Kinetics of a Gaussian random copolymer as a prototype for protein folding

Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

Volumn 54, Issue 4 SUPPL. B, 1996, Pages 4071-4086

  Timoshenko, E G ^b   Kuznetsov, Yu A ^a,b   Dawson, K A ^b  

^a INSTITUTE OF THEORETICAL AND EXPERIMENTAL BIOPHYSICS   (Russian Federation)

^b UNIVERSITY COLLEGE DUBLIN   (Ireland)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 4243875619     PISSN: 1063651X     EISSN: None     Source Type: Journal    
DOI: 10.1103/physreve.54.4071     Document Type: Article

References (80)

1. Protein Folding, edited by T. E. Creighton (Wiley, New York, 1992); New Developments in Theoretical Studies of Proteins, edited by R. Elber (World Scientific, Singapore, 1994).
2. Protein Folding, edited by T. E. Creighton (Wiley, New York, 1992); New Developments in Theoretical Studies of Proteins, edited by R. Elber (World Scientific, Singapore, 1994).
3. H. A. Scheraga, Pure Appl. Chem. 36, 1 (1973); N. Go and H. Taketomi, Proc. Natl. Acad. Sci. U.S.A. 75, 559 (1978); O. Ptitsyn and A. Finkelstein, Q. Rev. Biophys. 13, 339 (1980); P. Privalov, Adv. Protein Chem. 35, 1 (1982); O. Ptitsyn, J. Protein Chem. 6, 273 (1987); K. A. Dill, Biochemistry 24, 1501 (1985); 29, 7133 (1990); H. Frauenfelder, in Structure and Dynamics of Nucleic Acids, Proteins and Membranes, edited by E. Clementi and S. Chin (Plenum, New York, 1986); H. Frauenfelder, F. Parak, and R. Young, Annu. Rev. Biophys. Biophys. Chem. 17, 451 (1988); M. Matsumoto et al., J. Polym. Sci. B 30, 779 (1992); K. Minagawa, Y. Matsuzawa, K. Yoshikawa, A. R. Khokhlov, and M. Doi, Biopolymers 34, 555 (1994).
4. H. A. Scheraga, Pure Appl. Chem. 36, 1 (1973); N. Go and H. Taketomi, Proc. Natl. Acad. Sci. U.S.A. 75, 559 (1978); O. Ptitsyn and A. Finkelstein, Q. Rev. Biophys. 13, 339 (1980); P. Privalov, Adv. Protein Chem. 35, 1 (1982); O. Ptitsyn, J. Protein Chem. 6, 273 (1987); K. A. Dill, Biochemistry 24, 1501 (1985); 29, 7133 (1990); H. Frauenfelder, in Structure and Dynamics of Nucleic Acids, Proteins and Membranes, edited by E. Clementi and S. Chin (Plenum, New York, 1986); H. Frauenfelder, F. Parak, and R. Young, Annu. Rev. Biophys. Biophys. Chem. 17, 451 (1988); M. Matsumoto et al., J. Polym. Sci. B 30, 779 (1992); K. Minagawa, Y. Matsuzawa, K. Yoshikawa, A. R. Khokhlov, and M. Doi, Biopolymers 34, 555 (1994).
5. H. A. Scheraga, Pure Appl. Chem. 36, 1 (1973); N. Go and H. Taketomi, Proc. Natl. Acad. Sci. U.S.A. 75, 559 (1978); O. Ptitsyn and A. Finkelstein, Q. Rev. Biophys. 13, 339 (1980); P. Privalov, Adv. Protein Chem. 35, 1 (1982); O. Ptitsyn, J. Protein Chem. 6, 273 (1987); K. A. Dill, Biochemistry 24, 1501 (1985); 29, 7133 (1990); H. Frauenfelder, in Structure and Dynamics of Nucleic Acids, Proteins and Membranes, edited by E. Clementi and S. Chin (Plenum, New York, 1986); H. Frauenfelder, F. Parak, and R. Young, Annu. Rev. Biophys. Biophys. Chem. 17, 451 (1988); M. Matsumoto et al., J. Polym. Sci. B 30, 779 (1992); K. Minagawa, Y. Matsuzawa, K. Yoshikawa, A. R. Khokhlov, and M. Doi, Biopolymers 34, 555 (1994).
