Mechanical feedback in regulating the size of growing multicellular spheroids

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Abstract

The mechanism by which cells measure the dimension of the organ in which they are embedded, and slow down their growth when the final size is reached, is a long-standing problem of developmental biology. The role of mechanics in this feedback is considered important. Morphoelasticity is a standard continuum framework for modeling growing elastic tissues. However, in this theory, in the absence of additional variables, the feedback between growth and mechanical stress leads to either a collapse or unbounded growth of the tissue, but usually prohibits reaching a finite asymptotic size (‘size control’). In this article, we modify this classical setting to include an energetic cost associated with growth, leading to the physical effect of size control. The present model simultaneously provides a qualitatively correct residual stress profile and has a naturally emerging necrotic core, all of which have previously been experimentally established in multicellular spheroids. This is achieved through a local feedback mechanism derived from a thermodynamical framework. The model delivers testable predictions for experimental systems and could be a step towards the understanding of the role of mechanics in the multifaceted question of how growing organs attain their final size.

Introduction

In morphogenesis, living tissues change shape and size very rapidly. Morphogenetic events include spectacular shape changes such as self-inversion (Höhn et al., 2015), looping as in the case of the heart tube (Ramasubramanian et al., 2008), branching such as in lungs, kidney and vascular networks (Erlich et al., 2019b). As these examples illustrate, morphogenesis involves complex interactions between growth, non-linear mechanics, shape and size, calling for mathematical approaches that can model such interplay.

The determination of the appropriate form of evolution equations for growth and shape change has been the focus of much research (Epstein and Maugin, 2000, Lubarda and Hoger, 2002, DiCarlo and Quiligotti, 2002, Ambrosi and Guillou, 2007, Ganghoffer, 2010). With limited experimental data available, this research has employed thermodynamic arguments to motivate appropriate forms of the growth law that satisfy a dissipation inequality. The reasoning is typically to assume a single constituent theory in which every material point is in contact with a mass reservoir at an imposed chemical potential setting the tendency to grow. When the free energy depends only on the elastic deformation, by following a standard set of arguments and derivations, one arrives at a variant of the growth law ĠG1=K(SS)where G is a tensor representing the geometrical rearrangement of the material due to growth, S is an Eshelby stress, S is a homeostatic stress representing the mass reservoir, and K is a constant symmetric positive-definite matrix of growth rates. In particular, the hypothesis behind mechanical homeostasis (Ambrosi et al., 2019, Latorre and Humphrey, 2019, Erlich et al., 2019a, Goriely, 2017) is that in the state of homeostatic stress S=S, growth and shape change do not occur (Ġ=0), as the cellular processes of birth, death and rearrangement balance each other out. Experimental evidence and physical understanding of homeostatic stress have been demonstrated in arteries, where residual stress is used to homogenize transmural stresses under physiological loading to minimize tissue abrasion during its lifetime (Chuong and Fung, 1986). There is experimental evidence that some living systems, such as embryos (Beloussov and Grabovsky, 2003) and fibroblast cells (Ezra et al., 2010), maintain such a target, or homeostatic, stress.

The debate on the respective roles of mechanics and biochemistry in the study of how biological tissues control their size has undergone interesting developments in recent decades. How biological tissues control their size has been a decade-old mystery in mostly pure developmental biology research (Travis, 2013). Many hypotheses have been experimentally tested and proven wrong: Size is not determined, for example, solely by a cell clock or cell counting (Vollmer et al., 2017). As models based on reaction and diffusion of growth-promoting chemicals (morphogens) failed to explain numerous experimental observations (Day and Lawrence, 2000), mechanics has received increasing interest as a likely candidate for growth regulation in the developmental biology community (Shraiman, 2005, Hufnagel et al., 2007, Aegerter-Wilmsen et al., 2007, Aegerter-Wilmsen et al., 2012). For example, mechanics is fully accepted as a key ingredient in the growth of multicellular spheroids, a lab-made model system for the early stages of tissue expansion. Growth is affected by both the level of compression of the spheroid and the availability of oxygen (Stylianopoulos et al., 2012, Ambrosi and Mollica, 2004, Gao et al., 2016, Dolega et al., 2020). An appealing aspect of the model (1) it that it shifts the emphasis from biochemistry to mechanics, allowing a more nuanced understanding of the role of mechanics and residual stress in tissue growth and regulation, while still allowing a coarse-grained description of chemistry to enter through the external chemical potential S that can be refined as needed by, for instance, coupling growth to a diffusion process (Ambrosi and Guillou, 2007).

