Michaelis-Menten kinetics

Michaelis-Menten kinetics models substrate binding to an enzyme, complex dissociation and product formation. Under the relevant quasi-steady-state regime, its initial-rate law is the saturating relation given by the following equation.

Substrate replication competes with saturating enzyme clearance in this fast-slow model, where $K>\lambda >0$ and $ϵ\ll 1$. The physical strip $u\ge 0$, $0\le v\le 1$ is invariant. If $0<\eta <\lambda /K$, the origin is attracting and a positive saddle exists at $u_e=\lambda /\eta -K$, $v_e=1-\eta K/\lambda$. Eliminating the fast complex fraction gives $u^{\prime }\simeq \eta u(u-u_e)/(u+K)$, separating extinction from runaway. For initially zero complexes, the stable-manifold threshold is $u_{\ast }=u_e+ϵ(1-\eta )v_e+O({ϵ}^2)$, away from singular parameter limits. To derive the correction, write the stable manifold as $u=u_e+ϵ\varphi (v)$; its invariance equation gives ${\varphi }^{\prime }=-(1-\eta )$ and $\varphi (v_e)=0$. A Dulac factor $1/u$ has strictly negative weighted divergence, excluding a periodic alternative to the two basins. Saturated clearance and autocatalytic growth can therefore create an unstable concentration threshold rather than a stable nonzero balance.

A plot of $[S]/v$ against $[S]$ for a reaction modeled by Michaelis-Menten kinetics. The slope is $1/V_{\max }$, the vertical intercept $K_m/V_{\max }$ and the horizontal intercept $-K_m$; graphical transformation changes how measurement errors are weighted.

The Michaelis-Menten rate law isIt is linear at low substrate concentration and saturates at $V_{\max }$ at high concentration.

The substrate concentration giving half the limiting reaction rate in the Michaelis-Menten equation. In the elementary mechanism $E+S\rightleftharpoons ES\to E+P$, it is $K_M=(k_{-1}+k_{\mathrm{cat}})/k_1$ under the quasi-steady-state approximation, whereas the binding dissociation constant is $K_d=k_{-1}/k_1$.