Saving Ligand Efficiency?

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I’ll be concluding the series of posts on ligand efficiency metrics (LEMs) here so it’s a good point at which to remind you that the open access for the article on which these posts are based is likely to stop on September 14 (tomorrow). At the risk of sounding like one of those tedious twitter bores who thinks that you’re actually interested in their page load statistics, download now to avoid disappointment later. In this post, I’ll also be saying something about the implications of the LEM critique for FBDD and I’ve not said anything specific about FBDD in this blog for a long time.

LEMs have become so ingrained in the FBDD orthodoxy that to criticize them could be seen as fundamentally ‘anti-fragment’ and even heretical.  I still certainly believe that fragment-based approaches represent an effective (and efficient although not in the LEM sense) way to do drug discovery. At the same time, I think that it will get more difficult, even risky, to attempt to tout fragment-based approaches primarily on a basis that fragment hits are typically more ligand-efficient than higher molecular weight starting points for synthesis.  I do hope that the critique will at least reassure drug discovery scientists that it really is OK to ask questions (even of publications with eye-wateringly large numbers of citations) and show that the ‘experts’ are not always right (sometimes they’re not even wrong).

My view is that the LEM framework gets used as a crutch in FBDD so perhaps this is a good time for us to cast that crutch aside for a moment and remind ourselves why we were using fragment-based approaches before LEMs arrived on the scene.  Fragment-based approaches allow us to probe chemical space efficiently and it can be helpful to think in terms of information gained per compound assayed or per unit of synthetic effort consumed.  Molecular interactions lie at the core of pharmaceutical molecular design and small, structurally-prototypical probes allow these to be explored quantitatively while minimizing the confounding effects of multiple protein-ligand contacts.  Using the language of screening library design, we can say that fragments cover chemical space more effectively than larger species and can even conjecture that fragments allow that chemical space to be sampled at a more controllable resolution.  Something I’d like you to think about is the idea of minimal steric footprint which was mentioned in the fragment context in both LEM (fragment linking) and Correlation Inflation (achieving axial substitution) critiques.  This is also a good point to remind readers that not all design in drug discovery is about prediction.  For example, hypothesis-driven molecular design and statistical molecular design can be seen as frameworks for establishing structure-activity relationships (SARs) as efficiently as possible.

Despite criticizing the use of LEMs, I believe that we do need to manage risk factors such as molecular size and lipophilicity when optimizing lead series. It’s just I don’t think that the currently used LEMs provide a generally valid framework for doing this.  We often draw straight lines on plots of activity against risk factors in order to manage the latter.  For example, we might hypothesize that 10 nM potency will be sufficient for in vivo efficacy and so we could draw the pIC₅₀ = 8 line to identify the lowest molecular weight compounds above this line.   Alternatively we might try to construct line a line that represents the ‘leading edge’ (most potent compound for particular value of risk factor) of a plot of pIC₅₀ against ClogP.  When we draw these lines, we often make the implicit assumption that every point on the line is in some way equivalent. For example we might conjecture that points on the line represent compounds of equal ‘quality’.  We do this when we use LEMs and assume that compounds above the line are better than those below it.

Let’s take a look at ligand efficiency (LE) in this context and I’m going to define LE in terms of pIC₅₀ for the purposes of this discussion. We can think of LE in terms of a continuum of lines that intersect the activity axis at zero (where pIC₅₀  = 1 M).  At this point, I should stress that if the activities of a selection of compounds just happen to lie on any one of these lines then I’d be happy to treat those compounds as equivalent.  Now let’s suppose that you’ve drawn a line that intersects the activity axis at zero.  Now imagine that I draw a line that intersects the activity axis at 3 zero (where IC₅₀  = 1 mM) and, just to make things interesting, I’m going to make sure that my line has a different slope to your line.  There will be one point where we appear to agree (just like −40° on the Celsius and Fahrenheit temperature scales) but everywhere else we disagree. Who is right? Almost certainly neither of us is right (plenty of lines to choose from in that continuum) but in any case we simply don’t know because we’ve both made completely arbitrary decisions with respect to the points on the activity axis where we’ve chosen to anchor our respective lines.  Here’s a figure that will give you a bit more of an idea what I'm talking about.

