Link between truncated fractals and coupled oscillators in biological systems

Fractal dynamics of individual mitochondrial oscillators measure local inter-mitochondrial coupling

Mitochondrial inner membrane potentials in cardiomyocytes may oscillate in cycles of depolarization/repolarization when the mitochondrial network is exposed to metabolic or oxidative stress. The frequencies of such oscillations are dynamically changing while clusters of weakly coupled mitochondrial oscillators adjust to a common phase and frequency. Across the cardiac myocyte, the averaged signal of the mitochondrial population follows self-similar or fractal dynamics; however, fractal properties of individual mitochondrial oscillators have not yet been examined. We show that the largest synchronously oscillating cluster exhibits a fractal dimension, D, that is indicative of self-similar behavior with D=1.27±0.11, in contrast to the remaining network mitochondria whose fractal dimension is close to that of Brownian noise, D=1.58±0.10. We further demonstrate that fractal behavior is correlated with local coupling mechanisms, whereas it is only weakly linked to measures of functional connections between mitochondria. Our findings suggest that individual mitochondrial fractal dimensions may serve as a simple measure of local mitochondrial coupling.

How cytoskeletal proteins regulate mitochondrial energetics in cell physiology and diseases

Mitochondria are ubiquitous organelles that play a pivotal role in the supply of energy through the production of adenosine triphosphate in all eukaryotic cells. The importance of mitochondria in cells is demonstrated in the poor survival outcomes observed in patients with defects in mitochondrial gene or RNA expression. Studies have identified that mitochondria are influenced by the cell's cytoskeletal environment. This is evident in pathological conditions such as cardiomyopathy where the cytoskeleton is in disarray and leads to alterations in mitochondrial oxygen consumption and electron transport. In cancer, reorganization of the actin cytoskeleton is critical for trans-differentiation of epithelial-like cells into motile mesenchymal-like cells that promotes cancer progression. The cytoskeleton is critical to the shape and elongation of neurons, facilitating communication during development and nerve signalling. Although it is recognized that cytoskeletal proteins physically tether mitochondria, it is not well understood how cytoskeletal proteins alter mitochondrial function. Since end-stage disease frequently involves poor energy production, understanding the role of the cytoskeleton in the progression of chronic pathology may enable the development of therapeutics to improve energy production and consumption and slow disease progression.

This article is part of the theme issue ‘The cardiomyocyte: new revelations on the interplay between architecture and function in growth, health, and disease’.

High-dimensional dynamics in a single-transistor oscillator containing Feynman-Sierpiński resonators: Effect of fractal depth and irregularity

Fractal structures pervade nature and are receiving increasing engineering attention towards the realization of broadband resonators and antennas. We show that fractal resonators can support the emergence of high-dimensional chaotic dynamics even in the context of an elementary, single-transistor oscillator circuit. Sierpiński gaskets of variable depth are constructed using discrete capacitors and inductors, whose values are scaled according to a simple sequence. It is found that in regular fractals of this kind, each iteration effectively adds a conjugate pole/zero pair, yielding gradually more complex and broader frequency responses, which can also be implemented as much smaller Foster equivalent networks. The resonators are instanced in the circuit as one-port devices, replacing the inductors found in the initial version of the oscillator. By means of a highly simplified numerical model, it is shown that increasing the fractal depth elevates the dimension of the chaotic dynamics, leading to high-order hyperchaos. This result is overall confirmed by SPICE simulations and experiments, which however also reveal that the non-ideal behavior of physical components hinders obtaining high-dimensional dynamics. The issue could be practically mitigated by building the Foster equivalent networks rather than the verbatim fractals. Furthermore, it is shown that considerably more complex resonances, and consequently richer dynamics, can be obtained by rendering the fractal resonators irregular through reshuffling the inductors, or even by inserting a limited number of focal imperfections. The present results draw attention to the potential usefulness of fractal resonators for generating high-dimensional chaotic dynamics, and underline the importance of irregularities and component non-idealities.

Fractal aggregation of porcine fumarase induced by free radicals

The present study demonstrates that H(2)O(2) and OH(.-) cause fibril aggregation and catalytic inactivation of porcine fumarase. In the aggregated (oxidized) enzyme, modifications in both secondary and tertiary protein structure occur and the enzyme aggregation obeys to fractal geometry. We then collected information on the fractal dimension and on the size and shape of fumarase aggregates by using Synchrotron Radiation (SR) Small Angle X-ray Scattering (SAXS) analysis. The geometrical self-similarity assessment of aggregates has been revealed by both AFM and SEM measurements at different scale of magnification. Micrographs collected remarkably demonstrate that the oxidized enzyme shows dendritic fractal structure over a large range of sizes.