On Wed, 2014-12-03 at 09:50 -0500, Aron Broom wrote:
> Hi James,
>
> I've mostly ended up using the Quasi-Harmonic method for the entropy, as
> the NMA takes a very long time for each snapshot when you have a larger
> system. But when I did use it, I also saw a similar output. It seems like
> it wants to be run for more steps of minimization (probably because your
> system is larger than maybe the typical test systems were) and at this
> point hasn't stopped being able to reduce the energy. Under the &nmode
> namelist thing in the input file, you can give a value for maxcyc, which
> defaults to 10,000. Keep in mind, increasing the number of cycles, and
> adding more snapshots is going to mean a lot more computational time used.
> I would suggest, while that is running, to try out the quasi-harmonic
> option. It will give the same output, and also breakdown the entropy to
> the same terms, but will complete much faster. Also, from the minimal
> reading I've done on the subject, it didn't seem like the normal modes
> analysis was a clear cut winner in terms of the results, at least for
> protein sized systems. Maybe someone with more experience and
> understanding can comment on that?
In my understanding, the difference between the quasi-harmonic
approximation (QHA) and the normal mode approximation (NMA) is subtle,
but important. In the former, the eigenvectors of the mass-weighted
covariance matrix (think principal components) are treated as pseudo
vibrational modes, with the eigenvectors representing their frequencies.
For NMA, you minimize to the closest local minimum and compute the
Hessian matrix directly and get the vibrational modes and frequencies
from the eigenvectors and eigenvalues of the Hessian. NMA is analytical
(for analytical second-derivatives), QHA is not.
There are notable similarities -- i.e., both cases utilize the rigid
rotor approximation, assuming that you can separate the vibrational,
rotational, and translational modes (in reality, vibrational and
rotational modes are not orthogonal, since rapid rotation will "squash"
a globular protein and induce vibrations in the system). The _nature_
of the vibrations are inherently different between the two approaches,
though. Consider the following image of a reduced-dimensional free
energy surface (commonly seen in the theory of protein folding)
http://www.learner.org/courses/physics/visual/img_half/funnel.jpg.
To take a look at what you do for a NMA calculation, consider picking a
random point on that surface. What NMA does is drop that point to the
"closest" local minimum and compute the Hessian. The vibrational modes
you get correspond to a quadratic (harmonic) function that best fits
that minimum at the minimum (this is why you need to minimize *well* for
NMA). However, not all of these basins are the same shape (some are
steeper, with lower vibrational entropy, and others are shallower with
higher vibrational entropy). So you need multiple minima to get more
data, with the general assumption seeming (to me) to be that more of
those local minima resemble the "large" harmonic well than don't.
For QHA, though, the "harmonic well" is actually the entire sampled
space at that temperature (assuming "complete" sampling). So in the
figure I linked, the small wells are the ones from which NMA entropies
are approximated, whereas QHA will give you the entropy assuming the
entire funnel is one large harmonic well (unless I'm misunderstanding
exactly what the mass-weighted covariance matrix represents). This need
for "complete" sampling makes the QHA _very_ slow to converge in terms
of the number of independent snapshots you need (i.e., you need long
simulations in order to obtain the asymptotic limit of the QHA entropy).
NMA requires far fewer snapshots to "converge", but each snapshot takes
*much* longer due to the need to minimize thoroughly and compute the
Hessian. Since GPUs are making MD -- not NMA -- fast, I suspect that
QHA will start to win out on the grounds of computational cost.
So while they seem very similar in theory, the QHA and NMA have strong
differences in their theoretical underpinnings. Here I have to agree
with Aron -- it isn't clear to me which is "better" (given the
difficulty in computing entropy, they're probably both "bad" :).
As a fun exercise, let's try to unify them: if you sample from an ideal,
simple harmonic potential, there should be no difference between NMA and
the QHA. You can actually get pretty close to this scenario in real
biomolecules -- run your simulations at VERY low temperatures. When you
do that, the simulation will become trapped in the same local minimum
that you are computing the normal modes for, and will appear largely
harmonic if the temperature is low enough. In this case, the principal
components should be the same as your vibrational modes, and your
entropies should come out the same. One of my MMPBSA.py coauthors ran
one of these calculations 4-5 years ago at low temperature and indeed
found that the QHA converged rapidly and gave the same vibrational
entropies as NMA (the rotational and translational entropies are
analytically identical in both cases, if I recall correctly).
If I've perpetrated a misunderstanding with respect to physical
significane of the mass-weighted covariance matrix here, I hope someone
will correct me.
I hope this helps,
Jason
--
Jason M. Swails
BioMaPS,
Rutgers University
Postdoctoral Researcher
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Received on Fri Dec 05 2014 - 06:30:02 PST
