Kinetic Modeling of Covalent Inhibition: Effects of Rapidly Fluctuating Intermediate States

Abstract

There is increasing interest in the discovery of small-molecule inhibitors that form covalent bonds with their targets for therapeutic applications. Nevertheless, identifying clear rational design principles remains challenging because the action of these molecules cannot be understood as common noncovalent inhibitors. Conventional kinetic models often reduce the binding of covalent inhibitors to a two-step irreversible process, overlooking rapid complex dynamics of the associated unlinked inhibitor before the formation of the covalent bond with its target. In the present analysis, we expand the intermediate state into two conformations-reactive (E.I) and nonreactive (E..I). To illustrate the consequences of such simplification, the expanded kinetic model can be reduced to an effective two-step scheme expressed in terms of the equilibrium probability of the unlinked inhibitor to form either conformation. A mass-action-based numerical workflow is implemented to simulate time-dependent kinetics, overcoming the common limitations of empirical models. The numerical workflow helps relate microscopic states observed in molecular dynamics simulations to macroscopic observables like ${\text{EC}}_{50}$ and the apparent rate of covalent inhibition, showing the impact of transient intermediates on dissociation rates and potency. The proposed framework refines the interpretation of dose-response data, aiding medicinal chemists in optimizing covalent inhibitors and provides a quantitative platform for relating molecular conformational distributions to empirical parameters.

Figure 1:

Simulated time courses of the total occupancy from Scheme 2 (bottom) and its noncovalent version from Scheme 4 (top) for different values of $p_{\mathrm{r}}$ with $[\mathrm{E}{]}_0=1\text{nM}$, $[\mathrm{I}{]}_0=1\mu \mathrm{M}$, $k_{\text{on}}=1\times {10}^{-2}\mu {\mathrm{M}}^{-1}{\mathrm{s}}^{-1}$, $k_{\text{off}}=1\times {10}^{-2}{\mathrm{s}}^{-1}$, and $k_{\text{inact}}=1\times {10}^{-4}{\mathrm{s}}^{-1}$. The occupancy of a one-step scheme is also included (black-dashed) to illustrate the contribution of binding affinity, $k_{\text{off}}/k_{\text{on}}$.

Figure 2:

Noncovalent residence time $\left(\tau \right)$ of a bound ligand as a function of $k_{\text{off}}$ and $p_{\mathrm{r}}.[\mathrm{E}{]}_0=1\text{nM}$, $[\mathrm{I}{]}_0=1\mu \mathrm{M}$, and $k_{\text{on}}=1\times {10}^{-2}\mu {\mathrm{M}}^{-1}{\mathrm{s}}^{-1}$. The residence times calculated analytically from eq 8 or numerically from simulations of Scheme 4 are the same.

Figure 3:

Simulated time-responses of covalent occupancy $\left(\right[\mathrm{I}{]}_0=1\mu \mathrm{M})$ (top) and dose-rate (bottom) data for different values of $p_{\mathrm{r}}$ with $[\mathrm{E}{]}_0=1\text{nM}$, $k_{\text{on}}=1\times {10}^{-2}\mu {\mathrm{M}}^{-1}{\mathrm{s}}^{-1}$, $k_{\text{off}}=1\times {10}^{-2}{\mathrm{s}}^{-1}$, and $k_{\text{inact}}=1\times {10}^{-4}{\mathrm{s}}^{-1}$. The ODEs of Scheme 3 were solved numerically for the time-responses (top), and the fraction of the covalent complex was fit for $k_{\text{obs}}$ with eq 9 (bottom).

Figure 4:

Comparisons between eq 16 (darkened dashed line) and ODE-based numerical simulations of Scheme 3 (solid line) for three representative cases. In Case 2, the parameters are $t_{\text{DR}}=7.1\mathrm{h}$, $[\mathrm{E}{]}_0=1\text{nM}$, $k_{\text{on}}=1\times {10}^{-2}\mu {\mathrm{M}}^{-1}{\mathrm{s}}^{-1}$, $k_{\text{off}}^{\text{app}}=5\times {10}^{-3}{\mathrm{s}}^{-1}$, and $k_{\text{inact}}^{\text{app}}=5\times {10}^{-5}{\mathrm{s}}^{-1}$. The altered parameters for Case 1 are $k_{\text{on}}=5\times {10}^{-4}\mu {\mathrm{M}}^{-1}{\mathrm{s}}^{-1}$ and $k_{\text{inact}}^{\text{app}}=5\times {10}^{-3}{\mathrm{s}}^{-1}$. The altered parameters for Case 3 are $[\mathrm{E}{]}_0=1\mu \mathrm{M}$, $k_{\text{on}}=5\times {10}^{-4}\mu {\mathrm{M}}^{-1}{\mathrm{s}}^{-1}$, and $k_{\text{inact}}^{\text{app}}=5\times {10}^{-3}{\mathrm{s}}^{-1}$. The fitted values of $n$ for cases 1, 2, and 3 are 1.47, 1.03, and 1.11, respectively.

Figure 5:

The variation of dose-response data (simulated by numerically solving the ODEs of Scheme 3) as a function of $k_{\text{off}}$ (left column), $k_{\text{inact}}$ (middle column), and $p_{\mathrm{r}}$ (right column) when $t_{\text{DR}}$ is 0.7 (top row), 7.1 (middle row), and 71.1 (bottom row) hours. The default parameters are $[\mathrm{E}{]}_0=1\text{nM}$, $k_{\text{on}}=1\times {10}^{-2}\mu {\mathrm{M}}^{-1}{\mathrm{s}}^{-1}$, $k_{\text{off}}=1\times {10}^{-2}{\mathrm{s}}^{-1}$, $k_{\text{inact}}=1\times {10}^{-4}{\mathrm{s}}^{-1}$, and $p_{\mathrm{r}}=0.5$.