Zen and the Art of Motorcycle Maintenance: An Inquiry into Values

The structural heterogeneity of a biomolecular crystal structure is typically captured using atomic B-factors, determined during structure refinement. Here, the authors use molecular dynamics to show that this strategy is flawed, and that crystallographic B-factors underestimate structural heterogeneity.

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X-ray refinement significantly underestimates the level of microscopic heterogeneity in biomolecular crystals

The higher arithmetic (5th ed.), by H. Davenport. Pp 189. £12 hard covers, £5 paperback. 1982. ISBN 0 521 24422 6/28678 6 (Cambridge University Press)

Fully periodic Hartree–Fock and density functional theory calculations have been used to compute the anisotropic displacement parameters (ADPs) of molecular crystals at different temperatures by using the CRYSTAL code. Crystalline urea was adopted as a benchmark system to investigate the dependence on basis set and Hamiltonian. The results were compared with ADPs derived from neutron diffraction experiments. The approach can estimate the internal ADPs, corresponding to the contributions of high-frequency intramolecular vibrations, and for these internal contributions the results are almost independent of the basis set and Hamiltonian. Much larger variations and discrepancies from neutron diffraction experiments are seen for the external, low-frequency modes, which become dominant at higher temperatures. The approach was then tested on benzene and urotropine. Finally, ADPs of l-alanine were predicted at the B3LYP/6-31G(d,p) level of theory. The total ADPs, including low-frequency external modes, are underestimated, but can be brought into good agreement with the experimental ADPs by introducing a Gruneisen parameter, which partly accounts for anharmonicity of the potential energy surface, but likely also contains contributions from other deficiencies of the calculations.

Anisotropic displacement parameters for molecular crystals from periodic Hartree–Fock and density functional theory calculations

The translation–libration–screw-rotation (TLS) model of rigid-body harmonic displacements introduced in crystallography by Schomaker and Trueblood [On the rigid-body motion of molecules in crystals...

TLS from fundamentals to practice.

Two new anharmonic forms for the Debye-Waller factor, aimed at modelling curvilinear and asymmetric motion, have been introduced. These forms permit the refinement of structures with these types of anharmonic motion using a small number of additional parameters. Molecular-dynamics-derived numerical probability density functions (PDFs) have been used to assess the merit of these new functions in real space. The comparison is favourable particularly for the curvilinear PDF based on a parabolic coordinate system change of a trivariate Gaussian distribution. The initial results also suggest that high-order even terms from the Gram-Charlier series may be important for modelling methyl-group libration. The molecular-dynamics data sets provide useful insights into the nature of anharmonic thermal motion. Addressing the problem in real space allows intuitive PDFs to be developed but numerical methods may be necessary for these methods to be implemented in refinement programs as an analytical Debye-Waller factor cannot always be obtained.

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Using molecular‐dynamics simulations to understand and improve the treatment of anharmonic vibrations. II. Developing and assessing new Debye–Waller factors

Groups in modular arithmetic

Playing diffy with real sequences

85.55 What proportion of integers are the sum of two squares? Square numbers have always fascinated mathematicians. There is some evidence that Diophantus (250 AD) realised that every positive integer can be expressed as the sum of four or fewer squares. This was proved by Lagrange in 1770, building on earlier work by Euler. Although some integers need a full complement of four squares, others can be made with fewer squares. The table below shows how the first 20 integers can be represented as sums of squares, in each case using as few squares as possible. The different columns show how many squares are needed.

85.55 What proportion of integers are the sum of two squares

6 + 7 + ... + 69 = 180 (64 terms) 406 + ... + 917 = 5544 (512 terms) 1134 + ... + 2133 = 16830 (1000 terms) In general we can write down such an equation for k = v terms, where v is even and not a multiple of 3. Other possible values of k are 20 and 25. More generally still, we may seek values of k > 1 such that sum of k consecutive nth powers of positive integers is an nth power for n > 3. We know of no such example and it is possible that none exists.

K. Robin McLean

Papers

Using molecular‐dynamics simulations to understand and improve the treatment of anharmonic vibrations. II. Developing and assessing new Debye–Waller factors

Groups in modular arithmetic

Playing diffy with real sequences

85.55 What proportion of integers are the sum of two squares

89.33 Determinants of binomial coefficients