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Biophys J. 1995 Jan; 68(1): 46–53.
PMCID: PMC1281659

A fractional calculus approach to self-similar protein dynamics.


Relaxation processes and reaction kinetics of proteins deviate from exponential behavior because of their large amount of conformational substrates. The dynamics are governed by many time scales and, therefore, the decay of the relaxation function or reactant concentration is slower than exponential. Applying the idea of self-similar dynamics, we derive a fractal scaling model that results in an equation in which the time derivative is replaced by a differentiation (d/dt)beta of non-integer order beta. The fractional order differential equation is solved by a Mittag-Leffler function. It depends on two parameters, a fundamental time scale tau 0 and a fractional order beta that can be interpreted as a self-similarity dimension of the dynamics. Application of the fractal model to ligand rebinding and pressure release measurements of myoglobin is demonstrated, and the connection of the model to considerations of energy barrier height distributions is shown.

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Selected References

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  • Austin RH, Beeson KW, Eisenstein L, Frauenfelder H, Gunsalus IC. Dynamics of ligand binding to myoglobin. Biochemistry. 1975 Dec 2;14(24):5355–5373. [PubMed]
  • Frauenfelder H. Function and dynamics of myoglobin. Ann N Y Acad Sci. 1987;504:151–167. [PubMed]
  • Frauenfelder H, Parak F, Young RD. Conformational substates in proteins. Annu Rev Biophys Biophys Chem. 1988;17:451–479. [PubMed]
  • Goldberger AL, Bhargava V, West BJ, Mandell AJ. On a mechanism of cardiac electrical stability. The fractal hypothesis. Biophys J. 1985 Sep;48(3):525–528. [PMC free article] [PubMed]
  • Goldberger AL, Bhargava V. Comments on 1/f power spectrum of the QRS complex revisited. Biophys J. 1991 Nov;60(5):1301–1302. [PMC free article] [PubMed]
  • Iben IE, Braunstein D, Doster W, Frauenfelder H, Hong MK, Johnson JB, Luck S, Ormos P, Schulte A, Steinbach PJ, et al. Glassy behavior of a protein. Phys Rev Lett. 1989 Apr 17;62(16):1916–1919. [PubMed]
  • Millhauser GL. Reptation theory of ion channel gating. Biophys J. 1990 Apr;57(4):857–864. [PMC free article] [PubMed]
  • Montroll EW, Shlesinger MF. On 1/f noise and other distributions with long tails. Proc Natl Acad Sci U S A. 1982 May;79(10):3380–3383. [PMC free article] [PubMed]
  • Rabinovich S, Agmon N. Scaling and critical-like behavior in multidimensional diffusive dynamics. Phys Rev E Stat Phys Plasmas Fluids Relat Interdiscip Topics. 1993 May;47(5):3717–3720. [PubMed]
  • Straub JE, Thirumalai D. Exploring the energy landscape in proteins. Proc Natl Acad Sci U S A. 1993 Feb 1;90(3):809–813. [PMC free article] [PubMed]

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