Skip to main content

Finding the lowest free energy conformation of a protein is an NP-hard problem: Proof and implications

Bulletin of Mathematical Biology volume 55, pages 1183–1198 (1993)Cite this article

Abstract

The protein folding problem and the notion of NP-completeness and NP-hardness are discussed. A lattice model is suggested to capture the essece of protein folding. For this model we present a proof that finding the lowest free energy conformation belongs to the class of NP-hard problems. The implications of the proof are discussed and we suggest that the natural folding process cannot be considered as a search for the global free energy minimum. However, we suggest an explanation as to why, for many proteins, the native functional conformation maycoincide with the lowest free energy conformation.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Literature

  • Anfinsen, C. B. 1973. Principles that govern the folding of proteins chains.Science 181, 223–230.

    Google Scholar 

  • Anfinsen, C. B., E. Haber, M. Sela and F. H. White. 1961. The kinetics of formation of native Ribonuclease during oxidation of the reduced polypeptide chain.Proc. natn. Acad. Sci. U.S.A. 47, 1309–1314.

    Article  Google Scholar 

  • Bernstein, F. C., T. F. Koetzle, G. J. B. Williams, E. F. Meyer, M. D. Brice, J. R. Rodgers, O. Kennard, T. Shimanouchi and M. Tasumi. 1977. The protein data bank: a computer-based archival file for macromolecular structures.J. molec. Biol. 112, 535–542.

    Google Scholar 

  • Blundell, T. L., B. L. Sibanda, M. J. E. Sternberg and J. M. Thornton. 1987. Knowledge-based preduction of protein structures and the design of novel molecules.Nature 326, 347–352.

    Article  Google Scholar 

  • Brooks, C. L., M. Karplus and B. M. Pettitt. 1988.Proteins: A theoretical Perspective of Dynamics. Structure, and Thermodynamics (Advances in Chem. Physics, Vol. 71) New York: Wiley.

    Google Scholar 

  • Clore M. G. and A. M. Gronenborn. 1991. Comparison of the solution nuclear magnetic resonance and X-ray structures of human recombinant Interleukin-1β.J. mole. Biol. 221, 47–53.

    Article  Google Scholar 

  • Cook, S. A. 1971. The complexity of theorem-proving procedures. Proc. 3rd Ann. ACM Symp. on Theory of Computing, ACM New York, pp. 151–158.

  • Covell, D. G. and R. L. Jernigan. 1990. Conformation of folded proteins in restricted spaces.Biochemistry 29, 3287–3294.

    Article  Google Scholar 

  • Dill, K. A. 1990. Dominant forces in protein folding.Biochemistry 29, 7133–7155.

    Article  Google Scholar 

  • Fraenkel, A. S. 1990 Deexponentializing complex computational mathematical problems using physical and biological systems. TR. CS90-30, The Weizmann Institute of Sciences, Rehovot, Israel.

    Google Scholar 

  • Fraenkel, A. S. 1993. Complexity of protein folding.Bull. math. Biol. 55, 1199–1210.

    Article  MATH  Google Scholar 

  • Garey, M. R. and D. S. Johnson. 1979.Computers and Intractability: A Guide to the Theory of NP-Completeness (San Francisco, CA: Freeman.

    MATH  Google Scholar 

  • Garey, M. R., D. S. Johnson and L. Stockmeyer. 1976. Some simplified NP-complete graph problems.Theor. comput. Sci. 1, 237–267.

    Article  MATH  MathSciNet  Google Scholar 

  • Gething, M. J. and J. Sambrook. 1992. Protein folding in the cell.Nature 355, 33–44.

    Article  Google Scholar 

  • Herzberg, O. and J. Moult. 1991. Analysis of the steric strain in the polypeptide backbone of proteins.Proteins 11, 223–229.

    Article  Google Scholar 

  • Holm, L. and C. Sander. 1991. Database algorithm for generating protein backbone and side chain coordinates from aC α trace.J. molec. Biol. 218, 183–194.

    Article  Google Scholar 

  • Karplus, M. and D. L. Weaver. 1976. Protein folding dynamics.Nature 260, 404–406.

    Article  Google Scholar 

  • Kleene, S. C. 1952.Introduction to Metamathematics. Princeton, NJ: D. Van Nostrand.

