Theo F. Nonnenmacher

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.

Relaxation in filled polymers: A fractional calculus approach

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.

Generalized viscoelastic models: their fractional equations with solutions

Recently fractional calculus (FC) has encountered much success in the description of complex dynamics. In particular FC has proved to be a valuable tool to handle viscoelastic aspects. In this paper we construct fractional rheological constitutive equations on the basis of well known mechanical models, especially the Maxwell, the Kelvin-Voigt, the Zener and the Poynting-Thomson model. To this end we introduce a fractional element, in addition to the standard purely elastic and purely viscous elements. As we proceed to show, many of the fractional differential equations which we obtain by this construction method admit closed form, analytical solutions in terms of Fox H-functions of the Minag-Leffler type.

Fox Function Representation of Non-Debye Relaxation Processes

Applying the Liouville-Riemann fractional calculus, we derive and solve a fractional operator relaxation equation. We demonstrate how the exponentΒ of the asymptotic power law decay ∼t−β relates to the orderΝ of the fractional operatordv/dtv (0<Ν<1). Continuous-time random walk (CTRW) models offer a physical interpretation of fractional order equations, and thus we point out a connection between a special type of CTRW and our fractional relaxation model. Exact analytical solutions of the fractional relaxation equation are obtained in terms of Fox functions by using Laplace and Mellin transforms. Apart from fractional relaxation, Fox functions are further used to calculate Fourier integrals of Kohlrausch-Williams-Watts type relaxation functions. Because of its close connection to integral transforms, the rich class of Fox functions forms a suitable framework for discussing slow relaxation phenomena.

Space- and time-fractional diffusion and wave equations, fractional Fokker–Planck equations, and physical motivation

Abstract We investigate the physical background and implications of a space- and time-fractional diffusion equation which corresponds to a random walker which combines competing long waiting times and Levy flight properties. Explicit solutions are examined, and the corresponding fractional Fokker–Planck–Smoluchowski equation is presented. The framework of fractional kinetic equations which control the systems relaxation to either Boltzmann–Gibbs equilibrium, or a far from equilibrium Levy form is explored, putting the fractional approach in some perspective from the standard non-equilibrium dynamics point of view, and its generalisation.

Fractals in Biology and Medicine

Preface: Summary of the Symposium.- The Significance of Fractals for Biology and Medicine. An Introduction and Summary.- Fractal Geometry and Biomedical Sciences7.- A Fractals Lacunarity, and how it can be Tuned and Measured.- Spatial and Temporal Fractal Patterns in Cell and Molecular Biology.- Chaos, Noise and Biological Data.- Fractal Landscapes in Physiology & Medicine: Long-Range Correlations in DNA Sequences and Heart Rate Intervals.- Fractals in Biological Design and Morphogenesis.- Design of Biological Organisms and Fractal Geometry.- Fractal and Non-Fractal Growth of Biological Cell Systems.- Evolutionary Meaning, Functions and Morphogenesis of Branching Structures in Biology.- Relationship Between the Branching Pattern of Airways and the Spatial Arrangement of Pulmonary Acini - A Re-Examination from a Fractal Point of View.- Multivariate Characterization of Blood Vessel Morphogenesis in the Avian Chorioallantoic Membrane (CAM): Cell Proliferation, Length Density and Fractal Dimension.- Phyllotaxis or Self-Similarity in Plant Morphogenesis.- Fractals in Molecular and Cell Biology.- Evolutionary Interplay Between Spontaneous Mutation and Selection: Aleatoric Contributions of Molecular Reaction Mechanisms.- Error Propagation Theory of Chemically Solid Phase Synthesized Oligonucleotides and DNA Sequences for Biomedical Application.- Fractional Relaxation Equations for Protein Dynamics.- Measuring Fractal Dimensions of Cell Contours: Practical Approaches and their Limitations.- Fractal Properties of Pericellular Membrane from Lymphocytes and Leukemic Cells.- Cellular Sociology: Parametrization of Spatial Relationships Based on Voronoi Diagram and Ulam Trees.- A Fractal Analysis of Morphological Differentiation of Spinal Cord Neurons in Cell Culture.- Fractal Dimensions and Dendritic Branching of Neurons in the Somatosensory Thalamus.- Fractal Structure and Metabolic Functions.- Organisms as Open Systems.- Transfer to and across Irregular Membranes Modelled by Fractal Geometry.- Scaling and Active Surface of Fractal Membranes.- Structure Formation in Excitable Media.- Colony Morphology of the Fungus Aspergillus Oryzae.- Estimation of the Correlation Dimension of All-Night Sleep EEG Data with a Personal Super Computer.- Fractals in Pathology.- Changes in Fractal Dimension of Trabecular Bone in Osteoporosis: A Preliminary Study.- Use of the Fractal Dimension to Characterize the Structure of Cancellous Bone in Radiographs of the Proximal Femur.- Distribution of Local-Connected Fractal Dimension and the Degree of Liver Fattiness from Ultrasound.- Fractal Dimension as a Characterisation Parameter of Premalignant and Malignant Epithelial Lesions of the Floor of the Mouth.- Modelling.- Modelling HIV/AIDS Dynamics.- Morphological Diagnosis Turns from Gestalt to Geometry.- Fluorescence Recovery after Photobleaching Studied by Total Internal Reflection Microscopy: An Experimental System for Studies on Living Cells in Culture.- Anomalous Diffusion and Angle-Dependency in the Theory of Fluorescence Recovery after Photobleaching.- List of Speakers.- List of Participants.

