Hydromechanical field theory of plant morphogenesis
Hadrien Oliveri Ibrahim Cheddadi















































mpipz,mpicbg,csbd]Hadrien Olivericor1

inst2,rdp]Ibrahim Cheddadicor2


 [cor1] mailto:holiveri@mpipz.mpg.deholiveri@mpipz.mpg.de

[cor2]mailto:ibrahim.cheddadi@univ-grenoble-alpes.fribrahim.cheddadi@univ-grenoble-alpes.fr

[mpipz]organization=Max Planck Institute for Plant Breeding Research,
            
            city=Cologne,
            postcode=50829, 
            
            country=Germany

[mpicbg]organization=Max Planck Institute of Molecular Cell Biology and Genetics,
            
            city=Dresden,
            postcode=01307, 
            
            country=Germany


[csbd]organization=Center for Systems Biology Dresden,
            
            city=Dresden,
            postcode=01307, 
            
            country=Germany
            
[inst2]organization=Université Grenoble Alpes, CNRS, Grenoble INP, TIMC,
            
            city=Grenoble,
            postcode=38000, 
            
            country=France

[rdp]organization=Laboratoire de Reproduction et Développement des Plantes, Université de Lyon, ENS de Lyon, UCBL, INRAE, CNRS, Inria,
            
            city=Lyon,
            postcode=69364, 
            
            country=France

§ ABSTRACT The growth of plants is a hydromechanical phenomenon in which cells enlarge by absorbing water, while their walls expand and remodel under turgor-induced tension. In multicellular tissues, where cells are mechanically interconnected, morphogenesis results from the combined effect of local cell growths, which reflects the action of heterogeneous mechanical, physical, and chemical fields, each exerting varying degrees of nonlocal influence within the tissue. To describe this process, we propose a  physical field theory of plant growth. This theory treats the tissue as a poromorphoelastic body, namely a growing poroelastic medium, where growth arises from pressure-induced deformations and osmotically-driven imbibition of the tissue. From this perspective, growing regions correspond to hydraulic sinks, leading to the possibility of complex non-local regulations, such as water competition and growth-induced water potential gradients. More in general, this work aims to establish  foundations for a mechanistic, mechanical field theory of morphogenesis in plants, where growth arises from the interplay of multiple physical fields, and where biochemical regulations are integrated through specific physical parameters.












plant mechanics morphogenesis growth  elasticity morphoelasticity poroelasticity





[2020] 74B20 74F10 74F20 92B99
























Einfach schlief in dem Samen die Kraft; ein beginnendes Vorbild
 Lag, verschlossen in sich, unter die Hülle gebeugt,

Blatt und Wurzel und Keim, nur halb geformet und farblos; 
 Trocken erhält so der Kern ruhiges Leben bewahrt, 
 Quillet strebend empor, sich milder Feuchte vertrauend,

Und erhebt sich sogleich aus der umgebenden Nacht.Johann Wolfgang von Goethe, Die Metamorphose der Pflanzen, 1798




§ INTRODUCTION





 §.§ Modelling the growth of plant tissues

Morphogenesis is the fundamental process by which biological organisms establish their shape. This phenomenon relies upon multiple genetic, chemical, mechanical, and physical regulations, which are typically multiscale, thermodynamically open, out-of-equilibrium, and coupled in a so-called complex system. 

To achieve a rational understanding of morphogenesis, dedicated models are indispensable; in particular, a grand challenge is to build mechanistic continuum theories that describe the emergent physical behaviour of living tissues at the macroscopic scale.

In plants, the key phenomenon driving morphogenesis is growth, the general process by which a physical body deforms by gaining mass <cit.>. Indeed, in plant tissues, adjoining cells are mechanically interconnected and constrained by an extracellular cell wall matrix which prevents cell migration and topological rearrangements <cit.>. Therefore, complex organ shapes result from the anisotropic and heterogeneous growth (and division) of the cells, generating large solid deformations of connected tissue regions <cit.>.


The theory of morphoelasticity is a modern mechanical theory of growth, which has emerged historically from the nonlinear field theories of elasticity and elastoplasticity  <cit.>. 


This framework has been adapted with remarkable success to a wealth of biological systems <cit.>, and provides a natural framework to model plant growth at the tissue scale <cit.>. Historically, an important problem in growth theory has been the characterisation of growth-induced mechanical instabilities, which refer to the loss of stability and  change in configuration undergone by a growing tissue as a result to growth-induced internal stresses <cit.>. In this context, the growth deformation is treated typically as a constant bifurcation parameter controlling the onset of instability. In this respect, the focus lies on the static  elasticity problems arising from growth, rather than the physical origin and kinetics of growth itself. 
In contrast, the so-called problem of morphodynamics consists in studying growth as a variable of a dynamical system whose evolution reflects the physical state of the tissue, reflecting the inherently dynamic nature of developmental processes  <cit.>.


From a biological point of view, plant growth is controlled dynamically through a multitude of genes and hormones that form heterogeneous patterning fields <cit.>. To address the role of genetic patterning in morphogenesis, continuum computational models have been proposed, based on the so-called notion of specified growth, detailed in <cit.> and applied in various context, see 
<cit.> <cit.>. Specified growth refers to the intrinsic growth a volume element undergoes when isolated from the rest of the tissue, reflecting some local biological identity (e.g. gene expression, polarity or, more abstractly, the action of morphogens) which control the kinematics of growth. Then, given a specified growth field, a compatible deformation can be obtained in a second step, by minimising the body's elastic energy. 
This approach has been instrumental in demonstrating how spatially heterogeneous gene expression patterns control the emergence of complex shapes in three-dimensions. However, from a biophysical perspective, specified growth remains a fundamentally phenomenological and essentially kinematic representation of growth, as its precise link to cellular mechanics–specifically the explicit physical action of genes–is not fully characterised.





In contrast, the aim of biophysics is to explain growth from physical and mechanical principles. In this context, physically-based discrete models  describe tissues at the cellular level, incorporating detailed cellular structure and geometry, mechanical anisotropies, and fine mechanisms underlying growth   <cit.>. The goal in these models is then to reproduce the growth phenomenon from first principles, thereby allowing for rigorous testing of hypotheses at the cellular scale through  specific biophysical parameters.  However, this type of approach relies on a discretised representation of the tissue, thus it is often doomed to remain computational. While cellular models offer detailed insights, they do not enjoy the generality, minimalism, scalability, and analytic tractability of a continuum mathematical theory.



 §.§ The hydromechanical basis of cell growth


To build such a theory on physical grounds, we start with the basic physics of cell expansion. Plant cells grow by absorbing water from their surrounding—a process driven by their relatively high osmolarity <cit.>. This phenomenon is captured by the historical model of <cit.>, later expanded in a somewhat similar manner by <cit.> and <cit.> <cit.>. This model describes the elongation of a single cylindrical cell of length ℓ. The expansion rate ℓ̇ of such cell can be expressed in terms of the volumetric influx of water through <cit.>

    ℓ̇= k_c ℓπ - p ,

with k_c  the effective hydraulic conductivity the cell;  π and p respectively the excess osmotic and hydrostatic pressures with respect to the outside; and where the overdot denotes differentiation w.r.t. time t. The quantity ψ = p-π, the water potential of the cell relative to the outside, characterises the capacity of the cell to attract water. Note that (<ref>) does not define a closed system, as the pressure p is related to the mechanics of the cell wall via the balance of forces. The growth of plant cells is a complex process that involves loosening, yielding under tension, and remodelling of the cell walls <cit.>. This process is  modelled as a Maxwell-type viscoelastoplastic creep. In an elongating cylindrical cell, stress can be explicitly written in terms of pressure, and this rheological law can be  written as

    ℓ̇/ℓ  = Φp - y + ṗ/E ,

with xmaxx,·  the ramp function; y a threshold pressure above which growth occurs; Φ the so-called extensibility of the cell; and E the effective Young's modulus of the cell. Equations (<ref>) now form a closed system for ℓ and p, illustrating that the turgor pressure is not a control parameter of the rate of growth, but a dependent variable of the cell expansion process. Indeed, in the steady growth regime (ṗ = 0), the pressure is fully determined by the three parameters y, π and α_c   k_c / (k_c + Φ), and can be eliminated from the system:

    ℓ̇/ℓ = α_cΦπ - y,    
        p = α_cπ +  (1-α_c)y

(with y≤ p ≤π).







Recently, the Lockhart model served as a starting point for a vertex-based, multicellular hydromechanical model of plant morphogenesis, including water movements between cells, and mechanically-driven cell wall expansion 
<cit.>. This generalisation of the single-cell model generates several additional emergent spatiotemporal effects that arise from the coupling between wall expansion and water movements within a large developing tissue, such as hydraulic competition effects and water gradients. 





Overall, the relationship between water and growth, as highlighted by the Lockhart-Ortega-Cosgrove model and its multidimensional generalisation, is central to the physical functioning of plant living matter, and the basic notion that water must travel from sources (e.g. vasculature) to growing regions is a fundamental principle  of development.




 §.§ Hydromechanical field theory for plant growth

Building on this notion, we here develop a physical field theory of plant morphogenesis including tissue mechanics, wall synthesis, and hydraulics. Viewing the tissue as a network of cells exchanging water, a natural formalism is the theory of porous media, which describes the deformations of fluid-saturated materials. The modern theory of porous media has a complex history originating in the seminal works of Fick and Darcy on diffusion, and Fillunger, Terzaghi and Biot on soil consolidation, formalised later within the context of the theory of mixtures, notably based on contributions by Truesdell and Bowen  <cit.>. There is now a rich body of literature exploring the application of these concepts to biology in various contexts, including biological growth. 
However, despite the clear significance of water in plant mechanics <cit.>, the role of hydromechanics in the context of plant growth modelling has but marginally been examined. Several notable  works have featured linear poroelastic descriptions of plant matter, occasionally including growth <cit.>; however, these studies are limited to simple geometries and have not culminated in a general three-dimensional theory of growth. While the limiting role of water fluxes within plant tissues has long been debated <cit.> and remains challenging to assess experimentally, recent years have seen a resurgence of interest in this problem, with new studies supporting the potential for hydromechanical control in tissue development <cit.>. 
Building on these recent advancements, we propose a hydromechanical model for plant tissues seen as morphoelastic, porous materials encompassing the solid matrix of the cells and their fluid content. 
Technically speaking, our approach is analogous to previous models, for instance for tumour growth <cit.> or brain oedema <cit.>, however the specificities of plant living matter demand a dedicated treatment. Overall, this work aims to bridge the conceptual gap between cell biophysics and growth at the tissue scale, by describing growth as a consequence of simultaneous mechanical deformations and imbibition of the cells, thereby extending the seminal analysis of Lockhart to the framework of continuum media.  

