Separating Expression Changes From Composition Changes

Why a Spatial Pattern Can Have More Than One Explanation

A gene can appear more abundant in one tissue region than another for at least two distinct reasons. The region may contain different proportions of cell types, or cells of the same type may express the gene differently across regions. Both mechanisms can operate together. A spatial map of spot-level abundance alone does not distinguish them: a spot is a mixture, so its value can change even when expression within every cell type remains fixed.

The aim here is to make that distinction explicit with a mixture model and a numerical decomposition. The quantities in the examples are hypothetical. They represent comparable, exposure-adjusted gene abundance on a common scale, not unadjusted sequencing counts.

This equation is an expectation for a mixture: if a cell is drawn from the spot’s cell population, its expected gene abundance is the proportion-weighted average of the cell-type means. For raw spot counts, an additional factor such as the number of cells represented and capture or sequencing exposure affects the expected total. Consequently, comparing \(q_s\) requires that differences in such exposure have been accounted for or are equal across the comparison. Earlier material on spot counts and linear mixture models provides the setting for this representation; the focus here is interpreting changes in the mixture itself.

The cell proportions in this definition are proportions of cells, not automatically proportions of captured RNA. If cell types differ greatly in RNA content, or if normalization changes the scale, a mixture based on RNA contribution may be more appropriate than a cell-fraction mixture. The meaning of the weights must match the quantities being averaged.

Separating a Difference Between Two Regions

Consider two regions, \(A\) and \(B\). Let \(p_{Ak}\) and \(p_{Bk}\) be their cell-type proportions, and let \(\mu_{Ak}\) and \(\mu_{Bk}\) be their cell-type-specific means for the gene. Their mixture means are \(q_A=\sum_k p_{Ak}\mu_{Ak}\) and \(q_B=\sum_k p_{Bk}\mu_{Bk}\). The observed mixture difference is \(\Delta q=q_B-q_A\). It is useful to split this difference into a composition component and an expression component.

The composition component asks how the mixture would shift as the proportions change, averaging the two regions’ cell-type means. The expression component asks how the mixture would shift as cell-type means change, averaging the two regions’ proportions. The symmetric averaging is useful because it does not privilege either region as the sole baseline.

Why the Decomposition Adds Up

For each cell type, abbreviate the changes by \(\Delta p_k=p_{Bk}-p_{Ak}\) and \(\Delta\mu_k=\mu_{Bk}-\mu_{Ak}\). The contribution of type \(k\) to the mixture difference can be written as \(p_{Bk}\mu_{Bk}-p_{Ak}\mu_{Ak}\). Since \(p_{Bk}=p_{Ak}+\Delta p_k\) and \(\mu_{Bk}=\mu_{Ak}+\Delta\mu_k\), expansion gives \(p_{Ak}\Delta\mu_k+\mu_{Ak}\Delta p_k+\Delta p_k\Delta\mu_k\). This expression includes an interaction term: changing both proportion and mean makes the total different from simply adding changes evaluated at region \(A\).

The symmetric decomposition splits that interaction evenly between the two components. In fact, \(\Delta p_k(\mu_{Ak}+\mu_{Bk})/2=\mu_{Ak}\Delta p_k+\Delta p_k\Delta\mu_k/2\), and \(\Delta\mu_k(p_{Ak}+p_{Bk})/2=p_{Ak}\Delta\mu_k+\Delta p_k\Delta\mu_k/2\). Adding these expressions recovers the expanded contribution exactly. Summing over cell types proves \(C+E=q_B-q_A\). This is an accounting identity for the specified mixture means; it does not establish that either component is causal.

Another valid approach is to choose region \(A\) as a reference and first change proportions while holding means at \(\mu_A\), then change means at proportions \(p_B\). That gives a composition term \(\sum_k\Delta p_k\mu_{Ak}\) and an expression term \(\sum_k p_{Bk}\Delta\mu_k\), which also add exactly to \(\Delta q\). Reversing the order assigns the interaction differently. The symmetric definition avoids that order dependence by sharing the interaction equally.

Worked Examples

Assumptions, Diagnostics, and Interpretation

The decomposition is only as meaningful as its inputs. The cell types should form a mutually exclusive and sufficiently comprehensive partition of the cells represented in the spots. Within each region, the proportions should be nonnegative and sum to one. The cell-type means and the mixture values must refer to the same gene, compatible measurement scale, and comparable exposure adjustment. If a cell type is absent from one region, its mean there may not be estimable from that region’s observations; a decomposition using a borrowed or imputed mean must say so explicitly.

Check the following before interpreting the components:

• Proportion checks. Verify that each proportion lies between zero and one and that the proportions sum to one, allowing only small discrepancies from rounding. Large violations may indicate an estimation or cell-type definition problem.
• Scale checks. Confirm that the cell-type means are comparable across regions. Different library-size adjustments, transformations, or reference scales can produce apparent differences that are not biological changes in per-cell expression.
• Mixture reconstruction. Calculate \(\sum_k p_{sk}\mu_{sk}\) for each region and compare it with the chosen exposure-adjusted spot-level quantity. A large discrepancy signals that the mixture inputs do not represent that quantity well.
• Reference and classification checks. Misclassified cells, omitted cell types, or cell states that vary across regions can shift both estimated proportions and estimated means. Treating a heterogeneous type as uniform can make a within-type change look like composition, or the reverse.
• Uncertainty checks. Estimated proportions and cell-type means are not known exactly. The arithmetic decomposition is exact conditional on those estimates, but that does not make the component estimates precise. When inputs are uncertain, report that uncertainty rather than treating the decomposition as fixed truth.

A composition component is a standardized descriptive contrast: it quantifies how the mixture mean changes with the observed proportions under the specified cell-type means. The expression component similarly summarizes changes in the observed type-specific means under averaged proportions. Neither is automatically a causal effect. Spatially patterned unmeasured factors, tissue processing, cell-state shifts, and errors in deconvolution can affect the inputs. Also, a small total difference can conceal large opposing components; reporting only \(\Delta q\) can miss that cancellation.

How This Differs From Other Summaries

A raw spot-level comparison describes the net difference in the measured spot quantity, but by itself does not explain whether it arises from composition or cell-type-specific expression. Comparing estimated means within each cell type can reveal expression contrasts, but alone it does not quantify how the changing proportions contribute to the spot-level difference. Standardization brings both pieces onto the same mixture scale and makes their contributions add to the observed contrast.

A composition-only counterfactual can be useful: hold all cell-type means fixed at a chosen reference and recalculate the mixture using the other region’s proportions. Its answer depends on the selected reference. The symmetric decomposition instead averages over both regions, though it still relies on valid proportions and means. These are descriptive alternatives, not substitutes for a design that supports causal interpretation or for an analysis that accounts for uncertainty and count variability.

Spatial smoothness or clustering in spot-level abundance does not, by itself, identify which mechanism is responsible. A spatial pattern may track tissue architecture and hence changing composition, within-type expression, or both. The mixture calculation clarifies the possible sources of a pattern; it does not replace checking whether the inputs and comparison are suitable.