Growth phases of an active tissue: determinate, indeterminate, and proportionate

Abstract: Growth may cease at a target size or continue throughout life: the determinate and indeterminate phenotypes. We develop an active viscoelastic continuum model of a tissue growing along one axis, in which cell division and death generate active stresses. We find two asymptotic states: one in which the tissue reaches a relative size fixed by its material parameters, and one in which it elongates linearly without bound. Which state is realised is set by the ratio of active stress to elastic modulus. The transition originates in a bound on the elastic stress the tissue can support: a sufficiently large activity can never be balanced. In a tissue made of parts with different material properties, the growing phase settles into fixed length proportions, set by the mechanical impedances of the parts rather than inherited; matching impedances to initial lengths preserves the proportions the tissue began with. Determinate, indeterminate and proportionate growth thus appear as regimes of one continuum mechanical framework.

• The paper shows that determinate growth saturates while indeterminate growth becomes asymptotically linear when active or applied stress crosses the tissue’s maximum elastic stress, with the threshold controlled by a dimensionless stress-to-modulus ratio.
• The model predicts that large tissues grow mainly at their edges, while heterogeneous tissues settle into length fractions governed by mechanical impedances rather than their initial proportions.
• The framework extends to autonomous tissues with logistic cell turnover, reproduces the same growth transition, and identifies low-turnover oscillations and multidirectional growth as important areas for future research.

Overview

Watwani, Kumar, and Vasan develop a minimal active viscoelastic continuum model of a tissue elongating along one axis, in which cell division and death generate active stresses rather than being prescribed through a growth law. The central result is that determinate growth (saturation at a finite size) and indeterminate growth (perennial, asymptotically linear elongation) are two regimes of the same mechanical framework, selected by a single dimensionless control parameter: the ratio of driving stress to elastic modulus, $\sigma_b/E$ under an applied boundary traction or $\zeta f(\rho_0)/E$ when the drive is generated internally by cell density. The same framework addresses proportionate growth of heterogeneous tissues, showing that length fractions freeze at values set by mechanical impedances rather than inherited from initial conditions. The paper is accompanied by extensive supplementary calculations that establish the kinematic and constitutive basis for these results (2608.20091).

Continuum formulation

The tissue is modeled as an overdamped viscoelastic continuum on a deforming domain $\Omega_t$, with material points obeying $\dot{\mathbf{x}} = \mathbf{v}$ and displacement field evolving via $D_t\mathbf{u} = \mathbf{v}$. Momentum balance reduces to quasi-static force balance $$\nabla \cdot \boldsymbol{\sigma} + \mathbf{F}_{\mathrm{ext}} = 0$$, with stress decomposed into elastic (moduli $K$, $E$), viscous ($\mu$, $\eta$), and active contributions, coupled to substrate friction $\zeta f(\rho_0)/E$0. Friction introduces two organizing scales: the hydrodynamic length $\zeta f(\rho_0)/E$1 and the impedance $\zeta f(\rho_0)/E$2.

An important structural observation is that within affine displacements $\zeta f(\rho_0)/E$3, only two asymptotic states exist: static ($\zeta f(\rho_0)/E$4) and perennially deforming ($\zeta f(\rho_0)/E$5). The latter is degenerate: at $\zeta f(\rho_0)/E$6 the deformation gradient $\zeta f(\rho_0)/E$7 is singular and all memory of the initial configuration is lost. Indeterminate growth is thus doubly indeterminate—unbounded size and erasure of reference-configuration information. This state is approached only asymptotically; at any finite time the Lagrangian map remains invertible provided $\zeta f(\rho_0)/E$8 is bounded.

Two modeling restrictions are stated plainly. First, material parameters are frozen in time—the model does not couple growth to morphogen fields. Second, until the density field is introduced, the growing tissue rarefies without any notion of local cell number. Additionally, the constitutive law assumes the elastic stress relaxes slowly compared with growth timescales, complementary to the fluidization limit of Ranft et al., where division and apoptosis relieve stress at rate $\zeta f(\rho_0)/E$9.

