Subject: Absolute binding free energy

From: "Dr. Jorge Seminario" <[email-masked]>
Date: 06 Oct 1995 12:23:12 -0600 (CST)

Dear CCL's: I am posting the following query from a 
subscriber of the "water-list" who is not in the CCL.
Please, if you answer to the list also send a copy to
his address.
Regards
Jorge Seminario
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From: [email-masked]

Calculated "Absolute" binding free energies: what are they?
I posed this question to the Water Science Network on 15 Sept 95.
I have a tentative answer, based on further thought and sharpened by useful
responses (thanks to Drs. Berendsen, Cramer, Stouton).  Comments welcome.

What do computationalists mean when they report the "ABSOLUTE free
energy of binding" and compare this with a simulation energy?
Example:  J. Aqvist et al, Protein Engineering 7(3), 385-391 (1994);
        also papers by Kollman, Hermans, Warshel, etc.
These calculations are always based on averages of some sort, in which
no volume or density appears explicitly. 

But "absolute" free energy of binding should correspond somehow to
a dissociation constant which has units of CONCENTRATION: 
		   [A(free)][R(free)] / [A:R(bound)]
Taking  logarithm ln() of Kd requires us to choose some arbitrary unit of
concentration. (Molar, 1 M,  is based on radius of earth and density of water.
Nanomolar, 1 nM, is based on these quantities plus # of fingers on 2 hands.)

Free energy of TRANSFER  between two homogeneous media (solvents
or gas phase) corresponds to a ratio of concentrations.
RELATIVE binding free energy is also a ratio of concentrations (Kd's).
Neither one raises any difficulty in princple.  The simulation yields 
an average energy which gives this ratio:  delta-G = -kT ln (c2/c1). 

How can a free-energy calculation "build in" the units of concentration
when calculating 'absolute' binding free energy, but not contain any 
unit of concentration when calculating relative free energies?

Tentative answer: 
  a) Most computational papers dealing with 'absolute' binding free energy
  don't address this question.  The simulation reports an average energy;
  the authors workers either do or do not adjust this energy.  Then they
  compare the result with experimental dissociation constants in units 1M. 
  b) There's no problem in principle when simulating free energy of transfer.
  by "turning on" interactions of A with the homogeneous solvent.  Any
  assumption of standard concentration or volume affects both the interacting
  and the noninteracting state equally, and cancels out of the final result.
  The same holds for transfer between solvents.
  c) There's no problem when simulating relative binding free energy (B:R-A:R)
  because it involves 'growing' A to B in the context of the receptor R and
  also 'growing' A to B in homogeneous solvent.  The simulation within the
  receptor is insensitive to the size of the box; and as before, the simulation
  within homogeneous fluid can also be made independent of box size.
  [Remarks (b) and (c) but apply also to 'particle insertion' or 'thermodynamic
  perturbation' or any other AVERAGING method.]
  d) Simulations of absolute binding free energy are subtly different
  from either absolute TRANSFER free energy or RELATIVE binding free energy.
  They involve growing "null" to "A" to "B" in the context of the receptor.
  Workers treat this formally as the same as growing "A" to "B".  But
  at the very endpoint of the simulation, as ligand "A" becomes "null" or is
  created from "null", the ligand gains (or loses) the ability to wander
  through all of the simulation volume.  If the simulation is done
  carefully enough, this last stage is a hugely expensive estimate of the
  the ratio of volumes of simulation 'box' to the binding site. 
  The logarithm of this ratio is an entropic free energy. Thus,
=>  e) the implicit 'standard state' of an absolute binding free energy simula-
  tion is (1 ligand per simulation box), if the end point is sampled well.
  f) The state "1M" happens correspond to 1 ligand in a box of side 12A.
  Even if workers forget to adjust for the size of the simulation 'box'
  (typically about 30A on a side) the result, compared to Kd in molar,
  appears sensible within an order of magnitude, and doesn't raise any 'flags'.

I haven't seen this argument spelled out, though Ben-Naim's careful discussions
of 'cratic' and 'unitary' entropy certainly address the same issues in the
context of transfer free energies. 

Your comments are welcome.
/ Bruce/
[email-masked]   Merck Research Labs, Rahway NJ 07065

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