Minimal models for proteins and RNA from folding to function.

We present a panoramic view of the utility of coarse-grained (CG) models to study folding and functions of proteins and RNA. Drawing largely on the methods developed in our group over the last twenty years, we describe a number of key applications ranging from folding of proteins with disulfide bonds to functions of molecular machines. After presenting the theoretical basis that justifies the use of CG models, we explore the biophysical basis for the emergence of a finite number of folds from lattice models. The lattice model simulations of approach to the folded state show that non-native interactions are relevant only early in the folding process - a finding that rationalizes the success of structure-based models that emphasize native interactions. Applications of off-lattice $C_{\alpha}$ and models that explicitly consider side chains ($C_{\alpha}$-SCM) to folding of $\beta$-hairpin and effects of macromolecular crowding are briefly discussed. Successful application of a new class of off-lattice model, referred to as the Self-Organized Polymer (SOP), is shown by describing the response of Green Fluorescent Protein (GFP) to mechanical force. The utility of the SOP model is further illustrated by applications that clarify the functions of the chaperonin GroEL and motion of the molecular motor kinesin. We also present two distinct models for RNA, namely, the Three Site Interaction (TIS) model and the SOP model, that probe forced unfolding and force quench refolding of a simple hairpin and {\it Azoarcus} ribozyme. The predictions based on the SOP model show that force-induced unfolding pathways of the ribozyme can be dramatically changed by varying the loading rate. We conclude with a discussion of future prospects for the use of coarse-grained models in addressing problems of outstanding interest in biology.

[1]  P. Anderson Basic Notions of Condensed Matter Physics , 1983 .

[2]  S. Woodson,et al.  The effect of long-range loop-loop interactions on folding of the Tetrahymena self-splicing RNA. , 1999, Journal of molecular biology.

[3]  K Schulten,et al.  VMD: visual molecular dynamics. , 1996, Journal of molecular graphics.

[4]  Reinhard Lipowsky,et al.  Kinesin's network of chemomechanical motor cycles. , 2007, Physical review letters.

[5]  V. Muñoz,et al.  Folding dynamics and mechanism of β-hairpin formation , 1997, Nature.

[6]  C. Brooks,et al.  From folding theories to folding proteins: a review and assessment of simulation studies of protein folding and unfolding. , 2001, Annual review of physical chemistry.

[7]  Tao Pan,et al.  RNA folding: models and perspectives. , 2003, Current opinion in structural biology.

[8]  Mark J. Schnitzer,et al.  Single kinesin molecules studied with a molecular force clamp , 1999, Nature.

[9]  D. Thirumalai,et al.  Role of counterion condensation in folding of the Tetrahymena ribozyme. II. Counterion-dependence of folding kinetics. , 2001, Journal of molecular biology.

[10]  Steven M. Block,et al.  Kinesin Moves by an Asymmetric Hand-OverHand Mechanism , 2003 .

[11]  J. Doudna,et al.  Metal-binding sites in the major groove of a large ribozyme domain. , 1996, Structure.

[12]  Changbong Hyeon,et al.  Extracting stacking interaction parameters for RNA from the data set of native structures. , 2005, Journal of molecular biology.

[13]  D Thirumalai,et al.  Factors governing the foldability of proteins , 1996, Proteins.

[14]  D Thirumalai,et al.  Mechanisms and kinetics of beta-hairpin formation. , 2000, Proceedings of the National Academy of Sciences of the United States of America.

[15]  Changbong Hyeon,et al.  Charge density of divalent metal cations determines RNA stability. , 2007, Journal of the American Chemical Society.

[16]  W. Eaton,et al.  Protein folding studied by single-molecule FRET. , 2008, Current opinion in structural biology.

[17]  R. Gamet,et al.  POLARIZED TARGET ASYMMETRY FOR PI-ZERO PHOTOPRODUCTION AT 4 GeV. , 1972 .

[18]  Lisa J. Lapidus,et al.  Fast kinetics and mechanisms in protein folding. , 2000, Annual review of biophysics and biomolecular structure.

[19]  J. Onuchic,et al.  DIFFUSIVE DYNAMICS OF THE REACTION COORDINATE FOR PROTEIN FOLDING FUNNELS , 1996, cond-mat/9601091.

[20]  Changbong Hyeon,et al.  Dynamics of allosteric transitions in GroEL , 2006, Proceedings of the National Academy of Sciences.

