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Review: How dynamic prestress governs the shape of living systems, from the subcellular to tissue scale

Cells and tissues change shape both to carry out their function and during pathology. In most cases, these deformations are driven from within the systems themselves. This is permitted by a range of molecular actors, such as active crosslinkers and ion pumps, whose activity is biologically controlled in space and time. The resulting stresses are propagated within complex and dynamical architectures like networks or cell aggregates. From a mechanical point of view, these effects can be seen as the generation of prestress or prestrain, resulting from either a contractile or growth activity. In this review, we present this concept of prestress and the theoretical tools available to conceptualise the statics and dynamics of living systems. We then describe a range of phenomena where prestress controls shape changes in biopolymer networks (especially the actomyosin cytoskeleton and fibrous tissues) and cellularised tissues. Despite the diversity of scale and organisation, we demonstrate that these phenomena stem from a limited number of spatial distributions of prestress, which can be categorised as heterogeneous, anisotropic or differential. We suggest that in addition to growth and contraction, a third type of prestress -- topological prestress -- can result from active processes altering the microstructure of tissue.


Introduction
A peculiarity of living systems is their ability to constantly rearrange their structure in order to perform biological function. Cells, for example, transition from static to migrating while changing shape, or split in the process of cell division. At the tissue scale, specific shapes are acquired during development and maintained at adult age to accomplish organ function, but can be lost as in the case of cancer. These rearrangements are in most cases the result of active 1 processes taking place within the cells or tissues themselves, rather than being imposed from the exterior through boundary conditions. Across the scales and specificities of systems, one finds a number of ways for these internal stresses to be generated, ranging from protein synthesis or pumping of ions that give rise to osmotic pressure gradients to ATP hydrolysis-fuelled changes of conformation of crosslinkers within biopolymer networks. One common point of these mechanisms is that they are controlled by biological pathways, and that they can be triggered or modulated dynamically, enabling systems to change shape or state. In this review, we aim to show how these very different force-generation mechanisms can be usefully understood with a common concept of prestress (and prestrain). While this concept has been useful in describing stable tissue shape in adult tissues [Taber, 1995], here we focus on how dynamic changes in prestress can alter shape. Often, prestress generation interplays with other less specific properties of living systems like their complex rheological properties, thin-sheet geometry and foam or network architecture. In some cases, it appears challenging to distinguish between phenomena stemming from dynamic prestress and those stemming from these complex material properties. An example developed below is the cell neighbour exchange (also called T1 transition) occurring in epithelia, which can have either active or passive origins.
In section 2 of the review, we will present the concept of prestress and the theoretical frameworks available to describe prestress in statics and dynamics. Following on, we will briefly present in section 3 the strategies available to experimentalists to identify and measure prestress in living systems. Then, in the final two sections, we will describe strategies of prestress generation in biological systems and the macroscopic effects obtained. On our way, we will show that common categories of spatial regulation of prestress (heterogeneous, anisotropic or differential) are used in systems with different compositions and scales.
Two key concepts will be used throughout this review: growth and contraction. By growth, we mean any active process that leads to increase the equilibrium size of some of the components of a system. This can be due to the addition of material within the system by some out-ofequilibrium process, e.g. the polymerisation of filaments fed by monomers diffusing into it. This can also be more simply due to osmotically-driven attraction of more water into the system. By contraction, we mean any active process that leads to the reduction of the equilibrium size of some of the components of a system.
In section 4 of the review, we will focus on crosslinked networks of semiflexible biopolymers, either within cells-they are in this case part of the cytoskeleton-or external to them-then part of the extracellular matrix (ECM). These networks can be remodelled and crosslinked by other proteins which have out-of-equilibrium dynamics, fuelled by active processes. We will show how contraction and growth of these networks govern shapes and deformations of subcellular compartments, cells and fibrous tissue. Within the cytoskeleton, one network of particular interest is the one formed by actin and the crosslinker (and molecular motor) myosin.
Finally, in section 5, we will describe how prestress affects the shape of cellularised tissue. Here, the material is formed by cells of regulated volume and mechanically connected through adhesions, molecular bonds joining their plasma membranes. The adhesions are formed of transmembrane complexes, allowing tension to be transmitted between the cytoskeletons of neighbouring cells. In this section, the elements experiencing growth and contraction interact with these other elements, making the material heterogeneous and bringing additional effects. Prestress can also be built through topological changes of the cell contours. We refer to this type of prestress as topological prestress. In particular, we suggest that, from a mechanical point of view, morphogenesis through cell proliferation is conveniently described if one distinguishes within the effect of cell division between growth prestress (increase of volume of cells) and topological prestress (due to the apprearance of new cell-cell junctions following cytokinesis 2 events).

Concept of prestress in living systems
In engineering, generally, "pre" in "prestress" refers to the fact that it is due to an operation done before establishing the system, for instance imposing boundary conditions such as tension on a structure such as wires, called tendons, before putting them in parallel with a compressionbearing material, e.g. by casting concrete. This results in a system whose reference configuration is not compatible with the reference configuration of each of its components, which are thus prestressed. In systems of linear elastic components, one can equivalently consider that they are prestrained, the prestress and prestrain being simply related by the elastic modulus.
The source of prestress is not necessarily an externally imposed force applied to a component of the system, but can also be due to an internal change in the system. Prestress can thus arise in a system which is already connected and is originally stress-free, if one of the components, which we will call the active component, changes its equilibrium configuration. As an example, a porous material (passive component) whose pores are occupied by a liquid will become prestressed if that liquid (active component) changes volume due to freezing or crystallisation. In that case, "pre" is not understood anymore as referring to a process in time, but rather as making reference to the fact that prestrain corresponds to the deformation between the initial configuration and a virtual configuration where the active component has assumed its new equilibrium shape. However, this stress-free configuration is virtual because the shapes of the active and passive components are not compatible anymore.
The actual configuration in the absence of external load results from the mechanical balance between the active and the passive components, neither of which will be stress-free: the observed stress field in the absence of external loads is called residual stress [Fung, 2013, Taber, 1995, Goriely, 2017. While residual stress has this narrow definition, the term prestress is somewhat broader. In the engineering community, prestress is typically due to external loads [Gower et al., 2017]. The prestressed configuration is then used as a reference configuration, from which another elastic deformation (for instance the wave propagation in the prestressed body) is studied [Parnell, 2012]. In biophysical models, prestress can also be of active origin, for instance due to a morphogenetic event [Ben Amar et al., 2018]. We take advantage from this relative freedom to define prestress so that it corresponds to the notion of active stress which was defined in the context of actomyosin systems . Indeed, if one chooses to describe the prestressed material with respect to its original shape, that is, the configuration it would have in the absence of prestress, as a reference configuration, then one finds that there is now a stress field associated to it, which for this choice is a prestress field [Salençon, 1994]. Since this prestress does not need to preexist the system, it can be adjusted at any time by nonmechanical processes, such as biological pathways, so that it can drive dynamics.
We now illustrate the concept using elementary mechanical elements. In order to describe the behaviour of individual cells within tissue, a leading approach is to use cell-based discrete models, which can capture cell-cell interactions and dynamical changes in topology. On the other hand, continuum descriptions offer the benefit of giving access to quantities such as the Young's modulus or Poisson's ratio, and offer a vast array of tools for simplification through mathematical analysis [Jones and Chapman, 2012]. For the sake of simplicity of presentation, we present ideas in terms of continuum models when possible, and discrete models when necessary.
Figure 1: Prestrain and prestress in simple 1D systems. (a) Illustration of deformation gradient decomposition for contractile prestrain. An active element (red) and a passive one (gray) are put in parallel in between force-bearing walls (black vertical lines). The active element is composed of a spring whose length is actively decreased (or increased) from L 0 to L, with an anelastic stretch, or prestrain, λ a (here λ a < 1) imposed via a crank. In the virtual configuration, both elements remain stress-free but the system's topology (dashed line connections) is not respected. In the current configuration, even at equilibrium (no net force on the walls), the structure is under stress. Operating the crank the other way, λ a > 1, gives the effect of growth prestrain. (b) A system equivalent to the one in a can be obtained by replacing the active crank and spring element by a stress generator element (pulleys and weight system) whose magnitude is the prestress. (c) For topological prestress, no active spring is necessary; the activity consists of disconnecting and reconnecting elements into a new network. Initially, springs k 1 and k 2 are in parallel, and the pair is connected in series with k 3 . The topological change reconnects the springs, such that k 1 and k 3 are in series, this pair connected in parallel with k 2 . Due to the change of topology (see directed graph insets) the initially stress-free structure becomes prestressed; spring k 2 is in tension, springs k 1 and k 3 are in compression. (d ) A viscous passive element (dashpot) in parallel with a stress generator gives a permanent regime of contraction at a constant strain rate. 4

