Regulation of plant cell wall stiffness by mechanical stress: a mesoscale physical model

Abstract

A crucial question in developmental biology is how cell growth is coordinated in living tissue to generate complex and reproducible shapes. We address this issue here in plants, where stiff extracellular walls prevent cell migration and morphogenesis mostly results from growth driven by turgor pressure. How cells grow in response to pressure partly depends on the mechanical properties of their walls, which are generally heterogeneous, anisotropic and dynamic. The active control of these properties is therefore a cornerstone of plant morphogenesis. Here, we focus on wall stiffness, which is under the control of both molecular and mechanical signaling. Indeed, in plant tissues, the balance between turgor and cell wall elasticity generates a tissue-wide stress field. Within cells, mechano-sensitive structures, such as cortical microtubules, adapt their behavior accordingly and locally influence cell wall remodeling dynamics. To fully apprehend the properties of this feedback loop, modeling approaches are indispensable. To that end, several modeling tools in the form of virtual tissues have been developed. However, these models often relate mechanical stress and cell wall stiffness in relatively abstract manners, where the molecular specificities of the various actors are not fully captured. In this paper, we propose to refine this approach by including parsimonious biochemical and biomechanical properties of the main molecular actors involved. Through a coarse-grained approach and through finite element simulations, we study the role of stress-sensing microtubules on organ-scale mechanics.

Appendices

Appendix A: The tensor ramp function

Here the tensor function $\left( \cdot \right) _+$ is defined on the set of the symmetric second order tensors. For such tensor $\varvec{T}$ (of dimension d), the tensor rank decomposition of $\varvec{T}$ reads:

$$\begin{aligned} \varvec{T} = \displaystyle \sum _ {n=1}^d \lambda _n \varvec{t}_n \otimes \varvec{t}_n \end{aligned}$$

where $\lambda _n$ are the eigenvalues of $\varvec{T}$, and $\varvec{t}_n$ are corresponding normalized eigenvectors. We define:

$$\begin{aligned} \left( \varvec{T} \right) _ + = \displaystyle \sum _ {n=1}^d \max \left( \lambda _n,0 \right) \varvec{t}_n \otimes \varvec{t}_n, \end{aligned}$$

which assures that $\left( \varvec{T} \right) _ +$ is a positive symmetric tensor.

Appendix B: The stiffness tensor as a function of the microfibril distribution

This appendix details Eq. (6), that establishes a relation between the fiber organization and the associated stiffness, as described in Cox (1952).

We consider that microfibrils behave like linear springs (of rest length $l_0$ and stiffness k). Moreover, by assuming affine deformation, their deformation reads:

$$\begin{aligned} \frac{\Delta l \left( \theta \right) }{ l_0 } \simeq \varvec{e}_{\theta }\cdot \left( \varvec{E}\cdot \varvec{e}_{\theta } \right) = \varvec{E}: \varvec{e}_{\theta } \otimes \varvec{e}_{\theta } = \varvec{E}: \varvec{\Theta }. \end{aligned}$$

where $\varvec{\Theta }=\varvec{e}_{\theta } \otimes \varvec{e}_{\theta }$ is the projector on direction $\theta$ (’$\otimes$’ depicts the tensor outer product). The previous expression yields the total stretching energy density per unit volume of fibers (summed over all directions):

$$\begin{aligned} u = \frac{1}{2} Y_{\text {f}}\int _ {\pi } \mathrm {d}\theta \rho \left( \theta \right) \left( \varvec{E}: \varvec{\Theta }\right) ^2 = \frac{1}{2} \varvec{E}: Y_{\text {f}}\int _ {\pi }\mathrm {d}\theta \rho \left( \theta \right) \varvec{\Theta }\otimes \varvec{\Theta }: \varvec{E}\end{aligned}$$

