# Elastic Instability of a Growing Yeast Droplet

Original entry: Nikolai Eroshenko, Fall 2009

## Contents

## Reference

**Elastic Instability in Growing Yeast Colonies**

Nguyen B, Upadhyaya A, Oudenaarden A, Brenner MP

Biophysics Journal **86**: 2740-2747 (2004)

PMID: 15111392 (Pubget)

## Summary

This paper tested and explored the differential adhesion hypothesis, which states that cells growing as an aggregate (such as a tissue) will aim to minimize the adhesive energy between the different cells. *Saccharomyces cerevisiae* (Baker's yeast) was the model system chosen for this study. Small colonies of *S. cerevisiae* are spherical, which suggests that they can be modeled as liquid droplets attempting to minimize surface energy; larger colonies, in contrast, have non-spherical shapes. This paper proposes that this size-induced change in colony morphology occurs from the competition between elastic deformations and surface energies.

## Experimental Aspects

If surface energy minimization is the sole factor driving colony morphogenesis, then by Young's law:

<math> \gamma_{SA} = \gamma_{CA} \cos\theta + \gamma_{CS} </math>

where:

- <math>\gamma_{SA}</math> is the solid surface free energy (as energy per unit area of the substrate)
- <math>\gamma_{CA}</math> is the adhesive energy between cells
- <math>\gamma_{CS}</math> is the free energy between the cells and the substrate
- <math>\theta</math> is the equilibrium contact angle

### Intercellular adhesive strength

<math>\gamma_{CA}</math> was varied by using different strains of *S. cerevisiae*, which differed in their expression of the adhesive protein FLO11. The strains, in order of increasing <math>\gamma_{CA}</math>, were:

- flo11<math>\Delta</math>
- wild-type (WT)
- sfl1<math>\Delta</math>

### Cell-substrate adhesive strength

<math>\gamma_{CS}</math> was varied by varying agar concentration between 1% and 3% (percent Bacto agar in YPD growth medium).

## Soft Matter Aspects

Until they get to a certain critical size yeast colonies grow as spherical colonies. In this spherical regime the colony's contact angle is a function of agar concentration and the *S. cerevisiae* strain, and is (somewhat surprisingly) independent of colony size. However, as cells divide the colony grows, and it eventually reaches a critical volume at which the spheroid morphology is replaced by a morphology from, as the paper puts it, "a zoo of contact-angle-dependent colony shapes". The threshold volume seems to depend primarily on the colony's pre-transition contact angle.

The authors argue that the transition can be predicted if we start taking into account the elastic energy of the system. A two-dimensional (x,z) colony of height <math>h \left(x \right)</math> has a displacement <math>\mathbf{u} \left (x,z \right) </math>. If we assume that a colony is incompressible, and that the characteristic length of the the colony in the x direction is significantly larger that it is in the z direction, then:

<math> G \frac{\partial^2 \mathbf{u}}{\partial z^2}=-\nabla p </math>

where G is the elastic modulus of the colony, and p is the pressure. If we take into account Gibb's condition <math>p \left (z \right) = -\gamma h* </math>, and if we further assume that the displacement at the agar-colony interface (z = 0) and the shear stress at the colony-air interface (z = h(x)) is zero, then the total energy of a colony can (supposedly) be computed to equal:*

<math>
E = \frac{2 \gamma^2}{3G} \int{\! h**\,^2 h^3 \, dx} + \gamma \int{\! \sqrt{1+h'\,^2} \, dx} + p_0 \int{\! h \, dx}**
</math>

in which the first term describes the elastic energy, the second term gives the surface energy, and the third term contains the Lagrange multiplier <math>p_0</math> which maintains the system at a constant volume. The paper uses this relationship to show that a critical volume exists at which the elastic energy component begins to dominate, creating non-spherical shapes.