 1. Introduction
 2. Density modification by reciprocalspacebased likelihood optimization
 3. Likelihoodbased density modification
 4. Likelihood function for an electrondensity map with errors
 5. Evaluation of maximumlikelihood density modification with model and real data
 6. Discussion
 References
 1. Introduction
 2. Density modification by reciprocalspacebased likelihood optimization
 3. Likelihoodbased density modification
 4. Likelihood function for an electrondensity map with errors
 5. Evaluation of maximumlikelihood density modification with model and real data
 6. Discussion
 References
research papers
density modification
^{a}Structural Biology Group, Mail Stop M888, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
^{*}Correspondence email: terwilliger@lanl.gov
A likelihoodbased approach to density modification is developed that can be applied to a wide variety of cases where some information about the electron density at various points in the ), Acta Cryst. D55, 1863–1871].
is available. The key to the approach consists of developing likelihood functions that represent the probability that a particular value of electron density is consistent with prior expectations for the electron density at that point in the These likelihood functions are then combined with likelihood functions based on experimental observations and with others containing any prior knowledge about structure factors to form a combined likelihood function for each A simple and general approach to maximizing the combined likelihood function is developed. It is found that this likelihoodbased approach yields greater phase improvement in model and real test cases than either conventional solvent flattening and histogram matching or a recent reciprocalspace solventflattening procedure [Terwilliger (1999Keywords: density modification; likelihood functions.
1. Introduction
The phase information obtained from experimental measurements on macromolecules using either multiple ; Bricogne, 1984, 1988; Cowtan & Main, 1993, 1996; Giacovazzo & Siliqi, 1997; Goldstein & Zhang, 1998; Gu et al., 1997; Lunin, 1993; Perrakis et al., 1997; Podjarny et al., 1987; Prince et al., 1988; Refaat et al., 1996; Roberts & Brünger, 1995; Rossmann & Arnold, 1993; Vellieux et al., 1995; Wilson & Agard, 1993; Xiang et al., 1993; Zhang & Main, 1990; Zhang, 1993; Zhang et al., 1997). The fundamental basis of densitymodification methods is that there are many possible sets of structurefactor amplitudes and phases that are all reasonably probable based on the limited experimental data, and those structure factors that lead to maps that are most consistent with both the experimental data and the prior knowledge are the most likely overall. In these methods, the choice of prior information that is to be used and the procedure for combining prior information about electron density with experimentally derived phase information are crucial parts.
or multiwavelength anomalous diffraction is often insufficient by itself for constructing a electrondensity map useful for model building and interpretation. Many densitymodification methods have been developed in recent years for improving the quality of electrondensity maps by incorporation of prior knowledge about the features expected in these maps when they are obtained at high or moderate resolution (2–4 Å). Among the most powerful of these methods are solvent flattening, averaging, histogram matching, phase extension, maximization and iterative model building (Abrahams, 1997Until recently, density modification, the combination of knowledge about expected features of an electrondensity map with experimental phase information, has generally been carried out in a twostep procedure that is iterated until convergence. In the first step, an electrondensity map obtained experimentally is modified in real space in order to make it consistent with expectations. This can consist of flattening solvent regions, averaging noncrystallographic symmetryrelated regions or histogram matching, for example. In the second step, phases are calculated from the modified map and are combined with the experimental phases to form a new phase set.
The disadvantage of this realspace modification approach is that it is not at all clear how to weight the observed phases with those obtained from the modified map. This is a consequence of the fact that the modified map contains some of the same information as the original map and some new information. This difficulty has been recognized for a long time and a number of approaches have been designed to improve the relative weighting from these two sources, recently including the use of maximumentropy methods and the use of weighting optimized using crossvalidation (Xiang et al., 1993; Roberts & Brünger, 1995; Cowtan & Main, 1996) and `solvent flipping' (Abrahams, 1997).
2. Density modification by reciprocalspacebased likelihood optimization
We have recently developed a very different approach to combinining experimental phase information with prior knowledge about expected electrondensity distributions in maps. Our approach is based on maximization of a combined likelihood function (Terwilliger, 1999). The fundamental idea is to express our knowledge about the probability of a set of structure factors {F_{h}} in terms of two quantities: (i) the likelihood of having measured the observed set of structure factors if this structurefactor set were correct and (ii) the likelihood that the map resulting from this structurefactor set {F_{h}} is consistent with our prior knowledge about this and other macromolecular structures.