Static description
We illustrate the concept of a virtual configuration in Fig. 1a based on a 1D example, meaning that we only consider deformations in the horizontal direction. An active element (red) is connected in parallel to a passive element (gray). The vertical black lines represent forcebearing walls. In the initial configuration, both elements have length L 0 , and the system is unstressed. A contraction reduces the length of the active element to L, acting effectively like a spring that reduces its rest length through an active process. We refer to this active stretch as λ a = L/L 0 . In this example, λ a < 1, since an active contraction occurs. In the virtual configuration, the system is still stress-free, but incompatible, since the two elements now have different rest lengths making their existing connection impossible [Eckart, 1948]. Compatibility is restored through building stress: in the current configuration, both elements have length l, which is the new equilibrium length. The elastic stretch of the active element is λ e = l/L. In this new equilibrium configuration, the active element is in tension (λ e > 1), and the passive spring is in compression (its elastic stretch is l/L 0 < 1). In this framework, growth can be treated the same way, with one crucial difference: the active element increases, rather than decreases, its rest length, so that the active stretch is λ a > 1.
It is useful to understand the relationship between prestrain and prestress, since it allows to make a link with the work on active stress in the context of contractile networks . To understand this relationship, let us consider one active element in isolation, with initial length L 0 , virtual length L, and current length l. The Hookean stress-strain relationship of this element is where E is Young's modulus. If we now deform the spring to the formerly stress-free state (l = L 0 ), we will be met with resisting stress σ equal to the prestress σ = σ a = E(λ −1 a − 1). In this simple case, there is thus an explicit relation that can be written between the prestress field σ a and the active stretch that characterises prestrain λ a = L/L 0 . How can this be made sense of in the context of the stress-strain relationship of the spring, Eq. (1)? It is possible to split (1) as follows: This shows a Hookean behaviour near the initial, formerly stress-free state, with the modified Young's modulusẼ = E + σ a . At the formerly stress-free configuration l = L 0 , we recover the prestress σ = σ a . Up to the change of elastic modulus, we thus see the equivalence between a system in which the active element is a growing (σ a < 0) or contracting (σ a > 0) elastic material and a system in which the active element is a "stress generator" of magnitude σ a , see Fig. 1b.
In order to describe more generally mechanical systems with prestress, and to allow the prestrain to be either due to growth or active contraction, we use a framework of anelasticity [Rodriguez et al., 1994, Lubarda, 2004, Epstein, 2012. At its core is the decomposition of the deformation gradient as F = F e F a , where F is decomposed into an anelastic part F a and an elastic part F e , see Fig. 1. The elastic deformation is taken to be hyperelastic and isotropic, for example neo-Hookean, captured by a strain-energy density W = W (F e ). The anelastic deformation is associated with an irreversible process that in some way modifies the microstructure: F a can mean active contraction, like energy-consuming contraction due to myosin, in which mass is conserved or reduced (i.e. transferred from the solid phase to an extracellular reservoir), det F a ≤ 1. Alternatively, it could mean growth, in which mass is added into the solid phase from an extracellular reservoir, det F a ≥ 1. Cauchy stress is then defined as σ = (det F e ) −1 (∂W/∂F e ) F T e and mechanical equilibrium is div σ = 0 in the absence of body forces and respecting that anelasticity Figure 2: Creating residual stress from different patterns of prestrain in the anelastic framework for a disk. The disk is incompressible (det F e = 1) and of neo-Hookean material. The boundary conditions are no traction, σe R = 0. We denote the components of the prestrain in polar coordinates F a = diag(γ R , γ θ ). Here we illustrate contraction, γ < 1, however equivalent situations are found for growth, γ > 1. (a) We consider spatially heterogeneous prestrain which is isotropic (γ := γ R = γ θ ). The initial configuration shows the undeformed, uncontracted, stress-free disk. In the virtual configuration, most contraction occurs towards the periphery, leading to an incompatible body. The prestrain is explicitly shown in the inset, where γ = 1 (no prestrain) at the center and γ = 0.5 (contraction) at the boundary of the disk. The result in the current configuration is a residually stressed body, with tensile hoop stress (σ θ > 0) at the disk boundary, and compressive stress (σ R < 0, σ θ < 0) at the disk center. Due to the tensile hoop stress at the boundary, the disk would open if incised in the periphery. (b) We consider the case of anisotropic (γ R = γ θ ) but spatially homogeneous (dγ R /dR = dγ θ /dR = 0) prestrain. This corresponds to pizza slice shaped pieces being cut out in the virtual configuration. The resulting stress field is qualitatively the same as in the heterogeneous case. (c) We consider the case in which the outer part of the disk has a different prestrain than the inner part, prestrain being isotropic. The scenario is a discrete version of the heterogeneous case. The hoop stress is discontinuous.
(active contraction or growth) occurs at time scales much larger than elastodynamics. In this view, prestress will exist in the system due to the incompatibility [Eckart, 1948, Skalak et al., 1997, Aharoni et al., 2016, Truskinovsky and Zurlo, 2019, Lee et al., 2021 of the anelastic strain F a : for instance, if the system is composed of multiple components with different incompatible reference configurations (F a in layer 1 is different from F a in layer 2), or anisotropy (F a in, e.g., radial direction does not match F a in hoop direction), or some heterogeneity in the anelastic strain (say, F a (x) = F a (x + ∆x)), which could capture a spatial gradient in growth or active contraction. Fig. 2 illustrates how residual stress from different patterns of prestress can be created from the anelastic point of view. A distinctly different possibility for building prestress is through changes in the microstructure due to rearrangements in the network. An example are T1 transitions, which are neighbour exchanges between cells in epithelial tissues, see section 5.3. We offer to name this type of prestress topological prestress. We define it as prestress that is added to or removed from an interconnected network of mechanical elements by breaking and reconnecting network elements rather than prestressing individual elements. For instance, a passive T1 transition relaxes the stress in the system purely by exchanging which cell is connected to which, and not through modifying the reference configuration of any of the cells.
The concept of a microstructural rearrangement leading to topological prestress is illustrated in Fig. 1c. A network of three springs is presented in the initial configuration, but breaking and reconnecting bonds changes the network topology in the virtual configuration (topology refers to the connectivity of a network). Such changes of connectivity, which are meant to describe networks of discrete elements such as cell adhesions or polymer crosslinks, are challenging to describe with a continuum field like F a that is meant to describe the larger tissue scale. For example, discrete deformations like slip lines in crystal plasticity have been successfully described with the continuum framework F = F e F a where F a describes the macroscopic plastic deformation, F e the elastic deformation, and F the total deformation Conti, 2014, Reina et al., 2016]. But for biological tissue, which is generally amorphic and has no crystalline structure, the definition of macroscopic topological prestrain has not fully been achieved yet, although it is an active area of research [Chenchiah and Shipman, 2014, Murisic et al., 2015, Erlich et al., 2020, Kupferman et al., 2020.