The goals are then (i) to establish theoretical foundations for a generic theory of plant growth within the framework of nonlinear continuum mechanics (<ref>); and (ii) to characterise the guiding principles that emerge from multiple couplings between mechanical, physical and hydraulic fields (<ref>).

 















§ GENERAL THEORY











 §.§ Geometry and kinematics







  §.§.§ Total deformation

We describe the geometry and kinematics of the evolving tissue viewed as a continuum domain embedded in the three-dimensional space <cit.>. We introduce the initial configuration ℬ_0 ⊂ℝ^3 at time t=0 of the body, and its current configuration ℬ⊂ℝ^3 at t≥ 0; see <ref>(a). The deformation of the body from ℬ_0 to ℬ is described by a smooth map χ: ℬ_0 ×ℝ→ℬ that sends material points X⃗∈ℬ_0 onto spatial points x⃗ = χ(X⃗,t)∈ℬ, at a given time t∈ℝ. 
The deformation gradient tensor is defined as F χ,



with  the material (or Lagrangian) gradient, with respect to the reference configuration. 






Finally, the Jacobian determinant of the deformation, J  F, measures the local volumetric expansion due to χ (i.e. v = J  V). 






Next, we describe the kinematics of the deformation by introducing the material and spatial (or Eulerian) velocities, respectively V⃗ and v⃗, defined as V⃗ (X⃗, t)=v⃗ (x⃗, t)= χ (X⃗, t)t.




A natural measure of the relative rate of deformation is then the gradient of spatial velocity,

    L  v⃗ = Ḟ F^-1,

with (·) =  F^-⊤ (·)  the Eulerian gradient with respect to the current configuration. 
We also recall the standard kinematic formulae
J̇ / J= v⃗ =
   L, 
with  the Eulerian divergence and  the trace operator. 



Here, the overdot denotes the material derivative J̇ = Jt, related to the Eulerian derivative Jt through the generic formula

    J̇ = Jt + v⃗· J.






  §.§.§ Growth


We model growth using the framework of morphoelasticity <cit.>. The fundamental postulate of morphoelasticity is the multiplicative decomposition of deformations into an elastic deformation tensor A and a growth tensor G   <cit.>: 

    F =  A   G ;
 
see <ref>(a).
Constitutively, the growth deformation G is assumed to be anelastic, meaning it does not contribute to the strain energy, and describes the configurational change in the local reference configuration of the body due to local mass addition. In contrast, the elastic deformation A generates stresses and enters the strain energy function. Note that there is no requirement for the growth deformation to be compatible, that is, in general there is no deformation embeddable in the Euclidean space of which 𝐆 is the gradient, unlike F which is compatible by definition <cit.>.  
We define the Jacobian determinants J_A  A and J_G  G, so that (<ref>)

    J = J_AJ_G;
 
and the rate of growth L_G Ġ G ^-1 and rate of elastic deformation L_A Ȧ A ^-1, with

    L =  L_A +    A  L_G  A^-1 .

Finally, we introduce Γ L_G = J̇_G/J_G, measuring the relative rate of volumetric expansion due to growth.

Note that, in the context of plant morphogenesis, most authors have restricted their attention to linear elasticity. In contrast, as we do not require A or G to be small here, our approach is geometrically exact and can capture arbitrarily large strains, and rich elastic behaviours such as strain-stiffening effects <cit.>.








 §.§ Balance laws




  §.§.§ Balance of mass


We model the growing tissue as a poromorphoelastic solid representing the cell matrices and the water content; see <ref>(b). Specifically, the tissue is viewed as a triphasic porous medium composed of a solid phase representing the extracellular cell wall matrix (`s'), a pure fluid phase (`f'), and the cytoplasmic osmolytes (`o'). We define v⃗_s, v⃗_f, and v⃗_o, the Eulerian velocities of the respective phases, with the solid velocity chosen as the reference:

    v⃗_s = v⃗.

We denote with ϕ_s, ϕ_f, ϕ_o the volume fractions of the solid, fluid and osmolyte components, respectively, in the current configuration (that is, the true volume occupied by each component per unit volume of  tissue). Assuming that the material is saturated, we have

    ϕ_s + ϕ_f + ϕ_o = 1.

The pore space volume occupancy is measured by the Eulerian porosity ϕ 1 - ϕ_s, which represents the ratio of pore space volume to current tissue volume. Similarly, the intermediate porosity Φ J_A ϕ measures the current pore space volume per unit intermediate volume. The Lagrangian porosity Φ^0  Jϕ= J_G Φ measures the current pore space volume per unit reference volume. 

Here, we assume that all phases are inherently elastically incompressible <cit.>. This assumption implies that in a  macroscopic elastic deformation, the pores will shrink or dilate so as to conserve the volume of the solid phase. Thus, we can relate the solid volume fraction in the unstressed configuration (Φ_s, the solid volume fraction observed upon relieving the stresses) to that in the current configuration ϕ_s, through

    Φ_s = J_Aϕ_s = J_A - Φ .






During growth and remodelling of the tissue, cells secrete new solid material, divide and form new cell walls that may affect the structure of the cell matrix locally,  thus, the porosity. To represent this effect, we assume generally that the solid volume fraction may evolve as

    Φ̇_s  = ℋ(x⃗, t, Φ_s,  G,…),

where ℋ is a function of the different variables that symbolises the local evolution of the matrix structure during growth. For example, in a scenario where cells do not divide but maintain a constant cell wall thickness, we can assume ℋ = - ΓΦ_s/3. We assume here that the tissue maintains locally a constant homeostatic cellular structure, so that ℋ=0 and Φ_s is a constant. 





 




We can rewrite (<ref>) using (<ref>) as





    ϕ_st+ ( ϕ_s v⃗_s)=  ℋ/J_G + Γϕ_s := ξ_s,

with ξ_s representing the source of solid material. 
 Here we assume that the availability of the solid material required for growth is non-limiting, i.e., we ignore the physiological aspects of biosynthesis underlying growth (photosynthesis, nutrient uptake).  

Similarly, the balance of fluid mass is expressed through

    ϕ_ft +    (ϕ_fv⃗_f) = ξ_f,
 




where ξ_f is a possible source of water that accounts, typically, for a bulk vascularisation of the tissue. A possible constitutive law for ξ_f is discussed later in this paper. 








Lastly, the balance of mass for the osmolytes is expressed through
 




    ϕ_ot + ϕ_ov⃗_o  =  ξ_o.

with ξ_o representing the synthesis/uptake of osmolytes. Summing (<ref>)  using (<ref>), we obtain a single volume balance equality  <cit.>:

    ϕ_sv⃗ + ϕ_fv⃗_s + ϕ_ov⃗_o = ξ_s + ξ_f + ξ_o.

















































  §.§.§ Balance of momentum


Since the growth and fluid transport happen at timescales (hours) that are much longer than the timescale of elastic relaxation (seconds), we  neglect inertia. 
Thus, we write the balance of momentum in terms of the Cauchy stress tensor T as

    T + b⃗ = 0⃗  ,   T  =  T^⊤,

with b⃗ a density of applied body loads (per unit current volume), typically self-weight in our context. Henceforth, we focus on the case b⃗=0⃗.  



The symmetry of the stress tensor results from the absence of applied torque and micropolar structure in the material. In a more general theory, asymmetric stresses may be introduced to reflect the possible local twist of the cells <cit.>.



  §.§.§ Balance of internal energy








To complete the theory, we discuss the thermodynamics of the system. The balance of energy for the total internal energy ℰ of a region Ω⊂ℬ encompasses internal work contributions 𝒫, heat 𝒬, and addition of new material 𝒮:

    ℰt  = 𝒫 + 𝒬 + 𝒮  ,

where

    ℰ = ∫_Ω e   v,

with e the internal energy per unit volume of mixture.








The work rate encompasses the work of internal forces,  the works required to make room for new fluid, osmolyte and  solid material (w_s) during growth, and the work w̅_o accounting for the active transport of the osmolytes, reflecting chemical processes that we do not address explicitly. Altogether, we have

    𝒫 =   ∫_ΩT⃗  :  L v   + ∫_Ω p  ξ_f   v- ∫_∂Ω p  j⃗_f ·a⃗+ ∫_Ω p  ξ_o   v- ∫_∂Ω p j⃗_o ·a⃗  + ∫_Ω    w_s ξ_s   v + ∫_Ωw̅_o v   ,

where j⃗_f = ϕ_f w⃗_f and j⃗_o = ϕ_o w⃗_o are the relative fluxes of fluid and osmolytes through the mixture; with w⃗_f v⃗_f - v⃗ the seepage velocity of the fluid and w⃗_o v⃗_o - v⃗; and where a⃗ and v denote respectively the outward normal surface element and the volume element for the integration.

The heat contribution encompasses a bulk source r and a flux q⃗ through the boundary:

    𝒬 = ∫_Ω r  v - ∫_∂Ωq⃗·a⃗.

Energy is also added to the system through addition of new  material. We note e_f, e_o and e_s the internal energy of the fluid, the osmolytes and the solid so that

    𝒮 = ∫_Ω e_s ξ_s   v + ∫_Ω e_f ξ_f v - ∫_∂Ω e_f j⃗_f ·a⃗+∫_Ω e_o ξ_o v- ∫_∂Ω e_o j⃗_o ·a⃗    .






Using the divergence theorem to eliminate the boundary terms, then the Leibniz rule for differentiation under the integral sign, and the localisation theorem,



we derive the local form of the balance of internal energy 

    et +   ( e v⃗  ) = T⃗  :  L + r   + h_f ξ_f  +  h_o ξ_o +  h_s ξ_s+ w̅_o-   h_f  j⃗_f + h_o j⃗_o + q⃗,

where we have introduced h_f =  e_f + p, h_o =  e_o + p and h_s = e_s + w_s the enthalpies per unit volume of pure fluid, osmolytes and solid material, respectively. Alternatively, it is convenient to express the balance of energy in reference configuration as

    Ė^0 = P⃗  : Ḟ + R^0  +  h_f Ξ_f^0 +   h_o Ξ_o^0 +  h_s Ξ_s^0   + W̅_o^0-   h_f J⃗_f^0 + h_o J⃗_o^0 + Q⃗^0,

where  denotes the Lagrangian divergence;  
E^0 = Je is the internal energy per unit reference volume; 
P = J T F^-⊤ is the Piola–Kirchhoff stress <cit.>; and Q⃗^0 = J F^-1q⃗, J⃗_o^0 = J F^-1j⃗_o and J⃗_f^0 = J  F^-1j⃗_f are the flux terms pulled back to the reference configuration. We define the quantities Φ_f^0 = Jϕ_f, Φ_o^0 = Jϕ_o,  and Φ_s^0 = Jϕ_s=J_GΦ_s, measuring respectively the  current volume of each component per unit reference volume, so that Ξ_s^0=Φ_s^0Γ, Ξ_o^0 = Jξ_o and Ξ_f^0 = Jξ_f are the local production of solid, osmolytes and fluid  volume per unit reference volume; R^0 = Jr is the heat source per unit reference volume; W̅_o^0 = Jw̅_o is the work due to active transport per unit reference volume. 





