🧬 biology

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

This paper presents a unified active viscoelastic continuum model demonstrating that determinate, indeterminate, and proportionate growth regimes emerge from the interplay between active cellular stresses and material properties, specifically governed by the ratio of activity to elastic modulus and the mechanical impedance matching of tissue components.

Original authors: Jigyasa Watwani, K. Vijay Kumar, Vishal Vasan

Published 2026-08-21

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Jigyasa Watwani, K. Vijay Kumar, Vishal Vasan

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ⚕️ This is an AI-generated explanation of a preprint that has not been peer-reviewed. It is not medical advice. Do not make health decisions based on this content. Read full disclaimer

In the living world, growth is a fundamental act, yet it plays out in two very different ways. Some organisms, like humans or birds, grow to a specific size and then stop; their bodies reach a target and hold steady, a pattern scientists call determinate growth. Others, such as certain worms or crustaceans, never seem to have a final size limit. They continue to grow throughout their entire lives, sometimes slowing down but never truly stopping, a behavior known as indeterminate growth. For decades, biologists have wondered if these two distinct behaviors require completely different biological machinery, or if they are simply two sides of the same coin. The question extends beyond total size to the shape of the organism itself. When a body grows, do its parts expand in perfect harmony to keep the original proportions, or do they drift apart? Understanding how a soft, living tissue manages to expand, stop, or maintain its shape is a puzzle that sits at the intersection of biology and physics, asking how the forces inside a cell translate into the form of a whole animal.

Researchers at the International Centre for Theoretical Sciences in India have approached this puzzle by treating a growing tissue not as a collection of individual cells, but as a continuous, stretchy material, much like a thick gel or a piece of rubber. They built a mathematical model that describes how this material moves and changes shape over time. In their view, the tissue is a viscoelastic continuum, meaning it has properties of both a solid that springs back and a fluid that flows. The model accounts for the fact that cells are constantly dividing and dying, which generates internal forces that push and pull on the tissue. By focusing on these mechanical forces rather than chemical signals, the team discovered that the difference between stopping growth and growing forever depends on a simple balance: the ratio of the internal pushing force to the stiffness of the material.

The study reveals that there are two possible outcomes for this growing tissue, determined entirely by whether the internal force is strong enough to overcome the material's resistance. If the tissue is stiff enough relative to the force generated by cell division, it will stretch to a specific size and then stop. This state mimics determinate growth, where the organism reaches a target size and holds it. However, if the internal force is too strong for the material's stiffness to contain, the tissue cannot find a stopping point. Instead, it enters a state of perpetual growth, elongating indefinitely. The researchers found that this transition is not a gradual slowing down but a sharp switch. There is a hard limit to how much elastic stress the tissue can support; once the active force exceeds this limit, the material simply cannot balance the pressure and must flow forever. This explains why some organisms grow to a fixed size while others do not, suggesting that the difference is not a complex biological program but a physical threshold.

The team also investigated what happens when a tissue is made of different parts with different properties, such as a limb with a soft joint and a stiff bone. They found that in a growing tissue, the relative lengths of these parts eventually settle into a fixed ratio. However, this final ratio is not determined by how the tissue started. Instead, it is dictated by the mechanical properties of the parts themselves, specifically a concept they call mechanical impedance, which combines how stiff a part is with how much it resists flow. In most cases, a tissue will drift away from its original proportions and settle into a new shape defined by these material properties. The only way to preserve the exact proportions the tissue began with is to carefully match the mechanical properties of the different parts to their initial lengths. If the parts are not matched this way, the tissue will naturally reconfigure itself as it grows, favoring the mechanical balance over the original blueprint.

This work suggests that the complex behaviors of growth—stopping at a certain size, growing forever, or maintaining a specific shape—can all emerge from a single physical framework. The researchers did not need to invent special rules for each type of growth or rely on chemical signals to dictate the outcome. Instead, they showed that the interplay between the forces generated by cells and the stiffness of the tissue is sufficient to produce all these patterns. The model predicts that growth is not uniform throughout the body; rather, the actual stretching happens mostly at the edges, while the center remains relatively still. This edge-focused growth is a direct consequence of the friction between the tissue and its surroundings. By demonstrating that determinate, indeterminate, and proportionate growth are just different regimes of the same mechanical system, the study offers a new way to think about how living things build themselves, suggesting that the rules of physics, rather than just biology, set the stage for how an organism grows.

Technical Summary: Growth Phases of an Active Tissue

Problem Statement
Biological growth manifests in two primary phenotypes: determinate growth, where tissues cease at a target size (common in vertebrates), and indeterminate growth, where tissues continue to grow throughout life, often linearly (observed in planaria and modular organisms). A central question in biophysics is whether these distinct phenotypes arise from fundamentally different mechanisms or represent different regimes of a single physical description. Furthermore, many organisms exhibit proportionate growth, maintaining fixed relative sizes of their parts despite heterogeneity in material properties. It is unclear how a purely mechanical model, without prescribed growth rules or chemical morphogen gradients, can account for both the transition between growth phases and the preservation of tissue proportions in inhomogeneous systems.

