© Z. Awada and B. Nedjar,, Published by EDP Sciences, 2023

1 Introduction

Every living organism is made up at the base of an element that can be though as a mini-factory, called cell. In the cell biology field, there exist a large diversity of items that share many commun functions and architecture. Saccharomyces cerevisiae yeast provides a powerful system to study eukaryotic cells. It is easily manipulated in laboratory and it grows very quickly. Due to these features, in addition to the advantage of being regarded as safe, this yeast remains the host cell to investigate humain processes. In particular, S. cerevisiae yeast is known by its more common names, baker’s yeast or brewer’s yeast. It is widely used in food industry for baking, winemaking and brewing [1,2]. This yeast is also beneficial in the medical sector. It serves for the production of insulin and vitamins [3]. Moreover, yeast extracts help to improve skin health, see [4].

This yeast, as other fungal, bacterial and plant cells, are characterized by a high internal pression, called turgor pressure, [5]. It results from the pressure difference between the interior and the exterior of the cell.

Over and above that, these species are surrounded by a tough and flexible structure; the cell-wall. The structure and composition of this latter are constantly adapted to accompany the permanent remodeling of cell architecture and the environmental changes. For example, a cell-wall expansion occurs simultaneously with the growth of the cell. The growth phenomenon corresponds to a replication of the content, increases of the total mass of the cell and it leads under ideal conditions to cell division.

In this work, we focus on the growth process of the S. cerevisiae yeast that poliferates by budding. During a cell cycle, only one bud can be formed at a time, giving rise to a cell daughter after division. The growth procedure could be stimulated by biological, chemical or mechanical factors, among others. Biological and biochemical stimulated growth have been previously studied. The variation of the cell-wall composition during growth has also been investigated. According to [6], at the base of the emerging bud, a chitin is formed.

Several studies have been conducted on growth driven by the turgor pressure. In [7], Basu et al. explored the role of this pressure in the fission of Schizosaccharomyces pombe yeast, while Rojas et al. worked on bacteries, see [8]. Over the same period, two dynamic models have been developed on S. cerevisiae growth under turgor pressure, see [9]. These authors assumed that the yeast is a shell structure with an external radius of 2.5 μm and a wall thickness of 115 nm. They also hypothesized that the shell is pressurized by a turgor pressure estimated to 0.2 MPa. Furthermore, a mechanical feedback on the relation between cell-wall expansion and assembly is achieved in [10]. The budding yeast was assumed as a shell with a wall thickness ~100 nm, very small with respect to the bud diameter ~1 μm. In the present work, the emphasis will be on the purely mechanical aspects. The growth is stimulated by the stress state as a result of turgor pressure. Once the wall is acted upon this pressure, inplane mechanical tensions will be created and lead to the cell-wall expansion.

The remainder of this contribution is as follow: in Section 2, we provide a summary of the quasi-Kirchhoff structural shell theory in the finite strain range together with the kinematic choice based on the multiplicative decomposition of the deformation gradient; into reversible and growth parts. Later on, constitutive equations for the hyperelastic energy and the evolution equation of growth are given in accordance with continuum thermodynamic requirements. Section 3 is devoted to numerical simulations where a representative numerical example is shown together with a parametric study of the perhaps most important parameters; the growth threshold and the growth characteristic time. We end this paper with conclusions and perspectives.

2 Basic equations

Yeasts can be regarded as thin-walled shell structures and, due to their soft nature, a formulation within the finite strain range is herein adopted. We first review the basic kinematics that we extend toward finite growth. We next derive the constitutive equations that will be used later on for numerical simulations.

2.1 Shell kinematics with growth

As a starting point, use is made of the quasi-Kirchhoff-type theory for thin shells of revolution that has been derived in [11]. Briefly, denoting by E t he Green-Lagrange strain tensor¹, its non-zero components reduce to the meridional, E₁, the circumferential, E₂, and the transverse shear, E₁₃, that are valid for finite rotations together with large strains. They can be split into membrane (m), bending (b) and shear (s) parts as:

(1)

where ξ is the through-the-thickness coordinate. The different terms are given by, see [12] for full details,

(2)

with the notations r = s sin θ, e_θ = (u sin θ − w cos θ)/r, and c₂ = (sin θ sin ß + cos θ cos ß) /r², s being the arc length which represents the initial position. The displacement components u and w are relative to the local coordinate system, and ß is the in-plane rotation angle of the director d, see the sketch of Figure 1.

Next, as the shear strain E13 is set to zero by a penalty term, i.e. quasi-Kirchhoff shell theory, E₁ and E₂ are then regarded as principal strains. In this case, the principal stretches λ_i relative to the deformation gradient F follow from equation (1) 1 by considering the relation,

(3)

together with λ₃ = 1 .