When set up in this way, the overlap of information that occurred in the realspace modification methods is not present because the experimental and prior information are kept separate. Consequently, proper weighting of experimental and prior information only requires estimates of probability functions for each source of information.
The likelihoodbased densitymodification approach has a second very important advantage. This is that the derivatives of the likelihood functions with respect to individual structure factors can be readily calculated in ).
by FFTbased methods. As a consequence, density modification simply becomes an optimization of a combined likelihood function by adjustment of structure factors. This makes density modification a remarkably simple but powerful approach, only requiring that suitable likelihood functions be constructed for each aspect of prior knowledge that is to be incorporated. We previously showed that such an approach could be applied to solvent flattening and that the resulting algorithm was greatly improved over methods depending on realspace modification and phase recombination (Terwilliger, 1999Here, we extend the idea of likelihoodbased density modification to include prior information on the electrondensity distribution from a wide variety of potential sources and demonstrate it on both the electron density in the solvent region and the region occupied by a macromolecule. First, we describe the mathematics of likelihoodbased density modification in a practical formulation that is modified somewhat from the one we used for reciprocalspace solvent flattening (Terwilliger, 1999). We then show how a likelihood function for a map that includes information on both the solvent and macromoleculecontaining regions can be constructed and used.
3. Likelihoodbased density modification
The basic idea of our likelihoodbased densitymodification procedure is that there are two key kinds of information about the structure factors for a crystal of a macromolecule. The first is the experimental phase and amplitude information. This can be thought of in terms of a likelihood (or loglikelihood) function LL^{OBS}(F_{h}) for each F_{h}, where the probability distribution for the p^{OBS}(F_{h}) is given by
For reflections with accurately measured amplitudes, the chief uncertainty in F_{h} will be in the phase, while for unmeasured or poorly measured reflections it will be in both phase and amplitude.
The second kind of information about structure factors in this formulation is the likelihood of the map resulting from them. For example, for most macromolecular crystals a set of structure factors {F_{h}} that leads to a map with a flat region corresponding to solvent is more likely to be correct than one that leads to a map with uniform variation everywhere. This maplikelihood function describes the probability that the map obtained from a set of structure factors is compatible with our expectations,
We then combine our two principal sources of information along with any prior knowledge of the structure factors to yield the likelihood of a particular set of structure factors,
where LL^{o}({F_{h}}) includes any structurefactor information that is known in advance, such as the distribution of intensities of structure factors (Wilson, 1949).
3.1. Approximating the likelihood function to simplify the procedure
In order to maximize the overall likelihood function in (3) we are going to need to know how the maplikelihood function changes in response to changes in structure factors. In the case of the maplikelihood function LL^{MAP}({F_{h}}) this can be thought of as two separate relationships, the response of the likelihood function to changes in electron density and the changes in electron density as a function of changes in structure factors. In principle, the likelihood of a particular map is a complicated function of the electron density over the entire map. Furthermore, the value of any affects the electron density everywhere in the map. To simplify the mathematics, we explicitly use a loworder approximation to the likelihood function for a map instead of attempting to evaluate the function precisely. As Fourier transformation is a linear process, each reflection contributes independently to the electron density at a given point in the cell. Although the loglikelihood of the electron density might have any form, we expect that for sufficiently small changes in structure factors, a firstorder approximation to the loglikelihood function would apply and each reflection would also contribute relatively independently to changes in the loglikelihood function.
Consequently, we construct a local approximation to the maplikelihood function, neglecting correlations among different points in the map and between reflections, expecting that it might describe reasonably accurately how the likelihood function would vary in response to small changes in structure factors.
By neglecting correlations among different points in the map, we can write the loglikelihood for the whole electrondensity map as the sum of the loglikelihoods of the densities at each point in the map, normalized to the volume of the ),
and the number of reflections used to construct it (Terwilliger, 1999Additionally, by treating each reflection as independently contributing to the likelihood function, we can write a local approximation to the loglikelihood of the density at each point. This approximation is given by the sum over all reflections of first few terms of a Taylor's series expansion around the value obtained with the starting structure factors used in a cycle of density modification, ,
where and are the differences between F_{h} and along the directions of and , respectively.
