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Enzyme Kinetics

Field: Enzymology

The study of the chemical reactions that are catalysed by enzymes.

Sequence of Expressions

Define the Michaelis constant KmK_m as the substrate concentration [S][S] that yields a reaction velocity vv equal to half of the maximum velocity VmaxV_{max}: Km=[S]such thatv=Vmax2 K_m = [S] \quad \text{such that} \quad v = \frac{V_{max}}{2}
Define the maximum reaction velocity VmaxV_{max} as the limiting rate of the reaction as the substrate concentration [S][S] approaches saturation, where the enzyme concentration [E]T[E]_T is the limiting factor: Vmax=lim⁡[S]→∞v=kcat[E]T V_{max} = \lim_{[S] \to \infty} v = k_{cat} [E]_T
Let vv be the initial reaction velocity, [S][S] be the substrate concentration, VmaxV_{max} be the maximum velocity, and KmK_m be the Michaelis constant. The rate law is given by:
Define the turnover number τ\tau (or kcatk_{cat}) as the maximum rate of product formation per active site, which relates the maximum velocity VmaxV_{max} to the total enzyme concentration [E]T[E]_T: τ=kcat=Vmax[E]T \tau = k_{cat} = \frac{V_{max}}{[E]_T}
Consider the double reciprocal plot of the initial velocity vv versus substrate concentration [S][S]. The relationship is linear in the form y=mx+by = m x + b, where y=1/vy = 1/v and x=1/[S]x = 1/[S]. The equation is derived from the Michaelis-Menten kinetics as:
Theorem

Rate Law

Define the reaction rate vv as the time derivative of the product concentration, v=d[P]dtv = \frac{d[P]}{dt}. For a general reaction involving reactants AiA_i and rate constant kk, the rate law is expressed as:\nd[P]dt=v=k∏i=1n[Ai]ni\frac{d[P]}{dt} = v = k \prod_{i=1}^{n} [A_i]^{n_i}
The catalytic efficiency, η\eta, is defined as the ratio of the turnover number (kcatk_{cat}) to the Michaelis constant (KmK_m), quantifying the enzyme's ability to convert substrate at low concentrations:\nη=kcatKm\eta = \frac{k_{cat}}{K_m}
For a weak acid HA\text{HA} and its conjugate base A−\text{A}^-, the relationship between the pH\text{pH} and the acid dissociation constant (KaK_a) is given by:\npH=pKa+log⁡([A−][HA])\text{pH} = pK_a + \log\left(\frac{[A^-]}{[HA]}\right)\nwhere pKa=−log⁡(Ka)pK_a = -\log(K_a).
Let [S]0[S]_0 be the initial substrate concentration. If [S]0≫Km[S]_0 \gg K_m, the Michaelis-Menten rate law simplifies to a pseudo-first-order rate constant kobsk_{obs}: \nv=kobs[S]v = k_{obs}[S]\nwhere kobs=Vmax[S]0k_{obs} = \frac{V_{max}}{[S]_0}.
For a general reversible enzyme-inhibitor interaction, the rate vv under inhibition by [I][I] is modified from the standard Michaelis-Menten equation. For competitive inhibition, the apparent KmK_m is increased:\nv=Vmax[S]Km(1+[I]KI)+[S]v = \frac{V_{max}[S]}{K_m(1 + \frac{[I]}{K_I}) + [S]}