Archbishop Ussher's guide to efficient selection of development candidates

One piece of advice I gave in NoLE is that “drug designers should not automatically assume that conclusions drawn from analysis of large, structurally-diverse data sets are necessarily relevant to the specific drug design projects on which they are working” and the L2021 study that I’m reviewing in this post will give you a good idea of what I was getting at when I wrote that. I see a fair amount of relatively harmless “stamp collecting” in L2021 but there are also some rather less harmless errors of the type that you really shouldn’t be making if cheminformatics is your day job.  

I’ll start the review of L2021 with annotation of the abstract:

"Physicochemical descriptors commonly used to define ‘drug-likeness’ and ligand efficiency measures are assessed for their ability to differentiate marketed drugs from compounds reported to bind to their efficacious target or targets. [I would argue that differentiating an existing drug from existing compounds that bind to the same target is not something that medicinal chemists need to be able to do. It is also incorrect to describe efficiency metrics such as LE and LLE as physicochemical descriptors because they are derived from biological activity measurements such as binding affinity or potency.] Using ChEMBL version 26, a data set of 643 drugs acting on 271 targets was assembled, comprising 1104 drug−target pairs having ≥100 published compounds per target. Taking into account changes in their physicochemical properties over time, drugs are analyzed according to their target class, therapy area, and route of administration. Recent drugs, approved in 2010−2020, display no overall differences in molecular weight, lipophilicity, hydrogen bonding, or polar surface area from their target comparator compounds. Drugs are differentiated from target comparators by higher potency, ligand efficiency (LE), lipophilic ligand efficiency (LLE), and lower carboaromaticity. [I may be missing something but stating that drugs tend to differ in potency from non-drugs that hit the same targets does rather seem to be stating the obvious. The same point can also be made about efficiency metrics such as LE and LLE since these are derived, respectively, by scaling potency with respect to molecular size and offsetting potency with respect to lipophicity (LLE).] Overall, 96% of drugs have LE or LLE values, or both, greater than the median values of their target comparator compounds.” [What is the corresponding figure for potency?]

I must admit to never having been a fan of drug-likeness studies such as L2021 (when I first encountered analyses of time dependency of drug properties about 20 years ago I was left with an impression that some senior medicinal chemists had a bit too much time on their hands) and it is now ten years since the term "Ro5 envy" was introduced in a notorious JCAMD article. My view is that the data analysis presented in L2021 has minimal relevance to drug discovery so I’ll be saying rather less about the data analysis than I’d have done had J Med Chem asked me to review the study.

The L2021 study examines property differences between marketed drugs and compounds reported to bind to efficacious target(s) of each drug. Specifically, the property differences are quantified by difference between the value of the property for the drug and the median of the values of property for the target comparator compounds. If doing this then you really do need to account for the spread in the distribution if you’re going to interpret property differences like these (a large difference in values of a property for the drug and the median property for the target may simply reflect a wide spread in the property distribution for the target).  However, I would argue that a more sensible starting point for analysis like this would be to locate (e.g., as a percentile) the value of each drug property within the corresponding property distribution for the target comparator compounds.

Let’s take a look now at how the authors of L2021 suggest their study be used.  

“This study, like all those looking at marketed drug properties, is necessarily retrospective. Nevertheless, those small molecule drug properties that show consistent differentiation from their target compounds over time, namely, potency, ligand efficiencies (LE and LLE), and the aromatic ring count and lipophilicity of carboaromatic drugs, are those that are most likely to remain future-proof. Candidate drugs emerging from target-based discovery programs should ideally have one, or preferably both, of their LE and LLE values greater than the median value for all other compounds known to be acting at the target.”

I would argue that the L2021 study has absolutely no relevance whatsoever to the selection of compounds for development since the team will have data available that enables them to rule out the vast majority of the project compounds for nomination.  A discovery team nominating a compound for development will have achieved a number of challenging objectives (including potency against target and in one or more cell-based assays) and the likely response of team members to a suggestion that they calculate medians for LE and LLE for comparison with nomination candidate(s) is likely to be bemused eye-rolling. In general, a discovery team nominating a development candidate has access to a lot of unpublished potency measurements (which won’t be in ChEMBL) and it’s usually a safe assumption that the development candidate will be selected from the most potent compounds (LE and LLE values for these compounds are also likely to be above average). In the extremely unlikely event that the discovery team nominates a compound with LE or LLE values below the magic median values then you can be confident that the decision has been based on examination of measured data (consider the likelihood of the discovery team members acting on a suggestion that they should pick another compound with LE or LLE value above the magic median values because doing so will increase the probability of success in clinical development).   

As the start of the post, I did mention some errors that you don’t want to be making if cheminformatics is your day job and regular readers of this blog will have already guessed that I’m talking about ligand efficiency (LE). I should point out l that the problem is with the ligand efficiency metric and not the ligand efficiency concept which is both scientifically sound and useful, especially in fragment-based design where molecular size often increases significantly in the hit-to-lead phase. 

The problem with the LE metric is that perception of efficiency changes when you express affinity (or potency) using a different unit and this is shown clearly in Table 1 in NoLE. Expressing a quantity using a different unit doesn’t change the quantity so any change in perception is clearly physical nonsense. That’s why I appropriate a criticism (it’s not even wrong) usually attributed to Pauli when taking gratuitous pot shots at the LE metric.  The change in perception is also cheminformatic nonsense and that’s why it’s rather unwise to use the LE metric if cheminformatics is your day job. L2021 does cite NoLE but simply notes the LE metric’s “scientific basis and application have provoked a literature debate”.

The L2021 study asserts that “the absolute LE value of a drug candidate is less important” but the problem is that even differences in LE change when you express affinity (or potency) using a different concentration unit. This is shown in Table 2 in NoLE and the problem is that there is no objective way to select a particular concentration unit as ‘better’ than all the other concentration units.  To conclude, can we say that a medicinal chemistry leader’s choice of concentration unit (1 M) is any better (or any worse) than that of Archbishop Ussher (4.004 μM)?  

Why fragments?

Paramin panorama

Crystallographic fragment screens have been run recently against the main protease (at Diamond) and the Nsp3 macrodomain (at UCSF and Diamond) of SARS-Cov-2 and I thought that it might be of interest to take a closer look at why we screen fragments. Fragment-based lead discovery (FBLD) actually has origins in both crystallography [V1992 | A1996] and computational chemistry [M1991 | B1992 | E1994]. Measurement of affinity is important in fragment-to-lead work because it allows fragment-based structure-activity relationships to be established prior to structural elaboration. Affinity measurement is typically challenging when fragment binding has been detected using crystallography although affinity can be estimated by observation of the response of occupancy to concentration (the ∆G° value of −3.1 kcal/mol reported for binding of pyrazole to protein kinase B was derived in this manner).

