The main research topic in our group **Self-Organization of Multicellular Systems** is the biophysics of cells and tissues (*nomen est omen*), but we have wider research interests in mathematical biology and soft matter physics.

Our main research focus is the mechanics of cells and tissues. In particular, we want to understand how robust development is compatible with mechanical constraints and biological variability. We therefore derive novel continuum theories for biological tissues and apply them to experimental data of the rich mechanical behaviour of tissues during development. Although our research is theoretical, we work in close collaboration with experimental groups at the MPI-CBG and beyond.

Deformations of cell sheets during morphogenesis are driven by developmental processes such as cell division and cell shape changes. In morphoelastic shell theories of development, these processes appear as variations of the intrinsic geometry of a thin elastic shell. However, morphogenesis often involves large bending deformations that are outside the formal range of validity of these shell theories. By asymptotic expansion of three-dimensional incompressible morphoelasticity in the limit of a thin shell, we have derived a shell theory for large intrinsic bending deformations and emphasize the resulting geometric material anisotropy and the elastic role of cell constriction. Taking the invagination of the green alga *Volvox* as a model developmental event, we have shown how results for this theory differ from those for a classical shell theory that is not formally valid for these large bending deformations and reveal how these geometric effects stabilize invagination. [Text and figure from Haas & Goldstein, Phys. Rev. E **103**, 022411 (2021).]

The shapes of epithelial tissues result from a complex interplay of contractile forces in the cytoskeleta of the cells in the tissue and adhesion forces between them. A host of discrete, cell-based models describe these forces by assigning different surface tensions to the apical, basal, and lateral sides of the cells. These differential-tension models have been used to describe the deformations of epithelia in different living systems, but the underlying continuum mechanics at the scale of the epithelium are still unclear. We have derived a continuum theory for a simple differential-tension model of a two-dimensional epithelial monolayer and study the buckling of this epithelium under imposed compression. The analysis reveals how the cell-level properties encoded in the differential-tension model lead to linear and nonlinear elastic as well as nonlocal, nonelastic behavior at the continuum level. [Text and figure from Haas & Goldstein, Phys. Rev. E **99**, 022411 (2019).]

The embryos of the green alga *Volvox carteri* are spherical sheets of cells that turn themselves inside out at the close of their development through a program of cell shape changes. This process of inversion is a model for morphogenetic cell sheet deformations; it starts with four lips opening up at the anterior pole of the cell sheet, flipping over, and peeling back to invert the embryo. Experimental studies have revealed that inversion is arrested if some of these cell shape changes are inhibited, but the mechanical basis for these observations has remained unclear. We have analyzed the mechanics of this inversion by deriving an averaged elastic theory for these lips and we have interpreted the experimental observations in terms of the mechanics and evolution of inversion. [Text and figure from Haas & Goldstein, Phys. Rev. E **98**, 052415 (2018).]

Variability is emerging as an integral part of development. It is therefore imperative to ask how to access the information contained in this variability. Yet most studies of development average their observations and, discarding the variability, seek to derive models, biological or physical, that explain these average observations. We have analysed this variability in a study of cell sheet folding in the green alga *Volvox*, whose spherical embryos turn themselves inside out in a process sharing invagination, expansion, involution, and peeling of a cell sheet with animal models of morphogenesis. We have generalised our earlier, qualitative model of the initial stages of inversion by combining ideas from morphoelasticity and shell theory. Together with three-dimensional visualisations of inversion using light sheet microscopy, we have obtained a detailed, quantitative model of the entire inversion process. With this model, we have shown how the variability of inversion reveals that two separate, temporally uncoupled processes drive the initial invagination and subsequent expansion of the cell sheet. This implies a prototypical transition towards higher developmental complexity in the volvocine algae and provides proof of principle of analysing morphogenesis based on its variability. [Figure and text from Haas *et al.*, PLoS Biol. **16**, e2005536 (2018).]

Deformations of cell sheets are ubiquitous in early animal development, often arising from a complex and poorly understood interplay of cell shape changes, division, and migration. We have explored perhaps the simplest example of cell sheet folding: the “inversion” process of the algal genus *Volvox*, during which spherical embryos turn themselves inside out through a process hypothesized to arise from cell shape changes alone. We have used light sheet microscopy to obtain the first three-dimensional visualizations of inversion *in vivo*, and have developed the first theory of this process, in which cell shape changes appear as local variations of intrinsic curvature, contraction and stretching of an elastic shell. Our results support a scenario in which these active processes function in a defined spatiotemporal manner to enable inversion.