[21]  P. Zarrinkar,et al.  Kinetic intermediates in RNA folding. , 1994, Science.

[22]  Andrew B. Martin,et al.  Single-molecule protein folding: diffusion fluorescence resonance energy transfer studies of the denaturation of chymotrypsin inhibitor 2. , 2000, Proceedings of the National Academy of Sciences of the United States of America.

[23]  Shin'ichi Ishiwata,et al.  Loading direction regulates the affinity of ADP for kinesin , 2003, Nature Structural Biology.

[24]  K. Dill,et al.  Intrachain loops in polymers: Effects of excluded volume , 1989 .

[25]  Ashwin,et al.  Dynamics of Random Hydrophobic-Hydrophilic Copolymers with Implications for Protein Folding. , 1996, Physical review letters.

[26]  Julio M Fernandez,et al.  Force-Clamp Spectroscopy Monitors the Folding Trajectory of a Single Protein , 2004, Science.

[27]  P S Kim,et al.  Kinetic role of nonnative species in the folding of bovine pancreatic trypsin inhibitor. , 1992, Proceedings of the National Academy of Sciences of the United States of America.

[28]  H. Chan,et al.  Polymer principles of protein calorimetric two‐state cooperativity , 2000 .

[29]  P. D. Gennes,et al.  Superconductivity of metals and alloys , 1966 .

[30]  Masahide Kikkawa,et al.  15 Å Resolution Model of the Monomeric Kinesin Motor, KIF1A , 2000, Cell.

[31]  Martin Gruebele,et al.  Exploring the energy landscape of a small RNA hairpin. , 2006, Journal of the American Chemical Society.

[32]  D. Thirumalai,et al.  Kinetics of Folding of Proteins and RNA , 1996 .

[33]  F. Hartl,et al.  Polypeptide Flux through Bacterial Hsp70 DnaK Cooperates with Trigger Factor in Chaperoning Nascent Chains , 1999, Cell.

[34]  D. Thirumalai,et al.  Magnesium-dependent folding of self-splicing RNA: exploring the link between cooperativity, thermodynamics, and kinetics. , 1999, Proceedings of the National Academy of Sciences of the United States of America.

[35]  D Thirumalai,et al.  Stretching single-domain proteins: phase diagram and kinetics of force-induced unfolding. , 1999, Proceedings of the National Academy of Sciences of the United States of America.

[36]  D Thirumalai,et al.  Theoretical predictions of folding pathways by using the proximity rule, with applications to bovine pancreatic trypsin inhibitor. , 1995, Proceedings of the National Academy of Sciences of the United States of America.

[37]  Eric J. Sorin,et al.  How well can simulation predict protein folding kinetics and thermodynamics? , 2005, Annual review of biophysics and biomolecular structure.

[38]  Bernard R Brooks,et al.  Effects of denaturants and osmolytes on proteins are accurately predicted by the molecular transfer model , 2008, Proceedings of the National Academy of Sciences.

[39]  E. Mandelkow,et al.  X-ray structure of motor and neck domains from rat brain kinesin. , 1997, Biochemistry.

[40]  T. Kiefhaber,et al.  The speed limit for protein folding measured by triplet-triplet energy transfer. , 1999, Proceedings of the National Academy of Sciences of the United States of America.

[41]  Jie Chen,et al.  Allosteric communication in dihydrofolate reductase: signaling network and pathways for closed to occluded transition and back. , 2007, Journal of molecular biology.

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

[43]  D. Thirumalai,et al.  Chaperonin-mediated protein folding. , 2001, Annual review of biophysics and biomolecular structure.

[44]  Changbong Hyeon,et al.  Forced-unfolding and force-quench refolding of RNA hairpins. , 2006, Biophysical journal.

[45]  J. Liphardt,et al.  Reversible Unfolding of Single RNA Molecules by Mechanical Force , 2001, Science.

[46]  Steven M Block,et al.  Backsteps induced by nucleotide analogs suggest the front head of kinesin is gated by strain. , 2006, Proceedings of the National Academy of Sciences of the United States of America.

[47]  S. Jackson,et al.  How do small single-domain proteins fold? , 1998, Folding & design.

[48]  M. Nadeau Proteins : Structure , Function , and Genetics , .

[49]  I. Tinoco,et al.  How RNA folds. , 1999, Journal of molecular biology.

[50]  P S Kim,et al.  Rapid formation of the native 14-38 disulfide bond in the early stages of BPTI folding. , 1996, Biochemistry.

