Each cell carries a driver gene modelled by the standard two-species splicing kinetics used in real single-cell RNA velocity analysis (La Manno et al., 2018): unspliced pre-mRNA u is transcribed at rate α(t) and converted to spliced mRNA s at rate β, which is then degraded at rate γ:
du/dt = α(t) − β·u
ds/dt = β·u − γ·s
The quantity ds/dt is the cell's instantaneous RNA velocity for that gene — it is positive while transcription is outrunning splicing (the gene is being induced) and negative once splicing/decay dominates (the gene is relaxing toward its steady state). At true steady state, u and s sit on the line s = (β/γ)·u; a cell sitting above that line is predicted to increase s (velocity arrow points "forward"), a cell below it is predicted to decrease.
Here every cell starts as an undecided progenitor moving along a shared trunk. Past a branch point (pseudotime d ≈ 0.32) each cell stochastically commits to Fate A or Fate B, switching on that fate's driver gene. Its RNA velocity ds/dt is used to advance its own pseudotime coordinate d, so cells whose driver gene is actively being induced move visibly forward along the bifurcating path — exactly the "future state" prediction RNA velocity gives biologists inspecting a real UMAP embedding of a differentiating tissue.
- α (transcription rate) — how strongly the fate-specific driver gene is induced after commitment; higher α means faster, more forceful differentiation.
- β (splicing rate) — how fast unspliced pre-mRNA is converted to mature spliced mRNA; sets the lag between transcriptional induction and the velocity signal.
- γ (degradation rate) — how fast spliced mRNA decays; sets the final steady-state expression level s* = (α/γ)·(β/(β+γ)) roughly, and how sharply cells relax once past the branch.
- Arrows — each visible arrow is the true 3D velocity vector: the tangent of the bifurcating path scaled by that cell's ds/dt, the same forward/backward prediction a real velocity field encodes.
Real-world relevance: this is the core computation behind tools like scVelo and velocyto.R, used to infer the direction of differentiation, cell-cycle progression or immune activation from a single snapshot of single-cell RNA-seq data, without needing a time-course experiment.