Although fragment-based approaches to lead discovery are widely used, it is less clear why fragment-based lead discovery works as well as it appears to. While it has been stated that “fragment hits form high-quality interactions with the target”, the concept of interaction quality is not sufficiently well-defined to be useful in design. I ran a poll which asked about the strongest rationale for screening fragments.  The 65 votes were distributed as follows: ‘high ligand efficiency’ (23.1%), ‘enthalpy-driven binding’ (16.9%), ‘low molecular complexity’ (26.2%) and ‘God loves fragments’ (33.8%). I did not vote.

The belief is that fragments are especially ligand-efficient has many adherents in the drug discovery field and it has been asserted that “fragment hits typically possess high ‘ligand efficiency’ (binding affinity per heavy atom) and so are highly suitable for optimization into clinical candidates with good drug-like properties”. The fundamental problem with ligand efficiency (LE), as conventionally calculated, is that perception of efficiency varies with the arbitrary concentration unit in which affinity is expressed (have you ever wondered why Kd , Ki or IC50 has to be expressed in mole/litre for calculation of LE?). This would appear to be an rather undesirable characteristic for a design metric and LE evangelists might consider trying to explain why it’s not a problem rather than dismissing it as a “limitation” of the metric or trying to shift the burden of proof is onto the skeptics to show that the evangelists’ choice of concentration unit for calculation of LE is not useful.

The problems associated with the arbitrary nature of the concentration unit used to express affinity were first identified in 2009 and further discussed in 2014 and 2019. Specifically, it was noted that LE has a nontrivial dependency on the concentration,  C°, used to define the standard state. If you want to do solution thermodynamics with concentrations defined then you do need to specify a standard concentration. However, it is important to remember that the choice of standard concentration is necessarily arbitrary if the thermodynamic analysis is to be valid. If your conclusions change when you use a different definition of the standard state then you’ll no longer be doing thermodynamics and, as Pauli might have observed, you’ll not even be wrong. You probably don't know it, but when you use the LE metric, you’re making the sweeping assumption that all values of Kd, Ki and IC50 tend to a value of 1 M in the limit of zero molecular size. Recalling the conventional criticism of homeopathy, is there really a difference between a solute that is infinitely small and a solute that is infinitely dilute?

I think that’s enough flogging of inanimate equines for one blog post so let’s take a look at enthalpy-driven binding. My view of thermodynamic signature characterization in drug discovery is that it’s, in essence, a solution that’s desperately seeking a problem. In particular, there does not appear to be any physical basis for claims that the thermodynamic signature is a measure of interaction quality.  In case you’re thinking that I’m an unrepentant Luddite, I will concede that thermodynamic signatures could prove useful for validating physics-based models of molecular recognition and in, in specific cases, they may point to differences in binding mode within congeneric series. I should also stress that the modern isothermal calorimeter is an engineering marvel and I'd always want this option for label-free, affinity measurement in any project.

It is common to see statements in the thermodynamic signature literature to the effect that binding is ‘enthalpy-driven’ or ‘entropy-driven’ although it was noted in 2009 (coincidentally, in the same article that highlighted the nontrivial dependence of LE on C°) that these terms are not particularly meaningful. The problems start when you make comparisons between the numerical values of ∆H (which is independent of C°) and T∆S° (which depends on C°). If I’d presented such a comparison in physics class at high school (I was taught by the Holy Ghost Fathers in Port of Spain), I would have been caned with a ferocity reserved for those who’d dozed off in catechism class.  I’ll point you toward an article which asserts that, “when compared with many traditional druglike compounds, fragments bind more enthalpically to their protein targets”. I have a number of issues with this article although this is not the place for a comprehensive review (although I’ll probably pick it up in ‘The Nature of Lipophilic Efficiency’ when that gets written).

While I don’t believe that the authors have actually demonstrated that fragments bind more enthalpically than ligands of greater molecular size, I wouldn’t be surprised to discover that gains in affinity over the course of a fragment-to-lead (F2L) campaign had come more from entropy than enthalpy. First, the lost translation entropy (the component of ∆S° that endows it with its dependence on C°) is shared over greater number of intermolecular contacts for structurally-elaborated compounds and this article is relevant to the discussion. Second, I’d expect the entropy of any water molecule to increase when it is moved to bulk solvent from contact with molecular surface of ligand or target (regardless of polarity of the molecular surface at the point of contact). Nevertheless, this is something that you can test easily by examining the response of (∆H + T∆S°) to ∆G° (best to not to aggregate data for different targets and/or temperatures when analyzing isothermal titration calorimetry data in this manner). But even if F2L affinity gains were shown generally to come more from entropy than enthalpy, would that be a strong rationale for screening fragments?

This gets us onto molecular complexity and this article by Mike Hann and GSK colleagues should be considered essential reading for anybody thinking about selecting of compounds for screening. The Hann model is a conceptual framework for molecular complexity but it doesn’t provide much practical guidance as to how to measure complexity (this is not a criticism since the thought process should be more about frameworks and less about metrics). I don’t believe that it will prove possible to quantify molecular complexity in an objective manner that is useful for designing compound libraries (I will be delighted to be proven wrong on this point). The approach to handling molecular complexity that I’ve used in screening library design is to restrict extent of substitution (and other substructural features that can be considered to be associated with molecular complexity) and this is closer to ‘needle screening’ as described by Roche scientists in 2000 than to the Hann model.

Had I voted in the poll, ‘low molecular complexity’ would have got my vote.  Here’s what I said in NoLE (it’s got an entire section on fragment-based design and a practical suggestion for redefining ligand efficiency so that perception does not change with C°):

"I would argue that the rationale for screening fragments against targets of interest is actually based on two conjectures. First, chemical space can be covered most effectively by fragments because compounds of low molecular complexity [18, 21, 22] allow TIP [target interaction potential] to be explored [70,71,72,73,74] more efficiently and accurately. Second, a fragment that has been observed to bind to a target may be a better starting point for design than a higher affinity ligand whose greater molecular complexity prevents it from presenting molecular recognition elements to the target in an optimal manner."