(17)

with:

$$\begin{aligned} Y_{\text {f}}= \frac{kl_0^2}{h} \end{aligned}$$

(where h is the wall’s thickness). Deriving twice the energy yields the stiffness tensor:

$$\begin{aligned} \mathbb {C}_{\text {f}}= \frac{\partial ^2 u}{\partial ^2 \varvec{E}} = Y_{\text {f}}\int _ {\pi } \mathrm {d}\theta \rho \left( \theta \right) \varvec{\Theta }\otimes \varvec{\Theta }\end{aligned}$$

or in the basis $(e_x, e_y)$:

$$\begin{aligned} \left( \mathbb {C}_{\text {f}}\right) _{ijkl} = Y_{\text {f}}\int _ {\pi } \mathrm {d}\theta \rho \left( \theta \right)&\cos \left( \theta \right) ^{\Delta _{ijkl}} \sin \left( \theta \right) ^{4 - \Delta _{ijkl} } \end{aligned}$$

with $\Delta _{ijkl} = \delta _{i,1} + \delta _{j,1} + \delta _{k,1} + \delta _{l,1}$, where ’$\delta$’ stands for the Kronecker delta function. Linearizing the cosine-sine products yields an expression of $\mathbb {C}_{\text {f}}$ involving the $0^{\mathrm{th}}$, $1^{\mathrm{st}}$ and $2^{\mathrm{nd}}$ order Fourier coefficients:

$$\begin{aligned} \left[ \mathbb {C}_{\text {f}}\right] =\frac{\pi Y_{\text {f}}\rho _0}{16} \left[ \begin{array}{ccc} 3+\frac{\rho _2+4\rho _1}{\rho _0} &{}\quad 1-\frac{\rho _2}{\rho _0} &{}\quad \frac{2\tilde{\rho }_1+\tilde{\rho }_2}{\rho _0} \\ 1-\frac{\rho _2}{\rho _0}&{}\quad 3+\frac{\rho _2-4\rho _1}{\rho _0} &{}\quad \frac{2\tilde{\rho }_1-\tilde{\rho }_2}{\rho _0}\\ \frac{2\tilde{\rho }_1+\tilde{\rho }_2}{\rho _0} &{}\quad \frac{2\tilde{\rho }_1-\tilde{\rho }_2}{\rho _0} &{}\quad 1-\frac{\rho _2}{\rho _0}\\ \end{array}\right] \end{aligned}$$

where, for any natural number n:

$$\begin{aligned} \hat{\rho }_n = \int _{\pi } \frac{\mathrm {d}\theta }{\pi } \rho \left( \theta \right) e^{-2 i n \theta } \quad \text {and} \quad \left( \rho _n, \tilde{\rho }_n\right) = 2 \left( \mathfrak {R}\left( \hat{\rho }_n \right) , -\mathfrak {I}\left( \hat{\rho }_n \right) \right) . \end{aligned}$$

Appendix C: Modeling cellulose deposition via CSC trajectories

We here develop a kinetic model for CSC-microtubule binding, providing a chemical background to the cellulose deposition model. Here, cellulose deposition is controlled through the orientation of the CSC trajectories depicted by an a priori non-uniform $\pi$-periodic distribution $\mu \left( \theta \right)$ that depicts the trajectories of proteins (regardless of their sense).

$$\begin{aligned} \mathrm {d}{ _t} \rho \left( \theta \right) = k_{\rho }^0 \mu \left( \theta \right) - k'_{\rho }\rho \left( \theta \right) , \end{aligned}$$

(18)

We assume that a constant pool of CSCs is available, characterized by a constant concentration $\mu _{\text {tot}}$. Proteins can be either free or bound to a microtubule. The free proteins are assumed to follow individual Brownian trajectories within the membrane bi-layer; for that the related cellulose deposition is isotropic, characterized by a concentration $\mu _{\text {free}}$. The contribution of the bound proteins is characterized by the distribution $\mu _{\text {bound}} \left( \theta \right)$ that depicts the trajectories of proteins on microtubules. We assume that CSCs are active regardless of their binding state, implying that cellulose deposition occurs even in the absence of microtubules (Emons et al. 2007). Microtubules solely introduce a bias in the distribution of trajectories $\mu$. The total concentration of CSCs reads:

$$\begin{aligned} \int _{\pi } \mu = \mu _{\text {free}} + \int _{\pi } \mu _{\text {bound}}=\mu _{\text {tot}}. \end{aligned}$$

