bioRxiv preprint doi: https://doi.org/10.1101/2020.06.25.171207; this version posted June 27, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC 4.0 International license. A two-point IC50 method for evaluating the biochemical potency of irreversible enzyme inhibitors Petr Kuzmicˇ BioKin Ltd., Watertown, Massachusetts, USA http://www.biokin.com Abstract Irreversible (covalent) enzyme inhibitors cannot be unambiguously ranked for biochemical potency by using IC50 values determined at a single point in time, because the same IC50 value could originate either from relatively low initial binding affinity accompanied by high chemical reactivity, or the other way around. To disambiguate the potency ranking of covalent inhibitors, here we describe a data-analytic procedure relying on two separate IC50 values, determined at two different reaction times. In the case of covalent inhibitors following the two-step kinetic mechanism E + I E·I ! EI, the two IC50 values alone can be used to estimate both the inhibition constant (Ki) as a measure of binding affinity and the inactivation rate constant (kinact) as a measure of chemical reactivity. In the case of covalent inhibitors following the one-step kinetic mechanism E + I ! EI, a simple algebraic formula can be used to estimate the covalent efficiency constant (kinact=Ki) from a single experimental value of IC50. The two simplifying assumptions underlying the method are (1) zero inhibitor depletion, which implies that the inhibitor concentrations are always significantly higher than the enzyme concentration; and (2) constant reaction rate in the uninhibited control assay. The newly proposed method is validated by using a simulation study involving 64 irreversible inhibitors with covalent efficiency constants spanning seven orders of magnitude. Key words: enzyme kinetics; inhibition; irreversible inhibition; covalent inhibition; mathematical model; algebraic model 1. Introduction reporting the I50 for an irreversible inhibitor without also spec- ifying the corresponding reaction time is essentially meaning- Many medicines currently in use to treat various human dis- less. An even more serious conceptual problem is that the same eases and symptoms are enzyme inhibitors. Furthermore, many value of I50 could originate either from relatively high initial important drugs and drug candidates are irreversible covalent binding affinity (low Ki) and relatively low chemical reactiv- inhibitors [1–4], which express their pharmacological effect by ity (low kinact), or the other way around. Therefore, two or more forming a permanent chemical bond with the protein target. inhibitors with disparate molecular properties could easily man- Probably the most well known representative of this class is ifest as having “the same” biochemical potency if the I50 assay acetylsalicylate, or Aspirin, an irreversible covalent inhibitor of is conducted at a single point in time. cyclooxygenase. Evaluating the biochemical potency of irre- This report presents a data-analytic procedure that relies on versible inhibitors in the process of pre-clinical drug discovery two separate I50 determinations, conducted at two different re- is exceedingly challenging. Even the task of simply arranging action times. If a given inhibitor follows a stepwise mecha- a list of potential drug candidates in order of their biochemi- nism of inhibition, involving a kinetically detectable noncova- cal potency presents a serious obstacle. The main challenge is lent intermediate, we show that it is possible to estimate both that the overall biochemical potency of irreversible enzyme in- Ki and kinact from the two time-dependent I50 measurements hibitors has two distinct components, namely, binding affinity alone. Many highly potent covalent inhibitors display a one- measured by the inhibition constant (Ki) and chemical reactiv- step kinetic mechanism, apparently without the involvement of ity measured by the inactivation rate constant (kinact). a clearly detectable noncovalent intermediate [6]. In those cases Medicinal chemists and pharmacologists involved in drug it is in principle impossible to measure the Ki and kinact sepa- discovery are accustomed to expressing the biochemical po- rately, but the data-analytic method described here allows for tency of enzyme inhibitors primarily in terms of the I [5]1. 