Origin of the transition

Specializing to one dimension under uniform boundary tension $\Omega_t$0 yields closed-form solutions. For $\Omega_t$1, the steady state has length

$\Omega_t$2

with linear stability analysis showing the approach rate vanishes as $\Omega_t$3 while the steady-state size diverges as $\Omega_t$4 — distinct exponents arising because the slowest relaxation rate depends on $\Omega_t$5 itself.

For $\Omega_t$6, the tissue grows with velocity

$\Omega_t$7

and asymptotic rate $\Omega_t$8: linear-in-time elongation. A notable result is that although material points near the midpoint separate exponentially initially, their total separation over the entire history of the tissue is bounded by the factor $\Omega_t$9, because the strain rate decays as the domain lengthens. For $\dot{\mathbf{x}} = \mathbf{v}$0, strain rate is confined to within $\dot{\mathbf{x}} = \mathbf{v}$1 of the edges; the interior becomes quiescent and growth occurs peripherally even though division continues throughout the bulk. This edge-localization is what makes composite-tissue growth rates additive across parts.

The threshold originates in a bound on the elastic stress the tissue can support. The strain measure used, $\dot{\mathbf{x}} = \mathbf{v}$2 (the $\dot{\mathbf{x}} = \mathbf{v}$3 Seth–Hill measure), saturates below unity as stretch $\dot{\mathbf{x}} = \mathbf{v}$4, so the Cauchy stress cannot exceed $\dot{\mathbf{x}} = \mathbf{v}$5 in extension. Within the Seth–Hill family, a threshold exists if and only if $\dot{\mathbf{x}} = \mathbf{v}$6; logarithmic, Biot, and Green measures admit no transition. The paper emphasizes this asymmetry honestly: no comparable bound exists under compression, which is why bounded de-growth exists but unbounded de-growth does not. It also concedes that the symmetrized-gradient strain is not strictly frame-indifferent at finite deformation, though both it and the objective Almansi measure are bounded in extension, so only the numerical threshold changes; and that the spring and dashpot refer to different strain measures, so the material is not Kelvin–Voigt at finite strain and the closure is phenomenological rather than hyperelastic.

Proportionate growth in heterogeneous tissues

For a two-part tissue with parameters $\dot{\mathbf{x}} = \mathbf{v}$7 ($\dot{\mathbf{x}} = \mathbf{v}$8), three regimes exist, separated by $\dot{\mathbf{x}} = \mathbf{v}$9 and $D_t\mathbf{u} = \mathbf{v}$0: both parts stationary, both growing, or one of each. In the long-tissue limit each part exceeds its own hydrodynamic length, the edge speeds decouple:

$D_t\mathbf{u} = \mathbf{v}$1

with the interface drifting at $D_t\mathbf{u} = \mathbf{v}$2. Length fractions freeze at impedance-determined values shifted by $D_t\mathbf{u} = \mathbf{v}$3 — but the authors are careful to distinguish frozen fractions from preserved proportions. The frozen values bear no relation to initial proportions; a composite tissue generically drifts before settling (demonstrated numerically for initial ratios 1:1, 1:2, and 1:3, all converging to identical fractions). Preserving initial proportions requires the single scalar constraint $D_t\mathbf{u} = \mathbf{v}$4 on six material parameters, leaving a five-parameter family of solutions. Its most transparent realization is impedance matching with equal moduli: $D_t\mathbf{u} = \mathbf{v}$5, i.e., the more heavily damped part must be the initially shorter one.

Two validity conditions apply: positivity of fractions requires $D_t\mathbf{u} = \mathbf{v}$6, otherwise the softer part consumes the stiffer one; and the results are asymptotic, holding only once $D_t\mathbf{u} = \mathbf{v}$7. The authors also distinguish this mechanical notion from morphogen-scaling mechanisms [Aguilar-Hidalgo et al., PRL 120, 198102], noting the latter preserves positional information for cell identity respecification, whereas the present mechanism concerns lengths of already-specified parts — neither subsumes the other. The qualitative type of transition matches the critical point of the morphogen-growth feedback framework, but here arises from mechanics alone, with no chemical field and no prescribed growth rule.

