Michaelis-Menten Kinetics

Enzyme Catalysis

Enzymes are biological catalysts that accelerate reactions by binding substrates and lowering the activation energy. The Michaelis-Menten model describes the steady-state kinetics of a single-substrate enzyme reaction:

$E + S \rightleftharpoons ES \rightarrow E + P$

The Michaelis-Menten Equation

$v = \frac{V_{\max} [S]}{K_m + [S]}$

• v = reaction velocity (e.g., mM/s)
• V_max = maximum velocity (when enzyme is saturated)
• [S] = substrate concentration
• K_m = Michaelis constant (substrate concentration at v = V_max/2)

Key insight: When [S] = K_m, then v = V_max/2. A lower K_m means higher affinity for substrate.

Catalytic Parameters

Turnover number (k_cat): reactions per enzyme per second $k_{\text{cat}} = \frac{V_{\max}}{[E]_{\text{total}}}$

Catalytic efficiency: how well the enzyme works at low [S] $\frac{k_{\text{cat}}}{K_m} \quad (\text{units: M}^{-1}\text{s}^{-1})$

Diffusion-limited enzymes approach ~10⁹ M⁻¹s⁻¹ (the "catalytic perfection" limit).

Lineweaver-Burk Plot

The double-reciprocal form linearizes the data:

$\frac{1}{v} = \frac{K_m}{V_{\max}} \cdot \frac{1}{[S]} + \frac{1}{V_{\max}}$

x-intercept = −1/K_m, y-intercept = 1/V_max, slope = K_m/V_max.

Competitive Inhibition

A competitive inhibitor (I) competes with substrate for the active site, effectively raising the apparent K_m:

$v = \frac{V_{\max}[S]}{K_m\left(1 + \frac{[I]}{K_i}\right) + [S]}$

V_max is unchanged; K_m_app = K_m(1 + [I]/K_i).

Your Task

Implement three functions:

1. mm_velocity(S_mM, V_max_mM_s, K_m_mM) — Michaelis-Menten reaction velocity
2. catalytic_efficiency(k_cat_s, K_m_mM) — k_cat/K_m in M⁻¹s⁻¹ (convert K_m from mM to M)
3. mm_competitive_inhibition(S_mM, V_max_mM_s, K_m_mM, I_mM, K_i_mM) — velocity with competitive inhibitor