Skip to main content
NCBI home page
Biophysical Journal logoLink to Biophysical Journal
. 2001 Aug;81(2):751–766. doi: 10.1016/S0006-3495(01)75739-6

Are proteins well-packed?

J Liang 1, K A Dill 1
PMCID: PMC1301551  PMID: 11463623

Abstract

The average packing density inside proteins is as high as in crystalline solids. Does this mean proteins are well-packed? We go beyond average densities, and look at the full distribution functions of free volumes inside proteins. Using a new and rigorous Delaunay triangulation method for parsing space into empty and filled regions, we introduce formal definitions of interior and surface packing densities. Although proteins look like organic crystals by the criterion of average density, they look more like liquids and glasses by the criterion of their free volume distributions. The distributions are broad, and the scalings of volume-to-surface, volume-to-cluster-radius, and numbers of void versus volume show that the interiors of proteins are more like randomly packed spheres near their percolation threshold than like jigsaw puzzles. We find that larger proteins are packed more loosely than smaller proteins. And we find that the enthalpies of folding (per amino acid) are independent of the packing density of a protein, indicating that van der Waals interactions are not a dominant component of the folding forces.

Full Text

The Full Text of this article is available as a PDF (459.0 KB).

Selected References

These references are in PubMed. This may not be the complete list of references from this article.

