Estimates of triplet invariants given a model structure

Updating direct methods

The standard method of joint probability distribution functions, so crucial for the development of direct methods, has been revisited and updated. It consists of three steps: identification of the reflections which may contribute to the estimation of a given structure invariant or seminvariant, calculation of the corresponding joint probability distribution, and derivation of the conditional distribution of the invariant or seminvariant phase given the values of some diffracted amplitudes. In this article the conditional distributions are derived directly without passing through the second step. A good feature of direct methods is that they may work in the absence of any prior information: that is also their weakness. Different types of prior information have been taken into consideration: interatomic distances, interatomic vectors, Patterson peaks, structural model. The method of directly deriving the conditional distributions has been applied to those cases. Some new formulas have been obtained estimating two-, three- and four-phase invariants. Special attention has been dedicated to the practical aspects of the new formulas, in order to simplify their possible use in direct phasing procedures.

Solving proteins at non-atomic resolution by direct methods: update

Direct methods can be used to solve proteins of great structural complexity even when diffraction data are at non-atomic resolution. However, one of the main obstacles to the wider application of direct methods is that they reliably phase only a small fraction of the observed reflections, those with a sufficiently large value of the normalized structure factor amplitude. The subsequent phase expansion and refinement required for full structure solution are difficult. Here a new phase refinement procedure is described, which combines (1–2) difference Fourier synthesis with electron density modification techniques and thevive la differenceand Free Lunch algorithms. This procedure is able to solve data resistant to other direct space refinement procedures.

Solution of the phase problem at non-atomic resolution by the phantom derivative method

For a given unknown crystal structure (thetarget),nrandom structures, arbitrarily designed without any care for their chemical consistency and usually uncorrelated with the target, are sheltered in the same unit cell as the target structure and submitted to the same space-group symmetry. (These are called ancil structures.) The composite structures, whose electron densities are the sum of the target and of the ancil electron densities, are denotedderivatives. No observed diffraction amplitudes are available for them: in order to emphasize their unreal nature, the termphantomis added. The paper describes the theoretical basis by which the phantom derivative method may be used to phase the target structure. It may be guessed that 100–300 ancil structures may be sufficient for phasing a target structure, so that the phasing technique may be denoted as the multiple phantom derivative method. Ancil phases and amplitudes may be initially combined with observed target magnitudes to estimate amplitudes and phases of the corresponding phantom derivative. From them suitable algorithms allow one to obtain poor target phase estimates, which are often improved by combining the indications arising from each derivative. Probabilistic criteria are described to recognize the most reliable target phase estimates. The method is cyclic: the target phase estimates just obtained are used to improve amplitudes and phases of each derivative, which, in their turn, are employed to provide better target phase estimates. The method is a fullyab initiomethod, because it needs only the experimental data of the target structure. The term derivative is maintained with reference to SIR–MIR (single isomorphous replacement–multiple isomorphous replacement) techniques, even if its meaning is different: therefore the reader should think of the phantom derivative method more as a new method than as a variant of SIR–MIR techniques. The differences are much greater than the analogies. The paper also describes how phantom derivatives may be used for improving structure models obtainedviaotherab initioor non-ab initiotechniques. The method is expected to be insensitive to the structural complexity of the target and to the target experimental data resolution, provided it is better than 4–6 Å.

An alternative method for the calculation of joint probability distributions. Application to the expectation of the triplet invariant

This paper presents a completely new method for the calculation of expectations (and thus joint probability distributions) of structure factors or phase invariants. As an example, a first approximation of the expectation of the triplet invariant (up to a constant) is given and acomplexnumber is obtained. Instead of considering the atomic vector positions or reciprocal vectors as the fundamental random variables, the method samples over all functions (distributions) with a given number of atoms and given Patterson function. The aim of this paper was to explore the feasibility of the method, so the easiest problem was chosen: the calculation of the expectation value of the triplet invariant inP1. Calculation of the jointprobabilitydistribution of the triplet is not performed here but will be done in the future.