💻 bioinformatics

Scale-resolved correlation functions unify spatial omics across platforms and resolutions

The paper introduces SCOTE, a scalable computational framework that unifies spatial omics analysis across diverse platforms and resolutions by quantifying spatial correlations at all length scales, thereby distinguishing biological signals from platform-specific artifacts and enabling efficient, high-throughput processing of massive datasets.

Original authors: Ilin, M., Singh, Y., Mages, S. W.

Published 2026-10-16

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Ilin, M., Singh, Y., Mages, S. W.

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ⚕️ This is an AI-generated explanation of a preprint that has not been peer-reviewed. It is not medical advice. Do not make health decisions based on this content. Read full disclaimer

Imagine trying to understand the layout of a bustling city. You could count the people standing on every street corner, you could group them into neighborhoods, or you could simply note where every single person is standing at a specific moment. In the field of spatial biology, scientists are doing something similar, but instead of a city, they are mapping the inside of living tissue. For years, researchers have used different tools to take these maps. Some tools act like high-resolution cameras, snapping pictures of individual molecules of genetic material floating inside cells. Others act like a grid of buckets, catching and counting how many molecules land in each square. The problem is that these tools speak different languages. A map made by one camera cannot easily be compared to a map made by a different camera, or to a grid-based count, because the methods for analyzing the data are built specifically for each tool. This has made it difficult to see the big picture or to combine results from different studies to find universal truths about how tissues are organized.

A team of researchers at the Gene Center Munich has developed a new way to look at these maps that speaks a single, common language. They created a computational framework called SCOTE, which stands for Spatial COvariance TEnsors. Instead of forcing the data into pre-defined boxes or relying on imperfect guesses about where cell boundaries are, this method measures how often any two types of molecules appear near each other across every possible distance. It asks a simple question: if you pick a molecule of type A, how likely are you to find a molecule of type B at a distance of one micrometer, ten micrometers, or a millimeter away? By calculating this relationship for every pair of genes and at every scale, from the size of a single molecule to the size of an entire organ, the researchers found that the underlying patterns of tissue organization are actually the same, regardless of which machine was used to capture the data.

The power of this approach lies in its ability to strip away the noise created by the machines themselves. When the researchers compared maps of colorectal cancer tissue taken with two different high-tech platforms—one that tracks individual molecules and another that counts them in small squares—they found that the patterns of gene interaction were nearly identical. The two machines agreed on the spatial relationships between genes with a correlation of nearly 90 percent, a level of agreement that was far higher than the differences seen between different patients. This suggests that the biological reality of how cells arrange themselves is robust and can be measured consistently, provided the analysis method is right. The study also showed that the differences between the machines were not random errors but were confined to specific, identifiable scales. For instance, one machine struggled to see molecules that were extremely close together, while another missed signals from rare cell types because it relied on grouping molecules into cells first. By measuring these patterns directly, the new method could spot these machine-specific quirks and treat them as known features rather than hidden biases.

One of the most striking findings was that looking at individual molecules revealed biological changes that were completely invisible when scientists looked only at whole cells. In a study of mouse spleens, the researchers examined a genetic change that disrupts the organization of immune cells. When they analyzed the data by grouping molecules into cells first, the genetic change seemed to have no effect. However, when they looked at the raw positions of the molecules, a clear signal emerged: the immune cells were avoiding each other much more strongly in the genetically altered mice than in the normal ones. This subtle shift, which happened at a scale smaller than a single cell, was missed by traditional methods that rely on cell boundaries. The new approach showed that the biological signal was there all along, but it was hidden by the step of grouping molecules into cells, a step that often introduces errors because molecules from neighboring cells can spill over into each other.

The researchers also tackled the question of how much data is actually needed to get a reliable answer. A common assumption in the field is that sequencing deeper—finding more molecules in the same piece of tissue—will always lead to better results. The study proved this is not necessarily true. They found that the precision of the measurements depends far more on the total area of the tissue being examined and the number of separate tissue samples taken, rather than on the sheer number of molecules counted in a single sample. Doubling the number of molecules in a tiny piece of tissue did not significantly reduce the error, whereas looking at a larger area or adding more samples did. This insight offers a clear guide for future experiments: to understand tissue organization, it is better to spread the budget across more tissue area and more samples than to try to count every single molecule in a small spot.

Finally, the team demonstrated that this method is fast enough to handle the massive amounts of data being generated by modern technology. They ran the analysis on a single graphics card, a type of computer chip often used for video games, and processed a section of tissue containing over one billion molecules in just 26 minutes. This speed makes it possible to analyze entire genomes across whole tissue sections, a task that would have taken weeks or months with older methods. The result is a unified view of spatial biology where data from different machines, different resolutions, and different patients can be compared directly. By focusing on the relationships between molecules across all scales, the researchers have provided a way to see the true architecture of living tissue, separating the signal of biology from the noise of the tools used to measure it.