To be fair, those who advocate the use of LE and thermodynamic signatures in fragment-based design do not deny the importance of molecular complexity. Let’s assume for the sake of argument that interaction quality can actually be defined and is quantified by the LE value and/or the thermodynamic signature for binding of compound to target. While these are massive assumptions, LE values and thermodynamic signatures are still effects rather than causes.

The last option for poll was ‘God loves fragments’ and more respondents (33.8%) voted for this than any of the first three options. I would interpret a vote for ‘God loves fragments’ in three ways. First, the respondent doesn’t consider any one of the first three options to be a stronger rationale for screening fragments than the other two. Second, the respondent doesn’t consider any of the first three options to be a valid rationale for screening fragments. Third, the respondent considers fragment-based approaches to have been over-sold.

This is a good place to wrap up. While I remain an enthusiast for fragment-based approaches to lead discovery, I do also believe that they have been somewhat oversold. The sensitivity of LE evangelists to criticism of their metric may stem from the use of LE to sell fragment-based methods to venture capitalists and, internally, to skeptical management. A shared (and serious) deficiency in the conventional ways in which LE and thermodynamic signature are quantified is that perception changes when the arbitrary concentration,  C°, that defines the standard state is changed. While there are ways in which this deficiency can be addressed for analysis, it is important that the deficiency be acknowledged if we are to move forward. Drug design is difficult and if we, as drug designers, embrace shaky science and flawed data analysis then those who fund our activities may conclude that the difficulties that we face are of our own making.     

Efficient trajectories

I'll examine an article entitled ‘Mapping the Efficiency and Physicochemical Trajectories of Successful Optimizations’ (YL2018) in this post and I should note that the article title reminded me that abseiling has been described as the second fastest way down the mountain. The orchids in Blanchisseuse have been particularly good this year and I’ll include some photos of them to break the text up a bit.

It’s been almost 22 years since the rule of 5 (Ro5) was published. While the Ro5 article highlighted molecular size and lipophilicity as pharmaceutical risk factors, the rule itself is actually of limited utility as a drug design tool. Some of the problems associated with excessive lipophilicity had actually been recognized (see Yalkowsky | Hansch) over a decade before the publication of Ro5 in 1997 and there’s also this article that had been published in the previous year. However, it was the emergence of high-throughput screening that can be regarded as the trigger for Ro5 which, in turn, dramatically raised awareness of the importance of physicochemical properties in drug design. The heavy citation and wide acceptance of Ro5 provided incentives for researchers to publish their own respective analyses of large (usually proprietary) data sets and this has been expressed more succinctly as “Ro5 envy”.

So let's take a look at YL2018 and the trajectories. I have to concede that ‘trajectory’ makes it all seem so physical and scientifically rigorous even though ‘path’ would be more appropriate (and easier to say after a few beers). As noted in ‘The nature of ligand efficiency’ (NoLE), I certainly believe that it is a good idea for medicinal chemistry teams to both plot potency (e.g. pIC50) against risk factors such as molecular size or lipophilicity for their project compounds and to analyze the relationships between potency and these quantities. However, it is far from clear that a medicinal chemistry team optimizing a specific structural series against a particular target would necessarily find the plots corresponding to optimization of other structural series against other targets to be especially relevant to their own project.

YL2018 claims that “the wider employment of efficiency metrics and lipophilicity control is evident in contemporary practice and the impact on quality demonstrable”. While I would agree that efficiency metrics are integral to the philatelic aspects of modern drug discovery, I don’t believe that YL2018 actually presents a single convincing example of efficiency metrics being used for decision making in a specific drug design project. I should also point out that each of the authors of YL2018 provided cannon fodder (LS2007 | HY2010 ) for the correlation inflation article and you might want to keep that in mind when you read the words “evident” and “demonstrable”. They also published 'Molecular Property Design: Does Everyone Get It?' back in 2015 and you may find this review of that seminal contribution to the drug design literature to be informative.

I reckon that it would actually be a lot more difficult to demonstrate that efficiency metrics were used meaningfully (i.e. for decision making rather than presentation at dog and pony shows) in projects than it would be to demonstrate that they were predictive of pharmaceutically relevant behavior of compounds. In NoLE, I stated:

"However, a depiction [6] of an optimization path for a project that has achieved a satisfactory endpoint is not direct evidence that consideration of molecular size or lipophilicity made a significant contribution toward achieving that endpoint. Furthermore, explicit consideration of lipophilicity and molecular size in design does not mean that efficiency metrics were actually used for this purpose. Design decisions in lead optimization are typically supported by assays for a range of properties such as solubility, permeability, metabolic stability and off-target activity as well as pharmacokinetic studies. This makes it difficult to assess the extent to which efficiency metrics have actually been used to make decisions in specific projects, especially given the proprietary nature of much project-related data."

YL2018 states, “Trajectory mapping, based on principles rather than rules, is useful in assessing quality and progress in optimizations while benchmarking against competitors and assessing property-dependent risks.” and, as a general point, you need to show you're on top of the physical chemistry if you're going write articles like this.

Ligand efficiency represents something of a liability for anybody claiming expertise in physical chemistry. The reason for this is that perception of efficiency depends on the unit that you use to express affinity and this is a serious issue (in the "not even wrong" category) that was highlighted in 2009 and 2014 before NoLE was published. While YL2018 acknowledges that criticisms of ligand efficiency have been made, you really need to say exactly why this dependence of perception is not a problem if you're going lecture about principles to readers of Journal of Medicinal Chemistry.

Ligand lipophilic efficiency (LLE) which is also known as ligand lipophilicity efficiency (LLE) and lipophilic efficiency (LipE) can be described as offset efficiency metric (lipophilicity is subtracted from potency). As such, perception of efficiency does not change when you use a different unit to express potency and, provided that ionization of ligand is insignificant, efficiency can be seen as a measure of the ease of transfer of ligand from octanol to its binding site. Here's a graphic that illustrates this:

LLE (LipE) measures ease of transfer of ligand from octanol to binding site

I'm not entirely convinced that the authors of YL2018 properly understood the difference between logP and logD. Even if they did, they needed to articulate the implications for drug design a lot more clearly than they have done. Here's an equation that expresses logD as a function of logP and the fraction of ligand in the neutral form at the experimental pH (assuming that only neutral forms of ligands partition into the octanol).

The equation highlights the problems that result from using logD (rather than logP) to define "compound quality". In essence the difficulty stems from the composite nature of logD which means that logD can be also be reduced by increasing the extent of ionization. While this is likely to result in increased aqueous solubility, it is much less likely that problems associated with binding to anti-targets will be addressed. Increasing the extent of ionization may also compromise permeability.    

