As regular readers here know, we often discuss metrics because everyone uses them. Last year, agent provcateur Pete Kenny unleashed a broadside against those who defended metrics. This seems to be like the corpse flower that blooms once a year, stinks the place up, yet everyone runs to go see it. Well, recently Pete posted in the LI group: "Ligand efficiency validated fragment-based design?"and asked whether or not people agreed with the statement. This of course has inspired a wave of comments. I disagreed with the statement, but not for the "metrics suck" argument. I strongly urge people to go read the thread, unless of course you have something better to do.
To me, this is not about the validity of metrics. [Let me add here, that I prefer the "LEAN" metric (pIC50/HAC) because it can be done in your head on the fly.] I think people have a good understanding of what they do, their limitations, and their strengths. I disagreed with the statement because of the use of the word "validated". In the development world, we talk about our assays very specifically: they are qualified or validated. A validated assay is one that has been shown to be accurate, specific, reproducible, and rugged for the analyte in the concentration range to be measured. Put plainly, this means that if you expect to measure analyte X at 5 uM, you have to show that for all samples it will be measured in you can identify it, measure 5 uM accurately, and do it every time. That's a validated measurement. When you are qualifying an assay, the bar is much lower. An assay is considered qualified if it has been demonstrated to be "fit for purpose". Fit for purpose means that it will do the job, but you haven't beat the sugar out of it to make sure it is "valid". To me, ligand efficiency is fit for purpose of driving medchem decisions; it is qualified for that purpose, but not validated (N.B. I am not saying "not valid".)
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Hi Teddy,
The broadside that we fired last year was at ligand efficiency metrcs and not at other metrics nor inviduals. I would regard an IC50 in an enzyme kinetic assays to be a 'fit for purpose' metric of affinity although I'd want to be sure that it had been measured under the same conditions if I were using it to assert that the affinity of one compound was greater than another.
Our criticism of LE (and other LEMs) is essentially that they are not fit for purpose. When you you write LEAN = pIC50/HA you're assuming that 1 M is somehow special. In more formal terms, you're setting ConcUnit in -log10(IC50/ConcUnit) to 1 M. Now it may be that this is really 'correct' but you need to be honest that this assumption is being made and you can't simply declare 1 M to be the truth in the manner that a Republican might declare a 6000 year-old to Earth to be the truth (OK you can but don't be offended if not everyone believes you). If you want people to believe you when you assert that LEAN is fit for purpose then the burden of proof is on you to either show that 1 M really is special or that the concentration unit is irrelevant.
When factors with units are simple multiplicative factors they matter less and you don't really need to know units of energy to know that that the affinity or one compound is greater than that of another. That said, it is extremely sloppy to discard units even when they are incorporated in simple multiplicative factors. When the units are 'inside' logarithms you need to be very careful and not do anything silly. It's OK to subtract values of log10(IC50/ConcUnit) as one might do when analysing SAR or assessing selectivity. It's also OK to multiply by a constant term as one might do to convert kcal/mol to kJ/mol. However, dividing log10(IC50/ConcUnit) by the value of a molecular property fits into the 'silly' category. The problem is not with the product of the mathematical transformation which is mathematically valid. The problems start when you assign meaning to what you've created. Now I don't know a priori whether or not your assumption that 1 M is 'special' is correct (and actually you don't know either). Although I don't know whether or not 1 M is 'special' I can show the relevance of assuming that it is. I do this by taking compound A that you say would is more ligand efficient than compound B. Then I change ConcUnit and show that B is actually more ligand efficient than A. Whose view is correct? Which metic is more fit for purpose? Like I said on LinkedIn, change of unit is an excellent bullshit detector.
It's also worth remembering that drug discovery business is not in the most robust of health and there is much hand-wringing about attrition and spiralling costs. If were using metrics that cause our perception to change when we use different units then our paymasters might start to ask whether the difficulties we face might be of our own making.
Pete's a smart guy, and provocateurs can play a useful function. There's a lot of verbiage on that LinkedIn thread, so I'll just summarize what to me is Pete's most striking assertion.
Unlike some critics, Pete acknowledges that ligand efficiency (LE) is mathematically valid, defined as:
LE = ∆G°/Nnon-hydrogen atoms
The reason Pete alleges that ligand efficiency is "not even wrong" is his rejection of standard state conditions as "arbitrary". Of course, by rejecting the standard state, he also rejects much of thermodynamics. Thus, according to Pete, one should not claim that the reaction of hydrogen with oxygen to form water is exothermic.
Absolute relativism can lead to strange places.
Pete has a PhD in verbiage - what a load of pointless lip-flapping. Back when I first started in fragment based design (2003, prior to formalised/published efficiency metrics) people used the concept as a rough guide, not some Nobel-worthy intricate mathematical theory.