Dynamic description
Both the active and passive elements of the system can exhibit a time-dependent behaviour. The growth rate and actomyosin contractility both depend on some nonmechanical processes, such as protein synthesis, nutrient intake, or ATP hydrolysis. All these are tightly regulated by biochemical pathways in physiological conditions, and in a large part the dynamics of the mechanical systems can be enslaved to biochemical clocks [Michaux et al., 2018, Nishikawa et al., 2017, Heer et al., 2017. Active elements are also sensitive to the mechanical context. For instance, individual molecular motors are known to stall beyond some maximal load [Liepelt and Lipowsky, 2009]. In the context of growth, the timescale is rather large, since it is the one of the cycle of cell division which takes place over hours or days. Therefore, the passive elements are considered to be always at equilibrium.
On the other hand, regimes of cell motility often rely on the dynamics of the passive component with a constant prestress. This may be required to achieve movements which are faster than the rates at which prestress can be created. This is the case e.g. for carnivorous plants [Forterre et al., 2005] where an elastic instability is being used to suddenly release elastic energy that had been stored by the slow build-up of prestress.
While the full complexity of interacting timescales between the active prestress one and the passive viscoelastic ones is encountered in some cases, we will focus here on models that describe cases where they are sufficiently well separated. In the case of growth, the system is often considered to be purely elastic and the models thus focus on the timescale of the evolution of prestrain. In the case of contractile networks, viscous dissipation in the microstructure is often important for the observed dynamics and sets their timescale, whereas the active stress is often assumed to be slowly varying.

Dynamics governed by the active component: the example of growth laws
The timescale of elastodynamics in soft tissue (i.e. wave propagation in soft elastic media) is on the order of milliseconds, and a typical viscous time scale due to internal friction is on the order of seconds to minutes. The timescale of growth, on the other hand, varies from minutes (doubling time of Escherichia coli ) to years (slow growing tumours). Growth is thus a case where the separation of timescales is generally sufficient to consider that the passive components are instantaneously reaching their equilibrium configuration [Goriely, 2017]. The dynamics is then governed by a law that prescribes the evolution of the active component's prestrain as a function of the current configuration. Thermodynamic arguments based on the entropy inequality or a dissipation principle motivate appropriate forms of such a law [Epstein and Maugin, 2000, Lubarda and Hoger, 2002, DiCarlo and Quiligotti, 2002, Ambrosi and Guillou, 2007, Ganghoffer, 2010. By following a standard set of arguments and derivations, one arrives at a variant of the growth laẇ where /ρ r is the Eshelby stress, W is the strain-energy density, and ρ r the density in the virtual configuration. The homeostatic Eshelby stress is σ * E and K is a positive-definite coefficient matrix. The principle of homeostasis states that organisms have the ability to self-regulate some of their properties so as to optimise function in a physiological state, such as the ability of mammals to maintain a constant body temperature. In the context of mechanics, homeostasis can be understood as a living tissue's ability to grow and remodel to accommodate a preferred (homeostatic) stress, i.e. to reshape itself to reduce the difference between its actual stress and the a priori known or genetically encoded homeostatic stress [Di-Carlo andQuiligotti, 2002, Erlich et al., 2019]. Growth laws of the type (3) which employ a homeostasis mechanism have been applied to morphogenesis problems, like sea urchin gastrulation [Taber, 2009], the formation of ribs in Ammonite's seashells [Erlich et al., 2018], and the intestinal crypt [Almet et al., 2021], as well as other applications such as wound healing [Bowden et al., 2015, Taber, 2009 and discrete networks such as plant cell networks [Erlich et al., 2020].

Dynamics governed by the passive component: the example of contractile networks
On the other hand, there are systems in which the limiting rate of strain is set by the passive component. Large amplitude motion, such as muscle contraction or intracellular retrograde flow, could not take place in a purely elastic medium. Indeed, in the elastic models of Fig. 1a, an obvious upper bound for the amplitude of strain is the magnitude of the prestrain. However, if the passive spring is replaced by a viscous dashpot (Fig. 1d ), then a constant rate of strain is achieved. This modelling is consistent for instance with the theory of sliding filaments for muscles, as delineated by [Huxley, 1957], where the regime of maximum contraction speed is explained in terms of a balance between active stress and a sliding friction. This compound 8 element is thus governed by an equation of the form: Here we find striking similarity with Eq.
(3), with the difference that the strain rate of the whole system appears directly. In both cases, as long as the stress differs from a given homeostatic stress (σ * ) or active stress (σ a ), an internal length, the virtual length L or a length of 'telescoping' of filaments, is being adjusted at a rate set by a parameter which has the dimensions of a viscosity. We will come back to the molecular-scale understanding of the sliding friction in section 4.1.2. In effect, this sliding friction can be likened to the fluidisation of any viscoelastic liquid beyond its relaxation time. In transiently reticulated networks, such as the actomyosin network, this relaxation time is related to the residence time of crosslinkers [Larson, 1999] and thus the maximum speed of actomyosin contraction can be related to these crosslinker dynamics [Étienne et al., 2015], yielding a Maxwell model for the passive component: whereε is the rate of strain tensor, E the elastic modulus of the crosslinked actin network, and τ a a characteristic relaxation time. The time derivative σ has to be an objective time derivative of the stress tensor σ, starting from rubber elasticity theory one obtains an upper-convected Maxwell derivative [Yamamoto, 1956[Yamamoto, ,Étienne et al., 2015. This can be related to the fact that the network structure is based on linear elements under stretch deformation [Hinch and Harlen, 2021]. Note that corotational derivatives are widely used in the field. This constitutive relation is consistent with the general framework of active gels , which provides a thermodynamic framework relating the active prestress σ a to the chemical potential difference associated with the myosin activity. At the molecular scale, the corresponding continuous injection of energy drives these system out-of-equilibrium and is at the origin of a spectacular violation of the fluctuation-dissipation relation [Mizuno et al., 2007], although effective equilibrium descriptions can be restored at higher scales [O'Byrne et al., 2022].
Contrary to growing systems, contractile ones generally deform while keeping a constant mass. In order to sustain a deformation rate that will in general not be volume-preserving, the density ρ of the network needs to be actively regulated to a constant value ρ 0 by a reaction term, which in its simplest expression writes, in one dimension: where v is the velocity in x direction. Here τ −1 n provides another bound for the maximum rate of sustained flow. On the other hand, the reaction term in the mass balance can itself be a source of growth-related prestress. Assuming a density-dependent rheology of the material, such as σ = −E(ρ − ρ * )/ρ * in its simplest form, with ρ * an equilibrium density. When density is close to this equilibrium, ρ ρ * , we find again Eq. (4) with η = τ n E and σ a = E(ρ * − ρ 0 )/ρ * [Putelat et al., 2018].
Finally, one situation which is encountered in several living systems is a stationary system size emerging from an enduring permanent internal flow regime. It can be easily seen e.g. that a closed system governed by Eq. (6), but with different regulation densities ρ a 0 < ρ b 0 in different geometrical regions a and b, will establish a flow from the region b to a. The system total size will adjust as the combination of the local growth and shrinkage, and there can be geometries and parameters for which this balances yields a constant total size. This sort of dynamic equilibrium will be exemplified below in actomyosin networks and cell spheroids.

9
Obtaining detailed and reliable expositions of the prestresses which shape living matter has presented a great technical and conceptual challenge. A major difficulty is the large number of components which are in fact represented by the simplified components of Fig. 1. In every cell and tissue, numerous stress-bearing and -generating elements are mechanically coupled in complex (and often unknown) arrangements. Current techniques typically allow us to probe only small subsets of those components at once, so we are liable to overlook the many connected parts which remain invisible. On top of this, of course, the reference configuration of the components of Fig. 1 cannot be deduced from the reference configuration of the ensemble alone, they can only be revealed through perturbations. But living matter has a great propensity to react and adapt to the perturbations we introduce in order to measure, so that there is always a real risk of measuring artefacts.
In order to face these challenges, in recent years, a multiplicity of experimental methods have been developed by biologists and physicists to study dynamic prestress. The most commonly used approaches are: • live imaging of molecular actors generating prestress and subsequent strain of the biological material; • biological perturbation of prestress generators via drugs or molecular loss of function; • prestress release by cutting and ablation followed by measurement of resulting strains; • insertion of or embedding into stress-sensing deformable elements, functioning from the tissue scale down to the subcellular scale.
These developments have been reviewed in detail elsewhere for both cell [Polacheck and Chen, 2016] and tissue scale measurements [Gómez-González et al., 2020] and will be described when required in the sections below.

