Models, Numerics and Simulations of Deforming Biological Surfaces

Thin, elastic surfaces are a fundamental building block in each biological system. Their main representative on the small scale are biomembranes; an important example on the larger scale are cell tissues. In both cases, the surfaces define a mechanical and chemical border, indispensable for the genesis and maintenance of each organism. An essential property of the surfaces is a lateral inhomogeneous composition of the surfaces themselves: without these inhomogeneities, the complexity of shapes, mechanochemical properties and dynamics would not be possible. In this thesis, we develop continuous mechanobiological models of membranes and tissues. Since these surfaces are experimentally often difficult to access, our approaches allow to investigate their behavior theoretically. The developed mathematical models are coupled nonlinear systems of partial differential equations (PDE) of fourth order. To enable simulations of these models, we significantly extend numerical algorithms for surface deformation based on the finite-element method (FEM). Extensive systematic simulations of the different models - in close comparison to recent experimental and theoretical studies on different scales - lead to new findings in membrane as well as tissue research. The key findings are the prediction and characterization of new mechanisms of communication between the two monolayers of a biomembrane, the investigation of the elusive role of the Gaussian rigidity in different fundamental membrane processes (like budding and lateral sorting), and moreover, the postulation and investigation of a new model for pattern formation in biological tissues, leading to experimental evidences for a new key mechanism for symmetry break in Hydra polyps

A continuous mechanobiological model of lateral inhomogeneous biological surfaces

Thin elastic surfaces containing molecules infuencing the mechanical prop- erties of the surface itself are wide spreaded structures of different scales in biological systems. Prominent examples are bilayer membranes and cell tis- sues. In this paper we present a continuous dynamical model of deforming lateral inhomogeneous surfaces, using the example of biological membranes. In agreement with experimental observations the membrane consists of dif- ferent molecule species undergoing lateral phase separation and influencing the mechanical properties of the membrane. The presented model is based on the minimization of a free energy leading to a coupled nonlinear PDE system of fourth order, related to the Willmore flow and the Cahn-Hilliard equation. First simulations show the development of budding structures from stochas- tic initial conditions as a result of the gradient flow, which is comparable to experimentally observed structures. In our model mechanical properties are described via macroscopic mechanical moduli. However, the qualitative and quantitative relationships of mechanical moduli and the local composi- tion of the membrane are unkown. Since the exact relationship significantly influences the emerging structures, this study motivates the development of techniques allowing for upscaling from the molecular scale

Migrating curlews on schedule: departure and arrival patterns of a long-distance migrant depend on time and breeding location rather than on wind conditions

Background However, few studies have tried to validate the significance of these three concepts simultaneously, and long-term, high-resolution tagging datasets recording individual movements across consecutive years are scarce. We used such a dataset to explore intraspecific and intra-individual variabilities in departure and arrival decisions from/to wintering grounds in relation to these three different concepts in bird migration.We equipped 23 curlews (Numenius arquata) wintering in the Wadden Sea with Global Positioning System data loggers to record their spatio-temporal patterns of departure from and arrival at their wintering site, and the first part of their spring migration. We obtained data for 42 migrations over 6 years, with 12 individuals performing repeat migrations in consecutive years. Day of year of departure and arrival was related to 38 meteorological and bird-related predictors using the least absolute shrinkage and selection operator (LASSO) to identify drivers of departure and arrival decisions.Curlews migrated almost exclusively to Arctic and sub-Arctic Russia for breeding. Curlews breeding further away in areas with late snowmelt departed later. Departures dates varied by only < 4 days in individual curlews tagged over consecutive years.These results suggest that the trigger for migration in this long-distance migrant is largely independent of weather conditions but is subject to resource availability in breeding areas. The high intra-individual repeatability of departure days among subsequent years and the lack of relationship to weather parameters suggest the importance of genetic triggers in prompting the start of migration. Further insights into the timing of migration in immatures and closely related birds might help to further unravel the genetic mechanisms triggering migration patterns

Beyond Turing: mechanochemical pattern formation in biological tissues

Background: During embryogenesis, chemical (morphogen) and mechanical patterns develop within tissues in a self-organized way. More than 60 years ago, Turing proposed his famous reaction-diffusion model for such processes, assuming chemical interactions as the main driving force in tissue patterning. However, experimental identification of corresponding molecular candidates is still incomplete. Recent results suggest that beside morphogens, also tissue mechanics play a significant role in these patterning processes. Results: Combining continuous finite strain with discrete cellular tissue models, we present and numerically investigate mechanochemical processes, in which morphogen dynamics and tissue mechanics are coupled by feedback loops. We consider three different mechanical cues involved in such feedbacks: strain, stress, and compression. Based on experimental results, for each case, we present a feedback loop spontaneously creating robust mechanochemical patterns. In contrast to Turing-type models, simple mechanochemical interaction terms are sufficient to create de novo patterns. Conclusions: Our results emphasize mechanochemical processes as possible candidates controlling different steps of embryogenesis. To motivate further experimental research discovering related mechanisms in living tissues, we also present predictive in silicio experiments. Reviewers: Reviewer 1 - Marek Kimmel; Reviewer 2 - Konstantin Doubrovinski (nominated by Ned Wingreen); Reviewer 3 - Jun Allard (nominated by William Hlavacek)

