The Unbearable Heat Capacity of Being

There is a peculiar connection in my mind; that between thermodynamics and Beethoven's 5th symphony. I was in my final year of high school and it was a rainy and stormy night outside. I had to desperately study thermodynamics for my final exam. The only light that was on was from my table lamp. I was also listening to Beethoven's 5th symphony for the 2nd or 3rd time. Somehow within the mystical shadows and strange shapes manifested by the light, the strains of the strings and the equations of entropy formed a hybrid meld in my mind that has never dissociated. After that night, whenever I read thermodynamics, I don't always remember Beethoven's 5th. But whenever I listen to Beethoven's 5th, I am immediately transformed to that night, into the middle of a fluid energy landscape if you will.

Since then thermodynamics has been an enduring interest of mine. Another reason why it has been an interest of mine is because I don't understand it very well. In my opinion thermodynamics is one of those difficult subjects like quantum mechanics, where a great deal of effort has to be put into understanding abstract concepts and even then concepts remain elusive. Maybe it's a feature of all those sciences that are intimately bound with the fabric of matter and life. It is relatively easy to colloquially grasp entropy as an increase in disorder- we can grasp this point every time we put ice in our drink even as we struggle to understand thermodynamic principles- but much harder to get the physical meaning of the derivative of the pressure with respect to the entropy or some similar expression. Enter the Maxwell relations.

Over the years I have found myself coming back to thermodynamics and repeatedly trying to understand its fine points. I have a long way to go but I am confident I am going to continue my frequently ineffectual efforts. There are some classic books which I have encountered on the way that have served as guides, sometimes strict and sometimes gentle- Enrico Fermi's "Thermodynamics" is a jewel still in print, the thermodynamics treatment in Alberty and Silbey's physical chemistry book is quite nice and Ken Dill's Molecular Driving Forces has the best treatment of statistical thermodynamics applied to chemical and biological systems that I am aware of. There's also an old book on thermodynamics which is gold- Samuel Glasstone's "Thermodynamics for Chemists".

I cannot deny the value of thermodynamics and what it has taught me. Thermodynamics has been immensely useful in understanding computational chemistry, conformational changes in biomolecules and especially protein-ligand binding. All that really matters for protein ligand binding and the orchestration of the actions of numerous naturally occurring ligands and drugs is the free energy change ∆G. More than any other, there is one overriding goal today among the groups of people who are in the business of prediction- to predict binding affinity from first principles. Free us they say, free us from the constraints of predicting free energy.

There is an all-pervasive equation relating ∆G to the equilibrium constant of a reaction- ∆G=-RTlnK. This is perhaps the single most compelling equation in biology. Why? Because it tells you that life lives within a roughly 3 kcal/mol energy window. All the jiggling that transmits signals, folds proteins, docks molecules, makes neurons buzz, mainly happens within a 3 kcal/world. That does not of course mean that no process can have a ∆G of more than 3 kcal/mol, but it does mean that fragile life is pretty tightly constrained and can call the shots only within a limited thermodynamic domain. The reason is that a difference in ∆G of 3 kcal/mol means that the favourable product in any reaction exists to the extent of 99.96%. The exponential dependence of K on ∆G takes care of this. 3 kcal/mol is all a protein needs to toss at a ligand to decisively shift the equilibrium to the side of the bound ligand. It can of course toss more but 3 is enough. One of the reasons why prediction of binding affinity is still so difficult is because 'small' errors of 1 kcal/mol or so translate into huge differences in equilibria. Nature with its fondness for exponentials has doomed life- and chemists- to operate in a straitjacket.

But this same fondness has also made it possible to modulate different reactions and binding events in living systems with exquisite precision. The 3 kcal/mole value perfectly encapsulates the workings of such critical interactions as hydrogen bonds and Van der Waals forces. Expulsions of water, making and breaking of salt bridges, dispersion interactions, peptide hydrogen bond formation; everything can take place within 3 kcal/mol. At the same time the magic number of 3 also ensures that these interactions can be fleeting and rapidly annihilated and molecular partners can dissociate whenever necessary. What reins us in also frees us to explore an ever-widening energy landscape of weak interactions that strike the precise balance. By consigning our lives to whimsical energetic windows, we have finally liberated ourselves from the temptation of falling for monstrous blooming thermodynamic calamities that would have snuffed life out. We can be fortunate that we are not asymptotically free.

But ∆G is like statistics (or some would say like skirts); it hides much more than it reveals. Most techniques can give you ∆G but unraveling the details of a molecular process can immensely benefit from the knowledge of ∆H and ∆S, two crucial components that make up another of biology's key equations- ∆G=∆H-T∆S. Contrast ligand binding with ballroom dancing- what matters is not only how steadily you can hold on to your partner but also how flexible you concomitantly are. The correct combination of motion and attraction in this case can provide a cascade of favourable events. Ditto for ligand binding. Techniques like calorimetry can provide these valuable details. Theoretically, an infinite combination of ∆H and ∆S can add up to a ∆G value, which is all the more reason for finding out the exact composition that makes up a particular value. Two isoenergetic processes need not be either isoenthalpic or isoentropic. In a future post, I will mention a review that explores this aspect; suffice it for now to say that subtle differences in structure may give us the same ∆G but very different decomposition of ∆H and ∆S. Generally intermolecular forces contribute the most to ∆H while hydrophobic effects and the freeing up of water contribute dominantly to ∆S.

And so life lives and breathes, supported on two stilts. These two equations, one endowing biological reactions with the correct equilibria and the other modulating biological action by injecting the precise dosage of two key quantities are like the Magi. They bring us great gifts of understanding and insight. They ask only that we give them a patient ear.