[51]  山川 裕巳,et al.  Modern theory of polymer solutions , 1971 .

[52]  D. Herschlag,et al.  Direct Measurement of the Full, Sequence-Dependent Folding Landscape of a Nucleic Acid , 2006, Science.

[53]  C. M. Jones,et al.  Fast events in protein folding initiated by nanosecond laser photolysis. , 1993, Proceedings of the National Academy of Sciences of the United States of America.

[54]  Klaus Schulten,et al.  Mechanical unfolding intermediates in titin modules , 1999, Nature.

[55]  A. R. Fresht Structure and Mechanism in Protein Science: A Guide to Enzyme Catalysis and Protein Folding , 1999 .

[56]  T. Cech,et al.  Self-splicing RNA: Autoexcision and autocyclization of the ribosomal RNA intervening sequence of tetrahymena , 1982, Cell.

[57]  J. Onuchic,et al.  Topological and energetic factors: what determines the structural details of the transition state ensemble and "en-route" intermediates for protein folding? An investigation for small globular proteins. , 2000, Journal of molecular biology.

[58]  Eric Westhof,et al.  Assembly of core helices and rapid tertiary folding of a small bacterial group I ribozyme , 2003, Proceedings of the National Academy of Sciences of the United States of America.

[59]  Eugene Shakhnovich,et al.  Protein folding thermodynamics and dynamics: where physics, chemistry, and biology meet. , 2006, Chemical reviews.

[60]  D. Thirumalai,et al.  Mechanical unfolding of RNA: from hairpins to structures with internal multiloops. , 2006, Biophysical journal.

[61]  D. Thirumalai,et al.  Mechanical unfolding of RNA hairpins. , 2005, Proceedings of the National Academy of Sciences of the United States of America.

[62]  D. Thirumalai,et al.  Deciphering the timescales and mechanisms of protein folding using minimal off-lattice models. , 1999, Current opinion in structural biology.

[63]  A. Zakharov,et al.  Basic notions of condensed matter physics , 1989 .

[64]  Jennifer A. Doudna,et al.  The chemical repertoire of natural ribozymes , 2002, Nature.

[65]  A. Horwich,et al.  The crystal structure of the asymmetric GroEL–GroES–(ADP)7 chaperonin complex , 1997, Nature.

[66]  D. Thirumalai,et al.  Maximizing RNA folding rates: a balancing act. , 2000, RNA.

[67]  S. Altman,et al.  Catalytic activity of an RNA molecule prepared by transcription in vitro. , 1984, Science.

[68]  Roger Cooke,et al.  A structural change in the kinesin motor protein that drives motility , 1999, Nature.

[69]  Michael E Fisher,et al.  Kinesin crouches to sprint but resists pushing. , 2005, Proceedings of the National Academy of Sciences of the United States of America.

[70]  D. Thirumalai,et al.  Virtual atom representation of hydrogen bonds in minimal off-lattice models of alpha helices: effect on stability, cooperativity and kinetics. , 1998, Folding & design.

[71]  D Thirumalai,et al.  Stiffness of the distal loop restricts the structural heterogeneity of the transition state ensemble in SH3 domains. , 2002, Journal of molecular biology.

[72]  D. Selkoe Folding proteins in fatal ways , 2003, Nature.

[73]  Thirumalai,et al.  Minimum energy compact structures of random sequences of heteropolymers. , 1993, Physical review letters.

[74]  Changbong Hyeon,et al.  Revealing the bifurcation in the unfolding pathways of GFP by using single-molecule experiments and simulations , 2007, Proceedings of the National Academy of Sciences.

[75]  Hiromi Yamakawa,et al.  Modern Theory of Polymer Solutions , 1971 .

[76]  D. Yee,et al.  Principles of protein folding — A perspective from simple exact models , 1995, Protein science : a publication of the Protein Society.

[77]  K. Dill,et al.  A lattice statistical mechanics model of the conformational and sequence spaces of proteins , 1989 .

[78]  D. Thirumalai,et al.  Counterion charge density determines the position and plasticity of RNA folding transition states. , 2006, Journal of molecular biology.

[79]  V Muñoz,et al.  Folding dynamics and mechanism of beta-hairpin formation. , 1997, Nature.

[80]  I. Tinoco,et al.  RNA folding and unfolding. , 2004, Current opinion in structural biology.

[81]  Joshua W. Shaevitz,et al.  Probing the kinesin reaction cycle with a 2D optical force clamp , 2003, Proceedings of the National Academy of Sciences of the United States of America.

