Calculating Ensemble Averaged Descriptions of Protein Rigidity without Sampling

Previous works have demonstrated that protein rigidity is related to thermodynamic stability, especially under conditions that favor formation of native structure. Mechanical network rigidity properties of a single conformation are efficiently calculated using the integer body-bar Pebble Game (PG) algorithm. However, thermodynamic properties require averaging over many samples from the ensemble of accessible conformations to accurately account for fluctuations in network topology. We have developed a mean field Virtual Pebble Game (VPG) that represents the ensemble of networks by a single effective network. That is, all possible number of distance constraints (or bars) that can form between a pair of rigid bodies is replaced by the average number. The resulting effective network is viewed as having weighted edges, where the weight of an edge quantifies its capacity to absorb degrees of freedom. The VPG is interpreted as a flow problem on this effective network, which eliminates the need to sample. Across a nonredundant dataset of 272 protein structures, we apply the VPG to proteins for the first time. Our results show numerically and visually that the rigidity characterizations of the VPG accurately reflect the ensemble averaged properties. This result positions the VPG as an efficient alternative to understand the mechanical role that chemical interactions play in maintaining protein stability.

[1]  Harold N. Gabow,et al.  Forests, frames, and games: algorithms for matroid sums and applications , 1988, STOC '88.

[2]  J E Jimenez-Roldan,et al.  Comparative analysis of rigidity across protein families , 2009, Physical biology.

[3]  Yunje Cho,et al.  The crystal structure of flap endonuclease-1 from Methanococcus jannaschii , 1998, Nature Structural &Molecular Biology.

[4]  Victoria A. Higman,et al.  Uncovering network systems within protein structures. , 2003, Journal of molecular biology.

[5]  S E Ealick,et al.  Ab initio structure determination and refinement of a scorpion protein toxin. , 1997, Acta crystallographica. Section D, Biological crystallography.

[6]  Bruce Hendrickson,et al.  Conditions for Unique Graph Realizations , 1992, SIAM J. Comput..

[7]  Eric V. Denardo,et al.  Flows in Networks , 2011 .

[8]  William M. Rand,et al.  Objective Criteria for the Evaluation of Clustering Methods , 1971 .

[9]  M. Thorpe,et al.  Constrained geometric simulation of diffusive motion in proteins , 2005, Physical biology.

[10]  Donald J Jacobs,et al.  Elucidating protein thermodynamics from the three-dimensional structure of the native state using network rigidity. , 2005, Biophysical journal.

[11]  Donald J. Jacobs,et al.  Nonadditivity in conformational entropy upon molecular rigidification reveals a universal mechanism affecting folding cooperativity. , 2011, Biophysical journal.

[12]  Ann M Stock,et al.  Crystal structure of the chemotaxis receptor methyltransferase CheR suggests a conserved structural motif for binding S-adenosylmethionine. , 1997, Structure.

[13]  G. Laman.  On graphs and rigidity of plane skeletal structures , 1970 .

[14]  Hepan Tan,et al.  Identification of putative, stable binding regions through flexibility analysis of HIV‐1 gp120 , 2009, Proteins.

[15]  Ming Lei,et al.  Sampling protein conformations and pathways , 2004, J. Comput. Chem..

[16]  Donald J. Jacobs,et al.  Structural rigidity in the capsid assembly of cowpea chlorotic mottle virus , 2004 .

[17]  Bruce Randall Donald,et al.  Design of Protein-Protein Interactions with a Novel Ensemble-Based Scoring Algorithm , 2011, RECOMB.

[18]  Hiroshi Imai,et al.  On combinatorial structures of line drawings of polyhedra , 1985, Discret. Appl. Math..

[19]  D. Jacobs,et al.  Hydrogen bond networks determine emergent mechanical and thermodynamic properties across a protein family , 2008, Chemistry Central journal.

[20]  N. Stanietsky,et al.  The interaction of TIGIT with PVR and PVRL2 inhibits human NK cell cytotoxicity , 2009, Proceedings of the National Academy of Sciences.

[21]  Holger Gohlke,et al.  Constraint counting on RNA structures: linking flexibility and function. , 2009, Methods.

[22]  Dennis R Livesay,et al.  Conserved quantitative stability/flexibility relationships (QSFR) in an orthologous RNase H pair , 2005, Proteins.

[23]  A J Rader,et al.  Coarse-grained models: getting more with less. , 2010, Current opinion in pharmacology.

[24]  Dennis R Livesay,et al.  Allosteric response is both conserved and variable across three CheY orthologs. , 2010, Biophysical journal.

