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Ligand Efficiency: A Response to Shultz

I’d like to throw a few more logs on the ligand efficiency fire. Chuck Reynolds of J&J (author of several papers on the subject, as aficionados know) left a comment to an earlier post that I think needs some wider exposure. I’ve added links to the references:

An article by Shultz was highlighted earlier in this blog and is mentioned again in this post on a recent review of Ligand Efficiency. Shultz’s criticism of LE, and indeed drug discovery “metrics” in general hinges on: (1) a discussion about the psychology of various metrics on scientists’ thinking, (2) an assertion that the original definition of ligand efficiency, DeltaG/HA, is somehow flawed mathematically, and (3) counter examples where large ligands have been successfully brought to the clinic.
I will abstain from addressing the first point. With regard to the second, the argument that there is some mathematical rule that precludes dividing a logarithmic quantity by an integer is wrong. LE is simply a ratio of potency per atom. The fact that a log is involved in computing DeltaG, pKi, etc. is immaterial. He makes a more credible point that LE itself is on average non-linear with respect to large differences in HA count. But this is hardly a new observation, since exactly this trend has been discussed in detail by previous published studies (here, here, here, and here). It is, of course, true that if one goes to very low numbers of heavy atoms the classical definition of LE gets large, but as a practical matter medicinal chemists have little interest in extremely small fragments, and the mathematical catastrophe he warns us against only occurs when the number of heavy atoms goes to zero (with a zero in the denominator it makes no difference if there is a log in the numerator). Why would HA=0 ever be relevant to a med. chem. program? In any case a figure essentially equivalent to the prominently featured Figure 1a in the Shultz manuscript appears in all of the four papers listed above. You just need to know they exist.
With regard to the third argument, yes of course there are examples of drugs that defy one or more of the common guidelines (e.g MW). This seems to be a general problem of the community taking metrics and somehow turning them into “rules.” They are just helpful, hopefully, guideposts to be used as the situation and an organization’s appetite for risk dictate. One can only throw the concept of ligand efficiency out the window completely if you disagree with the general principle that it is better to design ligands where the atoms all, as much as possible, contribute to that molecule being a drug (e.g. potency, solubility, transport, tox, etc.). The fact that there are multiple LE schemes in the literature is just a natural consequence of ongoing efforts to refine, improve, and better apply a concept that most would agree is fundamental to successful drug discovery.

Well, as far as the math goes, dividing a log by an integer is not any sort of invalid operation. I believe that [log(x)]/y is the same as saying log(x to the one over y). That is, log(16) divided by 2 is the same as the log of 16 to the one-half power, or log(4). They both come out to about 0.602. Taking a BEI calculation as real chemistry example, a one-micromolar compound that weighs 250 would, by the usual definition, -log(Ki)/(MW/1000), have a BEI of 6/0.25, or 24. By the above rule, if you want to keep everything inside the log function, then say -log(0.0000001) divided by 0.25, that one-micromolar figure should be raised to the fourth power, then you take the log of the result (and flip the sign). One-millionth to the fourth power is one times ten to the minus twenty-fourth, so that gives you. . .24. No problem.
Shultz’s objection that LE is not linear per heavy atom, though, is certainly valid, as Reynolds notes above as well. You have to realize this and bear it in mind while you’re thinking about the topic. I think that one of the biggest problems with these metrics – and here’s a point that both Reynolds and Shultz can agree on, I’ll bet – is that they’re tossed around too freely by people who would like to use them as a substitute for thought in the first place.

19 comments on “Ligand Efficiency: A Response to Shultz”

  1. bhip says:

    Can’t overstate the impact of “the psychology of various metrics on scientists’ thinking”. The brain dead metrics that are often/usually applied in Phama drug discovery tend to “dissuade” biologists & chemists from taking chances. God help you if you propose or (worse) act on an idea that is outside the safe confines of dogma (or even dogma-lite).

  2. Anonymous says:

    ((log x)/y) = log(x^(-y)) = log ((1/x)^y).

  3. Christian R says:

    I have to agree with Schultz on point 2. The value of log(Ki) will depend on the units you use for Ki; for instance it will change by +3 if you change units from M to mM. This does not matter as long as you just rank according to log(Ki), but when you divide with a number (HA) that varies between compounds, your ranking of compounds will (in some cases) depend on which unit you use for measuring Ki. That is not a mathematically sound construct.

  4. bhip says:

    Sorry- just to follow up. In case it wasn’t clear, my reference to “brain dead metrics” pertains to those applied to rate some poor schmucks’ performance at years end …..

  5. Wavefunction says:

    Metrics are crutches used by imperfect human minds to compensate for their failure to understand complex biological systems. That’s how it will always be.

  6. Anonymous says:

    #2: Sorry, you’ve mistaken negation (taking a negative) and inversion (taking a reciprocal). Your equations are incorrect, but Derek is right.
    (log x)/y = (1/y)*(log x) = log(x^(1/y))

  7. Anonymous says:

    I was just about to say, #2’s equation works only if y is imaginary…

  8. Anonymous says:

    But in any case, you can describe a correlation with any mathematical function you like, as long as the residual is small, and you don’t try to extrapolate too much beyond the existing data. Whether or not the function actually corresponds to the underlying physics is a different matter, but also quite irrelevant for the purposes of prediction, as long as the fit is good and based on enough data.