Prestress in the actin cytoskeleton
The cytoskeleton is made of three categories of dynamic filaments (namely actin, microtubules and intermediate filaments) but only actin and microtubules are found to interact with molecular motors, which are major actors in prestress generation. We focus here on the actin cytoskeleton, although the framework defined above can be applied to the microtubule network, for instance to understand the force-balance within the mitotic spindle [Gay et al., 2012]. The actin cytoskeleton is made of polar semiflexible filaments composed of G-actin monomers, which turnover within filaments in time-scales ranging from seconds to minutes depending on cell types and actin structures considered [Amato and Taylor, 1986, Elkhatib et al., 2014, Saha et al., 2016, Clément et al., 2017. Growth rate and geometry of the actin networks are regulated by a variety of actin-binding proteins which can either nucleate, elongate or sever actin filaments, or cap their ends [Pollard, 2016].
Actin also binds to a specific type of crosslinker, the myosin molecular motors, which walk along actin filaments by using ATP hydrolysis (Fig. 3b). Myosin II filaments in particular can attach to two actin filaments thanks to head domains at their two ends. Actin and myosins form together a zoology of network structures which range from the crystalline structure found in muscle sarcomeres to the less ordered actomyosin cortex, a thin actin gel lying underneath the cell  Figure 3: Examples of (a,b) subcellular and (c-g) tissue scale deformations due to active prestress related (a-e) to microscale rest shape change or (f,g) to a change of connectivity, which we refer to as topological prestress. (f ',f ") and (g',g") represent how the topological change in the tissue can be obtained by a subcellular active process, involving myosin prestress, however this level of detail can usefully be ignored when modelling tissue-scale deformations using the concept of topological prestress. 11 membrane. Intermediate in terms of organisation are linear bundles of actin enriched in myosin, such as the so-called stress fibres [Burnette et al., 2014] and junctional cortex in epithelia [Bertet et al., 2004]. The signalling pathways controlling the formation of these respective organisations is beyond the scope of this review and has been reviewed elsewhere [Tojkander et al., 2012].

Mechanical balance of prestressed actomyosin
Contractile prestress. Importantly, myosin generates contractile prestress within actin networks. The contractile nature of stress fibres was demonstrated by measuring the rate and amplitude of the viscoelastic recoil of individual stress fibres after laser ablation. These two quantities were shown to be reduced when Myosin II activity was inhibited by drug treatment [Kumar et al., 2006]. Furthermore, dose-dependent treatments of Blebbistatin (an inhibitor of Myosin II contractile activity) on single cells isolated in a parallel plates traction force apparatus revealed that cell-scale traction exerted by the actomyosin cortex is proportional to Myosin II ATPase activity, indicating that myosin is the main generator of contractile prestress in the cell cortex [Mitrossilis et al., 2009]. It can thus be established that the actomyosin meshwork is exerting a contractile active stress, proportional to the chemical potential of myosin , which can be understood as a prestress. How is it that contraction at the cell scale dominates over expansion despite the disordered nature of these networks? Various hypotheses have been proposed. The most documented one posits that because actin filaments buckle under compression, myosin activity will result only in a tensile contribution [Lenz, 2014, Belmonte et al., 2017, Koenderink and Paluch, 2018.
This contractile prestress can be balanced by the mechanical resistance of three types of other mechanical elements: the cell environment, the other cytoskeletal networks and the fluid component of the cytoplasm (or cytosol).
The tension-compression balance between tensile actomyosin and the external environment to which cells adhere was revealed thanks to the development of deformable substrates (elastomers, hydrogels, or micro-posts arrays). It was shown that adherent cells seemingly "at rest" apply tensile stresses radially directed towards the cell center [Harris et al., 1980, Pelham and Wang, 1997, Dembo and Wang, 1999, Tan et al., 2003]. This stress is transmitted to the substrate at the sites of focal adhesions, mechano-sensitive protein aggregates connecting cells to the ECM [Tan et al., 2003, Burridge andGuilluy, 2016]. Larger deformations are observed in the direction parallel to stress fibres [Mandal et al., 2014]. In line with this, the recoil of the cell substrate away from the site of incision after ablation of stress fibres demonstrated the mechanical connection between those dense actomyosin fibres and the ECM [Kumar et al., 2006].
Actomyosin tensile stress could also be balanced by compression of other cytoskeletal components. Among them, the microtubule network was suggested as a major mechanical actor bearing actomyosin tensile prestress, forming a biological illustration of the tensegrity model [Ingber, 1993, Ingber et al., 2014. Depolymerising microtubules using a drug increases traction forces on the substrate, suggesting that the compression borne by microtubules is transferred to the substrate [Stamenović et al., 2002]. Actomyosin pretension can however also be biochemically affected by microtubule depolymerisation [Rape et al., 2011].
Finally, the tension developed within the actomyosin cortex can be balanced by the cytosol, an incompressible fluid which permeates the whole cytoplasm of the cell [Salbreux et al., 2012]. The cytosol is restricted from escaping by the cell plasma membrane to which the actomyosin cortex is adhered via the ERM family of proteins [Mangeat et al., 1999]. This mechanical balance is spectacularly broken when this cortex or its adhesion with the plasma membrane is locally ruptured: in such occasions, a high-curvature spherical protrusion, called a bleb, forms at the wounded site and inflates with cytosol from the cell body [Charras et al., 2008, Tinevez et al.,  In podosomes, protruding forces applied by the actin core (growing at a rate v p ) onto the substrate are balanced by a contractile actomyosin network of prestress (σ a ), organised as a dome and attached to the substrate at the periphery via adhesion proteins [Labernadie et al., 2014]. (b) In adherent cells, actomyosin prestress (σ a ) is balanced by cytosol pressure (∆P) and substrate deformation. Cell shape is further refined by anisotropic and heterogeneous actomyosin network contraction. Here, orthoradial stress fibres are connected to radial stress fibres, which are attached to the substrate via adhesion proteins at the cell periphery [Burnette et al., 2014]. (c) In the zebrafish semicircular canal, pressure (∆P) is generated within the ECM via synthesis of hyaluronan, pumping in interstitial fluid. This deforms the overlying epithelium which is further shaped by an anisotropic prestress (σ a ) generated by actin-and cadherin-rich protrusions [Munjal et al., 2021].
2009]. Reducing myosin activity, e.g. with the fittingly-named drug Blebbistatin, decreases the volume and rate of expansion of blebs, evidencing that the pressure driving the flow is due to the prestressed actomyosin cortex.
The actomyosin cortex and stress fibres constitute a continuous network and both contribute to cell-scale prestress [Labouesse et al., 2015, Vignaud et al., 2021, although their mechanical properties and regulatory pathways are different [Labouesse et al., 2015]. The spatial arrangement of these prestressed structures in equilibrium with the passive elements described above give rise, for instance, to the typical three-dimensional shape of crawling cells. In these cells, a flat compartment is formed at the cell front and an inversion of curvature of the cell profile is observed at the junction between this compartment and the dorsal cortex (see Fig. 4b). This specific shape results from the presence of orthoradial fibres whose pretension opposes cytosol pressure. These fibres are then connected to radial stress fibres which are attached to the substrate near the cell edge via focal adhesions. Depolymerising the orthoradial fibres via biochemical treatment restores a constant curvature along the cell profile [Burnette et al., 2014].
Growth prestress. The actin cytoskeleton can also exert growth prestress. This was shown in vitro where actin networks nucleated by Arp2/3 under an AFM cantilever [Bieling et al., 2016] or at the surface of spaced magnetic cylinders [Bauër et al., 2017] were shown to generate compression within the network. This effect can be conceptualised as in Eq. (6) (also Fig. 3a), although the growth is often localised at the boundary. In vitro and in vivo, the mechanical activity of network growth is mechanosensitive as shown by a force-velocity relationship in line with an increase of the network density in response to load [Bieling et al., 2016, Mueller et al., 2017.
The generation of such growth prestress is involved in various biological functions. First, it is at the origin of the motility of the Listeria pathogen [Theriot et al., 1992]. Here, like on spherical beads immersed in actin in vitro [Marcy et al., 2004], growth occurs first homogeneously at the surface, which generates residual stress at the periphery of the network. The network ultimately fails via an elastic instability [John et al., 2008]. This symmetry breaking thus forms an anisotropic gel at the surface of the object resulting in a directional movement.
In mammalian cells, a length increase of actin filaments can increase cortical thickness and more importantly counteract tension generated by myosin in the network [Chugh et al., 2017]. The best known example of a cell function in which actin network growth is involved is cell protrusivity, where actin can form a variety of structures pushing the cell membrane forward. This topic is a field of research in itself and we refer the reader to reviews treating it specifically [Blanchoin et al., 2014].
Example: podosomes are shaped by growth and contractile prestress. An example of a prestressed structure combining growth and tensile prestress at the subcellular scale is the podosome, whose mechanics have recently been clarified (see Fig. 4a). Podosomes are micronscale structures present in various cell types (macrophages, cancer cells, endothelial cells) known to probe cell substrate mechanical properties and to be the site of ECM digestion. They are formed of a dense core of actin filaments, oriented normally to the substrate, and bound to a corona of radially-oriented filaments which are tangential to the substrate and adhere to it thanks to focal adhesion proteins [Luxenburg et al., 2012, van den Dries et al., 2019. Protruding forces generated by actin polymerisation within the podosome's core were proposed to be a major contributor to the compression of the substrate [Labernadie et al., 2014]. While this growthrelated prestress remains a possible player, recent findings show that the mechanical balance is dominated by the peripheral actomyosin filaments, which are exerting tensile forces between the tip of the core and the substrate, and hence press the core into the substrate [Jasnin et al., 2021]. This system shows that despite their complexity, the mechanical equilibrium of biological structures can be unravelled by combining force measurements, imaging and careful biological perturbation experiments.

