1. Statistical Mechanics for Free Energy Calculations
WARNING: MATH AHEAD !!!

2. Statistical mechanics
Some definitions
Hamiltonian
Microscopic state: point in phase space {r,p}
Canonical partition function Object at heart of statistical mechanics
Z and its derivatives give access to any property
Requires sampling of all configurations; fortunately major contributions are due to low energy states

3. Statistical mechanics
Probability to find a particular microstate (Boltzmann distribution)
The average of any property can be written as (ensemble average / thermodynamic average)
For example the internal energy

4. Statistical mechanics
Molecular dynamics samples phase space by following a trajectory in time {r(t),p(t)}
This gives access to dynamic averages
Ergodic hypothesis Used as justification to calculate thermodynamic averages from MD trajectories However: Proper sampling can be problematic

5. Statistical mechanics
Molecule with two conformations
Relative population of two states at room temperature

6. Statistical mechanics
Relative populations at different temperatures

7. Can we get free energy differences just by running MD long enough?
Only if the barriers are very low, and we visit each state many times.
Important: if you have total frames in your simulation, the MAXIMUM possible value of delta G you can observe at 300K is -0.6*ln(1/9999)=5.52 kcal/mol

8. Statistical mechanics
The absolute free energy: A: Helmholtz free energy, Canonical (N,V,T) ensemble G: Gibbs free energy, Isothermal-isobaric (N,P,T) ensemble
Difficult to compute because high energy states contribute and MD (or MC) samples low energy regions of phase space

9. Phase space
- Phase Space is the idea of describing the properties of a system in terms of its position and momentum coordinates, defining all possible states.
- The concept of phase space, is only about 100 years old and at the root of Statistical Thermodynamics
- A phase space description of a system is what we should work with but can not, for various reasons:
- Can‘t be computed
- Can‘t be visualized/understood
- Can‘t be measured

10. So we use Reaction coordinates
Typically one or two dimensional
Be aware of massively reduced dimensionality (effectively integrating over 1020 orthogonal DOF)
This allows us to define comprehensible states (i.e. folded vs. unfolded)
Note: The whole notion of phase space is at odds with quantum mechanics, we use a classical approximation here, bc we lack equivalent QM statistics tools

11. MD Simulations The purpose of MD simulations is to explore phase space
A single structure is only one microstate!
A trajectory approximates a (small) region of phase space
We are often interested in transitions and equilibria between states
Sadly, MD does not always give the desired results (in finite time)
At this point, Free Energy Calculations can be employed

12. Motivation for Free Energy Calculations
Free energy of binding
Binding could mean:
- Activation
- Disactivation
- Destruction
- competitive Inhibition …
"Corpora non agunt nisi fixata" (No compound is active unless it is bound by a receptor) Paul Ehrlich, 1913

13. Free energy calculations
Direct calculation of free energies not feasible
Quantity of interest is usually a free energy difference
Several approaches are in use:
Non-equilibrium Jarzynski's Equation
Equilibrium
Real coordinates Umbrella Sampling
Abstract coordinates FEP / Thermodynamic
Integration

14. Thermodynamic Cycles Useful for calculating relative free energies
Relative free energy of solvation
For instance, solvation of benzene versus toluene.

15. Thermodynamic Cycles Relative free energy of binding
Two different drugs binding to the same target.

16. Free Energy Perturbation

17. Free energy perturbation

18. Free energy perturbation
Called FEP although the result is formally exact; but it connects the perturbed system B to the reference system A
Popular and quick to implement
Potential statistics / sampling problems
A and B need to overlap in phase space  perturbation VAB must be small  transformation must be divided into small steps

19. FEP: Severe sampling problemsDF
Taking the logarithm of the average of the
exponential of a noisy function is a bad idea!
Often done in multiple small steps, mixing the potential functions

20. Free energy perturbation
Subdivide into N steps (“Computational Alchemy”)

21. Thermodynamic Integration and Softcore Potentials

22. Thermodynamic Integration
Often linear mixing is used: f(λ)=λ

23. Thermodynamic Integration
Integral has to be solved by numerical quadrature
Perform series of simulations corresponding to discrete values of λ and form the averages of the derivatives of the Hamiltonian

24. Practical Issues for TI/FEP
How many windows are needed ?
Where are the windows placed?
How long should each window be run?
Should windows be run simultaneously or consecutively
How to pick the region that changes

25. Ready for ΔΔGbind via FEP/TI?
Relative free energy of binding

26. Drug Design Applications
Potential Drug Design applications

27. Adding and Removing Atoms

28. Adding and removing atoms
for example vie “dummy atoms”
End state: Molecule is made of
"ghost" atoms
λ=1
Start state: Molecule exists
λ=0

29. Adding and removing atoms
The interaction between the molecule and the solvent
at λ=0 is
and at λ=1 is

30. Adding and removing atoms
But, as λ1, the repulsive wall is still infinite! This means that water, when it tries to occupy the empty space, cannot! This leads to serious convergence problems.

31. The origin singularity effect
red: LJ Potential
green: LJ at λ=0.99
It is very hard to converge and fluctuations are very large

32. A modified vdW Potential: Softcore99
A modified vdW Potential: Softcore
With this change, the calculation does not blow up at the end points.

33. Dual Topology Approach

34. Softcore TI: Mixing of Forces

35. Softcore TI: Modified mdout Style
A V E R A G E S O V E R S T E P S
NSTEP = TIME(PS) = TEMP(K) = PRESS =
Etot = EKtot = EPtot =
BOND = ANGLE = DIHED =
1-4 NB = EEL = VDWAALS =
EELEC = EHBOND = RESTRAINT =
DV/DL =
EKCMT = VIRIAL = VOLUME =
Density =
Ewald error estimate: E-03
Softcore part of the system: atoms, TEMP(K) =
SC_BOND= SC_ANGLE= SC_DIHED =
SC_14NB= SC_14EEL= SC_EKIN =

36. Softcore TI example: Solvated Toluene

37. Softcore TI example: Solvated Toluene
Resulting free energy curve
69 l-points
20 ps equilibration
2 ns data collection
ΔG(Solv) = kcal/mol
Published: kcal/mol λ

38. Softcore TI: data collection
data collection still very noisy, be cautious
Time

39. Softcore TI: AMBER input
See Chapter of AMBER 14 manual on Free Energy Calculations

40. Free energies along a defined reaction coordinate via Umbrella Sampling

41. Umbrella Sampling
How to obtain free energy changes associated with conformational changes?
How to force barrier crossings without compromising thermodynamic properties?
Very slow spontaneous transitions

42. Umbrella Sampling
Free energy profile along the indicated dihedral angle?
Define a reaction coordinate
This is called a potential of mean force, and requires a partition function integrated along all coordinates except the one we want to look at.

43. Umbrella Sampling
One could just run dynamics and wait until all space has been sampled.
Then, if one extracts ρ(xk) from the trajectory, the PMF can be written as:
This is called unbiased sampling
However, it takes forever to properly sample all conformations, and to jump over the barrier. The solution is to bias the system towards whatever value of the coordinate we want.