Are Homeostatic States Stable? Dynamical Stability in Morphoelasticity
Abstract
Biological growth is often driven by mechanical cues, such as changes in external pressure or tensile loading. Moreover, it is well known that many living tissues actively maintain a preferred level of mechanical internal stress, called the mechanical homeostasis. The tissuelevel feedback mechanism by which changes in the local mechanical stresses affect growth is called a growth law within the theory of morphoelasticity, a theory for understanding the coupling between mechanics and geometry in growing and evolving biological materials. This coupling between growth and mechanics occurs naturally in macroscopic tubular structures, which are common in biology (e.g., arteries, plant stems, airways). We study a continuous tubular system with spatially heterogeneous residual stress via a novel discretization approach which allows us to obtain precise results about the stability of equilibrium states of the homeostasisdriven growing dynamical system. This method allows us to show explicitly that the stability of the homeostatic state depends nontrivially on the anisotropy of the growth response. The key role of anisotropy may provide a foundation for experimental testing of homeostasisdriven growth laws.
Keywords
Nonlinear elasticity Continuum mechanics Biological growth Dynamical systems Ordinary differential equations Discretization scheme1 Introduction
Biological tissues exhibit a wide range of mechanical properties and active behavior. A striking example is biological growth in response to the tissues mechanical environment. Artery walls thicken in response to increased pressure (Berry 1974; Goriely and Vandiver 2010), axons can be grown by applying tension (Lamoureux et al. 1992; Recho et al. 2016), and plant growth is driven by various mechanical cues (Goriely et al. 2008; Boudaoud 2010). The general idea underlying these phenomena is that the internal stress state is a stimulus for growth. As stress is rarely uniform, mechanically induced growth often coincides with differential growth, in which mass increase occurs nonuniformly or in an anisotropic fashion. In turn, differential growth produces residual stress, an internal stress that remains when all external loads are removed, appearing due to geometric incompatibility induced by the differential growth. Residual stress has been observed in a number of physiological tissues, such as the brain (Budday et al. 2014), the developing embryo (Beloussov and Grabovsky 2006), arteries (Fung and Liu 1989), blood vessels (Fung 1991), solid tumors (MacArthur and Please 2004), and in a wealth of examples from the plant kingdom (Goriely 2017). In many cases, residual stress has been found to serve a clear mechanical function; for instance in regulating size and mechanical properties.
Many living tissues actively grow in order to maintain a preferred level of internal residual stress, termed mechanical homeostasis. This phenomenon is characterized by growth being induced by any difference between the current stress in the tissue and the preferred homeostatic stress. Mechanically driven growth toward homeostasis poses several interesting and important questions, at the biological, mechanical, and mathematical level. For instance, what determines the homeostatic stress state? At the cellular level, the growth response may be genetically encoded, with a homeostatic state manifest by differential cellular response to mechanical stimuli. From a continuum mechanics point of view, a residually stressed configuration is typically thought of as corresponding to a deformation from an unstressed configuration; however, it is not clear that such a deformation should exist to define a homeostatic state. Connected to this is a question of compatibility: is it actually possible for a system to reach mechanical homeostasis? For example, the boundary of an unconstrained tissue will always be traction free, and thus, if the homeostatic stress for those boundary cells is nonzero, then the system can never completely reach homeostasis. From a dynamics point of view, there is a natural question of stability: is the homeostatic state stable, i.e., if the system is perturbed from its homeostatic equilibrium, is it able to grow in such a way to return to this state? There is also a practical issue of connecting experiment to theory: how does one quantify the homeostatic state and form of growth response?
In this paper, we study mechanically driven growth in the context of growing tubular structures. One motivation for a cylindrical geometry is that such structures are ubiquitous in the biological world, from plant stems (Goriely et al. 2010) to axons and airways (Moulton and Goriely 2011a, b), and exhibit diverse mechanical behavior. Working within a constrained geometry will also enable us to gain qualitative insight into the dynamics of structures with growth driven by mechanical homeostasis and to formulate a basic framework for studying the stability of a homeostatic state. Even in an idealized geometry, the full growth dynamics still consists of a set of partial differential equations, with mechanical equilibrium requiring the solution of a boundaryvalue problem at each time step, and a highly nonlinear growth evolution for components of the growth tensor. There is no mathematical theory, yet, that allows for such an analysis. Our approach is therefore to devise a discretization through a spatial averaging scheme that converts the system to a much more manageable initialvalue problem, to which we can apply standard techniques from dynamical systems. The discretization we propose consists of defining annular layers of the tubular structure, such that growth is uniform in each layer, driven by averaged values of the stress components in a law of the form (1). While this approach enables us to study efficiently properties of the continuous (nondiscretized) system as the number of layers increases, for a smaller number of layers it is also a useful model of a multilayered tube commonly found in many biological systems.
