Anchor Pairs as Evidence for Shared Cell States

What an Anchor Pair Says

When two single-cell RNA-sequencing datasets contain some of the same biological cell states, a useful question is whether cells in one dataset can be paired with cells in the other as plausible representatives of those shared states. Such a pair can provide evidence for shared structure and can help carry information—such as a cell-state annotation—from one dataset to the other.

The evidence is indirect. Cells measured in different datasets are not repeated measurements of the same physical cell. Instead, an anchor pair proposes that two observed cells correspond biologically in some relevant respect, despite technical differences between datasets. Interpreting the pair therefore requires an integration target: the biological features or states intended to be shared, rather than every difference present in the data.

Definition: An anchor pair is a pair of cells from different datasets, selected as a plausible cross-dataset correspondence in a specified representation. The pair is evidence that the cells may represent the same shared biological state. It is not proof that the cells are identical, that all their expression values should be equal, or that their datasets differ only by a technical effect.

Write \(z_i^{(A)}\) for the coordinate vector of cell \(i\) from dataset \(A\), and \(z_j^{(B)}\) for the coordinate vector of cell \(j\) from dataset \(B\), in a representation chosen for comparing cells. A candidate anchor is the cross-dataset pair \((i,j)\). The representation might be a feature space used in an earlier MNN analysis; here the focus is on what a proposed pair means, not on how a particular representation is constructed.

A pair is informative only relative to assumptions. The datasets must contain a relevant shared state; the chosen representation must retain enough state-related information to distinguish it from other states; and technical differences must not overwhelm the biological similarity being used for matching. These assumptions are substantive, not guaranteed by a small distance. If a state occurs in only one dataset, there is no genuine cross-dataset counterpart to anchor it to.

Why a Close Pair Can Be Evidence

Suppose cells in a shared state cluster around a state-specific location in the comparison representation. If technical variation and within-state biological variation are modest, two cells from that state may be close even when they come from different datasets. If distinct states are well separated, a close cross-dataset pair is more compatible with a shared state than with two different states.

A simple geometric argument makes the idea precise. Let \(c_s\) denote the location of state \(s\) in the representation. Suppose every cell from state \(s\), in either dataset, lies within distance \(r\) of \(c_s\). By the triangle inequality, any two cells from that same state are at most \(2r\) apart. Now suppose two distinct state centers are at least \(\Delta\) apart. For cells from these different states, the reverse triangle inequality gives a distance of at least \(\Delta-2r\). Thus, if \(\Delta>4r\), the largest possible same-state distance, \(2r\), is smaller than the smallest possible different-state distance, \(\Delta-2r\). In that idealized setting, proximity is discriminating evidence.

\begin{aligned} \text{same state:}\quad \|z_i^{(A)}-z_j^{(B)}\| &\leq 2r,\\ \text{different states:}\quad \|z_i^{(A)}-z_j^{(B)}\| &\geq \Delta-2r,\\ \Delta>4r &\implies 2r<\Delta-2r. \end{aligned}

This argument is a justification for treating proximity as evidence, not a guarantee that real data meet the bounds. Cell-state distributions can overlap, vary across donors, or be distorted by measurement and preprocessing. A pair can also look close because the representation omits a gene program that separates two biologically different states. Anchor interpretation must therefore include the data-generating context and the consequences of a wrong match.

Worked Example: A Reciprocal Nearest-Neighbor Pair

Consider a toy two-dimensional representation for two datasets. The coordinates are hypothetical and dimensionless. Dataset \(A\) has cells \(A_1=(0,0)\), \(A_2=(-0.4,0)\), and \(A_3=(3,0)\). Dataset \(B\) has cells \(B_1=(0.1,0.2)\) and \(B_2=(3.2,0.1)\). For this example, use Euclidean distance and define a reciprocal nearest-neighbor pair as a cross-dataset pair in which each cell is the other cell's nearest cell in the opposite dataset.