6. H. A. Scheraga, Pure Appl. Chem. 36, 1 (1973); N. Go and H. Taketomi, Proc. Natl. Acad. Sci. U.S.A. 75, 559 (1978); O. Ptitsyn and A. Finkelstein, Q. Rev. Biophys. 13, 339 (1980); P. Privalov, Adv. Protein Chem. 35, 1 (1982); O. Ptitsyn, J. Protein Chem. 6, 273 (1987); K. A. Dill, Biochemistry 24, 1501 (1985); 29, 7133 (1990); H. Frauenfelder, in Structure and Dynamics of Nucleic Acids, Proteins and Membranes, edited by E. Clementi and S. Chin (Plenum, New York, 1986); H. Frauenfelder, F. Parak, and R. Young, Annu. Rev. Biophys. Biophys. Chem. 17, 451 (1988); M. Matsumoto et al., J. Polym. Sci. B 30, 779 (1992); K. Minagawa, Y. Matsuzawa, K. Yoshikawa, A. R. Khokhlov, and M. Doi, Biopolymers 34, 555 (1994).
7. H. A. Scheraga, Pure Appl. Chem. 36, 1 (1973); N. Go and H. Taketomi, Proc. Natl. Acad. Sci. U.S.A. 75, 559 (1978); O. Ptitsyn and A. Finkelstein, Q. Rev. Biophys. 13, 339 (1980); P. Privalov, Adv. Protein Chem. 35, 1 (1982); O. Ptitsyn, J. Protein Chem. 6, 273 (1987); K. A. Dill, Biochemistry 24, 1501 (1985); 29, 7133 (1990); H. Frauenfelder, in Structure and Dynamics of Nucleic Acids, Proteins and Membranes, edited by E. Clementi and S. Chin (Plenum, New York, 1986); H. Frauenfelder, F. Parak, and R. Young, Annu. Rev. Biophys. Biophys. Chem. 17, 451 (1988); M. Matsumoto et al., J. Polym. Sci. B 30, 779 (1992); K. Minagawa, Y. Matsuzawa, K. Yoshikawa, A. R. Khokhlov, and M. Doi, Biopolymers 34, 555 (1994).
8. H. A. Scheraga, Pure Appl. Chem. 36, 1 (1973); N. Go and H. Taketomi, Proc. Natl. Acad. Sci. U.S.A. 75, 559 (1978); O. Ptitsyn and A. Finkelstein, Q. Rev. Biophys. 13, 339 (1980); P. Privalov, Adv. Protein Chem. 35, 1 (1982); O. Ptitsyn, J. Protein Chem. 6, 273 (1987); K. A. Dill, Biochemistry 24, 1501 (1985); 29, 7133 (1990); H. Frauenfelder, in Structure and Dynamics of Nucleic Acids, Proteins and Membranes, edited by E. Clementi and S. Chin (Plenum, New York, 1986); H. Frauenfelder, F. Parak, and R. Young, Annu. Rev. Biophys. Biophys. Chem. 17, 451 (1988); M. Matsumoto et al., J. Polym. Sci. B 30, 779 (1992); K. Minagawa, Y. Matsuzawa, K. Yoshikawa, A. R. Khokhlov, and M. Doi, Biopolymers 34, 555 (1994).
