Optimizing enzyme inhibition analysis: precise estimation with a single inhibitor concentration

Abstract

Enzyme inhibition analysis is essential in drug development and food processing, necessitating precise estimation of inhibition constants. Traditionally, these constants are estimated through experiments using multiple substrate and inhibitor concentrations, but inconsistencies across studies highlight a need for a more systematic approach to set experimental designs across all types of enzyme inhibition. Here, we address this by analyzing the error landscape of estimations in various experimental designs. We find that nearly half of the conventional data is dispensable and even introduces bias. Instead, by incorporating the relationship between IC₅₀ and inhibition constants into the fitting process, we find that using a single inhibitor concentration greater than IC₅₀ suffices for precise estimation. This IC₅₀-based optimal approach, which we name 50-BOA, substantially reduces (>75%) the number of experiments required while ensuring precision and accuracy. Additionally, we provide a user-friendly package that implements the 50-BOA.

Fig. 1. The canonical approach for estimating inhibition constants.

a An inhibitor (I) can suppress the enzyme-catalyzed reaction from a substrate (S) to a product (P) by binding to a free enzyme (E) (competitive), enzyme-substrate complex (C) (uncompetitive), or both (Mixed), forming reversible complexes (Y and B). Here, $S_T=S+C+P+B$ and $I_T=I+Y+B$ denote the total substrate and inhibitor concentrations, respectively, and $K_M=\frac{k_2+k_{-1}}{k_1}$ denotes the Michaelis-Menten (MM) constant. The dissociation constants of each binding ($K_{ic}=\frac{k_{-3}}{k_3}$ and $K_{iu}=\frac{k_{-4}}{k_4}$), known as the inhibition constants, determine which inhibition type is dominant. b The canonical approach for estimating the inhibition constants and types involves the following steps: (i) The inhibitor concentration leading to half of the % control activity ($I{C}_{50}$) is estimated from experiments with a single $S_T$ and varying $I_T$. Here, the % control activity is the percentage ratio of the initial velocity with inhibitor to the initial velocity without inhibitor. (ii) For various combinations of $S_T$ values ranging from $\frac{1}{5}{K}_M$ to $5K_M$ and $I_T$ values ranging from $\frac{1}{3}I{C}_{50}$ to $3I{C}_{50}$, including $I_T=0$, initial velocities are measured. (iii) By fitting Eq. 1 to the initial velocity data, the inhibition constants are estimated. It is unclear whether the dataset used in the canonical approach is sufficient or necessary for precise estimation.

Fig. 2. Using a single inhibitor concentration larger than inhibition constants leads to precise estimation.

a The initial velocity data (red dots), obtained based on a true $K_{ic}$ and $K_{iu}$ pair with a single $I_T$ value and several $S_T$ values (see “Methods” for details), were compared with Eq. 1 of different parameter pairs (e.g., solid and dashed lines). b For each parameter pair, the fitting error between Eq. 1 and the data was calculated. The heatmap representing the fitting error landscape was plotted for a range of parameter pairs. c When $I_T=0.1$ μM was much smaller than the true parameter values ($K_{ic}=K_{iu}=1$ μM), a wide range of parameter pairs (e.g., ▲, ▼, ●) led to low fitting errors, indicating imprecise estimation. d The fitted curves with these parameter pairs matched the data well. e–f When $I_T=2$ μM was greater than the true $K_{ic}=1$ μM but smaller than the true $K_{iu}=10$ μM, the pairs of the same $K_{ic}$ but distinct $K_{iu}$ values (e.g., ▲, ●) led to accurate fitting, indicating precise $K_{ic}$ but not $K_{iu}$ estimation. g, h Conversely, when $I_T=2$ μM exceeded the true $K_{iu}=1$ μM but much smaller than the true $K_{ic}=10$ μM, the pairs of the same $K_{iu}$ but distinct $K_{ic}$ values (e.g., ▼, ●) led to accurate fitting, Indicating precise $K_{iu}$ but not $K_{ic}$ estimation. i, j When $I_T=10$ μM was greater than or comparable to both true parameter values ($K_{ic}=K_{iu}=1$ μM), precise estimation for both $K_{ic}$ and $K_{iu}$ is possible (●). For c-j, the true data were obtained with $S_T=0.2K_M$, $K_M$, and $5K_M$ ($K_M=1$ μM). The initial velocity data were normalized by $V_{\max }=0.1$ μM$/\min /\mathrm{mg}\mathrm{protein}$.

Fig. 3. Using an inhibitor concentration above IC₅₀ with IC₅₀ regularization allows for precise estimation.

a $I{C}_{50}$, measured with $S_T$, is a weighted harmonic mean of $K_{ic}$ and $K_{iu}$, with the weight of $\alpha =\frac{K_M}{S_T+K_M}$ (Eq. 2). b–d Since $I{C}_{50}$ is always greater than either $K_{ic}$ or $K_{iu},$ using $I_T\ge I{C}_{50}$ allows for precise estimation of $K_{ic}$ (b), $K_{iu}$ (c), or both (d) when $K_{ic}$ and $K_{iu}$ differ within 10-fold (See Supplementary Fig. 1 for other cases). e With known ${IC}_{50}$, the unknown $K_{ic}$ and $K_{iu}$ need to satisfy the weighted harmonic mean equation, $H\left(K_{ic},K_{iu}\right)={\mathrm{IC}}_{50}$, which can be incorporated into estimation via a regularization term (regularization error; Eq. 3). f–h The heatmap shows regularization error, where the low regularization error region (black region) corresponds to the $K_{ic}$ and $K_{iu}$ values satisfying the constraint $H\left(K_{ic},K_{iu}\right)=I{C}_{50}$. i–k Incorporating the regularization error into the fitting error (total error) allowed for precise estimation of $K_{ic}$ and $K_{iu}$ (dashed lines), reducing the low error region (black region) compared to unregularized cases (b–d). l $I{C}_{50}$-based optimal approach (50-BOA) uses only a single $I_T\ge {IC}_{50}$ along with the ${IC}_{50}$ regularization, considerably reducing the number of required experiments for precise and accurate estimation, compared to the canonical approach (Fig. 1b).

