Tweetorial: Irreversible Inhibitors, Part Deux
6 January 2021
This one started as a short response to a paper that Bob the Grumpy Med Chemist sent my way via Pfizer. This paper argued that, under the right conditions, it was kosher to use a time-dependent IC50 to rank order irreversible inhibitor compounds for SAR rather than the more tedious measurement of k_inact/KI.
It turns out that quite a bit of math — and an understanding of the underlying assumptions — is needed to assess the validity of the Pfizer paper’s argument. I had to start writing things down, doing algebra, even (gasp!) doing a bit of calculus, just to wrap my own brain around it all. By the time I had written enough to satisfy myself that I understood it, I figured it was probably worth sharing.
The TL;DR is that the Pfizer argument is valid provided that the time point and assay conditions are judiciously chosen. But be damn sure you choose wisely, because those conditions are not necessarily “typical”. For the longer version, read onward!
Bob @med_chemist sent me this orthodoxy-challenging paper from Pfizer over the weekend and promptly ruined most of my evenings this week reading it and several predecessor papers more closely. Breakdown follows. 2/
sciencedirect.com/science/articl…
For review, here’s the original tweetorial, where I made a case (as many others have made) for k_inact/KI ratios to characterize irreversible inhibitors rather than IC50s, which are time-dependent for irreversible inhibitors. 3/
Keith Hornberger @KRHornberger
Two key additional references up front. First, here’s a 2000 paper from Maurer et al. (who is also on the 2021 Pfizer paper) which has the mathematical derivations that the work in the 2021 paper rests on. We’ll be referring to this one quite a bit. 4/
Second, here is, to my knowledge, what is considered the gold standard paper on the relationship between time-dependent IC50s, k_inact, and KI from Krippendorff et al. We’ll be diving into this one too. 5/
(Aside: I actually referenced the Krippendorff et al. paper back in the original tweetorial, with a note that you could go there if you were a glutton for mathematical punishment. So I guess we’re going there.) 6/
Keith Hornberger @KRHornberger
Strong advance warning: this is going to get very math-y very fast. Some basic integral calculus is unavoidable to really get into the meat of this one. If you’d rather be spared most of that horror and want to take my word for it, skip on down to the TL;DR in post 22. 7/
So: both the Pfizer and Krippendorff papers make the point that one can extract k_inact and KI by measuring time-dependent IC50s, and derive math for doing so. Pfizer goes further and says you can do this with a single properly-chosen IC50(t) rather than a time series. 8/
Pfizer thus argues that, with assumptions, this single IC50(t) can be used to rank-order compounds without the labor of a typical k_inact and KI determination. They show examples where a time-dependent IC50 correlates nicely with an independently-measured k_inact/KI. 9/
But… when one looks at the equations used, and assumptions made, these two interpretations are not the same. So what happened? Math time! 10/
Things start out the same. Both papers derive identical relationships for the differential equation governing inactivation of the enzyme in the presence of both substrate and an irreversible inhibitor. I’ll spare you how they got to these; it’s fairly straightforward. 11/
![Maurer: d[E]f/dt = -[E]f[I]*k_inact/([I]+KI(1+([S]/Km)))](https://substackcdn.com/image/fetch/w_600,c_limit,f_webp,q_auto:good,fl_progressive:steep/https%3A%2F%2Fpbs.substack.com%2Fmedia%2FFIatSaKWUAAHZls.png)
Next, both groups integrate that differential equation over time to derive an equation for the concentration of functionally active enzyme remaining (E_cat(t) or [E]f) as a function of time. Despite different notation, these equations are still identical. 12/
![Maurer: [E]f=[E]f0*e^-([I]/[I]+KI(1+[S]/Km))*k_inact*t)](https://substackcdn.com/image/fetch/w_600,c_limit,f_webp,q_auto:good,fl_progressive:steep/https%3A%2F%2Fpbs.substack.com%2Fmedia%2FFIatiu-WUAENdcQ.png)
