Iso mechanisms

In an iso mechanism the enzyme passes through two distinct conformations during a single turnover. The substrate binds one conformation; catalysis converts the enzyme–substrate complex into the other conformation of the enzyme–product complex; the product leaves from that conformation; and the free enzyme must then isomerize back to the first conformation before it can bind substrate again. That free-enzyme isomerization — the "iso" step — is the defining feature.

An iso uni-uni example

A second enzyme conformation is written with a different conformation label — Eprime in place of E. Bound forms carry their conformation (Eprime(P)), and the free enzyme moves between conformations through an isomerization step such as Eprime <--> E. Here S binds E, catalysis turns E(S) into Eprime(P), P leaves Eprime, and the free enzyme isomerizes from Eprime back to E:

using EnzymeRates
m = @enzyme_mechanism begin
    substrates: S
    products:   P
    steps: begin
        E + S ⇌ E(S)
        E(S) <--> Eprime(P)
        Eprime(P) ⇌ Eprime + P
        Eprime <--> E
    end
end
print(rate_equation_string(m))

(; K_P_Eprime, K_S_E, k_ES_to_EprimeP, k_E_to_Eprime, k_Eprime_to_E, Keq, E_total) = params
(; S, P) = concs
# Haldane constraints:
k_EprimeP_to_ES = (1 / Keq) * K_P_Eprime * (1 / K_S_E) * k_ES_to_EprimeP * (1 / k_E_to_Eprime) * k_Eprime_to_E
v = E_total * (k_ES_to_EprimeP * k_Eprime_to_E * S / K_S_E - k_E_to_Eprime * k_EprimeP_to_ES * P / K_P_Eprime) / (k_E_to_Eprime + k_Eprime_to_E + k_E_to_Eprime * P / K_P_Eprime + k_EprimeP_to_ES * P / K_P_Eprime + k_ES_to_EprimeP * S / K_S_E + k_Eprime_to_E * S / K_S_E + k_ES_to_EprimeP * P * S / (K_P_Eprime * K_S_E) + k_EprimeP_to_ES * P * S / (K_P_Eprime * K_S_E))

The derived equation carries the signature of an iso mechanism: an S·P cross term in the denominator (the … P * S / (K_P_Eprime * K_S_E) terms above). An ordinary single-conformation Michaelis–Menten enzyme does not have an S·P term. Saturating an iso enzyme with substrate and product together can therefore slow turnover in a way a regular Michaelis–Menten enzyme cannot. Each conformation appears in the parameter names: K_S_E is S binding to E, K_P_Eprime is P binding to Eprime, and k_E_to_Eprime / k_Eprime_to_E are the isomerization rate constants.