Exact analytical loop closure in proteins using polynomial equations

Loop closure in proteins has been studied actively for over 25 years. Using spherical geometry and polynomial equations, several loop‐closure problems in proteins are solved exactly by reducing them to the determination of the real roots of a polynomial. Loops of seven, eight, and nine atoms are treated explicitly, including the tripeptide and disulfide‐bonded loop‐closure problems. The number of valid loop closures can be evaluated by the method of Sturm chains, which counts the number of real roots of a polynomial. Longer loops can be treated by three methods: by sampling enough dihedral angles to reduce the problem to a soluble loop‐closure problem; by applying the loop‐closure algorithm hierarchically; or by decimating the chain into independently moving rigid elements that can be reconnected using loop‐closure algorithms. Applications of the methods to docking, homology modeling and NMR problems are discussed. ©1999 John Wiley & Sons, Inc. J Comput Chem 20: 819–844, 1999

[1]  P. Flory,et al.  The Configuration of Random Polypeptide Chains. II. Theory , 1965 .

[2]  Shen Lin.  Computer solutions of the traveling salesman problem , 1965 .

[3]  Z. Alexandrowicz.  Analytical Treatment of Excluded Volume. II. Quasi Random‐Flight Behavior , 1967 .

[4]  Harold A. Scheraga,et al.  Calculation of the Conformation of the Pentapeptide cyclo-(Glycylglycylglycylprolylprolyl). I. A Complete Energy Map , 1970 .

[5]  N. Go,et al.  Ring Closure and Local Conformational Deformations of Chain Molecules , 1970 .

[6]  Harold A. Scheraga,et al.  Ring Closure in Chain Molecules with Cn,I, and S2nSymmetry , 1973 .

[7]  Harold A. Scheraga,et al.  Calculation of the Conformation of cyclo-Hexaglycyl , 1973 .

[8]  H. Scheraga,et al.  Energy parameters in polypeptides. VII. Geometric parameters, partial atomic charges, nonbonded interactions, hydrogen bond interactions, and intrinsic torsional potentials for the naturally occurring amino acids , 1975 .

[9]  H A Scheraga,et al.  Use of a symmetry condition to compute the conformation of gramicidin S1. , 1975, Macromolecules.

[10]  Harold A. Scheraga,et al.  Calculation of the Conformation of cyclo-Hexaglycyl. 2. Application of a Monte-Carlo Method , 1978 .

[11]  P. Gennes.  Scaling Concepts in Polymer Physics , 1979 .

[12]  Hiroshi Wako,et al.  Distance-constraint approach to protein folding. I. Statistical analysis of protein conformations in terms of distances between residues , 1982 .

[13]  Timothy F. Havel,et al.  The theory and practice of distance geometry , 1983, Bulletin of Mathematical Biology.

[14]  H. Scheraga,et al.  Energy parameters in polypeptides. 9. Updating of geometrical parameters, nonbonded interactions, and hydrogen bond interactions for the naturally occurring amino acids , 1983 .

[15]  Robert Langridge,et al.  A distance geometry study of ring systems , 1983 .

[16]  H A Scheraga,et al.  Conversion from a virtual‐bond chain to a complete polypeptide backbone chain , 1984, Biopolymers.

[17]  Robert E. Bruccoleri,et al.  Chain closure with bond angle variations , 1985 .

[18]  C. Pabo,et al.  Computer-aided model-building strategies for protein design. , 1986, Biochemistry.

[19]  T. A. Jones,et al.  Using known substructures in protein model building and crystallography. , 1986, The EMBO journal.

[20]  J. Moult,et al.  An algorithm for determining the conformation of polypeptide segments in proteins by systematic search , 1986, Proteins.

[21]  M Karplus,et al.  Three-dimensional structure of proteins determined by molecular dynamics with interproton distance restraints: application to crambin. , 1986, Proceedings of the National Academy of Sciences of the United States of America.

[22]  Wang,et al.  Nonuniversal critical dynamics in Monte Carlo simulations. , 1987, Physical review letters.

[23]  T. L. Blundell,et al.  Knowledge-based prediction of protein structures and the design of novel molecules , 1987, Nature.

[24]  W. Braun,et al.  Local deformation studies of chain molecules: Differential conditions for changes of dihedral angles , 1987, Biopolymers.

[25]  C. Levinthal,et al.  Predicting antibody hypervariable loop conformation. I. Ensembles of random conformations for ringlike structures , 1987, Biopolymers.

[26]  W. Braun,et al.  Distance geometry and related methods for protein structure determination from NMR data , 1987, Quarterly Reviews of Biophysics.

[27]  M. Karplus,et al.  Prediction of the folding of short polypeptide segments by uniform conformational sampling , 1987, Biopolymers.

[28]  Jiří Novotný,et al.  Structure of antibody hypervariable loops reproduced by a conformational search algorithm , 1988, Nature.

[29]  R A Abagyan,et al.  New methodology for computer-aided modelling of biomolecular structure and dynamics. 2. Local deformations and cycles. , 1989, Journal of biomolecular structure & dynamics.

[30]  S. Wodak,et al.  Modelling the polypeptide backbone with 'spare parts' from known protein structures. , 1989, Protein engineering.

[31]  Adam Liwo,et al.  Molecular mechanics calculations on deaminooxytocin and on deamino-arginine-vasopressin and its analogues , 1989, J. Comput. Aided Mol. Des..

[32]  D. Zwillinger.  Handbook of differential equations , 1990 .

[33]  M Karplus,et al.  Modeling of globular proteins. A distance-based data search procedure for the construction of insertion/deletion regions and Pro----non-Pro mutations. , 1990, Journal of molecular biology.

[34]  J. Greer.  Comparative modeling methods: Application to the family of the mammalian serine proteases , 1990, Proteins.

[35]  Harold A. Scheraga,et al.  Standard‐geometry chains fitted to X‐ray derived structures: Validation of the rigid‐geometry approximation. I. Chain closure through a limited search of “loop” conformations , 1991 .

[36]  Bruce W. Char,et al.  Maple V Language Reference Manual , 1993, Springer US.

[37]  Harold A. Scheraga,et al.  Standard‐geometry chains fitted to X‐ray derived structures: Validation of the rigid‐geometry approximation. II. Systematic searches for short loops in proteins: Applications to bovine pancreatic ribonuclease A and human lysozyme , 1992 .

[38]  A molecular mechanics study of the effect of substitution in position 1 on the conformational space of the oxytocin/vasopressin ring , 1993, J. Comput. Aided Mol. Des..

[39]  Qiang Zheng,et al.  Loop closure via bond scaling and relaxation , 1993, J. Comput. Chem..

[40]  C. DeLisi,et al.  Computing the structure of bound peptides. Application to antigen recognition by class I major histocompatibility complex receptors. , 1993, Journal of molecular biology.

[41]  G Vasmatzis,et al.  Predicting immunoglobulin‐like hypervariable loops , 1994, Biopolymers.

[42]  H. Scheraga,et al.  Energy parameters in polypeptides. 10. Improved geometrical parameters and nonbonded interactions for use in the ECEPP/3 algorithm, with application to proline-containing peptides , 1994 .

[43]  H. Scheraga,et al.  Energy minimization of rigid‐geometry polypeptides with exactly closed disulfide loops , 1997 .

[44]  Harold A. Scheraga,et al.  B‐spline method for energy minimization in grid‐based molecular mechanics calculations , 1998 .