Technical Summary: Scale-resolved correlation functions unify spatial omics across platforms and resolutions

Problem Statement
Spatial omics technologies generate data across a multidimensional parameter space, varying significantly in resolution (from sub-micrometer molecule coordinates to millimeter-scale spots), panel size, sensitivity, and modality (imaging vs. sequencing). Current analytical approaches typically operate on a single scale, often relying on segmented cell representations where molecules are assigned to cells, followed by clustering and centroid-based statistics. This pipeline introduces limitations: segmentation errors (e.g., transcript spillover) propagate to downstream analyses, and results are rarely comparable across platforms or preprocessing steps. Existing spatial statistics (e.g., Ripley's $K$ , pair-correlation functions) often scale quadratically with object count, restrict analysis to cells or subsamples, and lack a unified normalization or error model that allows direct quantitative comparison between different platforms, samples, and patients.

Methodology: SCOTE
The authors present SCOTE (Spatial COvariance TEnsors), a computational framework designed to quantify spatial relationships for all feature pairs across all length scales for both molecule coordinates and binned counts. The core of the method is the estimation of the normalized two-point correlation function, $g_{ab}(r)$ , which measures the frequency of feature pairs $a$ and $b$ at distance $r$ relative to a random-label reference.

Key methodological components include:

Unified Estimator: The function $g_{ab}(r)$ is defined identically for molecule coordinates and binned counts, allowing a single framework to process diverse data types. It uses a random-label reference (permuting labels among observed positions) to remove the effects of tissue geometry, holes, and density gradients without requiring a parametric model.
Log-Hierarchical Lattice: To handle the vast range of scales (nanometers to millimeters) efficiently, the framework aggregates molecules onto a log-hierarchical lattice where the pitch doubles at each level. Pair counts for all feature pairs within a distance shell are computed as a single sparse matrix product ( $P_k = X^T S_k X$ ), enabling the calculation of all pairs at all scales in one pass.
Statistical Uncertainty: A block bootstrap over 500 µm tiles is applied to estimate uncertainty. The authors derive a closed-form solution for the variance of $\log g_{ab}(r)$ , attaching a standard deviation and a 95% confidence interval to every value without the computational cost of resampling.
Implementation: The framework is implemented in Python, utilizing sparse matrix operations and optimized back-ends (SciPy, Numba, PyTorch) to run efficiently on both CPUs and GPUs.

Key Results
The authors validate SCOTE using colorectal cancer (CRC) serial sections profiled on Xenium, Visium HD, and Visium v2, as well as mouse spleen sections from the SpatialBench study (Xenium, Visium HD, MERSCOPE).

Cross-Platform Unification: Normalized two-point function patterns agree significantly more between different platforms profiling the same patient than between different patients. For CRC sections, the Pearson correlation of $\log g$ between Xenium and Visium HD ranged from 0.85 to 0.90 across scales from 10 µm to 1 mm. Platform-specific differences were identifiable as scale-localized features (e.g., resolution limits, gene-specific backgrounds) rather than global biases.
Resolution of Biological Effects: Molecule-level functions resolved a knockout effect in mouse spleen ($Tbx21$ deletion in B cells) that cell-level analyses missed. While cell-centroid and Squidpy co-occurrence analyses failed to detect significant differences between wild-type, control, and knockout groups, the molecule-level function revealed a significantly stronger B–T anti-correlation in knockouts at distances below the cell diameter (10 µm), with narrower confidence intervals.
Error Determinants: Statistical error is primarily set by tissue area and the number of replicates, not by molecule depth (sequencing depth). The standard deviation of $\log g$ scales with the sampled area fraction $f$ as $\sigma \propto f^{-\alpha}$ , where $\alpha \approx 0.42–0.50$ . Increasing sequencing depth on a fixed area yielded diminishing returns for error reduction at scales larger than ~20 µm.
Artifact Identification: Platform artifacts appear as distinct, scale-localized features. Examples include the loss of same-gene pairs below optical resolution (e.g., <1 µm on Xenium), gene-specific backgrounds (e.g., ACTA2 in tumor regions), and segmentation failures for rare, compact cell types (e.g., mast cells), which are detectable via molecule-level functions but lost in cell-level clustering.
Scalability: The framework is highly efficient. It processed a whole-transcriptome section with 1.04 billion molecules (18,028 genes) on a single NVIDIA H100 GPU in 26.2 minutes, returning all gene pair correlations with uncertainties. In contrast, extrapolated runtimes for existing tools (Squidpy, KD-tree references) on similar datasets would require days to weeks.

Significance and Claims
The paper claims that scale-resolved two-point functions provide a "common quantitative currency" for spatial omics. By measuring the same quantity with attached statistical errors across platforms, SCOTE separates measurement artifacts from biological signals. The authors assert that this approach enables:

Differential analysis with calibrated error estimates and built-in controls.
Cross-platform comparisons at stated normalization levels and scales.
Experimental design optimization, prioritizing tissue area and replicates over sequencing depth for most spatial questions.
Quality control by identifying and discarding bins where platform-specific artifacts act.

The authors note that the current implementation is two-dimensional, which may bias the first bins on thick samples, and that while the method can pinpoint artifacts, it does not automatically correct them. The framework is generalizable to any modality providing labeled positions or binned values, including protein data, though the current validation is focused on RNA.

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