# 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.

## Interested in joining us? Please see Open Positions for available positions.

**Recent research activities**

*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

*BetheSF: Avoiding permutations in the computation of the exact tagged-particle propagator in single-file systems*

We present an algorithm for computing the non-Markovian time-dependent conditional probability density function of a tagged-particle in a single-file of particles diffusing in a confining external potential. The algorithm implements an eigenfunction expansion of the full interacting many-body problem obtained by means of the coordinate Bethe ansatz.* *

The Bethe eigenspectrum is exact bit involves the generation and evaluation of permutations, which becomes unfeasible for single-files with an increasing number of particles . We exploit the underlying exchange symmetries between the particles to the left and to the right of the tagged-particle and thereby reduce the complexity of the algorithm from the worst case scenario O(N!) down to O(N). A C++ code is provided and solutions for selected simple model potentials are readily implemented. The program allows for implementations of arbitrary potentials under the condition that the user can supply solutions to the respective single-particle eigenspectra.

*Manifestations of Projection-Induced Memory: Rigorous Results and the Tilted Single File*

Non-Markovian stochastic dynamics and anomalous diffusion evolved to mainstream theory, which transgressed the realms of physics to chemistry, biology and ecology. Numerous phenomenological approaches emerged, which can more or less successfully reproduce or account for experimental observations. However, as far as their predictions are concerned these approaches are not unique, often build on conceptually orthogonal ideas, and are typically employed on an *ad-hoc* basis. It is thus timely and necessary to establish a systematic, mathematically unifying and clean approach starting from more fine-grained principles. In our work we analyze projection-induced ergodic non-Markovian dynamics, both reversible as well as irreversible, using spectral theory. We investigate dynamical correlations between histories of projected and latent observables that give rise to memory in projected dynamics, and rigorously establish conditions under which projected dynamics is Markovian or renewal. A metric is proposed for quantifying the degree of non-Markovianity. As a simple example we study single file diffusion in a tilted box, which, for the first time, we solve exactly using the coordinate Bethe ansatz. Our results provide a rigorous foundation for a deeper and more systematic analysis of projection-induced non-Markovian dynamics and anomalous diffusion.

*Extreme value **statistics** of diffusion in free energy landscapes – **served** **with a ‘sandwich-bound’ on top *

Extreme value functionals of stochastic processes are inverse functionals of the first passage time. In our work we establish a framework for analyzing extreme value statistics of ergodic reversible Markov processes in confining potentials on the hand of relaxation eigenspectra. We derived a chain of inequalities – which we coined a ‘sandwich bound’ -- bounding the long-time asymptotics of first passage densities, and thereby extrema, from above and from below. These bounds involve only a time integral of the return probability. As an illustration we analyzed extreme value statistics in the case of an Ornstein–Uhlenbeck process and a 3D Brownian motion confined to a sphere, also known as Bessel process. Our work provides a novel perspective on the statistics of extrema beyond the established limit theorems for sequences of independent random variables and for asymmetric diffusion processes beyond a constant drift.

*Duality between relaxation and first passage processes: a spectral theory of reaction kinetics in the few-encounter limit*

Relaxation and first passage processes are the pillars of theoretical descriptions of kinetics in condensed matter, polymeric and single-molecule systems. Yet, an explicit connection between relaxation and first passage time-scales so far remained elusive. In contrast to first passage processes, relaxation processes are not terminated upon reaching a certain threshold, but instead approach an equilibrium Boltzmann distribution if the underlying dynamics is ergodic and microscopically reversible.

In our work we prove a duality between them in the form of an interlacing of the respective spectra. In the basic form the duality holds for reversible Markov processes to effectively one-dimensional targets as well as for all reversible discrete Markov jump processes with arbitrary state-space. Based on the spectral interlacing we derive the full first passage time distribution from the corresponding relaxation eigenspectrum in form of a Newton’s series of almost triangular matrices. The exploration of a triple-well potential and a rugged energy landscape is analyzed to demonstrate how the duality allows for an intuitive understanding of first passage trajectories in terms of relaxational eigenmodes.

Going beyond single-molecule kinetics we use the duality to investigate many-particle first passage problems in the few-encounter limit, where a response is triggered by a few reactive events, e.g. in nucleation-limited phenomena and the misfolding-triggered pathological protein aggregation.

We show that even in the presence of a time-scale separation few-encounter kinetics require the entire first passage time distribution of individual particles. Moreover, we demonstrate that a robust increase of both speed and precision is in fact inherent to kinetics in few-encounter limit.

*Projection-induced memory and anomalous diffusion from a trajectory perspective*

We recently obtained rigorous results for fluctuations and two-tag correlations of general bounded additive functionals of ergodic Markov processes with a diagonalizable propagator. They relate the statistics of functionals on arbitrary time-scales to the relaxation eigenspectrum. As a first example we studied tagged particle local times – the time a tracer particle spends at some predefined location along a single trajectory up to a time t. Exact results were derived for one- and two-tag local times, which revealed how the individual particles’ histories become correlated at higher densities because each consecutive displacement along a trajectory requires collective rearrangements. Our results unveil the intricate meaning of projection-induced memory on a trajectory level, invisible to ensemble-average observables, and allow for a detailed analysis of single-file experiments probing tagged particle exploration statistics.

*First passage time theory: trajectory-to-trajectory fluctuations in biochemical reactions at low-copy numbers*

The *first passage time*, the time needed for some stochastic observable to reach a given preset value for the first time along a given trajectory, is central to our understanding kinetics across many disciplines, such as diffusion-controlled chemical reactions, biological signaling cascades, transport in disordered media, bacteria and animals foraging for food, up to the global spreading of diseases and stock market dynamics. Modern single-molecule tracking and imaging methods nowadays allow us to probe single binding and reaction events in living cells. These experiments reveal striking sample-to-sample fluctuations in binding/encounter times.** **It is therefore **timely and important to extend the traditional mean-rate paradigm in (bio)chemical reaction kinetics to the full distribution of first passage time.** In this spirit we recently obtained rigorous results on the **first passage time universality class** for dynamics in finite systems, and explained a universal proximity effect in the context of so-called **kinetics in the few-encounter limit**, which provides a simple explanation for the observed robustness of both the **speed and precision of transcription regulation in biological cells** [Godec 2016a]. In our work we also explained the importance of a broken translational symmetry of the medium [Godec 2016b], of the presence of disorder [Godec 2015] and of the existence of multiple transport channels [Godec 2017] on the statistics of first passage time.

*Physical limits to biochemical signaling coupled to active molecular motor-mediated transport*

We developed a linear response theory quantifying the fundamental mean field precision limit for receptor signaling coupled to active molecular motor-mediated intracellular transport. In the model we couple the so-called intermittent search model with reversible Markovian binding to the receptor. Our results demonstrate the existence of **active signal focusing** enabling a faster and more precise delivery of the molecular cue to the corresponding receptor site [Godec 2015]. In addition, we explain the **unconditional improvement of the signaling precision in effectively one-dimensional structures** such as neuronal cells in terms of **breaking of the recurrence** in the motion of the signaling molecule [Godec 2016].