Microtubules (MTs) are self-assembled hollow protein tubes playing important functions in live cells. Their building block is a protein called tubulin, which self-assembles in a particulate 2 dimensional lattice. We study the vibrational modes of this lattice and find Dirac points in the phonon spectrum. We discuss a splitting of the Dirac points that leads to phonon bands with nonzero Chern numbers, signaling the existence of topological vibrational modes localized at MTs edges, which we indeed observe after explicit calculations. Since these modes are robust against the large changes occurring at the edges during the dynamic cycle of the MTs, we can build a simple mechanical model to illustrate how they would participate in this phenomenon.

The research in topological materials and meta-materials has reached maturity and is gradually entering the phase of practical applications and devices. However, scaling down experimental demonstrations presents a major challenge. In this work, we study identical coupled mechanical resonators whose collective dynamics are fully determined by the pattern in which they are arranged. We call a pattern topological if boundary resonant modes fully fill all existing spectral gaps whenever the pattern is halved. This is a characteristic of the pattern and is entirely independent of the structure of the resonators and the details of the couplings. The existence of such patterns is proven using Ktheory and exemplified using a novel experimental platform based on magnetically coupled spinners. Topological meta-materials built on these principles can be easily engineered at any scale, providing a practical platform for applications and devices.

We develop a theoretical framework to describe the dielectric response of live cells in suspensions when placed in low external electric fields. The treatment takes into account the presence of the cell's membrane and of the charge movement at the membrane's surfaces. For spherical cells suspended in aqueous solutions, we give an analytic solution for the dielectric function, which is shown to account for the alpha- and beta-plateaus seen in many experimental data. The effect of different physical parameters on the dielectric curves is methodically analyzed.

The emergence of a fractal energy spectrum is the quintessence of the interplay between two periodic parameters with incommensurate length scales. crystals can emulate such interplay and also exhibit a topological bulk-boundary correspondence, enabled by their nontrivial topology in virtual dimensions. Here we propose, fabricate and experimentally test a reconfigurable one-dimensional (1D) acoustic array, in which the resonant frequencies of each element can be independently fine-tuned by a piston. We map experimentally the full Hofstadter butterfly spectrum by measuring the acoustic density of states distributed over frequency while varying the long-range order of the array. Furthermore, by adiabatically changing the phason of the array, we map topologically protected fractal boundary states, which are shown to be pumped from one edge to the other. This reconfigurable crystal serves as a model for future extensions to electronics, photonics and mechanics, as well as to quasicrystalline systems in higher dimensions.

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