Elasticity Coefficient Explorer

Now zoom in on a single, isolated enzyme obeying Michaelis–Menten kinetics, v = V_max·S/(K_M+S). The elasticity ε_S^v = K_M/(K_M+S) is a local property of this one enzyme — no other enzymes in a pathway are involved.

current point tangent line (slope = dv/dS, or ε_S^v on the ln–ln plot)

[S] (substrate) 2.0

K_M 2.0

v (V_max=1)

dv/dS (slope)

ε_S^v

S / K_M

0 — inelastic (saturated) 1 — elastic (first-order)

Price-elasticity analogy: when S << K_M, the enzyme is far from saturated and v tracks S almost proportionally (ε ≈ 1) — like a price-elastic product where small changes cause big responses. When S >> K_M, the enzyme is saturated and v barely responds to more S (ε ≈ 0) — like an inelastic product whose demand doesn't budge.

Try it yourself

1. Unsaturated (elastic) regime. Set S=0.5, K_M=10 (S << K_M). What is ε_S^v? Why does the text's approximation v ≈ (V_m/K_M)·S, valid when S<<K_M, predict an elasticity close to 1?

2. Saturated (inelastic) regime. Set S=20, K_M=1 (S >> K_M). What is ε_S^v now, and what does the flattened tangent line tell you about how sensitive this enzyme is to further increases in its own substrate?

3. The halfway point. Set S=K_M=5, then try a few other equal S/K_M pairs (e.g., 2 and 2, or 8 and 8). What value does ε_S^v always take when S=K_M, regardless of the actual numbers? Can you show this algebraically from ε=K_M/(K_M+S)?

4. Reading the ln–ln slope. With S=3, K_M=6, switch to the ln–ln view. Confirm that the tangent's slope on this plot matches the ε_S^v value shown in the stat panel, the same way the ln–ln slope matched the FCC in the first explorer.

5. Local vs. system properties. Notice that ε_S^v here depends only on this one enzyme's own S and K_M — you don't need to know anything about the rest of the pathway to compute it, unlike the FCC and CCC explorers. Why does the text call the elasticity coefficient a "local" property while flux and concentration control coefficients are "system" properties? What experiment could you do to measure an elasticity that you couldn't do to measure an FCC?