Combining (4) and (5), we can write an expression for the map loglikelihood function,
3.2. FFTbased calculation of the reciprocalspace derivatives of loglikelihood of electron density LL[ρ(x, {F_{h}})]
The integrals in (6) can be rewritten in a form that is suitable for evaluation by an FFTbased approach. Considering the first integral in (6), we use the chain rule to write that
and note that the derivative of ρ(x) with respect to for a particular index h is given by
Now we can rearrange and rewrite the first integral in (6) in the form
where the complex number a_{h} is a term in the Fourier transform of LL[ρ(x, {F_{h}})],
In space groups other than P1, only a unique set of structure factors need to be specified to calculate an electrondensity map. Taking spacegroup symmetry into account, (9) can be generalized (Terwilliger, 1999) to read
where the indices h′ are all indices equivalent to h owing to spacegroup symmetry.
A similar procedure can be used to rewrite the second integral in (6), yielding the expression
where the indices h′ and k′ are each all indices equivalent to h owing to spacegroup symmetry and where the coefficients b_{h} are again terms in a Fourier transform, this time of the second derivative of the loglikelihood of the electron density,
The third and fourth integrals in (6) can be rewritten in a similar way, yielding the expressions
and
The significance of (4) through (15) is that we now have a simple expression (6) describing how the maplikelihood function LL^{MAP}({F_{h}}) varies when small changes are made in the structure factors. Evaluating this expression only requires that we be able to calculate the first and second derivatives of loglikelihood of the electron density with respect to electron density at each point in the map and carry out an FFT. Furthermore, maximization of the (local) overall likelihood function (3) becomes straightforward, as every reflection is treated independently. It consists simply of adjusting each to maximize its contribution to the approximation to the likelihood function through (3) to (15).
In practice, instead of directly maximizing the overall likelihood function, we use it here to estimate the probability distribution for each ) and then integrate this probability distribution over the phase (or phase and amplitude) of the reflection to obtain a estimate of the Using (3) to (15), the probability distribution for an individual can be written as
(Terwilliger, 1999where, as above, the indices h′ and k′ are each all indices equivalent to h owing to spacegroup symmetry and the coefficients a_{h} and b_{h} are given in (10) and (13). Also as before, and are the differences between F_{h} and along the directions of and , respectively. All the quantities in (16) can be readily calculated once a likelihood function for the electron density and its derivatives are obtained.
4. Likelihood function for an electrondensity map with errors
A key step in likelihoodbased density modification is the decision as to the likelihood function for values of the electron density at a particular location in the map. For the present purpose, an expression for the loglikelihood of the electron density LL[ρ(x, {F_{h}})] at a particular location x in a map is needed that depends on whether the point x is within the solvent region or the protein region. In general, this function might depend on whether the point satisfies any of a wide variety of conditions, such as being at a certain location in a known fragment of structure or being at a certain distance from some other feature of the map. We discussed previously (Terwilliger, 1999) how one might incorporate information on the environment of x by writing the loglikelihood function as the log of the sum of conditional probabilities dependent on the environment of 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. The probability that x is in the protein or solvent regions is estimated by a modification of the methods of Wang (1985) and Leslie (1987) as described previously (Terwilliger, 1999). If there were more than just solvent and protein regions that identified the environment of each point, then (17) could be modified to include those as well.
In developing (13) to (15), the derivatives of the likelihood function for electron density were intended to represent how the likelihood function changed when small changes in one were made. Surprisingly, the likelihood function that is most appropriate for our present purposes in this case is not a globally correct one. Instead, it is a likelihood function that represents how the overall likelihood function varies in response to small changes in one keeping all others constant. To see the difference, consider the electron density in the solvent region of a macromolecular crystal. In an idealized situation with all possible reflections included, the electron density might be exactly equal to a constant in this region. The goal in using (16) is to obtain the relative probabilites for each possible value of a particular unknown F_{h}. If all other structure factors were exact, then the globally correct likelihood function for the electron density (zero unless the solvent region is perfectly flat) would correctly identify the correct value of the unknown Now suppose we had imperfect phase information. The solvent region would have a significant amount of noise and its value would no longer be a constant. If we use the globally correct likelihood function for the electron density, we would assign a zero probability to any value of the that did not lead to an absolutely flat solvent region. This is clearly unreasonable, because all the other (incorrect) structure factors are contributing noise that exists regardless of the value of this structure factor.
