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# Theory of mechanochemical patterning in biphasic biological tissues

Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved January 28, 2019 (received for review August 3, 2018)

## Significance

Pattern formation is a central question in developmental biology. Alan Turing proposed that this could be achieved by a diffusion-driven instability in a monophasic system consisting of two reacting chemicals. In this paper, we extend Turing’s work to a more realistic mechanochemical model of multicellular tissue, modeling also its biphasic and mechanical properties. Overcoming limitations of conventional reaction–diffusion models, we show that mechanochemical couplings between morphogen concentrations and extracellular fluid flows provide alternative, non-Turing, mechanisms by which tissues can form robust spatial patterns.

## Abstract

The formation of self-organized patterns is key to the morphogenesis of multicellular organisms, although a comprehensive theory of biological pattern formation is still lacking. Here, we propose a minimal model combining tissue mechanics with morphogen turnover and transport to explore routes to patterning. Our active description couples morphogen reaction and diffusion, which impact cell differentiation and tissue mechanics, to a two-phase poroelastic rheology, where one tissue phase consists of a poroelastic cell network and the other one of a permeating extracellular fluid, which provides a feedback by actively transporting morphogens. While this model encompasses previous theories approximating tissues to inert monophasic media, such as Turing’s reaction–diffusion model, it overcomes some of their key limitations permitting pattern formation via any two-species biochemical kinetics due to mechanically induced cross-diffusion flows. Moreover, we describe a qualitatively different advection-driven Keller–Segel instability which allows for the formation of patterns with a single morphogen and whose fundamental mode pattern robustly scales with tissue size. We discuss the potential relevance of these findings for tissue morphogenesis.

How symmetry is broken in the early embryo to give rise to a complex organism is a central question in developmental biology. To address this question, Alan Turing proposed an elegant mathematical model where two reactants can spontaneously form periodic spatial patterns through an instability driven by their difference in diffusivity (1). Molecular evidence of such a reaction–diffusion scheme in vivo remained long elusive, until pairs of activator–inhibitor morphogens were proposed to be responsible for pattern formation in various embryonic tissues (2⇓⇓⇓⇓⇓⇓–9). Interestingly, these studies also highlight some theoretical and practical limitations of existing reaction–diffusion models, including the fact that Turing patterns require the inhibitor to diffuse at least one order of magnitude faster than the activator (

In this article, we derive a general mathematical formulation of tissues as active biphasic media coupled with reaction–diffusion processes, where morphogen turnover inside cells, import/export at the cell membrane, and active mechanical transport in the extracellular fluid are coupled together through tissue mechanics. While encompassing classical reaction–diffusion results (1⇓⇓–4), for instance allowing import–export mechanisms to rescale diffusion coefficients and to form patterns with equally diffusing morphogens (11), this theory provides multiple routes to robust pattern formation. In particular, assuming a generic coupling between intracellular morphogen concentration and poroelastic tissue mechanics, we demonstrate the existence of two fundamentally different non-Turing patterning instabilities, respectively assisted and driven by advective extracellular fluid flows, explaining pattern formation with only a single morphogen with robust scaling properties and how patterning can be independent of underlying morphogen reaction schemes. Finally, we discuss the biological relevance of such a model and in particular its detailed predictions that could be verified in vivo.

## Results

### Derivation of the Model.

As sketched in Fig. 1*A*, we model multicellular tissues as continuum biphasic porous media of typical length l, with a first phase consisting of a poroelastic network made of adhesive cells of arbitrary shape and typical size *SI Appendix*.

#### Intracellular morphogen dynamics.

Morphogens enable cell–cell communication across the tissue and determine cell fate decisions. Importantly, most known morphogens cannot directly react together and, as such, have to interact “through” cells (or cell membranes) where they are produced and degraded (20). Concentration fields of two morphogens,

#### Extracellular fluid dynamics.

Next, we write a mass conservation equation for the incompressible fluid contained in the tissue interstitial space between cells,**2** with *SI Appendix*, section 1.A.3).

As detailed below, we assume that local cellular morphogen concentrations have an influence on the volume fraction ϕ which couples tissue mechanics to local morphogens concentration in our theory. At linear order, this coupling generically reads *SI Appendix*, section 1.A.4), with

#### Extracellular morphogen dynamics.

Morphogens, once secreted by cells, are transported by diffusion and advection in the extracellular fluid,*SI Appendix*, section 1.C). Note that one could also take into account, at the mesoscopic level, some effective nonlocal interactions such as cell–cell communication via long-ranged cellular protrusions (30). This may require one to consider spatial terms in Eq. **1** to introduce an additional characteristic length scale from nonlocal cell–cell transport.

