In order to investigate the interaction between auxin cell wall softening and collective tissue mechanics, we use a vertex model to describe the mechanical behaviour of the tissue and a compartment model to express auxin concentration and transport between adjacent cells.
Geometrical set-up of the tissue
The tissue is described by a tiling of two-dimensional space into M cells surrounded by their cell walls. Walls are represented as edges connecting two vertices each, positioned at \(\mathbf {x}_i = \left( x_i,y_i\right) \) , \(i\in [1,N]\). Here, we reserve Latin indices for vertex numbering and Greek ones for cells. Each cell wall segment has two compartments, one facing each cell. Therefore, we represent each cell wall with two edges of opposite direction, one for each compartment. The position of tissue vertices fully define geometrical quantities such as cell areas, \(A_\alpha \), cell perimeters, \(L_\alpha \), wall lengths, \(l_{ij}=l_{ji}\), and cell centroids, \(\mathbf {X}_\alpha \) (Fig. 3 top left). To simplify notation significantly, we also define for each cell the cyclically ordered set of all vertices around that cell, \(\mathcal {V}_\alpha \), arranged counterclockwise (ccw). Hence, we use \(\sum _{i\in \mathcal {V}_\alpha }\) to signify the sum over all vertices surrounding cell \(\alpha \) with an arbitrary start, where \(i+1\) and \(i-1\) mean, respectively, the next and previous ccw vertex. Similarly, we introduce \(\mathcal {N}_\alpha \) as the cyclically ordered (counterclockwise) set of all neighbouring regions around cell \(\alpha \), one for each edge of \(\alpha \) (Fig. 3 top right).
Tissue mechanics–tissue-wide coupling
Vertex models are a widely employed theoretical approach to describe mechanics of epithelial tissues and morphogenesis [9, 48,49,50,51,52,53]. The essence of vertex models is that cell geometry within a tissue is given as the mechanical equilibrium of the tissue. In the case of plant cells, the shape of a cell is a competition between the turgor pressure, \(T_\alpha \), all cells exert on each other and the cell’s resistance to deformation with stiffness, \(E_\alpha \). Strain acting on each cell will be described using the second moment of area of the corresponding cell in reference to its centroid, \(M_\alpha \), whose components are
$$\begin{aligned}&M_{\alpha _{xx}} = \sum _{i\in \mathcal {V}_\alpha } \frac{n_i}{12}\left( x^{\prime ^2}_i + x^\prime _i x^\prime _{i+1} + x^{\prime ^2}_{i+1} \right) , \end{aligned}$$
(1)
$$\begin{aligned}&M_{\alpha _{yy}} = \sum _{i\in \mathcal {V}_\alpha } \frac{n_i}{12}\left( y^{\prime ^2}_i + y^\prime _i y^\prime _{i+1} + y^{\prime ^2}_{i+1} \right) ,\ \end{aligned}$$
(2)
$$\begin{aligned}&M_{\alpha _{xy}} = M_{\alpha _{yx}} = \sum _{i \in \mathcal {V}_\alpha } \frac{n_i}{24}\left( x^\prime _i y^\prime _{i+1} + 2 x^\prime _i y^\prime _i + 2 x^\prime _{i+1}y^\prime _{i+1} + x^\prime _{i+1} y^\prime _i \right) , \end{aligned}$$
(3)
where the primed coordinates represent the translation transformation, \(\mathbf {x}_i^\prime = \left( x^\prime _i,y^\prime _i\right) = \mathbf {x}_i - \mathbf {X}_\alpha \), and \(n_i = x^\prime _i y_{i+1}^\prime - x_{i+1}^\prime y_i^\prime , i\in \mathcal {V}_\alpha \). Given a rest shape matrix, \(M^{(0)}_\alpha \), we define cell strain as the normalized difference between both matrices,
$$\begin{aligned} \varepsilon _\alpha = \frac{M_\alpha - M_\alpha ^{(0)}}{\mathrm {Tr}\left( M_\alpha ^{(0)}\right) }, \end{aligned}$$
(4)
and stress with \(\sigma _\alpha = E_\alpha \varepsilon _\alpha \). Having described the tissue mechanically (Fig. 3 bottom left), we define the energy for a single cell as the sum of work done by turgor pressure and elastic deformation energy, resulting in the tissue mechanical energy,
$$\begin{aligned} \mathcal {H} = \sum _{\alpha =1}^M \left[ \frac{1}{2} A_\alpha E_\alpha \frac{\left| \left| M_\alpha - M_\alpha ^{(0)}\right| \right| _2^2}{\mathrm {Tr}^2\left( M_\alpha ^{(0)}\right) } - A_\alpha T_\alpha \right] . \end{aligned}$$
(5)
Using this model, we obtain the shape of the tissue by minimizing \(\mathcal {H}\) with respect to vertex positions.
