The Snap Point: One Number That Decides Whether Growing Matter Acts Solid or Liquid

Pay attention to a growing bean sprout or a healing wound, and you'll see something strange: matter that grows, bends, coils, and snaps into new shapes without ever being consciously built. What physics governs these transformations? For decades, scientists have described growing biological material as either elastic — springy, like rubber — or viscous — flowing, like honey. But most living tissue is neither. It's viscoelastic: it does both. This careful preprint by physicist Valentin Slepukhin and his advisor Oskar Hallatschek at UC Berkeley finds that this in-between nature can dramatically change how things grow, and that the most dramatic behavior happens precisely when these two tendencies are locked in a tug-of-war.

The paper's central discovery is that the mechanics of growing matter is governed by a single number: the product of the growth rate $g$ and the viscoelastic relaxation time $\tau$ — call it $g\tau$. This is a comparison of two clocks. The growth clock $g$ says how fast new material is being added. The relaxation clock $\tau$ says how fast stress inside the material is dissipating away. When growth is fast compared to relaxation, the material behaves elastically. When growth is slow, it flows viscously. But at the crossover, $g\tau \sim 1$, neither clock dominates — and something qualitatively new happens. The growing matter snaps between configurations, jumps discontinuously, and even undergoes phase-transition-like behavior never seen in either limiting case. The study reveals a hidden threshold in living matter, and it could reshape how we think about everything from biofilms to tumors to the microscopic structures inside cells.

The Science

The authors begin with a thought experiment rooted in everyday observation. Many living things grow mostly in one direction: fungal hyphae, pollen tubes, the axons of neurons, filamentous cyanobacteria like spirulina, and growing microtubules. When such an object is unconstrained, it simply extends, staying straight and stress-free. But what happens when the ends are pinned down or fixed in place? The growing beam can no longer just elongate — it must bend to accommodate its new length.

To understand this, Slepukhin and Hallatschek model a growing viscoelastic beam (Figure 1). The mathematics is the classic physics of a slender elastic rod (Landau et al., 1986), but with a crucial modification: the torque at any point depends not just on the current curvature but on the history of how that curvature evolved, weighted by an exponentially decaying memory. Earlier curvatures contribute less and less to the present torque as time passes, with characteristic timescale $\tau$. Growth does not appear directly in these equations; it enters through a compressive force $F$, determined by the constraint that the distance between the beam's endpoints stays fixed while its arclength grows.

The key analytical step is extending this to full generality. The researchers build a "growth-compatible" theory in which unconstrained growth is intrinsically stress-free — a crucial conceptual point. Growth is fundamentally different from stretching: stretching produces stress, while growth produces deformation without stress (or should, if unconstrained). They decompose the deformation into a growth part and an elastic part, $\mathbf{F}={\mathbf{F}}_g{\mathbf{F}}_e$, so that stress depends only on the elastic part, factored out from the growth component. They then incorporate viscoelasticity through the Lodge equation, a standard and elegant formalism from polymer physics (Larson, 2013), which allows stress to be "forgotten" through exponential decay.

The result is a general model — an upper-convected Maxwell equation generalized to growing matter — that exactly recovers the three known limits: purely viscous growth, purely elastic growth, and viscoelastic behavior without growth. That it reduces correctly in all three limits is itself a strong validation of the framework.

What They Found

The results are organized around the dimensionless number $g\tau$, and they are striking.

The snapping beam. In the elastic limit ($g\tau \gg 1$, here $g\tau =100$), a growing beam adopts a teardrop shape and its tip angle stops growing at about 140 degrees. In the viscous limit ($g\tau \ll 1$, here $g\tau =0.1$), the angle grows smoothly and monotonically as the beam coils. But in the intermediate regime ($g\tau =1$), the tip angle traces a "ladder" pattern of discontinuous jumps — the beam snaps, rapidly flipping from one configuration to another, then coasts, then snaps again (Figure 1b). Each snap rotates the tip by between π/2 and π radians. The effective energy shows a sawtooth pattern: it climbs, sharp-drops at each snap as the beam relaxes to a lower-energy branch, then climbs again (Figure 1e). The snaps dramatically accelerate the coiling rate — faster than either limit. This is genuinely new behavior that appears in neither the viscous nor the elastic case.

The phase transition. For a material growing in a two-dimensional channel constrained by walls (Figure 2), the researchers find a sharp transition at $g\tau =1$. Below this critical value, the stress builds up gradually and settles at a stationary value. Above it, no stationary solution exists — the stress grows exponentially without bound, until the model's assumptions inevitably break down. This is not a smooth crossover but a bifurcation, more like a phase transition (Slepukhin & Hallatschek, 2026).

Growth amplifies what already exists. In growth through a porous medium, the model predicts permeability increases by a factor $1+2.5g\tau$ for small $g\tau$. Notably, this effect arises only from the combination of viscoelasticity and growth: neither a growing viscous fluid nor a non-growing viscoelastic fluid shows any permeability change. The two effects amplify each other.