Bird migration in space and time: chain migration by Eurasian curlew Numenius arquata arquata along the East Atlantic Flyway

Migration patterns in birds vary in space and time. Spatial patterns include chain, leapfrog and telescopic migration. Temporal patterns such as migration duration, number, and duration of stopovers may vary according to breeding latitude, sex, and season. This study aimed to verify these patterns in a long-distance migrant, the Eurasian curlew Numenius arquata arquata, and to provide a synopsis of spatio-temporal migration patterns in this species of concern throughout the East Atlantic Flyway. We tagged 85 adults with GPS-data loggers in Germany, Poland, France and Estonia between 2013 and 2019. We computed the distance flown, linear loxodromic distance, duration, stopover number, total stopover duration, mean stopover duration, departure time and arrival time for 177 out of 187 tracks. On average (± standard deviation), spring migration occurred from 4 to 14 April (10.2 ± 8.4 days), curlews flew 3.623 ± 1.366 km, and had 5.8 ± 3.6 stopovers, with a duration of 29.4 ± 38.2 h per stopover, while autumn migration occurred from 18 to 29 June (10.9 ± 9.9 days), curlews flew 3.362 ± 1.351 km, and had 5.4 ± 4.0 stopovers, with 31.8 ± 32.3 h per stopover. Curlews displayed chain migration because wintering curlews maintained the latitudinal sequence to their breeding sites. Southern curlews had a longer nesting period due to their earlier arrivals. While spring arrival at breeding sites did not differ between the sexes, in autumn females departed earlier than males. Migration duration and distance, as well as stopover number and duration, showed a significant increase with breeding site latitude but did not differ between the sexes or between spring and autumn migrations, suggesting that curlews took a comparable amount of time migrating during both seasons. The high site faithfulness in curlews suggests that rapid autumn migration allows them to return to defend their winter foraging areas

Post-Turing tissue pattern formation: Advent of mechanochemistry.

Chemical and mechanical pattern formation is fundamental during embryogenesis and tissue development. Yet, the underlying molecular and cellular mechanisms are still elusive in many cases. Most current theories assume that tissue development is driven by chemical processes: either as a sequence of chemical patterns each depending on the previous one, or by patterns spontaneously arising from specific chemical interactions (such as "Turing-patterns"). Within both theories, mechanical patterns are usually regarded as passive by-products of chemical pre-patters. However, several experiments question these theories, and an increasing number of studies shows that tissue mechanics can actively influence chemical patterns during development. In this study, we thus focus on the interplay between chemical and mechanical processes during tissue development. On one hand, based on recent experimental data, we develop new mechanochemical simulation models of evolving tissues, in which the full 3D representation of the tissue appears to be critical for obtaining a realistic mechanochemical behaviour. The presented modelling approach is flexible and numerically studied using state of the art finite element methods. Thus, it may serve as a basis to combine simulations with new experimental methods in tissue development. On the other hand, we apply the developed approach and demonstrate that even simple interactions between tissue mechanics and chemistry spontaneously lead to robust and complex mechanochemical patterns. Especially, we demonstrate that the main contradictions arising in the framework of purely chemical theories are naturally and automatically resolved using the mechanochemical patterning theory

A Mechanochemical Model for Embryonic Pattern Formation: Coupling Tissue Mechanics and Morphogen Expression

<div><p>Motivated by recent experimental findings, we propose a novel mechanism of embryonic pattern formation based on coupling of tissue curvature with diffusive signaling by a chemical factor. We derive a new mathematical model using energy minimization approach and show that the model generates a variety of morphogen and curvature patterns agreeing with experimentally observed structures. The mechanism proposed transcends the classical Turing concept which requires interactions between two morphogens with a significantly different diffusivity. Our studies show how biomechanical forces may replace the elusive long-range inhibitor and lead to formation of stable spatially heterogeneous structures without existence of chemical prepatterns. We propose new experimental approaches to decisively test our central hypothesis that tissue curvature and morphogen expression are coupled in a positive feedback loop.</p></div

Influence of different model parameters on patterns.

<p>A: Increasing diffusion by changing results in bigger morphogen patches. B–C: An increase of coupling between tissue curvature and morphogen expression via (B) results in more but still equally sized patches. In all simulations red and blue color depict high and low morphogen levels, respectively.</p