AT as the uncommitted populations (500 cells uncommitted). Genes that were upregulated in any group relative to all other groups (FDR <0.05) were determined using the FindMarkers function from the Seurat R package (v4.0; ref. 72). We removed ribosomal and mitochondrial genes and then evaluated the gene set enrichment. Cancer hallmark gene sets (73) overrepresented (FDR <0.1) in markers associated with each AT were determined with the one-sided Fisher exact test using the clusterProfiler R package (v3.16.1; ref. 74) and visualized using the ggplot2 package.

Spatial RNA Data Generation

Spatial transcriptomics was conducted using 10× Genomics Visium Spatial Gene Expression Slide and Reagent Kit, 16 rxns (PN-1000184), according to the protocol detailed in document CG000239RevD for the TNBCs and CG000239RevE for the xenografts, available in 10× Genomics–demonstrated protocols. Cryosectioning was done on optimal cutting temperature embedded and snap-frozen tissue samples at 10 μm thickness and placed on cold Visium slide arrays. The sections were adhered to by swiftly warming the backside of the slide. The slides were then kept at −80°C for <4 weeks before processing accordingly.

In short, the slides were first warmed at 37°C for 1 minute and then immersed in prechilled methanol (VWR EU, 20847.307) for 30 minutes at −20°C for fixation. Staining with hematoxylin and eosin was carried out by 1 minute of drying of the tissues with 500 μL of isopropanol (Thermo Fisher Scientific, A461-1) followed by air-drying until sections turned white. Around 1 mL of Mayer’s hematoxylin (Agilent, S23309) was pipetted onto the slides and treated for 4 minutes. The slides were then washed in nuclease-free water followed by a 2-minute incubation with bluing buffer (Agilent, CS702), washed again, and then counterstained using buffered eosin (Sigma-Aldrich, HT110216, 1:10 dilution in Tris-acetic acid buffer). The slides were air-dried for about 2 minutes and then warmed for 5 minutes at 37°C before mounting using 85% glycerol (Merck, 104094) and a coverslip.

Bright-field histology images were obtained using a 20× objective on Zeiss microscope using the Metafer VSlide system, and the images were processed using VSlide software. The images were extracted as jpgs for downstream analysis.

After imaging, the coverslip and remaining glycerol were washed off in Milli-Q water, and the slides were attached in plastic cassettes included in the reagent kit and first subjected to 20 minutes of permeabilization at 37°C to let the mRNA reach the probes on the slide surface for binding.

The protocol was then followed without deviations to create amplified libraries that in the end were individually indexed using the Dual Index Kit TT SetA, (PN-1000215, 10× Genomics) and quality controlled on a bioanalyzer instrument, and concentrations were measured using Qubit dsDNA HS Assay. The libraries were pooled equimolarly (2 nmol/L) and sequenced on NextSeq 2000 (Illumina platform). To reach the appropriate read depth, the recommended number of reads per ST Spot was applied according to the protocol.

Spatial RNA Preprocessing

The Space Ranger analysis pipeline (v2.0.0, 10× Genomics) was used to align the sequencing reads (FASTQ) to a pre-built reference genome (10× Genomics) containing both the human and mouse genomes (GRCh38 + the mm10 genome) and gene expression quantified in each cell. QC statistics were computed using the STUtility package (75). Voxels with a library size <3,000 UMI were removed and the remaining voxels manually curated to remove those that were disconnected from the main tissue section. Human and mouse genes were separated to create two datasets per sample, one measuring the expression of human genes at each voxel, pertaining to tumor cells, and the other measuring expression of mouse genes, pertaining to cells from the microenvironment. Genes were removed if they were detected in <10 voxels. Finally, each dataset was filtered to remove voxels with low library diversity for a given genome (<1,000 human genes detected and <300 mouse genes detected). After QC filtering, the human dataset comprising 14,819 genes was measured at 1,170 and 1,105 voxels for each tumor, whereas the mouse dataset comprising 12,119 genes was measured at 1,150 and 1,079 voxels for each tumor xenograft section. The remaining datasets were each normalized to 10,000 reads per voxel and log transformed. To correct for dropout (human dataset = 69.8% zeros and mouse dataset = 90.6% zeros), data were smoothed with the manifold smoothing method MAGIC. Cell-cycle phase assignment was performed using the CellCycleScoring function in Seurat (72) with previously defined S-phase gene lists and G2/M-phase gene lists (71). Spatial plots throughout the article were generated using STUtility.

