Turing’s Diffusive Threshold in Random Reaction-Diffusion Systems
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.
Morphoelasticity of large bending deformations of cell sheets during development
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.
The noisy basis of morphogenesis: Mechanisms and mechanics of cell sheet folding inferred from developmental variability
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. Combining ideas from morphoelasticity and shell theory and using three-dimensional visualisations of inversion using light sheet microscopy, we have obtained a detailed quantitative mechanical 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.
Rahul G. Ramachandran, Ricard Alert#, Pierre A. Haas# Buckling by disordered growth. Phys Rev E, 110(5) Art. No. 054405 (2024)
Open Access DOI
Buckling instabilities driven by tissue growth underpin key developmental events such as the folding of the brain. Tissue growth is disordered due to cell-to-cell variability, but the effects of this variability on buckling are unknown. Here, we analyze what is perhaps the simplest setup of this problem: the buckling of an elastic rod with fixed ends driven by spatially varying, yet highly symmetric growth. Combining analytical calculations for simple growth fields and numerical sampling of random growth fields, we show that variability can increase as well as decrease the growth threshold for buckling, even when growth variability does not cause any residual stresses. For random growth, we find numerically that the shift of the buckling threshold correlates with spatial moments of the growth field. Our results imply that biological systems can either trigger or avoid buckling by exploiting the spatial arrangement of growth variability.
Matthew J Bovyn#, Pierre A. Haas# Shaping epithelial lumina under pressure. Biochem Soc Trans, 52(1) 331-342 (2024)
Open Access DOI
The formation of fluid- or gas-filled lumina surrounded by epithelial cells pervades development and disease. We review the balance between lumen pressure and mechanical forces from the surrounding cells that governs lumen formation. We illustrate the mechanical side of this balance in several examples of increasing complexity, and discuss how recent work is beginning to elucidate how nonlinear and active mechanics and anisotropic biomechanical structures must conspire to overcome the isotropy of pressure to form complex, non-spherical lumina.
2023
Martine Ben Amar#, Pasquale Ciarletta#, Pierre A. Haas# Morphogenesis in space offers challenges and opportunities for soft matter and biophysics. Communications Physics, 6(1) Art. No. 150 (2023)
Open Access DOI
The effects of microgravity on soft matter morphogenesis have been documented in countless experiments, but physical understanding is still lacking in many cases. Here we review how gravity affects shape emergence and pattern formation for both inert matter and living systems of different biological complexities. We highlight the importance of building physical models for understanding the experimental results available. Answering these fundamental questions will not only solve basic scientific problems, but will also enable several industrial applications relevant to space exploration.
Maarten P Bebelman✳︎, Matthew J Bovyn✳︎, Carlotta Mayer, Julien Delpierre, Ronald Naumann, Nuno P Martins, Alf Honigmann, Yannis Kalaidzidis, Pierre A. Haas#, Marino Zerial# Hepatocyte apical bulkheads provide a mechanical means to oppose bile pressure. J Cell Biol, 222(4) Art. No. e202208002 (2023)
Open Access DOI
Hepatocytes grow their apical surfaces anisotropically to generate a 3D network of bile canaliculi (BC). BC elongation is ensured by apical bulkheads, membrane extensions that traverse the lumen and connect juxtaposed hepatocytes. We hypothesize that apical bulkheads are mechanical elements that shape the BC lumen in liver development but also counteract elevated biliary pressure. Here, by resolving their structure using STED microscopy, we found that they are sealed by tight junction loops, connected by adherens junctions, and contain contractile actomyosin, characteristics of mechanical function. Apical bulkheads persist at high pressure upon microinjection of fluid into the BC lumen, and laser ablation demonstrated that they are under tension. A mechanical model based on ablation results revealed that apical bulkheads double the pressure BC can hold. Apical bulkhead frequency anticorrelates with BC connectivity during mouse liver development, consistent with predicted changes in biliary pressure. Our findings demonstrate that apical bulkheads are load-bearing mechanical elements that could protect the BC network against elevated pressure.
2022
Pierre A. Haas#, Maria A. Gutierrez, Nuno M. Oliveira#, Raymond E. Goldstein# Stabilization of microbial communities by responsive phenotypic switching. Phys Rev Research, 4(3) Art. No. 033224 (2022)
Open Access DOI
Clonal microbes can switch between different phenotypes and recent theoretical work has shown that stochastic switching between these subpopulations can stabilize microbial communities. This phenotypic switching need not be stochastic, however, but could also be in response to environmental factors, both biotic and abiotic. Here, motivated by the bacterial persistence phenotype, we explore the ecological effects of such responsive switching by analyzing phenotypic switching in response to competing species. We show that the stability of microbial communities with responsive switching differs generically from that of communities with stochastic switching only. To understand the mechanisms by which responsive switching stabilizes coexistence, we go on to analyze simple two-species models. Combining exact results and numerical simulations, we extend the classical stability results for the competition of two species without phenotypic variation to the case in which one species switches, stochastically and responsively, between two phenotypes. In particular, we show that responsive switching can stabilize coexistence even when stochastic switching on its own does not affect the stability of the community.
2021
Pierre A. Haas#, Raymond E. Goldstein#, Diana Cholakova#, Nikolai Denkov#, Stoyan K. Smoukov# Comment on "Faceting and Flattening of Emulsion Droplets: A Mechanical Model". Phys Rev Lett, 126(25) Art. No. 259801 (2021)
Open Access DOI
Pierre A. Haas#, Raymond E. Goldstein# Turing's Diffusive Threshold in Random Reaction-Diffusion Systems. Phys Rev Lett, 126(23) Art. No. 238101 (2021)
Open Access DOI
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. Here, we ask 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 analyze 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⩽6, we find 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 diffusing species.
