Spatially aligned random partition models on spatially resolved transcriptomics data

This paper proposes Spatially Aligned Random Partition (SARP) models, a Bayesian nonparametric framework that hierarchically links cell-type-specific cluster parameters to identify spatially aligned co-localization patterns among immune, stromal, and tumor cells, demonstrating its effectiveness through simulations and application to colorectal cancer data.

The Gist

The Big Picture: A Detective Story in the Tumor City

Imagine a tumor isn't just a blob of bad cells. Instead, think of it as a bustling, chaotic city. In this city, there are three main groups of residents:

1. The "Tumor" Citizens: The bad guys causing the trouble (epithelial cells).
2. The "Immune" Police: The body's defenders trying to stop the bad guys.
3. The "Stromal" Construction Crew: The support staff that builds the city's infrastructure.

For a long time, scientists could look at a map of this city (spatial data) to see where people lived, or they could look at a DNA test (gene expression) to see who people were. But they struggled to answer a crucial question: How do these groups interact? Specifically, do certain types of "bad guys" recruit specific types of "police" or "construction workers" to hang out right next to them?

Existing tools were like taking a photo of the whole city and saying, "Look, there's a crowd here." They couldn't easily say, "The specific group of Tumor Citizens in the North District is specifically recruiting the Myeloid Police to stand guard right next to them."

The Solution: SARP (Spatially Aligned Random Partition)

The authors, led by Yunshan Duan and Peter Mueller, invented a new statistical tool called SARP. Think of SARP as a super-smart, magical matching game that connects the dots between these different groups of residents.

Here is how it works, broken down into simple concepts:

1. The "Two-Track" System

Imagine you have two separate lists of people:

• List A: The Tumor Citizens.
• List B: The Immune and Construction Crew.

Usually, scientists would try to group List A and List B together into one big mix. But SARP is smarter. It says, "Let's keep the lists separate, but we will look at their locations to see if they are dancing together."

• The Gene Expression (The DNA ID): This tells us who someone is. SARP treats this independently for each group. A Tumor Citizen's DNA is compared only to other Tumor Citizens.
• The Spatial Location (The Address): This tells us where someone is. This is where the magic happens. SARP looks at the addresses and asks, "Do the addresses of the Immune Police cluster around the addresses of the Tumor Citizens?"

2. The "Magnet" Analogy

Think of the Tumor Citizens as magnets.

• Some magnets are weak; they don't attract anyone.
• Some magnets are strong; they pull specific types of Immune Police right next to them.

SARP is the tool that detects these invisible magnetic fields. It doesn't just say "Police are near Tumor." It says, "The specific subgroup of Tumor Citizens in Cluster #5 is acting like a super-magnet for the specific subgroup of Myeloid Police in Cluster #2."

3. The "Pitman-Yor" Prior (The Flexible Clustering)

In statistics, you often have to guess how many groups (clusters) exist. SARP uses a special mathematical rule called the Pitman-Yor process.

• Analogy: Imagine a restaurant (the "Chinese Restaurant Process"). New customers (cells) walk in. They either sit at an existing table (join a known group) or start a new table.
• SARP tweaks the rules of this restaurant. It allows tables to form naturally based on how many people are already there, but it also adds a "discount" rule that prevents the restaurant from having too many tiny, meaningless tables. This ensures the groups it finds are biologically real and not just random noise.

How They Tested It (The Simulation)

Before using this on real cancer patients, the authors played a game with fake data.

• They created a digital city with known "truths" (e.g., "We know Group A is supposed to be near Group B").
• They ran SARP and other existing tools against this fake city.
• Result: SARP was much better at finding the correct "magnetic" connections than the other tools. It correctly identified which groups were hanging out together and which were just passing by.

The Real-World Application: Colorectal Cancer

They took this tool to real data from Colorectal Cancer (CRC) patients.

• The Discovery: They found that the tumor isn't just one big blob. It has different "neighborhoods" (subtypes).
• The Insight: In some neighborhoods, the Tumor Citizens are recruiting specific "Construction Crew" (Stromal cells) to build a protective wall around them. In other neighborhoods, they are recruiting "Myeloid Police" to help them hide from the immune system.
• Why it matters: If we know exactly which tumor neighborhood is recruiting which helper, doctors might be able to design drugs to break that specific alliance. It's like knowing exactly which gang is paying off which police officer, so you can cut that specific deal.

The "Secret Sauce" (What makes this paper special?)

