Scaling Spliced and Unspliced Counts

Why the Relative Scale Matters

RNA velocity uses the relationship between unspliced and spliced RNA to describe changes in spliced abundance. In the two-compartment kinetic model, unspliced RNA contributes to spliced RNA, while spliced RNA is degraded. A common form of the spliced-abundance velocity is \(v=\beta u-\gamma s\), where \(u\) and \(s\) are the unspliced and spliced abundances, and \(\beta\) and \(\gamma\) are the splicing and degradation rates. This expression makes a basic measurement issue consequential: \(u\) and \(s\) must be on compatible scales before their contributions can be compared.

Observed counts are affected by cell-specific sequencing depth and capture efficiency, as well as by biological abundance. Library-size normalization attempts to account for differences in measurement exposure between cells. But a separate scaling choice for spliced and unspliced counts can also change their ratio. Normalizing each count matrix is not automatically equivalent to putting both matrices on a shared molecular scale.

Definition: Let \(U_{ig}\) and \(S_{ig}\) be the observed unspliced and spliced counts for cell \(i\) and gene \(g\). Let \(q_i^u\) and \(q_i^s\) be the corresponding library-size or exposure factors. Normalized measurements are \(u_{ig}=U_{ig}/q_i^u\) and \(s_{ig}=S_{ig}/q_i^s\). If a common factor is used, \(q_i^u=q_i^s=q_i\). A relative scaling mismatch occurs when the effective scales for the two modalities differ in a way that is not accounted for in the kinetic model.

The factors may be defined with a reference library size rather than as raw totals; the essential point is that counts are divided by a positive exposure estimate. The factors should be estimated in a way that reflects technical sampling differences, not simply selected to produce a desired velocity pattern. Whether the two count types share an exposure factor depends on the measurement and processing scheme. If they are independently processed or normalized, their relative scale requires particular scrutiny.

What Scaling Does to the Kinetic Relationship

Suppose \(U\) and \(S\) denote abundances on a common scale and \(u=c_u U\), \(s=c_s S\) are their scaled versions, where \(c_u\) and \(c_s\) are positive constants. The kinetic equation for the unscaled spliced abundance is \(dS/dt=\beta U-\gamma S\). If the scale factors are constant over time, then

\frac{ds}{dt} =c_s\frac{dS}{dt} =\beta\frac{c_s}{c_u}u-\gamma s.

Thus, under separate scaling, the coefficient multiplying \(u\) in the scaled velocity is \(\beta c_s/c_u\), not necessarily \(\beta\). The degradation term retains coefficient \(\gamma\) under this constant rescaling. A fitting procedure that treats the scaled values as if they retained the original common scale can therefore attribute a scale mismatch to altered kinetic rates or to a different phase-portrait relationship.

A shared constant multiplier \(c\) applied to both compartments preserves their ratio and multiplies the velocity by \(c\): \(u=cU\), \(s=cS\) implies \(ds/dt=c(\beta U-\gamma S)\). The direction of that gene-level velocity is unchanged, but its magnitude is now expressed in scaled units. This does not mean that all normalization is harmless: if the multiplier differs between cells or changes along a trajectory, it can alter comparisons across cells or introduce an apparent change over time.

Key result: A fixed, common scale for spliced and unspliced abundance preserves their relative scale. A modality-specific scale changes the coefficient linking them, and a cell-varying scale can affect both cross-cell comparisons and apparent temporal derivatives. Any velocity interpretation is conditional on the scale used for its inputs.

Worked Example: Applying a Shared Library-Size Factor

Consider a hypothetical gene measured in two cells. Cell A has \(U=80\) unspliced counts and \(S=400\) spliced counts, with a library-size factor \(q_A=2\). Cell B has \(U=50\), \(S=250\), and \(q_B=1\). For this illustration, assume the factors are valid exposure corrections and are shared by the two count types.

The normalized counts are:

u_A=\frac{80}{2}=40,\qquad s_A=\frac{400}{2}=200, \qquad u_B=\frac{50}{1}=50,\qquad s_B=\frac{250}{1}=250.

Before normalization, both cells have an unspliced-to-spliced ratio of \(0.20\): \(80/400=0.20\) and \(50/250=0.20\). After normalization the ratios remain \(40/200=0.20\) and \(50/250=0.20\). The shared factor changes each cell’s abundance scale but does not change the ratio within either cell. This calculation does not establish that the cells have equal biological RNA content; it only shows what follows from applying the stated common factors.

Worked Example: Separate Scales Change the Phase-Portrait Slope

Suppose hypothetical unscaled steady-state measurements for a gene are \((U,S)=(10,20)\) and \((20,40)\). The points lie on \(S=2U\), so the slope in this spliced-versus-unspliced representation is \(2\). Now suppose the values are rescaled as \(u=2U\) and \(s=0.5S\).

The transformed points are \((u,s)=(20,10)\) and \((40,20)\). Their slope is:

\frac{\Delta s}{\Delta u} =\frac{20-10}{40-20} =\frac{10}{20} =0.5.

The scale ratio is \(c_s/c_u=0.5/2=0.25\), so the transformed slope is \(0.25(2)=0.5\), matching the direct calculation. In the kinetic model, a steady state satisfies \(0=\beta U-\gamma S\), or \(S=(\beta/\gamma)U\). After these separate rescalings, the apparent slope is \((c_s/c_u)(\beta/\gamma)\). Therefore a fitted slope cannot be interpreted as the original rate ratio unless the scaling relationship is accounted for.

Cell-Varying Scales and Apparent Velocity

The constant-scale derivation has an important condition: the scale factors must not change with time along the trajectory being interpreted. If a scaled spliced measurement is \(s(t)=c_s(t)S(t)\), differentiation gives

\frac{ds}{dt} =c_s(t)\frac{dS}{dt} +\frac{dc_s(t)}{dt}S(t).

