**Project Heads**

*Felix Höfling, Carsten Hartmann*

**Project Members**

Arthur Straube (FU, from 07/19 to 12/19), Upanshu Sharma (FU, from 10/19 to 12/20)

**Project Duration**

01.01.2019 – 31.12.2021

**Located at**

FU Berlin

Diffusion in cellular environments, consisting of a variety of interacting entities, is a multiscale process. The project’s objective is to infer effective stochastic models and to quantify memory, using novel stochastic modelling and data assimilation techniques, based on data from experiments and simulations.

Macromolecular crowding is characterised by a dense and heterogeneous packing of differently sized, mobile and immobile components. From the interaction between these entitites, complex behaviour can emerge on large scales, including long time scales and persistent memory. Experiments on labelled molecules frequently show non-Markovian or non-Gaussian behaviour or both, depending on the window of time scales probed [1].

A description by an idealised Brownian motion is, at best, approximate, and may lead to significant modelling errors with respect to, e.g., spatial particle distributions and their reaction kinetics. Accurate and efficient modelling demands linking complex spatio-temporal data to suitable stochastic models, which should be able to accomodate the feature-rich experimental observations, yet allow for the robust estimation of parameters.

Non-Gaussian transport, yet with a linearly growing mean-square displacement (MSD) has attracted recent interest, whereas previous research almost exclusively focussed on the striking observation of (transient) subdiffusion [2]. A stochastic model that partially captures the former treats the particle’s effective diffusivity as a stochastic mean-reversion process, justified by the constantly rearranging environment [3]. In the most recent version of this “diffusing diffusivity (DD)” model, the molecule’s displacement 𝑋 obeys a non-degenerate diffusion that bears some resemblance with the well-known Cox-Ingersoll-Ross (CIR) process in mathematical finance. The full DD model is Markovian. However, the dynamics of 𝑋 alone may display significant memory effects or non-Markovianity, unless the system exhibits a clear time scale separation.

One possibility to describe non-Markovian effects is by means of the generalised Langevin equation (GLE) that, in its most frequently used Gaussian form, is specified by the autocorrelation function (ACF) of the noise increments or, equivalently, its memory kernel. A model-free alternative studied in this project uses higher-order memory functions that allow for the consistent interpolation from short to long time scales. The approach is based on a Fourier representation of the ACF of a stationary process in terms of a spectral density (e.g. [1]). The key quantity here, which is experimentally accessible, is the characteristic function of the process (i.e. the Fourier transform of its spectral density) which after systematic short-time approximations captures non-Markovian processes with persistent memory (long-time anomalies) as well as non-Gaussian transport with spatiotemporal memory or non-Gaussian noise, beyond the standard GLE.

A scientific novelty is that a completely data-driven framework is developed to study memory and crowding for diffusing particles that incorporates many available standard approaches, such as the GLE framework, and that rests on very few generic assumptions. The project explores the potential and limitations of the class of DD models, based on results of complex analysis and on actual data. The analysis of ACF including the frequency dependence is still in its infancies. Estimating spatiotemporal memory from trajectory data is new and goes beyond standard GLE approaches.

[1] F. Höfling and T. Franosch. Anomalous transport in the crowded world of biological cells. Rep. Prog. Phys., 76:046602, 2013.

[2] A. V. Chechkin, F. Seno, R. Metzler, and I. M. Sokolov. Brownian yet non-gaussian diffusion: From superstatistics to subordination of diffusing diffusivities. Phys. Rev. X, 7:021002, 2017.

[3] Y. Lanoiselée and D. S. Grebenkov. A model of non-Gaussian diffusion in heterogeneous media. J. Phys. A: Math. Theor., 51:145602, 2018.

**Project Webpages**

**Selected Publications
**

- C. Hartmann, L. Neureither, and U. Sharma,

*Coarse-graining of non-reversible stochastic differential equations: quantitative results and connections to averaging,*

SIAM J. Math. Anal., 52(3):2689–2733, 2020*(*preprint). - A. V. Straube, B. G. Kowalik, R. R. Netz, and F. Höfling,

*Rapid onset of molecular friction in liquids bridging between the atomistic and hydrodynamic pictures,*

Commun. Phys., 3:126, 2020. - D. Frömberg and F. Höfling,

*Generalized master equation for first-passage problems in partitioned spaces,*

J. Phys. A: Math. Theor., 54:215601, 2021 (preprint). - Z. Mezdoud, C. Hartmann, M. R. Remita, and O. Kebiri.

*α-hypergeometric uncertain volatility models and their connection to 2BSDEs,*

Bull. Inst. Math., Acad. Sin., 16(3):263–288, 2021 (preprint). - H. Bouanani, C. Hartmann, and O. Kebiri,

*Model reduction and uncertainty quantification of multiscale diffusions with parameter uncertainties using nonlinear expectations,*

submitted manuscript, arXiv:2102.04908, 2021. - T. Breiten, C. Hartmann, L. Neureither, and U. Sharma,

*Stochastic gradient descent and fast relaxation to thermodynamic equilibrium: a stochastic control approach*,

J. Math. Phys. 62(12):123302, 2021 (preprint). - U. Sharma and W. Zhang,

*Non-reversible sampling schemes on submanifolds.*

SIAM J. Numer. Anal., 59(6):2989–3031, 2021 (preprint). - F. Legoll, T. Lelièvre, and U. Sharma,

*An adaptive parareal algorithm: Application to the simulation of molecular dynamics trajectories,*

SIAM J. Sci. Comput., 44(1):B146–B176, 2022 (preprint). - R. I. A. Patterson, D. R. M. Renger, and U. Sharma.

*Variational structures beyond gradient flows: a macroscopic fluctuation-theory perspective,*

submitted manuscript, arXiv:2103.14384, 2021.

**Selected Pictures
**

Spectral analysis of memory shows that the smooth yet chaotic motion of atoms in a liquid leads to a rapid onset of friction below a characteristic frequency. More information can be found in Straube et al., Commun. Phys., 3:126, 2020.

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