YL2018 is clearly a long article and I'm going to focus on two of the ways in which the authors present values of efficiency metrics. The first of these is the "% better" statistic which is used to reference specific compounds (e.g. optimization endpoints) to sets of compounds (e.g. everything synthesized by project chemists). The statistic is calculated as the fraction of compounds in the set for which both LE and LLE values are greater than the corresponding values for the compound of interest. The smallest values of the "% better" statistic are considered to correspond to the most optimal compounds. The use of the "% better" statistic could be taken as indicating that absolute thresholds for LE and LLE are not useful for analyzing optimization trajectories..

The fundamental problem with analyzing data in this manner is that LE has a nontrivial dependence on the concentration unit in which affinity is expressed (this is shown in Table 1 and Fig. 1 in NoLE). One consequence of this nontrivial dependence is that both perception of efficiency and the "% better" statistic vary with the concentration unit used to express efficiency.

The second way that the authors of YL2018 present values of efficiency metrics is to plot LE against LLE and, as has already been noted, this is a particularly bone-headed way to analyze data. One problem is that the plot changes in a nontrivial manner if you express affinity in a different unit. This makes it difficult to explain to medicinal chemists why they need to convert the micromolar potencies from their project database to molar units in order for The Truth to be revealed. Another problem is that LE and LLE are both linear functions of pIC50 (or pKi) and that means that the appearance of the plot is heavily influenced by the (trivial) correlation of potency with itself.

A much better way to present the data is to plot LLE against number of non-hydrogen atoms (or any other measure of molecular size that you might prefer). In such a plot, expressing potency (or affinity) in a different unit simply shifts all points 'up' or 'down' to the same extent which means that you no longer have the problem that the appearance of plot changes when you change units. The other advantage of plotting the data in this manner is that there is no explicit correlation between the quantities being plotted. I have used a variant of this plot in NoLE (see Fig. 2b) to compare some fragment to lead optimizations that had been analyzed previously.

I think this is a good point to wrap things up. Even if you have found the post to be tedious, I hope that you have at least enjoyed the orchids. As we would say in Brazil, até mais!

    

The nature of ligand efficiency

"First they ignore you, then they laugh at you, 

then they fight you, then you win"

(Sometimes attributed to Mahatma Gandhi)

It's been a bit of a slog but the 'The nature of ligand efficiency' (NoLE) has just been published in Journal of Cheminformatics. As detailed in the previous post, the manuscript proved too spicy for two of the reviewers assigned by Journal of Medicinal Chemistry who did seem rather keen that the readers of that journal not be exposed to the meaninglessness of the ligand efficiency metric. One of the great things about preprints is that we no longer need to take shit from reviewers and, after spicing up the manuscript a bit more, I uploaded it as my first contribution to ChemRxiv. The exercise did teach me a number of lessons that should serve me well when I get round to writing 'The nature of lipophilic efficiency'...

Proofs for NoLE on my desk at Berwick-on-Sea in Blanchisseuse

I believe that NoLE may be the second scientific publication from the village of Blanchisseuse on the north coast of Trinidad (although I'll be happily be proven wrong on this point) and it follows an article in which I gave the editors of a number of ACS journals some unsolicited advice on assay interference. Sir Solomon Hochoy, who was Governor General of Trinidad and Tobago for ten years after independence, grew up in Blanchisseuse and had a house at the far end of the beach where I swim. I met him on the beach a couple of times before he passed away in 1983 and, after that, I would often chat with Lady Hochoy who was an expert in the use of the cast net (she said that it was the only way that her cats would eat and, evidently, they were either very numerous or extremely large). I suspect that she would have been able to teach us a thing or two about high throughput screening although I never summoned the courage to ask her exactly how many cats she had.

Steps leading to the Hochoy house

This part of of Blanchisseuse has a long been associated with picong being given in print and my two articles merely follow an established tradition. The journalist BC Pires is married to one of our next door neighbors and he achieved international notoriety for a 1994 column in which he gently poked fun at the ears of a touring England fast bowler:

"He's a bowler with bat ears. How can he bowl fast with ears like that? The wind resistance must be a bitch to overcome, which probably accounts for the odd look of his first four steps. If he had his ears tucked, he'd outpace Ambrose. A less determined man would have opted to bowl spin. I'm amazed that he can walk head-on into the wind, far less run in at a seamer's speed. If, just as he reached the end of his run-up, he tripped at the bowling crease, he would probably glide to the other wicket. He could stump the batsman himself off his own delivery. He could deliver his delivery. He's a gentleman though; he's the only bowler I know who walks with his own sight screen attached to his head."

Needless to say BC was forced to make a grovelling apology and, if forced to make a grovelling apology for any of my scientific articles, I shall certainly consult my old schoolmate who is a (the?) world-leading expert and thought leader in the making of grovelling apologies.

"But Mr Caddick has demanded an apology and so I feel that I must, with utmost sincerity, say that I am very sorry indeed that Andy Caddick has big ears."

"Talk to me, Basil, I'm all ears" (with BC in Barbados)

Like both BC and me, my late father was taught by the Holy Ghost Fathers and the title of his final book was inspired by a rather bizarre episode that had taken place a couple of years before BC was accosted in the commentary box at Bourda by a fast bowler who is 20 cm taller than me (take another look at the photo above as you try to imagine that scene). For decades, a weather vane in the form of a sea serpent had sat harmlessly on the roof of the Red House which is currently being renovated but at the time, housed Parliament. Some years previously, the calypsonian Sugar Aloes had sung that the sea serpent (commonly believed to be a dragon) was a evil omen and, when the People's National Movement (PNM) were re-elected, it was decided that the dragon simply had to go.

The dragon was duly replaced with a white dove in a nocturnal operation that was personally supervised by the Minister for Works. The reason given for installing the dove at night was that they wanted to minimize disruption of traffic and this must be the only occasion on which a government in Trinidad and Tobago has been concerned about disrupting traffic. Although some considered the dove (with an olive branch in its beak) to be masterpiece, others were unconvinced. In particular, there was something that just didn't look right. It was my late father, Professor of Zoology at the University of the West Indies, who identified the problem in a letter to the Daily Express.

"Examination of your photograph shows an aerofoil of mixed parentage. The inner segment resembles that of any small bird with a generalised wing such as a dove a keskidee. The outer segment with its greater width is clearly that of a soaring bird such as a corbeau.