9. H. A. Scheraga, Pure Appl. Chem. 36, 1 (1973); N. Go and H. Taketomi, Proc. Natl. Acad. Sci. U.S.A. 75, 559 (1978); O. Ptitsyn and A. Finkelstein, Q. Rev. Biophys. 13, 339 (1980); P. Privalov, Adv. Protein Chem. 35, 1 (1982); O. Ptitsyn, J. Protein Chem. 6, 273 (1987); K. A. Dill, Biochemistry 24, 1501 (1985); 29, 7133 (1990); H. Frauenfelder, in Structure and Dynamics of Nucleic Acids, Proteins and Membranes, edited by E. Clementi and S. Chin (Plenum, New York, 1986); H. Frauenfelder, F. Parak, and R. Young, Annu. Rev. Biophys. Biophys. Chem. 17, 451 (1988); M. Matsumoto et al., J. Polym. Sci. B 30, 779 (1992); K. Minagawa, Y. Matsuzawa, K. Yoshikawa, A. R. Khokhlov, and M. Doi, Biopolymers 34, 555 (1994).
10. H. A. Scheraga, Pure Appl. Chem. 36, 1 (1973); N. Go and H. Taketomi, Proc. Natl. Acad. Sci. U.S.A. 75, 559 (1978); O. Ptitsyn and A. Finkelstein, Q. Rev. Biophys. 13, 339 (1980); P. Privalov, Adv. Protein Chem. 35, 1 (1982); O. Ptitsyn, J. Protein Chem. 6, 273 (1987); K. A. Dill, Biochemistry 24, 1501 (1985); 29, 7133 (1990); H. Frauenfelder, in Structure and Dynamics of Nucleic Acids, Proteins and Membranes, edited by E. Clementi and S. Chin (Plenum, New York, 1986); H. Frauenfelder, F. Parak, and R. Young, Annu. Rev. Biophys. Biophys. Chem. 17, 451 (1988); M. Matsumoto et al., J. Polym. Sci. B 30, 779 (1992); K. Minagawa, Y. Matsuzawa, K. Yoshikawa, A. R. Khokhlov, and M. Doi, Biopolymers 34, 555 (1994).
11. H. A. Scheraga, Pure Appl. Chem. 36, 1 (1973); N. Go and H. Taketomi, Proc. Natl. Acad. Sci. U.S.A. 75, 559 (1978); O. Ptitsyn and A. Finkelstein, Q. Rev. Biophys. 13, 339 (1980); P. Privalov, Adv. Protein Chem. 35, 1 (1982); O. Ptitsyn, J. Protein Chem. 6, 273 (1987); K. A. Dill, Biochemistry 24, 1501 (1985); 29, 7133 (1990); H. Frauenfelder, in Structure and Dynamics of Nucleic Acids, Proteins and Membranes, edited by E. Clementi and S. Chin (Plenum, New York, 1986); H. Frauenfelder, F. Parak, and R. Young, Annu. Rev. Biophys. Biophys. Chem. 17, 451 (1988); M. Matsumoto et al., J. Polym. Sci. B 30, 779 (1992); K. Minagawa, Y. Matsuzawa, K. Yoshikawa, A. R. Khokhlov, and M. Doi, Biopolymers 34, 555 (1994).
12. H. A. Scheraga, Pure Appl. Chem. 36, 1 (1973); N. Go and H. Taketomi, Proc. Natl. Acad. Sci. U.S.A. 75, 559 (1978); O. Ptitsyn and A. Finkelstein, Q. Rev. Biophys. 13, 339 (1980); P. Privalov, Adv. Protein Chem. 35, 1 (1982); O. Ptitsyn, J. Protein Chem. 6, 273 (1987); K. A. Dill, Biochemistry 24, 1501 (1985); 29, 7133 (1990); H. Frauenfelder, in Structure and Dynamics of Nucleic Acids, Proteins and Membranes, edited by E. Clementi and S. Chin (Plenum, New York, 1986); H. Frauenfelder, F. Parak, and R. Young, Annu. Rev. Biophys. Biophys. Chem. 17, 451 (1988); M. Matsumoto et al., J. Polym. Sci. B 30, 779 (1992); K. Minagawa, Y. Matsuzawa, K. Yoshikawa, A. R. Khokhlov, and M. Doi, Biopolymers 34, 555 (1994).