This paper is structured as follows. In Sect. 2, we discuss the general deformation and growth dynamics for a tubular structure that is homogeneous in the axial direction. In Sect. 3, we focus on a tubular system made of two layers, illustrating the main ideas of our discretization approach and illustrating the rich dynamics of this system. In Sect. 4, we generalize from two to N layers. Here we find a rapid convergence of behavior as the number of layers increases and investigate how the anisotropy of the growth affects the stability.
2 Continuous Growth Dynamics in Cylindrical Geometry
2.1 Kinematics
2.2 Mechanics
2.3 Growth Law
2.4 Discretization Approach
For given homeostatic stress values and components of \({{\varvec{K}}}\), the growth dynamics is fully defined, with the growth components evolving according to (17). Even in the simplified cylindrical geometry, this comprises a system of nonlinear partial differential equations. Moreover, viewing the dynamics as a discrete process is still complicated by the fact that at each time step updating the growth requires knowing the stress components, which requires integration of (15), which requires integration of (7), which cannot be done analytically for general spatially dependent \(\gamma ^R\) and \(\gamma ^\theta \).
However, as stated above, for constant \(\gamma ^R\) and \(\gamma ^\theta \), the integrals determining stress may be computed analytically. This suggests a discretization process whereby the annular domain is divided into discrete layers, each with constant growth, and such that the growth in each layer evolves according to averaged values of the stress. In this way, analytical expressions may be determined for both the stress and the average stress, and hence, the dynamics is reduced to a set of ordinary differential equations for the growth components. The inhomogeneity of the full model is replaced by a piecewise homogeneous model. This preserves the key idea of inhomogeneity (allowing, for instance, circumferential growth to be higher near the nucleus than away from it), but is more analytically tractable and allows for precise statements about the longterm dynamics, stability, and qualitative investigation such as the influence of radial versus circumferential stress to the growth dynamics.
3 Growth Dynamics for 2Layer System
3.1 Kinematics
3.2 Mechanics
The expressions \(T_{1}^{RR}\) and \(T_{2}^{RR}\) as well as \(T_{1}^{\theta \theta }\) and \(T_{2}^{\theta \theta }\) can be determined analytically as functions of \(A_{0}\), \(A_{1}\), \(A_{2}\), \(\mu \), \(\gamma _{1}\) and \(\gamma _{2}\), though the exact expressions are long and have been suppressed here.
We note that the radial stress component is continuous at the interface between layers. This can be seen by evaluating \(T^{RR}_2\) at \(A_1\) in (21), which gives \(T^{RR}_1(A_1)=T^{RR}_2(A_1)\). The circumferential stress, however, is discontinuous at the interface unless the growth rates of adjacent layers are equal, \(\gamma _1=\gamma _2\). See also Fig. 4.
Sample stress profiles for varying values of \(\gamma _1\) (with \(\gamma _2=1\)) are given in Fig. 4. With \(\gamma _1>1\), the inner layer grows uniformly; hence, its reference state is a uniformly expanded annulus; however, it is constrained by attachment to the core and to the ungrowing outer layer. Thus the inside of the inner layer is in radial tension (the inner edge is “stretched” radially to match the core), the outside is in radial compression, and the entire layer is in compression in the hoop direction. The outer layer, on the other hand, is forced to expand circumferentially to accommodate the growing inner layer and is in circumferential compression; this is balanced by a compression in the radial direction. The inverse effect occurs with \(\gamma _1<1\).
3.3 Growth Law
3.4 Stability Analysis
To investigate the behavior of the growth dynamics, we can now apply standard techniques of dynamical systems to (27); that is we seek equilibria satisfying \({\dot{\gamma }}_{1}=0\) and \({\dot{\gamma }}_{2}=0\) and compute their stability. Let \(\left\{ \gamma _{1}^{\text {eq}},\gamma _{2}^{\text {eq}}\right\} \) denote an equilibrium state. The nonlinear nature of the dependence of \(\overline{T_{1}^{RR}}\), \(\overline{T_{2}^{RR}}\), \(\overline{T_{1}^{\theta \theta }}\) and \(\overline{T_{2}^{\theta \theta }}\) on \(\gamma _{1}\), \(\gamma _{2}\) makes it difficult to compute analytically the number and location of equilibrium states as a function of the parameters \({\tilde{K}}\) and \(T^{*}\) and we shall use numerical methods to this end.
3.5 Bifurcation Diagram

Region I has four equilibrium states, of which one is a stable node, two are saddles, and the fourth is either an unstable node or an unstable focus.

Region II has four equilibrium states: two are saddles and the other two are either stable nodes or a stable focus and stable node. A Hopf bifurcation at the interface of Regions I & II transforms the unstable focus into a stable focus.