Mapping of ATs to Spatial Transcriptomics with scMMGAN

To map ATs identified in the single-cell data from primary tumors to voxels in the spatial transcriptomic samples, we used scMMGAN (42). Smoothed expression data were zero-centered, unit scaled each dimension, and were reduced to 50 principal components that were used as input to the scMMGAN generator, with the combined primary tumor scRNA-seq data described above being used as the input for the discriminator layer. scMMGAN was run using a generator consisting of three internal layers of 128, 256, and 512 neurons with batch norm and leaky rectified linear unit activations after each layer and a discriminator consisting of three internal layers with 1,024, 512, and 256 neurons with the same batch norm and activations except with minibatching after the first layer. We used the geometry-preserving correspondence loss with a coefficient of 10, cycle-loss coefficient of 1, learning rate of 0.0001, and batch size of 256. This network was used to generate a single cell–like representation of each spatial voxel. These generated single-cell values were then embedded into the AAnet latent space that was trained using the combined scRNA-seq dataset. The affinity of each voxel to each AT was then determined on the basis of the distance of its generated single-cell values to each AT in the trained AAnet latent space. Each voxel was then assigned to an AT to which it had the highest affinity or uncommitted in case the maximum affinity corresponded to more than one AT. As each voxel contains more than one cell (estimated between 3 and 10 cells per voxel), ATs represent the dominant expression state among cells in the voxel. We also scored voxels based on the expression of the marker gene sets associated with each AT in single-cell data using the first principal component of gene set expression, analogous to the eigengene metric used to summarize gene coexpression networks (76). Scores were calculated on the basis of expression of the top 50 marker genes with the highest log fold enrichment per AT, excluding mitochondrial and ribosomal genes.

Microenvironment Mapping and Enrichment

We used data aligning to the mouse genome at each voxel to analyze the microenvironment associated with each AT. Based on the archetypal assignment of voxels using tumor data, as described above, we identified differentially expressed genes between microenvironment spots spatially colocalized with each AT using the FindMarkers function in Seurat (log fold change (LFC) >0.1; FDR <0.05; ref. 72). Enrichment of Gene Ontology biological processes and cell-type markers associated with “connective tissue,” “immune system,” “smooth muscle,” “epithelium,” “vasculature,” “blood,” “mammary gland,” or “skeletal muscle” in the PanglaoDB (v27/03/2020; ref. 77) among differentially expressed genes were determined with the one-sided Fisher exact test using the clusterProfiler R package (78). Putative ligand–receptor interactions between ATs were identified using CellPhoneDB (v2; ref. 45). Human orthologs to mouse genes were identified in biomaRt, and count matrices were merged on the basis of gene ID prior to running CellPhoneDB using the parameters: iterations, 1,000 and threshold, 0.2, and identifying significant interactions (FDR <0.05). The metabolism between ATs and the microenvironment was compared on the basis of expression of key enzymes involved in glycolysis and the TCA cycle in each AT and associated microenvironment. Glycolytic enzymes were designated as either hypoxic or not based on their enriched expression (LFC >0) in voxels assigned to the hypoxic AT. Heatmaps were generated on the basis of the mean of scaled values for voxels associated with each AT (human genes) or microenvironment (orthologous mouse genes).

Cell Culture

MDA-MB-231 and MDA-MB-231-luciferase-GFP cells were cultured in DMEM supplemented with 10% (v/v) FBS and 1% (v/v) penicillin–streptomycin. HCC1806 and HCC38-CD44^Hi cells were cultured in RPMI media. Media was supplemented with 10% (v/v) FBS (GE Healthcare) and 1% (v/v) penicillin–streptomycin. Cell lines that were procured from the ATCC were routinely tested to confirm the absence of Mycoplasma contamination using the MycoAlert Mycoplasma Detection Kit (Lonza).

siRNA-Mediated Knockdown of SLC2A3

siRNA knockdown was done using validated Silencer Select siRNA and nontargeting control (Silencer Select Negative Control No. 1; Thermo Fisher Scientific) using RNAiMAX transfection reagent (Thermo Fisher Scientific) as per the manufacturer’s protocol. siRNA complexes were added to cells to attain a final concentration of 20 nmol/L. Although cells were exposed to siRNA complexes in complete media from 24 to 96 hours for in vitro studies unless indicated, siRNA complexes were aspirated 24 hours after transfection and were replaced with complete media for in vivo studies.