#### Mechanical behavior of the cellular phase.

To complete our description, we need to specify a relation linking cell volume fraction to interstitial fluid velocity. For this, we use a poroelastic framework, whose applicability to describe the mechanical response of biological tissues has been thoroughly investigated in various contexts (31, 32). Taking a homogeneous tissue as a reference state, poroelastic properties imply that a local change of the cell volume fraction creates elastic stresses in the cellular phase which translate to gradients of extracellular fluid pressure p. Such gradients of pressure in turn drive extracellular fluid flows, which can advect morphogens, and we show (*SI Appendix*, section 1.A.7) that this effects results in a simple Darcy’s law between cell volume fraction and fluid flow (29):**3**, with κ the tissue permeability, K the elastic drained bulk modulus, and η the fluid viscosity. The hydrodynamic length scale *SI Appendix*, section 1.H) the role of growth and plastic cell rearrangements and show that they can be readily incorporated in our model, leading to different types of patterning instabilities. However, we highlight here that the results presented thereafter are all robust to small to intermediate levels of tissue rearrangements.

### Model of an Active Biphasic Tissue.

Eqs. **1**–**4** define a full set of equations describing the chemo-mechanical behavior of an active biphasic multicellular tissue (*SI Appendix*, section 1.B). To provide clear insights on the biophysical behavior of the system, we focus on a limit case where *SI Appendix*, section 1.C). Summing both internal Eq. **1** and external Eq. **3** conservation laws, we obtain a simplified description of the system (*SI Appendix*, section 1.C):**5** is controlled by a few nondimensional parameters:

### Orders of Magnitude on Morphogen Transport.

In the simplest limit of the model, the cell fraction remains constant, *SI Appendix*, section 1.F).

In Fig. 1*B*, we depict scaling arguments for the changes in effective diffusion coefficient at various time/length scales, associated with both tissue structure and import/export kinetics (11). At small time scales, diffusion is characterized by a local Fickian diffusion coefficient, theoretically expected to be of the order of *SI Appendix*, section 1.H.

Overall, although our model in its simplest limit (

### Turing–Keller–Segel Instabilities.

To assess the regions in parameter space where stable patterns can form in our mechanochemical framework, we perform a linear stability analysis on Eq. **5**. Here, we consider a classical Gierer–Meinhardt activator–inhibitor scheme (2),

In the phase diagram in Fig. 2*A*, we show that two distinct instabilities can be captured by this simplified theory. The first instability, identified here as “Turing patterns,” corresponds to a classical Turing instability, where diffusive transport of morphogens dominates over their advection by interstitial fluid (*A*) which, as expected, is always true regardless of the value of *A* (36). The physical origin of the resulting pattern is here similar to active fluid instabilities (15, 17, 37⇓⇓–40): If stochastic local changes in morphogen concentration result in an increase in cell volume fraction, fluid must be pumped inside cells. This causes local elastic deformations in the tissue which generate large-scale extracellular fluid flows from regions of low to high morphogen concentration, resulting in a positive feedback loop of morphogen enrichment (Fig. 3*A*) and steady-state patterns. Interestingly, such an instability can occur even for a single morphogen. In this limit, patterning occurs if *A*, which captures well the phase boundary in the limit *B* can be predicted by linear analysis (*SI Appendix*, section 1.D) because they are chosen close to the onset of instability.

Thus, coupling tissue mechanical behavior to morphogen reaction–diffusion provides, via the generation of advective fluid flows, a route to stable pattern formation with a single morphogen. Moreover, this instability has two remarkable features. First, it requires only the presence of a single morphogen (*SI Appendix*, section 1.G) which could correspond to many practical situations where an activator/inhibitor pair has not been clearly identified, for instance the role of Wnt in the antero-posterior pattern of planarians (41). Second, it possesses spatial scaling properties regarding to its fundamental mode, compared with a Turing instability. Indeed, when morphogen turnover rate is small compared with its effective hydrodynamic and Fickian diffusion (*SI Appendix*, section 1.G.2), whereas in the case of a Turing instability, this would require fine-tuning and marginally stable reaction kinetics. We illustrate such a scaling property in Fig. 3. This mechanism could potentially apply to situations where a binary spatial pattern is independent of system size such as dorso-ventral or left–right patterns in early vertebrate embryos (7, 9) or planarian antero-posterior patterns (41, 42). If so, it could provide a simpler alternative to previously proposed mechanisms involving additional species or complex biochemical signaling pathways (7, 42).

Importantly, simple estimates can be used to demonstrate the biological plausibility of such mechanical effects during morphogenetic patterning. A key parameter driving Keller–Segel instabilities is the hydrodynamic diffusion coefficient

### Cross-Diffusion Turing Instabilities.

Finally, we investigate the behavior of our model (Eq. **5**), when cell fraction sensitivity to morphogen concentration is negative (**5**. This relates to a realistic biological situation, where cell volume fraction relaxes rapidly after perturbation and depends weakly on morphogen levels, yielding*SI Appendix*, section 1.E). Such a scenario has been studied in the framework of monophasic reaction–diffusion systems with ad hoc cross-diffusion terms (43), which arise generically in various chemical and biological systems (44). Our work thus provides a particular biophysical interpretation of these terms in multicellular tissues, which we show to originate from intrinsically mechanochemical feedbacks between morphogen dynamics and tissue mechanics.