Certain details of the form of the growth law (1) are not entirely agreed upon. The presence of Eshelby stress as a driving force of growth has been widely employed (Epstein and Maugin, 2000, Ambrosi and Guana, 2005, Ambrosi and Guillou, 2007, Ganghoffer, 2010, Gao et al., 2016). The Eshelby stress was originally introduced to describe point forces due to elastic singularities, due to dislocations in crystal lattices (Eshelby, 1951, Eshelby, 1957). It emerges naturally in the context of growth (Ambrosi and Guana, 2005). However, some authors make the assumption that the free energy of the incoming material matches the free energy of the pre-existing material, which leads to the presence of Mandel or Cauchy stress instead of Eshelby stress in the growth law (1) (Goriely, 2017, Taber, 2008, Taber and Eggers, 1996). The difference of the two approaches has been contrasted in Buskohl et al. (2014). In a similar vein, there is some disagreement on the form of the coefficient K in (1). A thermodynamical treatment requires this coefficient to be a positive semi-definite matrix to ensure that the dissipation inequality is satisfied (Ambrosi and Guana, 2005). However, certain authors who emphasize the role of unknown biochemical processes in thermodynamical treatments choose fourth-order coefficient tensors instead (Taber, 2009, Goriely, 2017, Erlich et al., 2019a). This allows for cross-couplings in the growth dynamics that would be impossible in the classical treatment (Ambrosi and Guana, 2005), such as (in a system with spherical symmetry) the radial growth rate being coupled to hoop stress.

A number of issues with (1) have received relatively little attention:

  • 1.

    The first point concerns the homeostatic state itself. A conceptual problem raised by (1) is that if at the tissue boundary the homeostatic stress does not match the boundary condition (which might be a prescribed hydrostatic pressure for instance), growth never stops at the boundary, making an equilibrium impossible. Several authors found a way around this problem by hypothesizing further evolution equations for S which adapt in a delayed response to the boundary conditions of the system (Taber, 2008, Taber, 2009), or by postulating that the homeostatic stress is compatible with boundary conditions (Erlich et al., 2018). However, such choices are rather arbitrary in the context of biological tissue growth, where it is unclear why the reservoir of nutrients should be linked to the imposed mechanical boundary conditions.

  • 2.

    The second point concerns a larger question in biology: Is the purely mechanical feedback mechanism (1) sufficient to encode a final asymptotic size of the tissue? This ties into a larger debate in biology about how cells in an organ know what overall size the organ has, and how they “decide” when to stop dividing once the organ has reached the right size, and which role mechanics plays in such regulation (Buchmann et al., 2014, Eder et al., 2017, Vollmer et al., 2017, Irvine and Shraiman, 2017, Gou et al., 2020). It is questionable whether the system (1) will reach the same size or not depending on different initial conditions: (Ambrosi et al., 2015, Pettinati et al., 2016) hypothesized that it should not, but gave no proof or numerical example. While there have been some recent studies investigating the dynamics of (1) with methods of dynamical systems theory (Vandiver and Goriely, 2009, Erlich et al., 2019a, Latorre and Humphrey, 2019), to our knowledge no investigation of the final size exists to date.

In this paper we will address both issues of (1) by proposing a modification of the standard approach that overcomes the two issues mentioned. The idea is to penalize the growth process in the free energy of the system. While the classical theory neglects that growth has an energetic cost, even if the cell material building blocks are readily available in the extracellular fluid, it costs energy (Alberts et al., 2002, Phillips and Milo, 2009) to get them through the cell membrane and assemble or disassemble them into the solid cell components that constitute the cell dry mass (Zhou et al., 2009). An energetic cost is also involved in the ion pumping mechanism that is necessary to control cell volume and screen out the apparent osmotic imbalance between the cell inside and outside due to the presence of the macromolecules trapped in the cell (Cadart et al., 2019), generally leading to a certain level of control of the cell mass density during growth (Grover et al., 2011). We show that using this concept, there is no longer a need for the homeostatic stress to match or adapt to the boundary conditions. Further, we show that this model creates the final size robustly, independent of variations in initial conditions, something which the classical model (1) does not achieve, as we shall also demonstrate numerically. The final model qualitatively matches important known experimental observations about growing multicellular spheroids, namely that they reach an asymptotic size in the presence of external pressure (Helmlinger et al., 1997, Alessandri et al., 2013), that the residual hoop stress near the periphery of the spheroid is tensile as consistent with cutting experiments (Stylianopoulos et al., 2012, Colin et al., 2018, Guillaume et al., 2019), and that larger spheroids experience an inflow of material towards the core, known as a necrotic core (Franko and Sutherland, 1979, Delarue et al., 2013). While mechano-chemical models captured some of these experimental observations (Ambrosi et al., 2017b, Xue et al., 2016, Walker et al., 2023), to our knowledge, no existing model matches all these observations simultaneously.