Proliferation Assay

About 24 hours after siRNA treatment, the cells were trypsinized and re-seeded at 10³ cells/well in a 96-well plate. Wells were imaged every 24 hours for a 96-hour duration, and proliferation was evaluated using average confluency measures on the Incucyte S3 system (Sartorius). About 12 wells/condition were used for subsequent analysis. For proliferation assays conducted under hypoxic conditions, cells were placed into a hypoxic incubator (2% O₂) immediately after plating. Plates were removed from hypoxic conditions for imaging for the minimal time possible.

Tumorsphere Formation Assay

After overnight siRNA transfection, cells were trypsinized, counted, and resuspended in tumorsphere media [1% w/v methylcellulose and 20 ng/mL (w/v) bFGF, 20 ng/mL (w/v) EGF, 4 μg/mL heparin, and 0.2% (v/v) B27 (50×, Life Technologies; in basal cell culture media) to facilitate cell suspension and three-dimensional growth. Cell density was optimized for all cell lines, and cells were subsequently seeded at 200 to 300 cells per well, allowing tumorspheres to grow over a period of 2 to 3 weeks. To ensure single-cell suspension, cells were either thoroughly resuspended or passed through a filter. Individual cell seeding was confirmed with a bright-field microscope upon seeding. Tumorspheres were topped up every 5 days with an additional 50 μL of tumorsphere media per well.

For hypoxia experiments, tumorspheres were plated as above and then immediately placed into a hypoxic incubator (2% O₂).

Glucose Uptake

Cells were transfected as described above. At 24 hours after transfection, cells were re-seeded into a 96-well plate and allowed to adhere for another 24 hours. Culture medium was removed, and cells were thoroughly washed using PBS to remove any excess glucose. Glucose uptake was measured using the Glucose Uptake-Glo Assay (Promega) as per the manufacturer’s instructions. Luminescence was recorded using a plate reader (FLUOstar Omega, BMG Labtech), and the results were normalized to overall cell confluency (Incucyte S3 system, Sartorius) to account for any difference in cell number.

Seahorse Glycolytic Rate Assay

siRNA transfections were performed as described above. Cells were trypsinized after 24 hours and re-seeded into the Seahorse XF Cell Culture Microplate (Agilent Technologies) at 2 × 10⁴ cells per well. Standard Seahorse XF Glycolytic Rate Assay protocol was performed as per the manufacturer’s instructions, and data were analyzed using Seahorse Analytics software.

Animal Experiments

Mice were 8 to 9 weeks of age at the time of injections. All experiments were approved and conducted in accordance with the National Health and Medical Research Council Statement on Animal Experimentation; the requirements of New South Wales State Government legislation; and the rules for animal experimentation of the Biological Testing Facility of the Garvan Institute of Medical Research, as well as the Victor Chang Cardiac Research Institute (protocol #18/12).

Retro-Orbital Surgery

NSG mice were anesthetized using 4% isoflurane. Alcaine 0.5% anesthetic eye drops were administered prior to and after injection. About 4 × 10⁵ cells were resuspended in 100 μL PBS and injected into the retro-orbital vein using a 25- to 29-gauge syringe per mouse. Mice received thorough postsurgical monitoring for 3 days after surgery. Colonization of injected cells to the lungs was monitored through live in vivo bioluminescence imaging on days 0, 2, 7, and 10 or days 0, 2, 7, 12, 15, and 18 using PerkinElmer IVIS Spectrum In Vivo Imaging Systems. During each imaging session, a dosage of 10 μL luciferin/gram mouse body weight (D-luciferin potassium salt stock, 7.5 mg/mL; Sigma-Aldrich, LUCK-2G) was injected subcutaneously. Imaging was performed 5 minutes after D-luciferin administration to animals (autoexposure and 1-second C and B magnification) to maintain uniform photon flux. Statistical significance was measured using two-way ANOVA.