The second term is caused by the changing scale, not by the underlying kinetic equation. The same issue arises when comparing cells: if normalization factors vary with cell state or library composition, differences in normalized values may combine biology with differences in the transformation. A snapshot analysis does not directly observe a time derivative, but kinetic inference uses relationships among cells to estimate dynamics; systematic scale variation can therefore be mistaken for a biological pattern.

Worked Example: A Modality-Specific Factor Can Create a Nonzero Estimate

Consider a deliberately simplified hypothetical comparison in which the underlying abundances are the same for two cells: \(U=20\) and \(S=100\). Suppose the model uses \(\beta=1\) per time unit and \(\gamma=0.2\) per time unit. On the common abundance scale, the velocity is:

v=\beta U-\gamma S=(1)(20)-(0.2)(100)=20-20=0.

Now suppose the spliced count for the second cell is divided by an estimated factor of 2, but its unspliced count is divided by 1. The inputs become \(u=20\), \(s=50\). If the same coefficients are then applied without adjusting for the changed relative scale, the calculated value is:

v_{\mathrm{scaled}}=(1)(20)-(0.2)(50)=20-10=10 \quad\text{scaled abundance units per time unit.}

The nonzero result is created by the mismatch in this illustration, not by a change in the underlying abundances. In a real analysis, the rate estimates might also change during fitting, so this calculation is not a prediction of the exact bias in every pipeline. It demonstrates why scale compatibility must be checked before interpreting the sign or magnitude of a velocity.

Assumptions, Diagnostics, and Alternatives

Normalization is useful when differences in captured library size act approximately like differences in measurement exposure. Interpreting normalized counts as comparable abundances also requires that the chosen factors adequately address those exposure differences and do not remove the biological signal of interest. In particular, ordinary library-size normalization is compositional: it compares a gene’s count relative to a library total or reference. If total RNA content changes biologically, normalization by total counts can partly remove that global change. No size-factor calculation alone guarantees absolute molecule counts.

Several checks help reveal scaling problems:

Inspect factors and totals separately. Compare the distributions of spliced and unspliced library sizes, size factors, and detected counts across cells. Extreme values or systematic differences between modalities warrant investigation.
Check the relative scale. Examine spliced–unspliced ratios and phase-portrait patterns before and after normalization. A large systematic change caused only by separate normalization should be explained by the measurement design or calibration.
Test sensitivity. Refit or recalculate key gene-level quantities under defensible alternative scaling choices. Large changes in velocity sign or magnitude indicate that the conclusion depends on scale assumptions.
Look for dependence on library depth or cell state. If estimated velocities track library size, the normalization may not have removed technical exposure differences, or the normalization may be entangled with biology.
Check the kinetic units. State whether inputs are raw counts, normalized counts, or calibrated abundances. Rate and velocity units must match those inputs.

A common alternative is a single cell-specific factor applied to both modalities. This maintains their within-cell ratio under the normalization transformation, although it cannot correct a genuine modality-specific capture difference. Another option is modality-specific calibration, for example using appropriate external standards or a measurement model that explicitly represents separate exposures. Such calibration can address differing capture scales, but depends on the standards or model being suitable. Count-based models with exposure offsets provide another route: rather than treating normalized counts as directly comparable abundance measurements, they incorporate library exposure in the model. These approaches answer different measurement problems; none removes the need to justify assumptions about capture, composition, and scale.

Scaling also cannot repair every source of kinetic-model error. A correct relative scale does not ensure that the two-compartment model fits the gene, that rates are identifiable, or that counts are measured without substantial noise. Conversely, an apparent poor fit should not automatically be attributed to kinetics before scale compatibility has been checked.

Common Mistake

Incorrect reasoning: “Both count matrices were library-size normalized, so the spliced-to-unspliced relationship is comparable across cells.” Separate normalization can apply different factors, changing the relative scale even if each matrix is internally normalized consistently.

Correction: Record the factors used for each modality and establish whether their relative scaling is justified. If the same fixed factor is applied to both, the within-cell ratio is preserved; if factors differ, account for that difference in the kinetic interpretation. Also check whether factors vary systematically across cells, since a common but cell-varying factor can still affect cross-cell comparisons.

Key takeaway: Library-size normalization adjusts counts for measurement exposure; it does not by itself guarantee that spliced and unspliced values share a comparable scale. Velocity estimates are meaningful only relative to explicit, defensible scaling assumptions.

Check Your Understanding

Try each problem, then compare your reasoning with the solutions below.

If \(u=cU\) and \(s=cS\) for a fixed positive constant \(c\), what happens to the velocity \(dS/dt=\beta U-\gamma S\) when expressed as \(ds/dt\)?
Unscaled steady-state data have slope \(S/U=3\). They are transformed using \(u=0.5U\) and \(s=2S\). What is the slope \(s/u\)?
Why can separate spliced and unspliced normalization factors change the sign of a calculated velocity?
What additional term appears in \(ds/dt\) if \(s(t)=c_s(t)S(t)\) and the scale factor varies with time?

Solutions. (1) Since \(s=cS\), \(ds/dt=c\,dS/dt=c(\beta U-\gamma S)=\beta u-\gamma s\); the velocity is multiplied by \(c\) and expressed in scaled abundance units per time. (2) The transformed slope is \((c_s/c_u)(S/U)=(2/0.5)(3)=12\). (3) Separate factors alter the relative contributions of \(u\) and \(s\) in \(\beta u-\gamma s\); if the terms are close, changing their relative scale can change which is larger and therefore reverse the sign. (4) The product rule gives \(ds/dt=c_s(t)dS/dt+[dc_s(t)/dt]S(t)\); the second term reflects changing scale rather than underlying kinetic change.