  9. Anonymous says:

    A bit of a math lesson:
    Here’s the four main log transformations-
    x^(log base x of y) = y
    log(x) + log(y) = log(x*y)
    log(y)/log(x) = log base x of y
    x * log(y) = log(y^x)
    The main thing you cannot do is add two logs of unlike bases. You also need to watch out is to see whether the default log(x) has 10 or e as the base.

  10. Hap says:

    If differences between quantities are important, than relating the quantity on a linear axis makes sense (because differences between the quantities are represented the same no matter what the size of the quantity is).
    With binding constants/IC50 values, though, people generally aren’t interested in differences between the quantities but in ratios between them. In those cases, you want equivalent ratios to be shown equivalently whatever the magnitude of the underlying quantities, rather than equivalent differences being shown equivalently. Log scales allow one to show equivalent ratios equivalently.

  11. Anonymous says:

    “The main thing you cannot do is add two logs of unlike bases.”
    Sure you can! You just need to take care to include the appropriate “conversion factor”.

  12. dlib says:

    Sadly, if you use the binding constant in won’t prevent the natural inclination of med chemists to rigidify their molecules….dH/HA in conjunction might help a bit.

  13. anon the II says:

    @ #8
    No, just no. When parameter choices exceed data points, almost always true in MedChem, correlation is easy and prediction is futile.

  14. Anonymous says:

    @13: If you bothered to read the end of my post you would see that we are saying the same thing: “as long as the fit is good and based on enough data.” In other words, a good fit, but without over-fitting too few data with too many variables.

  15. gippgig says:

    One micromolar is .000001 M (one less 0).

  16. Anonymous says:

    Of course the log of the Kd or IC50 is really taken because it correlates linearly with ∆G which is obviously the correct frame of reference to analyse your data.

  17. phil says:

    How much more axe does Michael Shultz have left to grind? He should just publish one perspective with seven words: “Please think hard while designing a drug”

  18. Anon says:

    The criticism that Derek elevated in his post ignores another axis of Shultz’s argument which, in my mind, is the most important one: LipE (or LLE if you prefer) has been repeatedly and consistently correlated with successful optimization campaigns, whereas the same cannot be said of the other metrics. Those who would say that LE is useful in “hit-to-lead” type optimizations should refer back to another key Shultz point that such milestones are company-specific and therefore no universal guideline for when they should and shouldn’t be used can be proposed. In contrast, LipE has been shown to increase over the entire course of a successful optimization.
    For my own part as a practicing medicinal chemist, I have successfully used LipE to drive optimizations across multiple projects, chemotypes, and therapeutic areas. I believe in its utility because in my own experience it works consistently, whereas the other metrics do not.
    As many others have pointed out in the comments, no efficiency metric is a substitute for critical thinking. But if one wants to keep an eye on a metric, Shultz’s argument remains compelling that LipE is the right one and the others are not.

  19. shultz says:

    I’m afraid I might not have made the point very well so hopefully this clarifies.
    I agree with Reynolds that there is no rule that precludes dividing a log by an integer (I never asserted this) but to this point there are several results that we apparently all agree upon with regard to LE:
    1) At low (and non-zero) heavy atom integer numbers, the amplitude of LE becomes ridiculously large and independent of potency.
    2) At high (but still relevant) heavy atom integer numbers, the amplitude of LE becomes insignificant and independent of potency.
    3) A log/integer plot is nonlinear and only approximates linearity when the delta of the integers is very low. (restatement of Reynolds’ comment above).
    LE is valid for comparing molecules of identical size but invalid for comparing molecules of different sizes where the error increases with the delta of heavy atoms. Therefore, the validity and utility of LE are inversely related to one another because of this non-linearity!* A simple change in the formula (LE* = pIC50-#HA) eliminates all the size dependency, etc. This isn’t axe grinding, it is exactly what Reynold’s stated – ‘a natural consequence of ongoing efforts to refine, improve, and better apply a concept that most would agree is fundamental to successful drug discovery’. I think the previous versions of LE are flawed and invalid, I provided the rationale behind this assertion and a revised equation to mitigate these flaws. It could be further refined (as I don’t think it’s perfect) but it does, IMHO, have mathematical integrity.
    So while the facts are non-original (LE does not normalize size for potency) it is the interpretation of why LE is still used to normalize size for potency (Point 1 from Reynolds) that I was unable to find precedent.
    I guess the last observation is a reiteration of #18’s comments. If there is an assertion that the LE metric improves the probability of success, then those who advocate its use should provide statistical rather than anecdotal evidence.
    * The same is true for differences in potency but since this is a log scale the effective ranges are less significant but demonstrable. Since molecules are usually of different size and potency, this error becomes harder to see during practical use.

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