(19)

The kinetics of CSC-microtubule binding is modeled by a multiplicative coupling between the concentration of microtubules in a given direction $\theta$ and the concentration of available free proteins, along with an isotropic liberation rate:

$$\begin{aligned} \mathrm {d}{ _t} \mu _{\text {bound}} \left( \theta \right) = k_{\mu } \mu _{\text {free}} \phi ^{\infty } \left( \varvec{S}, \theta \right) - k'_{\mu }\mu _{\text {bound}} \left( \theta \right) \end{aligned}$$

(20)

where $k_{\mu }$ and $k'_{\mu }$ are kinetic constants. Eqs. (19) and (20) lead to the steady-state expression:

$$\begin{aligned} \mu _{\text {bound}} ^{\infty } (\theta ) =\mu _{\text {tot}} \cdot \frac{ K_{\mu }\phi ^{\infty } (\varvec{S},\theta ) }{ 1+ K_{\mu }\int _ {\pi } \phi ^{\infty }} \end{aligned}$$

(21)

where $K_{\mu } = {k_{\mu }}/{k_{\mu }'}$ depicts the affinity of CSCs for microtubules. By combining the previous steady-state distribution of CSCs with the expression of cellulose deposition (Eq. (18)), we obtain:

$$\begin{aligned} \mathrm {d}{_t}\rho (\theta ) = k_{\rho }^0 \mu _{\text {tot}}\cdot \frac{1+K_{\mu }\phi ^{\infty } (\varvec{S},\theta )}{ 1+K_{\mu } \int _ {\pi } \phi ^{\infty }} - k'_{\rho }\rho (\theta ) \end{aligned}$$

(22)

that can be roughly simplified to obtain Eq. (7) by assuming that for all $\theta$, $K_{\mu }\phi ^{\infty } (\varvec{S},\theta ) \gg 1$ (strong affinity):

$$\begin{aligned} \mathrm {d}{_t}\rho (\theta ) = k_{\rho }^0 \mu _{\text {tot}} \cdot \frac{\phi ^{\infty } (\varvec{S},\theta )}{ \int _ {\pi } \phi ^{\infty }} - k'_{\rho }\rho (\theta ). \end{aligned}$$

(23)

The product $k_{\rho }^0 \mu _{\text {tot}}$ defines the kinetic constant $k_{\rho }$ used in the main text.

Appendix D: Details about the specific expression of the microtubule depolymerization probability

Eq. (10) derives from Arrhenius law: a chemical reaction rate ($k_{\text {A}}$ hereafter) is a direct function of its activation energy, i.e. the energetic barrier ($E_{\text {A}}$) the system has to overcome to go from one chemical state to another:

$$\begin{aligned} k_{\text {A}}=k_{0}e^{-\frac{E_{\text {A}}}{RT}}. \end{aligned}$$

The product RT corresponds to the thermal energy of the considered molecules are embedded in (R and T respectively stand for the Arrhenius constant and the absolute temperature).

This general experimental law has been amended latter on in the context of biochemistry and cell adhesion in the following way (Bell 1978). The energy barrier molecules have to overcome can be lowered or raised when mechanical forces are applied to them. The resulting expression reads:

$$\begin{aligned} k_{\text {B}}=k_0 e^{-\frac{E_{\text {A}} \pm \alpha f}{RT}} \end{aligned}$$

where f depicts the force applied to the considered molecules and $\pm \alpha$ stands as a coupling parameter. Clearly, the sign of this coupling parameter depends on the action of the force: if the latter eases the reaction, then the coupling parameter must be negative, so the activation barrier is lowered and the transition rate is increased.