Region III has two equilibrium states, one of which is a stable node, the other a saddle node. At the interface between Regions II and III, a saddle node bifurcation occurs that annihilates the stable node and saddle node in Region II.

Region IV has no equilibrium states.
As is evident in Fig. 5, there is a wealth of possible dynamical behavior exhibited in this system. That an idealized twolayer model with isotropic growth and equivalent homeostatic values in each layer has such a rich structure highlights a more generic complex nature of mechanically driven growth. Our intent is not to fully categorize the behavior; rather this system should be seen as a paradigm to illustrate complex dynamics. Nevertheless, several observations are in order.
One observation from the phase portraits in Fig. 5b is that unbounded growth is not only possible but “common,” at least in the sense that many parameter choices and initial conditions lead to trajectories for which \(\gamma _i\rightarrow \infty \). Perhaps the most natural initial condition is to set \(\gamma _1=\gamma _2=1\), which corresponds to letting the system evolve from an initial state with no growth. Examining the trajectories in Fig. 5b shows that points P1 and P2 would not evolve toward the single stable state, but rather would grow without bound.
Another point of interest is that while regions I, II and III contain stable equilibria, the stable states in Regions I and III satisfy \(\gamma _{1}^{\text {eq}}\gamma _{2}^{\text {eq}}<1\). These are equilibria for which one of the layers has lost mass (at least one of the \(\gamma _i<1\)). Growth in both layers requires both \(\gamma _i>1\), and we find that such an equilibrium only exists in a small subset of Region II, shaded dark blue in Fig. 5. We further see that \(T^*<0\) in the dark blue region, and \({\tilde{K}}\) approximately in the range 10–17. This implies that in order for a stable equilibrium to exist where both layers have grown, the homeostatic stress must be compressive in one or both components, and the system must respond more strongly to radial than to circumferential stress.
Included in Fig. 6b are three sample trajectories, with the size of each layer shown at different times, and illustrative of the variety of dynamical behavior. The green trajectory quickly settles to a stable state marked by significant resorption (both \(\gamma _i<1\)); the blue and red trajectories sit outside the basin of attraction of P1 and show an initial period of resorption followed by significant growth. The red trajectory is in the basin of attraction of the stable focus and thus oscillates between growth and decay as it approaches the stable point at P3, while the blue trajectory, just outside the basin of attraction, ultimately grows without bound, never reaching an equilibrium state.
4 Growth of Discrete N Layer System
Next, we generalize the dynamical system of the previous section from two to N layers where growth and stresses are constant throughout each layer. If N is sufficiently large, a system of N layers can be used as a suitable spatial discretization of a continuous growth profile on which precise statements can be obtained. In this case, we can generalize Eq. (33) to N coupled ODEs. We will analyze the stability of this system near a homeostatic equilibrium and show to what extent the results obtained for \(N=2\) remain unchanged as the discretization is refined (N increases), which informs the stability of the continuous (\(N\rightarrow \infty \)) system.
A major difference compared to the twolayer model is the method to obtain homeostatic values. Previously, homeostatic values were prescribed via the homeostatic growth values \(\gamma _{1}^{*}\), \(\gamma _{2}^{*}\). In the present model, homeostatic values are obtained by assuming the existence of a prescribed continuous homeostatic growth profile \(\gamma ^{*}\left( R^{0}\right) \). The homeostatic values \(\left\{ \gamma _{i}^{*}\right\} \) are then obtained through local averaging of the prescribed profile \(\gamma ^{*}\left( R^{0}\right) \) over an interval by generalizing Eq. (23). These values are admissible by construction.
Since growth is taken as constant in each layer, the stresses can be determined fully analytically and a stability analysis can then be performed. The stability analysis will inform under which conditions the dynamical system will either relax to a homeostatic state after a small perturbation or lead to an instability.
4.1 Kinematics
4.2 Mechanics
4.3 Generating a Homeostatic State from a Prescribed Growth Profile.
We obtain the discrete homeostatic stress profile \(\left\{ \gamma _{i}^{*}\right\} \) from the continuous profile \(\gamma ^{*}\left( R^{0}\right) \) by computing the average according to (36). The homeostatic stress \({\mathbf {T}}\left( \varvec{\gamma }^{*}\right) \) is computed from the discrete homeostatic stress profile \(\left\{ \gamma _{i}^{*}\right\} \) according to (44) and (45). The homeostatic values \(\overline{{\mathbf {T}}}\left( \varvec{\gamma }^{*}\right) \) are obtained as averages according to (47) and (49). It is important to note that the homeostatic stress is generated by prescribing a growth profile (51), which by definition ensures that the homeostatic stress is admissible.
4.4 Growth Dynamics
We consider a growth law that generalizes (33) to N layers. The main difference with (33) is that the values for homeostatic stress are obtained by the linear growth profile.
