Steady-State RNA Velocity

Why a Steady-State Relationship Is Useful

Single-cell RNA sequencing records RNA abundances at sampled cells, not the time derivative of RNA abundance. RNA velocity uses a kinetic model to interpret the balance between unspliced and spliced RNA as evidence about whether spliced RNA is increasing or decreasing. The steady-state relationship gives a simple reference for that interpretation: for fixed splicing and degradation rates, it identifies combinations of unspliced and spliced abundance at which spliced RNA has zero instantaneous velocity.

This relationship is useful only with a clear account of what “steady state” means. The condition for spliced-RNA velocity to be zero is not, by itself, the condition that the entire two-compartment system is at equilibrium. That distinction matters when using a gene’s position relative to the relationship to infer a direction of change.

Model and Steady-State Relationship

Let \(u(t)\) denote unspliced RNA abundance and \(s(t)\) denote spliced RNA abundance for one gene at time \(t\). In the two-compartment kinetic model, transcription adds unspliced RNA, splicing transfers RNA from the unspliced to the spliced compartment, and degradation removes spliced RNA. With transcription rate \(\alpha\), splicing rate \(\beta\), and spliced-RNA degradation rate \(\gamma\), the model is

The rates \(\beta\) and \(\gamma\) are positive and have units of inverse time. The abundances \(u\) and \(s\) have the same abundance units, so each term in either derivative has units of abundance per time. For the sign interpretation below, assume the model describes the gene in the cell of interest, and that the same positive \(\beta\) and \(\gamma\) apply to the compared cells. The equations describe instantaneous rates; they do not require that transcription be constant over all time.

To derive the relation, set the spliced-RNA derivative in the model equal to zero:

This is a linear relationship through the origin, with slope \(\gamma/\beta\) when \(u\) is expressed as a function of \(s\). More importantly, subtracting its right-hand side from the observed \(u\) gives a direct expression for the velocity:

Because \(\beta>0\), the sign of \(v_s\) is the sign of the difference \(u-(\gamma/\beta)s\). If \(u\) exceeds the value on the zero-velocity relation for the given \(s\), then \(v_s>0\). If it is below that value, then \(v_s<0\). On the relation, \(v_s=0\). This conclusion follows from the assumed model; it is not a causal claim about what would happen if RNA abundances were experimentally changed.

Zero Spliced Velocity Is Not Necessarily Full Equilibrium

Full equilibrium requires both compartments to have zero derivative. If \(\alpha\) is constant and \(\beta,\gamma>0\), the equilibrium equations give

The equilibrium point \((u^*,s^*)\) lies on the zero-spliced-velocity relation, but the relation contains other points too. At one of those other points, spliced RNA can momentarily have zero velocity while unspliced RNA is changing. Thus, “on the steady-state line” is shorthand for being on the spliced-RNA zero-velocity relation; it does not establish that the cell is at full kinetic equilibrium.

Interpreting Deviations With Numerical Examples

The next examples use hypothetical abundances and rates. They assume that \(u\) and \(s\) are measured on a compatible abundance scale, and that the rates are positive and applicable to the cell being evaluated. A positive velocity means spliced abundance is increasing instantaneously under the model; a negative velocity means it is decreasing. It does not say how long that trend will continue.

Assumptions, Diagnostics, and Limitations

The sign rule is exact for the stated kinetic equation, but its application to single-cell measurements depends on whether that equation and its inputs are credible. In particular, the mathematical sign is determined by the true abundances and applicable rates. Sequencing measurements are noisy observations of RNA abundance, so a small apparent deviation from a reference relation does not guarantee that the true velocity has the same sign.

• Check the abundance scale. The \(u\) and \(s\) values must be expressed on compatible scales. If one compartment is transformed differently from the other, the numerical relation \(u=(\gamma/\beta)s\) cannot be applied without accounting for that transformation.
• Check the rate assumptions. The sign result uses positive \(\beta\) and \(\gamma\) and assumes the rates are appropriate for the cell being interpreted. If rates vary across cells or the model omits relevant processes, one shared reference relation may not describe all cells.
• Consider measurement noise and sparse counts. When counts are low, sampling variability can place a measured point on either side of the relation. A very small difference should not be treated as strong directional evidence without considering uncertainty.
• Do not infer duration from an instantaneous derivative. A positive \(v_s\) describes the local rate at the evaluated state. Changes in transcription, splicing, degradation, or abundance can alter that rate later.
• Do not equate a zero velocity with no system change. The worked example showed that \(ds/dt=0\) can coexist with \(du/dt\ne 0\).

The steady-state sign rule is more specific than simply ordering cells by an inferred trajectory: it uses the model’s balance of splicing input and degradation to interpret a gene’s spliced-RNA derivative. Its simplicity is also its limitation. A fuller dynamical approach can represent changes over time in the kinetic process, but a sign inferred from this relation alone does not establish the history that produced the observed abundances. The appropriate interpretation is therefore conditional: if the two-compartment model and its rates are adequate for the cell, the deviation gives the sign of instantaneous spliced-RNA velocity.

Common Mistake

A second error is to read a positive velocity as evidence that spliced RNA abundance is large. Velocity is a rate of change, not an abundance: a cell can have low \(s\) with positive \(v_s\), or high \(s\) with negative \(v_s\). Always report the sign and, when calculated, the abundance-per-time units of the derivative.