13. H. A. Scheraga, Pure Appl. Chem. 36, 1 (1973); N. Go and H. Taketomi, Proc. Natl. Acad. Sci. U.S.A. 75, 559 (1978); O. Ptitsyn and A. Finkelstein, Q. Rev. Biophys. 13, 339 (1980); P. Privalov, Adv. Protein Chem. 35, 1 (1982); O. Ptitsyn, J. Protein Chem. 6, 273 (1987); K. A. Dill, Biochemistry 24, 1501 (1985); 29, 7133 (1990); H. Frauenfelder, in Structure and Dynamics of Nucleic Acids, Proteins and Membranes, edited by E. Clementi and S. Chin (Plenum, New York, 1986); H. Frauenfelder, F. Parak, and R. Young, Annu. Rev. Biophys. Biophys. Chem. 17, 451 (1988); M. Matsumoto et al., J. Polym. Sci. B 30, 779 (1992); K. Minagawa, Y. Matsuzawa, K. Yoshikawa, A. R. Khokhlov, and M. Doi, Biopolymers 34, 555 (1994).
14. E. Shakhnovich and A. M. Gutin, Nature 346, 773 (1990); A. Sali, E. Shakhnovich, and M. Karplus, ibid. 369, 248 (1994).
15. E. Shakhnovich and A. M. Gutin, Nature 346, 773 (1990); A. Sali, E. Shakhnovich, and M. Karplus, ibid. 369, 248 (1994).
16. C. Tanford, The Hydrophobic Effect (Wiley, New York, 1980); K. A. Dill, Biochemistry 29, 7133 (1990); S. P. Obukhov, J. Phys. A 19, 3655 (1986).
17. C. Tanford, The Hydrophobic Effect (Wiley, New York, 1980); K. A. Dill, Biochemistry 29, 7133 (1990); S. P. Obukhov, J. Phys. A 19, 3655 (1986).
18. C. Tanford, The Hydrophobic Effect (Wiley, New York, 1980); K. A. Dill, Biochemistry 29, 7133 (1990); S. P. Obukhov, J. Phys. A 19, 3655 (1986).
19. P. G. Wolynes, in Spin Glass Ideas in Biology, edited by D. Stein (World Scientific, Singapore, 1991).
20. M. Mezard, G. Parisi, and M. Virasoro, Spin Glass Theory and Beyond (World Scientific, Singapore, 1987).
21. M. Mezard and G. Parisi, J. Phys. (France) I 1, 809 (1991).
22. J. D. Bryngelson and P.G. Wolynes, Proc. Natl. Acad. Sci. U.S.A. 84, 7524 (1987); P. G. Wolynes, J. N. Onuchic, and D. Thirumalai, Science 267, 1619 (1995).
23. J. D. Bryngelson and P.G. Wolynes, Proc. Natl. Acad. Sci. U.S.A. 84, 7524 (1987); P. G. Wolynes, J. N. Onuchic, and D. Thirumalai, Science 267, 1619 (1995).
24. K. F. Lau and K. A. Dill, Macromolecules 22, 3986 (1989); H. S. Chan and K. A. Dill, ibid. 22, 4559 (1989); J. Chem. Phys. 95, 3775 (1991); 96, 768 (1992); 99, 2116 (1993); K. A. Dill, K. M. Fiebig, and H.S. Chan, Proc. Natl. Acad. Sci. U.S.A. 90, 1942 (1993).
25. K. F. Lau and K. A. Dill, Macromolecules 22, 3986 (1989); H. S. Chan and K. A. Dill, ibid. 22, 4559 (1989); J. Chem. Phys. 95, 3775 (1991); 96, 768 (1992); 99, 2116 (1993); K. A. Dill, K. M. Fiebig, and H.S. Chan, Proc. Natl. Acad. Sci. U.S.A. 90, 1942 (1993).
