# Modulation of tissue growth heterogeneity by responses to mechanical stress

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Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved December 17, 2018 (received for review September 17, 2018)

## Significance

The two hands of most humans almost superimpose. Likewise, flowers of an individual plant have similar shapes and sizes. This is in striking contrast with growth and deformation of cells during organ morphogenesis, which feature considerable variations in space and in time, raising the question of how organs and organisms reach well-defined sizes and shapes. To link cell and organ scales, we built a theoretical model of growing tissue with fiber-like structural elements that may account for animal cytoskeleton or extracellular matrix, or for the plant cell wall. We show that the response of fibers to growth-induced mechanical stress may enhance or reduce cellular variability of growth, making it possible to modulate the robustness of morphogenesis.

## Abstract

Morphogenesis often yields organs with robust size and shapes, whereas cell growth and deformation feature significant spatiotemporal variability. Here, we investigate whether tissue responses to mechanical signals contribute to resolve this apparent paradox. We built a model of growing tissue made of fiber-like material, which may account for the cytoskeleton, polar cell–cell adhesion, or the extracellular matrix in animals and for the cell wall in plants. We considered the synthesis and remodeling of this material, as well as the modulation of synthesis by isotropic and anisotropic response to mechanical stress. Formally, our model describes an expanding, mechanoresponsive, nematic, and active fluid. We show that mechanical responses buffer localized perturbations, with two possible regimes—hyporesponsive and hyperresponsive—and the transition between the two corresponds to a minimum value of the relaxation time. Whereas robustness of shapes suggests that growth fluctuations are confined to small scales, our model yields growth fluctuations that have long-range correlations. This indicates that growth fluctuations are a significant source of heterogeneity in development. Nevertheless, we find that mechanical responses may dampen such fluctuations, with a specific magnitude of anisotropic response that minimizes heterogeneity of tissue contours. We finally discuss how our predictions might apply to the development of plants and animals. Altogether, our results call for the systematic quantification of fluctuations in growing tissues.

Variability has emerged as an inherent feature of many biological systems (1, 2), spanning molecular scales—such as in cytoskeletal dynamics (3)—to tissular scales—such as in organ expansion (4). For instance, cell growth was found to be spatially heterogeneous (5⇓⇓⇓–9), cell cycle length may appear random (10), and there is extensive evidence of stochastic gene expression (11, 12). Such variability has been hypothesized to be required for the emergence of complex shapes since it favors symmetry breaking (13) and self-organization (14) during development. Nevertheless, growth variability would need to be restrained to ensure robust morphogenesis. In plant tissues, an increase in the spatial correlations of growth fluctuations was shown to reduce the robustness of floral organ size and shape (15). In animal tissues, work on the wing imaginal disk of the fruit fly indicates that robust wing development involves cell competition and requires the modulation of cell division and apoptosis (16, 17).

Mechanical signals are natural candidates for the regulation of growth variability because spatial differences in growth or in deformation rates induce mechanical stress (18⇓–20). In animals, a mechanical feedback affecting the rate of cell divisions was hypothesized (21) and then supported by experiments in *Drosophila* and in zebrafish (22⇓⇓–25). Actomyosin cables are reinforced by mechanical tension in the wing imaginal disk of *Drosophila* (26). In plants, mechanical sensing is required to prevent growth fluctuations in roots (27). The deposition of cellulose fibers, the main load-bearing component of the cell wall, depends on wall tension (28, 29), which stiffens the cell wall in the direction of maximal tensile stress (30).

Previous theoretical studies have modeled how mechanical feedback regulates proliferation (21) and how transitions in tissue rheology are induced by proliferation and apoptosis (31, 32). Here, we build upon such studies; in addition, we account for small sources of stochasticity and investigate the consequences on large-scale tissue growth. We focus on generic aspects of tissue growth, so that our results may be broadly applicable to active matter (33).

