Jason,
That is a great write-up! It's very valuable to have some practical pros
and cons for these methods, and a sense of when one might be more useful
than another. Previously my comprehension was limited to QHA approximates
the everything has one harmonic, and NMA does them individually, but I'd
never expanded this to understand what that might mean in terms of what was
needed. Though I see the point about them both being "bad", in fact, I'm
often tempted to just use the "enthalpy" (it does include the solvent
entropy at least) when doing MMPBSA and hope the vibrational (or I guess
configurational?) entropies are roughly similar.
Is there an inherent reason that the calculations NMA performs wouldn't be
a good fit for the GPU? In particular, couldn't energy minimization be GPU
enhanced just as readily as MD (I don't recall at the moment if AMBER does
this anyway when using PMEMD.cuda?).
Thanks,
~Aron
On Fri, Dec 5, 2014 at 8:59 AM, Jason Swails <jason.swails.gmail.com> wrote:
> 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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>
--
Aron Broom M.Sc
PhD Student
Department of Chemistry
University of Waterloo
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Received on Fri Dec 05 2014 - 07:30:03 PST
