For the scheme

PHYSICS · Multi-part · Fermi University Physics · Biophysics & Medical Physics · Molecular machines, transport and bioenergetics · Molecular machines, transport and bioenergetics

Forces, Machines and Signals, Ch Enzymes and molecular motors

Problem

For the scheme E+S⇌k1k−1ES→k2E+PE+S\underset{k_{-1}}{\overset{k_1}{\rightleftharpoons}}ES\overset{k_2}{\rightarrow}E+P, someone argues: "In steady state the complex concentration does not change, so the rate at which ESES forms equals the rate at which it breaks apart by the reverse reaction: k1[E][S]=k−1[ES]k_1[E][S]=k_{-1}[ES]. Hence [ES]=[E][S]/Kd[ES]=[E][S]/K_d with Kd=k−1/k1K_d=k_{-1}/k_1, and using [E]0=[E]+[ES][E]_0=[E]+[ES], v=k2[ES]=k2[E]0[S]/(Kd+[S])v=k_2[ES]=k_2[E]_0[S]/(K_d+[S]), so KM=Kd=k−1/k1K_M=K_d=k_{-1}/k_1 in general." ((a)) Find the error in this working. ((b)) Give the correct steady-state expression for KMK_M. ((c)) State the condition on the rate constants under which the faulty conclusion KM=KdK_M=K_d happens to be numerically correct.

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