International
Tables for
Crystallography
Volume F
Crystallography of biological macromolecules
Edited by M. G. Rossmann and E. Arnold

International Tables for Crystallography (2006). Vol. F, ch. 18.2, pp. 375-377   | 1 | 2 |

Section 18.2.3. The target function

A. T. Brunger,a* P. D. Adamsb and L. M. Ricec

aThe Howard Hughes Medical Institute, and Departments of Molecular and Cellular Physiology, Neurology and Neurological Sciences, and Stanford Synchrotron Radiation Laboratory, Stanford Universty, 1201 Welch Road, MSLS P210, Stanford, CA 94305-5489, USA,bThe Howard Hughes Medical Institute and Department of Molecular Biophysics and Biochemistry, Yale University, New Haven, CT 06511, USA, and cDepartment of Molecular Biophysics and Biochemistry, Yale University, New Haven, CT 06511, USA
Correspondence e-mail:  axel.brunger@stanford.edu

18.2.3. The target function

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Crystallographic refinement is a search for the global minimum of the target [E = E_{\rm chem} + w_{\rm X{\hbox{-}}ray}E_{\rm X{\hbox{-}}ray} \eqno(18.2.3.1)] as a function of the parameters of an atomic model, in particular, atomic coordinates. [E_{\rm chem}] comprises empirical information about chemical interactions; it is a function of all atomic positions, describing covalent (bond lengths, bond angles, torsion angles, chiral centres and planarity of aromatic rings) and non-bonded (intramolecular as well as intermolecular and symmetry-related) interactions (Hendrickson, 1985[link]). [E_{\rm X{\hbox{-}}ray}] is related to the difference between observed and calculated data, and [w_{\rm X{\hbox{-}}ray}] is a weight appropriately chosen to balance the gradients (with respect to atomic parameters) arising from the two terms.

18.2.3.1. X-ray diffraction data versus model

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The traditional form of [E_{\rm X{\hbox{-}}ray}] consists of the crystallographic residual, [E^{\rm LSQ}], defined as the sum over the squared differences between the observed [(|{\bf F}_{o}|)] and calculated [(|{\bf F}_{c}|)] structure-factor amplitudes for a particular atomic model: [E_{\rm X{\hbox{-}}ray} = E^{\rm LSQ} = \textstyle{\sum\limits_{hkl\in {\rm working \; set}}} (|{\bf F}_{o}| - k|{\bf F}_{c}|)^{2}, \eqno(18.2.3.2)] where hkl are the indices of the reciprocal-lattice points of the crystal and k is a relative scale factor.

Minimization of [E^{\rm LSQ}] can produce improvement in the atomic model, but it can also accumulate systematic errors in the model by fitting noise in the diffraction data (Silva & Rossmann, 1985[link]). The least-squares residual is a limiting case of the more general maximum-likelihood theory and is only justified if the model is nearly complete and error-free. These assumptions may be violated during the initial stages of refinement. Improved targets for macromolecular refinement have been obtained using the more general maximum-likelihood formulation (Bricogne, 1991[link]; Pannu & Read, 1996[link]; Adams et al., 1997[link]; Murshudov et al., 1997[link]). The goal of the maximum-likelihood method is to determine the likelihood of the model, given estimates of the model's errors and those of the measured intensities.

A starting point for the maximum-likelihood formulation of crystallographic refinement is the Sim (1959)[link] distribution, i.e., the Gaussian conditional probability distribution of the `true' structure factors, F, given a partial model with structure factors [{\bf F}_{c}] and the model's error (Fig. 18.2.3.1)[link] (Srinivasan, 1966[link]; Read, 1986[link], 1990[link]) (for simplicity we will only discuss the case of acentric reflections), [P_{a} (\hbox{{\bf F}; {\bf F}}_{c}) = (1/\pi \varepsilon \sigma_{\Delta}^{2}) \exp [- ({\bf F} - D{\bf F}_{c})^{2} / \varepsilon \sigma_{\Delta}^{2}], \eqno(18.2.3.3)] where [\sigma_{\Delta}] is a parameter that incorporates the effect of the fraction of the asymmetric unit that is missing from the model and errors in the partial structure. Assuming a Wilson distribution of intensities, it can be shown that (Read, 1990[link]) [\sigma_{\Delta}^{2} = \langle |{\bf F}_{o}|^{2}\rangle - D^{2} \langle |{\bf F}_{c}|^{2}\rangle, \eqno(18.2.3.4)] where D is a factor that takes into account model error: it is unity in the limiting case of an error-free model and it is zero if no model is available (Luzzati, 1952[link]; Read, 1986[link]). For a complete and error-free model, [\sigma_{\Delta}] therefore becomes zero, and the probability distribution, [P_{a}(\hbox{{\bf F}; {\bf F}}_{c})], is infinitely sharp.