 
























































  §.§.§ Imbalance of entropy






The second law of thermodynamics is expressed through the Clausius-Duhem inequality, expressed in terms of the entropy density S^0 (entropy per unit initial volume) as <cit.>





    Ṡ^0    ≥R^0  /θ+ s_f Ξ_f^0 + s_oΞ_o^0    + s_fΞ_s^0  + S̅_o^0 - Q⃗^0  /θ +  s_f   J⃗_f^0 + s_o J⃗_o^0  ,

with θ denoting the thermodynamic temperature; and where s_f, s_o and s_s are the volumetric entropy densities of the added fluid, osmolyte and solid materials, respectively; and S̅_o^0 denotes the entropy contribution from active transport. Henceforth, we assume isothermal conditions for simplicity, so we take Q⃗^0 = 0⃗ and R^0=  0.  Introducing the Helmholtz free energy
Ψ^0 = E^0 -  θ S^0 and applying the energy balance equation (<ref>), we can reformulate the previous inequality as
































    Ψ̇^0  ≤P⃗  : Ḟ + g_f Ξ_f^0  +  g_o Ξ_o^0 +  g_s Ξ_s^0  + G̅_o^0 -   g_f J⃗_f^0 + g_o J⃗_o^0   ,

where g_s = h_s - θ  s_s, g_f = h_f - θ s_f and g_o = h_o - θ s_o denote the Gibbs free energy densities of the solid, fluid and solute material, respectively; and G̅_o^0 W̅_o^0 - θS̅_o^0. As detailed in the next section, the inequality (<ref>) is useful to derive constitutive laws for growth and material transport. 

 















































 §.§ Constitutive laws







  §.§.§ Coleman-Noll procedure

To close the system we need to formulate constitutive laws for the material. Therefore, we follow the approach of <cit.> and apply the Coleman-Noll procedure to the dissipation inequality (<ref>) to constrain thermodynamically the constitutive laws  <cit.>. Firstly, we decompose the free energy Ψ^0 into a solid part Ψ_s^0 and a fluid part Ψ_f ^0 =  Φ_f^0 g_f +Φ_o^0  g_o   - Φ^0  p <cit.>; so that Ψ^0 = Ψ_s^0 +Φ_f^0 g_f +Φ_o^0  g_o   - Φ^0  p.  For an inherently incompressible solid component, the porosity Φ ^0 = J_G (J_A - Φ_s ) is determined by the deformation. 
We introduce the mechanical free energy of the material in the form 























    W^0 ( A,  G, p) 
        = Ψ_s^0( A,  G)  - p J_G  (J_A - Φ_s),

encompassing an elastic contribution and a hydrostatic contribution. 
The energy per unit intermediate volume is

    W( A, p) = J_G^-1 W^0 ( A,  G, p) = Ψ_s ( A)   - p    ( J_A - Φ_s),

where Ψ_s( A) = J_G^-1Ψ_s^0 ( A,  G) defines the strain energy density of the solid, per unit intermediate volume.




Then, combining (<ref>) with the balance of mass relations Φ̇^0_f  =    Ξ_f^0 - J⃗_f^0 and Φ̇^0_o  = Ξ_o^0 - J⃗_o^0 (<ref>), and the Gibbs-Duhem equality ṗΦ^0 =  ġ_fΦ_f^0 +ġ_o Φ_o^0, 
we obtain 





    Ẇ^0  ≤P⃗  : Ḟ    - ṗΦ_f^0 +  
      G̅_o^0 - J⃗_f^0 · g_f - J⃗_o^0 · g_o   .


Finally, expanding Ẇ^0 as


Ẇ_0 J_G ^-1 =  W G^-⊤:Ġ + W A : Ȧ + Wpṗ







and substituting into (<ref>) using (<ref>) provides





    0  ≤    P  G^⊤  -  J_GW A:Ȧ     +    A^⊤ P  + g_s Φ_s^0 - J_G W G^-⊤:Ġ
      
    + G̅_o^0 - J⃗_f^0 · g_f - J⃗_o^0 · g_o   .


By the argument of <cit.>, the inequality (<ref>) must hold for any admissible process. In particular, since Ȧ is arbitrarily prescribed, the following standard equalities must hold universally

    
    P  =  W^0 A G^-⊤  ⇔   T = J_A^-1 A W A

which provides the standard stress-strain relation for a hyperelastic material. 





 The resulting dissipation inequality,

    0     ≤ A^⊤ P  + g_s Φ_s^0 - J_G W G^-⊤:Ġ
    + G̅_o^0 - J⃗_f^0 · g_f - J⃗_o^0 · g_o ,

highlights the different modes of dissipation in the system, coming from distinct biophysical processes, specifically growth and transport. Thus, these contributions must  satisfy (<ref>) individually, hence








    g_s Φ_s  1-  B:  L_G  ≥ 0 ,


    G̅_o^0 -J⃗_f^0 · g_f    - J⃗_o^0 · g_o  ≥ 0.



Here, B⃗  W 1-J_A  A^⊤ T A^-⊤ denotes the Eshelby stress tensor, which appears as an important quantity for growth <cit.>.


















  §.§.§ Growth law


We discuss the constitutive law for the growth of the cell matrix.  Focusing on the solid component, we first decompose the stress additively into a solid and fluid part as T =   T_s   -  ϕ p   1 where T_s is the partial stress due to the solid component <cit.>:





    T_s 
      =  J_A^-1 A Ψ_s A - ϕ_s p  1.






We can  rewrite (<ref>) in terms of the partial Eshelby stress B_s  Ψ_s 1 -   ϕ_s J_A  A^⊤ T_s A^-⊤ (= B) as

    g_sΦ_s 1 -   B_s: L_G ≥ 0 .



This dissipation inequality expresses a thermodynamic constraint on the growth law. In particular, the case g_s=0, where the solid is added with no energy to the matrix, defines a passive growth process <cit.>. 
Generally, from (<ref>), a natural path is then to adopt the Eshelby stress B_s as a driving force for growth. A common approach then is to postulate the existence of a homeostatic stress B^* such that B^*: L_G = Φ_sg_sΓ, so a natural growth law can be chosen of the general form L_G = ℋ :  ( B^* -  B_s) with  ℋ a fourth-order extensibility tensor such that the inequality holds for any B_s <cit.>. In plants however, it is unlikely that such homeostatic stress is actively maintained. Instead, the growth of  plant cell walls is generally seen as a passive, plastic-like process <cit.> which involves complex  anisotropies and nonlinear threshold effects  <cit.>. Whether simple growth laws based on the Eshelby stress can reproduce  basic properties of plant matter remains unclear to us. 

Thus, we reserve this problem for another occasion and instead adopt a simpler, phenomenological approach. Indeed, strain-based growth laws  of the form L_G = 𝒢( A) have been adopted <cit.>, where typically, the cell walls expand once they surpass a certain strain threshold. These growth laws are relatively simple to parameterise and capture elegantly some observed phenomenological properties of plant matter, for example the commonly-accepted fact that growth is slower in rigid tissues or along the direction of load-bearing cellulose microfibrils (all other things being equal). 












Following previous strain-based models we postulate a strain-based growth law where we assume that during growth, the strain is maintained close to a strain threshold E^*. Therefore, we posit a linear growth law of the form

    𝒢( A) =   𝒦: E -  E^*,

with 𝒦 a fourth-order tensor controlling the rate and anisotropy of the growth (with unit inverse time); E  ( A^⊤ A -  1)/2 the symmetric Lagrangian elastic tensor; E^* is a threshold strain tensor; and · is an invariant extension of the ramp function to symmetric tensors, that is, for any symmetric tensor S with eigenvalues S_i and corresponding normalised eigenvectors s⃗_i, we define
Smax0,S_i s⃗_i ⊗s⃗_i.
For comparison, (<ref>) is a generalisation of the growth law introduced by <cit.>. 

Note that  
there is no guarantee anymore that the dissipation inequality (<ref>),




    g_s Φ_s  ≥ B_s :𝒢( A)/𝒢( A),


will be always satisfied with g_s = 0. 
(However, in practice, the r.h.s. appears to be indeed negative in many cases studied here.) 
We conclude that strain-based growth processes may not be universally passive. 
We note that, recently, some authors have proposed that material insertion and pectin expansion may also drive growth, independently of turgor <cit.>. This hypothesis, albeit contentious <cit.>, implies that growth in plant cell walls may be–at least partly–an active process.























 





























































  §.§.§ Transport

Next, we discuss the second inequality (<ref>) for the material fluxes. Using J⃗_o^0 = Jϕ_o   F^-1w⃗_o and J⃗_f^0 = Jϕ_f  F^-1w⃗_f and postulating an active transport contribution of the form g̅_o = ϕ_o f̅⃗̅_o ·w⃗_o with f̅⃗̅_o an effective force sustaining active transport, (<ref>) becomes 

    -ϕ_fw⃗_f  · g_f  + ϕ_f w⃗_o  · ( f̅⃗̅_o -  g_o ) ≥ 0.


To make progress, we introduce Onsager's reciprocal relations <cit.>


    - ϕ_f  g_f = L_fo (v⃗_f - v⃗_o) + L_fs (v⃗_f - v⃗) = - L_fow⃗_o + ( L_fs + L_fo) w⃗_f,


    ϕ_of̅⃗̅_o - ϕ_o  g_o = L_of (v⃗_o - v⃗_f) + L_os (v⃗_o - v⃗) = - L_ofw⃗_f + ( L_os + L_of) w⃗_o .


with L_of=L_fo. Indeed, substituting (<ref>) into (<ref>) gives

    L_fs/L_fow⃗_f^2 +   L_os/L_fow⃗_o^2 +   (w⃗_f - w⃗_o)^2 ≥ 0 ,

which is universally satisfied.
Inverting (<ref>) and using the Gibbs-Duhem equality ϕ p = ϕ_f  g_f + ϕ_o ϕ_o <cit.> we obtain

    
    w⃗_f = f̅⃗̅_o L_of +L_osϕ_o  g_o-ϕ  (L_of+L_os)  p/L_of (L_fs+L_os)+L_fs L_os,


    w⃗_o = f̅⃗̅_o (L_of+L_fs)-L_fsϕ_o  g_o-L_ofϕ p/L_of (L_fs+L_os)+L_fs L_os.


In the limit of high osmolyte-solid drag L_os≫ L_fs,  L_of considered here we find

    w⃗_f ≈1/L_of+L_fsϕ_o  g_o-ϕ p,  w⃗_o  ≈f̅⃗̅_o/L_os.

