# Mathematical bioPhysics

The group pursues a mathematical physics-approach to study phenomena in biophysics.

Our main research focus is the **non-equilibrium statistical mechanics of single molecules/particles and the collective behavior of larger molecular and up to sub-cellular assemblies**. In particular, we aim at a **trajectory-based description of macromolecular conformation dynamics** as well as of **their spatial transport, binding, and reactions**. From a fundamental perspective we are focusing on **relaxation phenomena of Markovian and non-Markovian observables from far-from-equilibrium quenches **as well as the **generic physical origin and understanding of broken time-translation invariance**. In the analysis of **stochastic many-body systems** we aim at an **understanding of emerging collective effects** **from a trajectory perspective**. In our work we employ rigorous analysis corroborated by computer simulations. Please see **Recent research activities** below for more details.

**Recent research activities**

## Emergent memory and kinetic hysteresis in strongly driven networks

In elastic materials, friction is known to cause a phenomenon called hysteresis – more energy is required to stretch a rubber band than is released during unloading. As a result, the rubber band becomes warm. On microscopic scales where thermal fluctuations are important, such as in molecular motors operating far from equilibrium, hysteresis emerges in the form of a broken time-reversal symmetry. In our paper we report on a novel form of hysteresis that is of a purely kinetic nature and emerges as soon as a system does not locally equilibrate in metastable states. This gives rise to memory in the observed dynamics; that is, state changes depend on past states.

Kinetic hysteresis arises from the finite duration of transition paths, which are completed transitions between pairs of metastable states. We derive, for the first time, a network theory that preserves the microscopic dissipation and accounts for the finite duration of transition paths. We unravel three ingrained symmetries of the dynamics that determine the characteristics of the observed memory. These symmetries reveal hidden, vital information about the dynamics even in the limit where the hysteresis may become negligibly small. We use the theory to explain the counterintuitive “catch-bond” phenomenon, where a larger mechanical load prolongs the lifetime of an adhesion bond.

Our results pave the way toward a deeper understanding of violations of time-reversal symmetry in the presence of memory. In practice, the symmetries unraveled in our work allow for robustly detecting the coexistence of multiple transition pathways in a measured time series without resolving them individually.

## Criticality in cell adhesion

Cells, the fundamental building blocks of living organisms, stick to other cells via a process called cell adhesion. For multicellular organisms cell adhesion plays a crucial role in the immune response, wound healing, and cancer development. Cell adhesion is mediated by so-called adhesion bonds. The breaking and formation of individual adhesion bonds is coupled via thermal fluctuations of the anchoring cell membrane. The last two decades of experiments and theory have shown that cell adhesion may in fact be accurately described by a model originally developed to describe ferromagnetism: the Ising model.

In our work we generalize the Ising model in order to explain how membrane fluctuations and external forces on the cell induce many-body effects, and how these in turn affect the equilibrium behavior and dynamical properties of cell adhesion. Strikingly, changes in the membrane rigidity can prolong or shorten the mean time a cell adheres to a stiff substrate, which in the thermodynamic limit of many adhesion bonds leads to the notion of a novel kind of dynamical critical point. The existence of a dynamical critical point is equivalently implied in the context of magnetization reversal times in ferromagnetic materials.

**11**, 031067 (2021)

## Thermodynamic limits to anomalous diffusion

Thermal, Brownian fluctuations in physical systems were found to be bounded by a universal physical relation called the "thermodynamic uncertainty relation". Concurrently, experiments tracking individual particles or molecules frequently show fluctuations that deviate strongly from the Gaussian laws of Brownian motion – a phenomenon commonly referred to as "anomalous diffusion".

In our work we show, for the first time, that the two seemingly disjoint if not orthogonal fields of research – stochastic thermodynamics and anomalous diffusion – may be fruitfully combined to derive a new universal physical bound on the experimentally often unknown temporal extent of anomalous diffusion.

The paper was covered in a **Viewpoint** in **Physics**: https://physics.aps.org/articles/v14/116

## Toolbox for quantifying memory in dynamics along reaction coordinates

Memory effects in time-series of experimental observables are ubiquitous and have important consequences for the interpretation of kinetic data. Under given circumstances they may even affect the function of biomolecular nanomachines such as enzymes.

We propose a set of complementary model-free methods for quantifying conclusively the magnitude and duration of memory in a time series of a reaction coordinate. The toolbox is general, robust, easy to use, and does not rely on any underlying microscopic model.

## Signatures of memory in the barrier-crossing of a tagged-particle in a single file

We investigate memory effects in barrier-crossing in the overdamped setting. We focus on the scenario where the hidden degrees of freedom relax on exactly the same time scale as the observable. As a prototypical model, we analyze tagged-particle diffusion in a single file confined to a bi-stable potential. We identify the signatures of memory and explain their origin. The emerging memory is a result of the projection of collective many-body eigenmodes onto the motion of a tagged-particle. Notably and somewhat unexpectedly, at a fixed particle number, we find that the higher the barrier, the stronger the memory effects are. The fact that the external potential alters the memory is important more generally and should be taken into account in applications of generalized Langevin equations. Our results can readily be tested experimentally and may be relevant for understanding transport in biological ion-channels.

## Introducing: Time-average statistical mechanics

Many experiments on soft and biological matter probe individual trajectories. It is typically not feasible to repeat these experiments sufficiently many times in to apply the traditional concepts of (ensemble) statistical mechanics. It is, however, straightforward to analyze such data by means of time-averaging along individual realizations. However, a correct rationalization and interpretation of such time-averaged results requires the framework of “time-average statistical mechanics”.

In our work we develop a spectral-theoretic approach to describe fluctuations of time-average observables evolving from general (incl. non-equilibrium) initial conditions, and consider both, reversible and irreversible (i.e. driven) dynamics. Our results are directly applicable to a diverse range of phenomena underpinned by time-average observables in physical, chemical, biological systems, such as single-particle tracking and single-molecule spectroscopy, and may also find important applications in econophysics.

## An unforeseen asymmetry in relaxation to equilibrium: Nanoscale warming is faster than cooling

According to elementary physics cooling and warming rates should be identical if conditions are the same. A thermodynamic system generally evolves, or “relaxes,” to minimize its free energy, and if the total free energy difference between the initial and final conditions is the same in both cases, warming and cooling should be equally fast. As a result of a subtle imbalance in how the probability distribution of any system evolves under conditions of warming or cooling we find, however, that relaxation happens faster “uphill” than “downhill”.

We prove that near stable minima and for all quadratic energy landscapes it is a general phenomenon that also exists in a class of non-Markovian observables probed in single-molecule and particle-tracking experiments. The asymmetry is a general feature of reversible overdamped diffusive systems with smooth single-well potentials and occurs in multiwell landscapes when quenches disturb predominantly intrawell equilibria. Our findings may be relevant for the optimization of stochastic heat engines.

The paper was covered in **Physics Focus**: https://physics.aps.org/articles/v13/144