Competitive Inhibition — Stochastic Enzyme-Population Kinetics (2D)

Reaction Conditions

Substrate [S] 40 µM

Michaelis constant K_m 20 µM

Inhibitor affinity K_i 5.0 µM

Inhibitor [I] 0 µM

Enzyme Population (N=36)

Free (E) Substrate-bound (ES) Inhibitor-bound (EI) Turnover burst

Measured v/V_max

—

Analytic v/V_max

Apparent K_m

20.0 µM

IC₅₀ (Cheng–Prusoff)

— µM

Free / ES / EI now

36 / 0 / 0

Turnovers (total)

How this 2D model differs

The 3D version renders substrate and inhibitor as spheres orbiting one enzyme and picks a winner of a position-weighted race each frame. This 2D version instead simulates a population of 36 independent enzyme molecules, each one a genuine continuous-time Markov chain over three states — free (E), substrate-bound (ES) and inhibitor-bound (EI) — stepped with real first-order and second-order rate constants, not a spatial capture radius.

E  --k_on,S·[S]-->  ES  --k_cat-->  E + product   (turnover)
ES --k_off,S-->     E                              (unproductive release)
E  --k_on,I·[I]-->  EI  --k_off,I--> E              (inhibitor blocks, no reaction)

Km = (k_off,S + k_cat) / k_on,S     Ki = k_off,I / k_on,I
Steady state:  v = k_cat·[S] / ( Km·(1 + [I]/Ki) + [S] )     (Vmax = k_cat, normalised to 1)
IC50 = Ki·(1 + [S]/Km)                                        (Cheng–Prusoff, competitive case)

Every dot below is one enzyme molecule; its colour is its actual current CTMC state, not a visual stand-in. The measured v/V_max box is the ensemble's live turnover rate — literally counted product-release events divided by population size and elapsed time — which converges onto the analytic curve as the population averages out its own thermal noise, exactly like a real single-molecule kinetics assay would.

[S], Km, Ki, [I] sliders — set the true rate constants (see equations); every enzyme's transition probabilities update live.
Enzyme grid — grey = free, blue = substrate bound, red = inhibitor bound; a green burst marks a real turnover (product release) event, with the ejected product particle obeying simple 2D drag-and-fade motion.
Dose-response curve — analytic v/Vmax vs log[I] at the current [S]; the marker follows the [I] slider and the dashed line marks IC₅₀. Compare it against the measured stat box to see the stochastic ensemble tracking the deterministic law.

Free enzyme ES complex EI complex

[I] = 0.01·KiDose–response v/Vmax vs log[I][I] = 100·Ki