As shown in ref. 43, such cross-diffusion terms result in a dramatic broadening of the phase space for patterns. In particular, any two-morphogen reaction scheme can now generate spatial patterns and not just the classical activator–inhibitor schemes. For instance, it becomes possible to obtain patterns with activator–activator or inhibitor–inhibitor kinetics similar to those observed in numerous gene regulatory networks or signaling pathways involved in cell fate decisions (45). We illustrate this result by considering an inhibitor–inhibitor kinetic scheme, which cannot yield patterns in the classical Turing framework, and demonstrate analytically and numerically the existence of a region of stable patterns (from Eq. **5**), where a cross-diffusion–driven Turing instability can develop (Fig. 4).

## Discussion

In this paper, we have introduced a generalization of Turing’s work on pattern formation in biological tissues by coupling equations describing the structure and mechanical properties of multicellular tissues with a classical reaction–diffusion scheme. In particular, our work highlights two important features of multicellular tissues, as of yet largely unexplored in this context: their biphasic nature, i.e., the fact that morphogen production/degradation is controlled by cells while transport takes place extracellularly requiring active membrane exchanges [effectively rescaling diffusion (9, 11)], and the possibility for active large-scale flows to develop within the tissue interstitial space. We demonstrate that coupling tissue cell volume fraction to local morphogen levels [based on the dual role of morphogens in patterning and cell growth/volume regulation (23, 24)] provides a biophysically realistic route toward two qualitatively different modes of patterning instability. Extracellular fluid flows can have two important consequences on patterning. First, as the Turing instability is rooted in the cross-effects between a stable chemical reaction of two morphogens and their diffusion, the conditions of such instability are deeply affected by active hydrodynamic transport which can create cross terms in the effective diffusion matrix. This causes a drastic widening of the phase space of Turing patterning, rendering it robust and only weakly dependent on the morphogen reaction scheme. Second, extracellular fluid flows can also create an instability of a different nature (Keller–Segel), when these flows have an antidiffusive structure, spontaneously creating morphogen gradients. Here, chemical reactions between morphogens are setting only the number of patterns, and if such reactions are sufficiently slow, the spatial pattern of the morphogen always coarsens to the fundamental mode of instability and has robust scaling properties compared with conventional Turing models. This could have interesting implications concerning recent experimental evidence for robust scaling of the Nodal/Lefty pattern in the early zebrafish embryo (46).

In this respect, our approach, which has the advantage of parsimony, taking into account the manifest biphasic nature of multicellular tissues, is complementary to others which have been proposed to solve limitations of Turing’s model by introducing additional morphogen regulators (42, 47) and also displays connections with recent developments in the mechanochemical descriptions of active fluids such as the cell cytoskeleton (15, 16). Nevertheless, although our hypothesis of cell volume fraction gradients driving large-scale flows is generic to biphasic tissues, further quantitative experiments would be needed to test the relationship between morphogen concentration and cell volume fraction, as well as probe the role of transmembrane import/export kinetics or similar phenomena such as transmembrane signaling (11), morphogen adsorption/desorption on the cell surface (9), and long-distance cellular protrusions (30), on effective morphogen diffusion rates. Systems such as digits patterning, where the cell volume fraction spatial pattern appears concomitant to morphogen patterns (26), or planarian antero-posterior patterning, where activator/inhibitor pairs have not been clearly identified (41), provide possible testing grounds for our model. Interestingly, large-scale extracellular fluid flows have been increasingly observed during embryo development, not only in the classical case of cilia-driven flows (48), but also due to mechanical forces arising from cellular contractions as well as osmotic and poro-viscous effects (49, 50), calling for a more systematic understanding of passive vs. active transport mechanisms during embryonic pattern formation. Whether biological examples of Turing patterning instabilities, such as left–right or dorso-ventral patterning, digits pattern formation, or skin appendage patterns, are causally associated with concomitant changes in cell volume and/or cell packing remains a result to be experimentally investigated.

## Methods

Linear stability analysis was performed numerically using Mathematica, while numerical integrations of the model equations were performed using a custom-made Matlab code.

## Acknowledgments

The authors warmly thank Benjamin Simons, Anna Kicheva, David Jörg, Tom Hiscock, Pau Formosa-Jordan, Erik Clark, and Lev Truskinovsky for useful discussions. P.R. acknowledges the support of a CNRS-Momentum grant. A.H. is supported by a Wellcome Trust Junior Interdisciplinary Research Fellowship and the Wellcome Trust Grant 098357/Z/12/Z. A.H. also acknowledges the support of a David Crighton Fellowship of the University of Cambridge and of the core funding to the Gurdon Institute and the Cambridge Stem Cell Institute. E.H. acknowledges the support of the Austrian Science Fund (FWF) [P 31639].

## Footnotes

↵

^{1}P.R., A.H., and E.H. contributed equally to this work.- ↵
^{2}To whom correspondence may be addressed. Email: edouard.hannezo{at}ist.ac.at, pierre.recho{at}univ-grenoble-alpes.fr, or ah691{at}cam.ac.uk.

Author contributions: P.R., A.H., and E.H. designed research, performed research, and 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.1813255116/-/DCSupplemental.

- Copyright © 2019 the Author(s). Published by PNAS.

This open access article is distributed under Creative Commons Attribution License 4.0 (CC BY).

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