In flight a bird's tail feathers would trail and not be partly spread. If however it was landing the tail fan would be widely spread and the long axis of its body inclined sharply upward to the flight path. In flight, no bird would spread its legs in this way but would trail them. The bird in photograph is clearly not flying or landing normally. 

Zoologically the only circumstances under which this configuration of spread legs, partially spread tail fan and spread primaries is possible is when a bird defecates in flight."            

Masters of picong

No thought leaders, key opinions, cricketers or avifauna were harmed during the production of this blog post.

Reviewing the reviewers

I recently published The Nature of Ligand Efficiency (NoLE) as a ChemRxiv preprint and this was featured (for all the right reasons) in a post at In The Pipeline. The material had been previously submitted to J Med Chem but it proved a bit too spicy for two of the three reviewers. I'll review the J Med Chem reviewers in this blog post and I hope that the feedback will be useful in the event of the journal being presented with similarly flavored material in the future. NoLE was my second publication from Berwick-on-Sea in the village of Blanchisseuse on the north coast of my native Trinidad and I'll include some photos from there to break up the text a bit.

Gate at Berwick-on-Sea in Blanchisseuse. The house was built (quite literally) by my late father (who would have been 89 today) and was named for my mother's home town of Berwick-upon-Tweed which has changed hands between England and Scotland on a number of occasions and may even still be at war with Imperial Russia.

The selection of reviewers for manuscripts that criticize previous studies presents a dilemma for journal editors. While it is prudent to consult those with a stake in what is being criticized, these may not the best people to ask about whether or not the criticism should be made. In particular, a reviewer using his/her position as a reviewer to suppress criticism of something in which he/she has a stake raises ethical questions. A stake in ligand efficiency (LE) could take any of a number of forms. First, one could have introduced a metric for LE. Second, one could have written articles endorsing ligand efficiency metrics or asserting their validity. Third, one could have enthusiastically promoted the LE metric at one's institution (e.g. by mandating that LE values be quoted when presenting project updates at the dog and pony shows that are an essential part of modern drug discovery). Fourth, one might be a devout member of the Fragment Cult (for whom the Doctrinal Correctness of LE is an Article of Faith).

There were three reviewers for my manuscript and I'll call them A, B and C since their numbers got scrambled between different rounds of review (also using the term 'Reviewer 3' might give some readers anxiety attacks). Reviewer A had nothing constructive to say and simply spat feathers. Reviewer B was very positive about the manuscript and made a number of  helpful suggestions. Reviewer C demanded that the manuscript be watered down to homeopathic levels (and that was never going to happen).

Here's my office at Berwick-on-Sea. That's a printout of NoLE on my desk (under the hanging beach towel).

The central theme of my manuscript is the argument that ligand efficiency is physically meaningless because perception of efficiency changes with the concentration unit in which affinity is expressed. This is actually a very serious criticism since since a change in perception resulting from a change in a unit would normally be regarded in physical science as an error in the "not even wrong" category.  It's not something that one can simply sweep under the carpet as a "limitation" of ligand efficiency. Despite their howls of protest, neither Reviewer A nor Reviewer C offered coherent counter-argument.

The tactic adopted by Reviewer C was to simply dismiss the physical arguments presented in the manuscript as "opinion" without presenting counter-argument. J Med Chem really does need to make it clear to reviewers that they need to do much better than this since it reflects badly on the journal.

Reviewer C. "'Physically meaningless' is at best an inflammatory opinion whereas the fact that other choices could have been made is often under-appreciated."

PWK. This criticism appears to be doctrinal rather than scientific and I note that Reviewer C has not offered counter-argument to the argument that LE is physically meaningless.

Here's a view of the Caribbean Sea. The 20 m drop from the gap in the vegetation is just as precipitous as you would expect although we've not (yet) lost any personnel or household pets over the edge.

Reviewer A struggled woefully with rudimentary physical chemistry throughout the review process and, given that I'd suggested a number of potential reviewers with the necessary expertise in molecular recognition and chemical thermodynamics, I was at a loss to understand why a reviewer who was so ill-equipped for the task at hand had been invited to review the manuscript.

Reviewer A. Reactions are considered to be spontaneous under standard conditions when the free energy is negative, but by changing the definition of C° in an arbitrary manner, any reaction can be said to be spontaneous or not. This is true in a trivial sense, but generations of researchers have found the concept of negative or positive free energies useful.

PWK. The flaw in this argument is that if you change the value of C° then you also change whether or not the reaction is spontaneous under the standard conditions. This is the basis of the law of mass action and it is also important to remember that KD values are not measured at single concentration. A chemical process (at constant temperature and pressure) by which the system changes from state A to state B will be spontaneous if DG[A®B]  is negative. Regardless of experiences of generations of researchers, medicinal chemists rarely (if ever) appear to use the sign of  DG  (e.g. for binding under assay conditions) when analyzing SAR or for making any other decisions.

This is the start to the path down to the lower deck

In one round of review, Reviewer C stated “I believe that it is incumbent on the author to argue that the choice of standard state used by medicinal chemists is not useful” and Reviewer A repeated the criticism in a subsequent round, noting that this was "the central problem with the manuscript". I thought this was a bit rich given that Reviewer A and Reviewer C had each accused me of using straw man tactics at different points in the review process. The more serious problem, however, is that we have two LE advocates each attempting to to transfer the burden of proof that (in science) one accepts as soon as one advocates that people take an action (e.g. use LE metrics). Reviewers A and C appeared to do this in order to evade their responsibility as reviewers to present counter-argument to the arguments in the manuscript. This would be like a thought leader (yes, there really are people who call themselves 'thought leaders') responding to criticism of a claim that AI was going to transform drug discovery by saying that it was incumbent on the critics to argue that AI was not useful. Imagine if they ran clinical trials like this?

At this point, Reviewer A did rather lose it and I was half expecting to have to fend off a counterattack by Steiner's division. Needless to say, the latest version of the manuscript now opens with "Ligand efficiency (LE) is, in essence, a good concept that is poorly served by a bad metric." and this can be considered the equivalent of a two-fingered gesture that is mistakenly attributed to the English and Welsh longbowmen at Agincourt.

Reviewer A. Dr. Kenny dodges this challenge by stating that the burden of proof should not be on him, but by arguing that LE is a “bad metric” despite its wide usage, he does in fact have to explain why free energy is also a “bad” concept. Not doing so makes the manuscript deeply misleading and therefore inappropriate for publication.