26. K. F. Lau and K. A. Dill, Macromolecules 22, 3986 (1989); H. S. Chan and K. A. Dill, ibid. 22, 4559 (1989); J. Chem. Phys. 95, 3775 (1991); 96, 768 (1992); 99, 2116 (1993); K. A. Dill, K. M. Fiebig, and H.S. Chan, Proc. Natl. Acad. Sci. U.S.A. 90, 1942 (1993).
27. K. F. Lau and K. A. Dill, Macromolecules 22, 3986 (1989); H. S. Chan and K. A. Dill, ibid. 22, 4559 (1989); J. Chem. Phys. 95, 3775 (1991); 96, 768 (1992); 99, 2116 (1993); K. A. Dill, K. M. Fiebig, and H.S. Chan, Proc. Natl. Acad. Sci. U.S.A. 90, 1942 (1993).
28. K. F. Lau and K. A. Dill, Macromolecules 22, 3986 (1989); H. S. Chan and K. A. Dill, ibid. 22, 4559 (1989); J. Chem. Phys. 95, 3775 (1991); 96, 768 (1992); 99, 2116 (1993); K. A. Dill, K. M. Fiebig, and H.S. Chan, Proc. Natl. Acad. Sci. U.S.A. 90, 1942 (1993).
29. K. F. Lau and K. A. Dill, Macromolecules 22, 3986 (1989); H. S. Chan and K. A. Dill, ibid. 22, 4559 (1989); J. Chem. Phys. 95, 3775 (1991); 96, 768 (1992); 99, 2116 (1993); K. A. Dill, K. M. Fiebig, and H.S. Chan, Proc. Natl. Acad. Sci. U.S.A. 90, 1942 (1993).
30. C. J. Camacho and D. Thirumalai, Phys. Rev. Lett. 71, 2505 (1993); Proc. Natl. Acad. Sci. U.S.A. 90, 6369 (1993); 90, 1277 (1995).
31. C. J. Camacho and D. Thirumalai, Phys. Rev. Lett. 71, 2505 (1993); Proc. Natl. Acad. Sci. U.S.A. 90, 6369 (1993); 90, 1277 (1995).
32. C. J. Camacho and D. Thirumalai, Phys. Rev. Lett. 71, 2505 (1993); Proc. Natl. Acad. Sci. U.S.A. 90, 6369 (1993); 90, 1277 (1995).
33. J. D. Honeycutt and D. Thirumalai, Biopolymers 32, 695 (1992); D. Thirumalai, J. Phys. (France) I 5, 1457 (1995).
34. J. D. Honeycutt and D. Thirumalai, Biopolymers 32, 695 (1992); D. Thirumalai, J. Phys. (France) I 5, 1457 (1995).
35. E. Shakhnovich, Phys. Rev. Lett. 72 3907 (1994).
36. P. G. de Gennes, Scaling Concepts in Polymer Physics, 3rd ed. (Cornell University Press, Ithaca, 1988); M. Doi and S. F. Edwards, The Theory of Polymer Dynamics (Oxford Science, New York, 1989); J. des Cloizeaux and G. Jannink, Polymers in Solution (Clarendon, Oxford, 1990).
37. P. G. de Gennes, Scaling Concepts in Polymer Physics, 3rd ed. (Cornell University Press, Ithaca, 1988); M. Doi and S. F. Edwards, The Theory of Polymer Dynamics (Oxford Science, New York, 1989); J. des Cloizeaux and G. Jannink, Polymers in Solution (Clarendon, Oxford, 1990).
38. P. G. de Gennes, Scaling Concepts in Polymer Physics, 3rd ed. (Cornell University Press, Ithaca, 1988); M. Doi and S. F. Edwards, The Theory of Polymer Dynamics (Oxford Science, New York, 1989); J. des Cloizeaux and G. Jannink, Polymers in Solution (Clarendon, Oxford, 1990).