[82]  J. Onuchic,et al.  Funnels, pathways, and the energy landscape of protein folding: A synthesis , 1994, Proteins.

[83]  D. Thirumalai,et al.  Kinetics of protein folding: Nucleation mechanism, time scales, and pathways , 1995 .

[84]  T. Creighton,et al.  Kinetic role of a meta-stable native-like two-disulphide species in the folding transition of bovine pancreatic trypsin inhibitor. , 1984, Journal of molecular biology.

[85]  X. Zhuang,et al.  A single-molecule study of RNA catalysis and folding. , 2000, Science.

[86]  Scott T. Brady,et al.  A novel brain ATPase with properties expected for the fast axonal transport motor , 1985, Nature.

[87]  T. Earnest,et al.  Crystal Structure of the Ribosome at 5.5 Å Resolution , 2001, Science.

[88]  Ruxandra I. Dima,et al.  Asymmetry in the shapes of folded and denatured states of proteins , 2003, q-bio/0310023.

[89]  K. Wilson The renormalization group and critical phenomena , 1983 .

[90]  D. Herschlag,et al.  Probing the folding landscape of the Tetrahymena ribozyme: commitment to form the native conformation is late in the folding pathway. , 2001, Journal of molecular biology.

[91]  Leslie L. Chavez,et al.  Topological frustration and the folding of interleukin-1 beta. , 2006, Journal of molecular biology.

[92]  E. Rhoades,et al.  Watching proteins fold one molecule at a time , 2003, Proceedings of the National Academy of Sciences of the United States of America.

[93]  F. Young Biochemistry , 1955, The Indian Medical Gazette.

[94]  H. Orland,et al.  Random hydrophilic-hydrophobic copolymers , 1994 .

[95]  Changbong Hyeon,et al.  Mechanical control of the directional stepping dynamics of the kinesin motor , 2007, Proceedings of the National Academy of Sciences.

[96]  D. K. Treiber,et al.  Exposing the kinetic traps in RNA folding. , 1999, Current opinion in structural biology.

[97]  Hendrik Dietz,et al.  Exploring the energy landscape of GFP by single-molecule mechanical experiments. , 2004, Proceedings of the National Academy of Sciences of the United States of America.

[98]  P. G. de Gennes,et al.  Exponents for the excluded volume problem as derived by the Wilson method , 1972 .

[99]  Shang‐keng Ma Modern Theory of Critical Phenomena , 1976 .

[100]  R. Rosenfeld Nature , 2009, Otolaryngology--head and neck surgery : official journal of American Academy of Otolaryngology-Head and Neck Surgery.

[101]  D. Thirumalai,et al.  ANALYTICAL THEORIES OF PROTEIN FOLDING , 1996 .

[102]  Eugene I. Shakhnovich Protein Folding Thermodynamics and Dynamics: Where Physics, Chemistry, and Biology Meet , 2006 .

[103]  F. Schluenzen,et al.  Structural basis for the interaction of antibiotics with the peptidyl transferase centre in eubacteria , 2001, Nature.

[104]  I. Bahar,et al.  Coarse-grained normal mode analysis in structural biology. , 2005, Current opinion in structural biology.

[105]  N. Pace,et al.  The RNA moiety of ribonuclease P is the catalytic subunit of the enzyme , 1983, Cell.

[106]  P. Wolynes,et al.  Optimal protein-folding codes from spin-glass theory. , 1992, Proceedings of the National Academy of Sciences of the United States of America.

[107]  E. Shakhnovich,et al.  Formation of unique structure in polypeptide chains. Theoretical investigation with the aid of a replica approach. , 1989, Biophysical chemistry.

[108]  Masahide Kikkawa,et al.  High‐resolution cryo‐EM maps show the nucleotide binding pocket of KIF1A in open and closed conformations , 2006, The EMBO journal.

[109]  M. Karplus,et al.  Protein Folding: A Perspective from Theory and Experiment. , 1998, Angewandte Chemie.

[110]  D Thirumalai,et al.  Kinetics and thermodynamics of folding in model proteins. , 1993, Proceedings of the National Academy of Sciences of the United States of America.

[111]  C. Tanford Macromolecules , 1994, Nature.

[112]  P G Wolynes,et al.  An elementary mode coupling theory of random heteropolymer dynamics. , 1997, Proceedings of the National Academy of Sciences of the United States of America.

