Likelihood-based modification of experimental crystal structure electron density maps
View Patent ↗A maximum-likelihood method for improves an electron density map of an experimental crystal structure. A likelihood of a set of structure factors {F h } is formed for the experimental crystal structure as (1) the likelihood of having obtained an observed set of structure factors {F h OBS } if structure factor set {F h } was correct, and (2) the likelihood that an electron density map resulting from {F h } is consistent with selected prior knowledge about the experimental crystal structure. The set of structure factors {F h } is then adjusted to maximize the likelihood of {F h } for the experimental crystal structure. An improved electron density map is constructed with the maximized structure factors.
1. A method for improving an electron density map of an experimental crystal structure, comprising the steps of:
(a) forming a model electron density map from known crystallographic information of an exemplary model crystal structure;
(b) forming model histograms of model electron densities in identified protein and solvent regions of the model electron density map;
(c) fitting a model probability distribution function defined by
p
(
ρ
T
)
=
∑
k
w
k
exp
{
-
(
ρ
-
c
k
)
2
2
σ
k
2
}
to the model histograms, where k is separately indexed over the protein and solvent regions of the model map, p(ρ T ) is a probability of an electron density at a point, w k is a normalization factor, ρ is electron density, c k is a mean value of ρ, and σ k is a variance of ρ, where the fitting determines the coefficients w k , c k , and σ k ;
(d) determining a set of experimental structure factors from x-ray diffraction data for the experimental crystal structure and forming an experimental electron density map;
(e) forming separate experimental histograms of experimental electron densities over protein and solvent regions of the model electron density map;
(f) fitting an experimental probability distribution function defined by
p
(
ρ
T
)
=
∑
k
w
k
exp
{
-
(
ρ
-
β
c
k
)
2
2
(
β
σ
k
2
+
σ
map
2
)
}
to separate protein and solvent regions of the experimental histograms, where β is an expectation that an experimental value of ρ is less than a true value and σ map is a variance, where the fitting determines the coefficients β and σ map ;
(g) determine the overall experimental log-likelihood of the electron density in the protein and solvent regions of the experimental map from the experimental probability distribution function
LL (ρ( x,{F h }))=ln[ p (ρ( x )| PROT ) p PROT ( x )+ p (ρ( x )| SOLV ) p SOLV ( x )]
where p PROT (x) is the probability that x is in the protein region and p(ρ(x)|PROT) is the conditional probability for ρ(x) given that x is in the protein region, and p SOLV (x) and p(ρ(x)|SOLV) are the corresponding quantities for the solvent region;
(h) determine how the experimental log-likelihood of the electron density of the protein and solvent regions of the structure factor experimental electron density map would change as each experimental structure factor changes to output a revised log-likelihood of any value of each experimental structure factor;
(i) forming from the revised log-likelihood of experimental structure factor values a new set of structure factors; and
(j) forming a revised experimental electron density map from the revised structure factors.
2. The method according to claim 1 , wherein step (a) further includes a step of selecting the model crystal structure to be similar in size, data resolution, and atomic displacement factors to the experimental crystal structure.
3. The method according to claim 1 , wherein step (b) further includes a step of identifying protein and solvent regions by designating all points within a selected distance of an atom as “protein” and all other points as “solvent”.
4. The method according to claim 2 , wherein step (b) further includes a step of identifying protein and solvent regions by designating all points within a selected distance of an atom as “protein” and all other points as “solvent”.
5. The method according to claim 1 , wherein step (h) includes steps of forming a Taylor's series expansion of the log-likelihood of the experimental electron density map and evaluating terms of the Taylor's series expansion using a Fast Fourier Transform.