Biologically, this assumption expresses the idea that water and osmolytes are exchanged by cells via distinct pathways, so that in the absence of active transport, the osmolytes remain confined within the cells. 


    L_fs = ϕ/k,    L_of = R_gθ c_o/D_o.

The Gibbs free energy g_o of the solutes in the solution can be related to the molar concentration of osmolytes c_o through

    g_o = R_gθ/v_oln c_o + cst,

with R the universal gas constant. Using the formula c_o = ϕ_o / v_aϕ, we obtain finally


















    j⃗_f ≈ϕ_f 1/k+R_gθ c_o/D_oϕ^-1  R_gθ c_o -  p

When the concentration c_o is low, we recover Darcy's law <cit.> Not quite

    j⃗_f =  -  K ψ,

where K is the second-order symmetric permeability tensor that characterises the macroscopic permeability of the mixture to the fluid (expressed in unit area per pressure per time), assumed to not depend on the elastic deformation A. Physically, the tensor K reflects  the overall effects of cell topology, wall and cell membrane permeabilities, as well as the permeability linked to other water routes (e.g. apoplasmic route).  Combining (<ref>) we can derive a single equation for the balance of mass of the mixture <cit.>:




    v⃗   + j⃗_o - K ψ  = ΓΦ_s / J_A  + ξ_o + ξ_f.


To make progress, we introduce Onsager's reciprocal relations <cit.>


    - ϕ_f  g_f = L_fo (v⃗_f - v⃗_o) +  L_fs (v⃗_f - v⃗) = - L_fow⃗_o + (  L_fs + L_fo 1) w⃗_f,


    ϕ_of̅⃗̅_o - ϕ_o  g_o = L_of (v⃗_o - v⃗_f) +  L_os (v⃗_o - v⃗) = - L_ofw⃗_f + (  L_os + L_of 1) w⃗_o ,


where L_of=L_fo is a fluid-osmolyte drag coefficient; and L_os and L_fs are positive-definite symmetric second-order tensors that characterise the anisotropic osmolyte-solid and fluid-solid drags, respectively, to reflect the anisotropic structure of the solid matrix. Indeed, substituting (<ref>) into (<ref>) gives
w⃗_f · ( L_fsw⃗_f ) +    w⃗_o · ( L_osw⃗_o ) +   L_fo(w⃗_f - w⃗_o)^2 ≥ 0, 
which is always satisfied. These relations may be viewed as a generalisation of Fick's law to several transported species.
Inverting (<ref>) and using the Gibbs-Duhem equality ϕ p = ϕ_f  g_f + ϕ_o ϕ_o  we obtain the relative velocities








    w⃗_f=  L_os L_fs + L_of ( L_os+ L_fs) ^-1ϕ_oL_fof̅⃗̅_o + L_osϕ_o  -ϕ L_os + L_of 1 p ,


    w⃗_o =  L_os + L_of 1^-1ϕ_of̅⃗̅_o - ϕ_o  g_o +  L_fow⃗_f.










In particular, in the limit of high osmolyte-solid drag L_os≫ L_fs,  L_of, we derive





    w⃗_f ≈ L_fs + L_of 1 ^-1ϕ_o     ϕ_o   - ϕ p,   
       w⃗_o  ≈ϕ_o L_os  ^-1f̅⃗̅_o -   g_o.

Biologically, this assumption expresses the idea that in the absence of active transport, the osmolytes remain mostly confined within the cells (with some slow diffusive leakage) and are not convected by the fluid. Following <cit.>, the drag coefficients are given as
L_fs = ϕ K^-1 and L_of =  R_gθ c_o/D_o,
with K the second-order symmetric permeability tensor that characterises the permeability of the mixture to the fluid (expressed in unit area per pressure per time); c_o is the molar concentration of osmolytes; D_o is the diffusivity of the osmolytes in the fluid; and R_g the universal gas constant.  

The  chemical potential g_o of the solutes can be related to their molar concentration c_o in the fluid through

    g_o = g_o^0 + R_gθ/v_olnc_o/c_o^0  ,

with v_o the molar volume of the osmolytes; and g_o^0 and c_o^0 are reference values, taken to be constant.

Using the formulae j⃗_f=ϕ_fw⃗_f and c_o = ϕ_o / v_oϕ with (<ref>), we obtain finally




















    j⃗_f ≈ϕ_f  K^-1 + R_gθ c_o/D_oϕ 1 ^-1R_gθ c_o   -  p.

For small concentration, more specifically for R_gθ c_o  ≪  D_oϕ K^-1, we recover a Darcy-type law

    j⃗_f ≈  - ϕ_f K ψ,

where ψ = p - π as in (<ref>); with the osmotic pressure π given by the van 't Hoff relation π R_gθ c_o.  Combining (<ref>) we can finally derive

    v⃗   + j⃗_o -1 - ϕ_o -  Φ_s/J_A K ψ  = ξ_s  + ξ_o + ξ_f.







  §.§.§ Elastic constitutive relation

Using Terzaghi's effective stress principle <cit.>
,  the effective Cauchy stress  T for the mixture can be expressed as the sum of the partial stresses T_s and T_f = -p 1 for the solid and the the fluid respectively (with 1 the identity tensor):

    T =    T_s   - p  1.

We then can write the balance of linear and angular momenta (<ref>) as

    T_s  =  p,   T_s =  T_s ^⊤.

Macroscopically, we assume that the solid  is hyperelastic with respect to the unstressed virtual state, with Gibbs free energy function W=W( A) expressing the macroscopic strain energy per unit volume (of the intermediate state). The Cauchy stress is obtained from the Clausius-Duhem inequality and is expressed as

    T_s = J_A^-1 A W_s A.







  §.§.§ Growth law

This section needs to be rewritten Lastly, we model the evolution of the growth tensor G, that is, we formulate a  growth law for the material. 






In the biophysics community, plant cell growth has been widely described as a dissipative process akin to plastic yielding  <cit.>, as initially described by Lockhart <cit.>. In this line, several authors have proposed multidimensional extensions to Lockhart's model, based on plasticity theory <cit.>, or in the form of phenomenological stress- or strain-based tensorial growth laws <cit.>. Following this idea, we make the explicit postulate that growth is a thermodynamically passive process. That is, we postulate a  growth law that satisfies the Clausius-Duhem inequality in the absence of external entropy sinks:

    M :  L_G  ≥ 0 .,

A natural choice for the growth law is then to take L_G as a function of M.  



Drawing on viscoplasticity theory, we   postulate that the yield of the material can be described through a yield function Ψ, such that the elastic domain (assumed to be convex), that is the region in M-space where the material behaves purely elastically, is given by

    Ψ( M) ≤ 0.

In other words, growth occurs if and only if Ψ( M) > 0.
Then, in the anelastic domain, the yield rate is derived from the so-called principle of maximum dissipation via the normality criterion

    L_G  =  1/ηΨ M  for  Ψ( M)>0,






with η> 0 a growth constant akin to a viscosity (expressed with units of pressure time).



For simplicity, we here restrict our attention to scenarios where A and T_s commute, then M is symmetric and we have M = J_A^-1 T_s. This is true for isotropic materials, but also for problems that have appropriate symmetries. 



















Since M is symmetric, we can  write its spectral decomposition in terms of its eigenvalues m_i (i=1,2,3) and an orthonormal family of eigenvectors m⃗_i (in corresponding order):

    M = ∑_i=1^3 m_i m⃗_i ⊗m⃗_i ,

with ⊗ denoting the tensor product. Similar to <cit.>, we assume that line elements grow along direction m⃗_i only when m_i exceeds a threshold value m^*≥0. Therefore, we define a threshold tensor M^*  m^* 1, along with the excess stress tensor Δ M  M -  M^*, where · extends the ramp function to symmetric tensors (<ref>). A minimal, convex yield function capturing the aforementioned growth phenomenology can be then given in terms of the second main invariant J_2 = m_1 - m^*^2+m_2 - m^*^2+m_3 - m^*^2 of Δ M:  

    Ψ ( M)=  J_2/2.

Using (<ref>), we obtain

    L_G =1/ 2ηJ_2 M = 1/η∑_i=1^3  m_i - m^*m⃗_i ⊗m⃗_i .

 
  Note that the terms growth and plasticity  have been used often interchangeably by the plant biophysics community. However, despite formal analogies between the two concepts, plasticity and  growth are fundamentally distinct  phenomena <cit.>
. While plasticity reflects microscopic reorganisation of materials (metals typically), plant tissues are virtually unable to rearrange at the cellular level, but can gain mass through wall secretion and cell divisions. For example, in (<ref>), L_G is positive semi-definite,
expressing the fact that there growth induces no contraction of the material in any direction. This behaviours differs substantially from  plastic materials, which typically undergo isochoric anelastic deformations with L_G=0. 











































In our context, a useful simplification comes from the fact that, in plants, the fluid content actually accounts for most of the volume of the mixture, thus, water mass balance is barely affected by the small volume of the osmolyte and the cell walls. Therefore, we next take ξ_s ≪ξ_f, j⃗_o ≪j⃗_f, ϕ_o, ϕ_s≪ϕ and T_s ≈ T + p 1.



 §.§ A closed system of equations































































Combining (<ref>), we obtain the following system of equations:





    v⃗- K ψ =    ξ_f ,


    ϕ_ot + ϕ_ov⃗ + j⃗_o  =  ξ_o,


    T_s  + b⃗=  p,


    T_s  =  T_s ^⊤ ,








    F =  A  G ,


    L_G  =  𝒦: E -  E^*,


    T_s  = J_A^-1 A Ψ_s A,


    ψ = p -   R_gθϕ_o / v_o.


In total, we have a closed system of thirty-three partial-differential equations for thirty-three variables: the nine components of A, the nine components of G, the three components of χ, the nine components of T⃗_s, and the three scalar variables p,  ψ and ϕ_o.


 §.§ Boundary conditions


This system must be equipped with appropriate boundary conditions. Defining the outer normal n⃗  to the boundary ∂ℬ, typical boundary conditions encountered in our context describing the flux of water through the boundary are Robin boundary conditions of the form

    j⃗_f ·n⃗ = -  K ψ·n⃗=  k (ψ^*  - ψ)     on ∂ℬ,

expressing a flux across the boundary due to a difference of water potential with the outside ψ^*, where k is the interfacial hydraulic conductivity. No-flux boundary conditions K ψ·n⃗ =  0
 are a particular case of (<ref>). 
Note that when k→∞, the condition (<ref>) simplifies to a Dirichlet constraint

    ψ  = ψ^*   on ∂ℬ.


Similarly, the mass balance equation for the osmolytes may be equipped with Dirichlet (fixing c_o at the boundary) or flux boundary condition. The usual boundary condition for the stress (<ref>) is

    Tn⃗ = t⃗_0    on ∂ℬ ,

with t⃗_0 the applied traction density.