## Results

### Growing Tissues as Mechanoresponsive Active Fluids.

We built a continuous 2D model of tissue growth. The tissue is assumed to be made of a material with a preferred orientation (i.e., fiber-like), accounting for its main mechanical elements: cytoskeleton, polar cell–cell adhesive junctions, and extracellular matrix (ECM) in animals and cellulose within the cell wall in plants. Hence, the state of the tissue is locally described by two order parameters, the density of fibers and the nematic field describing the orientation of fibers and their degree of alignment, which confer isotropic and anisotropic mechanical properties to the material, respectively. In our continuous description, we account for variations in density, orientation, and degree of alignment only at supracellular length scales, although the material may be patterned at smaller scales (subcellular or cellular). We account for fiber synthesis and remodeling, which may be modulated by responses to mechanical stress: reinforcement of actin stress fibers or of the ECM, enhancement of myosin activity, or fluidization by cell division in animals and increase in cell wall synthesis, cellulose synthesis, or cell division in plants. Synthesis has a small random component, considered as a stochastic, uncorrelated source. Stochasticity in synthesis induces growth heterogeneity, which results in mechanical stress and feeds back on synthesis. We use a viscous description of long-term tissue remodeling, so that we cannot account for short-term elastic tissue behavior, which would be better captured by an elastic model as performed in a parallel study (34). Formally, the model describes an expanding, mechanoresponsive, nematic, and active fluid.

#### A model of nematic viscous fluid.

We describe the fibers with a density ^{T} for the transpose of the preceding tensor.

We neglect diffusion of fibers in the tissue. The equations of continuity for density and nematic tensor are then**1** and **2** account for the dilution of material density and degree of alignment due to tissue expansion; the third term of Eq. **2** accounts for the rotation of fibers due to the flow.

Expansion of the tissue is assumed to be driven by a uniform and isotropic tension, p, which may correspond to turgor pressure in plants, or to a pressure induced by cell divisions in animals (31); this tension is one of the active components of our model. The mechanical stress,

In the following, we consider small fluctuations around an average state. The statistical averages of variables are denoted by brackets. For convenience, tensorial fields **3** and **4**, the first term accounts for viscous-like remodeling and the second term for tissue compressibility (changes in density and alignment under stress).

#### Activity: mechanical responses and fluctuations.

On the one hand, mechanical stress orients cell divisions (22, 23, 35) and plant cell wall reinforcement (30). On the other hand, synthesis of ECM or of cell wall and cytoskeleton polymerization are not uniform in space, having some level of randomness (3, 36, 37). The two types of phenomena are incorporated in the other active component of our model, namely synthesis. Without loss of generality, synthesis may be written at linear order in fluctuations as**5** and **6** describe the mechanical feedback on synthesis.

#### Dimensionless parameters.

We rescale all fields and variables as follows:**1**–**6** are given in *SI Appendix*.

This rescaling shows that the model has eight dimensionless parameters.

### Response to Perturbations in Synthesis.

The general formulation is given in *Materials and Methods* (we assume that the uniform state is stable). Here, we discuss tissue response to an isotropic perturbation that is localized in space—a disk of initial radius ℓ—and in time—a duration that is small with respect to all other time scales. Formally, the perturbation to density synthesis is *A*–*E* and are explicitly given in *SI Appendix*. An immediate consequence of the perturbation is to stiffen the tissue, which reduces expansion (*SI Appendix*, the tissue is hyporesponsive and all amplitudes evolve monotonously as a function of time and vanish at times that are long with respect to the correlation time

These dynamics occur on a time scale *SI Appendix*). The time scale *F* (see *SI Appendix*, Fig. S2 for the effect of other parameters). Isotropic mechanical feedback makes perturbations more persistent in time, because

### Growth Fluctuations.

We find that growth (areal strain rate in 2D) has long-range correlations, with a correlation function that spatially decays with an exponent *SI Appendix*). In practice, growth is measured at the scale of the spatial resolution of experimental measurements, which depends on the landmarks used and is often at cell scale. We therefore define a coarse-grained growth rate and we consider the time correlation function *Materials and Methods*) by *A* shows time correlations for high and low anisotropic mechanical feedback, respectively, corresponding to the hyporesponse and hyperresponse. The negative correlations for high feedback are related to underdamped relaxation of hyperresponsive tissues. The correlation function decays quickly to 0 with a characteristic time scale that is exactly the relaxation time, *A*. Areal growth mean-square deviation appears roughly scale invariant (Fig. 2*B*); it is exactly scale invariant for hyporesponse and oscillates around scale invariant for hyperresponse. Two regimes characterize the decay. In a weakly correlated regime, when *SI Appendix* for a rationale.