1. Asymmetry: The model understands that the relationship isn't equal. The Tumor cells are the "host" (the reference), and the Immune cells are the "guests" being recruited. The math reflects this one-way street.
2. Partial Connection: It connects the groups based only on location, not on their DNA. This is crucial because a Tumor cell and an Immune cell will never have the same DNA, but they can still be best friends in the same neighborhood.
3. Uncertainty: The model doesn't just give a "Yes/No" answer. It gives a probability. "There is a 96% chance these two groups are linked, but a 4% chance they are just neighbors." This honesty about uncertainty is vital for medical science.

Summary

In simple terms, this paper introduces a new way to map the social life of cancer cells. Instead of just looking at a map or a DNA list separately, SARP combines them to reveal who is recruiting whom in the tumor city. This helps scientists understand the "secret alliances" that help cancer grow, potentially leading to better ways to break those alliances and cure the disease.

Technical Summary

1. Problem Statement

The paper addresses a critical gap in the analysis of Spatially Resolved Transcriptomics (ST) and Single-Cell RNA Sequencing (scRNA-seq) data. While existing methods can cluster cells within a single dataset or jointly cluster genes and spatial regions, there is no principled Bayesian approach to infer the spatial co-localization between distinct sub-populations of different cell types (e.g., tumor, immune, and stromal cells).

Specifically, the authors aim to answer: How do specific subtypes of immune and stromal cells recruit or cluster around specific subtypes of tumor cells?

• Challenge: Standard dependent mixture models (e.g., Hierarchical Dirichlet Processes) often assume dependence across all features or require symmetric structures. However, biologically, the dependence is asymmetric (tumor cells act as a reference recruiting others) and partial (dependence exists only in spatial coordinates, while gene expression profiles remain independent across cell types).
• Goal: Develop a model that performs model-based clustering on separate sets of cells while introducing dependence only in the spatial sub-vector of the cluster-specific parameters.

2. Methodology: Spatially Aligned Random Partition (SARP) Models

The authors propose SARP, a Bayesian Nonparametric (BNP) framework that constructs dependent random partitions across multiple sets of experimental units.

A. Statistical Framework

The model treats cell subtypes as clusters generated from mixture models. Let $y_{ji}$ represent the feature vector for cell $i$ of type $j$ (where $j=1$ is the reference, e.g., tumor/non-immune, and $j=2, \dots, J$ are other types like immune/stromal).

• Observation Model: $y_{ji} = (s_{ji}, x_{ji})$, where $s_{ji}$ are spatial coordinates and $x_{ji}$ are gene expression features.
• Likelihood: The data is modeled as a mixture of kernels: $f(y_{ji} | \omega_{ji})$, where $\omega_{ji}$ are cluster-specific parameters.
• Independence Assumption: The gene expression features ($x$) are modeled with independent base measures across cell types. The spatial features ($s$) are the only source of dependence.

B. Prior Construction (The Core Innovation)

The dependence is introduced via the Pitman-Yor Process (PYP) priors on the mixing measures $G_j$.

1. Reference Type ($j=1$): The mixing measure $G_1$ is drawn from a PYP with a non-atomic base measure $G^{\epsilon}_1$.
   $$G_1 \sim \text{PYP}(a_1, \vartheta_1, G^{\epsilon}_1)$$
2. Dependent Types ($j > 1$): The mixing measure $G_j$ is drawn from a PYP conditional on the latent parameters of the reference type ($\omega_{1, 1:n_1}$).
   $$G_j | \omega_{1, 1:n_1} \sim \text{PYP}(a_j + b_j, \vartheta_j, G^{\epsilon}_j)$$
3. Dependent Base Measure: The base measure $G^{\epsilon}_j$ for the dependent types is constructed as a mixture:
   $$G^{\epsilon}_j = a_j \tilde{G}^{\epsilon}_j + b_j \tilde{f}_1$$
   Where $\tilde{f}_1$ is a kernel centered at the atoms of the reference type ($\omega_{1t}$).
   • Mechanism: This allows clusters of immune/stromal cells to either form at new locations (via $\tilde{G}^{\epsilon}_j$) or form in the vicinity of existing tumor cell clusters (via $\tilde{f}_1$).
   • Asymmetry: The model is inherently asymmetric; tumor cells recruit immune cells, but not vice versa.