Next, both groups substitute E(t) into the differential equation for product formation. This is the reaction velocity at a time point t, accounting for the depletion of active enzyme due to the inactivation reaction. Still identical. 13/
max*(e^-([I]/[I]+KI(1+[S]/Km))*k_inact*t)*S/[S]+Km(1+[I]/KI)](https://substackcdn.com/image/fetch/w_600,c_limit,f_webp,q_auto:good,fl_progressive:steep/https%3A%2F%2Fpbs.substack.com%2Fmedia%2FFIatvIKWYAUbGnE.png)
Now it gets tricky. In the Maurer interpretation, they follow the Cheng-Prusoff assumption that when [I] = IC50, then V(t) = Vo / 2. By setting up that equivalence, the equation can then be solved for IC50(t). Sparing you a great deal of algebra, here’s the answer. 14/
If one is then willing to make the further simplifying assumption that the inactivation reaction has progressed far enough, IC50(t) << IC50(0). Since IC50(0) (at t=0 when no inactivation has occurred) follows reversible Cheng-Prusoff, IC50(0) = Ki*(1 + [S]/Km). 15/
This means we can assume IC50(t) approaches zero everywhere in an equation where it’s additive or a ratio with IC50(0) or Ki*(1 + [S]/Km), which allows for a dramatic simplification of the IC50(t) equation by comparison to the IC50(0) as follows. (Skipping steps again.) 16/
![IC50(t) = (KI/k_inact*t)(1+[S]/Km)(0.693)](https://substackcdn.com/image/fetch/w_600,c_limit,f_webp,q_auto:good,fl_progressive:steep/https%3A%2F%2Fpbs.substack.com%2Fmedia%2FFIauOAWWQAEbmoQ.png)
Key to this interpretation is that the IC50 is determined by the *reaction velocity* being half-maximal at a point in time, which means it would have to be measured from time point t forward with no further interference from inactivation of the enzyme. 17/
Krippendorff goes in a different direction. Under most experimental conditions, what’s measured is cumulative product formation since the beginning of the reaction. The IC50(t) is the inhibitor concentration that gives half the *product formation* of no inhibitor at time t. 18/
For reversible inhibitors, the interpretation of product formation vs. reaction velocity is ~equivalent. OG Cheng-Prusoff uses velocities. For irreversible inhibitors, however, reaction velocity is not constant. It decreases with time due to enzyme inactivation. 19/
Krippendorff thus integrates the velocity equation to solve for product concentration as a function of time. They then set up a Cheng-Prusoff relationship defining the IC50 as [P](t) = [P](0) / 2, which gives a much gnarlier equation. Again skipping much algebra. 20/
Indeed, Krippendorff has some criticism of the Maurer interpretation, showing it can generate some nonsensical IC50 predictions and says a “characterization based solely on reaction velocities is questionable.” 21/
In the end, I think this comes down to: a) how do you define an IC50: product formation or reaction velocity? This matters because they are not equivalent for irreversible inhibitors; b) how is the experiment to measure the IC50 set up? 22/
The Pfizer approach is potentially nice because it allows for measuring a single IC50 as a proxy for k_inact/KI, which is less time- and labor-intensive than measuring IC50s at many times as in the Krippendorff approach. 23/
But the Pfizer approach must be done under conditions where one is measuring reaction velocity from the time point of interest forward, and at a time where there’s little additional interference from further covalent inactivation of the enzyme. 24/
The Krippendorff approach seems more general and more compatible with usual experimental conditions of measuring product formation cumulatively over time. But it’s experimentally more resource-intensive, and in pharma, where time is money, that might matter. 25/
As a chemist doing optimization though, am I really happy to just drive SAR on the single time-dependent IC50 as Pfizer suggests? That feels good, I’m sure, to just have one number to wrangle with, but it’s standing in for two parameters, k_inact and KI. 26/
When I’m tracking SAR and my compound is getting “better”, I’d generally like to know why. I can imagine an IC50 becoming quite a head-scratcher when you make structural changes that influence both k_inact and KI, perhaps in opposite (canceling out?) directions. 27/
