Kinetic groups

A kinetic group is a set of steps that share one kinetic parameter. The point is parsimony: when several steps are chemically the same — the same metabolite binding the enzyme, or the same chemical conversion happening in more than one place — there is no reason for each to carry its own constant. Putting them in one group makes the model say they behave identically, and the rate equation then carries a single parameter for the whole set instead of one per step.

By default @enzyme_mechanism keeps each step in its own group. To declare that several steps share a parameter, wrap them in parentheses — a parenthesized step-group binds the steps into one kinetic group:

steps: begin
    (E + Q ⇌ E(Q), E(A) + Q ⇌ E(A, Q))   # both Q-binding steps share one constant
    ...
end

This is a modeling choice, distinct from the thermodynamic reduction (Thermodynamic constraints). The reduction also removes parameters, but only when a Haldane or Wegscheider relation forces two constants to be equal. Kinetic grouping is a decision you make: it ties together steps the thermodynamics leaves independent.

An example: a two-substrate, two-product mechanism

Consider a random-order reaction A + B ⇌ P + Q with two abortive complexes: E(A, Q), where the enzyme binds substrate A together with the product Q of the other half-reaction, and E(B, P), its mirror. Like the catalytic complexes, each abortive complex forms either way — E(A, Q) by Q binding E(A) or A binding E(Q). Every metabolite therefore binds the enzyme in several places: A binds free E, the E(B) complex, and E(Q), and the same holds for B, P, and Q.

With every step in its own group, each of those bindings gets its own constant:

using EnzymeRates
ungrouped = @enzyme_mechanism begin
    substrates: A, B
    products:   P, Q
    steps: begin
        E + A ⇌ E(A)
        E + B ⇌ E(B)
        E(A) + B ⇌ E(A, B)
        E(B) + A ⇌ E(A, B)
        E(A, B) <--> E(P, Q)
        E(P, Q) ⇌ E(P) + Q
        E(P, Q) ⇌ E(Q) + P
        E(P) ⇌ E + P
        E(Q) ⇌ E + Q
        E(A) + Q ⇌ E(A, Q)
        E(Q) + A ⇌ E(A, Q)
        E(B) + P ⇌ E(B, P)
        E(P) + B ⇌ E(B, P)
    end
end
print(rate_equation_string(ungrouped))

(; K_A_E, K_A_EB, K_A_EQ, K_B_E, K_B_EP, K_P_E, K_P_EQ, K_Q_E, k_EAB_to_EPQ, Keq, E_total) = params
(; A, B, P, Q) = concs
# Wegscheider constraints:
K_B_EA = (1 / K_A_E) * K_A_EB * K_B_E
K_P_EB = (1 / K_B_E) * K_B_EP * K_P_E
K_Q_EA = (1 / K_A_E) * K_A_EQ * K_Q_E
K_Q_EP = (1 / K_P_E) * K_P_EQ * K_Q_E
# Haldane constraints:
k_EPQ_to_EAB = (1 / Keq) * (1 / K_A_EB) * (1 / K_B_E) * K_P_EQ * K_Q_E * k_EAB_to_EPQ
v = E_total * (k_EAB_to_EPQ * A * B / (K_A_E * K_B_EA) - k_EPQ_to_EAB * P * Q / (K_P_E * K_Q_EP)) / (1 + A / K_A_E + B / K_B_E + P / K_P_E + Q / K_Q_E + A * B / (K_A_E * K_B_EA) + A * Q / (K_A_E * K_Q_EA) + B * P / (K_B_E * K_P_EB) + P * Q / (K_P_E * K_Q_EP))

There are nine independent constants, with form-specific names: K_A_E, K_A_EB, and K_A_EQ for A on three different forms, and similar families for the others. Four more bindings are not fit at all but fixed by Wegscheider relations (K_B_EA, K_P_EB, K_Q_EA, K_Q_EP), since each catalytic and abortive loop closes a thermodynamic cycle.

Now group every binding of a given metabolite together — all A-binding steps in one group, all B-binding in another, and likewise for P and Q. Each metabolite then has a single binding constant:

grouped = @enzyme_mechanism begin
    substrates: A, B
    products:   P, Q
    steps: begin
        (E + A ⇌ E(A), E(B) + A ⇌ E(A, B), E(Q) + A ⇌ E(A, Q))
        (E + B ⇌ E(B), E(A) + B ⇌ E(A, B), E(P) + B ⇌ E(B, P))
        E(A, B) <--> E(P, Q)
        (E(P, Q) ⇌ E(Q) + P, E(P) ⇌ E + P, E(B) + P ⇌ E(B, P))
        (E(P, Q) ⇌ E(P) + Q, E(Q) ⇌ E + Q, E(A) + Q ⇌ E(A, Q))
    end
end
print(rate_equation_string(grouped))

(; K_A_E, K_B_E, K_P_E, K_Q_E, k_EAB_to_EPQ, Keq, E_total) = params
(; A, B, P, Q) = concs
# Haldane constraints:
k_EPQ_to_EAB = (1 / Keq) * (1 / K_A_E) * (1 / K_B_E) * K_P_E * K_Q_E * k_EAB_to_EPQ
v = E_total * (k_EAB_to_EPQ * A * B / (K_A_E * K_B_E) - k_EPQ_to_EAB * P * Q / (K_P_E * K_Q_E)) / (1 + A / K_A_E + B / K_B_E + P / K_P_E + Q / K_Q_E + A * B / (K_A_E * K_B_E) + A * Q / (K_A_E * K_Q_E) + B * P / (K_B_E * K_P_E) + P * Q / (K_P_E * K_Q_E))

The parameter list collapses from nine constants to five — one binding constant per metabolite (K_A_E, K_B_E, K_P_E, K_Q_E) plus the catalytic k_EAB_to_EPQ:

(ungrouped = parameters(ungrouped), grouped = parameters(grouped))

(ungrouped = (:K_A_E, :K_A_EB, :K_A_EQ, :K_B_E, :K_B_EP, :K_P_E, :K_P_EQ, :K_Q_E, :k_EAB_to_EPQ, :Keq, :E_total), grouped = (:K_A_E, :K_B_E, :K_P_E, :K_Q_E, :k_EAB_to_EPQ, :Keq, :E_total))

The denominator becomes symmetric: the two abortive complexes read A * Q / (K_A_E * K_Q_E) and B * P / (K_B_E * K_P_E), with no form-specific suffixes. The Wegscheider section is gone, too — once each metabolite has a single binding constant, the loop-closing relations become identities and drop out. Grouping did the collapsing the thermodynamics could not: it is what keeps the parameter count at the lowest physically meaningful value, and the mechanism enumeration starts there, splitting groups back apart only as the data warrant (see The enumeration engine).