Tissue Embedding and Antigen Retrieval

Formalin-fixed, paraffin-embedded tissue blocks were used for IHC preparation (at the Histopathology Facility of the Garvan Institute of Medical Research). Then, 4-mm-thick formalin-fixed, paraffin-embedded tissue samples were deparaffinized using xylene and rehydrated with a series of ethanol washes in decreasing concentrations by using the BOND Autostainer (Leica). Heat-induced epitope retrieval was performed using Bond Epitope Retrieval Solution 1 (Leica Biosystems) at 93°C for 30 minutes.

Immunofluorescence Staining

After heat-induced epitope retrieval, Intercept Blocking Buffer (LI-COR Environmental) was added to each tissue section for 30 minutes to 1 hour. Sections were washed three times with ice-cold TBS, and 200 mL primary antibody solution (rabbit anti-Hif1a, B100-499, Novus Biologicals; mouse anti-Glut3, sc-74497, Santa Cruz Biotechnology; and rat anti-CD31, DIA-310, Taylor Biomedical; diluted in 3% BSA/TBS) was added per section and incubated overnight at 4°C. After primary antibody incubation, sections were washed three times with ice-cold TBS and secondary antibodies: anti–rabbit Cy3 (111-166-003, Stratech Scientific), anti–mouse 488 (A11001, Thermo Fisher Scientific), anti–rat 647 (A-21247, Thermo Fisher Scientific) at 1:400, and DAPI (D3106, Thermo Fisher Scientific) at 1:1,000 in 3% BS/TBS) were added for 2 hours at room temperature. ProLong Gold Antifade Mountant (Invitrogen) was used to mount a glass coverslip on each slide. Slides were allowed to dry for 24 to 48 hours prior to imaging. Imaging was performed using the Akoya PhenoCycler-Fusion scan system.

Cell Detection/Positive Cell Detection Analysis

All tissue IHC and immunofluorescence analyses were performed using QuPath software. Cell detection/positive cell detection parameters were optimized on a small representative area of tissue. Cell segmentation and threshold parameters were adjusted to accurately define individual cells within each sample. Cell detection/positive cell detection was subsequently run on each tissue section, excluding the edge region of the tumor.

DREMI and Correlation Analysis

Cell detection was performed as described. DREMI scores and conditional Density Rescaled Visualization heatmaps were generated using the knn-DREMI function from the scprep package (github.com/krishnaswamylab/scprep) and graphed using Matplotlib. Pearson R values were calculated using the pearsonr function from the SciPy package, and regression plots were generated using Seaborn.

Nearest Neighbor Analysis

Positive cell detection was undertaken for GLUT3, CD31, and HIF1a. The cells were classified as GLUT3 positive, HIF1a positive, and/or CD31 positive based on their relative expression of each marker. The nearest neighbor analysis (“Detection Centroid Distances 2D”) command was run and the values exported. The average distance to next the HIF1A-positive and CD31-positive cell was then plotted for all GLUT3-positive cells.