 §.§ Apparent elasticity of a growing tissue


Given a solution to the system, an adscititious problem is to characterise the effective elasticity of the turgid tissue subject to residual stresses. Indeed, at timescales shorter than that of growth and water transport, water is effectively trapped in the tissue and the mixture can then be seen as a macroscopically incompressible hyperlastic material. Given a field of  elastic pre-deformation A and pressure p, the effective strain energy function of the mixture with respect to an incremental, superimposed deformation Â is 

    Ŵ( Â, p̂) = J_A^-1 W ( Â A, p )   - p̂ (Â - 1 ) ,
 where p̂ is an undetermined Lagrange multiplier that accounts for the incompressibility constraint Â = 1. The associated Cauchy stress is then





    T̂ = ÂŴÂ 
        =  T (Â A,p) -   p̂ 1.

 This expression is useful to describe the overall elastic response of the system to external forces, e.g. in the context of compression experiments performed on entire tissues <cit.>, or to explore mechanical stability under growth-induced differential stresses or external loads <cit.>. 



§ LONGITUDINAL GROWTH: HYDRAULIC COMPETITION AND GROWTH-INDUCED WATER GRADIENTS






 §.§ General problem



To illustrate the behaviour of the system, we first examine the simple scenario of a growing, straight cylindrical rod so we reduce the system (<ref>) to one dimension. We define s, S and S_0, the arc lengths in the current, intermediate and initial configurations respectively. The associated total, growth and elastic stretches are λsS_0, γSS_0 and αsS, with λ=αγ (<ref>). From (<ref>), we derive

    vs = α̇/α + γ̇/γ,

with v the longitudinal Eulerian velocity (positive towards increasing s).
Assuming that the rod is unloaded, the  balance of linear momentum (<ref>) for the axial partial stress t_s yields t_s=p. 






System (<ref>) then reduces to



    sv  - K ψs =  
        ξ_f ,


    ϕ_ot + sϕ_o  v +   j_o  =  ξ_o,


    vs =    1/2 τα^2 - α^*^2 +  αt + v/ααs ,


    p = Ψ_sα,


    ψ = p -  R_gθϕ_o /v_o,


where τ is the characteristic time of wall synthesis during growth (i.e. 𝒦 = 1/τ); K = K is the permeability coefficient; and j⃗_0=j_0 is the osmolyte flux. 


For simplicity, we focus in this section on infinitesimal elastic deformations and define ϵ = α - 1 the infinitesimal strain, with ϵ≪ 1, and the strain threshold ϵ^* = α^* - 1≥ 0. Thus, Hooke's law provides

    p  = E ϵ,

with E the effective Young's modulus of the solid. 












Here we  assume that the osmotic pressure π is maintained constant w.r.t s and t, so we elide  (<ref>) and substitute ψs=ps into (<ref>). Similarly, we take K,  ϵ^*, and τ constant and homogeneous. Indeed, the focus here is on the role of a spatially heterogeneous stiffness, which has been linked experimentally <cit.> and theoretically <cit.> to organ patterning in development. Therefore, we allow the rigidity to vary spatially as E(s,t) = E_0 f(s) with f a dimensionless function of order unit, and E_0 a characteristic Young's modulus of the rod. Finally, taking τ, E_0 and √(KE_0τ) as reference time, pressure and length, respectively, we can nondimensionalise the system through the substitutions




    ṽ =v √(τ/KE_0),    s̃ = s/√(KE_0τ),  t̃ = t/τ,   ξ̃_f =  ξ_f τ,   p̃ =  p/E_0.

In particular, the length ℒ_h √(KE_0τ) defines the characteristic  hydromechanical length of the system.   Altogether, we obtain to O(ϵ):





























    ṽs̃  =   p̃s̃
      + ξ̃_f,


    ṽs̃ =   ϵ- ϵ^* +  ϵt̃ +ṽϵs̃ ,


with p̃ = f  ϵ. 
To simplify notations, we drop the tildes and use nondimensionalised variables. For comparison, this system ressembles the static model of <cit.>, and extends the Lockhart-Cosgrove-Ortega model (<ref>) to a one-dimensional continuum. Note also that the r.h.s of (<ref>) is always negative or zero here, thus, by the argument of <ref>, our strain-based growth law is compatible with a passive growth process. In the next two sections, we study the effect of heterogeneous elastic moduli on the growth behaviour of the rod.






































































































 §.§ Material heterogeneity and hydraulic competition


Here we are interested in the boundary effects arising between two regions with different mechanical properties, specifically the effect of different stiffnesses. Therefore, we consider a rod with initial arclength S_0∈-L_0/2,L_0/2, divided in the middle into two part of different rigidity, that is, we posit the rigidity field given by 

    f (s(S_0)) =   1 - ηθ(S_0),

where 0 ≤η < 1 denotes the  stiffness step at the interface s=0; and with
θ the Heaviside step function (i.e. θ(x) = 0 if x<0 and  θ(x) = 1 if x≥ 0). The region S_0>0 is softened, with stiffness 1-η, with respect to the base stiffness equal to unity. Assuming s(0)=v(0)=0, without loss of generality (i.e. considering an observer located at the interface), we have f(s)= 1 - ηθ(s). Here we assume no-flux boundary conditions at both ends of the domain, however we allow water entry through a bulk source ξ_f = k (p^* - p), with k the effective permeability with the outside, and p^* a constant base effective pressure, encompassing the excess osmolarity relative to the outside, and the outer hydrostatic pressure.

To gain insight into the shape of the solutions, we focus on the steady regime, i.e. on self-similar stationary solutions  on an infinite line. On setting ϵt=0 in (<ref>), the problem reduces to:


    v'  =   p”
      + k(p^* - p),


    v' =   ϵ- ϵ^*    +  v  ϵ',


with the apostrophe denoting derivative with respect to s, and where p=fϵ. 
To make progress, we consider a small perturbative softening  η≪ 1. Therefore we expand all variables as power series of η, i.e. p = p_0 + η p_1 + …, ϵ = ϵ_0 + ηϵ_1  + … and v = v_0 + η v_1  + …,  











and then treat each order in (<ref>) separately. To ease calculations, it is also convenient here to ignore the threshold effect in (<ref>) and assume p^*≥ϵ≥ϵ^*. The base solution at O (1) has uniform pressure and velocity gradient and is given by


    p_0= ϵ_0 = ϵ^* + kp^*/1+k,


    v_0 = a s,   with   a =  p^*-ϵ^* /1+k^-1 .


Unsurprisingly, p_0 can be identified to the Lockhart pressure (<ref>). At Oη, we have


    v_1'  =   p_1” - k   p_1, 
      
       v_1'= ϵ_1  +  v_0   ϵ_1' .
















Eliminating v_1' and using p_1 =     ϵ_1 - ϵ_0 θ(s) and p_0=ϵ_0, we obtain a single equation for p_1,

    p_1” - as    p_1' -1+k  p_1   = p_0θ(s),

defined on both regions s<0 and s>0. A general solution to this equation can be expressed in terms of the Kummer confluent hypergeometric function _1F_1, the gamma function Γ and the Hermite polynomials (of the first kind) H_λ. By imposing the condition that pressure must be bounded and continuously differentiable at s=0, we can determine the four integration constants and obtain the compound asymptotic solution  
   
   
   
   
   
   
   
   
    
    p  (s) = p_0 + η p_0 /k+1π( π c)/Γ( c/2 ) Γ(1-c) H_-c(s√(a/2))- θ(s) - θ(-s) h_-cs√(a/2)+ O (η ^2),

   
   
   
   
   
   
   
   
 
 
 
 
 
 
 
 
 
    
    with c k+1/a; and h_λ =   _1F_1(- λ/2; 1/2;x^2)  the Hermite functions (π denotes here the ratio of a circle's circumference to its diameter). The expansion rate v' is obtained directly by substituting this expression into (<ref>).
    
    
    

   
  Example solutions are shown in <ref>.  As can be seen, the softening of the right-hand-side region results in a pressure drop in that region, reflecting the reduced mechanical resistance of the cell walls to water entry. As pressure decreases smoothly at the junction between the soft to the stiff region,  the elongation rate v' jumps discontinuously, reaching a global minimum at s=0^- and a maximum at s=0^+. This jump directly results from the strain-based growth law, and the strain discontinuity at s=0. Due to the  pressure gradient, water seeps towards increasing s, i.e. from the stiff region to the soft region. The characteristic length of this seepage,  L_c  -p'(0)/2Δ p ≤ 1, increases when external water supply k decreases, as illustrated by the inset in <ref>(a). This result shows the emergence of hydraulic competition between the two regions. This competition concerns a larger portion of the tissue when external water supply becomes small. The presence of an external source of water k also tends to reduce the difference in pressure between the two regions (thus the seepage), as can be seen from the pressure drop Δ p = η p_0 / 1+k between the two asymptotes s= -∞ and s=+∞. This gap is maximal when k≪ 1, where water becomes scarce, with
   
    p(s)    =  ϵ^* -  η/2θ (-s)^s   + θ (s) (2-^-s)+ O (η ^2).

In this limit example however, the volume of water in the rod is conserved, thus as the right-hand side region grows along a region of effective size unity, the other side, which is actually under the growth threshold, has to shrink (thus this limit is actually nonphysical). This issue is addressed later in <ref>. For comparison, <ref> also shows the case where internal fluxes along the rod are suppressed (K=0),  illustrated by the dashed lines. In this case, the pressure and the velocity gradient are piece-wise continuous, with


    p(s)= p_0 - η p_0 θ(s)/k+1+ O (η ^2),
       
        v'(s) = a  + η  p_0θ(s)/1+k^-1 + O (η ^2),


corresponding to the asymptotes for the general case. In the general case where the two regions exchange water, this jump in expansion rate v' across the interface is amplified by a factor 1+k^-1 with respect to the situation with zero flux. In particular, in situations where water supply is impeded (i.e. k≪ 1) and the overall growth becomes slower, this amplification factor diverges as the effect of hydraulic competition becomes more visible. In other words, although one might initially expect that water movements along the rod would serve to smooth out heterogeneities, we show that, near the interface (| s|≲ L_c), the combined effects of growth, hydraulics and mechanics actually accentuates the effect of heterogeneous mechanical properties on the growth dynamics. Interestingly, such effects have been proposed to play a role in the initiation of organs in the shoot apical meristem <cit.> (see <ref>).


   
   
   
   

   









































 §.§ Water gradient in  apically growing organs 


Plant shoots and roots grow typically along an elongating apical region <cit.>, while the rest of the tissue stiffens and stops growing. Kinematically, this type of growth is akin (but not physically equivalent) to tip  growth. Here, we explore the existence of tip-like, self-similar solutions to our system. Therefore, we focus on the steady regime (<ref>), and define s, the arclength measured from the apex of the plant towards the base. In this scenario, v is then the velocity of the domain measured in the co-moving frame attached to the apex.