  1. Adler J, Meir Y, Aharony A, Harris AB. Series study of percolation moments in general dimension. Phys Rev B Condens Matter. 1990 May 1;41(13):9183–9206. doi: 10.1103/physrevb.41.9183. [DOI] [PubMed] [Google Scholar]
  2. Alexander P., Fahnestock S., Lee T., Orban J., Bryan P. Thermodynamic analysis of the folding of the streptococcal protein G IgG-binding domains B1 and B2: why small proteins tend to have high denaturation temperatures. Biochemistry. 1992 Apr 14;31(14):3597–3603. doi: 10.1021/bi00129a007. [DOI] [PubMed] [Google Scholar]
  3. Axe D. D., Foster N. W., Fersht A. R. Active barnase variants with completely random hydrophobic cores. Proc Natl Acad Sci U S A. 1996 May 28;93(11):5590–5594. doi: 10.1073/pnas.93.11.5590. [DOI] [PMC free article] [PubMed] [Google Scholar]
  4. Bae S. J., Sturtevant J. M. Thermodynamics of the thermal unfolding of eglin c in the presence and absence of guanidinium chloride. Biophys Chem. 1995 Aug;55(3):247–252. doi: 10.1016/0301-4622(94)00157-f. [DOI] [PubMed] [Google Scholar]
  5. Bromberg S., Dill K. A. Side-chain entropy and packing in proteins. Protein Sci. 1994 Jul;3(7):997–1009. doi: 10.1002/pro.5560030702. [DOI] [PMC free article] [PubMed] [Google Scholar]
  6. Chothia C. Principles that determine the structure of proteins. Annu Rev Biochem. 1984;53:537–572. doi: 10.1146/annurev.bi.53.070184.002541. [DOI] [PubMed] [Google Scholar]
  7. Chothia C. Structural invariants in protein folding. Nature. 1975 Mar 27;254(5498):304–308. doi: 10.1038/254304a0. [DOI] [PubMed] [Google Scholar]
  8. Delaney J. S. Finding and filling protein cavities using cellular logic operations. J Mol Graph. 1992 Sep;10(3):174-7, 163. doi: 10.1016/0263-7855(92)80052-f. [DOI] [PubMed] [Google Scholar]
  9. Demarest S. J., Zhou S. Q., Robblee J., Fairman R., Chu B., Raleigh D. P. A comparative study of peptide models of the alpha-domain of alpha-lactalbumin, lysozyme, and alpha-lactalbumin/lysozyme chimeras allows the elucidation of critical factors that contribute to the ability to form stable partially folded states. Biochemistry. 2001 Feb 20;40(7):2138–2147. doi: 10.1021/bi001975z. [DOI] [PubMed] [Google Scholar]
  10. Finney J. L. Volume occupation, environment and accessibility in proteins. The problem of the protein surface. J Mol Biol. 1975 Aug 25;96(4):721–732. doi: 10.1016/0022-2836(75)90148-5. [DOI] [PubMed] [Google Scholar]
  11. Gavish B., Gratton E., Hardy C. J. Adiabatic compressibility of globular proteins. Proc Natl Acad Sci U S A. 1983 Feb;80(3):750–754. doi: 10.1073/pnas.80.3.750. [DOI] [PMC free article] [PubMed] [Google Scholar]
  12. Gellatly B. J., Finney J. L. Calculation of protein volumes: an alternative to the Voronoi procedure. J Mol Biol. 1982 Oct 25;161(2):305–322. doi: 10.1016/0022-2836(82)90155-3. [DOI] [PubMed] [Google Scholar]
  13. Gerstein M., Chothia C. Packing at the protein-water interface. Proc Natl Acad Sci U S A. 1996 Sep 17;93(19):10167–10172. doi: 10.1073/pnas.93.19.10167. [DOI] [PMC free article] [PubMed] [Google Scholar]
  14. Gerstein M., Tsai J., Levitt M. The volume of atoms on the protein surface: calculated from simulation, using Voronoi polyhedra. J Mol Biol. 1995 Jun 23;249(5):955–966. doi: 10.1006/jmbi.1995.0351. [DOI] [PubMed] [Google Scholar]
  15. Griko Y. V., Makhatadze G. I., Privalov P. L., Hartley R. W. Thermodynamics of barnase unfolding. Protein Sci. 1994 Apr;3(4):669–676. doi: 10.1002/pro.5560030414. [DOI] [PMC free article] [PubMed] [Google Scholar]
  16. Harpaz Y., Gerstein M., Chothia C. Volume changes on protein folding. Structure. 1994 Jul 15;2(7):641–649. doi: 10.1016/s0969-2126(00)00065-4. [DOI] [PubMed] [Google Scholar]
  17. Ho C. M., Marshall G. R. Cavity search: an algorithm for the isolation and display of cavity-like binding regions. J Comput Aided Mol Des. 1990 Dec;4(4):337–354. doi: 10.1007/BF00117400. [DOI] [PubMed] [Google Scholar]
  18. Hobohm U., Sander C. Enlarged representative set of protein structures. Protein Sci. 1994 Mar;3(3):522–524. doi: 10.1002/pro.5560030317. [DOI] [PMC free article] [PubMed] [Google Scholar]
  19. Honig B. Protein folding: from the levinthal paradox to structure prediction. J Mol Biol. 1999 Oct 22;293(2):283–293. doi: 10.1006/jmbi.1999.3006. [DOI] [PubMed] [Google Scholar]
