Splicing kinetics ODE

中文導讀

這是所有 RNA velocity 方法的 kinetic backbone:單基因的 unspliced/spliced transcription–splicing–degradation ODE。重點在 identifiability——snapshot data 只定得出 β/γ 這類 ratio,定不了 absolute timescale(把 t 跟所有 rate 一起 rescale,(u,s) 分布不變)。 所以 direction identifiable、但 duration 不 identifiable,這正是 physical-time-grounding 的源頭。各方法差在 α(t) 怎麼處理:scVelo 假設 piecewise-constant,RegVelo 用 GRN 預測, metabolic-labeling 方法則用 label 把 absolute rate 釘住。

The model

For a single gene, with unspliced abundance u and spliced abundance s:

du/dt = α(t) − β·u        (transcription − splicing)
ds/dt = β·u  − γ·s        (splicing − degradation)

• α(t) — transcription rate (the term methods disagree about).
• β — splicing rate (unspliced → spliced).
• γ — degradation rate of spliced mRNA.

RNA velocity is the spliced time-derivative v = ds/dt = β·u − γ·s. Its sign (induction vs repression) is what positions a cell on the manifold; scVelo’s steady-state estimator fits γ as the slope of the u–s phase portrait with β≡1.

Steady states and the scale degeneracy

Induction steady state: u_ss = α/β, s_ss = α/γ. The steady-state u–s relation depends only on the ratio β/γ. Rescaling time τ = β·t leaves the dynamics depending only on α/β and γ/β — the common scale β (the factor converting latent time to real time) is free.

Consequences (the core of physical-time-grounding):

• Direction is identifiable from snapshot data.
• Absolute timescale / rate magnitude is not — known only up to a global (and, if rates drift, locally varying) factor.

Breaking this requires information snapshot data lacks: metabolic-labeling, true time series, or a calibrated reference clock.

How methods treat the terms

Method	α(t)	β, γ
scVelo (steady-state)	implicit; slope fit	ratio only (β≡1)
scVelo (dynamical)	piecewise on/off constant	gene-specific constants
veloVI	per-state, neural	gene-specific constants
RegVelo	α=h(|Ws+b|), GRN-driven	gene-specific constants
TopoVelo	ρ(z_i, neighbors), spatially coupled	gene-specific constants
dynamo (labeling)	estimable	absolute when labels present
GraphVelo	cell-context-specific α=u+du/dt	cell-context-specific γ=(u−ds/dt)/s

NOTE: nearly every method holds β, γ constant along the trajectory. Where they genuinely drift, the implied time mapping is biased — a recurring critique thread. Exceptions that relax constant rates: GraphVelo (manifold tangent-space) and cellDancer (per-cell DNN) recover cell-context-specific α, γ for multiple-rate-kinetics (MURK) genes — but at the velocity-correction level, still without an absolute-time anchor, so the scale degeneracy remains.

NOTE (missing term, full text): cell-growth-omission-2025 argues the velocity equation is mis-specified — it omits cell growth. In a growing population the homeostatic velocity is v = λx (positive, ∝ abundance), not zero, and the fitted degradation absorbs the growth rate: γ* ≈ γ + λ. The analysis targets labeling-based designs, so even metabolic-labeling /dynamo metric rates are growth-biased; residual artifacts make downstream trajectories point orthogonal/reversed, i.e. direction-wrong, not just scale-off. Fix: a growth-aware model + a measured growth rate λ as a second anchor (physical-time-grounding).

NOTE (basis discarded): TFvelo (tfvelo-2024) abandons this ODE altogether — it defines velocity as W·X − γ·y on total mRNA from a GRN of transcription factors, no unspliced/spliced. “Velocity” is becoming an umbrella over several different observables (see RNA velocity).

Related

RNA velocity · physical-time-grounding · latent-time · grn-informed-velocity · spatial-velocity · metabolic-labeling · manifold-consistent-velocity · velocity-skepticism · cell-growth-omission-2025 · velocyto · scVelo · veloVI · RegVelo · TopoVelo · dynamo · GraphVelo · cellDancer · UniTVelo · unitvelo-2022 · TFvelo · tfvelo-2024