MMTV-PyMT Model Data Preprocessing

Following the data processing steps from Jiang and colleagues (44), we first downloaded week-17 scRNA-seq raw data from the NCBI under accession code PRJNA762594. We then processed the raw data using Cell Ranger with default parameters for read alignment, filtering, barcode counting, UMI counting, and aggregation from different lanes. The reads were aligned to the mm10 reference genome from 10× Genomics. We further processed this dataset by normalizing and log-transforming counts, identifying highly variable genes, and scaling the data. As in the work of Jiang and colleagues (44), we clustered the cells and separated epithelial, immune, and fibroblast populations based on marker genes from the original work, in which each population was separately filtered for QC (epithelial: Epcam, Krt5, Krt14, Krt8, and Krt18; immune: Ptprc, Cd3d, Cd19, Klrd1, and Cd68; and fibroblasts: Dcn, Col3a1, Col1a2, and Col1a1). Epithelial cells and fibroblasts with >5,000 total counts, >2,000 expressed genes, and <8% proportion of mitochondrial gene counts were retained. Immune cells with >1,000 total counts, >500 expressed genes, and <8% proportion of mitochondrial gene counts were retained. Epithelial cells were then processed using the MAGIC method and then by PCA, and we ran AAnet on the resulting embeddings with four ATs based on the geodesic distance metric. We then scored cells with respect to the cancer hallmark gene sets for mice (73) and visualized the score for cells with affinity for each AT. Immune and fibroblast cells were visualized using PHATE and further subclustered and annotated on the basis of marker genes (fibroblasts: Dcn, Col3a1, Col1a2, and Col1a1; myeloid: Cd68, Cd86, Csf1r, Fcgr3, and Cd74; T cells: Cd3d, Cd81b1, Cd8a, Cd4, and Tcf7; and NK cells: Klrd1, Il2rb, Klrk1, and Ifngr1). Cross-talk was established by performing CellPhoneDB implemented in squidpy (79) with default parameters of the cells committed to each AT and immune and stromal cells (45). We subset to only cancer-immune or cancer-stromal ligand–receptor hits with P$\mathrm{<}0.05$.

Human scRNA-seq Data Preprocessing

Breast cancer samples for female patients were used. For experiments from Pal and colleagues (52), Bassez and colleagues (53), and Wu and colleagues (54), the datasets were first subset to cancer epithelial cells based on prior annotation, and only samples with ≥ 1,000 cells were analyzed with AAnet. We then further preprocessed the data by removing genes expressed in fewer than five cells, filtering the library size to between 1,500 and 60,000 UMI counts, and L1 normalizing for the library size. We then square root–transformed the data and filtered any remaining contaminating cells based on marker gene expression. Finally, we embedded the data using MAGIC and PCA before identifying ATs with AAnet. For experiments based on the PDX model, the same pipeline was followed, wherein additionally cells with <99% of reads aligning to human cells were removed, and mouse genes were removed.

After stringent filtering, we retained 15 samples from Pal and colleagues (52), nine from Wu and colleagues (54), eight from Bassez and colleagues (53), and two patient-derived xenograft (PDX) samples. These samples comprised 15 ER⁺ tumors, 13 TNBC tumors, and six HER2⁺ tumors and were distributed across all studies, with the exception of no HER2⁺ tumors retained from Wu and colleagues (54) and no ER⁺ and HER2⁺ tumors from the scPDX data. On average, we derived 4.66 ATs in ER⁺ samples, 4.33 ATs in HER2⁺ samples, and 4.53 ATs in TNBC samples, with no statistically significant difference in the distribution of the number of ATs between subtypes (KS P > 0.05 for all tests). On average, 44.85% of cells from ER⁺ tumors were uncommitted, 41.59% of cells from HER2+ tumors were uncommitted, and 39% of cells from TNBC tumors were uncommitted, with no statistically significant difference in distribution of proportion of uncommitted cells between subtypes (KS P > 0.05 for all tests). On average, over all ATs within each subtype, 12.01% of cells from ER⁺ tumors, 13.74% from HER2⁺ tumors, and 13.38% from TNBC tumors were committed.

For comparison of our AAnet-derived samples and human clinical samples, we anticipated lower cosine similarity overall due to differences between human and cell line data. Indeed, the hypoxic AT correlation had an IQR of (−0.18, 0.16) and the proliferative AT correlation had an IQR of (−0.29, 0.32). Therefore, for demonstrative and visualization purposes, we defined those human ATs similar to the hypoxic AT as having modest cosine similarity (with a threshold set to 0.25 due to lower overall similarity) and high dissimilarity to the proliferative AT (with a threshold set to −0.25). Changing this threshold (similarity of 0.1 and dissimilarity of −0.1) results in a similar overall trend (Supplementary Fig. S9B).

To determine if there was significant enrichment for a given cancer subtype and association with the MDA-MB-231–derived ATs, we performed overrepresentation analysis with a two-sample KS test, in which we ordered human ATs by similarity to each mouse AT and then performed the test between the overall ranking and the ranking of each subtype.

