No, you can only add and subtract B values because that is
mathematically equivalent to multiplication in reciprocal space (which
is equivalent to convolution in real space):
exp(-B1*s^2) * exp(-B2*s^2) = exp(-(B1+B2)*s^2)
Multiplying and dividing B values is mathematically equivalent to
applying fractional power-law or fractional root functions in reciprocal
space (and I don't even want to think about what that does in real space).
exp(-B1*B2*s^2) = ???
-James Holton
MAD Scientist
On 3/4/2013 11:19 AM, Bosch, Juergen wrote:
Yep, I agree calculate the average B per structure and divide each B
by this value, then multiply it by any value that is reasonable so you
can visualize color differences :-)
Jürgen
On Mar 4, 2013, at 2:16 PM, Jacob Keller wrote:
You only entertain addition+subtraction--why not use
multiplication/division to normalize the b-factors?
JPK
On Mon, Mar 4, 2013 at 2:04 PM, James Holton <jmhol...@lbl.gov
<mailto:jmhol...@lbl.gov>> wrote:
Formally, the "best" way to compare B factors in two structures
with different average B is to add a constant to all the B
factors in the low-B structure until the average B factor is the
same in both structures. Then you can compare "apples to apples"
as it were. The "extra B" being added is equivalent to
"blurring" the more well-ordered map to make it match the
less-ordered one. Subtracting a B factor from the less-ordered
structure is "sharpening", and the reason why you shouldn't do
that here is because you'd be assuming that a sharpened map has
just as much structural information as the better diffracting
crystal, and that's obviously no true (not as many spots). In
reality, your comparison will always be limited by the
worst-resolution data you have.
Another reason to add rather than subtract a B factor is because
B factors are not really "linear" with anything sensible. Yes,
B=50 is "more disordered" than B=25, but is it "twice as
disordered"? That depends on what you mean by "disorder", but no
matter how you look at it, the answer is generally "no".
One way to define the "degree of disorder" is the volume swept
out by the atom's nucleus as it "vibrates" (or otherwise varies
from cell to cell). This is NOT proportional to the B-factor,
but rather the 3/2 power of the B factor. Yes, 3/2 power. The
value of "B", is proportional to the SQUARE of the width of the
probability distribution of the nucleus, so to get the volume of
space swept out by it you have to take the square root to get
something proportional the the width and then you take the 3rd
power to get something proportional to the volume.
An then, of course, if you want to talk about the electron cloud
(which is what x-rays "see") and not the nuclear position (which
you can only see if you are a neutron person), then you have to
"add" a B factor of about 8 to every atom to account for the
intrinsic width of the electron cloud. Formally, the B factor is
"convoluted" with the intrinsic atomic form factor, but a
"native" B factor of 8 is pretty close for most atoms.
For those of you who are interested in something more exact than
"proportional" the equation for the nuclear probability
distribution generated by a given B factor is:
kernel_B(r) = (4*pi/B)^1.5*exp(-4*pi^2/B*r^2)
where "r" is the distance from the "average position" (aka the
x-y-z coordinates in the PDB file). Note that the width of this
distribution of atomic positions is not really an "error bar", it
is a "range". There's a difference between an atom actually
being located in a variety of places vs not knowing the centroid
of all these locations. Remember, you're averaging over
trillions of unit cells. If you collect a different dataset from
a similar crystal and re-refine the structure the final x-y-z
coordinate assigned to the atom will not change all that much.
The full-width at half-maximum (FWHM) of this kernel_B
distribution is:
fwhm = 0.1325*sqrt(B)
and the probability of finding the nucleus within this radius is
actually only about 29%. The radius that contains the nucleus
half the time is about 1.3 times wider, or:
r_half = 0.1731*sqrt(B)
That is, for B=25, the atomic nucleus is within 0.87 A of its
average position 50% of the time (a volume of 2.7 A^3). Whereas
for B=50, it is within 1.22 A 50% of the time (7.7 A^3). Note
that although B=50 is twice as big as B=25, the half-occupancy
radius 0.87 A is not half as big as 1.22 A, nor are the volumes
2.7 and 7.7 A^3 related by a factor of two.
Why is this important for comparing two structures? Since the B
factor is non-linear with disorder, it is important to have a
common reference point when comparing them. If the low-B
structure has two atoms with B=10 and B=15 with average overall
B=12, that might seem to be "significant" (almost a factor of two
in the half-occupancy volume) but if the other structure has an
average B factor of 80, then suddenly 78 vs 83 doesn't seem all
that different (only a 10% change). Basically, a difference that
would be "significant" in a high-resolution structure is "washed
out" by the overall crystallographic B factor of the
low-resolution structure in this case.
Whether or not a 10% difference is "significant" depends on how
accurate you think your B factors are. If you "kick" your
coordinates (aka using "noise" in PDBSET) and re-refine, how much
do the final B factors change?
-James Holton
MAD Scientist
On 2/25/2013 12:08 PM, Yarrow Madrona wrote:
Hello,
Does anyone know a good method to compare B-factors between
structures? I
would like to compare mutants to a wild-type structure.
For example, structure2 has a higher B-factor for residue X
but how can I
show that this is significant if the average B-factor is also
higher?
Thank you for your help.
--
*******************************************
Jacob Pearson Keller, PhD
Postdoctoral Associate
HHMI Janelia Farms Research Campus
email: j-kell...@northwestern.edu <mailto:j-kell...@northwestern.edu>
*******************************************
......................
Jürgen Bosch
Johns Hopkins University
Bloomberg School of Public Health
Department of Biochemistry & Molecular Biology
Johns Hopkins Malaria Research Institute
615 North Wolfe Street, W8708
Baltimore, MD 21205
Office: +1-410-614-4742
Lab: +1-410-614-4894
Fax: +1-410-955-2926
http://lupo.jhsph.edu