### Fluctuations of Organ Shape.

We are now interested in the effects of noise in synthesis on tissue contours or on organ shape. In a homogeneous and isotropic tissue, quantifying the fluctuation of contours is equivalent to determining the fluctuation of a vector joining two landmarks followed throughout growth of the tissue. Hence, we use a Lagrangian description and consider the position, *Materials and Methods*. Heterogeneity of contours is assessed using the coefficient of variation, *A*. Its asymptotic trend for long times depends on the value of the correlation time

We represent in Fig. 3*B* the maximal value of the coefficient of variation, normalized by *B* is qualitatively similar to the behavior of correlation time (*F*, indicating that the correlation time is a major determinant of heterogeneity because the correlation time sets how the tissue keeps the memory of its previous states.

## Discussion

We built a continuous viscous model of tissue growth, describing density and nematic order of the tissue, and modeled material synthesis and responses to mechanical stress. Here, the responses are characterized by two parameters,

In this study, we determined tissue response to a localized perturbation, depending on the mechanical feedback parameters *Arabidopsis* sepals (29). Here, we unraveled two possible regimes: hyporesponse at low anisotropic feedback—perturbations decay monotonously—and hyperresponse at high anisotropic feedback—perturbations oscillate before decaying, with a characteristic time that is minimal at the transition between the two regimes. This case study provides an assay of mechanical responses in both plant and animal systems, for instance by inducing clones with altered growth rate and quantifying the relaxation time scales in backgrounds with different levels of mechanical response.

We then investigated the statistical properties of tissue growth, unraveling long-range correlations, with slowly decaying correlation functions. To test this, it would be crucial to examine correlation functions in live imaging data of growing organs, e.g., refs. 15 and 44⇓⇓–47. Given that a larger correlation time (

Finally, we found that heterogeneity of contours and shapes is minimal for a well-determined level of anisotropic mechanical response. This generalizes a similar conclusion reached for local heterogeneity using a cell-based toy model (48). Here we also accounted for isotropic mechanical responses and considered heterogeneity at all scales. We identified the correlation time as a key parameter determining the extent of spatial correlations and the level of heterogeneity of organ shape. Based on our results, we make the following predictions. In plants (

## Materials and Methods

### Response to Perturbations in Synthesis.

We assume the tissue to be infinite, the perturbations to not induce rotation (*SI Appendix*), and the fields **7** is a sum over the values *SI Appendix*. We thus obtain the full response of the tissue to any perturbation of synthesis, in terms of the modified Fourier transform of the sources of density and of nematic order.

### Growth Fluctuations.

Using the linear response of flow velocity to synthesis perturbations, Eqs. **7** and **8**, we derived the velocity fluctuations, as detailed in *SI Appendix*. The correlation tensor of velocity fluctuations is proportional to the unit tensor,**7** and **8** and

### Fluctuations of Organ Shape.

We look for the probability **[9]**. We determined the asymptotic statistics of the Lagrangian flow by applying the saddle-point method (49) to P. P is maximized by the average trajectory *SI Appendix*)

## Acknowledgments

This work was supported by the AgreenSkills+ fellowship program, which has received funding from the European Union’s Seventh Framework Program under Grant Agreement FP7- 609398 (AgreenSkills+ contract, to A.F.), by the Department of Plant Biology and Breeding at Institut National de la Recherche Agronomique, and by Institut Universitaire de France (A.B.).

## Footnotes

- ↵
^{1}To whom correspondence may be addressed. Email: Arezki.Boudaoud{at}ens-lyon.fr or antoine.fruleux{at}ens-lyon.fr.

Author contributions: A.F. and A.B. designed research; A.F. performed research; and A.F. and A.B. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1815342116/-/DCSupplemental.

Published under the PNAS license.

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