Dynamics of cell wall elasticity pattern shapes the cell during yeast mating morphogenesis

The cell wall defines cell shape and maintains integrity of fungi and plants. When exposed to mating pheromone, Saccharomyces cerevisiae grows a mating projection and alters in morphology from spherical to shmoo form. Although structural and compositional alterations of the cell wall accompany shape transitions, their impact on cell wall elasticity is unknown. In a combined theoretical and experimental approach using finite-element modelling and atomic force microscopy (AFM), we investigated the influence of spatially and temporally varying material properties on mating morphogenesis. Time-resolved elasticity maps of shmooing yeast acquired with AFM in vivo revealed distinct patterns, with soft material at the emerging mating projection and stiff material at the tip. The observed cell wall softening in the protrusion region is necessary for the formation of the characteristic shmoo shape, and results in wider and longer mating projections. The approach is generally applicable to tip-growing fungi and plants cells.


Dynamic Cell Wall Models (DM)
We developed a computational model to simulate inhomogeneous elasticity and elasto-plastic growth due to internal turgor pressure in space and time. The approach is based on continuum mechanics, where plane stress for thin shells is assumed.
The cell wall is discretized into a triangular mesh using the finite element mesh generator Gmsh (Geuzaine and Remacle 2009) and simulations are performed with the general purpose finite element framework DUNE (Bastian et al. 2008). We simulated the model equations in different situations for a wide range of parameters.
Furthermore, the analytical results derived for the steady state model (SM) were compared and matched with the dynamic models DM1 and DM2.
First, we introduce the elastic properties of a single triangle in the discretization of the cell wall as suggested by Delingette (Delingette 2008). Let T be a triangle of the cell wall discretization, where the points of the undeformed triangle are denoted by 1 , 2 , 3 and the points of the elastically deformed triangle ̂ are denoted by 1 , 2 , 3 , as depicted in Fig. S10. In the remainder, we refer to the elastically undeformed and deformed also as relaxed and stretched triangle, respectively. The area of the relaxed and stretched triangle is denoted as and ̂, respectively. The evolution of the cell wall was described by a deformation function Φ( ⃗, ). The right Cauchy-Green deformation tensor was computed from and the Green-Lagrange strain tensor from These tensors were used to calculate the elastic energy, elastic forces and yield criteria. There are three important quantities that were expressed in terms of angles and length of the sides of the deformed and the relaxed triangle (Delingette 2008): tr( ) = 1 ( 1 2 cos ,1 + 2 2 cos ,2 + 3 2 cos ,3 ), tr(ℇ) = 1 (( 1 2 − 1 2 ) cos ,1 + ( 2 2 − 2 2 ) cos ,2 + ( 3 2 − 3 2 ) cos ,3 ), Here, , and , are the lengths and and ̂ are areas of the relaxed triangle and stretched triangle respectively. The angles of the relaxed triangle are denoted by , .
A sketch of the relaxed state and stretched triangle can be found in Fig. S10 The coefficients depend on the Young's modulus * for plane elasticity, the cell wall thickness d and the Poisson's ratio . The Young's modulus for plane elasticity is related to the 3D Young's modulus E by * = (1 − 2 ) .
Here, and are the Lamé parameters, which are given by The coefficient , can be interpreted as tensile stiffness and the coefficient , as angular stiffness. These coefficients read In this representation the coordinates of the stretched triangle are represented by ⃗⃗ , . The total force at each vertex was computed from the sum of the forces induced by the turgor pressure and the counteracting elastic force: The forces induced by the turgor pressure were computed from: The equation of motion for every vertex of the mesh reads ⃗̈= ∑ 1 ⃗ , .
with ∈ The mass was computed from the triangle area as = 1 3 , where A is the area of all triangles that share the vertex n.