For \(A_1\) and \(B_1\), the distance is \(\sqrt{(0-0.1)^2+(0-0.2)^2}=\sqrt{0.05}=0.2236\). The other distances from \(A_1\) to dataset \(B\) are \(d(A_1,B_2)=\sqrt{3.2^2+0.1^2}=3.202\). Thus \(B_1\) is \(A_1\)'s nearest cell in \(B\).

From \(B_1\), the distances to \(A_1\), \(A_2\), and \(A_3\) are, respectively, \(0.2236\), \(\sqrt{0.5^2+0.2^2}=0.5385\), and \(\sqrt{2.9^2+0.2^2}=2.907\). Therefore \(A_1\) is also \(B_1\)'s nearest cell in \(A\), so \((A_1,B_1)\) is a reciprocal nearest-neighbor pair. By contrast, \(A_3\) and \(B_2\) are also close: \(d(A_3,B_2)=\sqrt{0.2^2+0.1^2}=0.2236\), and are reciprocal nearest neighbors in their local group.

If independent biological knowledge indicates that \(A_1\) and \(B_1\) represent a shared cell state, the first pair is a plausible anchor for that state; the second may support a different shared state. The geometry alone does not supply those biological labels. Nor does the first pair establish that the two cells have identical expression profiles: it supports a correspondence in the selected representation.

Using Anchors to Transfer Shared Information

An anchor pair can connect information that is available in one dataset to a corresponding cell in another. For example, if cells in dataset \(A\) have carefully reviewed cell-state labels and dataset \(B\) does not, anchor correspondences can provide evidence for transferring those labels. The transferred label is a prediction based on a match, not a newly observed annotation.

For a small set of anchors, a transparent approach is to inspect the labels associated with the matches and report how often they agree. This is a descriptive summary of the proposed correspondences. It should not be interpreted as a calibrated probability that a transferred label is correct unless a separate validation procedure supports that interpretation.

Worked Example: Transferring a Label With Several Anchors

Suppose a reference dataset \(A\) has reviewed labels, and three cells in dataset \(B\) each have one proposed anchor in \(A\). The matched reference labels are T cell, T cell, and B cell. A simple plurality transfer assigns the label appearing most often among the three matched cells.

The T-cell label appears twice, while the B-cell label appears once. The transferred label for this small target group is therefore T cell by plurality. The agreement fraction is \(2/3=0.6667\), or \(66.67\%\). This fraction describes agreement among these three proposed matches; it does not mean there is a \(66.67\%\) probability that the transferred label is correct.

This conclusion relies on the reference labels being appropriate for the target's cell-state definitions, the anchor pairs linking genuinely comparable states, and the three matches not being misleading duplicates of one local feature. If the target cells are actually a distinct state absent from the reference, plurality can still produce a label, but that label may be biologically inappropriate. A responsible report therefore calls this a tentative transfer and preserves the possibility that no suitable reference state exists.

Anchors can also help interpret whether a proposed integration preserves shared biological structure: consistent correspondences across datasets are evidence that shared states have recognizable counterparts. But anchor pairs alone do not establish that all cells are correctly aligned, that state-specific biology has been preserved, or that a correction should be applied. Those are broader questions about integration and its validation.

Assumptions, Diagnostics, and Failure Modes

Before using anchors to support a biological claim, make the relevant assumptions explicit. At minimum, ask whether the intended state is plausibly present in both datasets, whether the comparison representation retains the features that distinguish it, and whether technical effects or sampling differences could create misleading proximity. These are the shared-state assumptions for batch alignment in operational form.

A crucial failure mode is matching cells that are close only because a biological distinction was lost. Consider a representation based on genes that capture a general activation program but omit a marker that distinguishes two related cell types. Cells from the different types may then appear nearly identical in the selected coordinates. A small distance would describe the representation, not guarantee that the cells share the biologically relevant state.