PWK. I only used the term “bad metric” in the conclusions where I wrote “Ligand efficiency is, in essence, a good concept served by a bad metric.” so it is incorrect to state that I have argued that LE is a “bad metric”. In any case, in the revised manuscript, I now question whether LE can accurately be described as a metric since neither its creators nor its advocates appear able (or willing) to say what it measures. Wide usage does not validate rules, guidelines or metrics and I note that, at one time, the prevailing view was that the sun orbited the earth. Once again, Reviewer A is making the serious error of assuming that everything that applies to free energy also applies to any function of free energy. The simple counter to Reviewer A’s challenge is that free energy is a state function and an integral part of the framework of thermodynamics. Although defined in terms of free energy, the LE metric is not is part of thermodynamics simply because it appears to require a privileged standard state.

I have occasionally stated that "useful is the last refuge of the scoundrel" and this tends to be misinterpreted as an assertion that utility of a model is unimportant. Nothing could actually be further from the truth and the statement is more a comment on the way that models can be 'validated' by simply labeling them as "useful". In some ways "useful" is analogous to the "God created it that way" statements that you will encounter if you are careless enough to become ensnared in arguments with Creationists. I should also point out that the manuscript did discuss the difficulties of demonstrating the utility of LE while neither A nor C presented any evidence (fervent belief does not usually constitute evidence in science) to support their assertion that the 1 M standard state is more useful than any other standard state.

Reviewer A appeared particularly aggrieved that one of The Great Unwashed should have the temerity to even question the value of LE and the toys were duly ejected from the pram. As my response below indicates, Reviewer A's comment is more what one might have expected from an inquisitor at a fifteenth century heresy trial than from an expert reviewer of a manuscript submitted to the premier medicinal chemistry journal. It is also worth pointing out that LE was touted as "useful" even as it was introduced in a 2004 letter to Drug Discovery Today and all three coauthors of that seminal contribution to the medicinal chemistry literature appeared to be blissfully unaware of the nontrivial dependency of their creation on the standard concentration. As such, I would argue that it would actually be a dereliction of duty not to question the utility of LE.

Reviewer A. Sixth, Dr. Kenny repeatedly questions the utility of LE; for example “The LE metric is claimed by advocates to be useful although it is rarely, if ever, shown to be predictive of pharmaceutically-relevant behavior” (p. 15) and “the LE metric is rarely, if ever, shown to be predictive of phenomena that are relevant to drug discovery” (p. 39).

PWK. This appears to be a doctrinal rather than scientific criticism.

Lower deck. I only swim from here if snorkeling because it's rocky.

Reviewer B was very positive about the "Molecular Size and Design Risk" section and made useful suggestions for its expansion. It's also worth mentioning that Derek quoted from this section in his post. However, Reviewer C suggested that the whole section be purged from the manuscript although it is possible that Reviewer C's underlying objective was to ensure that certain articles were not discussed. Reviewer C complained that my criticism of ref 48 was unfair although it may be that the reviewer considered ref 48 to be a liability (this post will give readers an idea why some LE advocates might consider ref 48 to be a liability). Another possibility is that the objection to criticism of ref 48 was actually a smokescreen and the real reason for suggesting that the section be purged was actually to avoid discussion of ref 45 (which might be considered to be an even greater liability by LE advocates).

Ref 58 and ref 59 are rare examples of articles that respond to criticism of LE and and a study such as NoLE really does need to discuss them (especially since both articles completely miss the point). The fundamental flaw that is common to both articles is that neither addresses the problems associated with the change in perception that results from using a different unit to express affinity. Reviewer C protested that it was gratuitous to single out ref 58 and even cited this 2014 post from Molecular Design in support of the charge that I was unfairly picking on ref 58. Reviewer C did seem rather rattled and also complained that I had quoted "non-scientific sections" of ref 59. I must confess to being unfamiliar with the concept that a scientific article can have non-scientific sections that can be declared off-limits for challenge. This was, perhaps, not Reviewer C's finest moment.

Reviewer A and Reviewer C both seemed rather keen that ref 94 not be discussed and they said that I should not be "attacking" fit quality (FQ) because it is rarely, if ever, used. I suspect the real reason was that both reviewers consider the metric (and ref 94) to be a significant liability from the LE perspective. I responded by noting that FQ had got its own box in the NRDD LE review and that ref 94 was cited in ref 58 (which asserts the validity of LE), suggesting that FQ may be of greater interest than Reviewer A and Reviewer C would have us believe. Another reason that Reviewer C might have preferred that the spotlight not be focused on FQ is that the discussion further exposes the illusion that fragments bind more efficiently than ligands of greater molecular size.

This is where I go swimming. It's a 5 minute walk from the house

So that concludes my review of the reviewers. I believe that the J Med Chem editors do need to think carefully about how (or even whether) they wish to have controversial topics addressed in their journal. Dr Eric Williams, the first Prime Minister of Trinidad and Tobago, suggested that his hearing impairment was an advantage in dealing with dissent because he could simply switch off his hearing aid. However, dealing with controversial topics in drug discovery might not be quite so simple. In particular, a journal needs to consider the potential vested interests of those from whom it seeks advice. For example, the Editors of a number of ACS journals may find it quite instructive to take a very close look at exactly how their journals came to endorse a frequent hitter model (trained on results from a panel of only six assays that all use the same readout) as a predictor of pan-assay interference...

I'll leave you with a selfie taken on the roof. A few minutes earlier I'd seen off a determined counter-attack by some jack spaniards (or should that be jacks spaniard?). Normally, I'd leave them alone but they were too close to where I needed to work. The technique is simple but its execution takes some nerve. First, arm yourself with a can of Baygon (don't forget to test it beforehand) and a broom. Second, with Baygon aimed, prod nest with broom. Third, spray a protective curtain of Baygon as the jack spaniards attack you (they are aggressive and they always attack). 

PWK one, jack spaniards nil 

Ligand efficiency and fragment-to-lead optimizations

The third annual survey (F2L2017) of fragment-to-lead (F2L) optimizations was published last week. Given that it was the second survey (F2L2016) in this series, that prompted me to write 'The Nature of Ligand Efficiency' (NoLE), I thought that some comments would be in order. F2L2017 presents analysis of data that had been aggregated from all three surveys and I'll be focusing on the aspects of this analysis that relate to ligand efficiency (LE).