    MATH  Google Scholar 

  • Levinthal, C. 1969. InMossbauer Spectroscopy in Biological Systems. Proceedings of a meeting held at Allerton House Monticello, IL. P. Debrunner, J. Tsibris and E. Munck (Eds), pp. 22–24. Urbana, IL: University of Illinois Press.

    Google Scholar 

  • Levitt, M. 1983a. Molecular dynamics of native proteins: analysis and nature of motion.J. molec. Biol. 168, 621–657.

    Google Scholar 

  • Levitt, M. 1983b. Protein folding by restrained energy minimization and molecular dynamics.J. molec. Biol. 170, 723–764.

    Google Scholar 

  • Levitt, M. and A. Warshel. 1975. Computer simulation of protein folding.Nature 253, 694–698.

    Article  Google Scholar 

  • Mazur, J. 1969. Non self intersecting random walks. InStochastic Processes in Chemical Physics. K. E. Shuler Ed.), pp. 261–280 New York: Interscience Publishers.

    Google Scholar 

  • Moult, J. 1989. Comparative modeling of protein structure: progress and prospects.J. Res. natn Inst. Stand Technol. 94, 79–84.

    Google Scholar 

  • Moult, J. and R. Unger. 1991. An analysis of protein folding pathways.Biochemistry 30, 3816–3824.

    Article  Google Scholar 

  • Ngo, J. T. and Marks, J. 1992. Computational complexity of a problem in molecular structure prediction.Protein Engng 5, 313–321.

    Google Scholar 

  • Ramakrishnan, C. and G. N. Ramachandran. 1965. Stereochemical criteria for polypeptide and protein chain conformation.Biophys. J. 5, 909–933.

    Article  Google Scholar 

  • Seetharamulu, P. and G. M. Crippen. 1991. A potential function for protein folding.J. math. Chem. 6, 91–110.

    Article  Google Scholar 

  • Serrano, L., A. Matouschek and A. R. Fersht. 1992. The folding of an enzyme: VI. The folding pathway of Baranase: Comparison with theoretical models.J. molec. Biol. 224, 847–859.

    Article  Google Scholar 

  • Svansson, L. A., J. Dill, L. Sjolin, A. Wlodawar, M. Toner, D. Bacon, J. Moult, B. Veerapandian and G. L. Gilliland. 1991. The crystal packing interactions of two different crystal forms of bovine Ribonuclease A.J. Cryst. Growth 110, 119–130.

    Article  Google Scholar 

  • Udgonkar, J. B. and R. L. Baldwin. 1988. NMR evidence for an early framework intermediate on the folding pathway of ribonuclease A.Nature 335, 694–699.

    Article  Google Scholar 

  • Unger, r., D. Harel, S. Wherland and J. L. Sussman. 1990. Analysis of dihedral angales distribution: The doublets distribution determines polypeptides conformations.Biopolymers 30, 499–508.

    Article  Google Scholar 

  • Wetlaufer, D. B. 1973. Nucleation, rapid folding, and globular interchain regions in proteins.Proc. natn Acad. Sci. U.S.A. 70, 697–701.

    Article  Google Scholar 

  • Wlodawer, A., M. Miller, M. Jaskolski, B. K. Sthayanarayana, E. Baldwin, I. T. Weber, L. M. Selk, L. Clawson, J. Schneider andS. B. H. Kent. 1989. Conserved folding in retroviral protease: crystal structure of a synthetic HIV-1 protease.Science 245, 616–621.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Center for Advanced Reserch in biotechnology, Maryland Biotechnology Institute, University of Maryland, 9600 Gudelsky Dr., 20850, Rockville, MD, U.S.A.

    John Moult

  2. Institute for Advanced Computer Studies, University of Maryland, 20742, College Park, MD, U.S.A.

    Ron Unger

Authors
  1. Ron Unger
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. John Moult
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Unger, R., Moult, J. Finding the lowest free energy conformation of a protein is an NP-hard problem: Proof and implications. Bltn Mathcal Biology 55, 1183–1198 (1993). https://doi.org/10.1007/BF02460703

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02460703

Keywords

  • Free Energy
  • Conformational Space
  • Hamiltonian Path
  • Polynomial Solution
  • Folding Process