Morphogenesis: Mathematical models with frills

The spectacular frill around the neck of the lizard Chlamydosaurus has its origins in a mechanical instability that arises during development.
  1. Pierre A Haas  Is a corresponding author
  1. University of Cambridge, United Kingdom

The development of tissues and organs involves a combination of mechanical forces and various chemical and genetic cues (Mammoto et al., 2013; Nelson, 2016). A remarkable example of the role of mechanical forces during development is the formation of ridges and grooves (termed gyri and sulci) in the cerebral cortex to maximize its surface area-to-volume ratio. This process, which is called gyrification, does not require increased cell proliferation at the location of the gyri or other spatial patterning. Instead, it relies on a mechanical instability: the cerebral cortex (the outer layer of grey matter) grows uniformly, but it grows more than the underlying white matter to which it is attached, causing it to buckle and hence to form ridges and grooves (Figure 1Tallinen et al., 2014; Nelson, 2016). An analogous mechanism of buckling due to differential growth has been proposed to explain the formation of the loops in the gut (Hannezo et al., 2011; Shyer et al., 2013).

Now, in eLife, Sophie Montandon, Anamarija Fofonjka and Michel Milinkovitch report that a similar instability plays a role in the development of the distinctive frill around the neck of Chlamydosaurus kingii, a lizard that is commonly called the ‘frilled dragon’ (Montandon et al., 2019). The frill is an outgrowth of cartilage and skin that, when erect for the purposes of defense or courtship, is rather akin to the ruffs that are found on the costumes of Elizabethan period dramas (Figure 1A). When at rest each of the two lobes of the frill settles into three folds (Figure 1B).

How frustrated growth can lead to the formation of frills.

(AChlamydosaurus kingii, the frilled dragon, posturing with erect frill. (B) Chlamydosaurus with frill at rest. Each lobe of the frill has pleated into three folds. (C) During development of the cerebral cortex, grey matter grows more than white matter, leading to the formation of ridges and grooves called gyri and sulci. This is an example of frustrated growth at a surface. (D) The formation of the folds in the frill of Chlamydosaurus is an example of frustrated growth at a boundary: the frill is fixed at one edge by its attachment to the neck, so it buckles and forms folds as it grows. (E) Buckling due to boundary frustration illustrated by a physical analog experiment. Panels A, B and E are from Montandon et al., 2019.

Montandon et al., who are based at the University of Geneva and the SIB Swiss Institute of Bioinformatics, showed that the three folds appear on each lobe during embryonic development. However, staining for cell proliferation did not reveal any localized differences in cell division rates, so spatial patterning of cell proliferation cannot be responsible for the formation of the folds. Therefore the researchers hypothesized that the folds form due to the growth of the frill being frustrated by its attachment to the neck. A similar boundary frustration causes the folds of a draped curtain (Cerda et al., 2004).

To test their hypothesis, Montandon et al. performed an elegant experiment in which a semicircle of a gel (representing the frill) was first clamped between two blocks of wood (mimicking the attachment to the neck) and then immersed in a chemical. This caused the gel to swell, thus mimicking the growth of the frill (Figure 1E). The gel buckled to form three folds similar to those of the frill of Chlamydosaurus. The researchers then turned to numerical simulations of the frustrated growth of a thin elastic sheet, first in a simplified geometry, then in more realistic geometries, and were able to reproduce the transition from two folds to three folds that is seen during embryonic development in Chlamydosaurus.

The frills in Chlamydosaurus and the gyrification seen in the cerebral cortex both result from frustrated growth, though they differ in that the former involves frustration along an edge and the latter involves frustration along a surface. While the work of Montandon et al. highlights the importance of frustrated growth for morphogenesis, other researchers have shown that the gyrification observed in the cerebellum in mice, though superficially similar to that seen in the cerebral cortex, cannot be explained by the same model of frustrated growth (Engstrom et al., 2018).

The success of mechanisms of frustrated growth at explaining very different developmental processes (at least to a first approximation) suggests that continuum models (that is, models of continuous biological materials that average over different cells and other structures) have much to offer to the field of development. However, the occasional failure of continuum models emphasizes the importance of understanding the links between these models and the underlying biological processes.

An important challenge for these models is to move from qualitative comparisons between experimental systems and continuum models to more quantitative comparisons. Developing continuum models for this purpose again requires understanding how they link to the underlying biological processes. For example, continuum models often assume that Hooke's law is valid (that is, that the extension of the material is proportional to the force applied), but one would not expect this law to apply for large deformations of biological materials. Indeed, Hooke's law does not hold for brain tissue (Mihai et al., 2015). Theoretical work has shown how to extend Hooke's law to biological tissues by deriving the nonlinear continuum limit of simple cell-based models (Haas and Goldstein, 2019), and by using a framework of 'active gels' to describe biological materials that are out of equilibrium (Prost et al., 2015). However, the development of equations that can describe large deformations of general biological materials remains a long way away.

