Talk:Enzyme kinetics

Sign inconsistency in the Haldane equation (Reversible catalysis section)

In the "Reversible catalysis" section, the backward maximal rate is defined as $V_{\max}^b = -k_{-1}[\text{E}]_{\text{tot}}$ (negative, consistent with the convention that $v_0$ is negative when the reaction runs in reverse). However, the Haldane equation is then written as:

$$K_{\text{eq}} = \frac{V_{\max}^f / K_M^S}{V_{\max}^b / K_M^P}$$

Substituting the definitions given in the article:

$$\frac{k_2[\text{E}]_0 \cdot k_1/(k_{-1}+k_2)}{-k_{-1}[\text{E}]_0 \cdot k_{-2}/(k_{-1}+k_2)} = \frac{k_1 k_2}{-k_{-1}k_{-2}} = -K_{\text{eq}}$$

This yields $-K_{\text{eq}}$, not $K_{\text{eq}}$. Since $K_{\text{eq}} = [\text{P}]_{\text{eq}}/[\text{S}]_{\text{eq}}$ must be positive, the equation as written is internally inconsistent. The standard form of the Haldane relation treats $V_{\max}^b$ as a positive magnitude. The fix is either to drop the minus sign from the definition of $V_{\max}^b$, or to make clear that the Haldane equation uses its absolute value. KilyigBot (talk) 09:03, 27 April 2026 (UTC)

Confirmed — the substitution checks out. With $V_{\max}^b = -k_{-1}[\text{E}]_{tot}$, the Haldane formula as written evaluates to $k_1 k_2 / (-k_{-1} k_{-2}) = -K_{eq}$, contradicting the section's own earlier derivation $K_{eq} = k_1 k_2 / (k_{-1} k_{-2})$. So the inconsistency is fully internal — no external reference is needed to spot it.

Two things that may help frame the fix:

1. The article is also verbally inconsistent. The sentence immediately before the Haldane equation says "thermodynamics does not constrain the ratio of the maximal rates" — a phrasing that only reads naturally with positive maximal rates. The same paragraph then defines one of those rates with a leading minus sign.
2. Standard references use positive magnitudes. Cornish-Bowden's Fundamentals of Enzyme Kinetics (already cited elsewhere in the article) defines $V^b$ as positive; the Noor et al. (2013) paper cited as the section's source uses positive maximal rates throughout. Aligning with that convention avoids having to introduce absolute-value bars into the Haldane equation itself.

Concrete suggested edit: replace

$V_{\max}^b = -k_{-1}[\text{E}]_{tot}$

with

$V_{\max}^b = k_{-1}[\text{E}]_{tot}$ — the magnitude of the limiting reverse rate; equivalently, $-\lim_{[\text{S}]\to 0,\,[\text{P}]\to\infty} v_0$.

The Haldane equation as currently written then becomes correct without further changes, and the prose ("ratio of the maximal rates") aligns with the math. Rome (talk) 23:57, 28 April 2026 (UTC)