RNAi Reaction–Diffusion Field (2D)

RNAi Pathway Controls

Active RISC (siRNA-AGO2) complexes 14

Transcription rate (new mRNA/s) 2.0

Seed-pairing mismatch tolerance 1

Cytoplasmic thermal motion 1.0×

Live Readouts

Target mRNA present

Gene knockdown

Cleavage events (total)

Cleavage rate

0.0 /s

Mean-field M steady-state

Stochastic vs mean-field

How this 2D model works

This is a genuinely 2D reaction-diffusion model, not a flattened view of the 3D scene. Left disk: RISC (siRNA·AGO2) and mRNA random-walk on a 2D cytoplasm cross-section and bind when within a 2D encounter radius, with the same seed-mismatch logistic rule as the mechanism itself:

P_bind(m) = 1 / (1 + exp[(m − tolerance) · 1.5])

But instead of a 3D collision cross-section (an area, πr²), two disks in a 2D gas collide at a rate set by a linear cross-section (a chord length, 2r) — standard 2D kinetic theory. That collision-rate constant, averaged over the mismatch distribution, feeds a mean-field pair of ODEs plotted live on the right as a time-series overlay against the actual particle count:

dM/dt  = k_tx − k_nat·M − a·(N_R − R_b)·M
dR_b/dt = a·(N_R − R_b)·M − R_b / DWELL
a = (2·r_bind·v_eff·⟨P_bind⟩) / Area

Fix vs. the 3D version: there, natural mRNA turnover (k_nat) only appeared in the knockdown-percentage formula, never in the actual particle step — so the population never really decayed on its own. Here k_nat is a real per-frame stochastic decay applied to every free transcript, matching the birth–death model it's plotted against.

Watch the "Stochastic vs mean-field" readout: it should hover within a few percent once the population settles, the same way a well-mixed chemical kinetics approximation tracks a spatial stochastic simulation at moderate particle counts.

Target mRNA RISC (siRNA·AGO2) Cleaved fragment Mean-field M(t)

RNAi Reaction–Diffusion Field — 2D Mean-Field Cross-Check