[113]  Changbong Hyeon,et al.  Pathways and kinetic barriers in mechanical unfolding and refolding of RNA and proteins. , 2006, Structure.

[114]  R. Tehver,et al.  Kinetic model for the coupling between allosteric transitions in GroEL and substrate protein folding and aggregation. , 2008, Journal of molecular biology.

[115]  J. D. Cloizeaux,et al.  The Lagrangian theory of polymer solutions at intermediate concentrations , 1975 .

[116]  M. Levitt,et al.  Computer simulation of protein folding , 1975, Nature.

[117]  D. Thirumalai,et al.  From Minimal Models to Real Proteins: Time Scales for Protein Folding Kinetics , 1995 .

[118]  D. Thirumalai,et al.  Emerging ideas on the molecular basis of protein and peptide aggregation. , 2003, Current opinion in structural biology.

[119]  Ignacio Tinoco,et al.  Unusual mechanical stability of a minimal RNA kissing complex , 2006, Proceedings of the National Academy of Sciences.

[120]  T. Cech,et al.  In vitro splicing of the ribosomal RNA precursor of tetrahymena: Involvement of a guanosine nucleotide in the excision of the intervening sequence , 1981, Cell.

[121]  D. Thirumalai,et al.  Modeling the role of disulfide bonds in protein folding: Entropic barriers and pathways , 1995, Proteins.

[122]  D Thirumalai,et al.  Probing the instabilities in the dynamics of helical fragments from mouse PrPC. , 2004, Proceedings of the National Academy of Sciences of the United States of America.

[123]  J. Onuchic,et al.  Theory of Protein Folding This Review Comes from a Themed Issue on Folding and Binding Edited Basic Concepts Perfect Funnel Landscapes and Common Features of Folding Mechanisms , 2022 .

[124]  Michael P. Sheetz,et al.  Identification of a novel force-generating protein, kinesin, involved in microtubule-based motility , 1985, Cell.

[125]  Stanley B. Prusiner,et al.  Nobel Lecture: Prions , 1998 .

[126]  A. Kolomeisky,et al.  Simple mechanochemistry describes the dynamics of kinesin molecules , 2001, Proceedings of the National Academy of Sciences of the United States of America.

[127]  R. Vale,et al.  The way things move: looking under the hood of molecular motor proteins. , 2000, Science.

[128]  Ignacio Tinoco,et al.  Identifying Kinetic Barriers to Mechanical Unfolding of the T. thermophila Ribozyme , 2003, Science.

[129]  H. Orland,et al.  DYNAMICS OF THE SWELLING OR COLLAPSE OF A HOMOPOLYMER , 1996 .

[130]  M. Karplus,et al.  Allostery and cooperativity revisited , 2008, Protein science : a publication of the Protein Society.

[131]  P. S. Kim,et al.  Reexamination of the folding of BPTI: predominance of native intermediates , 1991, Science.

[132]  I. Bahar,et al.  Gaussian Dynamics of Folded Proteins , 1997 .

[133]  D. Baker,et al.  Design of a Novel Globular Protein Fold with Atomic-Level Accuracy , 2003, Science.

[134]  D Thirumalai,et al.  The nature of folded states of globular proteins , 1992, Biopolymers.

[135]  D Thirumalai,et al.  Effects of crowding and confinement on the structures of the transition state ensemble in proteins. , 2007, The journal of physical chemistry. B.

[136]  Bohr,et al.  Magic Numbers in Protein Structures. , 1996, Physical review letters.

[137]  T. Steitz,et al.  The complete atomic structure of the large ribosomal subunit at 2.4 A resolution. , 2000, Science.

[138]  D. K. Treiber,et al.  Beyond kinetic traps in RNA folding. , 2001, Current opinion in structural biology.

[139]  C. Dobson Protein misfolding, evolution and disease. , 1999, Trends in biochemical sciences.

[140]  M. Fisher,et al.  Molecular motors: a theorist's perspective. , 2007, Annual review of physical chemistry.

[141]  Pernilla Wittung-Stafshede,et al.  Molecular crowding enhances native structure and stability of alpha/beta protein flavodoxin. , 2007, Proceedings of the National Academy of Sciences of the United States of America.

[142]  R. Vale,et al.  Kinesin Walks Hand-Over-Hand , 2004, Science.

[143]  E. Mandelkow,et al.  The Crystal Structure of Dimeric Kinesin and Implications for Microtubule-Dependent Motility , 1997, Cell.

