Biomechanics and the Thermotolerance of Development

Successful completion of development requires coordination of patterning events with morphogenetic movements. Environmental variability challenges this coordination. For example, developing organisms encounter varying environmental temperatures that can strongly influence developmental rates. We hypothesized that the mechanics of morphogenesis would have to be finely adjusted to allow for normal morphogenesis across a wide range of developmental rates. We formulated our hypothesis as a simple model incorporating time-dependent application of force to a viscoelastic tissue. This model suggested that the capacity to maintain normal morphogenesis across a range of temperatures would depend on how both tissue viscoelasticity and the forces that drive deformation vary with temperature. To test this model we investigated how the mechanical behavior of embryonic tissue (Xenopus laevis) changed with temperature; we used a combination of micropipette aspiration to measure viscoelasticity, electrically induced contractions to measure cellular force generation, and confocal microscopy to measure endogenous contractility. Contrary to expectations, the viscoelasticity of the tissues and peak contractile tension proved invariant with temperature even as rates of force generation and gastrulation movements varied three-fold. Furthermore, the relative rates of different gastrulation movements varied with temperature: the speed of blastopore closure increased more slowly with temperature than the speed of the dorsal-to-ventral progression of involution. The changes in the relative rates of different tissue movements can be explained by the viscoelastic deformation model given observed viscoelastic properties, but only if morphogenetic forces increase slowly rather than all at once.


Supplemental Text 1.1: 1D, small deformation approximation.
Blastopore closure is a complex, large-deformation problem (hence geometrically non-linear), involving materials that are likely to display non-linear material properties at sufficiently large deformations [1,2]. Furthermore material properties and force generation vary spatially [2]. A full model would require a very large number of parameters, many of which are unknown, and most of which are poorly characterized. Those parameters that have been measured are highly variable [1,2,3,4]. We seek a simplified model, with as few parameters as possible, to explore the effects of variation in viscoelasticity and timing of force generation on the temperature dependence of morphogenesis.
Here we argue that a simple linear, scalar (1D) model is a reasonable first approximation given the number of unknown parameters, and high levels of mechanical variation. This model captures expected behaviors such as: 1) increased forces lead to increased deformation; 2) usually (though not always) the rate of deformation declines with time after application of a load; and 3) compliance measured in the microaspirator should correlate with tissue responses to stresses driving blastopore closure (i.e. tissues that appear softer in microaspiration should behave as though they are softer during blastopore closure).
We also expect that if the tissue behaves like a solid in the microaspirator, it will behave like a solid during blastopore closure. Similarly if it behaves like a fluid in the microaspirator, it should also behave like a fluid during blastopore closure.
Consider the full 3D, non-linear, large-deformation model. The strain tensor (μ ij ) at time t and position X={x, y, z} is a function of the stress tensor σ and compliance tensor C: We assume zero strain at the beginning of gastrulation. C and σ are functions of both time and position in the embryo, and f is a function of C and σ over all points in time prior to t (represented by γ). Let us take σ kl =s kl *ϴ, where ϴ is a scalar, with f going to zero as ϴ goes to zero (i.e. when stress is zero at all times). Both ω and y are unknown even to sign, and y is chosen so that its contribution is as small as possible. Therefore, it is reasonable to take as a first approximation, the linear model: The scalar ε is now the only parameter that characterizes the progression of gastrulation, so we can map morphogenetic events (deformation states) to ε[t]. Hence, blastopore closure begins and ends at a particular values of ε (0 and ε c respectively).
Clearly, we have cut out a lot of important processes and parameters to get to this point. Furthermore, we have not accounted for changes in compliance over development [2,3]. However, given the unknowns, and the variation in the known parameters, this linear, scalar-valued model is a reasonable first approximation. It is also minimal: all parameters other than those that whose importance we wish to determine (v[t]), can be tied to measurements (J[t]), or cancel out (K).

Supplemental Text 1.2: An example of tolerance to variation in gastrulation.
A remarkable illustration of embryos' capacity to tolerate morphogenetic variation occurred in one clutch (a batch of embryos collected from the same female at the same time). In this clutch ( Figure   S1) gastrulation took a very different path than in others (7 of 8 embryos). Bottle cell formation began normally. However, at the point when superficial involution would normally begin at the dorsal blastopore edge, a tongue of material (most likely presumptive notochord) extended out across the blastopore from the dorsal side. Superficial involution then began at a point further up the dorsal side, and blastopore closure occurred over the tongue of material. Despite its abnormal trajectory, blastopore closure completed successfully and the resulting embryos looked essentially normal at later stages (neurula and early tadpole). Because blastopore closure was so unusual in this clutch, these results were not analyzed for shifts in developmental timing, yet they provide a clear example of how robust morphogenesis can be. Contrast and brightness were adjusted to optimize images; images in A -C were rotated dorsal side up. B shows embryo from Figure 2A.