Inspired by McCaskill's work on RNA structure prediction , the Boltzmann partition function can be found for a TMB by projecting the β-barrel structure onto a 2D lattice and solving a dynamic programming recursion. Although this problem is NP-Hard for 3D model, Istrail has shown that a 2D lattice can be solved in polynomial time . In a similar fashion, the individual residue-residue contact probability can be solved according to each contacts weight in the Boltzmann distribution. Finally, independent, random, structural conformations can be sampled from the ensemble of possible conformations, again according to their weight in the Boltzmann distribution. These techniques are inspired by work done by Ding and Lawrence .
Using our methods , a stochastic contact map can be created for any given TMB residue sequence, identifying likely structural conformations in the Boltzmann ensemble. In the figure below, both axis delineate the amino acid sequence, and darker regions in the map represent high-probability interactions according to our derived Boltzmann distribution. The minimum energy structure, as would be found be transFold is also shown in red. Initial results suggest that our residue-residue contact predictions are the best for transmembrane β-barrel proteins when compared to contacts found in x-ray crystal structures.
Since residue-residue contact probabilities can indicate regions of a TMB protein that are more flexible, we can compare the per-residue contact probability with the Debye-Waller factor (B-value) found in x-ray crystal structures. As seen in the figure below, sequential residue regions of high flexibility should maintain a low per-residue contact probability (in green). These low-probability regions sequence across the x-axis correspond very well with high experimental B-value regions which also indicate flexible and disordered structure.
 J. Waldispühl, B. Berger, P. Clote, and J-M. Steyaert. Predicting Transmembrane Beta-barrels and Inter-strand Residue Interactions from Sequence. In PROTEINS: Structure, Function and Bioinformatics, 65(1) pp. 61-74 (2006)
 J. Waldispühl, B. Berger, P. Clote, and J-M. Steyaert. transFold: a web Server for Predicting the Structure and Residue Contacts of Transmembrane β-barrels. In Nucleic Acids Research, 34 (Web Server issue) pp. W189-W193 (2006)
 J. Waldispühl and J-M. Steyaert. Modeling and Predicting all-Alpha Transmembrane Proteins Including Helix-Helix Pairing. In Theoretical Computer Science, special issue on Pattern Discovery in the Post Genome, pp. 67-92 (2005)
 J. S. McCaskill. The equilibrium partition function and base pair binding probabilities for RNA secondary structure. In Biopolymers, 29 pp. 1105-1119 (1990)
 I. Istrail. Statistical mechanics, three-dimensionality and NP-completeness: I. Universality of intractability of the partition function of the ising model across non-planar lattices. In Proceedings of the 32nd ACM Symposium on the Theory of Computing, pp. 87-96 (2000)
 Y. Ding, C. E. Lawrence. A statistical sampling algorithm for RNA secondary structure prediction. In Nucleic Acids Research, 31(24) pp. 7280-7301 (2003)
 J. Waldipühl, C. W. O'Donnell, N. Palmer, S. Devadas, P. Clote, B. Bergeer. Modeling Ensembles of Transmembrane β-barrel proteins. Under review.
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