Figure 11a shows a bifurcation diagram of the stability of the dynamical system (52) as a function of \({\tilde{K}}^{1}\) and \(C_{1}\) for \(N=9\) layers (note that unlike in Fig. 5, here we use the inverse of \({\tilde{K}}\) to focus on large circumferential stress). The regions are colored according to the largest real part of the eigenvalues \(\lambda _{i}\) of \({\mathbf {J}}\), that is \(\lambda =\text {Max(Re}\,\lambda _{1}, \text {Re}\,\lambda _{2}, \ldots \text {Re}\,\lambda _{N})\). There are three parameter regions: an unstable region (orange), a stable region (blue), and an undecidable region (green) for which \(\lambda \) is within a small tolerance of zero. This last region is included as it is typically within numerical error and its inclusion allows to make precise statements about stability. This relatively shallow region of \(\lambda \) is further explored in Fig. 12 and allows us to identify the clearly stable and clearly unstable regions of the diagram. Figure 11b–e shows that for increasing values of N (that is, a refinement of the discretization), the regions are practically unchanged (b–d), and that the largest eigenvalue of four selected points converges reliably to a finite positive (P1 and P2) or negative (P3 and P4) eigenvalue.
4.5 Solid Nucleus Versus Pressurized Cylinder
5 Conclusion
It is now well appreciated that growth can induce mechanical instabilities (Goriely and Ben Amar 2005; Ben Amar and Goriely 2005). The related problem that we have considered in this paper is the stability of a grown state through its slowgrowth evolution. The question is not therefore about mechanical instability but about the dynamic stability of a preferred homeostatic state. While the former is characterized by a bifurcation from a base geometry to a more complex buckled geometry, occurring on a fast elastic timescale, the latter involves the system evolving away from a given stress state on the slow growth timescale. In general the homeostatic state is not homogeneous; hence, the issue of stability requires the analysis of partial differential equations defined on multiple configurations with free boundaries. There are no standard mathematical tools available to study this problem even for simple nonhomogeneous systems. An alternative is to consider the stability of states that are piecewise homogeneous (in space). The problem is then to establish the stability of coupled ordinary differential equations describing locally homogenous states through the traditional methods of dynamical systems. Within this framework, we considered two relatively simple problems.
First, we considered the dynamical stability of a twolayer tube with different, but constant, growth tensors in each layer. We characterized the dynamics of the full nonlinear system and showed that the number of equilibria and their stability varies greatly and gives rise to highly intricate dynamics which we organized via several bifurcations. We identified a parameter region where the system is stable. We found that the growth dynamics of tubular structures in the neighborhood of the homeostatic equilibrium depends in a nontrivial way on the anisotropy of the growth response and that the equilibrium becomes unstable for highly anisotropic growth laws. This complexity of dynamics naturally raises the question about stability of homeostatic equilibria for more general systems.
Second, we showed that given a continuous law in a cylindrical geometry, we can introduce a suitable discretization of the problem that keeps all the characteristics of the continuous problem. We showed that for a linear growth law, there are clear regions where stability and instability persist independently of the discretization (for sufficiently large N). We expect that these regions represent the true behavior of the full inhomogeneous system. This result allows us to characterize the stability of a morphoelastic growing cylinder.
As our results (in particular Fig. 11) show, admissible homeostatic states can lead to either mechanically stable or unstable equilibria. This suggests a way to distinguish between physiological (stable) and pathological unbounded (unstable) growth. Indeed, our model also suggests a natural growth termination mechanism. The question of growth termination (what triggers a tissue to stop growing?) is a muchdebated question in developmental biology. It has been particularly well studied in the model system of the Drosophila melanogaster wing disk where the morphogen Decapentaplegic (Dpp) has been identified as the main regulator of growth. A number of models propose growth regulation and termination based on a combination of mechanical effects and Dpp concentration. Some of them are continuum models (AegerterWilmsen et al. 2007; Ambrosi et al. 2015), others are vertex models (Shraiman 2005; Hufnagel et al. 2007) but none of them is entirely satisfactory (Vollmer et al. 2017). We hope that the dynamical stability of homeostatic states offers an alternative way of looking at growth termination, which emerges naturally in our model as a stable equilibrium.
While we have only scratched the surface of the complex dynamic behavior that exists in such systems, the framework presented here provides a tool to explore growth dynamics and stability of homeostatic states and finally address some of the fundamental challenges of morphoelasticity (Goriely 2017): What growth laws, in general, would lead to dynamically stable homeostatic states? What is the final size of a growing organism for a given growth law? What are the conditions under which growth dynamics produces oscillatory growth?
Notes
Acknowledgements
We thank Dr. Thomas Lessinnes for many useful discussions in the early stages of this Project. The support for Alain Goriely by the Engineering and Physical Sciences Research Council of Great Britain under research Grant EP/R020205/1 is gratefully acknowledged.
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