# Elastic Instability of a Growing Yeast Droplet

Original entry: Nikolai Eroshenko, Fall 2009

## 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.

"Unstable" colonies that deviate from the spherical shape predicted by a simple surface energy minimization model. (A) Staircase, WT cells on 1.8% agar. (B) Dimple, sfl1$\Delta$ on 1.2% agar. (C) Dimple, sfl1$\Delta$ on 2% agar. Scale bar = 1 mm.

## Experimental Aspects

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

$\gamma_{SA} = \gamma_{CA} \cos\theta + \gamma_{CS}$

where:

• $\gamma_{SA}$ is the solid surface free energy (as energy per unit area of the substrate)
• $\gamma_{CA}$ is the adhesive energy between cells
• $\gamma_{CS}$ is the free energy between the cells and the substrate
• $\theta$ is the equilibrium contact angle
The contact angle of a cell droplet can be controlled either by varying the adherence of the cells to the substrate or by changing the agar concentration.

$\gamma_{CA}$ 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 $\gamma_{CA}$, were:

• flo11$\Delta$
• wild-type (WT)
• sfl1$\Delta$

$\gamma_{CS}$ 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 $h \left(x \right)$ has a displacement $\mathbf{u} \left (x,z \right)$. 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:

$G \frac{\partial^2 \mathbf{u}}{\partial z^2}=-\nabla p$

where G is the elastic modulus of the colony, and p is the pressure. If we take into account Gibb's condition $p \left (z \right) = -\gamma h$, 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:

$E = \frac{2 \gamma^2}{3G} \int{\! h\,^2 h^3 \, dx} + \gamma \int{\! \sqrt{1+h'\,^2} \, dx} + p_0 \int{\! h \, dx}$

in which the first term describes the elastic energy, the second term gives the surface energy, and the third term contains the Lagrange multiplier $p_0$ 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.