Estimating kinact and KI From Time-Dependent Inhibition Data

From Progress Curves to Inactivation Parameters

Time-dependent inhibition is not described by a single reduction in enzyme activity: the active enzyme fraction changes during exposure to inhibitor. A progress curve records product accumulation over that exposure, so its curvature contains information about the rate at which active enzyme disappears. Comparing that rate across inhibitor concentrations can distinguish the maximal inactivation rate from the inhibitor concentration required to approach it.

The distinction matters for interpretation and design. A set of curves collected only at low inhibitor concentrations may show clear time dependence while still being unable to estimate the two mechanism parameters separately. Conversely, concentrations that reach the inactivation-rate plateau can inform the maximal rate, but only if the sampling schedule is long and dense enough to resolve the time course.

This equation assumes that free inhibitor concentration remains effectively constant, reversible binding reaches equilibrium rapidly relative to inactivation, and active enzyme is inactivated through the bound state. In the simplest form, the assay condition makes substrate competition negligible. If substrate competes with inhibitor for enzyme, that competition must be accounted for; otherwise, the fitted \(K_I\) can be an assay-condition-dependent apparent value rather than the binding constant of interest.

For the progress-curve derivation below, assume no enzyme synthesis or turnover during the assay, constant assay conditions, and a product-formation rate proportional to active enzyme. Substrate depletion and product inhibition must be negligible over the fitted interval, or represented explicitly in the model. These assumptions let the inhibition kinetics be isolated from other causes of curve flattening.

Deriving the Progress-Curve Model

Under rapid reversible binding, the fraction of enzyme in the inhibitor-bound state is \([I]/(K_I+[I])\). Each bound enzyme molecule inactivates at rate \(k_{\mathrm{inact}}\). Therefore the effective first-order loss rate for active enzyme is the product of the bound fraction and the inactivation rate:

With constant inhibitor concentration, active enzyme \(E(t)\) follows a first-order decay. Separating variables and integrating from \(E(0)=E_0\) to \(E(t)\) gives the exponential time course:

If the initial product-formation rate is \(v_0\), the instantaneous rate later in the curve is \(v(t)=v_0e^{-k_{\mathrm{obs}}t}\). Integrating that rate gives accumulated product \(P(t)\):

Thus, when product is measured continuously, the curve approaches a finite asymptote under this idealized model. The rate parameter controls how quickly it bends toward that asymptote; \(v_0/k_{\mathrm{obs}}\) controls the amount of product accumulated above \(P(0)\). In practice, \(v_0\), baseline, and \(k_{\mathrm{obs}}\) may all need estimation, so a single endpoint is generally not enough to establish the curve shape.

Estimating the Concentration-Response Parameters

Once an observed rate has been estimated for each inhibitor concentration, the rates are related to concentration by a saturable curve. At low concentrations, the rate rises approximately in proportion to \([I]\). At high concentrations, it approaches \(k_{\mathrm{inact}}\). The transition between these regions provides information about \(K_I\).

The concentration range controls whether the parameters are separately identifiable. When \([I]\) is much smaller than \(K_I\), the denominator is approximately \(K_I\), so

In this range, the data mainly estimate the ratio \(k_{\mathrm{inact}}/K_I\), not both parameters independently. When \([I]\) is much larger than \(K_I\), \(k_{\mathrm{obs}}\approx k_{\mathrm{inact}}\), making the plateau informative about \(k_{\mathrm{inact}}\). Measurements near the bend of the curve help locate \(K_I\). A useful design therefore spans low concentrations, the transition region, and concentrations high enough to approach the plateau, subject to solubility, nonspecific effects, and assay limits.

Fitting Strategy, Diagnostics, and Alternatives

A practical two-stage strategy first estimates \(k_{\mathrm{obs}}\) from each progress curve, then fits the saturable concentration-response equation to those estimates. It is transparent and helps reveal concentration-specific problems. However, the second stage treats estimated rates as if they were measured without error unless their uncertainty is carried forward. Rates from poorly sampled curves can be much less precise than rates from well-resolved curves.

A global nonlinear fit instead estimates the shared \(k_{\mathrm{inact}}\) and \(K_I\), alongside curve-specific initial rates or baselines as needed, directly from all time-course observations. This can propagate information more coherently, but demands appropriate residual and error models. In either approach, inspect the fitted curves and residuals, not only the final parameter estimates. Systematic early-versus-late residual patterns can indicate a wrong time-course shape; concentration-specific departures can indicate a wrong concentration-response model or assay interference.

Reciprocal transformations are a tempting alternative because taking \(1/k_{\mathrm{obs}}\) yields a line in \(1/[I]\). This transformation changes the error structure and gives disproportionate influence to small rates, which are often the least precisely estimated. Nonlinear fitting on the original rate or observation scale is generally preferable. The two-stage method remains useful for visualization and diagnosis, while global fitting is often preferable when curve-level uncertainty and shared parameters need to be handled together.

Sampling times must cover the changing part of each curve. If all measurements are taken much earlier than the characteristic time \(1/k_{\mathrm{obs}}\), little enzyme loss occurs and the rate is difficult to estimate. If measurements begin only after nearly complete inactivation, the early curvature is missed. For example, at \(k_{\mathrm{obs}}=0.0100\ \mathrm{min}^{-1}\), the enzyme half-life is \(\ln(2)/0.0100=69.3\) minutes; ending sampling at 10 minutes captures only a small fraction of that decay. A range of early and later observations, with enough replicates to characterize measurement error, is more informative than dense sampling confined to one uninformative part of the curve.

Common failure modes include inhibitor depletion when enzyme concentration is not negligible relative to inhibitor, time-dependent loss of inhibitor activity, nonspecific enzyme loss, and assay signals that drift independently of enzyme inactivation. A control without inhibitor helps detect background loss. Replicate curves and residual plots help distinguish random assay noise from systematic misspecification. A plateau in product formation alone does not prove irreversible inactivation: substrate exhaustion or product inhibition can also flatten a curve.

Parameter uncertainty is especially important when the concentration range misses the transition or plateau. A narrow confidence interval from an optimizer does not guarantee identifiability if the model or error structure is wrong. Examine parameter correlation, profile-based uncertainty, or fits from plausible starting values, and check whether alternative parameter pairs give nearly indistinguishable predictions. The estimates are conditional on the assumed mechanism and assay conditions; these data alone do not establish that the proposed molecular mechanism is uniquely correct.