Thermodynamics and molecular interactions

So it’s #RealTimeChem week on twitter and I thought I’d get into the spirit with a blog post.  The article that I’ve selected for review focuses on non-additivity of functional group contributions to affinity.  The protein in question is Thrombin and ligand binding was characterised using protein crystallography and isothermal titration calorimetry.  

Before reviewing the article, it’s probably a good idea to articulate my position on the thermodynamics of ligand-protein binding.  Firstly, G, H and S are three state functions, each of which can be written in terms of the other two, but only one of which is directly relevant to the binding of ligands to proteins.  K_d is no less thermodynamic than ΔH° or ΔS° and the contribution of a particular intermolecular contact to ΔG° (or ΔH° for that matter) is not in general an experimental observable.  Thermodynamics with state functions is like accountancy in that if you over-pay one interaction, the other interactions will lose out.

Now back to the featured article.  One observation presented as evidence for non-additivity is that the slopes of plots of ΔG° against hydrophobic contact area differ according to whether X (see figure below) is H or NH2.  Inhibitors with the amino group bind with greater affinity than the corresponding inhibitors lacking the amino group which interacts (in its cationic form) with the protein.   However, the difference is not constant and actually increases with hydrophobic contact area.   I would certainly agree that something interesting is going on and if we’re ever going to understand ligand-protein then combining affinity measurements for structurally-related ligands with structural information will be very useful.   

About enthalpy measurements, I am a lot less sure.  Isothermal titration calorimetry (ITC) is a direct, label-free method for measuring affinity but the fact that you get ΔH° from the experiment at no extra cost doesn’t by itself make ΔH° useful.     If measured values of ΔH° lead to improved predictions of ΔG° or provide clear insight into the nature of the interactions between ligand and protein then I certainly agree that we should make use of ΔH°.  However, there is still the problem that there is no unique way to distribute ΔG°  (or, for that matter, ΔH°) over intermolecular contacts and biomolecular recognition also takes place in aqueous media.    The cohesiveness of liquid water that drives hydrophobic association in aqueous media is a consequence of strong, cooperative hydrogen bonds between water molecules and I like to think of the hydrophobic force as a non-local, indirect, electrostatic interaction.  This non-local nature of hydrophobic interactions complicates interpretation of affinity measurements in structural terms.

Let's see what the authors have to say:

"Analysis of the individual crystal structures and factorizing the free energy into enthalpy and entropy demonstrates that the binding affinity of the ligands results from a mixture of enthalpic contributions from hydrogen bonding and hydrophobic contacts, and entropic considerations involving an increasing loss of residual mobility of the bound ligands."

I'm going to put my cards on the table and say that I believe this statement represents an exercise in arm waving.    See what you think and ask yourself the question as to whether this statement would help you design a higher affinity Thrombin inhibitor?  It's also worth thinking carefully about the relationship between mobility and entropy.   One way of looking at entropy is as the degree to which systems are constrained and a more highly constrained system will be less mobile.    The authors state:

"The present study shows, by use of crystal structure analysis and isothermal titration calorimetry for a congeneric series of thrombin inhibitors, that extensive cooperative effects between hydrophobic contacts and hydrogen bond formation are intimately coupled via dynamic properties of the formed complexes."

I think that one needs to be very careful when talking about 'dynamic properties' in the context of equilibrium thermodynamics.  Ultimately entropy is determined by the characteristics of potential energy surfaces and Statistical Mechanics tells us that entropy (and other thermodynamic properties) can be calculated from the partition function.  It may be instructive to think how one would use the partition function to put these 'dynamic properties' on a quantitative basis.

I'm now going to change direction because there is one factor that the authors appear not to have considered  and I think that it could be quite important.  The ligand amino group (see figure above; it's protonated under assay conditions) interacts with the protein but it can also affect affinity in another way.  Have a look a this figure (showing the stricture of complex with 3e) from the article which shows how one of the inhibitors lacking the amino group binds.  Now I'd like you to look at one of the dihedral angles in the ligand structure and you can see how it is defined by looking at the substructures inset in the histograms below.  

