Combining Gene Velocities Across Cells

Why Combine Gene-Level Velocities?

For a cell, RNA velocity methods estimate a direction of change separately for each gene from its spliced and unspliced RNA measurements and a kinetic model. The estimates can differ in scale, precision, and sign. A cell-level summary is useful for asking whether the selected genes provide a coherent directional signal, but combining them is not simply a matter of adding their raw rates. A rate for a highly expressed gene can be numerically larger than a rate for a lowly expressed gene even when the relative changes are comparable.

The aim here is a summary of directional evidence across genes for one cell. It is not a substitute for retaining the gene-specific estimates, and a single summary cannot represent every gene program when genes move in opposing directions. In particular, a positive aggregate score means net positive evidence under the chosen scaling and weighting; it does not mean that every gene is increasing.

Put Gene Estimates on Comparable Scales

Before combining rates, choose a gene-specific scale \(a_g>0\). It might be a measure of typical variation in the gene’s spliced abundance across the analyzed cells, or another prespecified scale appropriate to the analysis. The purpose is to express each rate relative to a comparable amount of change for that gene. The scale must be defined consistently and must not be zero. Define the scaled estimate and its standard error by

If \(a_g\) is measured in normalized abundance units, \(z_{ig}\) is a relative rate, with units of inverse time. Scaling changes the estimand: the combined score summarizes relative rates, not raw changes in abundance. Consequently, results can depend on the scale chosen. Report it and use the same scale when comparing cells.

Do not confuse this step with a claim that genes have identical biology. Scaling makes the numerical values more comparable; it does not make each gene’s true rate equal. A pooled estimate is most directly interpretable when the scaled gene estimates are measurements of a shared underlying directional signal. When the genes genuinely have different directions or dynamics, the spread among them is part of the result, not merely nuisance noise.

Inverse-Variance Weighting and Its Justification

A weighted mean gives more influence to estimates with smaller uncertainty. For one cell, write \(w_{ig}=1/s_{ig}^2\), and define the inverse-variance weighted aggregate

Here \(G\) is the number of genes included. This formula is not automatically justified just because standard errors are available. Its basic model assumes that, after scaling, each gene estimates the same cell-level quantity \(\theta_i\), with independent, unbiased, approximately normal errors and known standard errors:

These assumptions are often a working approximation, not a biological fact. Gene-level errors may be correlated because estimates share cell-level measurements, normalization, or fitted model components. Also, different genes can have genuinely different dynamics, so a common \(\theta_i\) may not describe them well.

Under the stated independent-error model, the estimated variance of the weighted mean is

This standard error describes uncertainty from the gene-level estimates under the common-signal model. It does not account for model misspecification or biological disagreement among genes. A small standard error can therefore accompany a misleading aggregate if the assumptions fail.

Why the Weighted Mean Has This Form

The inverse-variance estimate is the value of \(\theta_i\) that minimizes the weighted sum of squared deviations, where estimates with higher uncertainty receive less weight. Under the common-signal model, consider

Differentiating with respect to \(\theta_i\) and setting the derivative to zero gives

For independent errors, the variance follows directly by treating the weights as fixed:

If errors are correlated, the independent-error variance formula is not valid. In matrix notation, let \(\mathbf{z}_i\) be the vector of scaled estimates, let \(\mathbf{1}\) be a vector of ones, and let \(\Sigma_i\) be their error covariance matrix. If \(\Sigma_i\) is known and the common-signal model still holds, the generalized least-squares estimate is

In practice, estimating the covariance matrix reliably can be difficult. Treating correlated gene estimates as independent can make the reported uncertainty too small, particularly when many genes carry largely redundant information.

Discordance Is a Diagnostic, Not a Detail to Average Away

A weighted mean can conceal disagreement. One diagnostic under the common-signal, independent normal-error model is the weighted residual sum of squares

When that model is correct and the variances are known, \(Q_i\) has approximately a chi-squared distribution with \(G-1\) degrees of freedom. A large value relative to that reference distribution signals more spread than expected from the stated measurement errors alone. It does not identify the cause: genuine gene-specific dynamics, unaccounted dependence, an inaccurate standard error, or model misfit can all produce excess spread. This diagnostic is not reliable evidence of agreement when its assumptions are poor.

Choosing and Reporting a Summary

Inverse-variance weighting is useful when the goal is to estimate a common scaled signal and when the uncertainty estimates and dependence assumptions are credible. Other summaries answer different questions:

A weighted sign vote can be defined by comparing the total weight on positive estimates with the total weight on negative estimates. It is a directional consensus measure, not an estimate of a common rate. A median may be informative when one or a few genes are extreme, but it cannot fix widespread biological disagreement. If the scientific target allows genuine variation among gene-level rates, a model that represents that variation is more appropriate than forcing a single common value; a simple practical option is to add a between-gene variance component to the uncertainty used for weighting. The resulting estimate answers a different, more heterogeneous-signal question.

For every cell-level aggregate, report the genes included, the scaling rule, the weighting rule, and a measure or display of between-gene agreement. Also report how many genes contributed. An aggregate based on a handful of genes is not directly comparable to one based on many genes unless the procedures and gene sets are comparable. Do not interpret a larger number of contributing genes as automatically stronger biological evidence: correlated estimates do not contribute as much independent information as the same number of independent estimates.

Common Pitfalls and Failure Modes

• Averaging raw rates across genes. The wrong reasoning is that all velocities have the same numerical unit, so their raw values are directly comparable. The correction is that a common unit does not imply a common biological scale; define and report a gene-specific scaling rule when relative changes are the target.
• Calling the weighted mean a consensus when genes disagree. A mean can be close to zero because positive and negative estimates cancel. Inspect the gene-level signs and dispersion, and describe the aggregate as a net summary rather than evidence that genes share one direction.
• Using inverse-variance weights as biological importance weights. Smaller standard error means greater estimated precision under the model, not greater biological relevance. Keep precision weighting distinct from any deliberate biological prioritization.
• Reporting an overly precise standard error. The usual formula assumes independent gene errors and a common signal. Shared data and fitted components can induce dependence, and genuine gene heterogeneity can add variation. Check those assumptions before interpreting the standard error.
• Treating a positive aggregate as proof of a cell state transition. The calculation summarizes model-based gene-level estimates. It does not establish a causal change or validate the kinetic model that produced them.