1. Evolutionary Biology
  2. Physics of Living Systems
Download icon

Morphogenesis: Mathematical models with frills

  1. Pierre A Haas  Is a corresponding author
  1. University of Cambridge, United Kingdom
  • Cited 0
  • Views 1,300
  • Annotations
Cite this article as: eLife 2019;8:e48520 doi: 10.7554/eLife.48520


The spectacular frill around the neck of the lizard Chlamydosaurus has its origins in a mechanical instability that arises during development.

Main text

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.


    1. Nelson CM
    (2016) On buckling morphogenesis
    Journal of Biomechanical Engineering 138:021005.

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
    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)


© 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.


  • 1,300
    Page views
  • 110
  • 0

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)

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

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

Further reading

    1. Evolutionary Biology
    2. Microbiology and Infectious Disease
    Emily R Ebel et al.
    Research Article

    The replication of Plasmodium falciparum parasites within red blood cells (RBCs) causes severe disease in humans, especially in Africa. Deleterious alleles like hemoglobin S are well-known to confer strong resistance to malaria, but the effects of common RBC variation are largely undetermined. Here we collected fresh blood samples from 121 healthy donors, most with African ancestry, and performed exome sequencing, detailed RBC phenotyping, and parasite fitness assays. Over one third of healthy donors unknowingly carried alleles for G6PD deficiency or hemoglobinopathies, which were associated with characteristic RBC phenotypes. Among non-carriers alone, variation in RBC hydration, membrane deformability, and volume was strongly associated with P. falciparum growth rate. Common genetic variants in PIEZO1, SPTA1/SPTB, and several P. falciparum invasion receptors were also associated with parasite growth rate. Interestingly, we observed little or negative evidence for divergent selection on non-pathogenic RBC variation between Africans and Europeans. These findings suggest a model in which globally widespread variation in a moderate number of genes and phenotypes modulates P. falciparum fitness in RBCs.

    1. Evolutionary Biology
    2. Genetics and Genomics
    Milton Tan et al.
    Research Article Updated

    Chondrichthyes (cartilaginous fishes) are fundamental for understanding vertebrate evolution, yet their genomes are understudied. We report long-read sequencing of the whale shark genome to generate the best gapless chondrichthyan genome assembly yet with higher contig contiguity than all other cartilaginous fish genomes, and studied vertebrate genomic evolution of ancestral gene families, immunity, and gigantism. We found a major increase in gene families at the origin of gnathostomes (jawed vertebrates) independent of their genome duplication. We studied vertebrate pathogen recognition receptors (PRRs), which are key in initiating innate immune defense, and found diverse patterns of gene family evolution, demonstrating that adaptive immunity in gnathostomes did not fully displace germline-encoded PRR innovation. We also discovered a new toll-like receptor (TLR29) and three NOD1 copies in the whale shark. We found chondrichthyan and giant vertebrate genomes had decreased substitution rates compared to other vertebrates, but gene family expansion rates varied among vertebrate giants, suggesting substitution and expansion rates of gene families are decoupled in vertebrate genomes. Finally, we found gene families that shifted in expansion rate in vertebrate giants were enriched for human cancer-related genes, consistent with gigantism requiring adaptations to suppress cancer.