C. Posterior Inference

The authors develop a blocked Gibbs sampler for posterior inference:

• Latent Variables: Cluster membership indicators ($z_{ji}$) and alignment indicators ($c_{jk}$) which determine if a cluster $k$ of type $j$ is aligned with a specific reference cluster $t$.
• Truncation: The PYP is truncated at a maximum number of clusters $K_{max}$ to facilitate computation.
• Conjugacy: The use of Gaussian kernels for spatial data and Normal-Inverse-Wishart for gene expression allows for closed-form full conditional distributions, enabling efficient MCMC sampling.

3. Key Contributions

1. Novel Dependence Structure: SARP introduces a specific type of dependence where only a sub-vector of parameters (spatial location) is shared/dependent across different experimental units, while other features (gene expression) remain independent. This aligns with biological reality where cell types have distinct transcriptomes but interact spatially.
2. Asymmetric Reference Framework: Unlike symmetric hierarchical models (e.g., HDP), SARP explicitly models one cell type as a "reference" that recruits others, formalizing the biological concept of tumor-driven immune recruitment.
3. Theoretical Properties: The paper establishes posterior contraction rates for the mixing measures under Finite Mixture (MFM) approximations and discusses the trade-off between the model's flexibility and the loss of Kolmogorov consistency (due to the dependence on sample size $n_1$ in the base measure of $G_2$).
4. Computational Efficiency: The construction allows for a convenient MCMC algorithm that treats atoms generated from the reference kernel as "pseudo-observations," simplifying the sampling steps.

4. Results

A. Simulation Studies

• Setup: The authors simulated bivariate data (spatial + gene expression) with known ground-truth co-localization patterns between immune and non-immune cells.
• Comparison: SARP was compared against a standard Gaussian Mixture Model (GMM) and SpaRTaCo (a state-of-the-art spatial clustering method).
• Findings:
  • In scenarios with well-separated clusters, both SARP and GMM performed well.
  • In semi-synthetic scenarios (using real CRC data distributions with added noise), SARP significantly outperformed GMM and SpaRTaCo in identifying the correct co-localization structure.
  • Metrics: SARP achieved higher True Positive Rates (TPR) and Accuracy (ACC) in recovering the cell-cell co-localization matrix, demonstrating robustness in high-dimensional settings with less distinct spatial boundaries.

B. Application to Colorectal Cancer (CRC) Data

• Data: Integrated scRNA-seq and ST data from a human colorectal cancer sample.
• Analysis:
  • Two-Type Model (Tumor vs. Immune): Identified 15 tumor and 15 immune clusters. The model successfully mapped specific immune subtypes (e.g., Myeloid, T-cells) to specific tumor regions.
  • Three-Type Model (Tumor, Stromal, Immune): Extended the model to $J=3$. This revealed finer resolution, showing how stromal and immune cells are spatially aligned with specific epithelial (tumor) subtypes.
• Biological Insights:
  • Differential Expression (DE): The authors performed DE analysis on the spatially defined epithelial subsets.
  • Validation: The model identified biologically meaningful patterns:
    • Epi-Stromal: High expression of CXCL1/2/3 (mediating epithelial-stromal crosstalk).
    • Epi-Myeloid: Enriched for AGR2 (macrophage-epithelial communication).
    • Epi-T: Upregulated CEACAM1/6 (immune evasion markers).
  • These findings recapitulate known biological interactions, validating that the spatial clusters identified by SARP correspond to functionally distinct microenvironments.

5. Significance and Conclusion

The SARP model provides a rigorous statistical framework for spatially resolved single-cell analysis. Its significance lies in:

• Biological Interpretability: It moves beyond simple spatial clustering to explicitly model the mechanism of cell recruitment (tumor $\to$ immune/stromal).
• Handling Heterogeneity: By decoupling spatial dependence from gene expression dependence, it avoids the "curse of dimensionality" where high-dimensional gene data might obscure spatial patterns.
• Clinical Relevance: The application to CRC data demonstrates the ability to discover novel therapeutic targets by identifying specific tumor-immune-stromal ecosystems (e.g., specific epithelial subtypes that recruit immunosuppressive myeloid cells).

Limitations: The authors note that the pipeline relies on imputed spatial coordinates (ignoring mapping uncertainty) and uses PCA for dimensionality reduction. Future work could explore repulsive priors to enforce cluster separation and direct modeling of high-dimensional gene expression.

In summary, SARP is a powerful tool for decoding the spatial architecture of the tumor microenvironment, offering a statistically principled way to understand how different cell types organize themselves to drive cancer progression.