Dynamics
As mentioned in section 2.2, the dynamics of the system can come either from an evolution of the prestress or from a passive relaxation of the material. With some notable exceptions [Fierling et al., 2022], the evolution of the prestress is generally slower than the relaxation of the material. Indeed, biopolymer networks are in general very dynamic as they are being constantly remodelled by processes of (de)reticulation and (de)polymerisation. While the time scales of ECM proteolysis and synthesis remain largely unknown, the rapid turnover time of actin filaments entails its long times liquid-like behaviour [Kruse et al., 2006]. Its effective viscosity scales like the product of the network short-times elastic modulus E and its characteristic turnover time τ a , see Eq. (5). In the muscle, the only crosslinkers between actin and thick filaments are the myosin heads themselves. In order to function as sliding filaments, myosin needs to cycle from attached to detached from the actin in order to perform repeated steps [Caruel and Truskinovsky, 2018], and the dissipation associated with maximal contraction velocity is due to an internal friction associated with the rate of detachment [Huxley, 1957]. This is also the case in most motile and tissue cells, where the turnover time of myosin was found to be close to that of actin monomers in actin filaments and of α-actinin (one of the major actin crosslinkers) [Khalilgharibi et al., 2019]. This explains persistent flows such as the retrograde flow observed in migrating and spreading cells, whose maximal rate of strain is thus set by the ratio of the prestress σ a to the viscosity, as in Eq. (4) [Kruse et al., 2006[Kruse et al., ,Étienne et al., 2015. This is also what sets the rate at which actomyosin-rich subcellular components or tissues straighten after having been buckled by external compressive forces [Tofangchi et al., 2016.
It is also of interest to consider the case in which macroscopic-scale deformation of the network is prevented by the boundary conditions. In those conditions, forces at the boundaries will need to balance the internal tension of the network. The energy input from myosin motors will then be dissipated internally by a microscopic scale creep, corresponding to the elastic energy loss incurred when elements in the network detach while under tension [Huxley, 1957[Huxley, ,Étienne et al., 2015. In this case, the stress measured at the boundaries will be equal to the prestress. Note that in energetic terms, there is work being continuously performed internally by the myosin in both the cases of zero external load or zero contraction of the system, the corresponding energy being respectively dissipated by internal friction and by internal creep [Étienne et al., 2015]. These observations demonstrate a peculiarity of actomyosin networks where prestress and dissipation possess common molecular origins. At the microscopic scale, active processes increase the intensity of fluctuations in the medium. This has been evidenced both for the outof-equilibrium (de)polymerisation of cytoskeletal filaments [Robert et al., 2010] and for myosin activity [Mizuno et al., 2007], by comparing the fluctuations to those that would be expected from thermal forces only.
The actomyosin cortex and myosin contractility also participate in dynamic equilibration of prestress when substrate stiffness is varied. Cell traction assays in parallel plate geometry performed in various contractile cell types revealed that the loading rate during traction increased with substrate stiffness, with the same trend as the maximum traction force [Mitrossilis et al., 2009, Lam et al., 2011. While the models most often evoked to explain the variations of cell tractions in response to changes in substrate stiffness at long time-scale rely on mechanotransduction and subsequent changes in biochemical activity, it was found that the loading rate adapted to real-time changes of cantilever stiffness in a sub-second time-scale [Mitrossilis et al., 2010, Crow et al., 2012, making purely mechanical explanations appealing [Fouchard et al., 2011]. Independently of the adaptation of myosin prestress to mechanical cues, a simple model such as Eq. (5) already predicts a biphasic behaviour with a maximal traction for very large stiffness of the exterior and a traction proportional to that stiffness when it is below a threshold [Étienne et al., 2015]. When the actomyosin cortex is continuously adhering to a deformable substrate, the interplay of this system with the elastic length scale of the substrate yields complex interactions which translate into a biphasic behaviour of the cell crawling velocity as a function of the elastic modulus of the substrate [Chelly et al., 2022].
The combination of growth at the leading edge of cells and active contraction to the back is the hallmark of cell crawling on a flat substrate [Mitchison and Cramer, 1996, Kruse et al., 2006, these two effects giving rise to a dynamic equilibrium setting the size of the system [Étienne et al., 2015, Ambrosi andZanzottera, 2016].

