Fractons in proteins: can they lead to anomalously decaying time-autocorrelations?

1. Fractons in proteins: can they lead to anomalously decaying time-autocorrelations? R. Granek, Dept. of Biotech. Eng., BGU J. Klafter, School of Chemistry, TAU

2. Outline • Single molecule experiments on proteins. • Fractal nature of proteins. Fractons – the vibrational normal modes of a fractal. • Time-autocorrelation function of the distance between two associated groups. • Conclusions

3. Single molecule techniques offer a possibility to follow real-time dynamics of individual molecules. • For some biological systems it is possible to probe the dynamics of conformational changes and follow reactivities. • Distributions rather than ensemble averages (adhesion forces, translocation times, reactivities)

4. Processes on the level of a single molecule • Dynamic Force Spectroscopy (DFS) of Adhesion Bonds • Translocation of ssDNA through a nanopore • Enzymatic activity(in collaboration with the groups of de Schryver and Nolte) • Protein vibrations

5. Dynamic Force Spectroscopy: Force(pN) Distance (nm) Force (pN) Distance (nm)

6. Processes: Mechanical response:

7. Maximal spring force: => F(V) ~ (lnV)2/3 as compared with

8. Single Stranded DNA translocation through a nanopore: One polymer at a time

9. Relevantsystems Individualmembranechannels: ionflux & biopolymerstranslocation A. Meller, L. Nivon, and D. Branton. Phys. Rev. Lett.86 (2001) J. J. Kasianowicz, E. Brandin, D. Branton and D. W. Deamer Proc. Natl. Acad. Sci. USA93 (1996) Translocationandconformationalfluctuation J. Li and H. A. Lester. Mol. Pharmacol. 55 (1999). O. Flomenbom and J. Klafter Biophys. J.86 (2004).

10. Lipase B From Candida Antarctica (CALB) Activity(The groups of de Schryver and Nolte) • The enzyme (CALB) is immobilized. • The substrate diffuses in the solution • During the experiment, a laser beamis focused on the enzyme, and the fluorescent state of a single enzyme is monitored. • The Michaelis-Menten reaction

11. Relevantsystems Chemicalactivity K. Velonia, etn al., Angew. Chem. (2005) O. Flomenbom, et al., PNAS (2005) L. Edman, & R. Rigler, Proc. Natl. Acad. Sci. U.S.A.,97 (2000) H. Lu, L. Xun, X. S. Xie, Science, 282 (1998)

12. Single molecule experiments in proteins:Fractons in proteins • Fluorescence resonant energy transfer (tens of angstroms). • Photo-induced electron transfer (a few angstroms) S. C. Kou and X. S. Xie, PRL (2004) W. Min et al., PRL (2005) R. Granek and J. Klafter, PRL (2005)

13. Autocorrelation function Small scale motion – VIBRATIONS?

14. Fluctuating Enzymes

15. Fractal nature of proteins. Mass fractality of proteins: Mass enclosed by concentric spheres of radius R centered at a backbone atom, in a single protein (1MZ5). Analysis covered over 200 proteins (!): M. B. Enright and D. M. Leitner, PRE (2005)

16. Manifold dimension D Linear polymers D=1 Membranes D=2 • Chemicallength – the length of the minimal path along the connecting springs. • Chemical length exponent • Or Flory exponent Manifold space Real space

17. Density of (eigen) states: – Spectral dimension Experiments (electron spin relaxation): for ~200-300 amino acids Computational studies involving ~60 proteins -> Molecular weight dependent : For ~100 amino acids For over 2000-3000 amino acids A. Vulpiani and coworkers (2002,2004)

18. Fractons mass Vibrations of the fractal Spring natural frequency displacement “name” of a point mass Normal modes (eigenmodes, eigenstates) – Fractons: Strongly localized eigenstates ! Yakubo and Nakayama (1989) S. Alexander and R. Orbach (1982)

19. Localization length in real space Localizationlength in manifold space Disorderaveraged eigenstate – Averaging over different realizations of the fractal, or over many localization centers:

20. Inequalities between the different broken dimensions: – Spectral dimension – Manifold dimension – Fractal dimension Remark: For folded proteins although the backbone is 1-dim. There are strong inter amino acid interactions, i.e. new “springs” connecting nearest-neighbor amino acids (in real space), even if they are distant along the backbone. Moreover, for the same reason we expect .

21. Landau-Peirels Instability – Amplitude of a normal mode Equipartitiontheorem Thermalfluctuations of the displacements ( ) – # of amino acids (“polymer index”) If , increases with increasing ! Large fluctuations may assist enzymatic/biological activity.

22. But: should not exceed the mean inter-amino acid distance, otherwise protein must unfold (or not fold). If evolution designed only folded proteins, should depend on . should approach the value of 2 for large proteins ! A. Vulpiani and coworkers (2002,2004)

23. Displacement difference time-autocorrelation function Two point masses, and . Positions in space and . Separation vector Equilibrium spacing Displacement difference vector Expansion in normal modes + disorder averaging

24. Two limits: 1) Undamped fractons (pure vibrations) The calculation involves a time-dependent propagation length If , motion of the two particles is uncorrelated. If , motion of the two particles is strongly correlated.

25. more precisely, for : numbers: Short-time exponent Long-time exponent

26. 2) Strongly overdamped fractons where is the friction. Therefore, the propagation length is

27. Conclusions • Novel approach for vibrations in folded proteins based on their fractal nature  Provides a description on a universal level, yet still microscopic in essence. • Slow power law decay of the autocorrelation function of the distance between two associated groups, even for pure vibrations. • In the case of pure vibrations, this powerlaw decay requires broken dimensions that obey the inequalities These inequalities do not hold for uniform lattices in all dimensions.