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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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In recent years the fractional calculus approach to describing dynamic processes in disordered or complex systems such as relaxation or dielectric behavior in polymers or photo bleaching recovery in biologic membranes has proved to be an extraordinarily successful tool. In this paper we apply fractional relaxation to filled polymer networks and investigate the dependence of the decisive occurring parameters on the filler content. As a result, the dynamics of such complex systems may be well–described by our fractional model whereby the parameters agree with known phenomenological models.

Anomalous diffusion in which the mean square distance between diffusing quantities increases faster than linearly in ''time'' has been observed in all manner of physical and biological systems from macroscopic surface growth to DNA sequences. Herein we relate the cause of this nondiffusive behavior to the statistical properties of an underlying process using an exact statistical model. This model is a simple two-state process with long-time correlations and is shown to produce a random walk described by an exact fractional diffusion equation. Fractional diffusion equations describe anomalous transport and are shown to have exact solutions in terms of Fox functions, including Lévy ␣-stable processes in the superdiffusive domain (1/2ϽHϽ1).