The Standard State in Binding Free Energy Calculations

By Emilio Gallicchio

Post date: Jul 30, 2019 11:10:01 PM

Phys. Chem. Intro

Q: What is the standard free energy of binding?

It is the free energy of binding between a ligand and a receptor when they are in an ideal solution at the standard concentration (C^◦=1 M). The standard free energy of binding is usually denoted by ΔG^◦_b

Q: how is ΔG^◦_b measured in practice?

There are many ways to measure a standard binding free energy, the most straightforward is to measure the binding constant K_b, or equivalently the dissociation constant K_d= 1/K_b

Q: But wait, the relation above does not make sense because it takes the log of a quantity in concentration units.

In Physical Chemistry equilibrium reaction constants are dimensionless. The dimensionless nature of the equilibrium constant for binding is clear from its definition:

where R is the receptor, L is the ligand and RL is the complex. Changing the units of concentration does not change the binding constant.

Q: yeah ... I am not sure, my friend works in this lab and they measured the K_d of a drug and he said it's 1.8 nM

In fields such as Biochemistry, Medicinal Chemistry, and Biology, equilibrium constants are very confusingly reported with units, sometimes very strange units. Unless they are doing something really odd, such as redefining the standard state, read the above as K_d = 1.8 x 10^-9, no units, and you'll be fine.

Q: really, I can change the standard state?

If you want, sure, but maybe don't call it "standard". A standard it's just a reference that people have agreed on, a bit like the origin of a coordinate system. You can certainly define your own reference solution state, I don't know ... 34,512 molecules per cubic millimeter. (As an example, for acid/base reactions, biologists like to use 10^-7 M as the standard concentration of the hydronium ion, causing endless confusion.) Anyway, if you do use your own standard do not expect to get the same "standard" free energy of binding and equilibrium constant as the rest of the world, which uses molar units, for the same reason that you can't expect to get the same distance from the origin if you change the origin.

Let's stick to the 1 M standard state, please ...

The Standard State in Binding Free Energy Calculations

Q: Why do I need to report the standard free energy of binding?

The binding free energy depends on the concentrations of receptor and ligand. For example, the binding free energy is zero at equilibrium concentrations. It is meaningless to report a free energy of binding without specifying the concentrations it corresponds to. By stating that a free energy of binding is "standard", we tell people that it corresponds to the standard state at 1 M concentrations.

Q: Do I need to do a simulation at the standard state? That is, do I need to place enough ligand and receptor molecules in the simulation box so that they are at 1 M concentration?

No. You can do a calculation at any concentration you want. As the simulation progresses, monitor the number of complexes, RL, and the number of free R and L to measure their equilibrium concentrations. That will give you the equilibrium constant and standard free energy of binding using the relation above.

However, the binding constant is rarely computed by "counting" in this way. We like to do computational alchemy. It is faster and more reliable.

Q: Okay, I did an alchemical calculation. I decoupled the ligand from the solution then I coupled it to the receptor. I fed the data into MBAR etc. which spit out a free energy. Can I call it a standard free energy of binding?

Generally, no. What you have calculated is the excess component of the free energy of binding. To turn it into a standard free energy of binding, add the ideal term:

where V_site is the volume of the binding "region".

Q: Okay, that sounds easy. Where does the quantity ΔG^◦_ideal come from?

ΔG^◦_ideal is the reversible work for transferring a ligand from an ideal solution at concentration C^◦ to the binding site of volume V_site . Read these papers:

• • 1. MK Gilson, JA Given, BL Bush, JA McCammon. The statistical thermodynamic basis for computation of binding affinities: a critical review, Biophysical Journal, 72, 1047 (1997).

    2. M Mihailescu, MK Gilson. On the theory of noncovalent binding. Biophysical Journal 87 (1), 23-36 (2004).

    3. E Gallicchio and RM Levy, Recent Theoretical and Computational Advances for Modeling Protein-Ligand Binding Affinities. Advances in Protein Chemistry and Structural Biology, 85, 27-80, (2011).

Q: But how do I get V_site?

Getting V_site is easy. You have set it yourself. It is whatever volume the ligand is allowed to explore during the process of alchemically coupling the ligand to the receptor.