However, there may just be a way out this sorry mess and the first thing that has to go is that arbitrary assumption that all IC₅₀ values tend to 1 M in the limit of zero molecular size.  Arbitrary assumptions beget arbitrary decisions regardless of how many grinning LeanSixSigma Master Black Belts your organization employs.  One way out of the mire is link efficiency with the response (slope) and agree that points lying on any straight line (of finite non-zero slope) when activity is plotted against risk factor represent compounds of equal efficiency.  Right now this is only the case when that line just happens to intersect the activity axis at a point where IC₅₀  = 1 M.  This is essentially what Mike Schultz was getting at in his series of (  1  | 2  |  3  ) critiques of LE even if he did get into a bit of a tangle by launching his blitzkrieg on a mathematical validity front. 

The next problem that we’ll need to deal with if we want to rescue LE is deciding where we make our lines intersect the activity axis.  When we calculate LE for a compound, we connect the point on the activity versus molecular size representing the compound to a point on the activity axis with a straight line.  Then we calculate the slope of this line to get LE but we still need to find an objective way to select the point on the activity axis if we are to save LE.  One way to do this is to use the available activity data to locate the point for you by fitting a straight line to the data although this won’t work if the correlation between risk factor and activity is very weak.  If you can’t use the data to locate the intercept then how confident do you feel about selecting an appropriate intercept yourself?  As an exercise, you might like to take a look at Figure 1 in this article (which was reviewed earlier this year at Practical Fragments) and ask if the data set would have allowed you to locate the intercept used in the analysis.

If you buy into the idea of using the data to locate the intercept then you’ll need to be thinking about whether it is valid to mix results from different assays.   It may be the same intercept is appropriate to all potency and affinity assays but the validity of this assumption needs to be tested by analyzing real data before you go basing decisions on it.  If you get essentially the same intercept when you fit activity to molecular size for results from a number of different assays then you can justify aggregating the data. However, it is important to remember (as is the case with any data analysis procedure) that the burden of proof is on the person aggregating the data to show that it is actually valid to do so. 

By now hopefully you’ve seen the connection between this approach to repairing LE and the idea of using the residuals to measure the extent to which the activity of a compound beats the trend in the data.  In each case, we start by fitting a straight line to the data without constraining either the slope or the intercept.  In one case we first subtract the value of this intercept from pIC₅₀ before calculating LE from the difference in the normal manner and the resulting metric can be regarded as measuring the extent to which activity beats the trend in the data.  The residuals come directly from the process and there is no need to create new variables, such as the difference between from pIC₅₀ and the intercept, prior to analysis. The residuals have sign which means that you can easily see whether or not the activity of compound beats the trend in the data.  With residuals there is no scaling of uncertainty in assay measurement by molecular size (as is the case with LE).  Finally, residuals can still be used if the trend in the data is non-linear (you just need to fit a curve instead of a straight line).  I have argued the case for using residuals to quantify extent to which activity beats the trend in the data but you can also generate a modified LEM from your fit of the activity to molecular size (or whatever property by which you think activity should be scaled by).

It’s been a longer-winded post than I’d intended to write and this is a good point at which to wrap up.  I could write the analogous post about LipE by substituting ‘slope’ for ‘intercept’ but I won’t because that would be tedious for all of us.  I have argued that if you honestly want to normalize activity by risk factor then you should be using trends actually observed in the data rather than making assumptions that self-appointed 'experts' tell you to make.   This means thinking (hopefully that's not too much to ask although sometimes I fear that it is) about what questions you'd like to ask and analyzing the data in a manner that is relevant to those questions. 

That said, I rest my case.