Prestress in fibrous tissues
The ECM is a biopolymer network which is made of assemblies of filamentous proteins (the most abundant being collagen I), proteoglycans and glycosaminoglycans (GAGs). It is the main component of fibrous tissues-also called soft connective tissues-and of the basement membrane on which bidimensional epithelial tissues lie. In most tissues, matrix content is dominated by collagen (especially type I collagen), which has the ability to self-organise into fibrils and fibres through physical bonds [Giraud-Guille et al., 2008]. These fibres can measure up to dozens of microns and often form a crosslinked gel in vivo. In soft fibrous tissues, the ECM is synthesised by cells of the fibroblast family which are embedded and mechanically connected to the matrix. Here, both cells and ECM lie in an interstitial fluid. Mechanical interactions between these three phases generate a large variety of architectures [Wershof et al., 2019] and mechanical properties [Levental et al., 2007], which vary from organ to organ and according to patho-physiological conditions.
The difficulty of performing live imaging in these three-dimensional systems, combined with their physical and biological complexity, has so far impaired the understanding of their active mechanical properties. Nevertheless, recent data show how prestress can be generated in fibrous tissues and affects tissue development and pathology. Despite being largely composed of inert protein, the ECM is not mechanically passive.
Contractile prestress. First, contraction of the ECM can result from variation in water content within the interstitial fluid. In particular, collagen molecules, known to resist tensile stresses in fibrous tissues, change conformation with decreasing water content of the surrounding medium. This effect induces high contraction of the network, which could be important for the function of the load-bearing tendon .
But the ECM is also made contractile through the traction forces that fibroblasts exert within it. Since fibroblasts are polarised mechanically and bound to the ECM through focal adhesions, they apply force dipoles on the collagenous network, acting like active crosslinkers. Isometric contraction of fibroblast assemblies self-organising in collagenous matrix could be measured in between parallel cantilevers and was shown to be dependent on myosin activity [Delvoye et al., 1991, Legant et al., 2009. In these systems, like in the actomyosin cortex, the nonlinear mechanical properties of collagenous networks resulting from the semiflexible nature of the fibres are thought to play a major role in the propagation and amplification of contractile stress at the tissue scale [Ronceray et al., 2016, Han et al., 2018. Because cell traction forces are sensitive to the stiffness of the extracellular environment [Discher et al., 2005, Mitrossilis et al., 2009, stiffening of the ECM fibres generated by cell traction forces triggers a positive feedback loop amplifying stiffening in fibrotic reactions [Calvo et al., 2013] or during directed cell migration [Van Helvert et al., 2018].
This mechanical activity of fibroblasts contracting the ECM influences tissue function during developement and in adulthood. First, in confirmation of a long-standing hypothesis [Harris et al., 1984], recent work shows that the patterning of multicellular aggregates in the chick dermis initiating feather follicles depends on fibroblast cell contractility, preceding differentiation of the epidermis [Shyer et al., 2017]. This indicates that fibroblast contractile activity can indeed act as an organising factor of fibrous tissues during development. Second, during lymph node physiological function, immune cells signal their arrival to the fibroblast reticular cells and tune their contractility in order to relax the tissue. This is thought to help maintaining lymph node integrity while the lymph node expands [Acton et al., 2014]. Finally, cancer-associated fibroblasts, which shape the fibrous tissue of the tumour microenvironment and are more contractile than normal fibroblasts [Sahai et al., 2020], could also apply active stresses at the global scale of the tumour. Along this line, it was shown that their collective contraction concomitant with an orthoradial assembly around tumour aggregates compress tumour compartments in vivo, as well as cylindrical micropillars in vitro [Barbazan et al., 2021].
Growth prestress. The interstitial fluid can generate growth prestress within the ECM. Indeed, the osmotic pressure within the ECM can be tuned in particular by the presence of GAGs and proteoglycans, thanks to their long, negatively charged chains. This property is used during developmental morphogenesis where localised synthesis of hyaluronan (a common GAG) contributes to the bulging of an epithelial monolayer laying on top of the swelling ECM [Munjal et al., 2021] (see Fig. 4c). Such water influx is then balanced by the matrix fibres under tensile load [Ehret et al., 2017]. Thus, in contrast to the poro-elastic behaviour observed in the cell cortex , tension relaxation correlates with expelled interstitial fluid [Ehret et al., 2017]. In the context of cancer, the deregulated fibrous tissue defining the mechanical properties of the tumour stroma (i.e the tumour micro-environment) is also affected by an increase in hyaluronan synthesis and subsequent elevated interstitial fluid pressure. In pancreatic adeno-carcinoma, this effect generates collapse of blood vessels, which could participate in organ loss of function and impairs delivery of therapeutic agents [Provenzano et al., 2012].

Prestress in cellularised tissue
At the tissue scale, a new structural unit becomes fundamental: the cell. From a purely mechanical view point, one crucial aspect is that cells tessellate the space occupied by a tissue into units of regulated volume, which have interactions via adhesion molecules. Being functional units, cells may have individual biochemical activity which can translate into mechanical activity, in turn giving rise to strains that can alter the tissue geometry. This biochemical activity can be patterned at the scale of single cells. For example, in epithelial tissues, which typically form thin sheets of a single cell layer, apical (cell 'top' surface), basal ('bottom') and lateral (where cells are in contact) surfaces are, to some extent, independent biological and hence mechanical units.
Since tight junctions between cells allow tissues to form impermeable layers, cells which actively and directionaly pump ions can in this way impose a pressure difference between apical and basal surfaces, which can generate and maintain a liquid-filled cavity called lumen. Lumens, like neighbouring tissues and the ECM network synthesised by cells, in turn impact tissue mechanics by boundary conditions that vary in space and time.
Cells also define a tissue topology via the neighbour relations created by cell-cell adhesion. This topology can evolve over the course of time as cell-cell junctions are assembled and disassembled, which can in turn give rise to further tissue-scale strain that can be modelled as the result of a topological prestress, as opposed to contractile or growth prestress.
With these differences in microstructure come processes with new timescales [Khalilgharibi et al., 2016]. While a single actin filament may turn-over in seconds, a cell-cell junction requires at least minutes to disassemble and reassemble when a cell changes neighbour [Tlili et al., 2020, Clément et al., 2017. The creation of a new junction during cell division similarly occurs over minutes. However, the full timescale of the cell cycle, which we may consider to encompass the prestress changes associated with cell growth and cytokinesis, varies greatly between animal cell types, from minutes to years. These cell-level processes may be synchronised throughout a tissue, generating tissue-level deformations at a similar timescale, or they may be asynchronous and so add up gradually over longer timescales.

Supracellular scale prestressed networks in interaction with their environment
Although actomyosin is restricted to the cell cytoplasm, it can form supracellular structures by means of adhesion molecules that mechanically connect one subcellular network to that of a neighbouring cell, forming multicellular structures under tension [Fernandez-Gonzalez et al., 2009. The emergence of such a mechanical continuum is illustrated during the reformation of a dissociated epithelial monolayer in vitro by an increase of apparent tissue stiffness, which is coincident with the development of cell-cell junctions and dependent on actomyosin activity [Harris et al., 2014]. The tissue-level prestress generated by the continuous actomyosin network can be measured directly in in vitro epithelial monolayers devoid of ECM and suspended between the arms of a force cantilever . Here, a ramp of compressive strain imposed at the tissue boundary results in a linear decrease of tissue stress. As would be the case for a prestressed thin elastic plate, the tissue then buckles when it reaches a compressed state. Strikingly, the dynamics of stress recovery upon a rapidly applied compressive strain match those of an isolated actomyosin network, indicating that, in this case, the cellularised structure has little impact on the overall mechanics . Such a response, consistent with a continuous model of an epithelium, has also been observed in vivo, in several Drosophila epithelia, where anisotropies of prestress were revealed by the recoil of circular regions of tissue after laser-cutting [Bonnet et al., 2012]. Just as epithelia can change length and generate prestress through their actomyosin networks, they can also be connected to an active element and play a passive role. Their challenge is then to bear the stress generated in order to maintain epithelial integrity [Bonfanti et al., 2022]. This is illustrated by the blisters that epithelia form under active ion pumping directed towards their basal side. In analogy with the pressure in the cell cytoplasm, the pressure within the blister is balanced by the tension in the actomyosin network. Under increased stress, cell deformations can reach hundreds of percents. When the actin pool is exhausted so that the filament network can no longer cover the cell surface area, the keratin network (an intermediate filament network connected through an other type of cell-cell junctions called desmosomes) takes over to resist mechanical stress [Latorre et al., 2018].
While the passive response of an epithelium leads to a dome shape in the system above, an active participation of the epithelium is required to generate a tubular shape, as is the case in the zebrafish inner ear [Munjal et al., 2021]. Here, anisotropic multicellular cables which are both contractile and adhesive form in the direction orthoradial to the cylinder axis, and so by breaking the symmetry of prestress allows an anisotropic shape to arise. Notably in this example, the element driving growth is an increase of osmotic pressure in the ECM actively regulated by cell synthesis of hyaluronan.

