Enzymes formulas

Enzymes: classification, catalysis, kinetics and inhibition. 10 formulas for GATE BT General Biology, each with when to use it. Each is explained in the topic notes.

All 10 Enzymes formulas

MICHAELIS-MENTEN EQUATION

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

Initial velocity of a single-substrate enzyme under steady state with $[S] ≫ [E]_{T}$ . $K_{m}$ is in concentration units (mM or $μ$ M), $V_{ma x}$ in concentration per time (for example $μ$ M/min).

MICHAELIS CONSTANT

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

For $E + S k_{- 1} ⇌ k_{1} E S \to k_{2} E + P$ . $K_{m}$ equals the substrate concentration at which $v = V_{ma x} /2$ . It equals the dissociation constant $K_{d} = k_{- 1} / k_{1}$ of ES only if $k_{2} ≪ k_{- 1}$ .

MAXIMUM VELOCITY AND TURNOVER NUMBER

V_{ma x} = k_{c a t} [E]_{T}

$k_{c a t}$ (units s $^{- 1}$ ) is the number of substrate molecules converted per active site per second at saturation. For the simple mechanism above $k_{c a t} = k_{2}$ .

CATALYTIC EFFICIENCY

catalytic efficiency = \frac{k _{c a t}}{K _{m}}

Units M $^{- 1}$ s $^{- 1}$ . It is the apparent second-order rate constant at $[S] ≪ K_{m}$ , where $v \approx (k_{c a t} / K_{m}) [E]_{T} [S]$ . Its upper limit is set by diffusion, about $1 0^{8}$ to $1 0^{9}$ M $^{- 1}$ s $^{- 1}$ .

FRACTION OF VMAX REACHED AT A GIVEN [S]

\frac{v}{V _{ma x}} = \frac{[ S ]}{K _{m} + [ S ]} \Rightarrow [S] = \frac{f}{1 - f} K_{m} for v = f V_{ma x}

For example $f = 0.5$ gives $[S] = K_{m}$ , and $f = 0.9$ gives $[S] = 9 K_{m}$ .

LINEWEAVER-BURK (DOUBLE RECIPROCAL)

\frac{1}{v} = \frac{K _{m}}{V _{ma x}} \cdot \frac{1}{[ S ]} + \frac{1}{V _{ma x}}

Plot $1/ v$ against $1/ [S]$ : slope $= K_{m} / V_{ma x}$ , y-intercept $= 1/ V_{ma x}$ , x-intercept $= - 1/ K_{m}$ .

EADIE-HOFSTEE

v = V_{ma x} - K_{m} \frac{v}{[ S ]}

Plot $v$ against $v / [S]$ : slope $= - K_{m}$ , y-intercept $= V_{ma x}$ , x-intercept $= V_{ma x} / K_{m}$ .

HANES-WOOLF

\frac{[ S ]}{v} = \frac{1}{V _{ma x}} [S] + \frac{K _{m}}{V _{ma x}}

Plot $[S] / v$ against $[S]$ : slope $= 1/ V_{ma x}$ , y-intercept $= K_{m} / V_{ma x}$ , x-intercept $= - K_{m}$ .

APPARENT CONSTANTS WITH A REVERSIBLE INHIBITOR

V_{ma x}^{a pp} = \frac{V _{ma x}}{α ^{'}}, K_{m}^{a pp} = \frac{α}{α ^{'}} K_{m}

General mixed case. Competitive: $α^{'} = 1$ . Uncompetitive: $α = 1$ . Pure non-competitive: $α = α^{'}$ ( $K_{i} = K_{i}^{'}$ ), so $K_{m}$ is unchanged.

HILL EQUATION FOR COOPERATIVE ENZYMES

lo g \frac{v}{V _{ma x} - v} = n_{H} lo g [S] - lo g K^{'}

Here $K^{'} = (K_{0.5})^{n_{H}}$ . A plot of the left side against $lo g [S]$ has slope $n_{H}$ , the Hill coefficient. $n_{H} > 1$ means positive cooperativity, $n_{H} = 1$ no cooperativity, $n_{H} < 1$ negative cooperativity. $n_{H}$ cannot exceed the number of substrate-binding sites.

Where this was asked: 2019 (1), 2020 (1), 2025 (2), 2026 (2). See the 6 previous-year questions on Enzymes. This shows what was asked, not what will be asked.

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Enzymes: the notes and 6 solved previous-year questions