39. C. D. Sfatos, A. M. Gutin, and E. I. Shakhnovich, Phys. Rev. E 48, 465 (1993).
40. E. I. Shakhnovich and A. M. Gutin, J. Phys. A 22, 1647 (1989); J. Chem. Phys. 93, 5967 (1990).
41. E. I. Shakhnovich and A. M. Gutin, J. Phys. A 22, 1647 (1989); J. Chem. Phys. 93, 5967 (1990).
42. V. S. Pande, A. Yu. Grosberg, and T. Tanaka, Phys. Rev. E 51, 3381 (1995).
43. A. V. Dobrynin and I. Ya. Erukhimovich, Pis'ma Zh. Éksp. Teor. Fiz. 53, 545 (1991).
44. G. Z. Archonts and E. I. Shakhnovich, Phys. Rev. E 49, 3109 (1994).
45. T. Garel and H. Orland, Europhys. Lett. 6, 307 (1988); 6, 597 (1988); G. H. Fredrickson, S. T. Milner, and L. Leibler, Macromolecules 25, 6341 (1992); C. D. Sfatos, A. M. Gutin, and E. I. Shakhnovich, Phys. Rev. E 51, 4727 (1995).
46. T. Garel and H. Orland, Europhys. Lett. 6, 307 (1988); 6, 597 (1988); G. H. Fredrickson, S. T. Milner, and L. Leibler, Macromolecules 25, 6341 (1992); C. D. Sfatos, A. M. Gutin, and E. I. Shakhnovich, Phys. Rev. E 51, 4727 (1995).
47. T. Garel and H. Orland, Europhys. Lett. 6, 307 (1988); 6, 597 (1988); G. H. Fredrickson, S. T. Milner, and L. Leibler, Macromolecules 25, 6341 (1992); C. D. Sfatos, A. M. Gutin, and E. I. Shakhnovich, Phys. Rev. E 51, 4727 (1995).
48. T. Garel and H. Orland, Europhys. Lett. 6, 307 (1988); 6, 597 (1988); G. H. Fredrickson, S. T. Milner, and L. Leibler, Macromolecules 25, 6341 (1992); C. D. Sfatos, A. M. Gutin, and E. I. Shakhnovich, Phys. Rev. E 51, 4727 (1995).
49. J. D. Bryngelson, J. Chem. Phys. 100, 6038 (1994).
50. K. Yue and K.A. Dill, Proc. Natl. Acad. Sci. U.S.A. 89, 4163 (1992).
51. E. I. Shakhnovich and A.M. Gutin, Proc. Natl. Acad. Sci. U.S.A. 90, 7195 (1993).
52. V. S. Pande, A. Yu. Grosberg, and T. Tanaka, J. Chem. Phys. 101, 8246 (1994); Macromolecules 28, 2218 (1995); J. Chem. Phys. 103, 9482 (1995).
53. V. S. Pande, A. Yu. Grosberg, and T. Tanaka, J. Chem. Phys. 101, 8246 (1994); Macromolecules 28, 2218 (1995); J. Chem. Phys. 103, 9482 (1995).
54. V. S. Pande, A. Yu. Grosberg, and T. Tanaka, J. Chem. Phys. 101, 8246 (1994); Macromolecules 28, 2218 (1995); J. Chem. Phys. 103, 9482 (1995).
55. A. Dinner, A. Sali, M. Karplus, and E. Shakhnovich, J. Chem. Phys. 101, 1444 (1994).
56. P. E. Leopold, M. Montai, and J. Onuchic, Proc. Natl. Acad. Sci. U.S.A., 89, 9721 (1992); N. D. Socci and J. N. Onuchic, J. Chem. Phys. 103, 4732 (1995).