Worked Example: A Close Pair With a Missing Biological Feature

Suppose two hypothetical cells have the following values for a retained activation coordinate and an omitted marker gene. Cell \(A_1\) has retained coordinate \(0.0\) and marker value \(10\); cell \(B_1\) has retained coordinate \(0.1\) and marker value \(1\). A different cell \(B_2\) has retained coordinate \(0.8\) and marker value \(10\). Assume, for illustration, that the two marker levels distinguish two relevant cell states.

If matching uses only the retained coordinate, \(A_1\) is closer to \(B_1\) than to \(B_2\): the distances are \(|0.0-0.1|=0.1\) and \(|0.0-0.8|=0.8\). Yet the omitted marker differs by \(|10-1|=9\) for \(A_1,B_1\), compared with \(|10-10|=0\) for \(A_1,B_2\). If that marker is biologically essential to the target, the nearest match is a poor state correspondence. The calculation does not show that the marker must always be included; it shows why a close anchor is conditional on what the representation preserves.

A practical diagnostic is to examine anchor pairs in context: compare known cell-state markers or other relevant biological measurements, look for pairs that contradict trusted annotations, and check whether proposed matches are plausible given the experimental design. These checks can reveal problems, but they cannot prove that every pair is valid. In particular, an unmeasured or omitted distinction cannot be diagnosed from the matching coordinates alone.

The logic differs from direct paired measurements. If the same biological unit is measured under two conditions, the pairing is part of the study design and supports within-unit comparisons. An anchor is an inferred correspondence between different cells in different datasets; it does not create a paired experiment or justify treating the cells as repeated measurements. It also differs from a nearest-neighbor match considered in isolation: reciprocal nearest neighbors provide a familiar way to propose pairs, while an anchor is interpreted as a cross-dataset correspondence relevant to shared biology.

Do not confuse a collection of anchors with a complete integration assessment. A set of plausible pairs may provide evidence for shared states while missing a rare state, overrepresenting abundant states, or failing for cells that have no counterpart. Conversely, forcing every cell to match something can create false correspondence for dataset-specific states. The conclusion should be limited to the shared structure supported by the observed pairs.

Common Mistakes

Calling an anchor an identical cell. The pair links different observed cells as plausible representatives of a shared state; it is not a physical-cell match.
Treating proximity as proof. Distance supports correspondence only under assumptions about the representation, shared states, and technical variation.
Reading a match as a probability. A close pair or a label-agreement fraction is not automatically a calibrated probability of correctness.
Assuming every cell must have an anchor. A state may be absent from one dataset, and an unpaired cell may be biologically meaningful rather than a matching failure.
Transferring labels without checking their meaning. A reference label is useful only if it describes the same biological target in the receiving dataset.

Key takeaway: An anchor pair is a proposed cross-dataset correspondence that can support claims about shared cell states and enable cautious transfer of information. Its evidential strength depends on the representation and biological assumptions; it is neither proof of identity nor a complete validation of integration.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions below.

Two cells from different datasets form a reciprocal nearest-neighbor pair. What does this establish, and what does it not establish?
In the geometric justification, why does \(\Delta>4r\) make same-state pairs closer than different-state pairs under the stated bounds?
Four anchor pairs transfer reference labels A, A, B, and A. What is the plurality label and agreement fraction? What does that fraction mean?
A target cell has no plausible anchor in a reference dataset. Give one biological explanation and one reason not to force a match.

Solutions. (1) It establishes reciprocal nearest-neighbor status in the chosen representation, which supports a candidate correspondence. It does not prove identity or shared biological state. (2) Same-state distances are at most \(2r\); different-state distances are at least \(\Delta-2r\). If \(\Delta>4r\), then \(\Delta-2r>2r\), so the bounds separate the two cases. (3) The plurality label is A, appearing three times. The agreement fraction is \(3/4=0.75\), or \(75\%\); it describes the share of these proposed matches with label A, not the probability that the transfer is correct. (4) The target may represent a state absent from the reference. Forcing a match could assign it to a biologically different state and create false correspondence.