Just as most medicinal chemists would say that a compound with an IC50 of 10 nM is equipotent with a compound with IC50 of 15 nM, so it goes with LE.
I can't believe how much garbage can be written - and that I even wasted time writing this.....so please Pete, shut your pie hole, the world doesnt care how clever you are.
[Request to moderator] Ad hominem attacks should be deleted. If people can't make their case without them, then they can't make a case.
Thanks for your comment, Noel, and I’m guessing that we may be in agreement that there is a ‘sound and fury’ aspect to the input from ‘Anonymous’. While tolerance for personal attacks like this does detract from the excellent reputation of this blog, I do have to admit that being attacked in this manner does feel, as Denis Healey famously observed, “like being savaged by a dead sheep”.
Dan, I do not believe that you have summarized the LinkedIn discussion accurately. Firstly, it is untrue to state that I reject the standard state and I challenge you to back up your charge with evidence. What I have stated repeatedly is that you can’t do chemical thermodynamics without standard states. The arbitrary nature of the standard states means that, when your perception of reality changes with the definition of the standard state, your perception is “not even wrong”. I use the term ‘voodoo thermodynamics’ to describe this sort thinking.
Your statement, “Thus, according to Pete, one should not claim that the reaction of hydrogen with oxygen to form water is exothermic” also misrepresents what I said. What I stated was that in the gas phase at sufficiently low pressure, the equilibrium for formation of water from hydrogen and oxygen can be shifted in the direction of the ‘reactants’. The reason for this is that formation of water from hydrogen and oxygen results in a net reduction in the number of molecules and entropy increases with dilution. I was merely restating Le Chatelier’s principle. I don’t find the term ‘exothermic’ particularly helpful and only use it for processes that are associated with a decrease in enthalpy or internal energy. In dilute solutions enthalpy is independent of concentration and it is the dependence of entropy on dilution that creates the need to define a standard concentration.
Hi Pete,
Apologies if you feel I have misrepresented you as that was certainly not my intention. I took these statements of yours from the LinkedIn thread as a rejection of the standard state:
However, the choice of 1 M as the standard concentration is arbitrary...
and
Put another way, let's focus on whether or not we are allowed to vary C° rather than worry at this stage about why we might want to do this.
Allowing C° to vary from 1 M is by definition changing the standard state, which as I understand is your chief objection to ligand efficiency. Changing C° could also alter whether any reaction in which the net number of species changes is exothermic or endothermic. Whether or not you find the term exothermic "helpful," isn't this true?
No worries, Dan, and no offence taken although please note that I was genuinely trying to inform in the LinkedIn discussion and tried to explore the thermodynamics from a number of different angles in the hope that one of them might have helped. It would be better to stay with the LinkedIn thread if you want to continue the discussion.
Probably best to make my position clear as succinctly as possible to prevent further misunderstanding. A standard concentration is necessary if you want to do solution thermodynamics with a specification of composition other than mole fraction. Using standard concentration entails the introduction of an arbitrary unit of concentration. The arbitrary nature of the standard concentration means that we cannot invoke thermodynamics in support of a perception of the system that changes with standard concentration. Equivalently, valid thermodynamic reasoning does not permit us to declare special privilege for a particular value of concentration.
Thanks Pete,
I prefer discussions here since I find LinkedIn threads harder to find later. Also, since you alleged that I misrepresented you here, this is the most appropriate place to respond.
I'm afraid I still don't understand how I have misrepresented your position. Simply put, changing the value of C° could make the reaction of hydrogen with oxygen to form water appear "endothermic".
More to the point, you could show that the binding of any small molecule to any protein is favorable or unfavorable by sufficiently altering C°.
Aren't these natural consequences of your reasoning?
OK, Dan, we can continue here. Firstly, please can we avoid using exothermic/endothermic because there is potential confusion about whether we're talking about enthalpy (which is constant for solutes in dilute solutions) or free energy (which has a logarithmic dependence on concentration in dilute solutions)?
Let's use the example of a ligand-protein complex with a Kd of 10 nM and I will describe the ligand and protein as 'reactants' and the 'protein-ligand' complex as 'product'. Please let me know whether or not you agree with the following statements:
(1) Complete conversion of reactants at 1M (ignoring the fact that solution is no longer dilute) results in a reduction of the free energy of the system
(2) Complete conversion of reactants at 1nM results in an increase in the free energy of the system
(3) Complete conversion of reactants at 10 nM to products results in no change in the free energy of the system
(4) The free energy changes associated with (1) to (3) are independent of the concentration used to specify the standard state.
Hi Pete,
First, I find it telling that you refuse to acknowledge - let alone answer - my question, which I will rephrase here as a simple statement with which you can either agree or disagree.