After minimizing (Eq. 5), we quantify the stress acting on each wall through the average strain acting on each cell given by (Eq. 4). Assuming that cell wall rest length is the same between two adjacent wall compartments then it follows that they are under the same longitudinal strain, which is, to first approximation, the average between the two cells surrounding them. Therefore, longitudinal average strain acting on a specific wall used here is
$$\begin{aligned} \bar{\varepsilon }_{\alpha \beta } = \bar{\varepsilon }_{\beta \alpha } \sim \hat{\mathbf {t}}_{\alpha \beta }^T \frac{\varepsilon _\alpha + \varepsilon _\beta }{2} \hat{\mathbf {t}}_{\alpha \beta }, \end{aligned}$$
(6)
where \(\mathbf {t}_{\alpha \beta }\) is a unit vector along the wall separating cell \(\alpha \) and cell \(\beta \). Note that this interpolation assumes a continuous strain field. Then the stresses acting on each compartment are by the constitutive equation of a linear elastic isotropic material with Poisson ratio \(\nu =0\),
$$\begin{aligned} \sigma _{\alpha \beta } = E_\alpha \bar{\varepsilon }_{\alpha \beta } \ne \sigma _{\beta \alpha } = E_\beta \bar{\varepsilon }_{\beta \alpha }. \end{aligned}$$
(7)
Note that we are only considering the longitudinal components with regards to the cell wall, which means that \(\bar{\varepsilon }_{\alpha \beta }\) and \(\sigma _{\alpha \beta }\) are scalar quantities. More details on the mechanical model used can be found in the supporting text. As argued in the supporting material, our choice of \(\nu =0\) does not impact the qualitative behaviour studied here.
Tissue mechanics–uncoupled tissue approximation
To assess the impact of collective mechanical behaviour within a tissue on auxin pattern self-organization, we approximate the tissue-wide mechanical model to a static tissue geometry where we approximate the effects of turgor pressure of each individual cells in the static tissue by a constant average stress \(\bar{\sigma }\) acting on it [10]. Again assuming that both wall compartments have the same rest length, we infer that the stress acting on a particular wall depends only on \(\bar{\sigma }\) and the stiffness of the adjacent cells. Effectively, the average longitudinal strain acting on a wall surrounded by cells \(\alpha \) and \(\beta \) would simply be
$$\begin{aligned} \bar{\varepsilon }_{\alpha \beta } = \bar{\varepsilon }_{\beta \alpha }=2\bar{\sigma }/\left( E_\alpha + E_\beta \right) . \end{aligned}$$
(8)
This way, instead of minimizing the full mechanical model (Eq. 5) given a set of turgor pressures \(T_\alpha \) and rest shape matrices \(M_\alpha ^{(0)}\) we can, in the static tissue, immediately compute stress with Eq. 7 yielding,
$$\begin{aligned} \sigma _{\alpha \beta } = \frac{2E_\alpha \bar{\sigma }}{E_\alpha + E_\beta }. \end{aligned}$$
(9)
Interestingly, Eq. 9 is valid for \(\nu \ne 0\) as demonstrated in the supporting material.
In order to compare the two models, we choose the value of \(\bar{\sigma }\) to be the same as the stress obtained through minimisation of (Eq. 5), for a given set of \(T_\alpha \) and \(M_\alpha ^{(0)}\), with the constraint of the same end geometry.
Note that not only can this approximation be interpreted as the tissue being mechanically coupled only to the nearest neighbours, disregarding the rest of the tissue, (Fig. 2), but also as an analogous non-mechanical auxin concentration feedback model.
Auxin transport–compartment model
Compartment models for auxin transport are well adapted to the context of plant development, since the prerequisite of a boundary of a plant cell is particularly well defined by courtesy of the cell wall.