  20. Hubbard S. J., Argos P. Cavities and packing at protein interfaces. Protein Sci. 1994 Dec;3(12):2194–2206. doi: 10.1002/pro.5560031205. [DOI] [PMC free article] [PubMed] [Google Scholar]
  21. Hubbard S. J., Gross K. H., Argos P. Intramolecular cavities in globular proteins. Protein Eng. 1994 May;7(5):613–626. doi: 10.1093/protein/7.5.613. [DOI] [PubMed] [Google Scholar]
  22. Jackson S. E., Fersht A. R. Folding of chymotrypsin inhibitor 2. 1. Evidence for a two-state transition. Biochemistry. 1991 Oct 29;30(43):10428–10435. doi: 10.1021/bi00107a010. [DOI] [PubMed] [Google Scholar]
  23. Kim S., Liang J., Barry B. A. Chemical complementation identifies a proton acceptor for redox-active tyrosine D in photosystem II. Proc Natl Acad Sci U S A. 1997 Dec 23;94(26):14406–14411. doi: 10.1073/pnas.94.26.14406. [DOI] [PMC free article] [PubMed] [Google Scholar]
  24. Kitamura S., Sturtevant J. M. A scanning calorimetric study of the thermal denaturation of the lysozyme of phage T4 and the Arg 96----His mutant form thereof. Biochemistry. 1989 May 2;28(9):3788–3792. doi: 10.1021/bi00435a024. [DOI] [PubMed] [Google Scholar]
  25. Kleywegt G. J., Jones T. A. Detection, delineation, measurement and display of cavities in macromolecular structures. Acta Crystallogr D Biol Crystallogr. 1994 Mar 1;50(Pt 2):178–185. doi: 10.1107/S0907444993011333. [DOI] [PubMed] [Google Scholar]
  26. Lee B., Richards F. M. The interpretation of protein structures: estimation of static accessibility. J Mol Biol. 1971 Feb 14;55(3):379–400. doi: 10.1016/0022-2836(71)90324-x. [DOI] [PubMed] [Google Scholar]
  27. Liang J., Edelsbrunner H., Fu P., Sudhakar P. V., Subramaniam S. Analytical shape computation of macromolecules: I. Molecular area and volume through alpha shape. Proteins. 1998 Oct 1;33(1):1–17. [PubMed] [Google Scholar]
  28. Liang J., Edelsbrunner H., Fu P., Sudhakar P. V., Subramaniam S. Analytical shape computation of macromolecules: II. Inaccessible cavities in proteins. Proteins. 1998 Oct 1;33(1):18–29. [PubMed] [Google Scholar]
  29. Liang J., Edelsbrunner H., Woodward C. Anatomy of protein pockets and cavities: measurement of binding site geometry and implications for ligand design. Protein Sci. 1998 Sep;7(9):1884–1897. doi: 10.1002/pro.5560070905. [DOI] [PMC free article] [PubMed] [Google Scholar]
  30. Liang J., McGee M. P. Hydration structure of antithrombin conformers and water transfer during reactive loop insertion. Biophys J. 1998 Aug;75(2):573–582. doi: 10.1016/S0006-3495(98)77548-4. [DOI] [PMC free article] [PubMed] [Google Scholar]
  31. Liang J., Subramaniam S. Computation of molecular electrostatics with boundary element methods. Biophys J. 1997 Oct;73(4):1830–1841. doi: 10.1016/S0006-3495(97)78213-4. [DOI] [PMC free article] [PubMed] [Google Scholar]
  32. Lim W. A., Sauer R. T. Alternative packing arrangements in the hydrophobic core of lambda repressor. Nature. 1989 May 4;339(6219):31–36. doi: 10.1038/339031a0. [DOI] [PubMed] [Google Scholar]
  33. Makhatadze G. I., Clore G. M., Gronenborn A. M., Privalov P. L. Thermodynamics of unfolding of the all beta-sheet protein interleukin-1 beta. Biochemistry. 1994 Aug 9;33(31):9327–9332. doi: 10.1021/bi00197a037. [DOI] [PubMed] [Google Scholar]
  34. Makhatadze G. I., Kim K. S., Woodward C., Privalov P. L. Thermodynamics of BPTI folding. Protein Sci. 1993 Dec;2(12):2028–2036. doi: 10.1002/pro.5560021204. [DOI] [PMC free article] [PubMed] [Google Scholar]
  35. Makhatadze G. I., Privalov P. L. Energetics of protein structure. Adv Protein Chem. 1995;47:307–425. doi: 10.1016/s0065-3233(08)60548-3. [DOI] [PubMed] [Google Scholar]
  36. Mateo P. L., Privalov P. L. Pepsinogen denaturation is not a two-state transition. FEBS Lett. 1981 Jan 26;123(2):189–192. doi: 10.1016/0014-5793(81)80284-0. [DOI] [PubMed] [Google Scholar]
  37. McGee M. P., Teuschler H., Liang J. Effective electrostatic charge of coagulation factor X in solution and on phospholipid membranes: implications for activation mechanisms and structure-function relationships of the Gla domain. Biochem J. 1998 Feb 15;330(Pt 1):533–539. doi: 10.1042/bj3300533. [DOI] [PMC free article] [PubMed] [Google Scholar]
  38. Murphy K. P., Gill S. J. Solid model compounds and the thermodynamics of protein unfolding. J Mol Biol. 1991 Dec 5;222(3):699–709. doi: 10.1016/0022-2836(91)90506-2. [DOI] [PubMed] [Google Scholar]
