Perhaps I was a bit gruff, but what I was trying to do was throw out a voice of caution. In crystallography, it is very tempting to "normalize" everything by setting the mean to zero and the RMS deviation to one, but this is a "linear transformation", and linear transformations only work when the things you are comparing are "linear". True, everything is linear to a first-order approximation, and you may well be "okay" comparing B factors between two crystals that are "not all that different" in average B, but the assumption of "linearity" with something fundamentally non-linear like B factors will break down at some point. I'm not really sure where, but I would suspect that "normalizing" B factors from a 5 A structure to those of a 2.0 A structure might be "ill advised".
My opinion is that multiplying and dividing B factors makes no physical sense. If you must "normalize" something, I'd suggest normalizing the "half-occupancy volume" (proportional to sqrt(B)^3), or the width of the half-occupancy volume (proportional to sqrt(B)). If you want to try to make a leap to "entropy", then if you assume the half-occupancy volume contains an ideal gas, then the entropy should be proportional to log(B). But B itself is proportional to the surface area of the half-occupancy volume, and I can't think of too many physical-chemical properties that would be "proportional" to that. Nevertheless, purely on theory, I predict you will much more easily hit "pitfalls" (or quicksand) by trying to normalize "B" directly instead of first converting it into something more tangible (like a volume). That's all I'm sayin' -James Holton MAD Scientist On Mon, Mar 4, 2013 at 12:31 PM, John Fisher <johncfishe...@gmail.com> wrote: > Seriously? > I believe this specific forum has become quicksand rather than a useful > tool. Normalization based simply on ratios of one struct to the other should > allow normalization. Correct?, or have I just simply lost my mind here? > J > > John Fisher, M.D./PhD > St. Jude Children's Research Hospital > Department of Oncology > Department of Structural Biology > W: 901-595-6193 > C: 901-409-5699 > > On Mar 4, 2013, at 1:26 PM, James Holton <jmhol...@lbl.gov> wrote: > > > 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> 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 > ******************************************* > > > ...................... > 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 > > > > >