To illustrate the role of chemical growth regulation in our framework, 
we here assume that growth is controlled by a growth hormone (e.g. auxin) with concentration ρ. We posit that the hormone is actively convected with velocity v_a from the apex (s=0) towards the base (s=∞), diffuses with diffusion coefficient D and is reabsorbed with rate constant β. In the steady regime, the concentration profile obeys

    Dρ” - ( v+v_a)ρ' - (v'+β) ρ = 0 .

In the convection-dominated regime D ρ ', vρ≪ v_aρ, the hormone concentration 
is given by 
ρ (s) = ρ_0 ^-s/σ,
where ρ_0 ρ(0) is the concentration at the apex; and where σβ/v_a sets the length scale for the hormonal regulation  <cit.>. In plants, transported hormones such as auxin are involved in the control of cell wall elastic properties <cit.>. Therefore, we assume that the effect of the hormone is to reduce the elastic modulus of the tissue. Assuming a small, linear effect of the hormone on the elastic modulus, we write 

    f≈1 - ηρ/ρ_0,
 
with η≪ 1. Note that an alternative mechanism producing a gradient in rigidity  could be a gradual stiffening of the tissue as the cells age and move away from the apex <cit.>. 

















Firstly, we assume that no growth  occurs in the absence of apical softening (i.e. if η = 0), thus we choose ϵ^*≥ p^*. Indeed, in <ref>, the choice ϵ^*<p^* resulted in exponential growth, precluding then the possibility of tip-like growth regimes. As previously, we solve (<ref>) asymptotically to first order in η. The base solution is p_0=ϵ_0=p^* and v_0=0. At O (η ), we have

    p_1” -  (1+k) p_1   = p_0^-s/σ.

Here k can be interpreted biologically as a rate constant characterising the effective permeability between the growing tissue and the organ's vascular bundle.



Note that (<ref>) is valid only as long as the strain is above the growth threshold, i.e. ϵ>ϵ^*. Away from the tip, under this growth threshold, the system is in the purely elastic regime and obeys

    p_1” -  k p_1   = 0.

The general solutions for (<ref>) can be obtained easily.
The difficulty is then to determine the unknown arclength Σ at which the threshold is attained. Enforcing the condition that pressure p is bounded and continuously differentiable, and using ϵ (Σ^-)=ϵ (Σ^+) = ϵ^* along with the no-flux condition at the tip p'(0)=0, we can express all the integration constants as functions of Σ. The latter is ultimately identified as the root of some (slightly uncomely) transcendental equation






which can be solved numerically, given  values of k, η, σ,  ϵ^*, and p^*. 


<ref> shows example solutions. As can be seen in <ref>(a), the softening of the distal region results in a drop in pressure there. In the non-growing region (i.e. beyond the intersection point between the solid line and the dashed line), the pressure converges to its base value p^* as ^-s√(k), i.e., perhaps surprisingly, there exists a water potential gradient of lengthscale k^-1/2 resulting from the softening of the shoot over a length  σ. 

We can compute the tip velocity with respect to the base as 
   v_∞ = v(Σ), which is, as expected, a growing function of k; see the inset in <ref>(b). 
In the limit case k≫ 1, with p=p^*, the apex grows along a region of maximal size 
Σ = σlogηϵ^* / ϵ^*- p^*. This length is positive only if η > 1 - p^*/ϵ^*, i.e. when the equilibrium pressure p^* is not too far from the growth threshold ϵ^*, given a certain level of softening η, otherwise the pressure is too low to produce any growth at all. The tip velocity is then maximal, given  by


    v_∞≈  p^* σp^*/ϵ^*-1+η+σ  (p^*-ϵ^*) log(ηϵ^*/ϵ^*-p^*).





Conversely, when k→ 0, the growth zone vanishes and v_∞→ 0. In this case, the base of the plant located at s→∞ is the only possible source of water but cannot sustain permanently the gradient in water potential necessary for growth. Both limit cases are plotted with solid black lines in <ref>.





























































































































    























































































































































 






















































































































































































































































































































































































































































































































































































































































































































































































































































































































 §.§ Parameter estimates


The biological relevance of hydraulic gradients is predicated upon the ratio of hydromechanical length ℒ_h = √(KEτ) to growth region size. To estimate this ratio, 





we consider a linear chain of identical Lockhart cells of average length ℓ (maintained constant through cell-division), hydraulically-insulated from their environment, but exchanging water with their two adjacent neighbours with membrane conductivity κ. From dimensional considerations, we expect the bulk conductivity for this chain of cells to be K ∼ℓκ. We take κ = 10^-8–10^-7.-1.-1 <cit.>. Taking the turgor pressure p to be of the order of 1  <cit.> and the strain ϵ =0.05, the bulk elastic modulus is E=p/ϵ=20 . Taking ℓ=10  and estimating τ = 10^1–10^3 , we obtain 

ℒ_h  ≈ 1–10 ℓ; namely, for this chain of cells, the length of interest for hydromechanical control is on the order of 1 to 10 cells. In practice, these values are very hard to measure with precision, and may vary a lot between different scenarios.





















§ GROWTH OF A CYLINDER: HYDRAULICS  AND RESIDUAL STRESS IN  STEM DEVELOPMENT



 

 §.§ Overview

 
We now move on to a fully three-dimensional, nonlinearly elastic scenario to study the growth of a cylindrical stem. Here, the focus is on the interplay between heterogeneous material properties, water fluxes, and growth in multiple dimensions,  illustrating how hydraulic effects and differential material properties can constrain the growth and internal stresses within a simple three-dimensional structure.

 

 §.§ Governing equations



























































































































































































































































































































































































We consider the growth of a cylinder of initial radius A and length L, and current radius a and length ℓ. The initial domain is parameterised by the system of cylindrical coordinates  (R,Θ,Z), with R the radial coordinate, Θ the azimuthal angle, and Z the axial coordinate (<ref>). In an axisymmetric deformation, a point R,Θ,Z in the reference configuration is moved to the location r,θ,z in the current configuration. Thus, the deformation χ is given explicitly by
r=  r(R,t), θ =Θ, and z = z(Z,t), with 

    r(0,t) = 0,    z(0,t) = 0
 
at R=0.
In virtue of the symmetry, we can identify the two orthonormal bases associated with both systems of cylindrical coordinates: E⃗_R=e⃗_r, E⃗_Θ =e⃗_θ and E⃗_Z = e⃗_z. 
Further assuming that the gradient of deformation is invariant by Z, the problem is effectively one-dimensional and depends only on r and  t. The deformation gradient is given by

    F = r' e⃗_r⊗E⃗_R + r/R e⃗_θ⊗E⃗_Θ +  ℓ/L e⃗_z⊗E⃗_Z,
 
where the apostrophe denotes differentiation w.r.t. R; and ⊗ is the tensor product. The Eulerian and Lagrangian velocities are given respectively by v⃗  = v  e⃗_r + z ζ̇ e⃗_z and V⃗  = V  e⃗_r +(Z ζ̇ℓ/L)  e⃗_z,  with ζ̇ℓ̇/ℓ the relative rate of elongation.  The growth and elastic tensors can be written in the cylindrical basis as
  G = γ_r,γ_θ,γ_z and A = α_r,α_θ,α_z. By (<ref>), we have r' = α_rγ_r, r/R=α_θγ_θ and ℓ/L = α_zγ_z. The gradient of velocity can be related to the growth and elastic stretch rates through (<ref>)


    vr =γ̇_r/γ_r + α̇_r/α_r ,  v/r = γ̇_θ/γ_θ + α̇_θ/α_θ ,
      ζ̇= γ̇_z/γ_z + α̇_z/α_z.













 
 
 By symmetry, the stress is also diagonal in the basis {e⃗_r, e⃗_θ,e⃗_z }, with the total and partial Cauchy stresses given by T=T_r,T_θ,T_z and T_s=t_r,t_θ,t_z, respectively. 
The only non-vanishing component in the balance of momentum (<ref>) is then

    t_rr + t_r-t_θ/r = pr.















Assuming that the end caps are free and not subject to any load, we obtain the boundary condition for the stresses at the end caps <cit.>,

    ∫_0^a (t_z - p) r  r= 0.






Further, we assume that the outer pressure at r=a is zero, so (<ref>) yields the   condition

    p(a) = t_r(a)   .
 















































The balance of mass (<ref>) reads

    1/rrr v - K r ψr +  ζ̇= ξ_f ,

where K = K 1.
The source ξ_f represents the water intake from a bulk vasculature  modelled as a single reservoir with water potential ψ^*, i.e. ξ_f = k(ψ^* - ψ). Assuming that the system is far from hydraulic equilibrium, i.e. ψ^* ≫ψ, we have ξ_f ≈  kψ^*, which is taken to be a function of r only. Thus (<ref>) can be integrated directly as







    v-K ψr
             =  𝒳_f - ζ̇r/2   , 
               with  𝒳_f (r) 1/r∫_0^r ξ_f(r') r' r'  ,

where we have used the no-flux regularity condition ψr = 0, and the geometric conditions r=0 and v=0 at R=0.
For simplicity, here we take ξ_f  constant so that 𝒳_f = ξ_f r /2. Similarly the osmotic pressure π is here assumed to not vary across the domain, so we elide the balance of osmolyte mass equation (<ref>) and we take simply ψr = pr. The cylinder exchanges water through its boundary at R=A, so we use the boundary condition (<ref>) 
    


    ψ(a) =   ψ_a  ⇔   p(a) = p_a,

with p_a π(a) + ψ_a. In the example treated next, we assume ξ_f=0 so that water  enters the domain only through the boundary (however, for the sake of generality, we keep ξ_f in the following derivations). In plants, the organisation of the vascular bundle may vary considerably between species.  The situation presented here mimics a type of architecture where the vascular bundle–xylem and phloem–is located near the epidermis, as illustrated in <ref>(b). For more complex vascular systems, we may assume a non-zero source ξ_f, or an additional water exchange point at R=0 (as in roots, where the vascular bundle is located generally near the centre
). More in general, it is  easy to extend the problem to the case of a vascular bundle placed at an arbitrary position r_v ∈0,a by treating the two problems r< r_v and r> r_v separately <cit.>.