Our working hypothesis is that mechanical forces applied to microtubules reduce their depolymerization rate; we therefore postulated a positive coupling parameters. Simplifying the notation as follows:

$$\begin{aligned} {\left\{ \begin{array}{ll} k'_{\phi }{^0}{:}{=}k_{0}e^{-\frac{E_{\text {A}}}{RT}}\\ \gamma {:}{=}\frac{\alpha }{RT} \end{array}\right. } \end{aligned}$$

and considering that $E_{\text {A}}$ is proportional to force, leads to Eq. (10).

Appendix E: Microtubule principal axis and anisotropy at steady regime

Generally, the microtubule distribution $\phi$ is summarized through the angle of its mean orientation ($\theta _{\phi }$) and through its anisotropy ($\alpha _{\phi }$), respectively given by:

$$\begin{aligned} \theta _{\phi }= - \frac{\arg \hat{\phi }_1}{2} ~~~\text {and}~~~ \alpha _{\phi }= \frac{| \hat{\phi }_1 |}{\hat{\phi }_0} \end{aligned}$$

(24)

where ’$\arg$’ is the complex argument, and $\{ \hat{\phi }_n \}_{n\in \mathbb {N}}$ depict the Fourier coefficients of $\phi$. We define $\theta _{\rho }$ and $\alpha _{\rho }$ in the same way, using the microfibril distribution $\rho$.

The steady-state Fourier sequence of microtubules is derived from Eq. (11) and read:

$$\begin{aligned} \forall n \in \mathbb {N}: \quad \hat{\phi }_n^{\infty }= \frac{c_0}{\pi } \frac{I_n \left( 2 \gamma | \hat{f}_1 | \right) }{\frac{e^{- \gamma f_0}}{\pi K_{\phi }}+I_0 \left( 2 \gamma | \hat{f}_1 | \right) } e^{- 2 i n\theta _{S}} \end{aligned}$$

(25)

where $\theta _{S}$ depicts the angle of main stress, and where functions $\left\{ I_n \right\} _{n\in \mathbb {N}}$ depict the modified Bessel functions of the first kind (Abramowitz and Stegun 1972):

$$\begin{aligned} \forall n \in \mathbb {N}, \forall x \in \mathbb {R}: I_n \left( x \right) = \int _ {\pi } \frac{\mathrm {d}\theta }{\pi } e^{x \cos \theta } \cos n \theta . \end{aligned}$$

By expressing the complex argument of $\hat{\phi }_1$ from the previous expression, one sees that $\theta _{\phi }^{\infty }=\theta _{S}$. This means that microtubules align with the main direction of stress. This alignment is moreover symmetric, according to this direction (with the parametrization $\theta \leftarrow \theta - \theta _{S}$, the imaginary part of the Fourier coefficients vanishes).

The anisotropy ratio $\alpha _{\phi }^{\infty }$ (Fig. 3b) reads:

$$\begin{aligned} \alpha _{\phi }^{\infty }= \frac{I_1 \left( 2\gamma | \hat{f}_1 | \right) }{I_0 \left( 2 \gamma | \hat{f}_1 | \right) }. \end{aligned}$$

(26)

which is a growing function of $| \hat{f}_1 |$ (if $\gamma > 0$). According to the previous expression, the steady-state anisotropy of microtubules depends on stress only, and not on the chemical kinetics.

Appendix F: Details on the simulation pipeline

We detail the numerical procedure employed in Sect. 3.3. All simulations have been performed in the framework presented in Boudon et al. (2015), based on the open-source softwares Sofa (Faure et al. 2012) and OpenAlea (Pradal et al. 2008). The numerical procedure employed in this paper is adapted from Boudon et al. (2015).