Data Availability

The data generated in this study are publicly available in Gene Expression Omnibus (GEO) under accession number GSE299393. Public CD8⁺ T-cell data (29) can be accessed at GEO under accession number GSE182509. MMTV-PyMT model raw data (44) can be downloaded from the NCBI under accession code PRJNA762594. Processed scRNA-seq data from Pal and colleagues (52) are available as GEO series GSE161529. Raw sequencing reads of all single-cell experiments from Bassez and colleagues (53) have been deposited in the European Genome-phenome Archive under study number EGAS00001004809. Processed scRNA-seq data from Wu and colleagues (54) are also available through the GEO under accession number GSE176078.

The source code is available at https://github.com/KrishnaswamyLab/AAnet.

Supplementary Material

Supplementary Table 1

Primary_replicate_differentially_expressed_genes

Supplementary Table 2

All tissues _differentially_expressed_genes

Figures S1-9

This files contains all supplementary data supporting the development and validation of AAnet

Authors’ Disclosures

S.E. Youlten reports personal fees from Compbiosphere outside the submitted work, as well as being the founder and director of Compbiosphere, a computational biology analytics and consulting company. D.B. Burkhardt reports other support from Cellarity, Inc. and NVIDIA and grants from Chan Zuckerberg Initiative outside the submitted work. A. Benz reports other support from Cellarity, Inc., outside the submitted work. J. Lundeberg reports grants from The Swedish Cancer Society during the conduct of the study, as well as personal fees from 10× Genomics, Inc. outside the submitted work. S. Krishnaswamy reports personal fees from Meta and Latent Alpha outside the submitted work. No disclosures were reported by the other authors.

Authors’ Contributions

A. Venkat: Conceptualization, data curation, software, formal analysis, validation, investigation, visualization, methodology, writing–original draft, writing–review and editing. S.E. Youlten: Conceptualization, data curation, software, formal analysis, validation, investigation, visualization, methodology, writing–original draft, writing–review and editing. B.P. San Juan: Conceptualization, data curation, formal analysis, validation, investigation, methodology. C.A. Purcell: Investigation, writing–review and editing. S. Gupta: Conceptualization, validation, investigation, writing–review and editing. M. Amodio: Investigation, visualization, methodology, writing–review and editing. D.P. Neumann: Conceptualization, data curation, software, formal analysis, investigation, visualization, methodology, writing–review and editing. J.G. Lock: Conceptualization, data curation, software, formal analysis, supervision, investigation, visualization, methodology, writing–original draft. A.E. Westacott: Conceptualization, data curation, software, formal analysis, supervision, investigation, visualization, methodology, writing–original draft, writing–review and editing. C.S. McCool: Conceptualization, software, formal analysis, investigation, visualization, methodology, writing–review and editing. D.B. Burkhardt: Conceptualization, resources, data curation, software, formal analysis, investigation, methodology, writing–original draft, project administration. A. Benz: Conceptualization, resources, data curation, software, formal analysis, investigation, methodology, writing–original draft, project administration. A. Mollbrink: Conceptualization, resources, data curation, software, formal analysis, supervision, investigation, methodology, project administration, writing–review and editing. J. Lundeberg: Conceptualization, resources, data curation, software, formal analysis, supervision, investigation, methodology, project administration, writing–review and editing. D. van Dijk: Conceptualization, resources, data curation, software, formal analysis, supervision, funding acquisition, investigation, methodology, project administration, writing–review and editing. J. Holst: Conceptualization, resources, data curation, software, formal analysis, supervision, funding acquisition, investigation, writing–original draft, project administration, writing–review and editing. L.D. Goldstein: Conceptualization, resources, data curation, software, formal analysis, supervision, funding acquisition, investigation, writing–original draft, project administration, writing–review and editing. S. Kummerfeld: Conceptualization, resources, data curation, software, formal analysis, supervision, funding acquisition, investigation, writing–original draft, project administration, writing–review and editing. S. Krishnaswamy: Conceptualization, resources, data curation, software, formal analysis, supervision, funding acquisition, investigation, visualization, writing–original draft, project administration, writing–review and editing. C.L. Chaffer: Conceptualization, resources, data curation, formal analysis, supervision, funding acquisition, investigation, visualization, writing–original draft, project administration, writing–review and editing.