1. 8/7/2018
Statistical Thermodynamics
Lecture 19: Free Energies in Modern Computational Statistical Thermodynamics: FEP and Related Methods
Dr. Ronald M. Levy

2. Free energy calculations
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Free energy calculations
Free energy is the most important quantity that describes a thermodynamic process
Two types of free energy calculations:
1. Path independent methods for calculation of free energies. e.g. Free
Energy Perturbation (FEP), Thermodynamic Integration (TI)
2. Path dependent methods for calculations of free energies. e.g. Potential of
Mean Force (PMF)

3. General perturbation theory
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General perturbation theory
Perturbation theory is one of the oldest and most useful, general techniques in applied mathematics.
Its initial applications to physics were in celestial mechanics to explain how the presence of bodies other than the sun perturbed the elliptical orbits of planets.
The main idea: One starts with an initial system, called the unperturbed or reference system. The system of interest, called the target system, is represented in terms of a perturbation to the reference system.

4. Statistical mechanics description of free energy
In the canonical ensemble:
N-particle reference system described by Hamiltonian H(x,p), function of 3N Cartesian coordinates x and conjugated momenta p.
Partition function:
Free energy difference :

6. Probability density function P0 of finding the reference system in a state defined by x and p is:
Substituting in expression of ΔA:
Fundamental FEP formula for the transformation :
If the kinetic term cancels out :

7. Free energy perturbation method
In this formula, ΔU = U1(x) – U0(x) is the difference in potential between target and reference system, and the average is over the ensemble of the initial state corresponding to reference system with potential U0(x).
Similarly, the free energy difference can also be written in terms of an average over the ensemble of the final state corresponding to target system with potential U1(x):

8. Free energy perturbation calculations continue
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Free energy perturbation calculations continue
According to the central limit theorem (CLT), ΔU would be Gaussian-distributed if U1 and U0 were the functions of a sufficient number of identically distributed random variables.
Therefore:
sigma,
where:
Analytical expression of ΔA:
Notice: use of this analytical expression can be successful only if P0(ΔU) is a narrow function of ΔU; this condition does not imply that the ΔA must be small.

9. Example: hydration free energy of benzene
reference state: solute has high probability to overlap with solvent in the gas phase
target state: solute has low probability to overlap with solvent in the aqueous phase
ΔA = kcal/mol
Although hydration free energy of benzene is small, this quantity cannot be successfully calculated directly using the FEP equation; because low-energy configurations in the target ensemble, which do not suffer from the overlap between the solute and solvent molecules, are not sampled in simulations of the reference state.

10. FEP calculation --- staging
The difficulty in applying FEP formula can be circumvented through staging strategy.
Construction of several intermediate states between reference and target state such that P(ΔUi,i+1) for two consecutive states i and i+1 sampled at state i is sufficiently narrow for the direct evaluation of the corresponding free energy difference ΔAi,i+1.
With N-2 intermediate states,
Intermediate states do not need to be physically meaningful, they do not have to correspond to systems that actually exist.

11. FEP calculation --- formula
The Hamiltonian can be considered to be a function of some parameter λ.
λ can be defined between 0 and 1, such that λ = 0 for reference state and λ = 1 for target state.
Dependence of hybrid Hamiltonian on λ :
ΔH is the perturbation Hamiltonian, equal to H1-H0.
If N-2 intermediate states are created to link the reference and target states, such that λ1 = 0 and λN = 1:
with Δλi =λi+1-λi
Total free energy difference:

12. Important concept in FEP: Order parameter
Order parameter: They are collective variables used to describe transformations from the reference system to the target one.
order parameter: distance
order parameter: dihedral
An order parameter may or may not correspond to the path along which the transformation takes place in nature, and would be called the reaction coordinate if such were the case.
There is more than one way to define an order parameter. The choice of order parameters may have a significant effect on the efficiency and accuracy of free energy calculations.

13. Application of FEP: binding free energy calculation
The binding free energy of two molecules, ΔAbinding, defined as the free energy difference between two molecules in the bound and free, unbound states, can be determined experimentally through the measurement of binding constants.
For direct calculation of ΔAbinding, it needs to define an order parameter that measures the separation between the ligand and the binding center of the protein. This may be difficult when the ligand is buried deep in the binding pocket.
Alternative route: double annihilation of the ligand through a thermodynamic cycle.

14. Thermodynamic cycle and double annihilation
Double annihilation involves the removal of the ligand molecule from aqueous solution in the unbound state and bound state, respectively.
Through thermodynamic cycle, the double annihilation free energies ΔA0annihilation and ΔA1annihilation were related to ΔAbinding
The ligand in the binding pocket is annihilated from a strongly constrained position, whereas the unbound ligand can translate and rotate freely during annihilation.
Proper corrections for the loss of translational, rotational and conformational entropies should be taken into account.
protein + ligand protein ligand
protein + nothing protein nothing
ΔArestrain
ΔA0annihilation
ΔA1annihilation
ΔAbinding

15. Example: Relative binding free energies for ligands
In the case of seeking potential inhibitors of a target protein, determining relative binding free energies for a series of ligands is required. This can be handled by repeating the absolute binding free energy calculation for each ligand of interest.
ΔAXLC = kcal/mol
Ligand XLC
Protein FXa
ΔAXLD = kcal/mol
Ligand XLD
Protein FXa
ΔΔAcal = ΔAXLD – ΔAXLC = kcal/mol. Match well with ΔΔAexp = kcal/mol

16. Alchemical transformations
protein + ligand A protein ligand
ΔA0mutation
ΔA1mutation
ΔAAbinding
protein + ligand B protein ligand
ΔABbinding
There is an alternate pathway for calculations of relative binding free energies. Mutation of ligand A into ligand B in both the bound and the free states, following a different thermodynamic cycle.
Example: In the mutation of ethane into methanol, the former serves as the common topology. As the carbon atom is transformed into oxygen, two hydrogen atoms of the methyl moiety are turned into non-interacting, “ghost” particles by annihilating their point charges and van der Waals parameters.

17. Thermodynamic integration method
If N-2 intermediate states are created to link the reference and target states, such that λ1 = 0 and λN = 1:
TI:
FEP:

18. Potential of mean force method
The reaction coordinate ξ, describes the path along which the transformation takes place in nature.
Potential of mean force (PMF) is a path dependent method for calculations of free energies based on the reaction coordinate ξ.
constraint force
Constraint force: force required to constrain the system at a fix reaction coordinate ξ; if this force can be extracted, free energy can be evaluated from integration of the mean force along the reaction coordinate.

19. Example: Single ion at liquid/vapor interface
Z direction ξ
Gibbs Dividing Surface
Zero Position
The reaction coordinate is the separation between ion and zero position along the z direction.
Iodide shows more stability than chloride around interface from PMF profile.

20. Summary
Different method for calculation free energy: free energy perturbation (FEP), thermodynamic integration (TI) and potential of mean force (PMF).
In the next lab, we will use FEP method, implemented in the Schrodinger package, to calculate the relative binding free energy between a pair of ligand XLC and XLD binding to the receptor protein Fxa.