Pierre A. Haas#, Raymond E. Goldstein# Morphoelasticity of large bending deformations of cell sheets during development. Phys Rev E, 103(2-1) Art. No. 022411 (2021)
Open Access DOI
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. Here, by asymptotic expansion of three-dimensional incompressible morphoelasticity in the limit of a thin shell, we derive 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 show 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.
2020
Pierre A. Haas#, Nuno M. Oliveira#, Raymond E. Goldstein# Subpopulations and stability in microbial communities. Phys Rev Research, 2(2) Art. No. 022036 (2020)
Open Access DOI
In microbial communities, each species often has multiple, distinct phenotypes, but studies of ecological stability have largely ignored this subpopulation structure. Here, we show that such implicit averaging over phenotypes leads to incorrect linear stability results. We then analyze the effect of phenotypic switching in detail in an asymptotic limit and partly overturn classical stability paradigms: Abundant phenotypic variation is linearly destabilizing but, surprisingly, a rare phenotype such as bacterial persisters has a stabilizing effect. Finally, we extend these results by showing how phenotypic variation modifies the stability of the system to large perturbations such as antibiotic treatments.
2019
Pierre A. Haas, Diana Cholakova, Nikolai Denkov, Raymond E. Goldstein, Stoyan K. Smoukov Shape-shifting polyhedral droplets. Phys Rev Research, 1(2) Art. No. 023017 (2019)
Open Access DOI
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. Here, combining experiment and theory, we analyze 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 reveal 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. The analysis thus shows that the phase transition mechanism underlies the observed “shape-shifting” phenomena.
Pierre A. Haas#, Raymond E. Goldstein# Nonlinear and nonlocal elasticity in coarse-grained differential-tension models of epithelia. Phys Rev E, 99(2) Art. No. 022411 (2019)
Open Access DOI
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. Here, we derive 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.
2018
Pierre A. Haas#, Raymond E. Goldstein# Embryonic Inversion in Volvox carteri: The Flipping and Peeling of Elastic Lips. Phys Rev E, 98(5) Art. No. 052415 (2018)
Open Access DOI
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 programme 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. Here, we analyse the mechanics of this inversion by deriving an averaged elastic theory for these lips and we interpret the experimental observations in terms of the mechanics and evolution of inversion.
Pierre A. Haas✳︎, Stephanie S M H Höhn✳︎, Aurelia R Honerkamp-Smith, Julius B Kirkegaard, Raymond E. Goldstein The noisy basis of morphogenesis: Mechanisms and mechanics of cell sheet folding inferred from developmental variability. PLoS Biol, 16(7) Art. No. e2005536 (2018)
Open Access DOI
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. Here, we analyse 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 generalise 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, this yields a detailed, quantitative model of the entire inversion process. With this model, we show 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.
2017
Pierre A. Haas, Raymond E. Goldstein#, Stoyan K. Smoukov#, Diana Cholakova, Nikolai Denkov Theory of Shape-Shifting Droplets. Phys Rev Lett, 118(8) Art. No. 088001 (2017)
Open Access DOI
Recent studies of cooled oil emulsion droplets uncovered transformations into a host of flattened shapes with straight edges and sharp corners, driven by a partial phase transition of the bulk liquid phase. Here, we explore theoretically the simplest geometric competition between this phase transition and surface tension in planar polygons and recover the observed sequence of shapes and their statistics in qualitative agreement with experiments. Extending the model to capture some of the three-dimensional structure of the droplets, we analyze the evolution of protrusions sprouting from the vertices of the platelets and the topological transition of a puncturing planar polygon.
2015
Pierre A. Haas, Raymond E. Goldstein Elasticity and glocality: initiation of embryonic inversion in Volvox. J R Soc Interface, 12(112) Art. No. 20150671 (2015)
Open Access DOI
Elastic objects across a wide range of scales deform under local changes of their intrinsic properties, yet the shapes are glocal, set by a complicated balance between local properties and global geometric constraints. Here, we explore this interplay during the inversion process of the green alga Volvox, whose embryos must turn themselves inside out to complete their development. This process has recently been shown to be well described by the deformations of an elastic shell under local variations of its intrinsic curvatures and stretches, although the detailed mechanics of the process have remained unclear. Through a combination of asymptotic analysis and numerical studies of the bifurcation behaviour, we illustrate how appropriate local deformations can overcome global constraints to initiate inversion.
Stephanie S M H Höhn, Aurelia R Honerkamp-Smith, Pierre A. Haas, Philipp Khuc Trong, Raymond E. Goldstein Dynamics of a Volvox embryo turning itself inside out. Phys Rev Lett, 114(17) Art. No. 178101 (2015)
Open Access DOI
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. Here, we explore 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 use light sheet microscopy to obtain the first three-dimensional visualizations of inversion in vivo, and develop 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.
2013
James W J Anderson, Pierre A. Haas, Leigh-Anne Mathieson, Vladimir Volynkin, Rune Lyngsø, Paula Tataru, Jotun Hein Oxfold: kinetic folding of RNA using stochastic context-free grammars and evolutionary information. Bioinformatics, 29(6) 704-710 (2013)
DOI
Many computational methods for RNA secondary structure prediction, and, in particular, for the prediction of a consensus structure of an alignment of RNA sequences, have been developed. Most methods, however, ignore biophysical factors, such as the kinetics of RNA folding; no current implementation considers both evolutionary information and folding kinetics, thus losing information that, when considered, might lead to better predictions.