57. P. E. Leopold, M. Montai, and J. Onuchic, Proc. Natl. Acad. Sci. U.S.A., 89, 9721 (1992); N. D. Socci and J. N. Onuchic, J. Chem. Phys. 103, 4732 (1995).
58. E. I. Shakhnovich and A. M. Gutin, J. Phys. A 22, 1647 (1989); E. I. Shakhnovich and A. M. Gutin, J. Chem. Phys. 93, 5967 (1990).
59. E. I. Shakhnovich and A. M. Gutin, J. Phys. A 22, 1647 (1989); E. I. Shakhnovich and A. M. Gutin, J. Chem. Phys. 93, 5967 (1990).
60. E. G. Timoshenko and K. A. Dawson, Phys. Rev. E 51, 492 (1995).
61. E. G. Timoshenko, Yu. A. Kuznetsov, and K. A. Dawson, J. Chem. Phys. 102, 1816 (1995).
62. Yu. A. Kuznetsov, E. G. Timoshenko, and K. A. Dawson, J. Chem. Phys. 103, 4807 (1995).
63. A. Yu. Grosberg, Yu. A. Kuznetsov, E. G. Timoshenko, and K. A. Dawson (unpublished).
64. E. G. Timoshenko, Yu. A. Kuznetsov, and K.A. Dawson (unpublished).
65. Note that modest adaptations permit us to incorporate the hydrodynamic effect. We have illustrated the method for homopolymer kinetics in a previous work in Ref. [36].
66. 3. However, this is a delicate question that requires a separate study for heteropolymers.
67. m . This expresses the inversion symmetry of Ref. [35] and allows one to establish the correct correspondence of a ring with N units to an open chain with N/2 units (see Ref. [36] for more detail).
68. E. G. Timoshenko, Yu. A. Kuznetsov, and K.A. Dawson, Phys. Rev. E 53, 3886 (1996).
69. Yu. A. Kuznetsov, E. G. Timoshenko, and K. A. Dawson, J. Chem. Phys. 104, 3338 (1996).
70. 3. The latter is considerably smaller than the former for the whole range of considered parameters, justifying the use of the weak-disorder expansion.
71. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C (Cambridge University Press, Cambridge, 1992).
72. q(t) exhibit the characteristic growth exponents similar to the treatments of spinodal decomposition by Cahn et al., but for special q's, not for special 3d momenta k. That is why we call this phenomenon the spinodal decomposition in the internal metric of the chain. This is discussed in more detail in Refs. [28,36].
73. q similarly to our heteropolymer work [35]. Thus a change of Ψ accompanied by fast changes of large-q modes (small distances along the chain) corresponds to the microphase separation, while a change of Ψ together with the small-q modes describes the overall restructuring of the hydrophobic and hydrophilic units. One can have a better feeling about both processes by using the visualization of individual conformations in the Monte Carlo method [29].
74. H. Sompolinsky and A. Zippelius, Phys. Rev. Lett. 45, 359 (1981); Phys. Rev. B 25, 274 (1982).
75. H. Sompolinsky and A. Zippelius, Phys. Rev. Lett. 45, 359 (1981); Phys. Rev. B 25, 274 (1982).
76. W. Götze and L. Sjögren, Z. Phys. B 65, 415 (1987); J. Phys. C 21, 3407 (1988).
77. W. Götze and L. Sjögren, Z. Phys. B 65, 415 (1987); J. Phys. C 21, 3407 (1988).
78. A. Moskalenko and K. A. Dawson, J. Chem. Phys. 103, 9886 (1995).
79. A. Yu. Grosberg and A. R. Khokhlov, Statistical Physics of Macromolecules (AIP, New York, 1994).
80. 1 describes the mean distance between two nearest neighbors along the chain. From Fig. 8 one can see that in the folded state [curve (e)] this differs only in approximately 1.3 times compared to the ideal coil, for which that distance is equal to 1 in our units by definition.