Now for the extension toward finite growth, use is made in this work of the multiplicative decomposition of the deformation gradient into an elastic part F^e and a growth part F^g as proposed in [13], i.e. the sketch of Figure 2, see also [14,15],

(4)

From equation (4), a multiplivative decomposition of the stretches is then deduced,

(5)

where and are the principal stretches related to F^e and F^g, respectively.

Furthermore, it proves convenient to define the logarithmic strains of the above quantities as,

(6)

and, as well, the decomposition (5) implies,

(7)

Fig. 1 Kinematics for geometrically nonlinear axisymmetric shells [12].

Fig. 2 Local decomposition of the deformation gradient for finite growth F = FeFg.

2.2 Constitutive relation for stress

Hiere we assume isotropy on the stress-free configuration defined by F^g. This implies that the model is invariant relative to rigid body motions with respect to the orientations on thia configuration. Therefore, the strain enerfy function, herein denoted by ψ, could depend on the part F^e of the deformation gradient via the elastic left Cauchy-Green tensor The notation (.)^T is used for the transpose operator. The stress is then given by the following form of the state law,

(8)

where τ= Jσ is the Kirchhoff stress tensor, σ being the true Cauchy stress tensor, and J = det[F] > 0 is the Jacobian of the (total) deformation gradient.

The energy function ψ can be of any form of known hyperelastic naodels. (Here, we choose the following incompressible N = 1-Ogden-type model written in terms of principal stretches as,

(9)

with , µ is the shear modulus, and α is the Ogden’s coefficient. Notice that by taking α = 2, a neo-Hooke-type model is retrieved.

Next, the principal Kirchhoff stresses (the principal values of t ) are given by the corresponding form deduced from the tensorial expression (8):

(10)

where ϖ is the pressure related to the material incom-pressibility. It can be obtained by using the plane stress assumption τ₃ = 0 as,

as . And when substituted into equation (10), we obtain the following expression for the two remaining principal stresses:

(11)

for γ = 1, 2. Tlie definition (6)₂ has been used for the second equality, (11) ₂.

2.3 Growth evolution equation

Concomitantly, a constitutive model for the growth evolution is to be specified. From the continuum thermodynamics point of view, the related reduced dissipation, that we denote here by D^g, e.g. [14], is given in terms of principal quantities as, see [16,17] for similarities with finite strain viscoelasticity,

(12)

There can exist many possible growth evolution equations that fulfil the restriction (12). Far from being arbitrary, they may be either of a simple or a complicated form. At this stage, we propose the following form:

(13)

wehere, fo_f convenience, we have introduced the notation G(τ) for the growth criterion. Here (.) + is the positive part function, i.e. , and the notation ‖·‖ is used for the norm of a second-order tensor quantity.

Two essential parameters are here involved: a true stress-like growth threshold σ_trs, and a growth characteristic time In equation (13), the growth path follows the stress state through the unit tensor term τ/‖τ‖ Here the stress norm is given by .

The model can be interpreted as follows: when ‖σ‖ ≡ ‖τ‖/J > σ_trs, growth takes place. It is proportional to the stress in excess of the threshold, and its rate is controlled by the parameter .

Notice further from equation (13), that the stress is here the (only) biomechanical factor that stimulates growth.

In summary, the present cell-wall growth model requires four mechanical parameters:

• µ, the shear modulus;

• α, the Ogden’s coefficient;

• σ_trs, the true stress-like growth threshold;

• , the growth characteristic time.

3 Numerical simulations and parametric studies

The proposed formulation has been implemented in an in-house finite element software through the coding of a new UMAT-like sub-routine. In this section, we provide a set of numerical simulations. We choose spherical cells with initial mid-plane diameter D = 5 μm and wall thickness h = 100 nm. These parameters are chosen based on the literature, e.g. [18,19] among others. The cell-wall is assumed homogeneous with a Young’s modulus E = 120 MPa as found in [20,21]. This corresponds to a shear modulus µ = 40 MPa because of the incompressibility. Further, we take α = 2 for Ogden’s coefficient in all trie following computations in order to work with its Neo-Hookean version for the sake of simplicity.

To generate cell shape changes, we must specify:

• A budding zone at the tip of the cell. We choose for this the top zone as shown in Figure 3;

• And non-homogeneous mechanical properties. Here the bud will expand while the rest of the cell remains elastic.

Last, we must specify the turgor pressure that creates in-plane cell-wall tensions leading to the expansion. Here it is fixed to , for instance as this was mentioned in [9]. Let us stress that this pressure is deformation-dependent, i.e. a follower load [22,23].

