This is the two-state (telegraph) model of gene expression, the standard model used to explain why gene expression is noisy and "bursty" rather than smooth (Peccoud & Ycart 1995; Raj & van Oudenaarden, Cell 2006). The promoter is a discrete Markov switch between an OFF (inactive chromatin) and ON (transcriptionally active) state:
OFF --k_on--> ON --k_off--> OFF
ON: d[mRNA]/dt = +k_tx - d_m·mRNA
all: d[Protein]/dt = +k_tl·mRNA - d_p·Protein
Each transition and each synthesis/decay event is simulated stochastically: in every small timestep dt, an event fires with probability rate·dt (a discretized continuous-time Markov / Gillespie process), so identical parameters never replay the same trajectory twice.
- kon, koff — how often the promoter switches on and how long each ON period lasts (mean burst duration = 1/koff).
- ktx — mRNA molecules produced per second while the promoter is ON; each firing spawns a transcript particle near the gene.
- ktl — proteins produced per mRNA per second; more mRNA (and longer-lived mRNA) means more ribosome passes and more protein.
- Expected burst size (mRNA per ON period) ≈ ktx/koff. Slow switching (small kon, koff) with high ktx produces large, sparse bursts — high noise; fast switching approximates constant (Poissonian) transcription — low noise.
- The Fano factor (variance/mean of the protein count, sampled over a rolling time window) is a standard noise metric: a value near 1 indicates Poisson-like (memoryless) fluctuations, while values well above 1 are the signature of bursty transcription — exactly what this simulator lets you dial in and observe directly.
Real-world relevance: transcriptional bursting has been directly observed by single-molecule mRNA FISH in bacteria, yeast and mammalian cells, and is a major reason genetically identical cells can have very different protein levels — a mechanism implicated in cell-fate decisions, drug tolerance in cancer, and viral latency.