Enzymes — lock-and-key vs induced fit; Michaelis-Menten kinetics

1. The active site & the two binding models

Lock-and-Key (Emil Fischer, 1894)

Induced Fit (Daniel Koshland, 1958)

Figure — Enzymes — lock-and-key vs induced fit; Michaelis-Menten kinetics

2. Michaelis–Menten kinetics — derived from scratch

Step 1 — The mechanism

$E + S \;\underset{k_{-1}}{\overset{k_1}{\rightleftharpoons}}\; ES \;\overset{k_2}{\longrightarrow}\; E + P$ Why this step? S must bind first (forming ES), then ES converts to product P, releasing free enzyme.

Step 2 — Rate of product formation

$v = k_2[ES]$ Why this step? Product only comes from the ES complex breaking down, so rate $\propto [ES]$ .

Step 3 — Steady-state assumption (Briggs–Haldane)

After a fast start, $[ES]$ stays roughly constant: formation = breakdown. $\underbrace{k_1[E][S]}_{\text{formation}} = \underbrace{(k_{-1}+k_2)[ES]}_{\text{breakdown}}$ Why this step? ES is a short-lived intermediate; its concentration barely changes during measurement, so $\frac{d[ES]}{dt}\approx 0$ .

Step 4 — Solve for $[ES]$

$[ES] = \frac{k_1[E][S]}{k_{-1}+k_2} = \frac{[E][S]}{K_m}, \qquad K_m \equiv \frac{k_{-1}+k_2}{k_1}$ Why this step? We defined the Michaelis constant $K_m$ to bundle the rate constants — it has units of concentration.

Step 5 — Conserve total enzyme

Free enzyme $[E]$ is unknown, but total is fixed: $[E]_T = [E] + [ES] \;\Rightarrow\; [E] = [E]_T - [ES]$ Substitute into Step 4 and solve: $[ES] = \frac{([E]_T-[ES])[S]}{K_m} \Rightarrow [ES]\,K_m = [E]_T[S]-[ES][S]$ $[ES] = \frac{[E]_T[S]}{K_m+[S]}$ Why this step? We can't measure free $[E]$ , but we do know total enzyme $[E]_T$ .

Step 6 — The Michaelis–Menten equation

Put Step 5 into $v=k_2[ES]$ , and note maximum rate $V_{max}=k_2[E]_T$ (all enzyme as ES):

The three regimes

3. Worked examples

4. Lineweaver–Burk (double reciprocal) — straightening the hyperbola

Flashcards

What does an enzyme do to activation energy and to equilibrium position?

Lowers

E_a

(speeds reaction); does NOT change equilibrium position.

Lock-and-key model: shape of active site?

Rigid, pre-shaped, complementary to substrate.

Induced-fit model: what happens on binding?

Active site is flexible and changes shape to wrap around the substrate.

Why is induced fit better than lock-and-key?

It explains catalysis (binding strains/orients S) and some substrate flexibility, not just recognition.

Write the Michaelis–Menten equation.

v_0 = \dfrac{V_{max}[S]}{K_m+[S]}

Define

K_m

in terms of rate constants.

K_m=\dfrac{k_{-1}+k_2}{k_1}

Physical meaning of

K_m

The

[S]

at which

v_0=V_{max}/2

; lower

K_m

= higher substrate affinity.

What is

V_{max}

in terms of

k_2

and enzyme?

V_{max}=k_2[E]_T

; proportional to total enzyme.

[S]\gg K_m

, what order is the reaction?

Zero order;

v_0\approx V_{max}

(saturated).

[S]\ll K_m

, what order?

First order;

v_0\approx (V_{max}/K_m)[S]

What assumption gives the MM equation?

Steady state:

d[ES]/dt\approx0

(formation = breakdown of ES).

Lineweaver–Burk y-intercept and x-intercept?

y-intercept

=1/V_{max}

; x-intercept

=-1/K_m

Recall Feynman: explain to a 12-year-old

Imagine a vending machine (enzyme) with one slot shaped for one coin (substrate). Lock-and-key: the slot is hard plastic — only the exact coin fits. Induced fit: the slot is soft rubber that squeezes around the coin to grab it tightly. Now, how fast does it give snacks? With few coins it's slow; as you feed more coins it speeds up — but there's only one machine, so once it's working flat-out, more coins just wait in line. That top speed is $V_{max}$ , and the number of coins needed to run at half-speed is $K_m$ . A machine that runs at half-speed with very few coins is a "greedy/grabby" machine — low $K_m$ , high affinity.

Connections

Proteins — enzymes are globular proteins; structure → active site.
Catalysis — enzymes vs inorganic catalysts; activation energy.
Chemical Kinetics — rate laws, order of reaction, rate constants.
Equilibrium — why catalysts don't shift $K_{eq}$ .
Enzyme Inhibition — competitive (↑ $K_m$ ) vs non-competitive (↓ $V_{max}$ ).
Vitamins and Coenzymes — cofactors that complete active sites.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, enzyme ek biological catalyst hota hai — yeh reaction ko fast karta hai by activation energy ko kam karke, lekin equilibrium ko shift nahi karta. Har enzyme ke paas ek special pocket hoti hai jise active site kehte hain, jahan substrate (reactant) aa ke fit hota hai. Ab sawaal: substrate fit kaise hota hai? Do model hain. Lock-and-key kehta hai active site bilkul rigid hai, jaise lock mein sirf sahi chaabi ghusti hai — isliye specificity samajh aati hai. Induced fit isse behtar hai: active site flexible hai, substrate aate hi enzyme apna shape badal ke uske around mould ho jaata hai, jaise haath ball ke around band hota hai. Yeh moulding hi catalysis mein madad karti hai.

Ab speed ki baat — Michaelis–Menten kinetics. Agar tum $v_0$ (rate) ko $[S]$ ke against plot karo, toh ek hyperbola milta hai: pehle tezi se badhta, phir saturate ho jaata hai (flat). Reason simple hai — enzyme limited hai, ek baar saare enzyme busy ho gaye toh aur substrate daalo bhi toh fayda nahi. Equation hai $v_0=\frac{V_{max}[S]}{K_m+[S]}$ .

Isme do star players: $V_{max}=k_2[E]_T$ matlab top speed, jo enzyme ki amount par depend karti hai. Aur $K_m$ — yeh wo $[S]$ hai jahan rate aadha $V_{max}$ ho jaata hai. Yaad rakho: chhota $K_m$ = high affinity (enzyme substrate ko strongly grab karta hai). Exam ka 80/20 yahi hai — "Km = half Vmax", "low Km = high affinity", aur teen regimes (low $S$ → first order, high $S$ → zero order). Itna pakka kar lo toh zyादातर questions ban jaate hain.

Enzymes — lock-and-key vs induced fit; Michaelis-Menten kinetics

1. The active site & the two binding models

Lock-and-Key (Emil Fischer, 1894)

Induced Fit (Daniel Koshland, 1958)

2. Michaelis–Menten kinetics — derived from scratch

Step 1 — The mechanism

Step 2 — Rate of product formation

Step 3 — Steady-state assumption (Briggs–Haldane)

Step 4 — Solve for $[ES]$

Step 5 — Conserve total enzyme

Step 6 — The Michaelis–Menten equation

The three regimes

3. Worked examples

4. Lineweaver–Burk (double reciprocal) — straightening the hyperbola

Flashcards

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Biomolecules

Connections