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

4.5.4 · Chemistry › Biomolecules

1. Active site aur do 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 — scratch se derive kiya

Step 1 — Mechanism

$E + S k_{- 1} ⇌ k_{1} E S ⟶ k_{2} E + P$ Ye step kyun? S pehle bind karna chahiye (ES banata hai), phir ES product P mein convert hota hai, free enzyme release karta hai.

Step 2 — Product formation ki rate

$v = k_{2} [E S]$ Ye step kyun? Product sirf ES complex ke breakdown se aata hai, isliye rate $\propto [E S]$ .

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

Ek fast start ke baad, $[E S]$ roughly constant rehta hai: formation = breakdown. $formation k_{1} [E] [S] = breakdown (k_{- 1} + k_{2}) [E S]$ Ye step kyun? ES ek short-lived intermediate hai; measurement ke dauran iska concentration bahut kam badalta hai, isliye $\frac{d [ E S ]}{d t} \approx 0$ .

Step 4 — $[E S]$ solve karo

$[E S] = \frac{k _{1} [ E ] [ S ]}{k _{- 1} + k _{2}} = \frac{[ E ] [ S ]}{K _{m}}, K_{m} \equiv \frac{k _{- 1} + k _{2}}{k _{1}}$ Ye step kyun? Humne Michaelis constant $K_{m}$ define kiya rate constants ko bundle karne ke liye — iske units concentration ke hain.

Step 5 — Total enzyme conserve karo

Free enzyme $[E]$ unknown hai, lekin total fixed hai: $[E]_{T} = [E] + [E S] \Rightarrow [E] = [E]_{T} - [E S]$ Step 4 mein substitute karo aur solve karo: $[E S] = \frac{([ E ] _{T} - [ E S ]) [ S ]}{K _{m}} \Rightarrow [E S] K_{m} = [E]_{T} [S] - [E S] [S]$ $[E S] = \frac{[ E ] _{T} [ S ]}{K _{m} + [ S ]}$ Ye step kyun? Hum free $[E]$ measure nahi kar sakte, lekin hum total enzyme $[E]_{T}$ jaante hain.

Step 6 — Michaelis–Menten equation

Step 5 ko $v = k_{2} [E S]$ mein daalo, aur note karo maximum rate $V_{ma x} = k_{2} [E]_{T}$ (sabhi enzyme ES ke roop mein):

Teen regimes

3. Worked examples

4. Lineweaver–Burk (double reciprocal) — hyperbola ko seedha karna

Flashcards

Enzyme activation energy aur equilibrium position ke saath kya karta hai?

E_{a}

kam karta hai (reaction speed up karta hai); equilibrium position nahi badalta.

Lock-and-key model: active site ki shape kaisi hoti hai?

Rigid, pehle se shaped, substrate ke complementary.

Induced-fit model: binding par kya hota hai?

Active site flexible hoti hai aur substrate ke around wrap hone ke liye apni shape badal leti hai.

Induced fit, lock-and-key se behtar kyun hai?

Ye catalysis explain karta hai (binding S ko strain/orient karti hai) aur kuch substrate flexibility, sirf recognition nahi.

Michaelis–Menten equation likho.

v_{0} = \frac{V _{ma x} [ S ]}{K _{m} + [ S ]}

K_{m}

ko rate constants ke terms mein define karo.

K_{m} = \frac{k _{- 1} + k _{2}}{k _{1}}

K_{m}

ka physical meaning?

[S]

jis par

v_{0} = V_{ma x} /2

; chhota

K_{m}

= zyada substrate affinity.

k_{2}

aur enzyme ke terms mein

V_{ma x}

kya hai?

V_{ma x} = k_{2} [E]_{T}

; total enzyme ke proportional.

[S] ≫ K_{m}

par reaction ki order kya hai?

Zero order;

v_{0} \approx V_{ma x}

(saturated).

[S] ≪ K_{m}

par kya order?

First order;

v_{0} \approx (V_{ma x} / K_{m}) [S]

Kaun sa assumption MM equation deta hai?

Steady state:

d [E S] / d t \approx 0

(ES ka formation = breakdown).

Lineweaver–Burk y-intercept aur x-intercept?

y-intercept

= 1/ V_{ma x}

; x-intercept

= - 1/ K_{m}

Recall Feynman: ek 12-saal ke bachche ko explain karo

Socho ek vending machine (enzyme) hai jisme ek slot hai ek coin (substrate) ke liye shaped. Lock-and-key: slot hard plastic ka hai — sirf exact coin fit hoti hai. Induced fit: slot soft rubber ka hai jo coin ke around squeeze hota hai usse tightly pakadne ke liye. Ab, ye snacks kitni tezi se deta hai? Kam coins se slow hai; jaise jaise zyada coins daalo ye speed up hota hai — lekin ek hi machine hai, isliye ek baar jab ye full speed par kaam kar rahi ho, zyada coins line mein wait karti hain. Wo top speed $V_{ma x}$ hai, aur half-speed par chalane ke liye kitni coins chahiye wo $K_{m}$ hai. Jo machine bahut kam coins se half-speed par chalti hai wo ek "greedy/grabby" machine hai — low $K_{m}$ , high affinity.

Connections

Proteins — enzymes globular proteins hain; structure → active site.
Catalysis — enzymes vs inorganic catalysts; activation energy.
Chemical Kinetics — rate laws, order of reaction, rate constants.
Equilibrium — catalysts $K_{e q}$ kyun shift nahi karte.
Enzyme Inhibition — competitive (↑ $K_{m}$ ) vs non-competitive (↓ $V_{ma x}$ ).
Vitamins and Coenzymes — cofactors jo active sites complete karte hain.