The only two equations that you should know

“Chemistry”, declared the Nobel laureate Roger Kornberg in an interview, “is the queen of all sciences. Our best hope of applying physical principles to the world around us is at the level of chemistry. In fact if there is one subject which an educated person should know in the world it is chemistry.” Kornberg won the 2006 Nobel Prize in chemistry for his work on transcription which involved unraveling the more than dozen complicated proteins involved in the copying of DNA into RNA. He would know how important chemistry is in uncovering the details of a ubiquitous life process.

I must therefore inevitably take my cue from Kornberg and ask the following question: What equation would you regard as the most important one in science? For most people the answer to this question would be easy: Einstein’s famous mass-energy formula, E=mc². Some people may cite Newton’s inverse square law of gravitation. And yet it should be noted that both of these equations are virtually irrelevant for the vast majority of practicing physicists, chemists and biologists. They are familiar to the public mainly because they have been widely publicized and are associated with two very famous scientists. There is no doubt that both Einstein and Newton are supremely important for understanding the universe, but they both suffer from the limitations of reductionist science that preclude the direct application of the principles of physics to the everyday workings of life and matter.

Take Einstein’s formula for instance. About the only importance it has for most physical scientists is the fact that it is responsible for the nuclear processes that have forged the elements in stars and supernova. Chemists deal with reactions that involve not nuclear processes but the redistribution of electrons. Except in certain special cases, Einstein therefore does not figure in chemical or biological processes. Newton’s gravitational formula is equally distant for most chemists' everyday concerns. Chemistry hinges on the attraction and repulsion of charges, processes overwhelmingly governed by the electromagnetic force. This force is stronger than the gravitational force by a factor of 10³⁶, an unimaginably large number. Gravity is thus too weak for chemists and biologists to bother with in their work. The same goes for many physicists who deal with atomic and molecular interactions.

Instead here are two equations which have a far greater and more direct relevance to the work done by most physical and biological scientists. The equations lie at the boundary of physics and chemistry, and both of them are derived from a science whose basic truths are so permanently carved in stone that Einstein thought they would never, ever need to be modified. The man who contributed the most to their conception, Josiah Willard Gibbs, was called "the greatest mind in American science" by Einstein. The science that Gibbs, Helmholtz, Clausius, Boltzmann and others created is thermodynamics, and the equations we are talking about involve its most basic quantities. They apply without exception to every important physical and chemical process you can think of, from the capture of solar energy by plants and solar cells to the combustion of fuel inside trucks and human bodies to the union between sperm and egg.

Two thermodynamic quantities govern molecular behavior, and indeed the behavior of all matter in the universe. One is the enthalpy, usually denoted by the symbol H, and roughly representing the quantity of energy and the strength of interactions and bonds between different atoms and molecules. The other is the entropy, usually denoted by the symbol S, and roughly representing the quality of energy and the disorder in any system. Together the enthalpy and entropy make up the free energy G, which roughly denotes the amount of useful work that can be extracted from any living or non-living system. In practical calculations, what we are concerned with are changes in these quantities rather than their absolute values, so each one of them is prefaced by the symbol ∆, indicating change. The celebrated second law of thermodynamics states that the entropy of a spontaneous process always increases, and it is indeed one of the universal facts of life, but that is not what we are concerned with here.

Think about what happens when two molecules – of any kind – interact with each other. The interaction need not even be an actual reaction, it can simply be the binding of two molecules to one another by strong or weak forces. The process is symbolized by an equilibrium constant K_e, which is simply the ratio of the concentrations of the products of the reaction to the starting materials (reactants). The bigger the equilibrium constant, the more the amount of the products. K_e thus tells us how much of a reaction has been completed and how much reactant has been converted to product. Our first great equation relates this equilibrium constant to the free energy of the interaction through the following formula:

∆G⁰ = -RT ln K_e

or, in other words

K_e = e^-∆G0/RT

Here ln is the natural logarithm to base e, R is a fundamental constant called the gas constant, T is the ambient temperature and ∆G₀ is the free energy change under so-called 'standard conditions' (a detail which can be ignored by the reader for the sake of this discussion). This equation tells us two major things and one minor thing. The minor thing is that reactions can be driven in particular directions by temperature increases, and exponentially so. But the major things are what's critical here. Firstly, the equation says that the free energy in a spontaneous process with a favorable positive equilibrium constant is always going to be negative; the more negative it is the better. And that is what you find. The free energy change for many of biology's existential reactions like the coupling of biological molecules with ATP (the “energy currency” of the cell), the process of electron transfer mediated by chlorophyll and the oxidation of glucose to provide energy is indeed negative. Life has also worked out ingenious little tricks to couple reactions with positive (unfavorable) ∆G changes to those with negative ∆G0 values to give an overall favorable free energy profile.

The second feature of the equation is a testament to the wonder that is life, and it never ceases to amaze me. It attests to what scientists and philosophers have called “fine-tuning”, the fact that evolution has somehow succeeded in minimizing the error inherent in life’s processes, in carefully reining in the operations of life to within a narrow window. Look again at that expression. It says that ∆G₀ is related to K_e not linearly but exponentially. That is a dangerous proposition because it means that even a tiny change in ∆G₀ will correspond to a large change in K_e. How tiny? It should be no bigger than 3 kcal/mol.

