Ammm:ProteinLigandBFE
Protein-Ligand Binding Free Energy (Elizabeth E. Wait) - WORK IN PROGRESS
What is binding free energy?
Binding free energy is the change (difference) in free energy between the unbound reactants and the bound complex.
Changes in free energy (such as binding free energy) relate to changes in potential energy.
The Zwanzig Equation, shown below, demonstrates how the change in free energy between two neighboring states relates to the change in potential energy between those two states.
R – Boltzmann's constant
T – Temperature
U – Potential Energy
r – Position
The potential energy of a system, such as a protein and a ligand, is the sum over all the atoms for all of the bonded and nonbonded interactions. In a perfect world, quantum mechanics would be used to define these interactions. Unfortunately, QM methods are too computationally expensive to use on systems larger than a few atoms and for times long enough to be useful. So instead, we need a help from classical mechanics, which we can use to approximate these interactions. What we call “force fields” in the field of MD are mathematical models used to describe these atomic interactions in a way that is easier to calculate, meaning the computer can get us an answer in a more reasonable amount of time. Despite the name, force fields describe the potential energy of the system, so force is actually the derivative (change) of the potential energy over the change in position.
Bonded interactions include things like bond stretching, angle bending, dihedral bending, out-of-plane bending, and stretch bending. Most force fields are very similar in these descriptions. We often refer to "parameters." In the context of MD force fields, parameters are the values we use for the constants in the mathematical models. For example, if we represent bond stretching with a term that looks similar to the potential energy of a spring, one of our parameters would be the spring constant. The values we use for parameters come from fitting to QM and/or experimental results.
The nonbonded interactions can include things like electrostatics, repulsion-dispersion, and dipole induction. These terms are where force fields tend to differ most from each other. The Ren Lab is involved in the development of the AMOEBA polarizable force field, which has a more complex description of electrostatics (more terms) that allows it to describe many systems and interactions with higher accuracy.
•This is really important for calculating binding free energy, as electrostatics are extremely important in protein-ligand interactions
•We have been able to calculate binding free energies of protein-ligand systems to within 1 kcal/mol of experimental results (aka chemical accuracy)
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•It also has terms for stretch-bend and other cross terms
•Mi = [qi, mui, Qi, …] where q, and Q are the monopole, dipole, and quadrupole components, respectively.
•T tensors come from extended derivatives of the inverse distance between the sites I and j, 1/R.
•these numbers were fit to noble gas (ne...ne) interaction where vdw is dominant.
Why do we care about calculating BFE?
•We can learn:
•How stable is complex?
•Complex more favorable than unbound?
•Is one binding conformation more favorable?
•Relative binding free energies
•Binding with one ligand more favorable than binding with another?
•Does one protein bind a ligand more favorably than another protein?
•What interactions are important for binding?
•But they are more accurate and our lab has even been able to describe absolute binding free energies between proteins and ligands at chemical accuracy (within 1 kcal/mol of experiment)
•MD has been used many times to investigate binding between a kinase and one or several inhibitors
Note: Absolute vs Relative (error cancellation and comparing to experiment)
•Protein-ligand
•Drug design (will mostly talk about in this context)
•Protein-ion
•Ion channels
•Rare earth metals (Lanthanum)
•New material – mimic protein pocket
•Does including a certain residue at a certain position relate to more stable binding of the ion
Kd relates to binding free energy
•Dissociation constant – ratio at equilibrium
•Kinetics – rate – determined by barrier height (activation energy – highest step is rate-determining)
•Kd doesn’t change if the height changes (higher barrier – slower but doesn’t impact equilibrium concentrations)
R – ideal gas constant
T – temperature
Kd – dissociation constant
•Free energy is a state function, meaning its value is not determined by the path you take to get there
•No matter what comes between here (point to line near uncomplexed) and here (point to line near complex), the binding free energy will be the same
•The path does impact kinetics, but the dissociation constant is determined by the ratio of forward and backward at equilibrium, rather than how long the reaction takes to get there
•we can relate binding free energy (change in energy from unbound complex to bound complex) to Kd – dissociation constant
•lower binding free energy corresponds with lower Kd and therefore higher affinity
•Dissociation constant – ratio at equilibrium
•Kinetics – rate – determined by barrier height (activation energy – highest step is rate-determining)
•Kd doesn’t change if the height changes (higher barrier – slower but doesn’t impact equilibrium concentrations)
•Free energy is a state function, meaning its value is not determined by the path you take to get there
•No matter what comes between here (point to line near uncomplexed) and here (point to line near complex), the binding free energy will be the same
•The path does impact kinetics, but the dissociation constant is determined by the ratio of forward and backward at equilibrium, rather than how long the reaction takes to get there
•we can relate binding free energy (change in energy from unbound complex to bound complex) to Kd – dissociation constant
•lower binding free energy corresponds with lower Kd and therefore higher affinity
Traditionally, you test a bunch of potential inhibitors with your target protein that causes a disease or something
You take the one that has the highest affinity and try to develop it from there
Want high affinity, so
Low Kd
Low binding free energy
Want Large separation of affinities for desired and undesired targets!