By altering C°, you could show that the binding of any small molecule to any protein is favorable or unfavorable.
Second, I'm happy to drop the terms exothermic/endothermic and stick with free energy, which I believe we both agree can be expressed as ΔG˚ = −RT ln K/C°. For the sake of clarity, a negative free energy is "favorable" in the italicized statement above, while a positive free energy is "unfavorable."
Finally, I don't know how to respond to your questions (which I explicitly acknowledge) since I don't know precisely what you mean by "complete conversion" in the context of reversible equilibrium-controlled reactions.
Hi Dan,
I’ll first reproduce what I said on LinkedIn about the hydrogen-oxygen-water equilibrium here:
“Now everybody knows that formation of water from hydrogen and oxygen is extremely favorable under ‘normal’ conditions (e.g. as existed at Lakehurst NJ on May 6 1937). However, the stoichiometry of the reaction means that at a sufficiently low pressure a mole of oxygen and two moles of hydrogen become more stable than two moles of water”
By making this statement, I was simply saying that we can displace a system from equilibrium by varying concentration in much the same way as we can increase occupancy by titrating a protein with ligand (e.g. ITC experiment).
Let’s leave my questions for now and address your question, the answer to which is no. The sign of ΔG˚ is simply a statement of whether or not the Kd exceeds standard concentration C° (which is arbitrary) and cannot used to categorize the interaction of a ligand with a protein as favorable or unfavorable. Analogously, would you use the numerical value of Kd to categorize the interaction of a ligand with a protein as favorable or unfavorable? The equation that you’ve written for ΔG˚(in which K is the association constant rather than the dissociation constant) provides you with the means to use a measured equilibrium constant to calculate ΔG˚ as a function of C°.
It is not correct make statements about whether or not binding of a ligand is favorable based on affinity (defined by ΔG˚ alone) and you also need to know concentration as well. At very least, you need to define a Kd threshold so that values of Kd below the threshold are considered to represent favorable interaction with the protein. One way to quantify the favorability of interaction of protein with ligand is to calculate the ratio of the ligand concentration to Kd (C/Kd) and you can see the concentration units ‘cancel’ when we calculate the ratio. If we want to talk about Kd and C then we still need a concentration unit like C° in which to express them in but we are free to use whatever unit of concentration that we want (e.g. M, nM). In thermodynamics it is really important that conclusions of analysis do not change with the value of the concentration used to specify the standard state since it is arbitrary.
Hi Pete,
Thanks for addressing the question, but I don't think I understand your answer. Let me break it down into a series of equations and propositions and see if we can figure out where we differ.
(1) Kd = [P][L]/[PL], where [P] = concentration of protein, [L] = concentration of ligand, and [PL] = concentration of complex at equilibrium
(2) ΔG˚ = −RT ln Kd/C°
(3) Let's take your example of a protein-ligand complex with Kd = 10 nM
(4) Under standard state conditions where C° = 1 M and T = 298 K, ΔG˚ = + 10.9 kcal/mol
(5) If we define C° = 1 nM, ΔG˚' = - 1.36 kcal/mol
(6) Thus, changing the definition of the standard state can change the sign of ΔG˚
Do you disagree with one or more of these?
Hi Dan,
These points provide a useful focus for discussion and I’ll respond to them in order.
(1) Agree
(2) Agree although on the understanding that the ΔG˚ in the equation is for the dissociation process. Normally we write this equation without the minus sign because we conventionally use a dissociation constant with an association free energy.
(3/4) Agree with the math (on the understanding that ΔG˚ is for the dissociation process) but disagree with inclusion of temperature in the definition of the standard state. Temperature is very important but is a very different to C° in the context of this discussion. If you change temperature, you really do change the system and you need to know about changes in enthalpy (and, in some cases, heat capacity) in order to use Kd measured at 298 K to calculate Kd at 310 K. It generally makes things clearer to write energies as RTln(ratio) in discussions like these although it’s not a big deal for what we’re talking about.
(5) Agree for T = 298 K and on the understanding that that ΔG˚ is for the dissociation process.
(6) Agree
Surely for Dan's argument above, if we change the standard state about to 1 nM, all other components must be converted to match, e.g. the gas constant R, which has units J 1/(K M) must now be J 1/(K nM) otherwise we'd be multiplying two this which were expressed as different orders of magnitude, and thus the value of the constant must be updated accordingly.
To use any of the arguments above we have failed to remember we treat all these solutions as ideal gases and have not converted the gas constant.
