A Generative Model for Doublets

Why Model a Doublet?

A barcode is usually interpreted as representing one cell, but a capture partition can contain two cells. RNA from both cells may then be assigned to the same barcode. The resulting row of the count matrix records one barcode, even though its RNA came from two sources. A generative model makes this distinction explicit: the observed counts are formed by combining two cell-specific count profiles.

This model explains two broad patterns. When the two cells have different expression profiles, their combined profile can contain signal from both. When their profiles are similar, the combined profile can retain an ordinary-looking composition while having more total counts than either source alone. Neither pattern uniquely establishes that a barcode is a doublet; singlets can also vary in total RNA and expression.

A Count-Level Generative Model

Suppose there are \(G\) genes. Let \(X_{kg}\) be the captured count for gene \(g\) from source cell \(k\), for \(k=1,2\). The observed count \(Y_g\) for gene \(g\) in the shared barcode is

Writing each cell’s gene counts as a vector gives \(\mathbf{Y}=\mathbf{X}_1+\mathbf{X}_2\). The total observed count is therefore the sum of the two source totals. Define \(N_k=\sum_{g=1}^{G}X_{kg}\) and \(N=\sum_{g=1}^{G}Y_g\). Then

A useful model for the shape of each profile conditions on its total count. Let \(\boldsymbol{\theta}_k=(\theta_{k1},\ldots,\theta_{kG})\) be the source cell’s gene-proportion vector, where \(\theta_{kg}\geq 0\) and \(\sum_g\theta_{kg}=1\). Conditional on \(N_k\) and \(\boldsymbol{\theta}_k\), model the source counts as

This is a simplified conditional model: the total \(N_k\) is treated as given, and the gene counts are allocated according to the cell’s proportions. It does not claim that all cells have the same total or the same gene proportions. Once two realized source profiles are combined, the normalized doublet profile is their total-count-weighted average:

Thus the profile is not generally the unweighted average of the two source profiles. The source contributing more captured counts has greater influence on the combined proportions. This is a statement about the realized counts; it does not mean that a doublet’s normalized proportions must look distinct from those of every singlet.

Assumptions and a Short Derivation

The additive model is useful when several conditions are reasonable. Both source profiles must use the same gene definitions and counting units; counts from the two sources must be assigned to the shared barcode; and the observed counts must represent molecules in a way that can be added across sources. The basic model also assumes that the barcode contains exactly two contributing cells and sets aside complications such as molecule-tag collisions, barcode errors, and ambient RNA. These are modeling simplifications, not guarantees about every experiment.

The weighted-profile result follows directly from the count model. Sum the gene-level equation \(Y_g=X_{1g}+X_{2g}\) over all genes to obtain \(N=N_1+N_2\). Divide the gene-level equation by \(N\), and multiply and divide each source term by its total:

The weights are nonnegative and sum to one, so the combined proportion for a gene lies between the two source proportions for that gene. This derivation concerns a profile normalized by its total; it does not imply that the total count is itself a reliable doublet label.

What the Model Implies for QC

A doublet can have more total counts than either source because the observed total is additive. It can also have more detected genes, but the increase depends on how much the source profiles overlap. If \(S_1\) and \(S_2\) are the sets of genes with positive counts in the two source profiles, then the detected genes in their sum are in \(S_1\cup S_2\). The number detected is

Genes detected in both cells are counted once in the combined barcode’s number of detected genes. Consequently, two source cells with many shared detected genes may produce a smaller increase in detected-gene count than two cells with little overlap. Neither total counts nor detected genes are guaranteed to increase by a fixed factor.

In practice, compare a barcode’s counts and detected genes with appropriate singlet expectations in its sample and cell context, rather than treating a single threshold as definitive. Inspect whether its gene profile contains a plausible combination of signals, while remembering that genuine biological variation can also produce broad or unusual expression. The earlier material on per-cell counts and detected genes provides the basic QC summaries; here, the generative model clarifies why elevated values can arise from two sources.

Diagnostics, Alternatives, and Limitations

The model is a description of how counts can be generated, not a complete detection procedure. Its simplifying assumptions can fail or be incomplete. For example, the two source cells may have different capture efficiencies, and repeated molecule tags within one barcode can prevent exact additivity in the observed deduplicated counts. A barcode may also contain RNA not adequately represented by the two-cell model. In those cases, the summed profile is an approximation to the recorded data rather than an exact physical accounting of every molecule.

A useful diagnostic question is whether several observations agree with the model: a barcode may have elevated total counts, additional detected genes, and a combination of expression signals that is plausible given the surrounding data. Agreement increases the plausibility of a doublet explanation, but it is not conclusive. Conversely, failure to see an obvious mixed profile does not rule out a doublet when the source profiles are similar or one source contributes much less captured RNA.

One alternative is to focus only on normalized proportions. This makes composition easier to compare across totals, but it discards the total-count information that can be a consequence of adding two profiles. Another is to treat every barcode as a single-cell profile and ignore the possibility of multiple sources; that is simpler, but can misrepresent the origin of the counts. The additive model keeps both composition and total in view, while remaining deliberately simpler than a full model of capture, sequencing, and technical errors.

A common mistake is to reason, “This barcode has twice the usual counts, so it must be a doublet.” The correction is that the model predicts a sum when two cells share a barcode, but a large count can also occur for a singlet; moreover, doublets need not have exactly twice a typical count. Source totals vary, capture is imperfect, and the two profiles can overlap. Counts are evidence to interpret in context, not a deterministic label.