 

The histograms show the distributions of (the absolute value of) this dihedral angle observed for the instances of substructure in the CSD. The histogram on the left suggests that the dihedral angle will tend to be 180° when the carbon next to the carbonyl carbon has two attached hydrogens. The histogram on the right shows how a substituent (X ¹ H) on that carbon shifts the distribution of dihedral angles.  Now let's go back to the structure of complex 3e, which can be downloaded from the PDB as refcode 2ZIQ, and we can see that the relevant dihedral angle is 117°.  Comparing the two histograms tells us is that a substituent (such as an amino group) on the carbon next to the carbonyl group will tend to stabilise the bound conformations of these inhibitors in addition to making direct interactions with the protein.  Do the ITC results tell us this?       

Literature cited

Baum, Muley, Smolinski, Heine, Hangauer and Klebe (2010) Non-additivity of functional group contributions by protein-ligand binding: A comprehensive study by crystallography and isothermal titration calorimetry. J Mol Biol 397:1042-1054  doi

Unknown knowns and binding kinetics

I hope that you all enjoyed the April Fool post but now it’s time to get back to business.  Before returning to Thermodynamics, I’d like to point you towards a couple of articles that you might find to be of interest.  In the first of these, Ben Davis and Practical Fragments’ Dan Erlanson take a look at some of what can go wrong when you screen fragments and it is noteworthy that they actively sought material from the fragment community before writing the review.  If you’re thinking about running a fragment screen, you do need to read this article which has already been featured in other blog posts ( 1 | 2 | 3).    

The article by Göran Dahl and Tomas Åkerud focuses on binding kinetics, specifically in the context of pharmacokinetics. In multistep processes, timescale determines relevance.  My take is that slow binding is equivalent to slow distribution and, at some point when the topic is raised, I will usually get round to asking whether slow distribution is something that you would want to design into a drug. The authors conclude that ‘the drug-target residence time appears to be limited in scope for prolonging duration of effect’ and the article provides a timely reminder that drugs act within a pharmacokinetic framework.

Literature cited

Davis & Erlanson (2013) Learning from our mistakes: The ‘unknown knowns’ in fragment screening. Bioorg Med Chem Lett In press DOI

Dahl & Åkerud (2013) Pharmacokinetics and the drug-target residence time concept. Drug Discov Today In press DOI

Isotopic enrichment by recrystallisation from water

Isotopic enrichment doesn’t normally fall under the molecular design remit.  However, a radically new approach has been recently discovered and predictive modelling was very important in showing that the method would indeed be feasible.  Isotopic enrichment is typically carried out in gas phase and if the elemental form is involatile it is necessary to prepare a compound which can be more readily persuaded into the gas phase. So if you want to enrich uranium you first make the relatively volatile hexafluoride (fluorine is good because it’s only got one natural isotope) and then stick it in a centrifuge and away you go.

The isotopic enrichment industry has been watching developments in aqueous solubility prediction with great interest.  As you’ll know aqueous solubility is a very important physicochemical property and a key determinant of the efficiency with which a drug is absorbed from the gastrointestinal tract.  By focussing on aqueous solubility you can get away from all the technological demands of working in gas phase.  You can do this in your garage.

Most pharmaceutical researchers are familiar with the idea that aqueous solublility decreases with molecular weight and there is large literature of many seminal articles on the subject.  One consequence of this anti-correlation is that the average solubility for compounds above a particular molecular weight cut off will always be lower than the average solubility for the compounds with molecular weights below the cut off.  The selection of the optimal cut off is a challenging problem and it was believed that the only option available was to design compounds such that the masses of two isotopic forms straddled the value of the molecular weight cut off.  Only recently has the seminal 1940 study of Bronshtein and Mercader, the result of a short collaboration in Mexico City, come to light.  In a nutshell, their study unequivocally demonstrated that the cut off could be set to the centroid of the isotopic masses for any compound.  The math in the article is truly formidable but the essence of the method is that diagonalisation of the reduced mass tensor is NKVD-complete.  Thus the cut off can be matched to compounds which is much easier than having to design compounds to the bracket the cut off.

Literature cited

Bronshtein and Mercader, Мисс Скарлетт с ледорубом в исследовании.  Dokl Akad Nauk 1940, 7, 432-456 doi