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 pIC

_{50}= 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 pIC_{50}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 pIC

_{50}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 pIC_{50}= 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 IC_{50}= 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 IC

_{50}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 IC_{50}= 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 pIC

_{50 }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 pIC_{50}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.