1.1 F.1 Mechanical equilibrium

1.1.1 F.1.1 Space discretization

1.1.1.1 Finite element procedure

We discretize the continuous model through triangular finite elements. Current material positions $\varvec{x}$ are expressed according to the mesh nodes $\varvec{x}_i$ based on a Lagrange $\mathbb {P}_1$ barycentric interpolation:

$$\begin{aligned} \varvec{x} = \sum _ i \xi ^i \left( \varvec{X} \right) \varvec{x}_i \end{aligned}$$

where $\xi ^ i \left( \varvec{X} \right)$ depicts the barycentric shape function associated with node i and evaluated at $\varvec{X}$, the 2D barycentric coordinates of the material point in the reference configuration. Spatial differentiation yields one $3\times 2$ barycentric deformation gradient tensors per triangle e:

$$\begin{aligned} \varvec{F}_e = \sum _ i \varvec{x}_ i \otimes \nabla _{\varvec{X}} \xi ^i \left( \varvec{X} \right) \end{aligned}$$

(27)

(operator $'\otimes '$ depicts the vector outer product, $\varvec{a} \otimes \varvec{b} = \varvec{a} \cdot \varvec{b}^T$). The 2D Green-Lagrangian strain tensor and the second Piola-Kirchhoff stress are derived from $\varvec{F}$ according to:

$$\begin{aligned} \varvec{E}_e = \frac{1}{2} \left( \varvec{F}_e^T \cdot \varvec{F}_e - \varvec{I} \right) \quad \text {and} \quad \varvec{S}_e = \mathbb {C}_{w,e} : \varvec{E}_e, \end{aligned}$$

yielding the internal energy of the system (that is only due to deformation):

$$\begin{aligned} U = \frac{h}{2} \sum _e \mathcal {A}_e \varvec{E}_e : \varvec{S}_e = \frac{h}{2} \sum _e \mathcal {A}_e \varvec{E}_e : \mathbb {C}_{w,e} : \varvec{E}_e, \end{aligned}$$

where h and $\mathcal {A}_e$ respectively stand for the wall thickness (that is uniform) and the surface of triangle e. Nodal inner forces are obtained by considering the infinitesimal work due to an infinitesimal displacement of the node, which, by virtue of the first law of thermodynamics (in adiabatic conditions), leads to:

$$\begin{aligned} \varvec{f}^{\text {int}}_i = - \frac{\partial U}{\partial \varvec{x}_i}. \end{aligned}$$

(28)

The inner forces are derived from pressure and expressed at each node v according to its adjacent triangles $\mathcal {T} (v)$:

$$\begin{aligned} \varvec{f}^P_v = \frac{P}{3} \sum _ {e \in \mathcal {T} (v)} \mathcal {A}_e \varvec{n}_e, \end{aligned}$$

where $\varvec{n}_e$ stands for the outer normal of triangle e.

Meshes All meshes have been generated through the open source software Blenderwww.blender.org. At the exception of the cylindrical stem (for which the original mesh was satisfactory), they have been enhanced via a surface smoothing procedure implemented in the non-commercial meshing tool Graphite (alice.loria.fr/software/graphite).

1.1.2 F.1.2 Time discretization

Computing the equilibrium (Eq. (3)) consists in minimizing the total energy U (see Boudon et al. 2015, for further details). This is performed dynamically by writing the quasi-static law of motion for each node i:

$$\begin{aligned} \varvec{f}^P_i + \varvec{f}^{\text {int}}_i \simeq \mu \frac{\mathrm {d}\varvec{x}_ i}{\mathrm {d}t} \end{aligned}$$

(where $\mu$ is a dam** coefficient that can be integrated into the time step). The previous equation is solved implicitly with the conjugate gradient algorithm.

To improve stability and to avoid excessive stiffness inhomogeneity, the tensor $\varvec{\varvec{S}}$ employed in Eq. 11 is modified on each element and integrates a weighted average of the neighboring stress tensors.

