Global minimum, free energy

[heiner -8 at 8- bio.vtt.fi]

 In response to an inquiry be K. Macdonald, P.B. Medawar wote
 >    One point is that "free energy" (so they tell me) is
 determined
 >    statistically by simulation, and takes account of entropy--which,
 >    for example, is greatly increased in a local minimum that is
 >   "narrow" in conformational space and therefore hard to reach
 >    in practice..
 >    It also seems to me that seeking the global minimum is a good bet,
 >    but not a certain one.  There may be many minima with practically
 >    the same energies but radically different conformations.  ......
 Thes statement seems sncorrectatement seems wrong, the 2nd oke, to me. Example:
 Consider a 1-D gaussian distribution of states. The entropy of this
 distribution is given by
           /
    S = -k | p ln(p) = -k { ln(h/sqrt(pi)) - 1/2},
           /
    where h is the inverse of the variance of the distribution,
    (1/sqrt(sigma))
    Now consider two distributions, one narrow (h1), one wide (h2). The entropy
    difference between the two is
    S1 - S2 = -kln(h1/h2) ==> S2 > S1.
    As the system strives for minimum free energy (or maximum entropy with
    the constraint of minimum energy), the wide distribution is the most
    stable one.
 Applying this to the case of protein folding, I would say that the global
 energy minimum conformation (MEC) is most the likely conformation of the
 protein, but when a nearby (local) MEC has a much wider distribution than
 the global one, the protein will adapt the 2nd conformation most of the time,
 being in the global MEC only part of the time. This will depend on the
 free energy differences between the two conformations. Hence there is no such
 thing as a SINGLE minimum free energy conformation, further, the AVERAGE of the
 DISTRIBUTION of low-energy conformations will determine the observed structure.
 The problem with current simulation techniques (MD, MC) is that the full
 distribution is hard to cover, partly due to long relaxation times of the
 system. This may yield the differences between experiment and theory.
 Whether one is able to simulate protein folding, depends on the force field
 used, R. MacDonald adds. However, taking into account electrostatics/vdWaals
 and hydrophobicity/exposed surface areas as a potential seems a bit overdone to
 me, as the latter contributions are phenomenological, the first two more
 elementary (but also with their errors, of course). I would say that, when
 the all interactions are included AT THE SAME LEVEL OF REFINEMENT, one should
 obtain the folded protein in the long run. Mixing contributions (To me, a
 thing like the hydrophobic potential does not exist; It is merely the effect
 of all explicit water molecules) seems not the right approach. In the end,
 we may end up simulating the protein folding ab initio, but covering all
 relevant states is the problem.
 Hope I made some sensible contribution
 Andrepeter Heiner (Physicist/Simulationist (?))
  -----------------------------------------------------------------------------
  | A.P. Heiner, Ph.D.             |  Phone      : +358-0-456 5105            |
  | VTT/Biotechnical Laboratory    |  Telefax    : +358-0-455 2028            |
  | P.O. Box 202 (Tietotie 2)      |  E-mail     : heiner -8 at 8- bio.vtt.fi
 |
  | FIN-02151 Espoo                |                                          |
  | FINLAND                        |                                          |
  -----------------------------------------------------------------------------