Fig. 4. The 50-BOA enables precise and accurate estimation for experimental mixed inhibition data.

a The normalized initial velocities ($\frac{V_0}{V_{\max }}$; $V_{\max }=1.02$ μM/min⁻¹/mg protein) of triazolam with its inhibitor ketoconazole ($I{C}_{50}=0.040$ μM when $S_T=50$ μM; $K_M=72$ μM) were measured for various triazolam ($S_T=10$, $25$, $50$, $100$, $250$, and $500$ μM) and ketoconazole ($I_T=0$, $0.01$, $0.025$, $0.05$, $0.1$, $0.25$, and $0.5$ μM) concentration combinations, and fitted with Eq. 1 (solid lines). b Heatmaps of the total errors without (upper row) and with (lower row) regularization for entire or single $I_T$ setups. Using the entire $I_T$ range (i.e., canonical condition) led to precise $K_{ic}$ and $K_{iu}$ estimation, identifying mixed inhibition. For single $I_T$ setups, when $I_T$ $<$ $I{C}_{50}$ (e.g., $I_T=0.01$ μM), estimation of $K_{ic}$ and $K_{iu}$ was imprecise, as indicated by the low-error markers at $K_{ic}=0.041$ μM, $K_{iu}=0.041$ μM (circle), $K_{ic}=0.041$ μM, $K_{iu}=100$ μM (triangle), and $K_{ic}=100$ μM, $K_{iu}=0.041$ μM (rectangle). However, precision became comparable to the canonical condition as $I_T$ increased through $I_T\approx I{C}_{50}$ (e.g., $I_T=0.05$ μM) to $I_T>I{C}_{50}$ (e.g., $I_T=0.5$ μM). Regularization enhanced precision to match the canonical condition. c, d Estimated parameters (dots and asterisks) and 95% confidence intervals (error bars). Utilizing a single $I_T$ lower than $I{C}_{50}$ led to inaccurate and imprecise estimations for $K_{ic}$ and $K_{iu}$ (i.e., out of the 1.5-fold range (dotted lines) of the estimated parameters (triangle)). On the other hand, the 50-BOA (e.g., $I_T=0.5$ μM; diamonds) allows for precise and accurate estimation of both parameters comparable to the canonical condition (c, d). e The 50-BOA (diamonds) requires only one seventh of the experiments required for the canonical condition (triangle) with comparable precision and accuracy.

Fig. 5. The 50-BOA allows for precise and accurate estimation for experimental competitive inhibition data.

a The normalized initial velocities ($\frac{V_0}{V_{\max }}$; $V_{\max }=11.77$ μM/min⁻¹$/\mathrm{mg}$ $\mathrm{protein}$) of chlorzoxazone with its inhibitor ethambutol ($\mathrm{I}{\mathrm{C}}_{50}=4$ μM when $S_T=50$ μM$;$ $K_M=39.1$ μM) were measured for various chlorzoxazone ($S_T=12.5$, $25$, $50$, $75$, and $100$ μM) and ethambutol ($I_T=0$, $0.41$, $1.23$, $3.7$, and $11.1$ μM) concentration combinations and fitted with Eq. 1 (solid lines). b Heatmaps of the total errors without (upper row) and with (lower row) regularization for entire $I_T$ or single $I_T$ setups. Using the entire $I_T$ (i.e., canonical condition) led to precise $K_{ic}$ estimation, while $K_{iu}$ estimation remained imprecise in a wide range even with regularization, indicating competitive inhibition. For single $I_T$ setups, the precision of the $K_{ic}$ estimation became comparable to the canonical condition, as $I_T$ increased from $I_T<\mathrm{I}{\mathrm{C}}_{50}$ (e.g., $I_T=0.41$ μM) through $I_T\approx \mathrm{I}{\mathrm{C}}_{50}$ (e.g., $I_T=3.7$ μM) to $I_T>\mathrm{I}{\mathrm{C}}_{50}$ (e.g., $I_T=11.1$ μM). c The $K_{ic}$ values (dots) with 95% confidence intervals (error bars) estimated with regularization. Using a single $I_T$ lower than $\mathrm{I}{\mathrm{C}}_{50}$ led to inaccurate or imprecise estimation (i.e., out of the 1.5-fold range (dotted lines) of the estimated $K_{ic}$ (triangle)), while the 50-BOA (diamond) enables precise and accurate estimation comparable to the canonical condition. d The asymptotic distribution of estimated $K_{iu}$ (see “Methods” for details) from the canonical condition is located at a substantially high value ($\approx {10}^{16}$ μM; dotted rectangle), reflecting the competitive inhibition type. Using a single $I_T$ lower than $\mathrm{I}{\mathrm{C}}_{50}$ led to an unexpected mode near $10$ μM (dashed rectangle), indicating potential misclassification of the inhibition type as mixed. Conversely, the 50-BOA provides $K_{iu}$ distribution consistent with the canonical condition, avoiding such misclassification. e The 50-BOA (diamond) requires only one fifth of the experiments required for the canonical condition (triangle) with comparable precision and accuracy.