Hello Anonymous, I'm assuming that you not the same Anonymous who commented earlier. Let's first take another look at the equation relating the standard Gibbs free energy change (for association) to the dissociation constant:
ΔG˚ = RTln(Kd/C°)
This equation shows that there is a clear separation between RT and the argument of the logarithm. You are free to express the gas constant in whatever valid units you want and in some studies the Boltzmann constant is multiplied by T to provide a unit of energy on a molecular (rather than molar basis). The flaw in your reasoning is to assume that expressing the gas constant using a particular unit of 'quantity' forces us to use the same unit of 'quantity' in the argument of the logarithm.
The equation tells us that for a given value of Kd, ΔG˚ is a function C°. Describing ΔG˚ < 0 as 'favorable biding' when the standard concentration is 1 M is simply stating that you associate Kd values of < 1M with favorable binding and Kd values of > 1M with unfavorable binding. The standard concentration is, unless you want to use mole fractions, an essential part of solution thermodynamics and it is very important to know that it is arbitrary. You are free to choose whatever concentration unit you want (e.g molecules per cubic Angstrom) even though it makes sense to stick with conventions.
The thermodynamic framework in most studies of protein-ligand binding is the dilute solution (enthalpy doesn't change with dilution) rather than the ideal solution (zero enthalpy of mixing).
Thanks for the reply Pete I am a different coward you are correct. I ran some quick simulations in between reading and found the same as you.
As for favorable and non-favorable binding, my interpretation of ΔG of binding was always the amount of energy needed to do useful work. In this instance we define useful as the ability half occupy a receptor. Thus is ΔG is -ve we have exceeded the energy requirement and the receptor will be >half occupied, if it is +ve we still need more energy (often we will supply this energy either by increasing the concentration or adding new interactions to between our receptor and ligand).
This energy is always calculated for the standard state, so we look at the energy needed for useful work with 1M ligand. If we change the standard state, as in Dan's example to less than the KD, it is logical the amount of energy needed will change. Dan gave a standard state as lower than the KD, and if we have defined useful work as half occupation of the receptor then of course we will have a +ve ΔG, we have in effect moved the goal posts. The enthalpy is constant, we have just not supplied enough energy in terms of concentration (will concentration ~ mass = energy).
I feel this has moved away from ligand efficiency however. 1M as a standard state seems to make sense to me as the vast majority of ligands we are interested in have KD values <1M. Either way as has been mentioned before HAC does not account for how the increase in potency is gained, although to say hydrophobicity=promiscuity=toxicity seems a bit of a reach.
Also, the difference between Trifluoromethanesulfonamide and Methanesulfonamide is three heavy atoms, but the potency change at CAII is >1000 fold for the former against the later. Has this made some new major interaction? No, it's just changed the pKa of a chelator. i.e. intrinsic enthalpy of the interaction is the same the observed observed will be different at a set pH.
I don't think ligand efficiency will ever be aerodynamically valid, but as a method for determining whether you are getting bang for your buck with simple changes it is 'fit for purpose' if the person who is employing it is willing to also consider how the changes in potency are achieved.
Hello again, Anonymous Too, if it’s OK with you I’m going to suggest that we move on from the thermodynamics because the real issue is basing our perception on an arbitrary 1 M concentration unit. Unless we can actually show that 1 M is the most appropriate concentration unit, LE is actually meaningless because our perception changes when we use a different concentration unit. You mention changes is potency associated with fluorination of methane sulfonamide and once you’re trying to interpret changes in potency it becomes a whole new ball game because changes in potency are independent of units. This is the basis of group efficiency (GE)and the folk who introduced it could have greatly strengthened the case for using it had they pointed out LE’s ‘units problem’. That said, GE is a bit unwieldy (I can explain why if you’re interested but I won’t do so here) for use in discovery projects but you can analyze your project’s potency data in ways that give you very similar information. For example, you can fit pIC50 to heavy atom count (or other risk factors like ClogP) and use the residual as a measure of the extent to which the activity of a compound beats the trend (i.e. 'bang for buck'). You can also regard the pIC50 intercept as defining a concentration unit for LE calculations although for reasons I won’t go into here I don’t think that would be a great way to do the data analysis. I used the Astex GE data to illustrate this approach to analyzing affinity data in ‘Ligand efficiency: nice concept, shame about the metric’ presentation: http://www.slideshare.net/pwkenny/ligand-efficiency-metrics-n
I see the main problem with LE (and some other metrics for drug discovery) is that people tend to use them instead of analyzing their data properly. Here’s the concluding paragraph from our critique of LE metrics:
“So why should we consider ligand efficiency metrics to be harmful? Most importantly, as shown in Table 1 and Figs. 1, 2, 4 they distort our perception of the relationship between activity and the risk factor(s) with which we choose to normalize it. Neither the scaling nor the offsetting transformations used for normalizing activity has any real physiochemical basis and excessive reliance on metrics inhibits more thorough examination of data. We hope this Perspective will stimulate debate and encourage others to challenge the assumptions and opinions that shape and constrain drug discovery.”
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