Efficient Voodoo Thermodynamics

So you’ve got a little data analysis problem.  You have some compounds with a range of IC₅₀ values and you’d like to explore the extent that molecular size contributes to potency/affinity.  Here’s one suggestion for starters.  Plot pIC₅₀ against your favourite measure of  molecular size which could be number of number of non-hydrogen atoms, molecular weight and look at the residuals which tell you how much each compound beats the trend (or is beaten by it).  Let’s start by defining  pIC₅₀ which you can calculate from:

 pIC₅₀ = -log(IC₅₀/M)                                                                              1

You might be asking why I’m not suggesting that you use the standard Gibbs free energy of binding which is defined by:

ΔG° = RTln(K_d/C°)                                                                                 2

The main reason for not doing this is that we can’t.  You’ll notice that ΔG° is calculated from K_d and not  IC₅₀ and the two are not the same thing even though they both have units of concentration.  Those of you who have worked on kinase projects may have even used IC₅₀ values measured at different ATP concentrations to get a better idea how much kick your inhibitors will have at physiological ATP concentration.  Put another way, you can measure the concentration of sugar in your coffee and plug this into equation 2 but that does not make what you calculate a standard Gibbs free energy of binding.   If you’ve measured IC₅₀ then you really should use  pIC₅₀ in this analysis.  Converting pIC₅₀ to ΔG° is technically incorrect and arguably pretentious since the converted pIC₅₀ can give the impression that it is somehow more thermodynamic than that from which it was calculated.  Converting pIC₅₀  to ΔG° also introduces additional units (of energy/mole) and there is always a degree of irony when these units, which may have been introduced to just make biological data look more physical, get lost when the results are presented.

So let’s get back to the data analysis problem.  Suppose that we’ve plotted  pIC₅₀ against number of heavy (i.e. non-hydrogen) atoms and the next step is to fit the data.  Best way to start is to fit a straight line although you could also fit a curve if the data justifies this.  Let’s assume that we’re fitting the straight line:

  pIC₅₀ = A + B×N_HA                                                                            3

I realise that using an intercept term (A) will cause a few eyebrows to become raised.  Surely the line of fit should go through the origin?   There is a problem with this line of thinking and it’s helpful now to talk instead in terms of affinity and ΔG° to develop the point a bit more.  You might be thinking that in the limit of zero molecular size a compound should have zero free energy of binding.  However, a zero free energy of binding corresponds to K_d being equal to the standard concentration and you’ll remember that the choice of standard state is arbitrary.  If you must derive insights from thermodynamic measurements then the very least that you can do is to ensure that any insights you derive are invariant with respect to the value of the standard concentration.  

When you use ligand efficiency (-ΔG°/N_HA) you’re effectively assuming that the value of ΔG° should be directly proportional to the number of heavy atoms in the ligand molecule.  One consequence of defining ligand efficiency in this manner is that relative values of ligand efficiency for compounds with different numbers of heavy atoms will change if you change (as thermodynamics tells us that we are allowed to do) the standard concentration used to define ΔG°.  I've droned on enough though and it's time to check out.  I will however, leave you with the question of whether it makes sense to try to correct ligand efficiency for the effects of molecular size. 

The wrong kind of free energy

It has been some time since I last blogged.   There has been the distraction of a little drug-likeness project and there is still much to do before I leave Brasil at the end of May.  I have, however, set up a twitter account (@pwk2013) which I use in futile attempts to engage minor celebrities (like God and Richard Dawkins) in conversation.   I’ll be taking a look at some aspects of thermodynamics in this post and I’ll start by writing the relationship between dissociation constant and the standard free energy of binding:

ΔG° = RTln(K_d/C°)

I realise that this might look a bit different to what you’re used to so I’ll try to explain why its been written like this.  First of all K_d has units of concentration and logarithms are only defined for numbers (i.e. unitless quantities) so you might want to consider the possibility that what you normally write is wrong (apologies for hideous pun).   The other thing that you need to remember is that the standard free energy of binding depends on the choice of standard state and writing the equation like this makes this connection explicit.   ΔG° is negative for a 10nM compound when the standard concentration is 1M.  However, if we were to define the standard concentration to be 1nM then   ΔG° would be positive.  If you consider the idea of a 1nM standard concentration to be offensive, think of a 1M solution of your favourite protein...

One of the misconceptions that drug discovery researchers tend to have is that you can’t call it thermodynamics unless you measure both enthalpy and entropy.  One reason for this is the use of isothermal titration calorimetry (ITC) to measure affinity.  ITC is a direct, label-free method for measuring affinity and you get the enthalpy as part of the package.  One of the important things to remember about thermodynamics is that you need to use the most appropriate thermodynamic quantity to describe the phenomenon that you’re interested in and for binding at constant pressure this is the Gibbs free energy.  In contrast, the engineer scaling up a synthetic reaction for manufacturing a drug will usually be more interested in the heat and volume (i.e. gaseous bi-products) generated by the reaction because failure to control these can result in the Big Kaboom.

One idea that has emerged as ITC has become more accessible in drug discovery is that enthalpy-driven binding is somehow better than entropy-driven binding.  I remain sceptical and ask the rhetorical question of how an isothermal system such as a human taking a drug senses the degree to which engagement of the drug’s target(s) is driven by enthalpy changes.   If measuring enthalpy of binding in addition to affinity helps us to predict affinity for compounds yet to be synthesised or gives us clear (i.e. keeping arms stationary) insights into the nature of binding then it makes sense to do it.  However, I’m not seeing this being done convincingly and sometimes wonder whether enthalpy optimisation is just a case of the wrong kind of snow...

We tend to interpret affinity in terms of contacts between protein and ligand although one must always remember that the contribution of a particular contact to affinity is not in general an experimental observable.   Proteins and their ligands associate in an aqueous environment and the non-local nature of the hydrophobic effect further complicates attempts to use structure to rationalise affinity.  I often hear hydrophobic interactions being described as non-directional and in my view this is complete bollocks since interactions are forces and forces are vectors which by definition have direction.  On this subject, I can't resist telling the tale of the Human Resources department getting the Scalar Quantity Award for having magnitude but no direction...

I think this is a good place to leave things and I'll try to be back in a week or some more specific stuff in a follow up to this post.