[144]  Dirar Homouz,et al.  Crowded, cell-like environment induces shape changes in aspherical protein , 2008, Proceedings of the National Academy of Sciences.

[145]  Changbong Hyeon,et al.  Internal strain regulates the nucleotide binding site of the kinesin leading head , 2007, Proceedings of the National Academy of Sciences.

[146]  A. E. Walter,et al.  Coaxial stacking of helixes enhances binding of oligoribonucleotides and improves predictions of RNA folding. , 1994, Proceedings of the National Academy of Sciences of the United States of America.

[147]  T. Creighton,et al.  Conformational restrictions on the pathway of folding and unfolding of the pancreatic trypsin inhibitor. , 1977, Journal of molecular biology.

[148]  T. Creighton,et al.  The disulfide folding pathway of BPTI. , 1992, Science.

[149]  José N Onuchic,et al.  Extracting function from a β-trefoil folding motif , 2008, Proceedings of the National Academy of Sciences.

[150]  John Karanicolas,et al.  The origins of asymmetry in the folding transition states of protein L and protein G , 2002, Protein science : a publication of the Protein Society.

[151]  P. Wolynes,et al.  Intermediates and barrier crossing in a random energy model , 1989 .

[152]  J. Onuchic,et al.  Protein folding funnels: a kinetic approach to the sequence-structure relationship. , 1992, Proceedings of the National Academy of Sciences of the United States of America.

[153]  D. Thirumalai,et al.  Low-frequency normal modes that describe allosteric transitions in biological nanomachines are robust to sequence variations , 2006, Proceedings of the National Academy of Sciences of the United States of America.

[154]  Kinetic and thermodynamic analysis of proteinlike heteropolymers: Monte Carlo histogram technique , 1995, chem-ph/9507003.

[155]  W. Greenleaf,et al.  Single-molecule studies of RNA polymerase: motoring along. , 2008, Annual review of biochemistry.

[156]  D. Thirumalai,et al.  Time Scales for the Formation of the Most Probable Tertiary Contacts in Proteins with Applications to Cytochrome c , 1999 .

[157]  D. Thirumalai,et al.  RNA and protein folding: common themes and variations. , 2005, Biochemistry.

[158]  P. Wittung-Stafshede,et al.  Molecular crowding enhances native structure and stability of α/β protein flavodoxin , 2007, Proceedings of the National Academy of Sciences.

[159]  D. Goldenberg,et al.  Amino acid replacement that eliminates kinetic traps in the folding pathway of pancreatic trypsin inhibitor. , 1993, Biochemistry.

[160]  A. Fersht,et al.  Protein Folding and Unfolding at Atomic Resolution , 2002, Cell.

[161]  T. Kiefhaber,et al.  Kinetic traps in lysozyme folding. , 1995, Proceedings of the National Academy of Sciences of the United States of America.

[162]  Nam-Kyung Lee,et al.  Folding of the Tetrahymena ribozyme by polyamines: importance of counterion valence and size. , 2004, Journal of molecular biology.

[163]  J. Sabina,et al.  Expanded sequence dependence of thermodynamic parameters improves prediction of RNA secondary structure. , 1999, Journal of molecular biology.

[164]  D. Thirumalai,et al.  Role of counterion condensation in folding of the Tetrahymena ribozyme. I. Equilibrium stabilization by cations. , 2001, Journal of molecular biology.

[165]  Mark J. Schnitzer,et al.  Kinesin hydrolyses one ATP per 8-nm step , 1997, Nature.

[166]  N. Wingreen,et al.  Emergence of Preferred Structures in a Simple Model of Protein Folding , 1996, Science.

[167]  Hongbin Li,et al.  Atomic force microscopy reveals parallel mechanical unfolding pathways of T4 lysozyme: Evidence for a kinetic partitioning mechanism , 2008, Proceedings of the National Academy of Sciences.

[168]  Christopher M. Dobson,et al.  Mutational analysis of acylphosphatase suggests the importance of topology and contact order in protein folding , 1999, Nature Structural Biology.

[169]  M. Daoud,et al.  Solutions of Flexible Polymers. Neutron Experiments and Interpretation , 1975 .

[170]  D. Thirumalai,et al.  LINKING RATES OF FOLDING IN LATTICE MODELS OF PROTEINS WITH UNDERLYING THERMODYNAMIC CHARACTERISTICS , 1998, cond-mat/9805061.