# Measuring protein diffusion: results too small?

From: Stephen Hicks (shicks_at_ccmr.cornell.edu)
Date: Wed Jun 03 2009 - 16:49:03 CDT

Hi,

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
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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