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The slope of $\log [\theta /(1-\theta )]$ against $\log [L]$ in a Hill equation description. A value above one supports positive cooperative binding under the model's assumptions. It does not by itself count occupied sites or identify a molecular mechanism.

Protein oligomerization associates several protein subunits into a defined assembly, such as a protein dimer. It can create cooperative binding sites, regulate activity or build structural units. A measured assembly depends on concentration and conditions: a dimer can dissociate on dilution. Determining subunit count therefore requires an assay model as well as a measured size or signal. ^[ref]

A sequential two-binding model can produce the reduced reaction rateHere $u=s/s_{\ast }$ is the free substrate concentration relative to a chosen scale $s_{\ast }$, $p$ is product concentration, and $A$ has units of concentration per time. The dimensionless parameters must be obtained from a specified enzyme kinetics mechanism; the rational function alone does not identify a mechanism involving allostery.

For example, suppose a single substrate species binds twice in sequence, producing three enzyme states $E,ES,E{S}_2$, and only $E{S}_2$ makes product. If binding relaxes rapidly compared with catalysis, take the approximate binding balancesThe total enzyme concentration $e_T=[E]+[ES]+[E{S}_2]$ givesMultiplying by the turnover number $k_{\mathrm{cat}}$ yields the stated law with $A=k_{\mathrm{cat}}{e}_T\beta /\gamma$. These rapid-binding balances are an additional model assumption within a quasi-steady-state approximation.

At low substrate concentration, $dp/dt\sim (A/\alpha )u^2$ because the active state requires two binding events. At high substrate concentration, the rate tends to $A\gamma /\beta =k_{\mathrm{cat}}{e}_T$. Such limiting behavior helps compare candidate mechanisms with measured reaction rates; evidence for cooperative binding requires examining how the binding constants depend on occupancy.