This is part 2 on free energies. (To return to part 1 ....)

Use the following free energy for the dynamics (Zwanzig, Berendsen, McCammon):

where UA describes molecule A (in solution, etc.) and UB describes molecule B (in solution, etc.). Then

More general form:

Non-linear coupling of Lennard-Jones energy ("inflate"):

and something similar for the electrostatic energy.

Equilibrium between two states: Water + Solute(vapor) and Solute-in-water

Use the following free energy function for the dynamics:

The ratio is the partition coefficient.

Protein + Ligand A <=> P.A has eqm.const. K_A and free energy DG_b,A

Protein + Ligand B <=> P.B has eqm.const. K_B and free energy DG_b,B

(e.g. two different inhibitors of an enzyme)

One computes this as the difference between the free energies for two transformations:

• solvated A <=> solvated B (solute alone)
• solvated PA <=> solvated PB (complex)

WT Protein + Ligand <=> P⁺.L, has eqm.const. K⁺ and free energy DG⁺

mutant Protein + Ligand => P^-.L has eqm.const. K^- and free energy DG^-

One computes this as the difference between the free energies for two transformations:

• solvated P⁺ <=> solvated P^- (protein alone)
• solvated P⁺L <=> solvated P^-L (complex)

Multiple-Minima problem

• hydrocarbon chain:
• 3 "conformers" per C-C single bond
• 3 "conformation states" per C-C single bond:
  (trans, gauche⁺, gauche^-)
• 3^M conformation states for an M+1 carbon chain
• circa 10^M conformation states for M amino acid residues
• A conformation "state" = the collection of conformations in the neighborhood of a free energy minimum in conformation space.
• Free dynamics:
• Infrequent transitions (low probability for barriers)
• Poor sampling
• Instead: compute the differences in free energy between different conformations directly.