So you want the ligand to have a high affinity for your target protein but low affinity for other proteins
You just want it to bind the protein of interest
Something kind of novel about my kinase inhibitor project is that I was looking at how my drug binds with other proteins, rather than testing a bunch of drugs to see which binds my protein
Getting binding free energies between the ligand and a bunch of proteins hopefully lets me predict binding affinity
So I can see if it looks like it may be more favorable for my drug to bind with proteins I don’t want it to than to my target, and see if I need to make some changes to fix that
That project is a good example of why absolute binding free energies are important and useful, because it is harder to get error cancellation comparing binding with a bunch of different proteins than for just changing the ligand
•Traditional: which inhibitor binds target protein with highest affinity
•Novel (my project): which protein binds inhibitor with highest affinity (hopefully target protein)
Can get affinities with assays / machine learning but won’t get mechanism underlying differences
No mechanism
•Binding Assays (expensive)
•Machine Learning (no physics)
High throughput, cheap, predict affinities and mechanism but NOT accurate
•Molecular Docking (crude model of physics)
•And yes, MD is not the only way to get binding free energies
•But a lot of the other methods are not great
•They are often expensive and time-consuming or not very accurate
•MD is also cool because we can get information on the binding mechanism and see what is underlying the differences in binding free energies between say a protein and two ligands
How do we calculate BFE?
Force Fields Intro
AMOEBA is cool Previous Results - Chemical Accuracy
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•I will calculate binding free energy using double decoupling and the Bennett Acceptance Ratio
•we find the free energy for hydration of the ligand and complexation between the ligand and protein
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•How do we do this?
•We scale down the interactions between the ligand and its environment in tiny steps and add up the changes in free energy along the way
•The complexation free energy minus the hydration free energy gives us the binding free energy
•All together, this is what’s referred to as our thermodynamic cycle – this is our roundabout way of getting binding free energy
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•We pretty much break all the contributions to it down into tiny steps which are easier to calculate
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••this is what’s referred to as the thermodynamic cycle
•We have a complex and we want to know the binding free energy
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•We start with that complex with full interaction between the ligand and protein
•We slowly scale down the strength of the interaction between ligand and protein in a bunch of tiny steps until the ligand is no longer interacting with its environment at all
•First we scale down electrostatic interactions, then we remove van der Waals
•We say that the ligand is a “ghost” when it is no longer interacting
•We calculate the change in potential energy at each tiny step and add them all up
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•We do the same thing for the ligand in solution – scale down the interactions in tiny steps then add them all up
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•We get the values for free energy of the ligand interacting with solution and the free energy of the ligand interacting with its host
•The complexation free energy minus the hydration free energy gives us the binding free energy
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•By scaling down interactions and seeing how that changes the energy, we get the energy of those interactions
•It sounds like a weird way to get there, but free energy is a state function so our path doesn’t matter
BAR
•It is really hard to get change in free energy between two states that are not close together, so that’s why we have to break things up and calculate everything as smaller steps with smaller differences
•For example, if we have state A and state B in this formula, it won’t be something like A is full interaction between ligand and protein and B is zero interaction
•It would likely be something like 100% electrostatic interaction and 90% electrostatic interaction
•So states A and B have to be close so we can get reliable difference in free energy
Restraints and Standard State Correction
Things to Think About
Entropy
Enthalpy
Induced Fit
Conclusion
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