Spatially patterned prestress and tissue bending
The generation of anisotropic prestress, as in the zebrafish inner ear, is one example of a common strategy of spatially organising prestress in order to control tissue shape. The most studied example is perhaps epithelial tissues where prestress, tangential to the plane of the tissue, varies along its transverse direction and drives bending via differential prestress. This aspect is particularly important during animal development and detailed reviews of this process have been made by others [Pearl et al., 2017, Tozluoǧlu andMao, 2020]. For the sake of this review, note that a variety of fine-tuned prestress regulation have been documented, including increase of prestress in the apical domain [Martin et al., 2009], increase or decrease of prestress in the basal domain [Sidhaye andNorden, 2017, Krueger et al., 2018], and increase in lateral prestress [Brodland et al., 2010, Gracia et al., 2019.
This differential prestress can be revealed by laser ablation in cultured suspended epithelia by measuring the spontaneous curvature generated orthogonal to tissue plane at a newly created free edge. This too allows for measurement of the out-of-plane forces involved by unfurling the curled tissue with a force cantilever .
Three-dimensional geometries other than linear folds can be produced through the same mechanism, via patterning of differential prestress throughout the plane of an epithelium. For instance in 2D-cultured gut organoids, three cell types are organised within the tissue plane into concentric circular islands. Traction force microscopy and laser ablations revealed that these regions display different mechanical behaviours, with apical-basal myosin polarisation in the central region leading to the doming of crypts (cup-shaped structures of the digestive tracts) . In 3D gut organoids, in which cells embedded in ECM form cysts, experiments revealed that crypt morphogenesis is also driven by membrane transporters which cause liquid transfer from the crypt cavity to the tissue [Yang et al., 2021]. Interestingly, the apical-basal polarisation of the crypt region was found to be large enough in comparison to that outside of the crypt region for the overall morphology to be robust to organoid volume changes [Yang et al., 2021].
Recent advances in optogenetics (optical activation of engineered biomolecules) have also allowed for the experimental control of this patterned differential prestress. For example, at the stage preceding gastrulation in Drosophila, localised apical activation of RhoA, an activator of myosin contractility, was shown to be sufficient to initiate folding in a variety of directions and locations where invagination does not normally occur [Izquierdo et al., 2018]. In vitro measurements showed that the spontaneous curvature generated by active differential prestress is so high (on the order of the inverse of tissue thickness) that a competition between in-plane elastic energy and bending energy takes place. In this way, tissue folding is continuously modulated by external tension and reciprocally . Nevertheless, differential prestress is not the only way to achieve folding. Recent data from the ventral furrow formation in Drosophila embryo showed the importance of the global ellipsoidal geometry of the embryo for an elongated ventral patch of myosin to achieve a fold along its long axis . Indeed, heterogeneneous prestress at the surface of a thin shell respecting this geometry leads to surface buckling initiating folding with the correct pattern of strain and dynamics [Fierling et al., 2022].

Growth prestress
Physiological growth in living tissues often results in material added (or lost) in a non-uniform manner, forcing neighbouring tissue to accommodate the newly added material through elastic deformation. These nonuniformities, as illustrated in Fig. 2, include heterogeneous, anisotropic and differential prestrain F a . As discussed in Section 2, this nonuniform prestrain is revealed by residual stress: an internal stress that remains when all external loads of an originally unloaded configuration are removed. Tissues actively build these internal stresses both during morphogenesis (when they rapidly change shape and add mass) and in the adult physiological state (when mass and volume changes serve the purpose of maintenance and are comparatively small).
A classic example is the residual stress in arteries, which has been theoretically and experimentally described by Fung and others [Fung, 1991, Vaishnav andVossoughi, 1987]. The observation is that arteries, when radially cut, open up due to compressive stress built in the hoop (circumferential) direction. The opening angle can be used to quantify residual stress [Fung, 1991]. Experiments also suggest that arteries are residually stressed in the axial direction [Goriely and Vandiver, 2010]. In a series of seminal studies, Fung and co-workers demonstrated residual stress in other cardiovascular systems such as the heart [Omens and Fung, 1990], veins [Xie et al., 1991], and the trachea [Han and Fung, 1991]. Residual stress was also identified in other physiological tissues and organs such as the brain [Budday et al., 2014] and bones [Yamada et al., 2011], as well as morphogenetic systems such as the optic cup [Oltean et al., 2016] and the developing embryo [Beloussov and Grabovsky, 2003]. It was also found in pathological tissue such as solid tumours [Ambrosi and Mollica, 2004]. Proliferation also produces compression within the core of tumours and orthoradial tension at the periphery as revealed by cutting of an excised tumour along its radius [Stylianopoulos et al., 2012]. In this context, growth-related prestress is referred to as 'solid stress'.
Apart from physiological systems, growth-induced prestress has been measured in a multitude of ways in cultured systems. Stress in growing multicellular spheroids has been measured with great precision: the growing cells were encapsulated inside permeable, elastic, hollow microspheres which deformed as the spheroids grew inside of them, allowing to reconstruct the traction forces [Alessandri et al., 2013]. The external pressure applied on the spheroid by the elastic coating leads to a steady-state size of the spheroid, in which there is an equilibrium between a necrotic core and a proliferating rim. The existence of a pressure at which spheroids reach a stationary size was theoretically proposed by Basan et al. [2009] and independently experimentally addressed by applying osmotic pressure to the spheroid [Montel et al., 2011].
A similar technique for measuring growth-induced stress, by the elastic deformation of the environment, was demonstrated for yeast cell colonies (which do not form cell-cell junctions) [Delarue et al., 2016]. A distinctly different method was used in [Cheng et al., 2009], where threedimensional aggregates of tumour cells were co-embeded with fluorescent micro-beads in agarose gels. The displacement of the beads allows a reconstruction of the spatial stress distribution.
Prestress generated by proliferation can also be observed in two-dimensional tissues, for example during the early development of the Drosophila wing disc. In a 3D finite element simulation of the wing disc as one heterogeneous layer, the growth rate and mechanical coupling to the elastic basement membrane of the tissue provoke the formation of spatially regulated folds [Tozluoǧlu et al., 2019]. The doming of a part of the wing disc, the wing pouch, was recently explained through a combination of differential growth and differential growth anisotropy between tissue layers [Harmansa et al., 2022]. Similarly, differential growth rates between adhered tissues has been shown to regulate the looping of the chick gut [Savin et al., 2011], the formation of villi in the chick gut [Shyer et al., 2013] and the gyrification of the brain [Tallinen et al., 2016].
Until here, we considered growth to be dictated by proliferation rate, but an increase of cell density can occur independently to proliferation. For instance, during the formation of the zebrafish optic cup, migration of cells from the outside of the organ is analogue to a local tissue growth. This process increases tissue curvature and is required to produce correct bending of the organ, in parallel to differential contractility [Sidhaye and Norden, 2017].
Finally, the removal of cells, through cellular processes such as apoptosis, can be seen as a reverse growth and can play equally important roles during tissue morphogenesis. For example, in a transient extra-embryonic epithelial tissue named the amnioserosa, which sits between two embryonic epithelia during Drosophila embryogenesis, the gradual apoptosis of amnioserosa cells drives the final stage of dorsal closure in which those two sheets are brought together [Pasakarnis et al., 2016].