  39. Peters K. P., Fauck J., Frömmel C. The automatic search for ligand binding sites in proteins of known three-dimensional structure using only geometric criteria. J Mol Biol. 1996 Feb 16;256(1):201–213. doi: 10.1006/jmbi.1996.0077. [DOI] [PubMed] [Google Scholar]
  40. Privalov P. L., Khechinashvili N. N. A thermodynamic approach to the problem of stabilization of globular protein structure: a calorimetric study. J Mol Biol. 1974 Jul 5;86(3):665–684. doi: 10.1016/0022-2836(74)90188-0. [DOI] [PubMed] [Google Scholar]
  41. Rashin A. A., Iofin M., Honig B. Internal cavities and buried waters in globular proteins. Biochemistry. 1986 Jun 17;25(12):3619–3625. doi: 10.1021/bi00360a021. [DOI] [PubMed] [Google Scholar]
  42. Renner M., Hinz H. J., Scharf M., Engels J. W. Thermodynamics of unfolding of the alpha-amylase inhibitor tendamistat. Correlations between accessible surface area and heat capacity. J Mol Biol. 1992 Feb 5;223(3):769–779. doi: 10.1016/0022-2836(92)90988-v. [DOI] [PubMed] [Google Scholar]
  43. Richards F. M. Areas, volumes, packing and protein structure. Annu Rev Biophys Bioeng. 1977;6:151–176. doi: 10.1146/annurev.bb.06.060177.001055. [DOI] [PubMed] [Google Scholar]
  44. Richards F. M., Lim W. A. An analysis of packing in the protein folding problem. Q Rev Biophys. 1993 Nov;26(4):423–498. doi: 10.1017/s0033583500002845. [DOI] [PubMed] [Google Scholar]
  45. Richards F. M. The interpretation of protein structures: total volume, group volume distributions and packing density. J Mol Biol. 1974 Jan 5;82(1):1–14. doi: 10.1016/0022-2836(74)90570-1. [DOI] [PubMed] [Google Scholar]
  46. Richmond T. J. Solvent accessible surface area and excluded volume in proteins. Analytical equations for overlapping spheres and implications for the hydrophobic effect. J Mol Biol. 1984 Sep 5;178(1):63–89. doi: 10.1016/0022-2836(84)90231-6. [DOI] [PubMed] [Google Scholar]
  47. Shortle D., Stites W. E., Meeker A. K. Contributions of the large hydrophobic amino acids to the stability of staphylococcal nuclease. Biochemistry. 1990 Sep 4;29(35):8033–8041. doi: 10.1021/bi00487a007. [DOI] [PubMed] [Google Scholar]
  48. Shrake A., Rupley J. A. Environment and exposure to solvent of protein atoms. Lysozyme and insulin. J Mol Biol. 1973 Sep 15;79(2):351–371. doi: 10.1016/0022-2836(73)90011-9. [DOI] [PubMed] [Google Scholar]
  49. Tiktopulo E. I., Privalov P. L. Papain denaturation is not a two-state transition. FEBS Lett. 1978 Jul 1;91(1):57–58. doi: 10.1016/0014-5793(78)80016-7. [DOI] [PubMed] [Google Scholar]
  50. Tsai J., Taylor R., Chothia C., Gerstein M. The packing density in proteins: standard radii and volumes. J Mol Biol. 1999 Jul 2;290(1):253–266. doi: 10.1006/jmbi.1999.2829. [DOI] [PubMed] [Google Scholar]
  51. Viguera A. R., Martínez J. C., Filimonov V. V., Mateo P. L., Serrano L. Thermodynamic and kinetic analysis of the SH3 domain of spectrin shows a two-state folding transition. Biochemistry. 1994 Mar 1;33(8):2142–2150. doi: 10.1021/bi00174a022. [DOI] [PubMed] [Google Scholar]
  52. Voorintholt R., Kosters M. T., Vegter G., Vriend G., Hol W. G. A very fast program for visualizing protein surfaces, channels and cavities. J Mol Graph. 1989 Dec;7(4):243–245. doi: 10.1016/0263-7855(89)80010-4. [DOI] [PubMed] [Google Scholar]
  53. Wintrode P. L., Makhatadze G. I., Privalov P. L. Thermodynamics of ubiquitin unfolding. Proteins. 1994 Mar;18(3):246–253. doi: 10.1002/prot.340180305. [DOI] [PubMed] [Google Scholar]
  54. Word J. M., Lovell S. C., LaBean T. H., Taylor H. C., Zalis M. E., Presley B. K., Richardson J. S., Richardson D. C. Visualizing and quantifying molecular goodness-of-fit: small-probe contact dots with explicit hydrogen atoms. J Mol Biol. 1999 Jan 29;285(4):1711–1733. doi: 10.1006/jmbi.1998.2400. [DOI] [PubMed] [Google Scholar]
  55. Yu Y., Makhatadze G. I., Pace C. N., Privalov P. L. Energetics of ribonuclease T1 structure. Biochemistry. 1994 Mar 22;33(11):3312–3319. doi: 10.1021/bi00177a023. [DOI] [PubMed] [Google Scholar]
  56. van der Marck SC Network Approach to Void Percolation in a Pack of Unequal Spheres. Phys Rev Lett. 1996 Aug 26;77(9):1785–1788. doi: 10.1103/PhysRevLett.77.1785. [DOI] [PubMed] [Google Scholar]

Articles from Biophysical Journal are provided here courtesy of The Biophysical Society

RESOURCES