Q: What do you mean? I did not restrain the ligand.

Then V_site is the volume of the simulation box. Did you run the simulation long enough so that the ligand visited the whole box? Probably not, eh!? Then the simulation is not converged. Even more troubling is that you also have implicitly defined the complex RL as any conformation in which R and L are in a region of solution of the same volume as the simulation box. This is probably not what you wanted. You probably wanted to measure the standard free energy of binding of the ligand to a specific region of the receptor, not for the whole receptor and even including the solvent. You should have restrained the ligand to that region. V_site is the volume of whatever that region is that you meant to consider.

Q: Right, right ... I forgot about it. I did apply a restraint potential to avoid the "wandering ligand" problem when the ligand and receptor are decoupled.

Ok, great. Than you have a V_site after all. Is the restraint potential based on the CM-CM distance between ligand and receptor atoms? If so, simply integrate the Boltzmann factor of the restraint potential over the three coordinates of space to get V_site. Or is the restraining potential something more complicated?

Q: Yeah ... more complicated. The orientation of the ligand was also restrained.

No problem. Integrate over the orientational angles as well and get the angular binding site volume Ω_site and add -k_B T ln Ω_site⁄8 π ² to the ideal standard state factor. Read the papers I told you about.

Q: No, no. I restrained the orientation of the ligand by tethering two atoms of the ligand to two receptor atoms.

That is not allowed. It would perturb the intramolecular conformational distributions of receptor and ligand when they are not coupled. Read this paper:

Boresch S, Tettinger F, Leitgeb M, Karplus M. Absolute binding free energies: A quantitative approach for their calculation. J. Phys. Chem. B. 107, 9535–9551 (2003)

Q: I see. But after coupling the ligand to the receptor I turned off the restraints. Does that change V_site?

The binding restraints are not meant to be turned off. They are on during the whole coupling alchemical leg and they stay on.

Q: But then I am not simulating the real thing! The real complex does not have restraints!

I never said that you are simulating the real thing. We are doing computational alchemy! The binding restraints define the complex. With the binding site restraints, you are essentially specifying the set of conformations of the receptor-ligand system that form what you call the "complexed" state. You can't change the definition of the complex depending on the alchemical state of the system.

Q: Okay, now I know you finally went off the deep end. You cannot be right because then the standard free energy of binding would depend on how I set the restraints!

Precisely. The free energy of the complex depends on how you defined it. This is not different from, say, having to define the ranges of φ and ψ backbone angles of the alpha helix state in order to compute the alpha-helix population of a peptide. Change the range of the angles, and the population, as well as the corresponding free energy, changes. You need to define the complexed state before you can measure its free energy!

Q: But then I do compare the calculated free energy to the experiment? The experimental system does not have restraints. What does the experiment measure?

Good question and a very interesting topic. Read paper no. 2 above.

Q: Okay, but now I am worried that I have used harmonic restraints centered on a specific pose of the ligand which I got from docking and I am not sure it is the correct bound pose. Is it a problem?

It could be. It is safer to use flat-bottom harmonic restraints for the binding site volume. That way you have more tolerance and a greater chance of including the "correct" pose.

Q: I am starting to get this. But I heard that imposing restraints and then releasing them can improve convergence. Is that wrong?

It is absolutely fine to impose and then release additional restraining potentials along the alchemical path to improve convergence. These do not need to obey the same strict rules for the binding site restraints discussed above. Just make sure that the binding site restraints stay on throughout the alchemical simulation.

Q: Now I understand why many people prefer to compute relative binding free energies (FEP) rather than the standard binding free energies! They do not need to worry about standard states, the binding site volume, and all this stuff!

Actually, all of this applies to relative binding free energies as well. The relative binding free energy is the difference between two standard binding free energies. Because these depend on the definition of the binding site, so does the relative binding free energy. In principle, without restraints, a ligand undergoing an FEP transformation will eventually dissociate and wander around the simulation box adversely affecting the relative free energy estimate. The longer the simulation the higher the chance that this will happen. You want your estimate to improve, not worsen, as you make the simulation longer. Binding site restraints should be used in FEP as well ... but people rarely do it.

Don't ask. I really don't know why.