The growth law (<ref>) corresponds to 

    γ̇_i/γ_i =  1/2τ_i(R)α_i^2-α^*^2,

with i∈{r,θ,z}; and  τ_r, τ_θ and τ_z denote the characteristic times of material synthesis in the three separate directions of the cylinder. In growing stems, it is well known that heterogeneous material properties result in mechanical tension within the epidermis, a phenomenon called tissue tension, manifesting the existence of residual stresses. These stresses emerge from differential growth of the tissue, where the core grows relatively faster than the epidermis   <cit.>. A classic experiment consists in peeling a stalk of rhubarb, which results in the detached cortex bending outward and shortening, along with rapid elongation of the exposed core when incubated in water <cit.>, revealing tension in the cortex and compression below. This difference in growth rate is likely to be due to higher  growth extensibility of
the cell walls of the inner tissues <cit.>. Therefore, we  assume that the characteristic time τ_z is larger near the epidermis than at the origin, so that, for equal strain levels, the epidermis undergoes slower growth. We posit  
τ_z(R) = τ_0 + ΔτR/A^2, where τ_0 is the value at the origin, and  Δτ≥ 0 defines the increment in τ_z between the core and the epidermis.  


Finally, the elastic constitutive equations read 
    t_i = α_i/J_AΨ_sα_i .

For simplicity, we here use an isotropic, compressible neo-Hookean strain energy function 

    Ψ_s(α_r, α_θ, α_z)    =μ/2(α_r^2+α_θ ^2+α_z^2-3-2 log (α_rα_θα_z))  +λ/2  (α_θα_r α_z-1)^2,

where the material coefficients μ and λ identify to the Lamé coefficients in the linear regime. Note that in the limit of incompressibility (λ≫ 1), no growth can occur since the solid cannot expand to absorb the fluid.  Henceforth we set λ = μ (corresponding to a Poisson ratio of 1/4 in the limit of linear elasticity). For simplicity, we assume that the elastic moduli are uniform <cit.>. In fact, note that 
a heterogeneity in elastic rigidity would not be sufficient to capture differential growth on its own. Indeed, in a scenario where only μ would be spatially heterogeneous, the axial stretch α_z (thus the growth rate) would still be uniform in a cylindrical deformation.



As in <ref>, we can nondimensionalise the system using √(Kμτ_0), τ_0 and μ as reference length, time and pressure, respectively. On eliminating γ̇_θ/γ_θ and γ̇_z/γ_z using (<ref>), re-expressing the problem in the reference configuration, and rearranging the terms, we obtain a closed system of seven equations for the seven variables γ_r, α_r, α_θ, α_z, V, r, p, defined on the fixed domain ℬ_0:



    r' =  α_rγ_r

    
    t_r ' - p'/r' =      t_θ - t_r/r ,



    p'/r' 
             = 1/2(ζ̇- ξ_f)   r   + V    ,





    γ̇_r/γ_r = α_r^2-α^*^2,



    α̇_r/α_r +  1/2τ_r α_r^2-α^*^2 =   V' /r'    ,


    α̇_θ/α_θ +   1/2τ_θα_θ^2-α^*^2= V/r ,
 

    α̇_z/α_z +  1/2τ_z α_z^2-α^*^2 = ζ̇,





where t_r and t_θ are given in terms of the elastic stretches via (<ref>).
This system is equipped with the boundary conditions (<ref>) and the integral constraint (<ref>) that is enforced via the undetermined parameter ζ̇. 





Note that (<ref>) has a geometric removable singularity at R=0 due to the boundary constraint r(0,t)=0, which generates computational difficulties. This issue is easily alleviated by considering a perturbed boundary condition r(ϵ,t) ≈ϵα_r(ϵ,t) γ_r(ϵ,t) at R=ϵ, where ϵ≪ 1. 























 §.§ Analysis of the solutions




  §.§.§ Steady regime



Before solving the full dynamical problem, we first restrict our attention to steady growth regimes for which we enforce V=α̇_r=α̇_θ=α̇_z=0 and 
γ_r=γ_θ= 1 (no radial growth). In this scenario, (<ref>) can be integrated directly, revealing that a permanent parabolic pressure profile is maintained across the elongating stem, as predicted by <cit.>:



























    p - p_a = 1/4(ζ̇- ξ_f) (r^2 - a^2).

In particular, this pressure is non-negative if 

    a^2 ≤4p_a/ζ̇- ξ_f;
 
otherwise, the radius is too large to allow for even distribution of water given the elongation rate ζ̇, and a zone of negative pressure forms at the centre of the stem. Eq.  (<ref>) may be viewed as a scaling constraint on growth, linking the  kinematics (ζ̇) and  geometry (a), for a given water supply (p_a, ξ_f).







   The rest of the system can be solved numerically (<ref>). 
<ref>(a–c) illustrate the steady solution and its dependency on the (nondimensionalised) stem radius A and the parameter Δτ. The density map in <ref>(a) shows the difference in axial stress T_z measured between the epidermis (R=A) and the origin (R=0), with red indicating states of epidermal tension where T_z(A)>T_z(0), and dashed line showing the level set T_z(A)=T_z(0). We show three cross-sections of the stem showing the stress distribution T_z in three cases: 1–uniform τ_z; 2–heterogeneous τ_z and thin stem; and 3–heterogeneous τ_z and thicker stem. <ref>(b, c) show respectively the elongation rate ζ̇ and mean pressure, in the A-Δτ space. <ref>(d) shows the profiles of pressure p and stresses T_z and t_z for each state labelled 1, 2 or 3 in <ref>(a).


For a uniform τ_z, i.e. when Δτ=0, the axial stress is compressive in the epidermis and tensile in the core, with T_z(A)<0<T_z(0), see case 1 in <ref>(a, d). This is purely a  hydraulic effect due to the higher pressure at the surface, p(A)=p_a. As can be seen, this heterogeneity in stress and pressure becomes more visible when the radius A increases. When a gradient in wall extensibility is introduced, i.e. for  Δτ>0, different scenarios are possible. For a relatively slender stem (i.e. above the  dashed line), a state of epidermal tissue tension is observed, where both the total stress T_z and the partial solid stress t_z are maximal at the epidermis, see case 2 in <ref>(a, d). In this example, the core is in compression (T_z(0)<0); however, the solid matrix is generally still subject to tensile stresses (t_z(R)>0 for all R), indicating that the cells remain turgid and that their walls are indeed under tension, despite the overall compressive stress. This observation challenges the  conception that the cell walls  should be compressed and possibly buckled due to overall tissue compression <cit.>. Overall, the distribution of stresses within a tissue is non trivial, in particular, the macroscopic stress–the one released upon cutting the tissue–is distinct from the stress experienced by the cell wall matrix. 

For thicker stems, below the dashed line in <ref>(a), epidermal tension can no longer be maintained as hydraulic effects override the prescribed heterogeneity in extensibility. Indeed, as can be seen in <ref>(c, d), the pressure becomes low near the origin, indicating a water deficit due to the increased distance to the source, as is especially visible in case 3 in <ref>(a, d). As a result, a lower elongation rate ζ̇ is observed; see <ref>(b). Here, even if the reduced epidermal extensibility would tend to promote epidermal tension, the epidermis is actually in compression, due to its better perfusion. A somewhat counterintuitive effect is observed where, unlike T_z, the axial stress t_z actually increases with R, i.e. the tension in the solid matrix is maximal in the epidermis notwithstanding the global compression. This effect can be interpreted in light of the heterogeneous pressure profile: While the epidermis has high turgor pressure, generating tension in the walls but overall compression within the tissue (since the epidermis is constrained by the core), the pressure near the origin is too low to generate much tension in the cell walls, and most part of the solid stress there is provided by the  epidermis. 


For even larger radii, i.e. below the solid line in <ref>(a), a central region appears where pressure at the origin is negative, i.e. the inequality   (<ref>) is violated. This extreme growth-induced effect results from the high deficit in water, and from axial tensile forces applied to the core by the epidermis, which  effectively create a suction. While it is unclear whether such growth-induced negative pressure can exist, insofar as it results from the saturation assumption (<ref>), we suspect that the relative water deficit within the core, and the associated tensile stresses, could potentially participate in cavity opening during stem hollowing, described mathematically by <cit.>. 





Following the developments of <ref> we also assess the mechanical resistance of the stem under axial loads (i.e. its axial linear elastic modulus). We denote by Â = ( α̂_r , α̂_θ, α̂_z) the incremental elastic deformation tensor, where α̂_r = r̂r and α̂_θ = r̂/r, with r̂ the radial coordinate in the incrementally deformed configuration.  
We first remark that the incompressibility condition
(α̂_zr̂/r ) r̂r = 1
is separable and can be integrated directly as
r̂ = r/  √(α̂_z)
from which we obtain 

    α̂_r = α̂_θ = 1 /  √(α̂_z).






































The stresses are given by (<ref>) as











































    T̂_i =   t_i (α̂_rα_r ,α̂_θα_θ ,α̂_zα_z)- p  - p̂ ,

with i∈{r,θ,z}.
Taking the first variation of (<ref>) around the base solution α̂_i = 1, we derive




    δT̂_i = 𝒯_i  δα̂_z   - δp̂,   with  𝒯_i t_iα_z -   1/2t_iα_r+t_iα_θ.


Then, on integrating the balance of momentum




    (δT̂_r) r = 𝒯_θ - 𝒯_r/r δα̂_z,





and using the identity δT̂_z - δT̂_r= (𝒯_z   - 𝒯_r)  δα̂_z, we obtain finally the total virtual reaction force of the whole stem




    δ F  2π∫_0^a δ T_z r r = ℳ_z δα_z ,

with ℳ_z the effective longitudinal  linear elasticity modulus given by

    ℳ_z  =  2π∫_0^a𝒯_z   - 𝒯_r -∫_r^a 𝒯_θ - 𝒯_r/r'  r'r r.

As can be seen, ℳ_z depends on the pre-existing stresses and stretches within the stem.
In particular, in the absence of pre-stress (α_r=α_θ=α_z=1, p=0), we recover the known value ℳ_z^0 = 3πμ B^2 for the linear response of an incompressible neo-Hookean tube under uniaxial load. <ref> shows the dependency of the relative stiffness ℳ_z / ℳ_z^0 on  A and Δτ. As can be seen by comparing the level sets of <ref> with those of <ref>(a), the variation in axial stiffness of the stem is related to the presence of axial residual stresses, measured by T_z(a) - T_z(0). This observation supports the hypothesis that tissue tension confers higher rigidity to the stem <cit.>.
However, unfortunately, 𝒦_z does not inform us directly on the resistance of the stem to buckling, even for a thin stem, insofar as the residually-stressed cylinder is effectively heterogeneous and anisotropic. To that end, a full and likely tedious  perturbation analysis including asymmetric modes is required <cit.>. 














  §.§.§ Dynamic regime

These different states of the system can be observed in the full dynamic problem, as illustrated in <ref>. Here, we simulate the three-dimensional growth of a stem with Δτ = 2 taken as constant.  To account for the rapid elongation of the stem, we also assume that the extensibility is smaller in the r and θ directions <cit.>. As predicted earlier, epidermal tension is maintained until a critical radius is reached, at which point pressure becomes too low  at the origin and epidermal compression appears. As the radius increases and the mean pressure decreases, the elongation rate ζ̇ also decreases. 

















