Talks

Cilia and flagella are micron-sized slender filaments that actively beat in a viscous medium with remarkable accuracy despite thermal fluctuations and other uncertainties. Such precise beating is essential for swift locomotion for microorganisms and for generating an efficient flow in a carpet of cilia in fluid media. To understand the role of the interplay between dissipation and cilia activity in achieving such a precise oscillation, we study a minimal model of cilia known as the rower model. Here, the complex beating of a filament is simplified by a one-dimensional periodic motion of a micron-sized bead between two positions (the amplitude) immersed in a viscous fluid. The bead performs Brownian motion in one of the two harmonic potentials and switches to the other once it reaches two specific positions with a pump of energy which is a measure of cilia activity. We quantify the precision using the quality factor and find a scaling law for the precision with activity and dissipation. Interestingly, for an optimal amplitude where the precision becomes maximum. The scaling and optimal behavior in the quality factor can be explained by studying the noise in the first passage time. Finally, we discuss the energy budget in achieving precision.

The diversity of diffusive systems exhibiting long-range correlations characterized by a stochastically varying Hurst exponent calls for a generic multifractional model. In this talk I will present a simple, analytically tractable model which fills the gap between mathematical formulations of multifractional Brownian motion and empirical studies. In the model, called telegraphic multifractional Brownian motion (TeMBM), the Hurst exponent is modelled by a smoothed telegraph process which results in a stationary beta distribution of exponents as observed in biological experiments. I will also discuss a methodology to identify TeMBM in experimental data and present concrete examples from biology, climate and finance to demonstrate the efficacy of the presented approach.

Mathlib is a vast and constantly growing library of formalized mathematics. As its size increases, it becomes increasingly easy to spend a significant amount of time formalizing a theorem, only to later discover that it was already present in the library. This can be both frustrating and discouraging. In this talk, we will introduce and demonstrate a variety of tools and techniques that can help users efficiently navigate Mathlib, search for existing results, and better understand the structure of the library.

Proteins are biopolymers, composed of repeating sequence of amino acids (AA). In a typical sequence, the constituting AAs have different charges, hydrophobicity, and capacities to form directional and non-directional interactions. Such heterogeneity can results in sequences lacking a stable three dimensional structure. This class of proteins are intrinsically disordered protein (IDPs). A deeper understanding of IDPs require appropriate characterization of the conformations. To this end, scattering and single molecular spectroscopic measurements often assign a single Flory exponent (equivalently fractal dimension) to the IDPs. In this talk, I highlight limitation of this method by enhanced sampling of atomistic resolution conformations of disordered \beta-casein. I will show that the underlying energy landscape of the IDP contains a global minimum along with two shallow funnels. Employing static polymeric scaling laws separately for individual funnels, we find that they cannot be described by the same polymeric scaling exponent. Around the global minimum, the conformations are globular, whereas in the vicinity of local minima, we recover coil-like scaling. To elucidate the implications of structural diversity on equilibrium dynamics, we initiated standard MD simulations in the NVT ensemble with representative conformations from each funnel. Global and internal motions for different classes of trajectories show heterogeneous dynamics with globule to coil-like signatures. Thus, IDPs can behave as entirely different polymers in different regions of the conformational space.

Run-and-tumble (RT) motion is commonly observed in flagellated microswimmers, arising from synchronous and asynchronous flagellar beating. In addition to hydrodynamic interactions, mechanical coupling has recently been recognized to play a key role in flagellar synchronization. To explore this, we design a macroscopic model system that comprises dry, self-propelled robots linked by a rigid rod to model a biflagellated microorganism. To mimic a low Reynolds number environment, we program each robot to undergo overdamped active Brownian (AB) motion. We find that such a system exhibits RT-like behavior, characterized by sharp tumbles and exponentially distributed run times, consistent with real microswimmers. We quantify tumbling frequency and demonstrate its tunability across experimental parameters. Additionally, we provide a theoretical model that reproduces our results, elucidating physical mechanisms governing RT dynamics.

Mathematicians have trained themselves to see objects from many points of view. When considering the real numbers, we can as easily see them as equivalence classes of Cauchy sequences, or Dedekind cuts, or the unique uniformly complete Archimedean field. The computer is not as forgiving and forces us to pick a particular definition that we must stick with. Using examples from the Lean mathematical library, Mathlib, I investigate why it is so important to choose the right definition when formalizing, what makes formal definitions look different from pen and paper definitions, and how we can design our definitions to make proofs flow smoothly.