[25]  Xin-She Yang,et al.  Introduction to Algorithms , 2021, Nature-Inspired Optimization Algorithms.

[26]  Donald J. Jacobs,et al.  Generic rigidity in three-dimensional bond-bending networks , 1998 .

[27]  I. Bahar,et al.  Folding core predictions from network models of proteins , 2004 .

[28]  Daniel W. Farrell,et al.  Generating stereochemically acceptable protein pathways , 2010, Proteins.

[29]  Ileana Streinu,et al.  Redundant interactions in protein rigid cluster analysis , 2011, 2011 IEEE 1st International Conference on Computational Advances in Bio and Medical Sciences (ICCABS).

[30]  Dennis R Livesay,et al.  Predicting the melting point of human C-type lysozyme mutants. , 2010, Current protein & peptide science.

[31]  Judith Klein-Seetharaman,et al.  Identification of core amino acids stabilizing rhodopsin. , 2004, Proceedings of the National Academy of Sciences of the United States of America.

[32]  M. Thorpe,et al.  Rigidity theory and applications , 2002 .

[33]  D. Jacobs,et al.  Protein flexibility predictions using graph theory , 2001, Proteins.

[34]  A G Murzin,et al.  SCOP: a structural classification of proteins database for the investigation of sequences and structures. , 1995, Journal of molecular biology.

[35]  Bruce R Donald,et al.  Predicting resistance mutations using protein design algorithms , 2010, Proceedings of the National Academy of Sciences.

[36]  Freddie R Salsbury,et al.  Molecular dynamics simulations of protein dynamics and their relevance to drug discovery. , 2010, Current opinion in pharmacology.

[37]  Donald J Jacobs.  Ensemble-based methods for describing protein dynamics. , 2010, Current opinion in pharmacology.

[38]  G. G. Wood,et al.  A flexible approach for understanding protein stability , 2004, FEBS letters.

[39]  Dennis R Livesay,et al.  Elucidating quantitative stability/flexibility relationships within thioredoxin and its fragments using a distance constraint model. , 2006, Journal of molecular biology.

[40]  R A Barrio,et al.  Current topics in physics : in honor of Sir Roger J. Elliott , 2005 .

[41]  Bruce Randall Donald,et al.  A Novel Minimized Dead-End Elimination Criterion and Its Application to Protein Redesign in a Hybrid Scoring and Search Algorithm for Computing Partition Functions over Molecular Ensembles , 2006, RECOMB.

[42]  Leslie A Kuhn,et al.  Modeling correlated main‐chain motions in proteins for flexible molecular recognition , 2004, Proteins.

[43]  Dennis R Livesay,et al.  New insight into long‐range nonadditivity within protein double‐mutant cycles , 2008, Proteins.

[44]  Dennis R Livesay,et al.  Helix/coil nucleation: a local response to global demands. , 2009, Biophysical journal.

[45]  Holger Gohlke,et al.  Protein rigidity and thermophilic adaptation , 2011, Proteins.

[46]  A. Rader,et al.  Identifying protein folding cores from the evolution of flexible regions during unfolding. , 2002, Journal of molecular graphics & modelling.

[47]  L. Chong,et al.  Reaching biological timescales with all-atom molecular dynamics simulations. , 2010, Current opinion in pharmacology.

[48]  Tiong-Seng Tay,et al.  Rigidity of multi-graphs. I. Linking rigid bodies in n-space , 1984, J. Comb. Theory, Ser. B.

[49]  Dennis R Livesay,et al.  Unifying mechanical and thermodynamic descriptions across the thioredoxin protein family , 2009, Proteins.

[50]  Amy C. Anderson,et al.  Computational structure-based redesign of enzyme activity , 2009, Proceedings of the National Academy of Sciences.

[51]  J. Troadec,et al.  Non-local and non-linear problems in the mechanics of disordered systems: application to granular media and rigidity problems , 1990 .

[52]  杉原 厚吉,et al.  On Redundant Bracing in Plane Skeletal Structures , 1980 .

[53]  R. Ladenstein,et al.  A protein disulfide oxidoreductase from the archaeon Pyrococcus furiosus contains two thioredoxin fold units , 1998, Nature Structural Biology.

[54]  Leslie A Kuhn,et al.  Protein unfolding: Rigidity lost , 2002, Proceedings of the National Academy of Sciences of the United States of America.

[55]  Leslie A. Kuhn,et al.  Flexible and Rigid Regions in Proteins , 2002 .

[56]  Jacobs,et al.  Generic rigidity percolation: The pebble game. , 1995, Physical review letters.