ON THE RIEMANN-LIOUVILLE FRACTIONAL CALCULUS AND SOME RECENT APPLICATIONS

When Benoit Mandelbrot discussed the problem of fractional Brownian motion in his classic book The Fractal Geometry of Nature, he already pointed out some strong relations to the Riemann-Liouville fractional integral and differential calculus. Over the last decade several papers have appeared in which integer-order, standard differential equations modeling processes of relaxation, oscillation, diffusion and wave propagation are generalized to fractional order differential equations. The basic idea behind all that is that the order of differentiation need not be an integer but a fractional number (i.e. dq/dtq with 0<q<1). Applications to slow relaxation processes in complex systems like polymers or even biological tissue and to self-similar protein dynamics will be discussed. In addition, we investigate a fractional diffusion equation and we present the corresponding probability density function for the location of a random walker on a fractal object. Fox-functions play a dominant part.

Fractional relaxation and the time-temperature superposition principle

Relaxation processes in complex systems like polymers or other viscoelastic materials can be described by equations containing fractional differential or integral operators. In order to give a physical motivation for fractional order equations, the fractional relaxation is discussed in the framework of statistical mechanics. We show that fractional relaxation represents a special type of a non-Markovian process. Assuming a separation condition and the validity of the thermo-rheological principle, stating that a change of the temperature only influences the time scale but not the rheological functional form, it is shown that a fractional operator equation for the underlying relaxation process results.

Fractional relaxation processes and fractional rheological models for the description of a class of viscoelastic materials

Abstract Following the modelling of Zener, we establish a connection between the fractional Fokker-Planck equation and the anomalous relaxation dynamics of a class of viscoelastic materials which exhibit scale-free memory. On the basis of fractional relaxation, generalisations of the classical rheological model analogues are introduced, and applications to stress–strain relaxation in filled and unfilled polymeric materials are discussed. A possible generalisation of Reiners Deborah number is proposed for systems which exhibit a diverging characteristic relaxation time.

Anomalous diffusion of water in biological tissues

This article deals with the characterization of biological tissues and their pathological alterations. For this purpose, diffusion is measured by NMR in the fringe field of a large superconductor with a field gradient of 50 T/m, which is rather homogenous and stable. It is due to the unprecedented properties of the gradient that we are able not only to determine the usual diffusion coefficient, but also to observe the pronounced Non-Debye feature of the relaxation function due to cellular structure. The dynamics of the probability density follow a stretched exponential or Kohlrausch-Williams-Watts function. In the long time limit the Fourier transform of the probability density follows a long-tail Lévy function, whose asymptotic is related to the fractal dimension of the underlying cellular structure. Some of the properties of Lévy walk statistics are discussed and its potential importance in understanding certain biophysical phenomena like diffusion processes in biological tissues are pointed out. We present and discuss for the first time NMR data giving evidence for Lévy processes that capture the essential features of the observed power law (scaling) dynamics of water diffusion in fresh tissue specimens: carcinomas, fibrous mastopathies, adipose and liver tissues.