[Figure 18.2.3.1]

Figure 18.2.3.1 | top | pdf |

The Gaussian probability distribution forms the basis of maximum-likelihood targets in crystallographic refinement. The conditional probability of the true structure factor, F, given model structure factors, is a Gaussian in the complex plane [equation (18.2.3.3)[link]]. The expected value of the probability distribution is [D\hbox{\bf F}_{c}] with variance [\sigma_{\Delta}], where D and [\sigma_{\Delta}] account for missing or incorrectly placed atoms in the model.

Taking measurement errors into account requires multiplication of equation (18.2.3.3)[link] with an appropriate probability distribution (usually a conditional Gaussian distribution with standard deviation [\sigma_{o}]) of the observed structure-factor amplitudes [(|{\bf F}_{o}|)] around the `true' structure-factor amplitudes [(|{\bf F}|)], [P_{\rm meas} (|{\bf F}_{o}|\hbox{;}\ |{\bf F}|). \eqno(18.2.3.5)]

Prior knowledge of the phases of the structure factors can be incorporated by multiplying equation (18.2.3.3)[link] with a phase probability distribution [P_{\rm phase} (\varphi) \eqno(18.2.3.6)] and rewriting equation (18.2.3.3)[link] in terms of the structure-factor moduli and amplitudes of [{\bf F} = |{\bf F}| \exp (i\varphi)].

The unknown variables [|{\bf F}|] and ϕ in equations (18.2.3.3)[link]–(18.2.3.5)[link] [link] have to be eliminated by integration in order to obtain the conditional probability distribution of the observed structure-factor amplitudes, given a partial model with errors, the amplitude measurement errors and prior phase information: [\eqalignno{P_{a} (|{\bf F}_{o}|\hbox{;}\ {\bf F}_{c}) &= (1/\pi \varepsilon \sigma_{\Delta}^{2}) \textstyle{\int} \hbox{d}\varphi \;\hbox{d} |{\bf F}|\; |{\bf F}| P_{\rm meas} (|{\bf F}_{o}|\hbox{;}\ |{\bf F}|)&\cr &\quad\times P_{\rm phase} (\varphi) \exp \left\{ - [|{\bf F}| \exp (i\varphi) - D{\bf F}_{o}]^{2}/\varepsilon \sigma_{\Delta}^{2}\right\}.\cr &&(18.2.3.7)}]

The likelihood, [{\cal L}], of the model is defined as the joint probability distribution of the structure factors of all reflections in the working set. Assuming independent and uncorrelated structure factors, [{\cal L}] is simply the product of the distributions in equation (18.2.3.7)[link] for all reflections. Instead of maximizing the likelihood, it is more common to minimize the negative logarithm of the likelihood, [E_{\rm X{\hbox{-}}ray} = {\cal L} = - \textstyle{\sum\limits_{hkl\in {\rm working \; set}}} \log [P_{a} (|{\bf F}_{o}|\hbox{;}\ {\bf F}_{c})]. \eqno(18.2.3.8)]

Empirical estimates of [\sigma_{\Delta}] [and D through equation (18.2.3.4)[link]] can be obtained by minimizing [{\cal L}] for a particular atomic model. It is generally assumed that [\sigma_{\Delta}] and D show relatively little variation among neighbouring reflections. Accepting this assumption, [\sigma_{\Delta}] and D can be estimated by considering narrow resolution shells of reflections and assuming that the two parameters are constant in these shells. Minimization of [{\cal L}] can then be performed as a function of these constant shell parameters while keeping the atomic model fixed (Read, 1986[link], 1997[link]). Alternatively, one can assume a two-term Gaussian model for [\sigma_{\Delta}] (Murshudov et al., 1997[link]) and minimize [{\cal L}] as a function of the Gaussian parameters. Note that individual atomic B factors are taken into account by the calculated model structure factors [({\bf F}_{c})].

