Advanced topics - free energy:
Entropy
(The following treatment is limited in scope!)
The entropy, S is a measure of the
variety of accessible states.
The entropy is large if the geometry of
a structure fluctuates a lot. We therefore consider fluctuations
about the mean geometry.
For a multivariate Gaussian distribution
of the fluctuations:
P (r) =
a exp[-(1/2)
(r - <r>)T s-1
(r - <r>)]
sij
= <(ri - <ri>) (rj -
<rj>)>
S = (1/2) Nd (k + ln 2p)
+ (1/2) ln s
where s is the determinant of s,
Nd the number of degrees of freedom, k Boltzmann's
constant, and
1. The extent
of deformation of the conformation due to thermal motion can be
determined in the harmonic approximation of a normal
mode calculation:
sij
= kT (F-1)ij
where F is the force matrix from the normal
mode calculation.
2. Karplus-Kushick
method: use MD to estimate
the fluctuations
(Karplus, Bray, Brooks, Kushick and Pettitt.
Proceedings of a Workshop "Molecular dynamics and protein
structure", 1984.)
When s
is obtained from MD a simulation, some of the non-Gaussian character
of the distribution is included.
Umbrella Sampling
a. Idea: Add
an artificial potential U* to the force
field such that
G(z)
+ U*(z)
= constant
Then, the effective probability will be even and
MD will sample all values of z.
But how to find G(z)?
b. Add an artificial
potential to the force field that looks
like
U*(q) = (1/2) K (q - qo)2
The dynamics will now sample a new probability distribution,
P*. This is related to the unperturbed probability distribution,
P, by
P* = (const) P exp[U*/kT]
From a series of simulations that produce overlapping§
distributions, P* one can construct the unperturbed distribution
P over a wide range of q, including portions where P is very small.
§ Overlap is needed to determine the value(s)
of the constant.
Thermodynamic
perturbation
Calculate a free energy difference in
discrete steps ("windows")
From statistical mechanics:
Write this for state A and also for state B:
This is the "ensemble average" of exp[-(UB
-UA)/kT] in a simulation of state A,
i.e.,
DAAB=
-kT<exp[-(UB -UA)/kT]>A
- In principle,
one might try to do this using results from a single simulation.
- For example: simulate a liquid and calculate
the free energy for converting the liquid to an ideal gas with
DAexcess
= kT ln < exp [Uintermolecular/kT]
>liquid
Uintermolecular
being equal to the potential energy difference between the two
states.
- Or, one might simulate an ideal gas and calculate
the free energy for converting the ideal gas to the liquid with
-DAexcess
= kT ln < exp [ -Uintermolecular/kT]
>gas
- In practice,
the two values of DA
obtained by these two simulations are equal only when the two
states are much more similar than liquid and gas.
- Accordingly, one divides the interval between
the two states into smaller sub-intervals (windows) and then calculates
the free energy difference for each.
- This means that one has to specify a path between
the two states, just as one did for the thermodynamic integration
method.