Why does fes significantly change with bias factor in Well-Tempered Meta-Dynamics?

[Ahmad F. Ghobadi]

I am running Lammps + plumed 2 to get free energy landscape of a coarse-grained peptide nucleic acid (PNA) duplex hybridization in implicit solvent. My CVs are reasonable (distance and angle between hybridizing single strands) and fes converges (almost) to a landscape that makes perfect sense based on my understanding of such systems. I use a bias factor of 5 along with some other meta dynamics parameters that is appropriate for my system (see below). 

Now, if I change bias factor to 10, I get a fes that has the same shape, but all values are shifted by ~50%  to more negative numbers comparing to the case of bias factor = 5.

My thought (based on reading the manual, Barducci et al PRL 2008 and Dama et al, PRL 2014) was that fes should not change substantially with bias-factor if I sample free energy wells correctly. Am I missing something?

Also, what happens if I set TEMP in plumed input file different from the temperature bath in my simulation (on lammps)? TEMP and bias factor should only impact the deltaT parameter nothing else, right?

It looks like fes changes with both TEMP and bias factor in a systematic way as if it is not re-scaled. But, this does not make sense because in Well-Tempered fes is rescaled already. I appreciate your help in advance. My plumed input file is below:

RESTART

UNITS LENGTH=nm ENERGY=kcal/mol TIME=fs

s1: COM ATOMS=3-14 NOPBC

s2: COM ATOMS=19-30 NOPBC

# fist cv, distance between two strands

dr: DISTANCE ATOMS=s1,s2 NOPBC

# Second cv, the angle between two strands

teta: ANGLE ATOMS=3,13,19,29

uwall: UPPER_WALLS ARG=dr AT=20.0 KAPPA=150

METAD ...

        LABEL=metad

        ARG=dr,teta

        TEMP=325

        SIGMA=0.5,0.5

        HEIGHT=1.0,1.0

        PACE=1000

        #BIASFACTOR=10

        BIASFACTOR=5

        GRID_MIN=0.0,0.0

        GRID_MAX=25.0,3.15

        GRID_SPACING=0.1,0.1

        GRID_WSTRIDE=1000

        GRID_WFILE=grids.txt

... METAD

PRINT ARG=dr,teta,metad.bias,uwall.bias STRIDE=1300

[James Krieger]

If you used sum_hills without the --negbias option then you are  rescaling twice. That might explain what you are seeing.

[Ahmad F. Ghobadi]

Thanks for your comment James. I added --negbias flag but free energy still scales with gamma parameter in well-tempered MetaD. In fact, the out put of sum_hills is consistent with the actual negative bias values that I dump to a file using PRINT command. Maybe I am overfilling the free energy landscape! Or maybe I should use --mintozero flag? 

Best,

[Massimiliano Bonomi]

> On Dec 7, 2015, at 17:29, Ahmad F. Ghobadi <ahm...@udel.edu> wrote:
>
> I am running Lammps + plumed 2 to get free energy landscape of a coarse-grained peptide nucleic acid (PNA) duplex hybridization in implicit solvent. My CVs are reasonable (distance and angle between hybridizing single strands) and fes converges (almost) to a landscape that makes perfect sense based on my understanding of such systems. I use a bias factor of 5 along with some other meta dynamics parameters that is appropriate for my system (see below).
>
> Now, if I change bias factor to 10, I get a fes that has the same shape, but all values are shifted by ~50% to more negative numbers comparing to the case of bias factor = 5.
> My thought (based on reading the manual, Barducci et al PRL 2008 and Dama et al, PRL 2014) was that fes should not change substantially with bias-factor if I sample free energy wells correctly. Am I missing something?

The free energy is defined modulo an additive constant, so the fact that the two fes have the same shape, just shifted,
means that you got identical results with bias factor equal to 10 or 5.

The bias factor regulates how fast the Gaussian height decreases. With a smaller bias factor, the Gaussian height
will decrease faster. Thus, if you compare two runs of the same length, but different bias factor, the total amount
of bias deposited in the two runs will be different. This is the reason why your run with bias factor 10 results
in a free energy shifted to more negative values.

Nothing wrong here.

Max

[Azade yazdan]

Hi,

Please excuse me since I am going to ask a question which is not really on plumed but more on the theory side of wt-mtd. 

I am very confused with the relationship between deposition rate, the height of Gaussians and bias factor used in plumed.

Here and in one other answer I find that:

- a bias factor of x reduces the barrier by a factor of x

- a smaller bias factor will decrease the heights faster

I cannot figure out how the bias factor is implemented in the original paper of wt-mtd [1], and then in plumed manual and what parameters the bias factor is changing. I would appreciate any explanations :-)

Thanks in advance.

Azade

[1] Barducci, A.; Bussi, G.; Parrinello, M. Well-Tempered Metadynamics : A Smoothly Converging and Tunable Free-Energy Method. Phys. Rev. Lett. 2008, 100, 020603:1-4 DOI: 10.1103/PhysRevLett.100.020603.

[Giovanni Bussi]

Hi!

In the original paper we defined DeltaT, in order to convey the message that the effective temperature of the CV is increased. Actually the CV is sampled as if it was at an effective temperature T'=T+DeltaT. In practice, this requires scaling the hills height with exp(-V/(k_BDeltaT))

However, in the plumed implementation we read the so-called bias factor, that was introduced in later papers, which tells you which factor you should multiply T to obtain T'. That is, T'=gamma*T.

Thus, gamma=(T+DeltaT)/T, and DeltaT=(gamma-1)*T. Hills height is scaled with exp(-V/(k_B*(gamma-1)*T))

gamma=1 (or DeltaT=0) implies unbiased sampling (hills become immediately infinitely small)

gamma=infinite (or DeltaT=infinite) implies non-welltempered metadynamics (no hills scaling)

Giovanni

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