Now for the budding zone, we assign the following growth parameters:

(14)

where the latter is of the order of timescales observed in budding kinetics, e.g. [10,24]. Table 1 b_elow summarizes the parameters used for a first simulation.

The axisymmetric analysis is performed with a mesh discretization using 480 uniformly distributed two-node linear shell elements (along the meridian), together with a reduced integration to avoid well-known shear locking, e.g. [12,23]. The boundary conditions only concern two nodes; the top and bottom nodes. For bottom node, the rotation and axiel displacement are prescribed to zero, while for the top node, only the rotation is imposed to zero. For the time discretization, the same time step of ∆t = 0.2 min has been used in all the simulations below.

Selected computed configurations are illustrated in Figure 4. As predicted, growth is initiated in the budding zone, Figure 4a, and continues with a large proliferation giving rise to a daughter cell, Figure 4b−c. Here, and during; all the growth evolution, the mother cell remains elastic _end almost undeformed aeter initiaily acnlying ehe turgor pressure.

A ^feries of compurgations is next performed by fixing the growth threshold to σ_trs = 1 MPa, and by using different values of the growth characteristic time, here , 45 and 60 min. The results are shown in the form of evolutions of the internal volume V_int relative to the initial one , see Figure 5. The first jump on the curves is due to the almost instantaneous application nf the pressure , here in 0.01 min, then maintained during the whole computations. Observe further that as the mother cell behaves elastically with low volume change, almost all the new volume is due to the growth in the daughter cell.

Likewise, a next set of computations is this time performed by fixing the characteristic time, here to , and with different values of the stress threshold, here with σ_trs = 0.9, 1, and 1.1 MPa. The results are plotted in Figure 6. One can observe that the present growth modeling framework clearly captures the influence of the stress excess with respect to the stress threshold on the growth kinetics. Notice that varying the stress threshold is equivalent to fixing this latter and varying the turgor pressure.

Fig. 3 Geometry of a spherical cell. The budding zone is shown at the top zone. Follower loads are used for the turgor pressure p.

Fig. 4 Typical deformed configurations with structural shell modeling after: (a) 40 min, (b) 70 min, and (c) 90 min. The material pa ram eters Eire those summarized in Table 1. The units of the graduations are scaled in (μm).

Fig. 5 Volume change due to growth for fixed stress growth threshold σtrs = 1 MPa, and with different growth characteristic times , 45 and 60 min.

4 Conclusions and perspectives

The aim of this contribution was to derive a mathematical modeling framework for growth in walled cells. On the one hand, a multiplicative decomposition of the deformation gradient has been adopted within the finite strain range and, on the other hand, a growth model has been proposed. Based on the purely mechanical aspects, this latter depends on perhaps the most important parameters; a stress threshold, and a growth characteristic time. The efficiency of the proposed framework has been highlighted through a set; of numerical examples with parametric studies.

We believe that this framework can trigger deeper research. For instance, this modeling can be enhanced by introducing further effects such as the polarization of the growth zones when coupled to more biological finds in the literature. In addition, the model is limited to the cell growth and does no include the separation of the cell daughter from the mother, which musí; be considered as a future research work. Moreover, an extension toward general shells will increase the field of applications, i.e. for any cell geometries.

Fig. 6 Volume change due to growth for fixed growth characteristic time min, and with different stress thresholds σtrs = 0.9, 1 and 1.1 MPa.

Funding

The Article Processing Charges for this article are taken in charge by the French Association of Mechanics (AFM)

Acknowledgments

The authors would like to thank the french ANR“Agence Nationale de la Recherche” for the financial support within the frame of Rheolife project (ANR-18-CE45-0012).

References

All Tables

All Figures

	Fig. 1 Kinematics for geometrically nonlinear axisymmetric shells [12].
In the text

	Fig. 2 Local decomposition of the deformation gradient for finite growth F = FeFg.
In the text

	Fig. 3 Geometry of a spherical cell. The budding zone is shown at the top zone. Follower loads are used for the turgor pressure p.
In the text

	Fig. 4 Typical deformed configurations with structural shell modeling after: (a) 40 min, (b) 70 min, and (c) 90 min. The material pa ram eters Eire those summarized in Table 1. The units of the graduations are scaled in (μm).
In the text

	Fig. 5 Volume change due to growth for fixed stress growth threshold σtrs = 1 MPa, and with different growth characteristic times , 45 and 60 min.
In the text

	Fig. 6 Volume change due to growth for fixed growth characteristic time min, and with different stress thresholds σtrs = 0.9, 1 and 1.1 MPa.
In the text