 




































































































§ GROWTH OF A SPHERE




As a final example, we study the full model (<ref>) including the transport of osmolytes, that is, instead of assuming a constant osmotic pressure as in <ref>, we here assume that the osmolytes can diffuse on the growing domain, here a thick spherical shell. This example  illustrates how physiochemical details of organ growth can be included, for instance in the context of fruit growth which involves complex transport of sugar, gas exchanges and water fluxes. Although we do not aim here to model the sheer physiological complexity of fruit growth and maturation, our approach provides a paradigm to generalise detailed zero-dimensional multi-compartment approaches–e.g. <cit.>–to a continuum.




We study the growth of a hollow sphere of inner and outer radius a and b in the current configuration (A and B respectively in the reference configuration). We introduce the system of spherical coordinates (r,θ,φ) in current configuration and  (R,Θ,Φ) in reference configuration; see <ref>(a). We assume the problem to be spherically symmetric so that, as in <ref>, the problem only involves the coordinates r and t.

For simplicity, we neglect active transport of the osmolytes (f̅⃗̅_o = 0) and we write L_os = (c_o R_gθ / D_os)  1 with D_os the diffusivity of the osmolytes in the solid, so that (<ref>) can be written as a standard Fickian flux
j⃗_o =  - D_osv_o c_or e⃗_r.
We assume that the outer hydrostatic pressure is zero on both faces of the shell. For the flux of material, we assume a Dirichlet condition (<ref>) at r=a. At the outer boundary r=b, we postulate an transpiration outflux associated with a water potential ψ_ev and conductivity k_b. Altogether, we have

    t_r(a,t) = p(a,t),  π(a,t) = π_a;   ψ(a,t) =  ψ_a,
      t_r(b,t) =p(b,t),  ψr(b,t)= k_b(ψ_ev - ψ),  πr(b,t)=0,

with ψ_a the water potential across the inner boundary; and π_a the osmotic pressure at r=a. We assume ξ_f = ξ_o = 0, so that the only source of water and osmolyte mass is the inner boundary. 
Following a procedure similar to that of <ref>, we derive the (nondimensionalised) governing equations for the growing sphere:










    
    t_rr + 2/rt_r-t_θ = pr,











    v-r(p-π )   = b^2/r^2ḃ - k_b(ψ_ev - p_b + π_b)  ,


    vπ -    D_os v_o  πr   =         b^2   ḃπ_b  /r^2   ,


    γ̇_r/γ_r =  1/2α_r^2 - α^*^2,  α̇_r/α_r +  1/2α_r^2 - α^*^2 = vr,
      α̇_θ/α_θ +   1/2α_θ^2 - α^*^2 = v/r,


    t_r = 1/α_θ^2Ψ_sα_r,   t_θ = 1/α_rα_θΨ_sα_θ,





where we have used the approximated van 't Hoff relation π≈ Rθϕ_o / v_o, the two no-flux boundary conditions (<ref>) to integrate (<ref>), and the neo-Hookean strain energy (<ref>); and where π_b π(b) and p_b  p(b) are the undetermined osmotic and hydrostatic pressures at the outer boundary. We have assumed that the osmolyte diffusion is fast  and in the quasi-steady regime, so that we  set ϕ_ot=0 in (<ref>). All parameters are taken to be uniform across the domain and constant in time. 
<ref>(b) illustrates the evolution of pressure in a typical simulation. We see that during growth, the pressure drops in regions located far from the central source (similar to <ref>). This simulation illustrates again the dynamic nature of pressure in the context of growth and osmotic regulations with multiple interfaces.
This approach may be beneficial to integrate spatiotemporal details of physiological and mechanical regulations in fruit development, as well as nonlocal couplings, which cannot be captured using more basic zero-dimensional models.











































































































































































































































































§ DISCUSSION
 

While many continuum models of plant tissues morphogenesis have relied on phenomenological kinematic specifications of the  growth behaviour, the establishment of a mechanistic theory of morphogenesis requires modelling growth in relation to more fundamental physical and mechanical fields  <cit.>. 
In plants, it is widely accepted that cellular growth results from mechanical deformations of the cell walls driven by cell osmolarity. However, the explicit connection between pressure and growth has not been systematically integrated in continuum models. To bridge this gap, we proposed a formulation of plant developmental processes that extends the paradigm of Lockhart, Cosgrove and Ortega to a hydromechanical continuum description of the growth phenomenon. 



 
 

The role of pressure have been considered in numerous multicellular discrete models which have described growth as a result of turgor-induced cell wall expansion. However, typically, these models have treated pressure as a prescribed biological parameter, thereby neglecting the fundamentally mechanical nature of pressure. Physically, this modelling simplification originates in the assumption that  growth is primarily limited by the cell wall extensibility Φ, thus, that water exchanges across cell membranes are instantaneous (i.e.  p≈π). 
While this assumption may hold true in a first approximation, such simplification does not allow for a complete representation of the physical events occurring during growth. Recent reevaluation of water conductivity contributions in various systems has also revealed a  nuanced role of hydraulic effects, which may be non-negligible <cit.>. These advancements motivate the development of new models based on a proper formulation of turgor. Indeed, the fundamental interplay between pressure, fluxes, osmolarity and cell mechanics is critical for a proper understanding of the notion of turgor, with profound experimental and conceptual implications. Thus, we assert that a robust physical description of growth should be grounded in clear balance relations. 
Our poroelastic theory  captures the growth of a tissue as the result of simultaneous mechanically-induced cell wall expansion and  water fluxes. Thus, pressure is treated as a dependent mechanical variable that  mediates  the  growth  deformations indirectly, through the balance of forces. This integrated perspective is directly in line with the original philosophy of Lockhart's approach.
 
 
 
 
 
 

 
  

 



 
A key mass balance principle can be stated as follows: For growth to occur, water must flow to fill the expanding volume, irrespective of the cause of cell wall expansion. In other words, growing regions correspond to hydraulic sinks and fast-growing regions are associated with lower water potential, as exemplified in <ref>. Such sink is characterised by the development of growth-induced gradients of water potential and a water flux directed towards the growing region, an idea which has surfaced in the literature in particular in the work of Boyer and coworkers <cit.>. This flux, captured here through a Darcy-type law (<ref>), introduces a fundamental growth-induced hydromechanical length ℒ_h =√(KEτ_0) which reflects the combined effects of growth (τ), tissue permeability (K), and elasticity (E). For growing avascular domains larger than this typical length, interesting nonlocal couplings may emerge. For example, in <ref>, we predicted that a fast growing region will absorb water from its neighbourhood, thereby hindering its growth. This idea is reminiscent of patterning mechanisms based on lateral inhibition <cit.>. Such phenomenon has been predicted in previous theoretical study <cit.> and was recently observed experimentally in Arabidopsis thaliana shoot apices, where peripheral cells adjacent to incipient organs exhibit a shrinkage consistent with a water deficit <cit.>. Here we  characterise mathematically the magnitude and spatial extent of this inhibition (<ref>). In particular, we show that inhibition will be amplified and more spatially extended in  weakly vascularised tissues  (k≪ 1), where the inhibition zone is expected to have a width  ∼ℒ_h.  


The role of hydromechanical effects is also illustrated in three dimensions in our model of a growing stem. Firstly, we showed that the classic epidermal tension hypothesis could be naturally captured by a gradient in cell extensibility; reproducing the phenomenology of previous continuum models <cit.>. Secondly, for thick stems where the distance to the vascular bundle increases,  the distribution of residual stresses was greatly perturbed by hydraulic scaling effects when pressure at the core became very low. 
Naturally, in reality, the vascular architecture of plants may evolve as part of the developmental process to maintain adequate vascularisation. Therefore, we anticipate that our predictions may not apply universally, and the model must be adapted to capture more realistic scenarios. Overall, this work aims to exhibit guiding principles of hydromechanical control of plant morphogenesis and illustrate the complex, nonlocal and fundamentally dynamic behaviours that emerge from integrating fundamental growth mechanisms in space and time. 



































Several  questions remain open. The most pressing problem is to formulate a constitutive law describing growth, specifically the process by which the cell walls expand and solid mass is added to the system. 
 Although a strain-based growth law could capture the basic phenomenology of plant growth, its mechanistic interpretation is not fully established, in particular, it remains  unclear which specific mechanical quantity should be adopted as a driver of growth, and whether an appropriate growth law can be derived from rational mechanics considerations (<ref>). Furthermore, a realistic growth law should also be able to capture the multiscale link between local cell structures and anisotropies and the overall growth of the tissue at the continuum level. In this context, promising efforts to derive continuum representations  from the cellular structures using multiscale analysis are emerging <cit.>, and we hope that our work will motivate further advances in this area.
 
 


Another question concerns the link between the effective permeability tensor and the microscopic details of water routes within the tissue. In the context of direct cell-to-cell water transport  <cit.>, an approach would consist of  deriving the effective hydraulic conductivity of a periodically-repeating representative cell network  using two-scale analysis, as described by  
<cit.>. More in general,  the biological details of water transport in different tissues are still an active subject of research, thus the precise biophysical interpretation of the effective conductivity is yet to be better characterised. For example, an interesting extension to our model would be to treat the apoplasmic route and the transmembrane exchanges between cell vacuoles separately <cit.>. 

Lastly, a natural extension to this work is to model the role of  morphogens (e.g. hormones, such as auxin, or genes), using regular advection-diffusion equations, or more complex, nonlinear reaction-diffusion-advection-systems <cit.>. In the spirit of our approach,  these morphogens should regulate specific physical and mechanical properties of the system, such as the rigidity, the osmolarity, the growth threshold or the cell extensibility, thereby indirectly influencing growth.

Overall, this work lays the foundations of a field theory of plant morphogenesis–a closed mathematical framework in which the growth phenomenon emerges as the product of multiple, coupled physical, chemical and mechanical fields acting more or less nonlocally. The construction of such theories is a formidable challenge in plant biomechanics and, in general, in the study of active living materials. 
In this broader context, the unassuming plant provides an interesting paradigm to build a general theory of living tissues.  



tocsectionAcknowledgements


§ ACKNOWLEDGEMENT

 I.C. acknowledges support from the Institut rhônalpin des systèmes complexes (IXXI), and the Agence Nationale pour la Recherche through the research project HydroField. The authors are grateful to Alain Goriely, Andrea Giudici, Christophe Godin, and Arezki Boudaoud for insightful discussions.



























 













§ NUMERICS AND IMPLEMENTATION

We integrate the system using a  relaxation method (backward time, centred space); see <cit.>. All simulations were implemented in the software https://www.wolfram.comWolfram Mathematica 14.0. Source code is available upon request.




§ TABLES OF MAIN QUANTITIES


We recapitulate in <ref> the main quantities used in the paper. Finish































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