This situation is very similar to the one encountered in structure e.g. Terwilliger & Berendzen, 1996; Pannu & Read, 1996). Similarly, the appropriate likelihood function for electron density for use in the present method is one in which the overall uncertainty in the electron density arising from all reflections other than the one being considered is included in the variance.
of macromolecular structures where there is a substantial deficiency in the model. The errors in all the other structure factors in the present discussion correspond to the deficiency in the macromolecular model in the case. The appropriate variance to use as a weighting factor in includes the estimated model error as well as the error in measurement (A likelihood function of this kind for the electron density can be developed using a model in which the electron density arising from all reflections but one is treated as a random variable (Terwilliger & Berendzen, 1996; Pannu & Read, 1996). Suppose that the true value of the electron density at x was known and was given by ρ_{T}. Then consider that we have estimates of all the structure factors, but that substantial errors exist in each one. The expected value of the estimate of this electron density obtained from current estimates of all the structure factors (ρ_{OBS}) will be given by 〈ρ_{OBS}〉 = βρ_{T} and the expected value of the variance by 〈(ρ_{OBS} − βρ_{T})^{2}〉 = . The factor β represents the expectation that the calculated value of ρ will be smaller than the true value. This is for two reasons. One is that such a estimate may be calculated using figureofmerit weighted estimates of structure factors, which will be smaller than the correct ones. The other is that phase error in the structure factors systematically leads to a bias towards a smaller component of the along the direction of the true This is the same effect that leads to the D correction factor in (Pannu & Read, 1996).
A probability function for the electron density at this point that is appropriate for assessing the probabilities of values of the
for one reflection can now be written asIn a slightly more complicated case, where the value of ρ_{T} is not known exactly but rather has an uncertainty σ_{T}, (18) becomes
Finally, in the case where only a probability distribution p(ρ_{T}) for ρ_{T} is known, (18) becomes
4.1. Likelihood function for solvent and macromoleculecontaining regions of a map
Using (19) and (20), we are now in a position to use a histogrambased approach (Goldstein & Zhang, 1998; Lunin, 1993; Zhang & Main, 1990) to develop likelihood functions for the solvent region of a map and for the macromoleculecontaining region of a map. The approach is simple. The probability distribution for true electron density in the solvent or macromolecule regions of a crystal structure is obtained from an analysis of model structures and represented as a sum of Gaussian functions of the form
If the values of β and σ_{MAP} were known for an experimental map with unknown errors but identified solvent and protein regions, then using (19) we could write the probability distribution for electron density in the each region of the map as
with the appropriate values of β and σ_{MAP}. In practice, the values of β and σ_{MAP} are estimated by a leastsquares fitting of the probability distribution given in (22) to the one found in the experimental map. This procedure has the advantage that the scale of the experimental map does not have to be accurately determined. Then (22) is used with the refined values of β and σ_{MAP} as the probability function for electron density in the corresponding region (solvent or macromolecule) of the map.
5. Evaluation of density modification with model and real data
To evaluate the utility of ). The first test case consisted of a set of phases constructed from a model with 32–68% of the volume of the taken up by protein. The initial effective figure of merit of the phases overall [〈cos(Δφ)〉] was about 0.40. In our previous tests, we showed that both realspace and reciprocalspace solvent flattening improved the quality of phasing considerably. In the current tests, the realspace density modification included both solvent flattening and histogram matching to be as comparable as possible to the density modification we have developed.
density modification as described here, we carried out tests using the same model and experimental data that we previously analyzed using reciprocalspace solvent flattening and by realspace solvent flattening (Terwilliger, 1999Table 1 shows the the quality of phases obtained after each method for density modification was applied to this model case. In all cases, density modification of this map resulted in phases with an effective figure of merit [〈cos(Δφ)〉] higher than any of the other methods. When the fraction of solvent in the model was 50%, for example, density modification yielded an effective figure of merit of 0.83, while realspace solvent flattening and histogram matching resulted in an effective figure of merit of 0.62 and reciprocalspace solvent flattening gave an effective figure of merit of 0.67.