We divide the manuscript as follows. In Section 2, we introduce our notation and state the balance laws and the first and second principle of thermodynamics of a growing solid in finite elasticity. Our main idea, which distinguishes this work from classical works like Rodriguez et al. (1994) and Ambrosi and Guana (2005), is that the free energy depends on the determinant of the growth tensor, |G|, in addition to the traditional dependence on the elastic deformation gradient A. In Section 3, we propose a concrete form of this free energy to derive a growth law which offers a crucial modification compared to the classical law (1). The consequences of our modification are explored in Section 4 based on two examples. Firstly, in Section 4.1, we consider a compressible uniaxially growing neo-Hookean bar. We find that our growth law enables the possibility of size control, and removes the need to prescribe the homeostatic pressure in the bar ad hoc, thus fixing both issues with the classical law discussed in the introduction. Secondly, in Section 4.2 we consider a growing compressible neo-Hookean spheroid, showing how in addition to size control, residual stress can also be built up in this system through growth. A residual stress profile consistent with experiments can be controlled by the anisotropy of the homeostatic stress tensor S. In Section 5, we explore how the anisotropy influences the spheroid’s ability to achieve size control. Large spheroids are subject to a flow of material towards the center, and we demonstrate how the necrotic core forms in our model in Section 6. Finally, in Section 7, we round up this work with a discussion.

Section snippets

Kinematics

From a mechanical perspective, a biological tissue of mammalian cells is typically constituted of cells interconnected directly by protein bonds or via some extracellular matrix. This system is permeated by an extracellular solvent which contains the nutrients and the building blocks necessary for its growth through biosynthesis and swelling of the cells followed by their division when reaching a critical added mass (Cadart et al., 2019). We model the tissue as a single continuum which initial

The energy cost due to growth

We now consider the following expression of the energy density W, WA,|G|=WelA+Wg|G|=G2I132log|A|+κ2|A|12+χ2|G|1|G|+12.The first part of the energy, Wel, is that of a compressible neo-Hookean material (Pence and Gou, 2015), where G is the shear modulus and κ the bulk modulus. The last term, Wg, weighted by the growth penalty χ, is an energetic penalty due to growth, which is not present in most current treatments of growth (Epstein and Maugin, 2000, Lubarda and Hoger, 2002, DiCarlo and

Application to a 1D and 3D scenario

In this section, we explore the consequences of the modification to the free energy density (18) and the growth law (21) that follows from it. We start in Section 4.1, with a uniaxially growing bar, which is restricted to homogeneous deformations. In this system, deformations and stresses are spatially uniform, making the mathematical analysis transparent. Next, in Section 4.2, we study a growing spheroid. This system permits anisotropy in growth and stress, which leads to the buildup of

Residual stress and size regulation

When the homeostatic stress tensor has no deviatoric part (σ̂=0), the deformation field in the spheroid is spatially homogeneous, i.e. the stress in the spheroid is isotropic and the same everywhere, matching the externally imposed pressure for all times. This case is illustrated in a 7A, in which the collapse region (red) and the unbounded growth region (yellow) are separated by a “V”-shaped size control region (blue). The region boundaries for this case σ̂=0 are worked out semi-analytically

Transient dynamics and the necrotic core

The presence of a necrotic core in the center of a growing spheroid is generally attributed to a nutrient depletion within the core (Tracqui, 2009). The classical explanation is that avascular tumor growth relies on the delivery of oxygen and nutrients from surrounding host tissue and their consumption by tumor cells. The nutrient availability affects cell proliferation and death, leading to the formation of a concentric structure with a necrotic center, a dormant middle layer, and a highly

Discussion

The key idea in this article is that we make a modification of the free energy which we identify with the physical effect of size regulation. The modification is led by the insight that for a cell to grow, or shrink, there is an energetic cost. Some active processes that consume chemical energy from ATP hydrolysis are directly involved in the control of the local tissue swelling such as the cells ion pumping and the process of endocytosis. This energy consumption can be maintained thanks to the

Computational codes

The numerical implementation was done in Mathematica 12.1. The notebooks generating all the figures can be downloaded at: https://github.com/airlich/size-control/.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

We thank Lev Truskinovsky, Larry Taber and Davide Ambrosi for insightful individual discussions on growth models, as well as each for their detailed feedback on our manuscript. We also thank Giuseppe Zurlo for in-depth discussions on growth and size regulation.

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    All authors equally contributed to this work.

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