As noted in NoLE, perception of efficiency changes when affinity is expressed in different concentration units and I have argued that this is an undesirable feature for a quantity that is widely touted as useful for design. At very least, it does place a burden of proof on those who advocate the use of LE in design to either show that the change in perception of efficiency with concentration unit is not a problem or to justify their choice of the 1 M concentration unit. One difficulty that LE advocates face is that the nontrivial dependency of LE on the concentration unit only came to light a few years after LE was introduced as "a useful metric for lead selection" and, even now, some LE advocates appear to be in a state of denial. Put more bluntly, you weren't even aware that you were choosing the 1 M concentration unit when you started telling medicinal chemists that they should be using LE to do their jobs but you still want us to believe that you made the correct choice?

I'm assuming that the authors of F2L2017 would all claim familiarity with the fundamentals of physical chemistry and biophysics while some of the authors may even consider themselves to be experts in these areas. I'll put the following question to each of the authors of F2L2017: what would your reaction be to analysis showing that the space group for a crystal structure changed if the unit cell parameters were expressed using different units? I can also put things a bit more coarsely by noting that to examine the effect on perception of changing a unit is, when applicable, a most efficacious bullshit detector.

The analysis in F2L2017 that I'll focus on is the the comparison between fragment hits and leads. As I showed in NoLE, it is meaningless to compare LE values because LE has a nontrivial dependency on the concentration unit used to express affinity. LE advocates can of course declare themselves to be Experts (or even Thought Leaders) and invoke morality in support of their choice of the 1 M concentration unit. However, this is a risky tactic because physical science can't accommodate 'privileged' units and an insistence that quantities have to be expressed in specific units might be taken as evidence that one is not actually an Expert (at least not in physical science).

So let's take a look at what F2L2017 has to say about LE in the context of F2L optimizations.

"The distributions for fragment and lead LE have also remained reasonably constant. On average there is no significant change in LE between fragment and lead (ΔLE = 0.004, p ≈ 0.8). Figure 5A shows the distribution of ΔLE, which is approximately centered around zero, although interestingly there are more examples where LE increases from fragment to lead (40) than where a decrease is seen (25). Some caution is warranted when interpreting these data, as our minimum criterion for 100-fold potency improvement may have introduced some selection bias. Nevertheless, there is no clear evidence in this data set that LE changes systematically during fragment optimization. Although the average change in LE from fragment to lead is small, Figure 5B shows that the correlation between fragment and lead LE is modest (R2 = 0.22), with a mean absolute difference between fragment and lead LE of 0.08."

This might be a good point at which to remind the authors of F2L2017 about some of the more extravagant claims that have been made for LE. It has been asserted that “fragment hits typically possess high ‘ligand efficiency’ (binding affinity per heavy atom) and so are highly suitable for optimization into clinical candidates with good drug-like properties”.  It has also been claimed that "ligand efficiency validated fragment-based design".  However, the more important point is that it is completely meaningless to compare values of LE of hits and leads because you will come to different conclusions if you express affinity using a different concentration unit (see Table 2 in NoLE). It is also worth noting that expressing affinity in units of 1 M introduces selection bias just as does the requirement for 100-fold potency improvement. 

Had I been reviewing F2L2017, I'd have suggested that the authors might think a bit more carefully about exactly why they are analyzing differences between LE values for fragments and leads. A perspective on fragment library design (reviewed in this post) correctly stated that a general objective of optimization projects is “ensuring that any additional molecular weight and lipophilicity also produces an acceptable increase in affinity". If you're thinking along these lines then scaling the F2L potency increase by the corresponding increase in molecular size makes a lot more sense than comparing LE for the fragments and leads. This quantifies how efficiently (in terms of increased molecular size) the potency gains for the F2L project have been achieved. This is not a new idea and I'll direct readers toward a 2006 study in which it was noted that a tenfold increase in affinity corresponded to a mean increase in molecular weight of 64 Da (standard deviation = 18 Da) for 73 compound pairs from FBLD projects. This is how group efficiency (GE) works and I draw the attention of the two F2L2017 authors from Astex to a perceptive statement made by their colleagues that GE is “a more sensitive metric to define the quality of an added group than a comparison of the LE of the parent and newly formed compounds”.

The distinction between a difference in LE and a difference in affinity that has been scaled by a difference in molecular size becomes a whole lot clearer if you examine the relevant equations. Equation (1) defines the F2L LE difference and first thing that you'll notice is that is that it is algebraically more complex than equation (2). This is relevant because LE advocates often tout the simplicity of the LE metric. However, the more significant difference between the two is that the concentration that defines the standard state is present in equation (1) but absent in equation (2). This means that you get the same answer when you scale affinity difference by the corresponding molecular size difference regardless of the units in which you express affinity.

So let's see how things look if you're prepared to think beyond LE when assessing F2L optimizations. Here's a figure from NoLE in which I've plotted the change in affinity against the change in number of non-hydrogen atoms for the F2L optimizations surveyed in F2L2016. The molecular size efficiency for each optimization can be calculated by dividing the change in affinity by the change in in number of non-hydrogen atoms. I've drawn lines corresponding to minimum and maximum values of molecular size efficiency and have also shown the quartiles.

So now it's time to wrap things up. A physical quantity that is expressed in a different unit is still the same physical quantity and I presume that all the authors of F2L2017 would have been aware of this while they were still undergraduates. LE was described as thermodynamically indefensible in comments on Derek's post on NoLE and choosing to defend an indefensible position usually ends in tears (just as it did for the French at Dien Bien Phu in 1954). The dilemma facing those who seek to lead opinion in FBDD is that to embrace the view that the 1 M concentration unit is somehow privileged requires that they abandon fundamental physicochemical principles that they would have learned as undergraduates.   

Confessions of a Units Nazi

Regular readers (both of them) of this blog will know that I have an interest, which some might term an obsession, with units. At high school in Trinidad, we had the importance of units beaten into us by the Holy Ghost Fathers and, for some of the more refractory cases, the beating was quite literal. I was taught physics by the much loved, although somewhat highly-strung, Fr. Knolly Knox (aka Knox By Night) who, as Dean of the First Form, used to give 'licks' with a cane of hibiscus (presumably chosen for its tensile properties). You quickly learned not to mess with The Holy Ghost Fathers, especially the Principal, Fr. Arthur Lai Fook (aka Jap), and it was a brave student who responded to the request by Fr. Pedro Valdez to define the dyne by answering, "Fah, it what happen after living". Fr. Pedro was a gentle soul although his brother, Fr. Toba, who taught me Latin, would lob a blackboard eraser with reproducible inaccuracy at any student who had the temerity to doze off during the Second Punic War while Hannibal and his elephants were steamrollering the hapless legions of Gaius Flaminius into Lake Trasimene. At least we didn't have detention at my school. Actually we did have detention only it was called 'penance'. Each and every student also had a Judgement Book in which was entered a mark (out of 10) for each subject each and every week. A mark of 5 (or less) or a failure to return one's Judgement Book, duly signed by parent or guardian, by Wednesday morning earned the transgressor a corrective package of Licks and Penance.  As a thoughtful child, I managed to shield my parents from this irksome bureaucracy and, in any case, it was simply safer that The Holy Ghost Fathers were never given the opportunity to familiarize themselves with the authentic parental signatures.

