# Re: Measuring protein diffusion: results too small?

From: Axel Kohlmeyer (akohlmey_at_cmm.chem.upenn.edu)
Date: Wed Jun 03 2009 - 17:53:53 CDT

On Wed, 2009-06-03 at 17:49 -0400, Stephen Hicks wrote:
> Hi,

hi stephen,

> I'm attempting to measure the diffusion constant of a more or less
> spherical protein, the C-terminal domain of HIV capsid protein 1AUM.
> It has 70 amino acids and I estimate the radius to be about a=1.3nm.
> At 310K I estimate the viscosity of water to be anywhere from
> eta=.25cP to .7cP (dependent on the model - I understand TIP3P is much
> lower than experiment). So I ran NVT simulations in a large TIP3P
> water box (10nm to a side) with a 0.8fs timestep (langevin thermostat,
> rigid bonds) and measured D as follows:
>
> D = <(x(t+t')-x(t'))^2>/6t

>
> where x is the (3-vector) position of the protein and the <..> are
> averaging over times t'. For the t part, I calculate the average as a
> function of t and fit the part near zero to a line to compute the
> slope, which is my diffusion constant. But when I do this, my result
> is always on the order of about 3e-7 cm^2/s, while I expect, using

actually, it might be better to use the time derivative of your
expression or fit a straight line to the large time part (last
third or so) of your MSD vs. time data. you definitely don't want
to fit so that D(0)=0, because the einstein relation only holds
for large t. have a look at, e.g. page 10 and beyond in.

you also have to factor in system size effects. there should be
a paper by gerhard hummer deriving an estimate formula for that,
but i forgot exactly where.

cheers,
axel.

> Einstein/Stokes, that
>
> D = kT/(6*pi*eta*a) ~ 2.5e-6 cm^2/s (using .7cP) or 7e-6 cm^2/s (using .25cP).
>
> So at best (using the experimental viscosity rather than the TIP3P
> viscosity) I'm off by nearly an order of magnitude. I've done this at
> different box sizes (4.3nm sides) and gotten the same D, so I'm
> confident that it's not a finite-size effect. I also tried another
> approach, applying a constant force of F=1.093kcal/mol*A =
> (6*pi*.25cP*1.3nm)*12m/s in the +x direction, distributed evenly (by
> mass) among all the atoms in the protein (i.e. F_i = F * m_i/m_total),
> and found that the protein was drifting with a velocity of about
> 0.8m/s - again about an order of magnitude too small!
>
> (On the other hand, I've integrated a box with nothing but water and
> got more or less the correct experimental self-diffusion constant,
> 3.5e-5 cm^2/s -- which may be only a factor of 2-3 too small if TIP3P
> self-diffusion is supposed to be larger than experiment, but then my
> protein results are off even more, by the same factor. I've also
> tried to use TIP4P but so far I haven't been able to get NAMD to run
> properly with it)
>
> I've seen a variety of papers that claim to measure the translational
> diffusion of proteins with MD, so my question is, is there any known
> effect that would be throwing off my results? Why can't I reproduce
> reasonable numbers here? Am I missing something in my integration, or
> my water, or the way I'm measuring?
>
> Any help is greatly appreciated!
> Steve Hicks
> Ph.D. Candidate, Henley Group
> Laboratory of Atomic and Solid State Physics
> Cornell University

```--
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Axel Kohlmeyer   akohlmey_at_cmm.chem.upenn.edu   http://www.cmm.upenn.edu
Center for Molecular Modeling   --   University of Pennsylvania
Department of Chemistry, 231 S.34th Street, Philadelphia, PA 19104-6323
tel: 1-215-898-1582,  fax: 1-215-573-6233,  office-tel: 1-215-898-5425
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