A brief digression to appreciate how small this value is. Energies in chemistry are usually expressed as kilocalories per mole. A bond between two carbon atoms is about 80 kcal/mol. A bond between two nitrogen atoms is 226 kcal/mol: this is why nitrogen can be converted to ammonia by breaking this bond only at very high temperatures and pressures and in the presence of a catalyst. A hydrogen bond - the "glue" that holds biological molecules like DNA and proteins together - is anywhere between 2 and 10 kcal/mol.

3 kcal/mol is thus a fraction of the typical energy of a bond. It takes just a little jiggling around to overcome this energy barrier. The exponential, highly sensitive dependence of Ke on ∆G₀ means that changing ∆G from close to zero to 3 kcal/mol will translate to changing K_e from 1:99.98 in favor of products to 99.98:1 in favor of reactants (remember that K_e is a ratio). This is a simple mathematical truth. Thus, a tiny change in ∆G₀ can all but completely shift a chemical reaction from favoring products to favoring reactants. Naturally this will be very bad if the goal of a reaction is to create products that are funneled into the next chemical reaction. Little changes in the free energy can therefore radically alter the flux of matter and energy in life’s workings. But this does not happen. Evolution has fine-tuned life so well that it has remained a game played within a 3 kcal/mol energy window for more than 2.5 billion years. It's so easy for this game to quickly spiral out of hand, but it doesn’t. It doesn’t for the trillions of chemical transactions which trillions of cells execute everyday in every single organism on this planet.

And it doesn’t happen for a reason; because cells would have a very hard time modulating their key chemical reactions if the free energies involved in those reactions had been too large. Life would be quickly put into a death trap if every time it had to react, fight, move or procreate it had to suddenly change free energies for each of its processes by tens of kilocalories per mole. There are lots of bonds broken and formed in biochemical events, of course, and as we saw before, these bond energies can easily amount to dozens of kcals/mol. But the tendency of the reactants or products containing those bonds to accumulate is governed by these tiny changes in free energy which nudge a reaction one way or another. In one sense then, life is optimizing small changes (in free energy of reactions) between two large numbers (bond energies). This is always a balancing act on the edge of a cliff, and life has managed to be successful in it for billions of years. It's one of the great miracles of the universe.

The second equation is also a relationship between free energy, enthalpy and entropy. It's simpler than the first, but no less important:

∆G = ∆H - T∆S

The reason this equation is also crucial to the operation of the universe is because it depicts a fine dance between entropy and enthalpy that dictates whether physical processes will happen. Note that entropy is multiplied by the temperature here and the sign is negative. So if it decreases in a process then ∆S becomes negative and the overall product (T∆S) becomes positive. In that case the change in enthalpy needs to be negative enough to compensate, otherwise the free energy will not be negative and the process won't take place. 

For instance, consider the schoolboy experiment of oil and water not mixing. When oil is put into water, the water molecules have to order themselves around oil molecules, leading their entropy to decrease and become negative. The attraction between water and oil on the other hand is weak, so the change in enthalpy does not compensate for the change in entropy, and oil does not mix. This is called the hydrophobic effect. It's a fundamental effect governing a myriad of critical phenomena; drugs interacting with signaling proteins, detergents interacting with grease, food particles attracting or repelling each other inside saucepans and human bodies. On the other hand, salt and water mix easily; in this case, while the entropy is still unfavorable because of the ordering of water molecules around salt molecules, the enthalpy is overwhelmingly favorable (negative) because the positive and negatively charged sodium and chloride ions strongly attract water.

Because temperature is part of the equation it too plays an important role. For instance consider a phenomenon like a chemical reaction in which the change in entropy is favorable but quite small. We can then imagine that this reaction will be greatly accelerated if T is high, making the product of it and the entropy large. This explains why the free energy of chemical reactions can be made much more favorable at high temperatures (there is a subtlety here, however: making the free energy more favorable is not the same as accelerating the reactions, it's simply making the products more stable. The difference is between thermodynamics and kinetics).

Even the origin of life during which the exact nature of molecular interactions was crucial in deciding which ones would survive, replicate and thrive was critically dependent on enthalpy and entropy. When little oily molecules called micelles repelled water molecules because of the unfavorable entropy and enthalpy described above, they sequestered themselves into tiny bags inside which fragile molecules like DNA and RNA could safely isolate themselves from the surrounding water. These DNA and RNA molecules could then experiment with copying themselves at leisure, not having to worry about being hydrolyzed by water. The ones with higher fitness survived, kickstarting the process which, billions of years later, finally led to this biped typing these words on his computer.

That's really all there is to life. We all thus hum along smoothly, beneficiaries of a 3 kilocalorie energy window and of the intricate dance of entropy and enthalpy, going about our lives even as we are held hostage to the quirks of thermodynamic optimization, walking along an exponential energy precipice.

And all because K_e = e^-∆G₀/RT