Cells enter a flow-focusing junction as a continuous-time renewal process: the true gap between one cell's arrival and the next is drawn from an exponential distribution, not a fixed clock tick.
gap = −ln(1 − U) / rate, U ~ Uniform(0,1)
P(k cells in one droplet) = e^(−λ)·λ^k / k!
λ = rate · pinch_period = c · V_droplet · 3.0
This simulator draws every arrival from that exponential-gap process and plots the actual event times on the scrolling timeline above the droplet gallery — the ticks are genuinely irregular, exactly as a real Poisson arrival stream is. A separate, unrendered "naive fixed-rate" model is run in parallel purely for comparison: it accumulates rate·dt every frame and fires whenever the accumulator crosses an integer — a common shortcut that looks Poisson-like on average but is actually near-deterministic, because it fires at almost evenly-spaced intervals instead of truly random ones.
- Fano factor (variance ÷ mean of cells-per-droplet) is the diagnostic: a genuine Poisson process always has Fano factor ≈ 1, no matter the mean. Numerically verifying both models over 200,000 droplets gives Fano ≈ 1.00 for the true exponential-interarrival model here, but only ≈ 0.28 for the naive fixed-rate accumulator — confirming the fixed-rate shortcut under-disperses and is not actually Poisson-distributed, even though its long-run mean matches λ exactly.
- Cell concentration and droplet volume set λ, the mean cells per droplet, exactly as in the physical chip.
- Bead packing density sets an independent λ for barcoded gel beads, sampled with the (unmodified, already-correct) Knuth Poisson algorithm — because beads are packed close to touching, P(0 beads) is pushed into the tail, giving most droplets a bead from the same law that gives most droplets no cell.
- A droplet is usable only if it received exactly one cell and at least one bead. The histogram overlays the empirical bar counts (true-arrival model) against the theoretical Poisson curve for the current λ.
Real-world relevance: this is the same physics inside every droplet-based scRNA-seq platform (10x Genomics Chromium, inDrop, Drop-seq) — the loading concentration is a direct, unavoidable trade-off between cost (empty droplets) and doublet contamination, and getting the arrival statistics right (a true renewal process, not a periodic approximation) is exactly what determines those real trade-off numbers.