Mechanogenetic coupling of Hydra symmetry breaking and driven Turing instability model

Abstract

The freshwater polyp Hydra can regenerate from tissue fragments or random cell aggregates. We show that the axis-defining step ("symmetry breaking") of regeneration requires mechanical inflation-collapse oscillations of the initial cell ball. We present experimental evidence that axis definition is retarded if these oscillations are slowed down mechanically. When biochemical signaling related to axis formation is perturbed, the oscillation phase is extended and axis formation is retarded as well. We suggest that mechanical oscillations play a triggering role in axis definition. We extend earlier reaction-diffusion models for Hydra regrowth by coupling morphogen transport to mechanical stress caused by the oscillations. The modified reaction-diffusion model reproduces well two important experimental observations: 1), the existence of an optimum size for regeneration, and 2), the dependence of the symmetry breaking time on the properties of the mechanical oscillations.

Figure 1

(A) Sketch of the different Hydra sphere preparations and tissue manipulations (see Methods for details). (B) Mechanical oscillations during regeneration of a small fragment of Hydra tissue, showing the evolution of the average radius of the sphere and the ratio between the minor and major axes as a function of time. Phase I and phase II denote the stages of isotropic and anisotropic motion, respectively.

Figure 2

Dependence of the symmetry breaking time, t_SB, on the size of the sphere, r₀. Each point is an average over 7–15 spheres of similar size. The curve is a parabolic fit of the form t_SB = t_min + 152.3 [(r₀/r_c − 1]², with $t_{\text{min}}\simeq 21.4$ h and $r_c\simeq 138$μm.

Figure 3

(A) Examples of inflation-contraction cycles for gradually higher osmotic concentration, ΔC = C_out − C_in, with C_inHydra medium and C_out sucrose in Hydra medium. The swelling rate dr/dt is obtained as linear fits during the inflation stage. (B) The average swelling rate, s, decreases linearly with ΔC, and at ΔC ≥ 100 mM, $s\simeq 0$. Each point is an average over two to five cycles per sphere, and over six spheres. (C) For the same spheres, the corresponding symmetry breaking time (black dots) increases with the concentration difference, ΔC, whereas the time difference between symmetry breaking and regeneration (gray squares) is insensitive to ΔC.

Figure 4

Dependence of the symmetry breaking time, t_SB, on the swelling rate, s, and the frequency of oscillations, f. (A) Main plot. The variation of t_SB as a function of s for Hydra spheres with different osmotic concentrations, ΔC, is well described by a fit of the form a/[(bs)^γ − 1] (gray curve), where $a\simeq 97$h, $b\simeq 49$h μm⁻¹, and γ = 0.3. Each point is an average over two to five cycles per sphere, and over six spheres. (Inset) A similar scaling with the same exponent, γ, is obtained for a set of Hydra spheres with fixed ΔC = 0 and size $r_0\simeq 150$μm. Each point is an average over two to five cycles of the same sphere. (B) The variation of t_SB as a function of f for different ΔC also follows the dependence a/[(bf)^γ − 1], with $a\simeq 106$h, $b\simeq 2142$h, and γ = 0.3. Each point is an average over two to five cycles per sphere, and over six spheres.

Figure 5

(A) Distribution of swelling rates and the relative change between consecutive contours during phase I (upper) and phase II (lower). The swelling statistics are obtained from 30 spheres of similar size, in the range r₀ = 130–150 μm, with a total of 150 cycles during phase I and 130 cycles during phase II. For the relative change between two consecutive contours, black and gray indicate inflation and deflation, respectively. The thickness of the color band is proportional to the variation between contours. (B) Time evolution of the amount of HZO-1 phosphorylation (squares) compared to the relative change of the period of the oscillations, τ, for spheres with $r_0\simeq 150$μm (circles, average over 20 experiments). The vertical line indicates the average symmetry breaking time. (C) Relative change in the period of the oscillations for spheres with $r_0\simeq 150$μm, and with staurosporine treatment (up triangles, average over three experiments), organizer incorporation (down triangles, six experiments), and control (black, 20 experiments). Lines are a guide to the eye. The inset shows the average symmetry breaking time for the different treatments.

Figure 6

(A) The model for Hydra symmetry breaking consists of a reaction-diffusion equation given on a periodic domain of length L. The domain with coordinate x corresponds to the circumference of the Hydra cell ball. The mechanical oscillations together with the Turing instability break the initial isotropy of the system. This triggers further biochemical steps that permanently transform the sphere into an elongated ellipsoid, displaying a nonhomogenous distribution of activator and inhibitor due to the reaction-diffusion instability. (B) The role of oscillations in our model is to create a periodic tangential extension of the cell layers, which in turn modifies the transport rate of morphogens. (C) Cells have viscoelastic properties that can be modeled by combinations of viscous dash-pots and elastic springs. The shear moduli depend on the frequency of oscillatory deformations for small frequencies.

Figure 7

Schematic representation of the evolution of radius, stress, and diffusion coefficient during phase I oscillations. (A) The system becomes unstable when a certain threshold (horizontal dashed line) is exceeded. Strong oscillations (solid black lines) drive the system into the unstable regime (gray areas). If the oscillations are too weak (dotted line), the system does not remain in the unstable region for sufficient time to develop the instability. (B) Corresponding evolution of the stress during the pulses. Microrheological studies (33) showed a power law dependence of shear moduli with frequency. Hence, the differential stress is smaller for slow oscillations (thin black lines in A and B) than for fast oscillations (thick black lines) if the differential radius is equal. Since we assume a linear relation between inhibitor transport and stress, the plots also represent the modulation of diffusion coefficient μ. μ_c is the critical diffusion coefficient above which Eqs. 1 and 2 are in the Turing unstable regime.