In later work, we have analysed the mechanics of the initial invagination, reminiscent of similar events during the development of higher organisms. Through a combination of asymptotic analysis and numerical studies of the bifurcation behaviour, we have illustrated how appropriate *local* deformations (i.e. cell shape changes represented as changes of the intrinsic geometry of the cell sheet) can overcome *global* constraints to initiate inversion.

[Text and figures from Höhn et al., Phys. Rev. Lett. **114**, 178101 (2015) and Haas and Goldstein, J. R. Soc. Interface **12**, 20150671 (2015).]

Turing instabilities of reaction-diffusion systems can only arise if the diffusivities of the chemical species are sufficiently different. This threshold is unphysical in most systems with *N*=2 diffusing species, forcing experimental realizations of the instability to rely on fluctuations or additional nondiffusing species. We have asked whether this diffusive threshold lowers for *N*>2 to allow "true" Turing instabilities. Inspired by May's analysis of the stability of random ecological communities, we have analyzed the probability distribution of the diffusive threshold in reaction-diffusion systems defined by random matrices describing linearized dynamics near a homogeneous fixed point. In the numerically tractable cases *N*<7, we have found that the diffusive threshold becomes more likely to be smaller and physical as* N * increases and that most of these many-species instabilities cannot be described by reduced models with fewer species. [Text and figure from Haas & Goldstein, Phys. Rev. Lett. **126**, 238101 (2021).]

In microbial communities, each species often has multiple, distinct phenotypes, but studies of ecological stability have largely ignored this subpopulation structure. We have shown that such implicit averaging over phenotypes leads to incorrect linear stability results. We have gone on to analyze the effect of phenotypic switching in detail in an asymptotic limit and have partly overturned classical stability paradigms: Abundant phenotypic variation is linearly destabilizing but, surprisingly, a rare phenotype such as bacterial persisters has a stabilizing effect. We have extended these results by showing how phenotypic variation modifies the stability of the system to large perturbations such as antibiotic treatments. [Text and figure from Haas *et al.*, Phys. Rev. Research **2**, 022036(R) (2020).]

Cooled oil emulsion droplets in aqueous surfactant solution have been observed to flatten into a remarkable host of polygonal shapes with straight edges and sharp corners, but different driving mechanisms—(i) a partial phase transition of the liquid bulk oil into a plastic rotator phase near the droplet interface and (ii) buckling of the interfacially frozen surfactant monolayer enabled by a drastic lowering of surface tension—have been proposed. We have explored the simplest geometric competition between this phase transition and surface tension in planar polygons to recover the observed sequence of shapes and their statistics in qualitative agreement with experiments, and to explain the formation of protrusions sprouting from the droplet vertices and the puncturing of planar polygonal droplets.

In later work, combining experiment and theory, we have analyzed the initial stages of the evolution of these “shape-shifting” droplets, during which a polyhedral droplet flattens into a polygonal platelet under cooling and gravity. Using reflected-light microscopy, we have revealed how icosahedral droplets evolve through an intermediate octahedral stage to flatten into hexagonal platelets. This behavior is reproduced by a theoretical model of the phase transition mechanism, but the buckling mechanism can only reproduce the flattening if the deformations are driven by buoyancy. This requires surface tension to decrease by several orders of magnitude during cooling and yields bending modulus estimates orders of magnitude below experimental values.

Others have since shown that the observed phenomena are also consistent with an "augmented" elastic mechanism based on the intrinsic curvature of the droplet surface, but we have commented that this interplay between surface tension and intrinsic curvature is in fact *mathematically* equivalent to the *physically* very different phase-transition mechanism. The mathematical models cannot therefore distinguish between the two mechanisms, and we have argued that differences in experimental suggest that the observed phenomena are therefore physically very different realizations of a general "shape-shifting" mechanism based on the inter-play of positive surface tension and negative edge tension in faceted droplets. [Text and figures from Haas et al., Phys. Rev. Lett. **118**, 088001 (2017), *iid.*, Phys. Rev. Research **1**, 023017 (2019), and *iid.*, Phys. Rev. Lett. **126**, 259801 (2021).]

Most of the many computational methods that have been developed for the prediction of RNA secondary structure and the prediction of the consensus structure from an alignment of sequences ignore the biophysical factors - including kinetic and co-transcriptional effects during RNA folding - that affect this secondary structure. In the framework of stochastic context-free grammars, we have developed an iterative algorithm, punningly named "Oxfold", that combines a simple model of the kinetics of RNA folding with an evolutionary model. By showing that the model outperforms previous non-kinetic grammatical models and compares favourably to thermodynamic models that do not model folding kinetics, we have established the importance of these kinetic effects for RNA secondary structure.