1.2 F.2 Dynamical resolution of Eq. 7

The dynamics of the microfibril distribution is computed by an iterative update of the first three Fourier coefficients of $\rho$ through the following forward Euler time scheme with constant time step $\Delta t$ (Eq. 7 and Fig. 8):

$$\begin{aligned} \forall n: \quad \hat{\rho }_n^{t+\Delta t}= \left( 1 - k'_{\rho }\Delta t \right) \hat{\rho }_n^{t} +\frac{2k_{\rho }\Delta t}{\pi } \frac{\hat{\phi }_n^{\infty }\left( \varvec{S}^t\right) }{\phi _0^{\infty }\left( \varvec{S}^t\right) }. \end{aligned}$$

(29)

The stiffness tensor $\mathbb {C}_{\text {w}}$ is subsequently updated according to Eqs. 4 to 6 and 25.

At $t = t_0$, the system is set to its null-stress equilibrium, derived from Eqs. 11 and 14:

$$\begin{aligned} \phi \left( \theta , t_0 \right) = \frac{c_0 K_{\phi }}{1+ K_{\phi }\pi }~~~\text {and}~~~\rho \left( \theta , t_0 \right) = \frac{K_{\rho }}{\pi }. \end{aligned}$$

(30)

Fig. 8

Simulation pipeline. The initial state corresponds to the null stress state (no residual stress and no loading). The elastic equilibrium is computed after loading the structure with pressure $P>0$. At each elastic equilibrium, the mechanical properties are updated, which brings the systems out of equilibrium

1.3 F.3 Visualization of the microfibril distributions in Figs. 5 to 7

We adapted the concept of nematic tensor from the physics of liquid crystals (De Gennes and Prost 1995) by defining:

$$\begin{aligned} \varvec{Q}_{\phi }= 4 \int _{\pi } \frac{\mathrm {d}\theta }{\pi } \phi \left( \theta \right) \varvec{\Theta }= \phi _0 \left[ \begin{array}{cc} 1 +\frac{\phi _1}{\phi _0}&{}\quad \frac{\tilde{\phi }_{1}}{\phi _0} \\ \frac{\tilde{\phi }_{1}}{\phi _0} &{}\quad 1 - \frac{\phi _1}{\phi _0} \end{array}\right] . \end{aligned}$$

(31)

Notice that, as stated in 3.1, the previous definition builds a formal equivalence between the definition of anisotropy based on Fourier coefficients (Eqs. (13) and (24)) and the nematic-based ratio employed in Boudaoud et al. (2014) (Eq. (12)). In fact, the matrix expression of $\varvec{Q}_{\phi }$ becomes diagonal in the basis associated with the rotation that zeroes the odd Fourier coefficient $\tilde{\phi }_1$ (in this basis $| \hat{\phi }_1| = \hat{\phi }_1 = \phi _1 / 2 \in \mathbb {R}^+$). Since $| \hat{\phi }_1|$ and $\phi _0$ are base-invariant, the two eigenvalues of $\varvec{Q}_{\phi }$ are equal to $\phi _0 \pm 2 | \hat{\phi }_1|$, which gives:

$$\begin{aligned} \frac{Q_{\phi , 1} - Q_{\phi , 2}}{Q_{\phi , 1} + Q_{\phi , 2}} = \frac{2 | \hat{\phi }_1| }{\phi _0} = \frac{ | \hat{\phi }_1| }{ \hat{\phi }_0} \end{aligned}$$

(32)

$\varvec{Q}_{\phi }$ is equivalent to an ellipse with main axis oriented by $\theta _{\phi }= - \nicefrac {1}{2} \arg \hat{\phi }_1$ (Eq. 24), and with aspect ratio and semi-major axis respectively given by $\left( 1 - \alpha _{\phi }\right) /\left( 1 + \alpha _{\phi }\right)$ (with $\alpha _{\phi }={| \hat{\phi }_1 |}/{\hat{\phi }_0}$) and $\phi _0 \left( 1 + \alpha _{\phi }\right)$ (see Fig. 9).

Fig. 9

Ellipse-based visualization of the nematic tensor associated with the distribution $\phi$