Using Clonal Labels to Constrain Lineage Models

Why Clonal Labels Add a Different Kind of Evidence

Expression measurements describe the molecular state of a cell. A barcode or fate-mapping label instead records information about ancestry: cells with the same reliable label may descend from a shared marked ancestor. In a branching lineage model, these sources of information are useful together, but they are not interchangeable. Expression can support assigning a cell to a branch; a shared label links cells through a common clone and can make their branch assignments statistically dependent.

The central modeling choice is how a clone can be distributed across branches. A label introduced after a branchpoint may be expected to appear in descendants of only one branch. A label introduced earlier can be inherited by descendants that later enter different branches. Thus, the observation that two cells share a label is evidence of shared ancestry under the assay assumptions, not proof that they must occupy the same terminal branch.

Here, \(Z_i\) takes one of \(K\) terminal-branch values, \(1,\ldots,K\). The expression model supplies a likelihood \(f(Y_i\mid Z_i,T,\theta)\), where \(\theta\) denotes expression-model parameters. The clonal component supplies a distribution for the assignments of cells sharing a label. This tutorial focuses on combining those components, not on selecting among alternative tree structures.

A Clone-Level Model for Branch Assignments

One flexible model gives each clone \(c\) a branch-propensity vector \(\boldsymbol{\pi}_c=(\pi_{c1},\ldots,\pi_{cK})\). Conditional on that vector, cells in the clone receive branch assignments independently. The vectors vary around a population-level vector \(\boldsymbol{\pi}\), allowing clones to differ in their branch representation.

The shared vector \(\boldsymbol{\pi}_c\) induces dependence between assignments within a clone. The concentration \(\kappa\) controls how much clone-specific propensities vary: large \(\kappa\) makes them concentrate near \(\boldsymbol{\pi}\), while smaller \(\kappa\) allows stronger clone-to-clone variation. This model is a statistical representation of shared clone-level tendencies; it is not, by itself, a complete biological model of when a barcode was introduced or how a cell migrated through the tree.

For clone \(c\), let \(n_{ck}\) be the number of its cells assigned to branch \(k\), and \(n_c=\sum_k n_{ck}\). Integrating out the shared vector gives the probability of a particular ordered set of branch assignments:

This is the Dirichlet-multinomial assignment probability. It depends on the group of cells sharing the label, not just on the total number of cells assigned to each branch across the entire data set. The full likelihood for observed expression and labels, conditional on the clone groupings, sums over possible latent assignments:

The sum is over assignment configurations for each clone. It preserves uncertainty: a cell with ambiguous expression need not be forced onto one branch before the clonal information is used. The clonal factor and expression factor contribute distinct parts of the joint model.

Why the Integrated Assignment Probability Has This Form

The derivation follows by multiplying the conditional categorical probabilities by the Dirichlet density and integrating over the clone-specific propensities. For a fixed assignment configuration, the conditional probability is \(\prod_k \pi_{ck}^{n_{ck}}\). The Dirichlet density has normalizing constant \(\Gamma(\kappa)/\prod_k\Gamma(\kappa\pi_k)\) and kernel \(\prod_k\pi_{ck}^{\kappa\pi_k-1}\). Their product has powers \(\kappa\pi_k+n_{ck}-1\). Integrating over the simplex uses the Dirichlet integral, whose normalizing constant for those powers is \(\prod_k\Gamma(\kappa\pi_k+n_{ck})/\Gamma(\kappa+n_c)\). Multiplying by the original density's normalizing constant yields the displayed formula.

This derivation assumes that labels correctly identify clone membership, that clone assignments follow the specified shared-propensity model, and that expression observations are conditionally independent given their branch assignments and parameters. It also treats the observed clone grouping as given. If labels collide, are misread, or are missing selectively, that grouping may be wrong or incomplete; the likelihood above does not automatically correct those errors.

Combining Expression and Clonal Evidence

For a fixed tree, an expression likelihood may favor one branch for a cell while leaving substantial uncertainty for another. The clonal term then connects those uncertain assignments through their shared label. The model does not simply “vote” once for expression and once for lineage: it combines probability contributions under explicit assumptions. The resulting branch probabilities depend on both the expression likelihoods and the clone-level assignment model.

Notice that clonal evidence is not a substitute for expression evidence. If expression strongly distinguishes branches, it can dominate the assignment for a cell. Conversely, when expression is weakly informative, a reliable clone grouping can alter the joint assignment probabilities. In either case, interpretation should identify which component is providing support.

Assumptions, Diagnostics, and Alternatives

A clonal constraint is credible only to the extent that the label measures shared ancestry. Before interpreting results, verify that the marking mechanism is heritable over the relevant divisions, that barcode collisions are sufficiently rare or modeled, and that sequencing and cell assignment errors are understood. The timing of marking matters: a label applied before a branchpoint can be shared across descendant branches, whereas a later label may imply a more restricted distribution. The model's \(\kappa\) and \(\boldsymbol{\pi}\) should reflect the intended biological assumptions, not just produce convenient assignments.

Useful diagnostics include examining clone sizes and branch-assignment counts, checking whether a few large clones dominate the result, and inspecting cells whose expression-based and clone-informed assignments differ. Large unexplained discordance may indicate label error, an inadequate expression model, a mismatch between labeling time and assumed clone behavior, or an overly restrictive concentration model. Very small clones provide little information about clone-level branch proportions. If clone labels are uncertain, treating them as error-free can give misleadingly confident assignments.

There are three common modeling choices. An expression-only model uses \(f(Y_i\mid Z_i,T,\theta)\) and ignores shared labels; it avoids assumptions about clone behavior but discards ancestry information. A hard constraint forces every cell in a clone onto one branch; it can be appropriate only when the experimental timing and biology justify that restriction, and is unsuitable for labels expected to precede divergence. A shared-propensity model such as the one derived here allows within-clone dependence and cross-branch descendants, but adds assumptions about the distribution of clone propensities. None of these approaches makes clonal association causal evidence that a particular expression state caused a fate.

A frequent mistake is to argue, “These cells share a barcode, so they must be on the same branch.” The correction is to ask when the barcode was introduced and what branch patterns are possible under that marking scheme. Shared labels support common ancestry under assay assumptions; branch identity remains an inference combining expression, lineage structure, and the model for clone distribution. A second mistake is to count every cell with the same label as independent lineage evidence. Their assignments are linked by shared clone membership, so the likelihood must represent that dependence.