Wrinkles at their biggest in the middle. When a thin growing layer (a stand-in for a wrinkling biofilm) is confined, the wrinkle amplitude is non-monotonic in $g\tau$, reaching a maximum in the intermediate regime $g\tau \sim 1$ (Figure 5). At that peak, the amplitude can be several times larger than in either the purely elastic or purely viscous limits. Again, the middle is not a smooth blend of the extremes — it's a peak all its own.

Beam snaps only in the intermediate regime

Label	Value
gτ = 0.1 (viscous)	0
gτ = 1 (intermediate)	13
gτ = 100 (elastic)	0

Data read from Figure 1d: snaps occur at intermediate gτ and are essentially absent in both the viscous (gτ ≪ 1) and elastic (gτ ≫ 1) limits.

Wrinkles peak at intermediate gτ

Label	Value
gτ = 0.1	100
gτ = 1	180
gτ = 100	100

Schematic of the paper's finding from Figure 5: intermediate gτ ~ 1 produces the largest wrinkles, exceeding the viscous and elastic limits by several times (about 50% in the paper's parameters).

A phase transition at gτ = 1

Label	Value
gτ = 0.8 (below)	1
gτ = 1.0 (critical)	2
gτ = 1.2 (above)	4

Schematic of the channel growth phase transition (Eqs. 13–14): stationary stress below gτ = 1, diverging at gτ = 1, and unbounded exponential growth above.

Why This Changes Things

The deepest implication is about what "growth" means mechanically. The paper draws a clean conceptual line between growth and stretching: growth creates deformation without stress, stretching creates stress even for identical deformation. This distinction, once made explicit and built into the model, produces consequences that matter for real biology.

Consider biofilms — the slimy communities of microbes that coat surfaces, teeth, and medical implants. Biofilms are viscoelastic, and their relaxation times span a wide range, from seconds to several minutes (Peterson et al., 2013; Klapper et al., 2002). The slowest viscoelastic mode has been measured at about $\tau =18$ minutes, remarkably universal across multiple species (Shaw et al., 2004). When combined with typical biofilm growth rates — slower than well-mixed cultures because making extracellular matrix is metabolically costly — the dimensionless parameter $g\tau$ reaches values around 0.1. That's still below unity, but the effects are already significant. In flow through a channel, permeability is modified by roughly 25%; for a wrinkling beam, the maximal amplitude rises by about 50% for the paper's parameters.

The wrinkle result is particularly evocative. A growing biofilm confined at its edges develops folds and ripples. The study finds these wrinkles grow largest when $g\tau$ is intermediate — that is, when growth and relaxation operate on comparable timescales. This has a subtle engineering implication: a biofilm might maximize its surface area (for nutrient uptake or gas exchange) precisely because it sits in this intermediate mechanical regime. The wrinkles are not just a passive consequence of growth; they're a feature tuned by the interplay of the two timescales.

The theory also explains why different regions of the same tissue can behave entirely differently. Growth is often non-homogeneous, depending on nutrient concentration, so one patch of a biofilm might sit at $g\tau \sim 1$ while another sits at $g\tau \sim 0.1$ — meaning they mechanically behave in qualitatively different ways. And because real materials have multiple relaxation times, the total stress is a sum over several viscoelastic modes, each with its own ${\tau }_i$, creating a spectrum of local behaviors.

There's a broader conceptual lesson about modeling. The field has historically oscillated between treating growing tissue as elastic (for organ development, following Rodriguez et al., 1994) or as viscous (for biofilms, modeled with Stokes equations, following Giometto et al., 2018). This paper argues that both are incomplete, and that the interesting physics lives in the neglected middle. The single parameter $g\tau$ tells you which description applies — and when neither does.

What's Next

The work is a preprint, not yet peer-reviewed, and the authors are appropriately careful about its scope. The single-component Maxwell model is the simplest possible viscoelastic description; real biological materials are more complex, with multiple relaxation timescales and nonlinear effects. The authors note, for instance, that an alternative theory uses a relaxation term quadratic in elastic strain, in which the relaxation rate itself grows with accumulated strain (Zieger et al., 2026) — a difference with real consequences near the snap threshold.

The most tantalizing direction is experimental. The theory makes sharp, falsifiable predictions: a growing beam should snap in a ladder pattern at intermediate $g\tau$; a confined biofilm should wrinkle most at intermediate values; a viscoelastic growing fluid should develop a "jet-like" flow in the middle of a cavity (Figure 7). These are all testable in carefully controlled experiments. The paper even notes that the instantaneous snaps in the model would, in reality, be slowed by inertia or friction — a detail that could itself become a probe of the material's properties.

Perhaps the deepest open question is whether this sharp threshold, $g\tau \sim 1$, marks a real transition in living systems. The paper shows it does for simple geometries. For tumors, for embryos, for the internal scaffolding of cells, the answer is not yet known — but the conceptual toolkit is now in place. One dimensionless number now stands between treating living matter as a solid thing or a flowing thing. As with so much of biology, the answer turns out not to be "either/or" but "it depends" — and now we know exactly what it depends on.