ALBERT

Notes: Abstract Calculation in unnormalized arithmetic provides, as has been described elsewhere, an estimate of the error of each quantity, initial, intermediate, and final, in a calculation. In an acceptable computational algorithm, this error estimate should be close to the true probable error. (The probable error is defined as follows: each computed quantity is defined, in the statement of the problem, explicitly or implicitly as some function of the input data, which are however known only up to errors having some distribution. As the input data vary in their joint distribution, the value of this function varies in a distribution whose variance determines the probable error of the result.) Many computational algorithms fail to achieve this goal, because whenever two numbers are combined arithmetically (+, -, ×, or ÷) in a computer, anycorrelation of the errors of the two numbers is perforce ignored; in consequence, the error estimate of a computed result may be greater than, comparable with, or smaller than the probable error, in which case the algorithm is called “conservative”, “faithful”, or “liberal”, respectively. In this article, a matrix-inversion algorithm based on an assignment strategy is described, which is very close to faithful, as evidenced by extensive numerical tests, which are also described. A test matrix has elements with varying numbers of leading zeros, the remaining bits in a 43-bit pattern are regarded as meaningful; its inverse is computed. A series of perturbed matrices are formed by adding or subtracting a one-bit in the 31st-stage of each input matrix element; this corresponds to amaximum change in the last twelve bits; the statistical aspect is then in the choice of addition-subtraction. By comparing the results of the successive inversions, the magnitude of the probable error (as defined above) resulting from these perturbations can be rather closely determined. (These tests cover the case in which the errors of the input matrix elements are uncorrelated.) If the calculation is regarded as one in which the 30th bit is the last significant one and the remaining 13 bits are guard digits, then, for a faithful algorithm, the observed probable error should appear starting in the 31st place. This is found to be the case quite accurately for the algorithm here proposed, but not for other commonly used algorithms. It should be pointed out that the algorithm here described is optimal also for normalized arithmetic (in which errors are ignored), because the precision on the one hand cannot be increased by retaining insignificant digits and on the other hand is in any case limited by the true probable error, no matter what algorithm is used (unless the input data are infinitely precise). However, the results are likely to be less satisfactory in normalized arithmetic, because explicit use is made of the relative precisions of certain quantities at each pivoting in the matrix inversion, and the machine indication of this relative precision is falsified when numbers are unduly normalized.

Notes: The probabilistic theory of the three-phase structure invariants for a pair of isomorphous structures [Hauptman (1982). Acta Cryst. A38, 289-294] is reexamined. The analysis leads to distributions capable of estimating cosine invariants in the full range of -1 to +1. In particular, it is shown that heavy-atom substructure information can be incorporated easily into the distributions. The initial applications, using calculated diffraction data from the protein cytochrome c550, MR ∼ 14 500, and its PtCl2-4 derivative show that a remarkable increase in accuracy results from the use of the revised distributions, particularly after the incorporation of heavy-atom substructure information. Finally, it is shown that in the individual phase determinations the redundant cosine invariants play a role identical to that of the multiple isomorphous derivatives and thus provide the basis for the solution of the phase problem in the single isomorphous replacement case.

Notes: The theoretical basis for the integration of direct methods into the single-wavelength anomalous dispersion technique is reexamined. The analysis shows that the approximations responsible for the ability to obtain unique estimates of the two- and three-phase structure invariants [Hauptman (1982). Acta Cryst. A38, 632-641; Giacovazzo (1983). Acta Cryst. A39, 585-592] or twofold estimates of the three-phase structure invariants [Kroon, Spek & Krabbendam (1977). Acta Cryst. A33, 382-385] are also responsible for the substantial errors observed in the applications. It is shown that, in the general case, the method of joint probability distributions leads to twofold estimates of the two-phase invariants and eightfold estimates of the three-phase invariants. Finally, it is shown that more accurately determined three-phase invariant estimates can be obtained by the use of anomalous scatterer substructure information, when available, and the use of a strategy that recognizes cases in which the eight estimates are clustered around one or two values. These cases are then distinguished from those where the eight estimates are widely scattered by a weighting function.

Notes: [Auszug] IN recent years evidence has been produced that the steps of incorporation of amino-acids into proteins include: (i) an activation dependent on adenosine triphosphate of the amino-acids through the COOH group, with the formation of an amino-acid-adenosine monophosphate complex by means of enzymes ...

Notes: [Auszug] WHEN a monomolecular film of protein at an VV air/water interface is compressed beyond a certain critical surface pressure it can no longer be re-expanded to its original area, due to the formation of intermolecular linkages by the unfolded film protein. As the film is further compressed, more such ...