Although passive diffusion occurs across cell walls, the dominant players in auxin transport are membrane-bound carriers [22, 24]. Namely, efflux transporters of the PIN family are important due to their anisotropic positioning around a cell [16], which leads to a net auxin flow from one cell to the next. Let \(a_\alpha \) denote an non-dimensional and normalized average auxin concentration inside cell \(\alpha \). Following the model by [10], which is similar to previous mathematical models [25, 26, 29], auxin evolves according to auxin metabolism in the cell, passive diffusion between cells and active transport across cell walls via PIN,
$$\begin{aligned} \frac{\mathrm{d} {a}_\alpha }{\mathrm{d}t}= & {} \gamma ^* - \delta ^* a_\alpha + \mathcal {D} \sum _{\beta \in \mathcal {N}_\alpha } W_{\alpha \beta } \left( a_\beta - a_\alpha \right) \nonumber \\&+ \mathcal {P}\sum _{\beta \in \mathcal {N}_\alpha } W_{\alpha \beta } \left( p_{\beta \alpha } \frac{a_\beta }{K+a_\beta } - p_{\alpha \beta } \frac{a_\alpha }{K+a_\alpha }\right) , \end{aligned}$$
(10)
where \(\gamma ^*\) is the auxin production rate, \(\delta ^*\) is the auxin decay rate, \(W_{\alpha \beta } = l_{\alpha \beta }/A_\alpha \), with K, \(\mathcal {P}\), and \(\mathcal {D}\) as adjustable parameters. \(\mathcal {D}\) is the passive permeability of plant cells, whereas \(\mathcal {P}\) is permeability of the cell wall due to PIN-mediated transport of auxin, and K is the Michaelis–Menten constant for the efflux of auxin. More information on how this expression is derived can be found in the supporting text. Although this description ignores the auxin present within the extracellular domain and inside the cell wall, it has been shown that under physiological assumptions, this is a valid approximation [29]. The active transport term depends on the amount of bound PIN in each cell wall,
$$\begin{aligned} p_{\alpha \beta }= \frac{f_{\alpha \beta }}{1 + \sum _{\gamma \in \mathcal {N}_\alpha } \frac{l_{\alpha \gamma }}{L_\alpha } f_{\alpha \gamma }},\beta \in \mathcal {N}_\alpha , \end{aligned}$$
(11)
where \(f_{\alpha \beta },\beta \in \mathcal {N}_\alpha \) expresses the ratio between binding and unbinding rates of a particular wall (Fig. 3 bottom right). Note that \(p_{\alpha \beta }\) is different from wall to wall and from cell to cell. This means that in general, \(p_{\alpha \beta }\ne p_{\beta \alpha }\), or equivalently, \(p_{ij}\ne p_{ji}\). This is consistent with the fact that there are two compartments to a cell wall shared by two adjacent cells. Expression (Eq. 11) is based on the assumption that cell walls around a particular cell compete for the same pool of PIN molecules and that the amount of PIN scales with cell perimeter. This competition has been shown to be important in the polarization of PIN [29]. Alternatively, one could also scale the amount of PIN with cell size or not scale it at all. In the former case, smaller cells would be slightly preferred for auxin accumulation, whereas in the latter, larger cells would be preferred instead. Since we want to study the impact of stress patterns on the tissue, we want to decouple it from this effect as much as possible, choosing instead to scale the amount of PIN with perimeter.
The trivial fixed point of these dynamical equations is given by \(a_\alpha = \mu ^*/\delta ^*,\; \forall \alpha \), which also results in equal PIN density across all walls, provided turgor pressure \(T_\alpha \) and stiffness \(E_\alpha \) are the same across the tissue.
The feedback between tissue mechanics and auxin pattern unfolds as auxin transport affects tissue mechanics due to auxin, \(a_\alpha \), controlling cell wall stiffness, \(E_\alpha \), and in reverse tissue stress, \(\sigma _\alpha \), affects auxin transport by regulating PIN binding rates, \(f_{\alpha \beta }\), as hypothesized by [10, 11].