Autonomous growth from activity

The imposed-stress results carry over when driving is internal. Cell density evolves logistically,

$D_t\mathbf{u} = \mathbf{v}$8

and generates an isotropic active stress $D_t\mathbf{u} = \mathbf{v}$9. With traction-free boundaries, the steady state in $$\nabla \cdot \boldsymbol{\sigma} + \mathbf{F}_{\mathrm{ext}} = 0$$0 dimensions is exact for any birth–death rate:

$$\nabla \cdot \boldsymbol{\sigma} + \mathbf{F}_{\mathrm{ext}} = 0$$1

The threshold $$\nabla \cdot \boldsymbol{\sigma} + \mathbf{F}_{\mathrm{ext}} = 0$$2 is $$\nabla \cdot \boldsymbol{\sigma} + \mathbf{F}_{\mathrm{ext}} = 0$$3-dependent because a dilation of magnitude $$\nabla \cdot \boldsymbol{\sigma} + \mathbf{F}_{\mathrm{ext}} = 0$$4 contributes $$\nabla \cdot \boldsymbol{\sigma} + \mathbf{F}_{\mathrm{ext}} = 0$$5 to $$\nabla \cdot \boldsymbol{\sigma} + \mathbf{F}_{\mathrm{ext}} = 0$$6; notably, the shear modulus $$\nabla \cdot \boldsymbol{\sigma} + \mathbf{F}_{\mathrm{ext}} = 0$$7 drops out entirely under isotropic dilation, so isotropic growth cannot be used to infer $$\nabla \cdot \boldsymbol{\sigma} + \mathbf{F}_{\mathrm{ext}} = 0$$8. The growing-phase velocity acquires a density-gradient correction controlled by the Green's function $$\nabla \cdot \boldsymbol{\sigma} + \mathbf{F}_{\mathrm{ext}} = 0$$9 of $K$0; its sign-definiteness permits bounding this correction by $K$1, negligible when $K$2 holds density near setpoint. Finite-element simulations on deforming meshes confirm the transition and show the steady-state length independent of $K$3 while the growth rate depends on it non-monotonically; the choice of regulation function $K$4 affects results only quantitatively. At low turnover ($K$5) the tissue oscillates as a whole, driven by the delay between density excursions and the contractile–extensile active stress they produce — a state outside the affine class. The heterogeneous autonomous case maps exactly onto the imposed-stress problem with $K$6.

Limitations and open questions

The model's assumptions bound its scope. Material properties are time-independent, excluding coupling between growth and morphogen fields, though the discussion sketches how $K$7, $K$8, $K$9, $E$0 could depend on a signalling concentration — distinguishing level-following (phase switching under a step change in morphogen) from rate-following (a transient burst). The active stress is held independent of mechanical state; a homeostatic closure responding to pressure would determine the setpoint and give the steady state an absolute size rather than a mere growth factor — the route to compensatory behavior characteristic of determinate growth, which the present model, whose uniform dilation mode is neutrally stable, does not capture. Growth in multiple directions simultaneously is not attempted; once shape can change, the steady state ceases to be a single number and proportionate growth becomes a statement about shape similarity, identified as the natural next question. Finally, the fixed reference configuration is justified only in the dense, slowly-relaxing regime, and whether a dynamically resetting reference configuration — making the constitutive relation nominally fluid at long times — reproduces the same phase structure is explicitly left open.

Conclusion

This paper demonstrates that determinate and indeterminate growth need not invoke distinct mechanisms: they are phases of one active continuum, separated by whether the activity exceeds the maximum elastic stress the tissue can sustain, with the transition's existence contingent on bounded extensional response ($E$1 Seth–Hill measures) and absent under compression. Edge-localized growth at large sizes makes composite dynamics additive, yielding impedance-controlled proportions that freeze generically but preserve initial proportions only under a tunable parameter condition. The framework's principal strength is that closed-form results survive both externally driven and fully autonomous settings, with numerical work covering the intervening low-turnover regime where oscillations appear.