Methodology
The authors develop a minimal, coarse-grained active viscoelastic continuum model of a tissue growing along a single axis. Key features of the methodology include:

Continuum Framework: The tissue is modeled as a deformable material where cell division and death generate active stresses. The dynamics are overdamped, reducing momentum balance to a quasi-static force-balance equation ( $\nabla \cdot \sigma + F_{ext} = 0$ ).
Stress Components: The total stress $\sigma$ comprises elastic, viscous, and active contributions. The elastic stress depends on the displacement gradient, viscous stress on the velocity gradient, and active stress on the cell density (in the autonomous case) or is externally applied.
Growth Mechanism: Growth is implicit rather than prescribed. The model neglects growth anisotropy and signaling morphogens, focusing solely on mechanical feedback.
Scenarios: The study analyzes two driving conditions:
1. Applied Stress: Growth driven by a uniform external tension $\sigma_b$ at the boundaries.
2. Autonomous Growth: Growth driven internally by cell turnover (birth/death) generating active stress $\sigma_{active} = -\zeta f(\rho) I$ , coupled with a logistic equation for cell density $\rho$ .
Inhomogeneity: The model is extended to composite tissues with two parts possessing different material parameters (elastic moduli $E$ , viscosities $\eta$ , and friction coefficients $\gamma$ ).

Key Results
The analysis reveals that the growth phenotype is determined by the ratio of active stress (or applied stress) to the elastic modulus ( $E$ ).

Determinate vs. Indeterminate Phases:
- Determinate (Bounded) Growth: When the applied stress $\sigma_b$ (or active stress parameter $\zeta f(\rho_0)$ ) is less than the elastic modulus $E$ , the tissue reaches a steady-state size. The growth factor diverges as the stress approaches $E$ from below ( $L^*/L_0 \propto (1 - \sigma_b/E)^{-1}$ ), and the approach to this size slows significantly ( $\propto (1 - \sigma_b/E)^3$ ).
- Indeterminate (Unbounded) Growth: When the stress exceeds $E$ , the elastic stress cannot balance the driving force at any finite strain. The tissue enters a perennially growing phase where it elongates linearly without bound.
- Asymmetry: The model admits unbounded growth and bounded degrowth (shrinkage) but only bounded growth, reflecting the constitutive bound on tensile elastic stress.
Spatial Localization of Growth:
In the unbounded phase, growth is localized at the tissue edges. The velocity vanishes at the center, and the strain rate is confined to a hydrodynamic length scale $\ell = \sqrt{\eta/\gamma}$ near the boundaries. The interior becomes asymptotically quiescent.
Proportionate Growth in Heterogeneous Tissues:
- Frozen Fractions: In a composite tissue growing indefinitely, the length fractions of the parts ( $L_\alpha/L$ ) settle into fixed values determined solely by the mechanical impedances ( $Z = \sqrt{\eta \gamma}$ ) of the parts, not by their initial lengths. The tissue "drifts" away from its initial proportions to a state dictated by material parameters.
- Preservation of Proportions: To maintain the initial proportions (true proportionate growth), a specific condition must be met: the impedances must be matched inversely to the initial lengths ( $Z_l/Z_r = L_{r0}/L_{l0}$ ). If this condition is met, the interface does not drift, and the tissue grows while preserving its starting geometry.
Autonomous Dynamics:
When driven by internal cell turnover, the system exhibits the same phase transition. The steady-state size is independent of the birth-death rate $\kappa$ , while the asymptotic growth rate in the unbounded phase depends non-monotonically on $\kappa$ . At high turnover rates, the density remains close to the carrying capacity, and the results converge to the analytical solutions derived for the applied stress case.

Significance and Claims
The paper claims to demonstrate that determinate, indeterminate, and proportionate growth are not distinct biological mechanisms requiring separate explanations, but rather regimes of a single continuum mechanical framework.

Mechanism Unification: The transition between bounded and unbounded growth arises naturally from the constitutive limit of elastic stress, requiring no external "stop" signal or morphogen threshold.
Mechanical Basis for Proportionality: The study shows that proportionate growth can emerge from mechanical principles alone. While generic heterogeneous tissues drift to parameter-defined proportions, the specific tuning of mechanical impedances allows for the preservation of initial proportions, offering a mechanical explanation for developmental robustness.
Minimalism: By excluding morphogens and prescribing no growth laws, the model suggests that the fundamental physics of active viscoelastic materials is sufficient to explain complex growth phenotypes observed across the tree of life.

The authors note that while the model captures these regimes, it relies on a fixed reference configuration and assumes slow stress relaxation relative to growth. They acknowledge that extending this to multi-directional growth and coupling with morphogen fields remains a direction for future work.

Technical Summary: Growth Phases of an Active Tissue

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