Re: A question on Jarkynski equation

From: johan strumpfer (
Date: Tue Jun 14 2011 - 09:51:22 CDT

Hi Jun,

Following on a little from what Jerome and Yi nicely explained, you
can get a little more info on the effect of pulling speed and water
viscosity from Hsin and Schulten, Biophys, J., 100:L22-L24 (2011).

Also, although this doesn't really help that much with ligand
unbinding, but you can pull first from point A to point B, and then
from point B back to point A. If you are pulling in solution only
(i.e. not trying to reproduce a bound state) both trajectories should
give the same (positive) work values. You can then combine them using
the bi-directional sampling method (as in FEP - see Pohorille,
Jarzynski and Chipot, J. Phys. Chem B, 2010, 114, 10235–10253), which
should give a free energy difference of 0.


Johan Strumpfer (
Theoretical and Computational Biophysics Group
3115 Beckman Institute
University of Illinois at Urbana-Champaign
405 N. Mathews
Urbana, IL 61801, USA

On Tue, Jun 14, 2011 at 10:09 AM, Jérôme Hénin <> wrote:
> Yi, thanks for your remark. This emphasizes one point that I should
> have mentioned: the "rare event" problem is highly dependent on
> pulling speed. At slow speeds in a low-viscosity solvent like water,
> friction is small or comparable to Brownian forces, so-called 'second
> law violations' become fairly frequent and the exponential estimator
> can converge. Similarly, in a FEP calculation between very similar
> states, convergence is much improved (hence the use of staged
> transformations).
> Cheers,
> Jerome
> On 14 June 2011 15:11, Wang Yi <> wrote:
>> Hi Jun,
>> Dr. Hénin has pointed out the general concerns regarding SMD and JE. But for
>> processes like moving in water, you might not have to worry too much about
>> that. What I mean is:
>> The work done for dragging a ligand in a water box is mostly against the
>> "friction" from solvent. Since water molecules move in all directions, the
>> "frictions" the ligand is experiencing changes all the time. Thus, you will
>> notice the SMD force fluctuating roughly around zero (albeit with large
>> swings). Then after integrating the force curve, the positive work and
>> negative work will mostly cancel each other. And after applying JE, although
>> there exists the issue of "heavey-weight rare events", the final value would
>> be pretty small (compared to thermo energy). That's especially true in a
>> unbinding process simulation, where the unbinding part has a much larger
>> magnitude than the "free moving in water" part.
>> That is based on my understanding.
>> Best,
>> ___________________________
>> Yi (Yves) Wang
>> Duke University
>> 在 2011-6-14,上午4:43, Jérôme Hénin 写道:
>> Hi,
>> You put your finger on a very interesting question. The Jarzynski
>> equality is formally exact, hence the estimator will converge towards
>> zero, at least on paper. As you note, since the reversible work is
>> zero, all measured work is irreversible: we expect it to be positive
>> basically all the time. Then how can the average be zero?
>> Actually, statistical mechanics dictates that a few trajectories will
>> give negative work values, i.e. negative irreversible work, i.e. a net
>> decrease in entropy. This is why such situations are sometimes called
>> 'second law violations'. They are not really violations, just a
>> reminder that the second law applies to macroscopic systems only (even
>> if we in the molecular modeling community often stretch the notion of
>> macroscopic a little far). In the case of a solute moving through a
>> solvent, the 'negative work' case would be a trajectory where random
>> fluctuations in the solvent happen to push the solute along its path,
>> instead of slowing it down in a normal frictional behavior.
>> Such negative work values will be exceedingly rare, but they have a
>> huge weight in the exponential average of the Jarzynski formula. That
>> is why a few negative values are enough to make the average zero even
>> though almost all values are positive. Because these events are so
>> rare, numerical convergence will be awful, so it is unlikely that you
>> will manage to get a zero free energy value from a numerical
>> simulation. To some extent, the same can be said of any application of
>> the Jarzynski estimator, and is also true of FEP calculations with the
>> exponential formula: these averages are dominated by rare events,
>> which results in various degrees of convergence problems.
>> For more details on 'second law violations', see for example:
>> Cheers,
>> Jerome
>> On 14 June 2011 05:01, Jun Zhang <> wrote:
>> Hello Everyone:
>> I want to use Jarkynski's equation combined with SMD to compute the binding
>> free energy of a protein and its ligand (eg. JCP, 120, 5946). However, I was
>> puzzled by some theoretical issues.
>> For example, a system composed of water and a ligand. if we move a single
>> ligand in aqueous for some distances, the work done cannot be zero, but the
>> free energy change should be zero since the state of the ligand has not
>> changed.
>> It seems somewhat strange, and may be a naive question. But I am really
>> puzzled by it, so I'm looking for help. Thank you in advance!
>> Cheers up!
>> Jun Zhang
>> Nankai University

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