50 the determination of the covalent efficiency constant keff, also However, in the case of irreversible inhibitors, the I by defi- 50 known as kinact=Ki, from any single measurement of I50. The nition decreases over time, until at asymptotically infinite time proposed method is validated by using a simulation study in- it approaches one half of the enzyme concentration. Therefore, volving 64 computer-generated inhibitors with molecular prop- erties (Ki, kinact, and kinact=Ki) spanning at least six orders of 1 magnitude. Throughout this manuscript, the conventionally used notation IC50 is ab- breviated as I50. BioKin Technical Note TN-2020-03 Revision 1.14 :: 24 June 2020 bioRxiv preprint doi: https://doi.org/10.1101/2020.06.25.171207; this version posted June 27, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC 4.0 International license. 2. Methods 2.2.2. Model discrimination analysis On the assumption that the enzyme assay proceeds kinet- This section describes the theoretical and mathematical meth- ically via the one-step inhibition mechanism C, the apparent ods that were used in heuristic simulations described in this re- ∗ second-order rate constant k1 is by definition invariant with re- port. All computations were performed by using the software spect to time, according to Eqn (1). Thus, in the idealized case package DynaFit [7, 8]. ∗(1) ∗(2) of zero experimental error, two values k1 and k1 determined at two different stopping times t(1) and t(2) should be exactly 2.1. Kinetic mechanisms of irreversible inhibition identical. However, in the realistic case of non-zero experi- In this report we will consider in various contexts the kinet- ∗ mental error, the two k1 values will always be ever so slightly ics mechanisms of substrate catalysis and irreversible inhibition different even if the one-step kinetic mechanism C is actually depicted in Figure 1. For further details see ref. [9]. ∗(1) ∗(2) operating. In order to decide whether or not k1 and k1 are sufficiently similar to warrant the acceptance of the one-step k1s k2s kinetic model, we use the geometric standard deviation defined E + S E•S E + P by Eqn (4), where µg is the geometric mean defined by Eqn (3). k-1s q µ = ∗(1) ∗(2) g k1 k1 (3) k1 k2 "A": E + I E•I EI tuv 0 ∗ 12 0 ∗ 12 k-1 B (1) C B (1) C B k1 C B k1 C σg = exp @Bln AC + @Bln AC (4) µg µg Ki k2 "B": E + I E•I EI The maximum acceptable value of σg will depend on the Ki = k-1 / k1 experimental situation. In the simulation study reported below, where the pseudo-random noise was equal to one percent of the σ < k1 maximum signal (e.g., fluorescence value), we found that g "C": E + I EI 1:25 was producing satisfactory results. Note that the geometric standard deviation σg is a dimensionless quantity measuring the Figure 1: Kinetic mechanisms of substrate catalysis (top) and “X-fold variation” associated with a group of numerical values. covalent inhibition (mechanisms A – C). σ < : ∗(1) In our situation, g 1 25 means that the two values k1 and ∗(2) k1 in a statistical sense differ by less than a factor of 1.25, or roughly by 25 percent in either direction (lower or higher). 2.2. Mathematical models ∗ ∗ 2.2.3. Determination of kinact and K from two values of I50 2.2.1. Determination of k1 from a single measurement of I50 i On the assumption that the enzyme assay proceeds kinet- On the assumption that the enzyme assay proceeds kinet- ically via the one-step inhibition mechanism C, the apparent ically via the two-step inhibition mechanism B, the apparent ∗ ∗ = = + = inhibition constant K = Ki (1 + [S]0=KM) can be computed second-order rate constant k1 k1 (1 [S]0 KM) can be com- i directly from any two measurements of I by using Eqn (5), puted directly from a single measurement of I50 by using Eqn 50 where t(1) < t(2) are the two reaction times used to determine (1), where t50 is the reaction time used to determine the ex- 50 50 (1) (2) perimental value of I50 and c is a constant depending on the I50 and I50 , respectively, and a is an empirical constant (see dimensions used to express time and concentration. When time below). is expressed in seconds and concentrations in micromoles per ( ) liter, c = 1:5936. The requisite algebraic derivation is shown in 1=a 1 − t(1)=t(2) the Supporting Information. Eqn (2) defines the second-order ∗ = 50 50 Ki ( ) = (5) covalent efficiency constant in the context of the one-step mech- (1) (2) 1 a 1 t =t anism C. − 50 50 (1) (2) I50 I50 2 0 1 3 c ∗ k∗ = (1) 1 6 B K C 7 1 k = 6a B i − C + b7 I50 t50 inact (2) exp 4 ln @ (2) 1A 5 (6) ! t50 I50 ∗ [S]0 ! k ff = k 1 + (2) e 1 K kinact [S]0 M keff = ∗ 1 + (7) Ki KM ∗ Once Ki is determined from Eqn (5), kinact can be computed (2) from it and from I50 by using Eqn (6), where a and b are em- 2 bioRxiv preprint doi: https://doi.org/10.1101/2020.06.25.171207; this version posted June 27, 2020.