This empirical approach to estimate [\sigma_{\Delta}] and D requires occasional recomputation of these values as the model improves. Refinement methods that improve the model structure factors, [{\bf F}_{c}], will therefore have a beneficial effect on [\sigma_{\Delta}] and D. Better estimates of these values will then enhance the next refinement cycle. Thus, powerful optimization methods and maximum-likelihood targets are expected to interact in a synergistic fashion (cf. Fig. 18.2.5.1[link]). Structure-factor averaging of multi-start refinement models can provide another layer of improvement by producing a better description of [{\bf F}_{c}] if the model shows significant variability due to errors or intrinsic flexibility (see below).

In order to achieve an improvement over the least-squares residual [equation (18.2.3.2[link])], cross validation was found to be essential (Pannu & Read, 1996[link]; Adams et al., 1997[link]) for the estimation of model incompleteness and errors ([\sigma_{\Delta}] and D). Since the test set typically contains only 10% of the diffraction data, these cross-validated quantities can show significant statistical fluctuations as a function of resolution. In order to reduce these fluctuations, Read (1997)[link] devised a smoothing method by applying restraints to [\sigma_{A}] values between neighbouring resolution shells where [\sigma_{A} = \left[1 - (\sigma_{\Delta} / \langle |{\bf F}_{o}|^{2}\rangle )\right]^{1/2}. \eqno(18.2.3.9)]

Pannu & Read (1996)[link] have developed an efficient Gaussian approximation of equation (18.2.3.7)[link] in cases of no prior phase information, termed the `MLF' target function. In the limit of a perfect model (i.e. [\sigma_{\Delta} = 0] and [D = 1]), MLF reduces to the traditional least-squares residual [equation (18.2.3.2)[link]] with [1/\sigma_{o}^{2}] weighting. In the case of prior phase information, the integration over the phase angles has been carried out numerically in equation (18.2.3.7)[link], termed the `MLHL' target (Pannu et al., 1998[link]). A maximum-likelihood function which expresses equation (18.2.3.7)[link] in terms of observed intensities has also been developed, termed `MLI' (Pannu & Read, 1996[link]).

18.2.3.2. A priori chemical information

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The parameters for the covalent terms in [E_{\rm chem}] [equation (18.2.3.1)[link]] can be derived from the average geometry and (r.m.s.) deviations observed in a small-molecule database. Extensive statistical analyses were undertaken for the chemical moieties of proteins (Engh & Huber, 1991[link]) and polynucleotides (Parkinson et al., 1996[link]) using the Cambridge Structural Database (Allen et al., 1983[link]). Analysis of the ever-increasing number of atomic resolution macromolecular crystal structures will no doubt cause some modifications of these parameters in the future.

It is common to use a purely repulsive quartic function [(E_{\rm repulsive})] for the non-bonded interactions that are included in [E_{\rm chem}] (Hendrickson, 1985[link]): [E_{\rm repulsive} = \textstyle{\sum\limits_{ij}} [(cR_{ij}^{\min})^{n} - R_{ij}^{n}]^{m}, \eqno(18.2.3.10)] where [R_{ij}] is the distance between two atoms i and j, [R_{ij}^{\min}] is the van der Waals radius for a particular atom pair ij, [c \leq 1] is a constant that is sometimes used to reduce the radii, and n = 2, m = 2 or n = 1, m = 4. van der Waals attraction and electrostatic interactions are usually not included in crystallographic refinement. These simplifications are valid since the diffraction data contain information that is able to produce atomic conformations consistent with actual non-bonded interactions. In fact, atomic resolution crystal structures can be used to derive parameters for electrostatic charge distributions (Pearlman & Kim, 1990[link]).

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