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Part of the book series: Fundamental Theories of Physics ((FTPH,volume 53))

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Abstract

The entropy dynamics method seeks maxima for the entropy of the electron density for N atoms in a crystal cell, when the Fourier amplitudes are fixed, but their phases are unknown. By analogy with molecular dynamics, the effective potential energy is the negative entropy V = −NS. The kinetic energy is proportional to the squared velocities of the electron densities at grid points in the map. It reduces to a sum of Fourier mode rotor energies. Each rotor angle experiences a couple equal to the phase gradient of S, and local dynamical equilibrium yields a Boltzmann distribution of S Trial calculations have been made of phase averages and correlations in a centrosymmetric projection of the membrane protein bacteriorhodopsin.

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References

  • Allen, M.P. Tildesley, D.J. (1987). Computer Simulation of Liquids. Oxford: University Press.

    MATH  Google Scholar 

  • Blundell, T.L. Johnson, L.N. (1976). Protein Crystallography. London: Academic Press.

    Google Scholar 

  • Bricogne, G. (1984). Acta Cryst A40, 410–445. Maximum Entropy and the Foundations of Direct Methods.

    Google Scholar 

  • Bricogne, G. Gilmore, C.J. (1990). Acta Cryst. A46, 284–297. A Multisolution Method of Phase Determination by Combined Maximisation of Entropy and Likelihood. I. Theory, Algorithms and Strategy.

    Google Scholar 

  • Brunger, A.T., Kuriyan, J. Karplus, M. (1987). Science 235, 458–460. Crystallographic R Factor Refinement by Molecular Dynamics.

    Google Scholar 

  • Bullough, P.A. Henderson, R. (1975). Biophys. J 58, 705–711. Phase Accuracy in High-Resolution Electron Microscopy of Trigonal and Orthorhombic Purple Membrane.

    Google Scholar 

  • Giacovazzo, C. (1980). Direct Methods in Crystallography. London: Academic Press.

    Google Scholar 

  • Gilmore, C.J., Bricogne, G. Bannister, C. (1990). Acta Cryst. A46, 297–308. A Multisolution Method of Phase Determination by Combined Maximization of Entropy and Likelihood. Il. Application to Small Molecules.

    Google Scholar 

  • Gull, S.F., Livesey, A.K. Sivia, D.S. (1987). Acta Cryst A43, 112–11.7. Maximum Entropy Solution of a Small Centrosymmetric Crystal Structure.

    Google Scholar 

  • Hamming, R.W. (1986). Coding and Information Theory, p. 97. Englewood Cliffs, NJ: Prentice-Hall.

    MATH  Google Scholar 

  • Henderson, R. (1977). Ann. Rev. Biophys. Bioeng. 6, 87–109. The Purple Membrane from Halobacterium Halobium.

    Google Scholar 

  • Karle, J. Hauptman, H. (1950). Acta Cryst 3, 181–187. The Phases and Magnitudes of the Structure Factors.

    Google Scholar 

  • Kirkpatrick, S., Gelatt, C.D. Vecchi, M.P. (1983). Science 220, 671–680. Optimization by Simulated Annealing.

    Google Scholar 

  • Kuriyan, J. Brunger, A.T., Karplus, M. Hendrickson, W.A. (1989). Acta Cryst. A45, 396–409. X-ray Refinement of Protein Structures by Simulated Annealing: Test of the Method on Myohemerythrin.

    Google Scholar 

  • Levine, R.D. Tribus, M. (Eds.) (1979). The Maximum Entropy Formalism. Cambridge Mass.: MIT Press.

    MATH  Google Scholar 

  • McGammon, J.A. Harvey, S.C. (1987). Dynamics of Proteins and Nucleic Acids. Cambridge: University Press.

    Book  Google Scholar 

  • McLachlan, A.D. (1989). A Statistical Potential for Modelling X-ray Electron Density Maps with Known Phases. In Maximum Entropy and Bayesian Methods. Edited by J. Skilling. p. 241–249. Dordrecht: Kluwer.

    Google Scholar 

  • Navaza, J. (1985). Acta Cryst A41, 232–244. On the Maximum Entropy Estimate of the Electron Density Function.

    Google Scholar 

  • Prince, E., Sjolin, L. Alenljung, R. (1988). Acta Cryst A44, 216–222. Phase Extension by Combined Entropy Maximisation and Solvent Flattening.

    Google Scholar 

  • Semenovskaya, S.V., Khachaturyan, K.A. Kachaturyan, A.G. (1985). Acta Cryst. A41, 268–273. Statistical. Mechanics Approach to the Structure Determination of a Crystal.

    Google Scholar 

  • Sheldrick. G.M. (1990). Acta Cryst A46, 467–473. Phase Annealing in SHELX-90: Direct Methods for Larger Structures.

    Google Scholar 

  • Subbiah, S. (1991). Science 252, 128–133. Low-Resolution Real-Space Envelopes: An Approach to the Ab Initio Macromolecular Phase Problem.

    Google Scholar 

  • Wilkins, S.W., Varghese, J.N. Lehmann, M.S. (1983). Acta Cryst. A39, 47–60. Statistical Geometry. I. A Self-Consistent Approach to the Crystallographic Inversion Problem Based on Information Theory.

    Google Scholar 

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© 1993 Springer Science+Business Media Dordrecht

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McLachlan, A.D. (1993). Crystal Phase Dynamics. In: Mohammad-Djafari, A., Demoment, G. (eds) Maximum Entropy and Bayesian Methods. Fundamental Theories of Physics, vol 53. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-2217-9_36

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  • DOI: https://doi.org/10.1007/978-94-017-2217-9_36

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-90-481-4272-9

  • Online ISBN: 978-94-017-2217-9

  • eBook Packages: Springer Book Archive

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