References

    1. Nelson CM
    (2016) On buckling morphogenesis
    Journal of Biomechanical Engineering 138:021005.
    https://doi.org/10.1115/1.4032128

Article and author information

Author details

  1. Pierre A Haas

    Pierre A Haas is in the Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Cambridge, United Kingdom

    For correspondence
    pah59@cam.ac.uk
    Competing interests
    No competing interests declared
    ORCID icon "This ORCID iD identifies the author of this article:" 0000-0002-6663-0393

Publication history

  1. Version of Record published: June 25, 2019 (version 1)

Copyright

© 2019, Haas

This article is distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use and redistribution provided that the original author and source are credited.

Metrics

  • 1,453
    Page views
  • 126
    Downloads
  • 0
    Citations

Article citation count generated by polling the highest count across the following sources: Crossref, PubMed Central, Scopus.

Download links

A two-part list of links to download the article, or parts of the article, in various formats.

Downloads (link to download the article as PDF)

Open citations (links to open the citations from this article in various online reference manager services)

Cite this article (links to download the citations from this article in formats compatible with various reference manager tools)

  1. Pierre A Haas
(2019)
Morphogenesis: Mathematical models with frills
eLife 8:e48520.
https://doi.org/10.7554/eLife.48520

Further reading

    1. Evolutionary Biology
    Alexei V Tkachenko, Sergei Maslov
    Research Article

    Life as we know it relies on the interplay between catalytic activity and information processing carried out by biological polymers. Here we present a plausible pathway by which a pool of prebiotic information-coding oligomers could acquire an early catalytic function, namely sequence-specific cleavage activity. Starting with a system capable of non-enzymatic templated replication, we demonstrate that even non-catalyzed spontaneous cleavage would promote proliferation by generating short fragments that act as primers. Furthermore, we show that catalytic cleavage function can naturally emerge and proliferate in this system. Specifically, a cooperative catalytic network with four subpopulations of oligomers is selected by the evolution in competition with chains lacking catalytic activity. The cooperative system emerges through the functional differentiation of oligomers into catalysts and their substrates. The model is inspired by the structure of the hammerhead RNA enzyme as well as other DNA- and RNA-based enzymes with cleavage activity that readily emerge through natural or artificial selection. We identify the conditions necessary for the emergence of the cooperative catalytic network. In particular, we show that it requires the catalytic rate enhancement over the spontaneous cleavage rate to be at least 102–103, a factor consistent with the existing experiments. The evolutionary pressure leads to a further increase in catalytic efficiency. The presented mechanism provides an escape route from a relatively simple pairwise replication of oligomers toward a more complex behavior involving catalytic function. This provides a bridge between the information-first origin of life scenarios and the paradigm of autocatalytic sets and hypercycles, albeit based on cleavage rather than synthesis of reactants.

    1. Cell Biology
    2. Evolutionary Biology
    Jonathan E Phillips, Duojia Pan
    Research Advance

    The genomes of close unicellular relatives of animals encode orthologs of many genes that regulate animal development. However, little is known about the function of such genes in unicellular organisms or the evolutionary process by which these genes came to function in multicellular development. The Hippo pathway, which regulates cell proliferation and tissue size in animals, is present in some of the closest unicellular relatives of animals, including the amoeboid organism Capsaspora owczarzaki. We previously showed that the Capsaspora ortholog of the Hippo pathway nuclear effector Yorkie/YAP/TAZ (coYki) regulates actin dynamics and the three-dimensional morphology of Capsaspora cell aggregates, but is dispensable for cell proliferation control (Phillips et al., 2022). However, the function of upstream Hippo pathway components, and whether and how they regulate coYki in Capsaspora, remained unknown. Here, we analyze the function of the upstream Hippo pathway kinases coHpo and coWts in Capsaspora by generating mutant lines for each gene. Loss of either kinase results in increased nuclear localization of coYki, indicating an ancient, premetazoan origin of this Hippo pathway regulatory mechanism. Strikingly, we find that loss of either kinase causes a contractile cell behavior and increased density of cell packing within Capsaspora aggregates. We further show that this increased cell density is not due to differences in proliferation, but rather actomyosin-dependent changes in the multicellular architecture of aggregates. Given its well-established role in cell density-regulated proliferation in animals, the increased density of cell packing in coHpo and coWts mutants suggests a shared and possibly ancient and conserved function of the Hippo pathway in cell density control. Together, these results implicate cytoskeletal regulation but not proliferation as an ancestral function of the Hippo pathway kinase cascade and uncover a novel role for Hippo signaling in regulating cell density in a proliferation-independent manner.