Topological prestress
The cellular microstructure of tissues offers a further mechanism through which to generate or relax prestress via changing the topology through cell-cell neighbour exchange events, termed intercalations (or T1 transitions in foam literature) , Blanchard, 2017. At the subcellular scale, this process requires the coordinated action of many biomolecular players to disassemble and reassemble cell-cell junctions and has often been found to depend on the active generation of stress to shrink or expand junctions [Bertet et al., 2004, Nestor-Bergmann et al., 2022. The resulting T1 defines an orientation, as specified by the orientation of the removed and of the added junctions. In some tissues, T1 transitions occur throughout tissues with no preferred orientation, in which case they relieve local cell packing stresses, allowing tissue ordering [Curran et al., 2017] or facilitating flow during migration [Tlili et al., 2020]. The transition from such a liquid-like state to a 'jammed' state has been shown theoretically to relate to cell density, junctional tension and fluctuations [Lawson-Keister and Manning, 2021]. However, the most dramatic examples occur when intercalations are globally aligned throughout a tissue, such as during Drosophila germband extension Wieschaus, 2004, Tetley et al., 2016]. This results in creating neighbour relations which, in the initial configuration of the tissue, strain the cells in the transverse direction: their relaxation thus entails a tissue-scale deformation in the longitudinal direction [Collinet et al., 2015], see Fig. 3g. Similarly, multilayered epithelia can expand (here isotropically in the plane) through 'radial intercalation' in which cells in lower layers intercalate into upper layers creating a precompression which is relaxed by expansion [Szabó et al., 2016]. Since these are relaxation processes following out-of-equilibrium topological changes, they can be conceptualised as topological prestress. A major challenge in the analysis of such deformations is to disentangle this process from boundary stresses that may also act on the tissue , Lye et al., 2015, Collinet et al., 2015.
Although epithelial topology is often represented as a two-dimensional network, a full threedimensional treatment revealed different connectivity in the apical and basal domains [Gómez-Gálvez et al., 2018]. Topological transitions were indeed observed along the apico-basal axis in a range of in vivo tissues, resulting in a three-dimensional cell shape named a scutoid. This solution is favourable when tissues are curved by different amounts with respect to their principle planar axes (i.e. tubular rather than spherical) unless the radius of curvature is large compared with the tissue thickness. It remains to be found whether these intercalations could drive bending itself, rather than simply relax stress.
A separate class of topological change which affects prestress is the introduction of a new junction into the network, which occurs during cytokinesis, see Fig. 3f. Again this is an oriented process, as the degree of freedom is the orientation of the new interface. Indeed, much has been discovered about the cell-and tissue-level signals read by a cell when choosing a division orientation. These signals include biochemical cues such as tissue polarisation [Gong et al., 2004, Gho andSchweisguth, 1998], as well as mechanical and geometrical signals such as tissue stress [Niwayama et al., 2019, Fink et al., 2011, Scarpa et al., 2018 and cell shape [Wyatt et al., 2015, Bosveld et al., 2016. In turn, the choice of division orientation alters prestress. A prestrained mother cell, depending on the choice of division orientation, could be divided into two daughter cells with either an increased or reduced anisotropy in shape. In tissues, the latter choice is usually observed, so as to homeostatically regulate cell shape anisotropy [Mao et al., 2013, Wyatt et al., 2015, Xiong et al., 2014. In the zebrafish, this alignment of division with cell shape (which coincides with the principal axis of tissue stress), was shown using laser ablations to reduce the stress that builds up as a cell layer migrates over the embryo's surface [Campinho et al., 2013]. Similarly, alignment of cytokineses along a given axis can contribute to driving directional growth [da Silva andVincent, 2007, Li et al., 2014].
Altogether, a complete description of tissue morphogenesis can be achieved by a linear combination of the active and passive mechanical modules we have described so far (cell stretching, oriented cell division, T1 transitions), provided that one has a good knowledge of the boundary conditions, which can often be dynamic in living systems [Etournay et al., 2015]. Example biophysical systems where mechanics is governed by heterogeneous, anisotropic or differential prestress of either sign, corresponding to contraction or growth. (a) Contractile actomyosin with local accumulation, in parallel with a length-regulating element and in continuous adhesion with a substrate, generates a friction pattern that enables motility . (b) Anisotropic pretension of the apical surface of cells regulates their shape [Burnette et al., 2014]. (c) Differential prestress between the weakly contractile apical (top) surface and strongly contractile basal (bottom) surface causes tissue curling  (d ) Residual stress due to heterogeneous growth is characterised by cutting experiments in tumours, revealing tensile hoop stress at the periphery [Stylianopoulos et al., 2012] (e) The core of podosomes grows within a confined space, generating anisotropic prestress [Labernadie et al., 2014] (f ) Arteries change curvature if cut, this is believed to be caused by differential growth and remodelling of concentric layers [Goriely and Vandiver, 2010] either contractile or growth activity. Figure 5 provides an example for each of the six typologies that thus emerge.
Additionally, we define topological prestress as the prestress that is added to, or removed from, an interconnected network of mechanical elements not through prestressing individual elements, but purely by breaking and creating connections between the network elements. This latter type of prestress corresponds to a higher level of phenomenology, since it does not in itself describe by which (active) process the connectivity is being changed, see the examples in Fig.  3f,g. It also has the property that it is always providing anisotropic prestress.
From an experimental point of view, characterising living systems as active materials is highly challenging. One difficulty is linked with the necessity to test systems while they are maintained in a state of function as close as possible to physiological conditions. For this, recent developments in organoid systems present great opportunities, since elements of tissue morphogenesis can now be recapitulated in an in vitro setting which is much simplified and much more amenable to experimental pertubations.
Observing the system simultaneously at different scales, especially combining global-scale stress and strain measurements with very local measurements could allow a better characterisation of how the microstructure dynamics give rise to emergent properties. For instance, tools measuring strain at the molecular scale, like Förster resonance energy transfer (FRET) [Borghi et al., 2012], or at the meso-scale, like micro-magnets [Laplaud et al., 2021], micro-droplets [Mon-gera et al., 2018] or optical tweezers [Han et al., 2018], could be coupled to cell-scale techniques, like parallel plate rheometry or TFM, to decipher the subcellular contributions to cell shape changes. On the other hand, cell-scale in parallel to tissue-scale mechanical testing appears necessary to understand the cellular origins of tissue flows [Moisdon et al., 2022].
In all cases, the interpretation of force-displacement relation measurements must be supported by careful modelling, as exemplified by the violation of the fluctuation-dissipation relation [O'Byrne et al., 2022]. This sheds light on the vital need for combining any experimental approach with a theoretical understanding and of testing this by employing multiple modalities. Notably, extending the existing models towards geometric or material nonlinearities is a necessity given the large deformations that are commonly encountered in real systems. How the prestrain and prestress fields relate in a nonlinear context also remains to be clarified. Solving mechanical models coupling different biological structures, in relevant geometries in three dimensions also remains an important challenge: for example, a full understanding of the three-dimensional mechanical balance of single or tissue cells, or of the dynamics of fibrous tissues which is governed both by cells and ECM are still lacking.
Another challenging theoretical task is to describe topological prestrain and prestress in a continuum framework. A missing intermediate step is the linking between the cellular and tissue scales. At the scale of several cells, vertex models [Farhadifar et al., 2007] are a highly studied family of models for biological tissue, both for their strength at capturing various tissue properties [Alt et al., 2017, Cheddadi et al., 2019, Latorre et al., 2018 and for their interesting physical behaviour [Bi et al., 2015, Farhadifar et al., 2007, Schoetz et al., 2013. However, the coarse-graining of vertex models to continuum descriptions, such as anelasticity, remains highly challenging, and is being tackled with approaches based on nearly periodic lattices [Murisic et al., 2015, Chenchiah and Shipman, 2014, Kupferman et al., 2020 and discrete calculus [Jensen et al., 2020, Nestor-Bergmann et al., 2018. These efforts may in the long term lead to the possibility of encoding topological transitions, like active or passive neighbour exchanges (T1 transitions), into a continuum field usable, for instance, in anelasticity approaches.
Seventy percent of the total cell volume is water, and there is growing evidence in favour of a coupling between cell mechanics and osmotic gradients controlling volume [Xie et al., 2018, Cadart et al., 2019. Cells maintain a prestress inside the membrane by carefully controlling the flow of water: water mobility in and out of cells relies on the permeation of water through the plasma membrane, which can be regulated by aquaporin channels [Kedem and Katchalsky, 1958], as well as ion pumps which actively consume energy. Therefore, there has been recently considerable interest in the modelling of water mobility. This has been approached via a fluid phase in the framework of poroelasticity [Ambrosi et al., 2017, Fraldi and Carotenuto, 2018, Xue et al., 2016, as well as by explicitly tracking water fluxes in a vertex model [Cheddadi et al., 2019]. In parallel, there are new developments coupling the electrochemistry of ion fluxes, mechanics of cell volume regulation, and active pumping [Cadart et al., 2019, Deshpande et al., 2021. Water mobility should ultimately add contributions to coarse-grained models such as anelasticity. From the experimental side, this requires the development of non-perturbative pressure sensors which is an important challenge for the future.