Elasto-plastic growth and yield criteria
The elasto-plastic growth depends on a yield stress criterion. In case the value calculated from the yield criterion was above a given yield limit, new material was inserted. In case the yield limit was not reached, we assumed only elastic deformations (see figure S10). Computationally this process was modeled by an irreversible deformation of the relaxed triangle. For these plastic deformations, we tested two yield criteria, yield stress (model DM1) and yield strain (model DM2).
As yield stress criterion we used the von Mises yield criterion for plane stress (Yu 2006): where 1 and 2 are the principal stresses. Assuming a spherical shmoo tip with typical radius 0.5 − 1.0 and a turgor pressure of 0.2 MPa the maximum and von Mises stress is approximately 0.5MPa to 1.0 MPa. It shall be noted, that in case of a cylinder the Maximum stress differs from the von Mises stress: For the dynamic model the stress criterion was computed from the linear stress tensor: The trace of S is given by and the determinant by Using these quantities, the von Mises stress criterion can be computed as follows The subsequent expansion rate was calculated from For the extensibility Φ a bell-shaped distribution around the tip was assumed The expansion of the cell wall was modeled as elongation of the edges of the relaxed triangle. Let and ′ be two adjacent triangles and , ′ the relaxed length of the edge that both triangles share. The expansion rates and ′ were calculated as above for the triangles and ′ , respectively. The elongation of the edge , ′ is given by For the DM2 we modelled cell wall expansion upon a yield strain. Here, we used the volumetric strain as a measure for yielding: The extensibility is in this case given by Φ * = (λ * λ) Φ ⁄ . Which leads to the expansion rate The elongation of the edge , ′ is then analogously given by

Steady State Cell Wall Model (SM)
In a mechanistic model for the cell shape, it is crucial to describe the distribution of forces due to turgor pressure, material insertion and the counterbalancing forces, which can be derived from the material properties of the cell wall. The first basic relationship connects the stresses (force per unit area) on the cell wall to the turgor pressure , the cell wall thickness and the local geometry. The latter is characterized by the principal curvatures. We assumed the cell shape to have a rotational symmetry and the shape of the cell was described as a surface of rotation.
The distribution of the corresponding stresses are expressed in terms of curvatures (Flügge 1973): ).
Here, the principal curvatures are given by the meridional curvature, , and the circumferential curvature, . The stresses were connected to the strains with a constitutive relationship for linear elasticity (Flügge 1973) are the meridional and circumferential strain, respectively. While denotes a small relaxed and the actual extend of a small surface patch in meridional direction, is the meridional distance of the relaxed shape measured from the base end. As for the dynamic model, the von Mises stress and volumetric strain are given by: From equations (28) and (29) we can derive a relationship between circumferential and meridional strain, which only depends on the geometry and the Poisson's ratio Using this relationship, we get the fomula (see also (Bernal, Rojas, and Dumais 2007)) which was used to identify points of the relaxed and natural shape of the cell starting from the end of the base. Therefore, strains and stresses are calculated from the parameters , and and the geometry of the relaxed and natural shape only. Note, that the strains are not calculated in the growth region, since we assume plastic growth at the shmoo tip. Inserting this relationship in the constitutive model we get: ) − 2 ( ) ).
For a given cellular geometry, the elasticity distribution was computed from (Bernal, Rojas, and Dumais 2007) We calculated the stress and elasticity distribution for the shmoo shape shown in Fig.   1 (see main text) and Fig. S2. For the numerical computations, cubic spline functions were used to characterize the shape. The NumPy and SciPy package in Python was used for this purpose. Calculations for different assumptions on relaxed and extended cell shape and strains are shown in Fig. S2.
In the special case of a sphere or a cylinder, the strains, the stresses and the elasticity can be computed analytically ( see Fig. S1). For a sphere, the circumferential and meridional strain are given by and the principal stresses are given by Assuming a spherical geometry at the base as well as tip, the Young's modulus the tip can be approximated by the following formula: * = (1 − ) 2 ( − ) .
This formula was used to compare the Young's Modulus obtained from AFM measurements and the Young's modulus estimated by osmotic shock experiments.
Figures S1 -S13 Figure S1. Geometrical considerations and principal in plane stresses and strains. A shows the used coordinates: circumferential angle , meridional distance s, radius r and shell thickness d. B shows principal stresses and strains for a given shell element. C Circumferential and meridional stresses ( , ) and strains ( , ) are equal for a sphere, while is twice as high as for the lateral surface of a cylinder. Additionally, exceeds , given that ≤ 0.5.     If the triangle is not in the defined growth zone or < , the triangle deforms purely elastically. 1 , 2 , 3 are the relaxed lengths of the unstressed triangle and and 1 , 2 , 3 are the elastically expanded length of the resulting triangle ′ .
Correspondingly, 1 , 2 , 3 represent the relaxed angles and 1 , 2 , 3 the angles of the deformed triangle. Additionally, triangles in the defined growth zone deform plastically if ≥ . Thereby, the relaxed lengths expand to new relaxed lengths,