Mechanical regulation of PIN binding
According to the hypothesis presented by [10, 11], mechanical cues up-regulate PIN binding. The distinction between whether these mechanical cues are strain or stress has been studied recently by [54], yet the exact nature remains unclear. Following the model presented by [10], we model the binding-unbinding ratio, \(f_{\alpha \beta }\), as being a power law on positive stress,
$$\begin{aligned} f_{\alpha \beta } = f\left( \sigma _{\alpha \beta }\right) = {\left\{ \begin{array}{ll} \eta \left( \sigma _{\alpha \beta }\right) ^n, &{}\sigma _{\alpha \beta } > 0,\\ 0, &{} \sigma _{\alpha \beta }\le 0, \end{array}\right. } \end{aligned}$$
(12)
where the stresses, \(\sigma _{\alpha \beta }\), follow from tissue mechanics after minimization of the full mechanical model (Eq. 5), or, in the averaged stress approximation, it is the stress load on that particular compartment given by (Eq. 9). Furthermore, n is the exponent of this power law, and \(\eta \) captures the coupling between stress and PIN. Effectively, this mechanical coupling to PIN parameter corresponds to the sensing and subsequent response to stress, loosely translating into how much resources the cell needs to spend for processing stress cues.
Auxin-mediated cell wall softening
Auxin affects the mechanical properties of a cell wall via methyl esterification of pectin [6, 7], resulting in a decrease of the stiffness of the cell wall. We assume that all cell wall compartments surrounding cell \(\alpha \) share the same stiffness, \(E_\alpha \). To capture this effect, we model stiffness with a Hill function [10],
$$\begin{aligned} E_\alpha = E\left( a_\alpha \right) = E_0 \left( 1 + r \frac{1-a_\alpha ^m}{1+a_\alpha ^m}\right) , \end{aligned}$$
(13)
where \(r\in [ 0,1[\) which we define as the cell wall loosening effect, m is the Hill exponent of this interaction, and \(E_0\) is the stiffness of the cell walls when its auxin concentration is \(a_\alpha =1\). At low values of auxin, \(E_\alpha \) approaches the value \(\left( 1 + r \right) E_0\), whereas at high auxin concentration, \(E_\alpha \) approaches \(\left( 1 - r\right) E_0\). Given a distribution of auxin, we can compute the wall stiffness in (Eq. 5) from (Eq. 13), or the stress acting on a specific compartment in (Eq. 9) for the approximated model.
Integrating auxin transport and tissue mechanics
At each time step, \(\varDelta t\), starting from an auxin distribution, we compute the stiffness of each cell according to (Eq. 13). Then, with the input of all turgor pressures, we minimize (Eq. 5) to obtain tissue geometry and stresses acting on each wall. Auxin concentration in each cell will evolve according to (Eq. 10), where the active transport term will be regulated by stress according to (Eq. 12) via (Eq. 11). A new auxin distribution will result at the end of this iteration, and we will be ready to take another time step (Fig. 4). We repeat this process until \(t=t_\mathrm {max}\).
Implementation
We implemented this model with C++ programming language, where we have used the Quad-Edge data structure for geometry and topology of the tissue [55], implemented in the library Quad-Edge [56]. In order to minimize the mechanical energy of the tissue, we have used a limited-memory Broyden–Fletcher–Goldfarb–Shanno algorithm (L-BFGS) [57, 58], implemented in the library NLopt [59]. For solving the set of ODEs presented in the compartment model, we used the explicit embedded Runge–Kutta–Fehlberg method (often referred to as RKF45) implemented in the GNU Scientific Library (GSL) [60]. We wrapped the resulting classes into a python module with SWIG. For additional details regarding the parameters used for the simulations of the following section, consult Table S1 in the supporting material.
Observables
In order to quantify the existence of auxin patterns, we compute the difference between an emerging auxin concentration pattern and the trivial steady state of uniform auxin concentration pattern defined as \(a_\alpha = \gamma ^*/\delta ^*,\; \forall \alpha \). To account for a large range of orders of magnitude of auxin concentration, we consider as an order parameter,
$$\begin{aligned} \varphi = \frac{\left\langle \ln ^2\left( a_\alpha \right) \right\rangle _M}{\delta ^2 + \left\langle \ln ^2\left( a_\alpha \right) \right\rangle _M}, \end{aligned}$$
(14)
where \(\left\langle \cdot \right\rangle _M\) denotes an average over all cells within the tissue. This way, \(\varphi \approx 0\) means that there are no discernible patterns, whereas \(\varphi \approx 1\) implies prominent auxin patterning. The term \(\delta ^2\) defines the sensitivity of this measure, such that an average deviation of \(\delta \) yields \(\varphi \approx 1/2\) (for small \(\delta \)). We will choose \(\delta =0.1\), i. e. , a 10% deviation from the trivial steady state.
We also keep track of the average of auxin above basal levels in order to gauge the potential degree of modulation of auxin-mediated cell behaviour.
