t 0 s (real time)Product 0Substrate left 0Occupied 0 %

substrate product halves inhibitor. Orange cleft = active site, gold dimple = side (allosteric) site. Drag to orbit, wheel or pinch to zoom.

Enzyme Active-Site Diorama

10 enzyme molecules in a 0.1 µM solution. Every binding, release and conversion you see is one event of an exact stochastic simulation (Gillespie).

Run

Slow-motion factor

Presets

Inhibitor

Inhibitor concentration [I]

Dissociation constant K_i

Substrate and enzyme

Initial substrate [S]₀

Enzyme molecules

Binding rate k₁

Unbinding rate k_-1

Turnover number k_cat

Live readings

Product vs time

Rate vs substrate (Michaelis-Menten)

Lineweaver-Burk plot

Press the button: the rate equations are solved at seven substrate levels with and without the inhibitor, and Km and Vmax are fitted from the straight lines.

Independent checks (live)

The science

Mechanism. E + S ⇄ ES → E + P with rate constants k₁, k_-1, k_cat. In the steady-state limit the rate is v = V_max [S] / (K_m + [S]) with V_max = k_cat [E]₀ and K_m = (k_-1 + k_cat)/k₁. K_m is the substrate concentration at half V_max; it equals the dissociation constant only when k_cat << k_-1.

Inhibition with α = 1 + [I]/K_i. Competitive: the inhibitor takes the active site, K_m,app = α K_m, V_max unchanged (lines meet on the 1/v axis; excess substrate wins). Non-competitive: the inhibitor binds a different site on E and ES, V_max,app = V_max/α (lines meet on the 1/[S] axis in the textbook limit). In this exact kinetic model K_m,app = (k_-1 + k_cat/α)/k₁, which equals K_m only when k_cat << k_-1. Uncompetitive: the inhibitor binds only ES, both K_m,app and V_max,app fall by α (parallel lines).

Lineweaver-Burk. 1/v = (K_m/V_max)(1/[S]) + 1/V_max. It is a clear picture of the mechanisms but a poor way to fit noisy data (it over-weights the smallest rates); modern practice is a non-linear fit.

Stochastic simulation. The Gillespie algorithm draws the waiting time to the next event from an exponential distribution with mean 1/Σa_j and picks the event j with probability a_j/Σa. Second-order propensities are k [A][B] / Ω with Ω = 100 molecules per µM. With only 10 enzymes the counts fluctuate visibly around the rate-equation curve; the ensemble mean agrees with it.

Simplifications, stated honestly. Well-mixed reaction (the diffusing molecules are a picture; the events are drawn from the exact kinetics, not from collisions), irreversible catalytic step, one substrate, no product inhibition, inhibitor exchange fast compared with catalysis for the analytic lines, the rate-equation curve of the same scheme is shown for comparison. The rate constants are round illustrative numbers in the range of real enzymes, not data for a named enzyme. Time is slowed by the factor you choose; the clock in the header is real (un-slowed) reaction time.

Educational model. Molecules were modelled in Blender and baked to textures.