OK, so lets suppose there is this bond in your structure that is
stretched a bit. Is that for real? Or just a random fluke? Let's say
for example its a CA-CB bond that is supposed to be 1.529 A long, but in
your model its 1.579 A. This is 0.05 A too long. Doesn't seem like
much, right? But the "sigma" given to such a bond in our geometry
libraries is 0.016 A. These sigmas are typically derived from a
database of observed bonds of similar type found in highly accurate
structures, like small molecules. So, that makes this a 3-sigma outlier.
Assuming the distribution of deviations is Gaussian, that's a pretty
unlikely thing to happen. You expect 3-sigma deviates to appear less
than 0.3% of the time. So, is that significant?
But, then again, there are lots of other bonds in the structure. Lets
say there are 1000. With that many samplings from a Gaussian
distribution you generally expect to see a 3-sigma deviate at least
once. That is, do an "experiment" where you pick 1000 Gaussian-random
numbers from a distribution with a standard deviation of 1.0. Then, look
for the maximum over all 1000 trials. Is that one > 3 sigma? It probably
is. If you do this "experiment" millions of times it turns out seeing at
least one 3-sigma deviate in 1000 tries is very common. Specifically,
about 93% of the time. It is rare indeed to have every member of a
1000-deviate set all lie within 3 sigmas. So, we have gone from one
3-sigma deviate being highly unlikely to being a virtual certainty if
you look at enough samples.
So, my question is: is a 3-sigma deviate significant? Is it significant
only if you have one bond in the structure? What about angles? What if
you have 500 bonds and 500 angles? Do they count as 1000 deviates
together? Or separately?
I'm sure the more mathematically inclined out there will have some
intelligent answers for the rest of us, however, if you are not a
mathematician, how about a vote? Is a 3-sigma bond length deviation
significant? Or not?
Looking forward to both kinds of responses,
-James Holton
MAD Scientist
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