The utility of et al., 1998). IF5A crystallizes in I4, with unitcell parameters a = 114, b = 114, c = 33 Å, one molecule in the and a solvent content of about 60%. The structure was solved using based on three Se atoms in the at a resolution of 2.2 Å. For purposes of testing densitymodification methods, only one of the three selenium sites was used in phasing here, resulting in a starting map with a to the map calculated using the final refined structure of 0.37. The resulting electrondensity map was improved by realspace density modification using solvent flattening and histogram matching with dm (Cowtan & Main, 1996), by realspace density modification using solvent flipping (Abrahams, 1997) and after density modification. The `experimental' map, the dmmodified map and the map are shown in Fig. 1. As anticipated, the realspace modified map obtained with dm is improved over the starting map; it has a of 0.65. Density modification including solvent flipping yielded a similar improvement, with a of 0.61 to the model map. The maximimumlikelihood modified map was much more substantially improved, with a to the map based on a refined model of 0.79.
density modification was also compared with realspace density modification and with reciprocalspace solvent flattening using experimental multiwavelength (MAD) data on initiation factor 5A (IF5A) recently determined in our laboratory (Peat6. Discussion
We have shown here that a ). The reason the approach works so well is that the relative weighting of experimental phase information and of expected electrondensity distributions is taken care of automatically by keeping the two sources of information clearly delineated and by defining suitable probability distributions for each.
approach can be used to carry out density modification on macromolecular crystal structures and that this approach is much more powerful than either conventional density modification based on solvent flattening and histogram matching or our recent reciprocalspace solventflattening procedure (Terwilliger, 1999The e.g. Bricogne, 1984, 1988; Lunin, 1993). The importance of the present work and of our recent work on reciprocalspace solvent flattening (Terwilliger, 1999) is that we have developed a simple, effective and general way to carry it out.
approach to improvement of crystallographic phases has been developed extensively by Bricogne and others (Although we have demonstrated here only two sources of expected electrondensity distributions (probability distributions for solvent regions and for proteincontaining regions), the methods developed here can be applied directly to a wide variety of sources of information. For example, any source of information about the expected electron density at a particular point in the ) can be used in our procedure to describe the likelihood that a particular value of electron density is consistent with expectations.
that can be written in a form such as the one in (22Sources of expected electrondensity information that are especially suitable for application to our method include ) with a value of ρ_{T} equal to the at all noncrystallographically equivalent points in the cell. The value of σ_{T} could be calculated based on their variance and the value of σ_{MAP}. In the case of knowledge of locations of fragments in the this knowledge can be used to calculate estimates of the electrondensity distribution for each point in the neighborhood of the fragment. These electrondensity distributions can then in turn be used just as described above to estimate ρ_{T} and σ_{T} in this region. An iterative process, in which fragment locations are identified by crosscorrelation or related searches (density modification) is applied and additional searches are carried out to further generate a model for the electron density, could even be developed, in an extension of the iterative chaintracing methods described by Wilson & Agard (1993). Such a process could potentially even be used to construct a complete probabilistic model of a macromolecular structure using structurefactor estimates obtained from with fragments of macromolecular structures as a starting point. In all these cases, the electrondensity information could be included in much the same way as the probability distributions we used here for the solvent and protein regions of maps. In each case, the key is an estimate of the probability distribution for electron density at a point in the map that contains some information that restricts the likely values of electron density at that point. The procedure could be further extended by having probability distributions describing the likelihood that a particular point in the is within protein, within solvent, within a particular location in a fragment of protein structure, within a noncrystallographically related region and so on. These probability distributions could be overlapping or nonoverlapping. Then for each category of points, the probability distribution for electron density within that category could be formulated as in (22) and our current methods applied.
and the knowledge of the location of fragments of structure in the In the case of the probability distribution for electron density at one point in the can be written using (22The procedure described here differs from the reciprocalspace solventflattening procedure described previously (Terwilliger, 1999) in two important ways. One is that the expected electrondensity distribution in the nonsolvent region is included in the calculations and a formalism for incorporating information about the electrondensity map from a wide variety of sources is developed. The second is that the probability distribution for the electron density is calculated using (22) for both solvent and nonsolvent regions and values of the scaling parameter β and the map uncertainty σ_{MAP} are estimated by a fitting of model and observed electrondensity distributions. This fitting process makes the whole procedure very robust with respect to scaling of the experimental data, which otherwise would have to be very accurate in order that the model electrondensity distributions be applicable.
Software for carrying out Resolve') and complete documentation is available on the WWW at http://resolve.lanl.gov .
density modification (`Acknowledgements
The author would like to thank Joel Berendzen for helpful discussions and the NIH and the US Department of Energy for generous support.
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