GAUSSIAN NETWORK MODEL

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GAUSSIAN NETWORK MODEL

• Sebnem Essiz
• Chem2440, Spring 2003

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Outline

• A brief Description of GNM
• Basic approximations in the model
• Theory behind the model, and the calculation of
  mean square fluctuations
• Application of model to apomyoglobin
• Summary

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GNM

• A coarse grained model to study vibrational
  dynamics of proteins in the folded state.
• Interactions between residues are replaced by
  linear springs, in analogy with the elasticity
  theory of random polymer networks.
• The above approximation is based on a Gaussian
  Distribution of inter-atomic distances about
  their equilibrium values.

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Basic approximations in GNM

• No distinction is made between different types of
  amino acids, so that a generic force constant g
  is adopted for the interaction potential between
  all pair of residues which are sufficiently
  close. a single parameter harmonic potential
• All fluctuations are assumed to be isotropic no
  directional preferences. This makes the number of
  independent modes as N-1, instead of 3N-6 in 3-D
  description.

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Schematic Representation of fluctuations

• Ri0 and Rj0 are equilibrium positions, and their
  instantaneous values are Ri and Rj. sij0 and sij
  are the equilibrium and instantaneous separation
  vectors. sij - sij0 Rj- Ri

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Overall Conformational Potential

• Overall conformational potential of the structure
  is
• Here DR is N-dimensional vector whose elements
  are the fluctuation vectors DRi of individual
  residues.g is the single parameter force
  constant.
• The above potential can also be seen as a
  Hamiltonian whose integration will give
  configurational partition function ZN

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Kirchhoff Matrix of contacts

• rc is the cutoff separation defining the range of
  interaction of non-bonded a carbons. A reasonable
  cutoff distance is 7.0 Ao.
• The ith diagonal element of G characterizes the
  coordination number of residue i.

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Fluctuations

• Fluctuations in a folded protein are assumed to
  obey Gaussian distribution
• With analogy to Gaussian networks,
  con-figurational partition function will be

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Fluctuations

• We can find the cross-correlations between
  residue fluctuations
• (3kBT/g)G-1
• To find the mean square fluctuations of
  individual residues take ij

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Results

• The GNM results for mean square fluctuations is
  nearly full agreement with x-ray crystallographic
  results o. In the following graph you will find
  only the results for apomyoglobin.
• (The mean square fluctuations of Ca of
  apomyoglobin, ltRi2gt as a function of residue
  index i. The dashed line is experimental results
  for temperature factor,Bi8p2ltDRi . DRi gt/3)
• If you want to see more results about GNM,
  you can find 12 more protein results in Folding
  Design, 2173-181(1997)

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Summary

• On a coarse-grained scale molecular motions can
  be approximated by normal fluctuations with
  rescaled force constants, even if the motions of
  individual atoms depart from harmonicity.
• It yields an analytical solution, devoid of
  sampling inaccuracies found in most other types
  of simulations. The computer time is limited by
  the inversion of the Kirchhoff matrix, and this
  time is several time shorter than MD or NMA
  calculations.
• The major drawback in GNM is about isotropic
  fluctuations. In reality the fluctuations are in
  general anisotropic and the direction of these
  motions are directly relevant to biological
  function and mechanism.

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REFERENCES

• I.Bahar, A.R. Atilgan, and B. Erman, Folding
  Des.2,173(1997)
• T. Haliloglu,I. Bahar, and B. Erman, Phys.Rev.
  Letters.79, 3090(1997)
• I. Bahar and R.L. Jernigan, J.Mol.Bio.
  281,871(1998)
• I. Bahar, B. Erman, R.L. Jernigan, A.R. Atilgan,
  and D. Covell, J. Mol.Bio. 285, 1023(1999)
• A.R. Atilgan,S.R. Durell, R.L Jernigan, M.C.
  Demirel, O.Keskin, and I.Bahar, Biophys. J.
  80,505(2001)