Furthermore, to characterize cells with regards to PIN localization we introduce the magnitude of the average PIN efflux direction,
$$\begin{aligned} F_\alpha = \left| \left| \sum _{i\in \mathcal {V}_\alpha }\frac{l_{ii+1}}{L_\alpha } p_{ii+1} \hat{\mathbf {n}}_{i i+1}\right| \right| , \end{aligned}$$
(15)
where \(\hat{\mathbf {n}}_{ii+1}\) is the unit vector normal to the wall pointing outwards from \(\alpha \).
Aside from a global measure of auxin patterning, it is also important to locally relate auxin to tissue mechanics. Namely, for auxin we are interested in auxin concentration, \(a_\alpha \), and auxin local gradient, obtained by interpolation,
$$\begin{aligned} \nabla a_\alpha = \frac{1}{2A^*_\alpha }\sum _{\gamma \in \mathcal {N}_\alpha } \begin{pmatrix} Y^\prime _{\gamma +1} &{} -Y^\prime _\gamma \\ -X^\prime _{\gamma + 1} &{} X^\prime _\gamma \end{pmatrix} \begin{pmatrix} a_\gamma - a_\alpha \\ a_{\gamma +1} - a_\alpha \end{pmatrix}, \end{aligned}$$
(16)
where \(\mathbf {X}^\prime _\gamma = \left( X^\prime _\gamma ,Y^\prime _\gamma \right) = \mathbf {X}_\gamma - \mathbf {X}_\alpha \) and
$$\begin{aligned} A^*_\alpha = \frac{1}{2}\sum _{\gamma \in \mathcal {N}_\alpha } \left( X^\prime _\gamma Y^\prime _{\gamma +1} - Y^\prime _\gamma X^\prime _{\gamma +1}\right) . \end{aligned}$$
(17)
In fact, the quantity \(\left| \nabla a_\alpha \right| \) can be used as an indicator of whether there is an interface between auxin spots and the rest of the tissue.
With regards to tissue mechanics, the local quantities we quantify are the isotropic component of stress,
$$\begin{aligned} P_\alpha = \frac{1}{2}\mathrm {Tr}\left( \sigma _\alpha \right) , \end{aligned}$$
(18)
and the stress deviator tensor projected along the direction of the auxin gradient,
$$\begin{aligned} D_\alpha = \frac{\nabla a_\alpha ^T \sigma '_\alpha \nabla a_\alpha }{\left| \nabla a_\alpha \right| ^2}, \end{aligned}$$
(19)
where \(\sigma ^\prime _\alpha = \sigma _\alpha - I P_\alpha \), and I is the identity matrix. Therefore, \(P_\alpha \) is a measure if a cell is being compressed (\(P_\alpha <0\)), or pulled apart (\(P_\alpha >0\)), and \(D_\alpha \) translates into if a cell is more compressed along the auxin gradient than perpendicular to it (\(D_\alpha <0\)), or vice-versa (\(D_\alpha >0\)).
Finally, to measure the disruption of an auxin pattern we approximate entropy by means of a Riemann sum,
$$\begin{aligned} S\left[ \varPi \right] = -\sum _{i=-\infty }^\infty \varPi \left( i \varDelta a\right) \varDelta a\ln \left( \varPi \left( i\varDelta a\right) \varDelta a\right) , \end{aligned}$$
(20)
where \(\varPi \left( a\right) \) is the probability density function of auxin and \(\varDelta a\) the partition size. Note that it is only meaningful to compare entropy measures obtained with the same partition size \(\varDelta a\). Here, the probability density function of auxin concentration is obtained by applying a kernel density estimation on the resulting tissue auxin values. Note that \(\varPi \left( a\right) \) is a continuous function. In order to infer it from simulation data, for each auxin value in the tissue, \(a_\alpha \), we add Kernel functions \(K_w(a)\), obeying \(\int _{-\infty }^\infty K_w(a) da = 1\) and \(K_w(a) = K_w(-a)\). Then we can estimate
$$\begin{aligned} \varPi \left( a\right) \sim \frac{1}{M}\sum _{\alpha =1}^M K_w(a-a_\alpha ), \end{aligned}$$
(21)
where w is a smoothing parameter defining the width of the Kernel, this parameter is sometimes called bandwidth. This statistical tool is called kernel density estimation (KDE) [61]. We use the Epanechnikov kernel because it is bounded and we can force \(\varPi \left( a\right) = 0, a\le 0\).