I used to think that 'Virtus et Scientia' was Latin for 'Licks and Penance'  (17-Feb-2018 update)

What we learned from the Holy Ghost Fathers was that most physical quantities have dimensions and if the quantities on the opposite sides of the 'equal sign' in an equation have different dimensions then it is a sign of an unforced error rather than a penetrating insight. For example the dimensions of force are MLT^-2 (M = mass; L = length; T = time) and you are free to express forces in newtons, dynes or poundals as you prefer. You can think of a physical quantity as a number multiplied by a unit and, without the unit, the number is meaningless. Units are extremely important but at the same time they are arbitrary in the sense that if your physical insight changes when you change a unit then it is neither physical nor an insight. Here's a good illustration of why dimensional analysis matters.

I have blogged ( 1 | 2 | 3 ) about how building the a concentration unit into the definition of ligand efficiency (LE) results in a metric that is physically meaningless (even though it remains a useful instrument of propaganda) and, for the masochists among you, there's also the LE metric critique in JCAMD. The problem can be linked to a lack of recognition of the fact that logarithms can only be calculated for numbers (which lack units). However, LE has another 'units issue' which is connected with the fact that it is a molar energy that is scaled in the definition of LE rather than pIC₅₀ or pK_d. This needn't be an issue but, unfortunately, it is. LE is defined by dividing a molar energy by the number of non-hydrogen atoms in the molecular structure and there is nothing in the definition of LE that says that the energy has to be expressed in any particular unit. This means that you can define LE using any energy unit that you want to. Some 'experts' appear to believe that dividing a molar energy by number of non-hydrogen atoms relieves them of the responsibility to report units. I'm referring, of course, to the practise of multiplying pIC₅₀ or pK_d by 1.37 when calculating LE. You might ask why people do this, especially given that 'experts' tout the simplicity of LE and they don't multiply pIC₅₀ or pK_d by 1.37 when they calculate LipE/LLE. Don't ask me because I'm neither expert nor 'expert'.

Let's take a look at this NRDD article on LE metrics and I'd like you to go straight to Box 1 (Ligand efficiency metrics). Six numbered equations are shown in Box 1 and it is stated towards the end of the first paragraph that "each equation corresponds to a mathematically valid function".  This statement is incorrect because the first equation (1) in Box 1 is not a mathematically valid function. The reason for this is that the logarithm function cannot take as its argument a quantity, such as K_d, that has units. Equation (5), which defines LLE_AT, is mathematically valid although it differs from the mathematically ambiguous equation that was originally used to define LLE_AT. 

To be honest, I think that Box 1 is probably beyond repair by conventional erratum and I'll back this opinion with an example:

"Assuming standard conditions of aqueous solution at 300K, neutral pH and remaining concentrations of 1M,
 –2.303RTlog(K_d/C°) approximates to –1.37 × log(K_d) kcal/mol." 

At my school in Trinidad this would have been called a 'ratch' and, once detected, it would have earned its perpetrator a corrective package of Licks and Penance. I don't think even the Holy Ghost Fathers could have exorcised a concentration unit quite this efficiently. 

In some physical chemistry literature, K_d is defined as a dimensionless quantity by including C° in the definition of K_d. However, in the literature of biochemistry, biophysics and medicinal chemistry,  K_d  is usually quoted in units of concentration. Binding free energy has the same value and same dependence on C° regardless of  which of the two conventions is used to define K_d . 

(Update 17-Feb-2018) 

I'd now like to talk a bit about the 'p' operator that we use to transform IC₅₀ and K_d values into logarithms. This makes it much easier to perceive structure-activity relationships and provides a better representation of measurement precision than when the IC₅₀ and K_d values themselves are used. To calculate pK_d,, first express K_d in molar concentration units, dump the units and calculate minus the logarithm of the number. I realize that this may come across as arm waving but the process of converting  K_d, to  pK_d, can actually be expressed exactly in mathematical terms as follows:

 pK_d = –log₁₀(K_d/M)

The 'p' operator has a 1 M concentration built into it. Although this choice of unit is arbitrary, it doesn't cause any problems if you're doing sensible things (e.g. subtracting them from each other) with the pK_d values. If, however, you're doing silly things (e.g. dividing them by numbers of non-hydrogen atoms) with the pK_d values then the plot starts to unravel faster than you can say 'Brexit means Brexit'. 

I'd like you take a look at another article which also has a Box 1 although I won't bother you with another tiresome 'spot the errors' quiz. The equation that I'll focus on is:

pK_d = pK_{H +} pK_S 

This equation describes the decomposition of affinity into enthalpic and entropic contributions and you might think this means that you can write:

K_d = K_H × K_S 

As Prof. Pauli would have observed, this is an error in the 'not even wrong' category and it is clear that a difference in opinion as to the importance of units was as much responsible for the unraveling of the Austro-Hungarian empire as that unfortunate wrong turn in pre-SatNav Sarajevo. The 'p' operator implies that each of K_d, K_H and K_S has units of concentration. However, multiplying two such quantities will give a quantity that has units of concentration squared. 

It is actually possible to decompose K_d into enthalpic and entropic contributions a valid manner but you need to be thinking carefully about the meaning of the standard state. As noted previously DG° depends on the concentration used to define the standard state. This is a consequence of the dependence of DS° on the standard concentration and DH is independent of the standard concentration (the standard state is assumed to be a dilute solution). This suggests defining K_S as quantity with units of concentration and K_H as a quantity without units.

This is probably a good point to wrap things up. My advice to all the authors of the featured NRDD and FMC articles is that they read (and make sure that they understand) the section of this article that is entitled '8. Ligand Efficiency and Additivity Analysis of Binding Free Energy'. This advice is especially relevant for those of the authors who consider themselves to be experts in thermodyamics.

May I wish all readers a happy, successful and metric-free 2017.