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