AA1 – Life Science



Receptor Dynamics and Interactions in Complex Geometries: An Inverse
Problem in Particle-Based Reaction-Diffusion Theory

Project Heads

Paolo Annibale, Martin Lohse, Christof Schütte

Project Members

Horst-Holger Boltz

Project Duration

01.01.2021 − 14.02.2022

Located at



Building on an existing incubator project, we extend our spectroscopy of single molecule interaction times, a highly relevant inverse particle-based reaction diffusion problem in cell biology, to more complex membrane geometries as well as dynamics of the particles on the membrane. This reflects (i) the often peculiar geometries encountered in more specialized cell types, and (ii), the substantial compositional heterogeneity of the cell membrane, leading to complex and diverse types of motion. In particular, we shall investigate how membrane curvature and anisotropy thereof can affect the dynamics and downstream coupling of seven transmembrane receptors (GPCRs).

We aim to push the analytical description of particle-based reaction-diffusion (PBRD) processes by explicitly accounting for the non-trivial geometry imposed by the cell membrane on which these receptors are diffusing. Curvature, particular spatial variation in curvature, afflicts the dynamics on both aspects: 1) the diffusion is no longer spatially homogenous and the long-time expected distribution non-uniform and 2) the interaction becomes effectively modulated, e.g. two particles that are located in the same “pit” are more likely to interact.

Equipped with these modeling tools, we set out to tackle the inverse problem of inferring suitable model parameters from actual biological data (single-molecule imaging movies, e.g. using TIRF), i.e. stroboscopic observations of the system with coarse precision.  Technically, we can phrase this as an parameter estimation problem

    \[\theta_\ast = \text{argmin}_\theta\: F(\theta), \quad F(\theta)=\mathbb{E}\left(\sum_i ||\Phi(x_{t_i}(\theta)) - \phi_{t_i}||+ R(\theta)\right)\]

in which Φ is a suitable observable and R is a regularization that can encode our biological expertise about the parameters. Finding good choices for either (as well as the question of a suitable norm) is an open problem. In prior work (see literature), we were able to infer relative affinity differences (or changes in free enthalpy) from the distribution of overlap times, which relies on prior particle tracking and imaging.

The residual function is highly non-convex making global minimization a challenge. Furthermore, any minimizer (say Gauss-Newton methods) has to be complimented by Bayesian optimization anduncertainty quantification approaches

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Related Publications

Related Pictures

Sketch of typical scenarios of curved cell membranes. Left: t-tubuluar network (red) in cardiomyocytes (green), the muscle cells of the heart. Right: Curvature gradients lead to hetero-generous geometry, here exemplified by a nearly spherical endocytic clathrin coated pit. During endocytosis membrane receptors are internalized by the cell. (Image credit see image itself)